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Theory of photoelectron counting statistics: An essay
THEORY OF PHOTOELECTRON COUNTING STATISTICS: AN ESSAY
Richard BARAKAT and Julian BLAKE Division of Applied Sciences, Harvard University, Cambridge, MA 02138, US.A.
I
NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Review Section of Physics Letters) 60, No. 5 (1980) 225 -~340. NORTH-HOLLAND PUBLISHING COMPANY
THEORY OF PHOTOELECTRON COUNTING STATISTICS: AN ESSAY R. BARAKAT* and J. BLAKE Division of Applied Sciences, Harvard University, Cambridge, MA 02138, U.S.A. Received September 1979
Contents: 1. Introduction
227
2 Summary of the Glauber theory of quantum optics 2.1. Coherent states and correlation functions 2.2. P-representation 2.3. The n-atom detector 3. Photoelectron generating functions 3.1. Single-fold photoelectron generating function 3.2. Mixed Poisson process 3.3. N-fold photoelectron generating function 3.4. Eigenvalue problem for generating functions 4. Single-fold statistics for Gaussian light 4.1. Evaluation of Q(~’.) 4.2.P(m,T) for completely chaotic case 4.3. Eigenvalue problem for Q(X) 4.4. Lorentz spectrum 4.5. Rectangular spectrum 4.6. Numerical solution of integral equation 4.7. Long-time asymptotics 4.8. Variance of photoelectron counts 4.9. Superposition of coherent and multiple mode chaotic radiation 4.10. P(m,T) for partially polarized chaotic light 4.11. P(m,T) for statistically independent fields Appendix 4.A: Derivation of Glauber’s second asymptotic approximation 5. Two-fold statistics for Gaussian light 5.1. Correlation and clipped correlation functions for photoelectrons 5.2. Solution for short counting times 5.3. Solutions for arbitrary counting times 5.4. Q(X1, X2) for Lorentz spectrum 5.5. Higher order photoelectron statistics
228 228 230 233 234 234 237 239 241 244 245 245 247 248 252 254 257 261 262 267 271 271 273 273 275 277 282 287
Appendix 5.A: Q(X
1, ?.2) for partially polarized light Appendix 5.B: Evaluation of the photoelectron correlation function 6. Time interval statistics 6.1. Derivation of time interval probability densities 6.2. Short counting time, partial polarization 6.3. Lorentz spectrum 6.4. Numerical schemes for V(I) and P(T) 7. Non-Gaussian statistics 7.1. Introduction 7.2. Fixed number of scatterers 7.3. Random number of scatterers 7.4. Moments of W(fflIV) and LV(f~I(N>) 7.5. Coefficients of skewness and excess for W 7.6. Variance ofP 7.7. Time interval statistics 7.8. Photoelectron correlation function 7.9. Clipped photoelectron correlation function Appendix 7.A: Formulas for scattered intensity Appendix 7.B: Asymptotics of W(~IN)and P(mIN) 8. The inversion problem 8.1. Introduction 8.2. Inversion via singular value decomposition 8.3. Parameter estimation 9. Influence of spatial correlations 9.1. Heuristics of spatial coherence 9.2. Generating function 9.3. Photoelectron correlation function 9.4. Clipped photoelectron correlation function Acknowledgments References
Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 60, No. 5 (1980) 225—340. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publishers. Orders must be accompanied by check. Single issue price Dfl. 48.00, postage included. *Also: Bolt Beranek and Newman Inc., Cambridge, MA 02138.
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R. Barakat and 1. Blake, Theory of photoelectron counting statistics: an essay
227
1. Introduction One of the most useful methods of studying the statistical and spectral properties of light is by means of photoelectron counting detectors. The detection process is, of course, intrinsically quantum mechanical. The simplest photoelectron detector, for example, is a single atom free to undergo photoemission after which the ejected electron is observed. An important first step was carried out by L. Mandel [1] who derived a photoelectron counting formula (that is, a relation between the intensity and the probability of photodetection). The development employed semiclassical techniques in that the radiation field is treated classically and the interaction with the electrons in the detector is treated quantum mechanically. The quantum theory of coherence phenomena (which subsums photoelectron counting) was mainly originated by R.J. Glauber in the early 1 960s. However, a detailed account of the theory did not appear until the publication of his Les Houches Lectures [2] in 1966. Further amplifications and elaborations of the theory were subsequently published [3, 4, 5]. Included in Glauber’s work is a complete quantum derivation of the photoelectron counting formula which has served as the basic “building block” for subsequent research. Although the semiclassical and quantum approaches employ similar mathematics the underlying physics is different. The contrasting sentiments of the two camps are probably best summarized in an exchange between Mandel—Woif [61 for the semiclassical and Jakeman—Pike [7] for the full quantum protagonists. There now exist several volumes devoted in part to photoelectron counting: Klauder and Sudarshan [8], Perina [9], Loudon [10], Crosignani, DiPorto and Bertolotti [11] and Saleh [12] as well as review articles of which the most pertinent are: Arecchi [131, Lax [14], Jakeman [1 5] and Pike and Jakeman [16], reference is also made to Mehta [17] for the semiclassical viewpoint. The purpose of the present essay is to provide a detailed analysis of those theoretical aspects of photoelectron counting which are capable of experimental verification. Most of our interest is in the physical phenomena themselves, while part is in the mathematical techniques. Many of the mathematical methods used in the analysis of the photoelectron counting problem are generally unfamiliar to physicists interested in the subject. For this reason we have developed the essay in such a fashion that, although primary interest is focused on the physical phenomena, we have also taken pains to carry out enough of the analysis so that the reader can follow the main details. We have chosen to present a consistently quantum mechanical version of the subject, in that we follow the Glauber theory throughout. In fact, we assume the reader to be familiar with [2]. No attempt has been made to provide an exhaustive list of articles written on the subject, rather, our references are somewhat eclectic and are chosen to reinforce the main text. Saleh’s recent text [12] contains a reference list of 521 items (!), sufficient to whet the appetite of any photoelectron counting bibliophile. Prominent among our references are the publications of the Royal Radar Establishment (Jakeman, Oliver, Pike) whose viewpoint we generally share. The plan of the essay is as follows. Section 2 is devoted to a summary of those aspects of Glauber’s theory directly pertinent to photoelectron counting. The general theory is continued in section 3, with emphasis upon the development of N-fold photoelectron generating function which appears here for the first time. Initially, the central theme, as conceived by Glauber, was the determination of the single-fold statistics of the photoelectrons for Gaussian light. Section 4 summarizes the various aspects of the problem and is based in part upon an unpublished manuscript by Barakat and Glauber [18] as well as unpublished work by Barakat and Blake. Some of this work was independently carried out and published by Jakeman and Pike [19].
8
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
Although the determination of the one-fold statistics is interesting, the information that is obtaine insufficient for many purposes. It is necessary to develop and exploit the formalism appropriate ~rtwo-fold statistics (especially correlation functions). Section 5 covers this aspect of the problem regards Gaussian light. The elegant paper by Jakeman [20] giving an exact solution to the -oblem for a Lorentz line is covered in full, especially since the analytical details are essentially nitted from the original paper. We are indebted to Dr. Jakeman for helpful correspondence. npublished work by us on the double generating function for partially polarized light is also .mmarized. Time interval statistics form the subject of section 6. Practically all the work is based on published Ld unpublished work by Glauber and Barakat, and by Blake and Barakat. To this point we have ~ated the case of Gaussian light. In many problems, such as scattering by a small number of trticles, the scattered light is non-Gaussian. Thus, the first and second moments are insufficient to Laracterize the distribution and higher moments are needed. Section 7 reexamines the various oblems discussed in sections 4, 5 and 6 for non-Gaussian light including the Chen, Tartaglia and isey [211 version of the non-Gaussian photoelectron correlation function. The problem of termining the spectrum of the incident light from measurements of the quantities discussed in evious sections is briefly discussed in section 8. Finally, the influence of spatial correlations on totoelectron counting, as outlined in Kelly [22] , is summarized in section 9. A number of topics have been omitted, such as dead time corrections, processing techniques, mpling scheme, etc., see [15, 23, 24] for such information. The thrust of this essay is on the incipal underlying physical ideas and not necessarily on applications of these to technologically eful applications. Summary of the Glauber theory of quantum optics This section is a summary of Glauber’s full quantum description of the interaction of the electroignetic field with the n-atom detector. Of particular importance, insofar as photoelectron counting itistics is concerned, are the coherent state description of the electromagnetic field and the representation of the density operator. t. Coherent states and correlation functions Let us review briefly Glauber’s coherent state description of the electromagnetic field. The umption of finite boundary conditions of a general sort permits us to describe the field as conting of a discrete set of modes, each analogous to a single harmonic oscillator. If ak is the nihilation operator for a photon in the kth mode, and its adjoint, a~ the corresponding creation erator, then these operators obey the commutation rules [ak,a~] [ak,ak’]
=
ókk
[a~i,a~J
=
0.
(2.1)
The stationary states for the kth mode are those consisting of precisely n photons. They can be istructed from the vacuum state I ~ by writing In)k
(n!)1”2 (a~)n I0)~~, n
=
1,2,
-
.
-
.
(2.2)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
229
The n-photon state is an eigenstate of the number operator (2.3)
Nk —akak
since NkIn)k =nln)k.
(2.4)
Though complete and orthonormal, these n-photon states are not a particularly useful basis when the quantum numbers describing the field are large. In this limit, a more useful basis is the set of eigenstates of the annihilation operator, ak: =o~kI{ak})
akl{ak})
(2.5)
where the ~‘~k} are arbitrary complex numbers. These states, introduced by Glauber, are the coherent states. They may be expressed as a sum of the n-photon states by writing (2.6) Ictk)=exp {—~IakI2} (n!)”~ (ak)’~In>k for the kth mode, and then forming the direct product over all the various modes:
I{ak})=~Ia)k.
(2.7)
The average number of photons in the kth mode, = =
,Tk,
is given by
({ak}INkI {o~k}) IakI exp{—IctkI2} ~ ~9~I~1çI = I°~kI.
(2.8)
Similarly the probability that a given coherent state has exactly n photons in a particular mode is a Poisson distribution
I(flkl{~})I = I(flkl Hk exp {—
=
=
(ñ~)’1e
—
2(ak)”k’ Ink’)k’I
2Iak’I}
~
(n~’Y”
(2.9)
n~
The reason for the name coherent state is that these states satisfy the classical condition for nth order coherence; we will demonstrate this shortly. Finally, we note that the coherent states form an overcomplete set, and are not orthogonal. Consequently the expansion of an arbitrary state vector is not completely straightforward. It is useful to express the electric field operator as the sum of two nonhermitian operators: E
(Pt) = ~
(Pt) + E~—~ (Pt).
(2.10)
The positive frequency part ~ contains all terms which vary as exp (—iwt), the negative frequency part ~ all those which vary as exp (iwt). Such a decomposition is useful because we may write E(+) entirely in terms of annihilation operators and E~)entirely in terms of creation operators. For example, if we employ box normalization we have ~
(Pt)
=
i ~ (~hw,~)”~ akUk(r) exp ~—iwkt}
where the mode functions, Uk(P), form a complete orthonormal set. Since
(2.11)
E(+) (Pt) is a linear
R. Barakar and J. Blake, Theory of photoelectron counting statistics: an essay
230
colnbination of the ak, we see that the coherent states are eigenstates of E(+) (Pt): E(~~(Ft)I {~k}~=~(Ft)I {ctk})
(2.12)
where ~ (Pt) is the c-number field derived from our expression for E(+) (Ft) when all tile
ak
are
replaced by ctk t (F,
t,
{ctk})
=
i ~ (~liw,,~)~ctkUk(P) exp {—iwkt}
~ c~ê,jPt).
(2.13)
In most physically realizable situations the field is not in a pure state; our knowledge of the parameters describing the field is incomplete. In making predictions of any measurements, then, we must average over an ensemble of fields prepared in a manner which makes them all consistent with whatever knowledge we do have. The most convenient way to handle this averaging is through the use of the density operator (2.14)
P~fI~3)(/3I}av
where I ~3)is one of tile possible field states. and {
}av
denotes an ensemble average. The expectation
value for any operator is given by (2.15)
{I~9I/3)}av=trp&.
We will find it useful to introduce a class of functions, called nth order correlation functions, defined through G~°~ (x1
..
.
x2~) tr [pEL) (x1)-
.
.
E~)(x0)E(~)(x~+ ~)
..
.
EW (x20)]
(2.16)
where x1 represents the /th space-time point, (F1, t1). The product of field operators is normally ordered, that is all the positive frequency factors precede the negative frequency factors when reading from right to left. Because of this normal ordering, the value of G(?1) (x1’ x20)in a coherent state becomes a product, each of whose factors depends upon only a single space-time point; .
G(°)(x1--x2~)—~ g*(x1 ~ak})’ ..g*(x0, {~k}) ~(x0~
i~~k})’’
g(x20, ~k}).
(2.17)
For fields which are not pure coherent states we will still be able to make use of this property to provide a useful alternate representation for ~ (x1’ ‘
2.2. P-representation
By using the overcompleteness of the coherent states, we may express any density operator by writing pHHI{ctk})({ctk}IpI{I3k,}>({I3k,}Id2akd2~k,.
(2.18)
For a large class of fields we may collapse this integral into a simpler expression: 2ak, (2.19) p=~P({ak})I{ak})({c~k}Ifld If a real-valued function, P({ctk}), exists so that it is possible to express pin this way, we say that the field has a P-representation. In such a case we may evaluate the average of any normally ordered
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
product of the operators E(+) and ~ becomes ~
(x1
231
easily. In particular, the nth order correlation function 2ak.
x2,~)= ~P({ak}) g*(x1 {ak})~ g*(~, {~h,c})~(Xn +
. . .
.
1~{a,~})-. . ~
{ak}) H d
(2.20) The P-representation for a pure coherent state is easily obtained. For simplicity we consider a single mode only. In this case the density operator for a pure coherent state is p1j3)( 131.
(2.21)
We now look upon eq. (2.19) as an integral equation for the unknown function P(a). The solution, by inspection, is P(ct) =
(2.22)
5(2) (a — 13)
where 6(2) (a) is the two-dimensional Dirac delta function 6(2)
(a)_ ö(Rea)6(Ima).
(2.23)
Generally, the integral equation cannot be solved by inspection and we must make use of the normally ordered characteristic function X N (s)
tr [p ~
e5*~~ 1
(2.24)
where N stands for normal ordering. Substituting the P-representation of p into this expression, we have XN (s)
=
f d2ctP(a) e(s~*
—
s*a)
(2.25)
This is a diagonalized form of the two-dimensional Fourier transform of the P-function. if the Fourier transform can be inverted, then P(ct)
=
~
~ d2s XN
(s) ~
— sa*)
(2.26)
which yields P in terms of p through the definition of XN. The conditions under which this formal inversion is possible have been described by Cahill and Glauber [25]. Fortunately this inversion is possible for the fields which interest us in photoelectron counting experiments. Perhaps the most important state from our point of view is the chaotic state, the most common state occurring in nature. Again restricting our consideration to a single mode, we define chaos to be the state with the maximum entropy S = —tr [p log p1
(2.27)
subject to the constraints of a fixed mean occupation number ñ—(n>—tr [pa~a],
(2.28)
and the normalizing condition trp= 1.
(2.29)
R. Baralcat and J. Blake, Theory of photoelectron counting statistics.’ an essay
232
Following the usual procedures of the calculus of variations we introduce two unknown Lagrange multipliers m~?72, and set the variation of S equal to zero. This requires that pexp{—r~1 —~2a~a}.
(2.30)
The Lagrange multipliers are determined from the two constraint equations. We find, then, that I
I (n)
laa
~i +(n)i
(2.31)
-
Following the procedure for determining the function P(cs), we first must evaluate XN (s). This can be shown to be 2}. (2.32) XN(s) exp{—(n)Is( Using eq. (2.26) for the inverse Fourier transform, we have P(a)
ir(n>
exp{—1a12/(n)}.
(2.33)
Thus tile P-function describing a single mode chaotic state is a Gaussian with variance proportional to (11). Thermal equilibrium states, typically generated by the usual thermal light sources (mercury arc lamps, or lasers operating below threshold) are special cases of the general chaotic state in which the mean occupation number for each mode is given by the Planck law (nk)
=
[exp (/lwkIkBT)
—
1] —1
(2.34)
The P-representation obeys a convolution property: if P 1 (a), P2 (a) describe fields which would be produced by two independent sources, then the P-function of the sum is P(a) ~P1(a1)P2(a_a’)d2a’.
(2.35)
This convolution property is useful because we can use it to generate the description of new fields. Consider, for example, N independent chaotic modes, each with mean occupation number (nk). By repeated application of the convolution integral it can be shown easily that the P-function for the total field is multivariate Gaussian 2/(nk)}. P({ak})
k~1~k>
(2.36)
exp{—IakI
We will need also the P-function for the sum of a chaotic state and a coherent state. We have ~ 62(ct’_13) ~_~.~~exp{_Ia_&I2/(n)}d2a?
ir(n)
=
~exp{_Ia_13I2/(n>}. ~r(n)
The P-function is still Gaussian, but with nonzero mean 13.
(2.37)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
233
2.3. The n-atom detector Let us consider a detector which functions by absorbing photons from the incident light. Upon absorbing a photon, the detector displays an observable change which we can use as a signal that a photon has arrived. The simplest example of such a detector is a single atom which undergoes ionization by photoabsorption. We first examine the behavior of this simplest detector, and then discuss the behavior of a collection of these atoms, a model which is a good approximation to the photocathode of the photomultiplier usually used in photon counting experiments. We assume that the atom is much smaller than the wavelength of the incident radiation. If, additionally, the field is weak, we may describe the effect of the field by writing the interaction Hamiltonian in the electric dipole approximation: H1 = —e
i~ E(ft),
(2.38)
where i~is the coordinate operator of the nth electron of the atom relative to its nucleus, and the sum is taken over the valance electrons in the atom. Glauber has shown, using first order time-dependent perturbation theory, that the probability of a photoabsorption during an observation time t is given by pO) (t) =
~-~--
JJ
G(’) (Pt’, Pt”) dt’dt”
s(w) exp{i(t” — t’)w}dw,
(2.39)
where s(w) is the sensitivity of the detector. s(w) contains the matrix element of q between initial and final atomic states, and may be frequency dependent. We shall assume that s(w) is essentially constant over the frequency range of the incident light, which is an excellent approximation in most photon counting experiments. The o.,-integral then reduces to a delta function, and eq. (2.39) becomes 1~ (Pt’, Pt’) dt’. ~(l)
(t) = s
(2.40)
G~
The counting rate, the probability per unit time of detecting a photon, is w(t)
d =
p(’~(t)
=
sG~’~ (Pt, Pt).
(2.41)
This is proportional to G~’~ (Pt, Pt), the first order correlation function, which is the average intensity of the light. In deriving eq. (2.39) from the interaction Hamiltonian, eq. (2.38), it is necessary to make one further assumption which warrants some discussion. We have pictured our detector as functioning by absorbing a photon. Actually, transitions which do not conserve energy — transitions in which a photon is emitted during ionization — are possible as well with the form of the interaction Hamiltonian given in eq. (2.38). If we replace E(Pt) with the sum of the negative and positive frequency parts, we find that the matrix elements of the negative frequency part, E(), oscillate very rapidly. Since the observation time is necessarily much longer than these oscillations, the contribution of ~ is negligible. Recalling that ~ is composed entirely of annihilation operators, we see that neglecting ~ is equivalent to ignoring those occasions on which a photon is emitted
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
234
during ionization. In deriving eq. (2.39) we thus impose conservation of energy in a way not strictly compatible with the uncertainty principle. Were we not to make this approximation, we would lose tile nornial ordering in eq. (2.39) and the problem would become untractible. We next consider a detector consisting of an array of n atoms, each atom behaving independently, and in exactly the same way we have pictured our one atom detector. If we expose the detector to tile field at t = 0, we may ask tile probability that the first photoabsorption occurs during the time interval (0, t1), the second during (0,t2). and so on. As long as the total number of photoabsorptions is much smaller than the total number of atoms, this compound probability is given by
f J
tO
tl
p(~~) (t1,’~ ‘t0)
s°
‘‘‘
G~°~ (F1 ti’’
‘Fflt~, F0t;~.‘‘Fi
t~)fl dt
(2.42)
where F1 is the position of the jth atom. Tile derivation of this result, due to Glauber, follows tile same approach used in analyzing the single atom detector. As before, we can ask for the average rate at which the photons are counted, in this case, the rate at which this compound event occurs. This is given by W(t1
- ..
t0)
=
ut1
a°~ ‘
ut0
p~°~ (t1’’
‘t0)
=
s°G~°~ (F1 t1~ F~t0,f~t0~’’ F~t~). . .
(2.43)
3. Photoelectron generating functions This section is devoted to the derivation of photoelectron generating functions using the tools developed in the previous section. 3.]. Single-fold photoelectron generating function
Let us consider a photoelectron detector which is in tile form of a plane perpendicular to the direction of propagation of a collimated beam of plane waves. We further assume that the large number, N, of atoms in the photosurface all experience the same field; that is the detector subtends less than one coherence angle. In this case all atoms are equivalent, and each contributes equally to the dynamics of the detection process. While this is far from the most general geometry, it approximates that used in most experiments. The difficult question of the effects of including more than one coherence angle will be treated in a subsequent section. Each atom of the photodetector has only two states of interest to us: the ground state, and an excited state in which the atom has absorbed a photon and emitted a photoelectron. This suggests defining an operatorz1 with the property: = =
0, if jth atom is in its ground state 1, if jth atom has emitted a photoelectron.
We shall ignore those few occasions on which an excited atom decays to its ground state during the time of observation. The expectation value of z1(t) is then a random function of time which is initially zero and jumps discontinuously to unity at some unknown later time. By using the operators we can describe the counting statistics of the n-atom detector in a much more useful way than
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
simply stating the value of the joint probability that atoms i,j, k, times t1, t1, tk,
235
have emitted photoelectrons by
~-
The photoelectrons emitted by different atoms are indistinguishable. Accordingly we can ask questions only about the total number of photoelectrons emitted during some time interval, t. Let us define the random operator Z(t)_ ~ z1(t),
(3.1)
whose expectation value is the total number of photoelectrons emitted by all the atoms in the detector. If we expose the detector to light for a fixed time period (by means either of a mechanical shutter or an electronic gate) we will find that Z(t) has taken on a particular value, and that this value will be different each time we repeat the experiment. What we must evaluate then, is the single-fold probability distribution ofZ(t). Specifying the values of z~(t)for / = 1, N uniquely determines the final state of the detector. The most complete description we can hope to calculate, however, is the probability distribution of these values, p [z1 (t), z2 (t), , ZN (t)1, which we abbreviate p(z1, , ZN, t). Once this distribution is determined, we can compute any observable function of z(t) by evaluating the ensemble average: . . -
...
~4
(f[z(t)1)
..
~ zk(t)] p(z~,. ~,zN,t).
allz1
(3.2)
.
k
Our task now is to express the distribution p(z1, ZN, t) as a function of the joint probabilities p~(t1,~ t,,) which we evaluated in the last section. Consider first the marginal distribution for the /th atom alone: .
. -
p1(Z1,
t)
p(z1,
=
,ZN,
t).
(3.3)
all zk (k * I)
1)(t) can be related to this marginal distribution by noticing
The one-atom transition probability p( that p(1)(t)—p 1(l,t)
~
(3.4)
Z1p~(Z1,t).
Zj
Similarly, the n-atom transition probability, p(fl) (t1, be expressed as p(~)(t1,~
.
, t~)5
tnt
=
Z1Z2
~
ZN p(Z1,z2
tn) evaluated
,ZN, t).
with t1
=
t2
=
=
tn
=
t
can
(3.5)
The marginal probabilities make it easy to express ensemble averages over equivalent final states of the detector: (Z(t))~~
p1(1,t)
(3.6)
and in particular, the single-fold generating function Q(X) defined by m p(m,t)((l \)m) Q(X)~ ~ (1 —X)
(3.7)
Z1(t~
j~1
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
236
becomes (1 _X)Z(t)
(3.8)
p(Z1’’.ZNt)
allzk
The operators z~(t)commute since they refer to different atoms, therefore (I
—
X)z(t)
Further, as (1
—
N
(3.9)
t)
fl (1
= j=1
—
X)ZJ(
can take only the values one or zero:
X)z/(s)
1
=
—
Xz
(3.10)
1(t).
Substituting this expression into eq. (3.8) yields Q(X)
=
~
[II
allzj~
jl
(1
—
~~)] p(z1,~
,ZN,
(3.11)
t).
We now expand the product and obtain Q(X) as a power series in X Q(X)
=
~
(—X)° ~
n0
=
~
allzk
~
Z1
‘Z1
-
p(Z1,’
“
,ZN, t)
~
(—X)°~p~”~(ti,’’~,t0)I
(3.12)
~
where .fF indicates that the sum is to be taken over all the n-fold combinations. All the atoms in the detector are exposed to the same field by our original assumptions, and so the sum over the n-fold combinations is just the number of possible ways of grouping n objects from a total collection of N: (t1,’’
.
~
=
=‘=~~=~
Since the number of excited atoms, we can write N! n!(N—n)!
n! (N— n)! p(fl) (t1,’’ ,t0)j~
ii,
‘
=...=
~
(3.13)
is far less than the total number of atoms in the detector, N,
N° n!
(3.14)
Finally, substituting eq. (2.42) which gives p(0) (t1, function, G~°~ (t1,ta), we have
.
.
.
, t~)interms of the nth order correlation
. .
Q(X) = ~
...
~
(t~’ ‘t~,t~---t)
U dt.
(3.15)
Actually the sum should terminate at N, but since N is so large we commit little error in letting N be infinite.
R. Barakat and I. Blake, Theory of photoelectron counting statistics.’ an essay
237
This sum looks almost like the exponential function series. However, ~ generally involves a product of noncommuting operators and as it is not possible to collapse the series in its present form. If the field has a P-representation, then ~ can in fact be expressed as a product of c-numbers, as we saw in section 2.2. In fact, eq. (2.20) may be generalized to the following identity (Ns)~
. -.
G(~)(t~
.
.
.
t~,t~ ..
.
t~)Hdt;
=
~P(~ak}) ~(t,
{ak})
H d2ak
(3.16)
where (Ns)
I1(t’,
{ak})I2dt’
(3.17)
is the time integrated intensity of the random field amplitudes and is, of course, itself a random process in time. Thus for fields which can be described by the P-representation, we may write the expression for the generating function in a very compact form: Q (X)
P( {ak}) exp ~—X~2(t,{ak}) } H d2 ak.
=
(3.18)
Equation (3.18) resembles a Laplace transform and we will find it advantageous to define the function W(~2) ~P({ak}) 6[~2 — ~2’(t, {ak})] H ~
(3.19)
We can now write the generating function as Q(X)
=
~ W(~2,t)e ~ dH
(3.20)
so that Q and W form a Laplace transform pair. Upon differentiating Q(X, t) in accordance with eq. (1.4) we have -
P(m,t)j
W(12)
m!
d~’Z.
(3.21)
The usefulness of this result is obvious because P(m, t) can be obtained by integration rather than by repeated differentiation of Q(X). The factorial moments are easily shown to be (m-~.(m—/+ 1))m(m(/))’~ W(~2)~’2’d~.
(3.22)
3.2. Mixed Poisson process The integral representation of P(m, t) given in eq. (3.21) is more than just a useful computational device. Suppose that W(fZ) were itself a continuous probability density function with respect to ~2
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
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over the interval (0,oo). Then m(t) is termed a mixed Poisson process (the terms compound and generalized are also used). The interpretation is simple. The real variable ~‘1~ in eq. (3. 1 7) is nonnegative since it is confined to the interval (0,oo). The factor ~m exp (—&Z)/m! in the integrand is a Poisson distribution with respect to ,~2,the mean of the distribution. Consequently P(rn, t) can be interpreted as a probability distribution in which the mean, is a random process having first order probability density W(~).Since &Z is non-negative it must be a non-Gaussian random process, even though it is a Gaussian random process. Thus eq. (3.21) connects the statistics of the continuous random process ~2(t) to the statistics of the discrete random process m(t). Necessary and sufficient conditions on W(&2) for it to be a probability density function over the interval (0,oo) with respect &2 are that it be a real function and simultaneously satisfy ~
1. ~ W(2)d~2—1, forallt 2. W(~)~ 0,
for all ~2and t.
The first condition is automatically satisfied irrespective of the second condition by setting X = 0 in eq. (3.7) and noting that ~ in
P(m,t)
1.
(3.23)
0
The non-negative constraint imposed by condition 2 must be handled on an individual basis. The mere fact that P(m, t) is expressible in the form given in eq. (3.21) does not imply that rn(t) is a mixed Poisson process because W(~2)can be negative over a finite interval and still satisfy condition 1. Reference is made to Schell and Barakat [261 for a physical situation where W takes on negative values and thus rn(t) is not always a mixed Poisson process. We can anticipate that m(t) is mixed Poisson and a formal proof will be given shortly. Since W(f~)is thus a probability density function, then the first moments of m(t) and cZ(t) are connected by m
(rn)
d~’2W(&2)e n
=
_______=
~1(mU!
0
J
&2W(~)df~= (~2).
(3.24)
0
The average number of photoelectron counts is thus equal to the ensemble average of ~2. The relation between the variances of m and &2 can be evaluated in similar fashion and is varm
(m)+var~
(3.25)
where var ~
J
2 [~‘~ —
(~)12 W(~2)d&~=
(~72)—
(3.26)
(rn)
This expression is important because it shows that the variance of the probability distribution of the number of photoelectrons is composed of two terms: the first term on the right-hand side of eq.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
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(3.25) is the variance of a Poisson process and would be the only term if the photons showed no tendency to cluster; the second term is the contribution of the mixing density W(~2).Since this second contribution is positive, the variance of a mixed Poisson process is greater than the variance of a simple Poisson process having the same mean. This second term, var ~ is a manifestation of the fact that photons, when in the thermal equilibrium, obey Bose statistics so that the photon arrival times are not statistically independent of one another but are correlated. 3.3. N-fold photoelectron generating function Almost all experiments designed to extract spectral information from photoelectron arrivals make use of two-fold or second order statistics: conditional probabilities, correlation functions, or clipped correlation functions. The reason for this is that the details of the spectral profile are much more strongly reflected in these two-fold distributions than in the single-fold quantities. As the generalization to three- and higher-fold statistics follows easily from the two-fold case, we will treat only the two-fold problem in order to keep the notation as simple as possible. Our interest is now in the joint probability P(m1, t1 rn2, t2) that m1 photoelectrons are counted during a time interval designated by t1, and subsequently m2 are counted during the interval r2. Notice that we are using t1 and t2 to specify an interval in time, and not necessarily that range t = (0, t1). As before, it is useful to define a two-fold generating function by m’ (1 (1 —X1)
Q(A1,X2)= in1
_A
P(m1,t1m2,t2).
2)2
(3.27)
Ut2
1 and Z2, whose expectation values are the number of Let us define a pair of random Z photoelectrons counted during operators, t 1 and t2, respectively. Each may in turn be written as a sum over the contributions of the individual atoms in the detector: Z’ = ~
4,
4
4
where
4 4
=
and
2
=
z
4
(3.28)
are random operators defined by
1
if the /th atom emits a photoelectron during time interval t
0 1 0
otherwise; if the jth atom emits a photoelectron during time interval t2 otherwise.
1 = = =
Thus, 2) Q(X1,X2)=
((1
—
=(INII \j=1
4
X1)z’ (1
— A)Z
(1 —zJXi)
N
\
II (1 —z~X
2)). k1
Following the development in subsection 3.1, let us define the probability distribution of as p(z~ Z~, t1 z~ z~,t2), and note that this probability distribution is related to the . .
(3.29)
/
4
and
R. Barakatand J. Blake, Theory of photoelectron countingstatistics: an essay
240
(rn1 + rn2)th-order correlation function through ~ ~ {z)}
{4~
(5) ~
+
~
~dt~-- Jdt’ _______
~dt~
_______J
+ 1
~dt’in,
‘‘‘
+
G(m,
in,
+
in,)
(t~
-
+ m,;
~
+ ~,
. . -
t~).
_____________
‘.—~-
tl
(3.30)
t2
The expectation value in eq. (3.29) for Q(X1,X2) can now be expressed as an explicit sum over the probability distribution of 4 and 4: Q(X1,X2)=~ ~ ,
{(_Xl)m1(_X2)m2
z~
~
rn,
.
LF,
-
~ z~’.‘z~ p(z~’ z~,ti;z.z~,t2)}(3.3l)
-z~
where,~indicates a sum over all permutations of z~ z’,~,..As all the atoms see the same field, this sum over the permutations is just N!/(m~!(N rn,)!) times any of the single terms, and as N rn~, we may replace this factor with N’°i/m~!.Thus the generating function becomes -
~‘-
m’ (—X ~
Q(X1,X2)
~
(—X1N)
in,
~ z~ -z~ p(z~-‘-z~,t1z~”’z~,t2)
2N)
‘
(ml)!(m2)!
in2
(—X~Ny~’ (_X2N)in2
=
in,
2
{z)}{Z~}
~dt’1--- ~dt~
~
m, ________
_)
______________
..—
t2
tl
(3.32) where we have employed eq. (3.30) to express p(zi,- z~,t2) in terms of the correlation function. By expanding G(m, + in,) (t~ ~fl + in~ + ~, t~)in the P-representation, one may prove the following identity (a generalization of eq. (3.16)): - -
. . -
(Ns)in,+m, Jdt~---~dt~, Jdt,,+1 ________
_)
‘—
Jdt’in,
..
..
.
+
in2
G(m1+m2)(t~
-t~,+~
tl
...t’~ ‘3
t2
2ak fP({ak})~m,
t,,.~
______________
(t1,’{ak})~’2(t2,
(3.33)
{ak})Hd k
where &2(t 1, {ak})
Ns ~ I~(t’,{ak}) 12 dt’.
(3.34)
ti
Thus,
m2
Q(X1, X2) = m,,rn,
(rn
1
~ P({ak})~in1 (t,,{ak})~
(t
2ak 2, ~ak}) H d
1)!(m2)’
= (exp {—X,~2(t1, {aAj)}
exp {—X2~’2(t2,{ak})).
(3.35)
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In this form Q(X1, A2) is an obvious generalization of eq. (3.7). The single-fold generating function Q(X) is obtained by setting A2 = 0. For this reason, it will be more efficient in the subsequent section to treat initially the evaluation of Q(A1, A2) and thus obtain Q(A) automatically. Having computed Q(X1, A2), we can then prove by induction that the n-fold generating function is ~
(exp{—A,~(t1,{ak})}
~~n)
..
~exp{—A~&~(t~,{ak})}).
(3.36)
3.4. Eigenvalue problem for generating functions The generating function can be evaluated by solving an eigenvalue problem involving an infinite matrix whose eigenvalues and eigenvectors determine Q(A1 ,A2). However, a useful computational approach for the completely chaotic case, where only the eigenvalues are needed (more on this point shortly), is to convert the infinite matrix eigenvalue problem into an integral equation whose eigenvalues are identical to those of the original problem. The P-representation for completely chaotic field is given in eq. (2.36) where (nk> is the mean occupation number for the kth mode. From the central limit theorem, any light source which we can in principle divide into many independent parts will radiate chaotic light. The set of all (nk) display the spectral profile of the radiation, 1(w), through the expression 1(w)
= hw(nk)
(3.37)
where w is the frequency. Upon substituting eq. (2.36) into eq. (3.35) for the two-fold generating function we obtain: 2 dt’ Q(A1,A2) ~exp ~ IakI — X,Ns ~ S(t’, {ak})1 k (nk) d2a A 2 dt’ H Ic (3.38) 2Ns I e’(t’, {ak})1 ) k in the form of a multiple integral. This expression can be simplified by defining the Hermitian matrix
f—
—
Mkk~ Ns {nk
J
[(x
1~
dt’ + X2 ~ dt’) e~(rt’)ek (rt’)](nk~)h12}
(3.39)
where ek(t) is given by eq. (2.18). Also set (3.40) With these definitions, we can rewrite eq. (3.38) in the succinct form Q(A1 ,A2) = ~exp {~+ (I + fi~} H
dyk
(3.41)
here ~ the unit matrix. Since M is Hermitian, it can be diagonalized by a unitary matrix U: CT~J1~UAdiag(m1,m2,-..)
U’~=~3.
(3.42)
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Performing the necessary change of variables, we can perform the integration explicitly: r d213 Q(A 3(I+ A)f3} Ii J k 1,X2) = ~exp{—1 -
~ exp{—13~(l~k)13k}
d13k
11(l
(3.43)
~‘~k)
where 1~k = 1~k(A
1, A2) are the eigenvalues of M. Consequently the problem shifts to the evaluation of these eigenvaiues. As we have just remarked there are two possible approaches. The first is to consider M itself by truncating it to finite size and evaluating the eigenvalues via any standard diagonalization scheme. The second is to transform the original problem into an equivalent eigenvalue problem involving an homogeneous Fredholm integral equation of the second kind having a symmetric kernel. Actually the methods complement each other in that the matrix formulation involves tile line shape 1(w) directly whereas the integral equation formulation involves the Fourier transform of the line shape, the correlation function, as its kernel. Although the matrix formulation is perfectly valid, the disadvantage is that the eigenvalues depend in a complicated way upon A~and A2 making the ensuing computations very difficult. (Nevertheless, we actually performed several calculations using tile matrix method to check the integral equation method.) With these remarks, we now outline a derivation of the integral equation formulation of Q(A1, A2) originally due to Jakeman [19]. Following tile formalism developed in subsection 2.2, we see that the statistical independence of the different modes requires that 6k, k, (3.45) (ar, ak2) = (nk) We will not work with the functions ek(t) appearing in eq. (2.28), which are orthonormal over tile time interval (_oo, 00), but with a set of functions ~ [5] which satisfy the equation -
(xi
J dt’ + A2 J
dt’) ~$(t’)Ø,(t’)
We further require that ek (t)
=
ek(t)
= 6~.
(3.46)
be related to ~1(t) by a unitary transform
~ Sk/ø/(t)
(3.47)
where ~S*SIc!
if
=Lö
-
ki’
~
~S =6 if-
4
k/Il
Corresponding to the coefficients, ak, the projection of a1
= [x1 ~ dt’ + A2
~ dl”]
~7(t’)I(t’, {ak }) =
I (t,{ak})
~ a1S,~J.
along a particular ~~(t) is given by (3.49)
This permits us to write A1 ~2(t1, {cz~}) + A2 ~(t2, ~ak })
= Ns ~
2. 1a11
(3.50)
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From the various sets of ~1(t), we assume that there is one particular set for which the a1 are independent
6~
(a1a1>
= (r1) 1
(3.51)
where the Kr1> are linear combinations of the (nk) whose explicit form we momentarily leave unspecified. The P-function for {a1} is 2/(r II
—~—-
exp{—1a11
1)}
(3.52)
and follows directly from eq. (2.33) and eq. (2.35). The double generating function can now be written directly in terms of Kr1> by following the approach we employed to derive eq. (3.43): 21 H ~d2a. = H 1 Q(A 2 ~ (3.53) r Ia.1 Kr 1, A2) = exp i. —NsIa,1 1> .i / ir(r1> / 1 + Ns(r1> .
—
The eigenvalues of the matrix M are NsKrk>. To prove this statement, we consider the matrix element (S~MS)kk’.We have 2 S/k’. (3.54) ($~I~IS)kk’ Ns~A1tl dt’ + A2 t dl”) ~S~1~(n,)hi’2 e7(t’)e1(t’)(n,)”
J
f
2
I,!
Upon expressing ek(t) in terms of~1(t)via eq. (3.47) and making use of the orthonormality of the 2
S,~S,~’S
(S’~MS)11. Ns i,j,r ~ (n)”
2 1~S11~(n,>”
= Ns ~ (n
1> S~Slk’= Ns ~ (a7a1)S~SIk’.
(3.55)
Next use eq. (3.49) 6kk’.
(S~MS)kk’= Ns ~ (ct7S7~ct~S(k’> = Ns(aak’> = Ns(rk> Consequently, m 1
(3.56)
= Ns(rk> are the eigenvalues of M.
To construct an integral equation with eigenvalues Kr1> we note that from eq. (3.49) we can derive the following expression for G~’~ (t’, t): G~’~ (t’, t)
=
=
(g*(~~ {ak})g(t,
{ak}))
~ (/~‘(t’)a1/~ (t)> = ~ (r~>~7(t’)~1 (t).
(3.57)
Next multiply both sides by Ø1(t’) and operate on them by the integral operator
(x,
f dt’ + A2 ~ dr’).
(3.58)
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The final result is the integral equation
(x1 ~ dt’ + A2
J
dt’) G(’~(t~t) = (i~)Ø~ (t).
(3.59)
1~ (t~t) becomes G~’~ (It’
—
tI). Use of the normalized
If the process is stationary in time, then G~ correlation function g(It’
—
tI) =
G~1~ (It’— tI)/G~’)(0)
(3.60)
and the average counting rate of the detector, w. wNsG(t,t)
(3.61)
permits us to rewrite eq. (3.59) in its final form (A
J
1
dt’ + A2 ~ dt’) g(It’
—
tI)01 (t’)
~! ~ (t).
(3.62)
We have thus found an integral equation whose eigen values are precisely the same as those of the matrix M, and it is this equation which forms a basis for the development of the next few subsections. By setting A2 = 0, we obtain an integral equation A
J g(t’
—
t)4~(t’) dt’ ~
~ (t),
(3.63)
whose eigenvalues enable us to evaluate the single-fold generating function Q (A). When the generating function has been evaluated as an analytic function of A1 and A2, the joint probabilities and factorial moments may be extracted by using the definition, eq. (3.27). Specifically, it is easy to show that P(m1t,;m2t2)=
(—l)”~’+in2 m1 !m2!
~3in,
~
ain, ~
Q(A1A~~
(3.64)
and
1
((m
a”
a
1) (m,
—
1)-~-(m1 —1+ l)(rn2)(m2
--
I)-- -(m2
—
k + l)>
(—l)’~”~ U14
Q(AX) x, =o N2
U!~.2
=
0
(3.65) Generalization of eqs. (3.64) and (3.65) may be derived easily for the n-fold problem.
4. Single-fold statistics for Gaussian light The determination of the single-fold statistics of the photoelectrons, namely the probability distribution of the photoelectrons, was initially the central theme. This section is devoted to
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
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summarizing the relevant analyses and discussing the physical consequences. Subsections 4~2—4.6 are based upon an unpublished report by Barakat and Glauber [18] , while subsections 4.9—4.10 are also based on unpublished work by Barakat and Blake. 4.]. Evaluation of Q(A) We now proceed to the evaluation of Q(A) via the integral representation of Q(A1 ,A2) in eq. (3.43) with A2 ~0: Q(A) = j’exp~_~(1+ft~)j~} H where M is given by eq. (3.39) with A2
(4.1) 0. In the case of completely chaotic light
Q(A)H(l +mk).
(4.2)
k
From eq. (3.63) we see that each of the eigenvalues mIc is directly proportional to A. This permits us to exhibit explicitly the dependence of Q(A) upon A in this singlefold case. Before doing so, however, we will rewrite eq. (3.63) in terms of dimensionless variables. The spectrum of the light will have some characteristic width, ‘-y, which will be more precisely defined for particular spectra. We take as our dimensionless unit of time 1 /y, setting T = ‘yt. In order to avoid a proliferation of symbols we will still use Tfor the counting time, but measure it in units of l/’~’.The corresponding dimensionless variable for average counting rate is v = w/’y. We may now rewrite eq. (3.63) as
—
r’)Ok(r’)
dr’
where we have defined bk
(4.3)
Ok(r)
mI/A. This allows us to write the generating function as
(1 +Ab1)’.
Q(A)=H
I
(4.4)
4.2. P(m, T)for completely chaotic case Assuming for the moment that the eigenvalues b1 are known, we can evaluate P(m, T) by first obtaining W(~2)and then utilizing eq. (3.16). W(~)can be expressed in terms of Q(A)by the usual complex inversion integral W(~)=
1
J
X0+ioo
—
2iri A0
=
1 2iri
—
Q(A) e~~’ dA
—
J
X0+ioo
No —
-
I~O
H (1 + Ab1)” ~
j=i
dA.
(4.5)
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It will be shown in the next subsection that all tile b1 are real and positive so that the poles of tile integrand are located at A, = (—b1)” and thus lie in the left hand side of the complex A plane. If in addition, the poles are distinct (i.e., b0 > b~> b2 > ) then all the poles are simple. Under these circumstances, we close the contour in tile left hand plane and a straightforward calculation of the residues results in the expression --
W(~)~
je~Ib/,
‘
~
(4.6)
where
çfi’
(1
(4.7)
_-.~-)~‘.
The prime indicates that the term 1 = / is to be omitted. Note that W(~2)is a non~negativefunction of~2. The single-fold probability distribution P(rn, T) follows directly by substituting eq. (4.6) into eq. (3.18) and integrating termwise: P(rn,T)
(b.)rn (1 +b1)m+l
j0
-
(4.8)
The probability distribution P(m, T) of the photoncounts is thus an infinite sum of power law distribution when the eigenvalues of M are distinct. However as the time of observation is increased, the simple poles will coalesce giving rise to multiple poles, in which case both eqs. (4.6) and (4.8) have to be modified (this will be done shortly). There are constraints on the C1 functions which are very useful as a check on the numerical computations. Since P(m, T) is a probability distribution, then ~
P(m,T)
1.
(4.9)
0
in=
If we substitute eq. (4.8) into this series, we find that ~ 1=
(4.10)
C~=l. 0
This result also follows from integration of the mixing density W(~)with respect to &2, i.e., ~ W(~)d~2=> ~
1.
(4.11)
Another constraint follows from (m)vT
(4.12)
where v is the average counting rate. We have ~ rn0
mP(m,T)
~
b1C1uT.
(4.13)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
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The factorial moments of P(m,T) can be shown to be given by
~ (b1Y’C,.
(4.14)
i=’o 4.3. Eigenvalue problem for Q(A) Before we can proceed any further we must consider the evaluation of the eigenvalues b1 as a function of the time of observation and spectral characteristics of the incident radiation. The eigenvalue problem for the infinite matrix M is (4.15) which as we saw in subsection 3.4 is equivalent to solving the integral equation, eq. (3.62). In a sense, eq. (3.62) is the “Fourier transform” of eq. (4.15) because the input spectrum hw(nk) occurs in the matrix description while its Fourier transform, the correlation function, appears in the integral equation description. For the single-fold case, the matrix elements appearing in eq. (4.15) are given by exp{i(wk D[hwk(nk>]
Mkk.
1/2
[hw1’ (n1)}
1/2
—
wk’)t}
(4.16)
—
Wk’)
where D is a real constant. Unfortunately, M is a rapidly oscillating function of the counting time which makes numerical calculations of b1 useful only for fairly short times. Although some numerical calculations were carried out on finite sections of M by Barakat and Glauber [181, a more useful computational approach is the integral equation formulation of eq. (3.63). Leaving aside the actual computation of the eigenvalues for the moment, let us state two theorems on Schmidt—Hilbert integral equations which will be employed in the subsequent analysis. Since g(r) is a correlation function, then it is known to be non-negative definite in the sense of Bochner [27]. The first theorem states: If g(T) is non-negative definite, then all the eigenvalues of eq. (3.63) are positive real numbers and can be sequentially ordered in the form b0 >~b,~ b2 ~s.. >0. As an immediate application of this theorem, if T ~ 1, then b0 ~ T with the remaining eigenvalues at least an order of magnitude smaller. The product functions C1 of eq. (4.7) become .
C~0
C0~l,
(j~l)
(4.17)
and the probability function P(m, T) degenerates to m/(l + nfl” + 1 (4.18) P(m, T) ~ (vT) This is the well-known Bose—Einstein (or power law) distribution expressed in terms of its mean (m> = uT. Note that the line width, y, is absent. Thus any spectral line shape will yield the same power law distribution in the short counting time approximation. The second theorem (Mercer’s theorem) states that if g(r) is non-negative definite, then it possesses an expansion of the form [28]
g(Ir’
—
r”I)
=
~
b,0 1(r’)0,~’(r”)
/=0
(4.19)
R. Bara/cat and J. Blake, Theory of photoelectron counting statistics: an essay
248
where 01(r) are tile orthonormalized eigenfunctions of eq. (3.62). If we set r’ = r” and integrate both sides of eq. (3.62) with respect to r’ (from 0 to T), we obtain
[g(o)
dT’
=T=
b1.
(4.20)
The sum of the eigenvaiues is equal to the normalized time interval of observation. The outcome of this result is that one has to contend with successively more eigenvalues as the time of observation is lengthened. As the time of observation T is increased even more, then the larger eigenvalues tend to coalesce (b0 = b1 > b2 >-- -> 0) because g(r) begins to act like a Dirac delta function. Consequently, eq. (3.62) will possess a number of eigenfunctions having the same eigenvalue and in the limit as T tends to infinity there will be one eigenvalue corresponding to an infinite number of eigenfunctions. It is precisely this feature of the problem which makes the rigorous analysis so complicated. Nevertheless, we will turn this degeneracy to our advantage shortly. The final general result we require is the bilinear series for the iterated kernel [281 g~(T’,r”) ~
b20,(T)07(r”)
(4.21)
1=0
where g~(Tcr”) = [g(T’,y)g~_
1 (t,T”)
dy,
r~ 2
(4.22)
g, (r, T”) = g(T’,r”). Let r’ = r” in eq. (4.21) and integrate with respect to r’ over the interval (0,r). Define a function Ir(T) by the equation:
Jr(fl~J~r(T”T’)dT’
=
-..
J g(T~y1)g(T~y2) g(~y . ..
-
i)g(yr
1’
r’)Hdy,
~
~tr
[Mn.
(4.23)
The usefulness of this expression lies in tile fact that the generating function can be expressed without the necessity of evaluating the eigenvalues individually through the relation Q(A) = exp
~r~1
~
1r(T)}
(4.24)
We will utilize this result shortly in our discussion of time interval statistics. 4.4. Lorentz spectrum Generally the eigenvalues of eq. (3.62) cannot be calculated explicitly and resort must be made to numerical techniques. However there are at least two spectra for which the eigenvalues can be written
R. Bara kat and J. Blake, Theory of photoelectron counting statistics: an essay
249
out explicitly; they are the Lorentz line and the rectangular line. Furthermore these two line shapes represent the extremes in behavior; the rectangular line has an infinitely sharp cutoff, while the Lorentz line decays very slowly. In fact, the Lorentz line has the slowest possible decay. These two cases are extremely useful because they serve as checks against the long-time asymptotic expressions to be discussed in subsection 4.6, as well as being important in their own right. The Lorentz spectrum is described by I(w)=hw(nk)=
2
(4.25)
2
(w—w0) +‘-y where A, half-width at half-height, serves as the characteristic width of the spectral line. In view of remarks made in the previous section concerning narrow, symmetric spectral lines, we can simply let w0 = 0. The Fourier transform of eq. (4.25) is the correlation function and when suitably normalized is g(T)e_ITI,
(T-’yt).
(4.26)
Kac and Siegert [29, 30] have shown that the eigenvalues of eq. (3.62) with eq. (4.26) as the displacement kernel are given by b. = 2 (1 + where
(4.27)
02)_I
are the positive roots of the transcendental equation
(I _0~)tan(T01)_2ij,1,
/=0,1,~.
(4.28)
See subsection 5.4 for details. Only the first eigenvalue b0 ~ T is of any importance in the short time region (T ‘~ 1). However, the eigenvalues decrease very slowly when T is outside this restricted range. In order to obtain a general idea as to the behavior of the Lorentz eigenvalues, see table 4. 1. Note the slowness with which the eigenvalues decay as T is increased. The practical outcome of this turn of events renders the direct evaluation of C1 almost impossible if we were so foolhardy as to evaluate the infinite product for T> 4. Table 4.1 First eight eigenvalues of Lorentz spectrum as a function of normalized time T (from Barakat and Glauber, unpublished report of 1966)
/
T0.5
T=4
T=8
0 1 2 3 4
0.42664 0.04159 0.01199 0.00549 0.00312
1.55049 0.86588 0.46308 0.26776 0.16923
1.81827 1.41548 1.01634 0.71643 0.51257
5
0.00201
0.11503
0.37674
6
0.00140 0.00103
0.08271 0.06210
0.28502 0.22149
7
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
250
This apparent difficulty can be circumvented by utilizing the identity C= /
b. ‘ 1 (A)/dAI~ dQ
=
(4~)9)
-bJ’
In the special case of a Lorentz line, Q(A) is known in closed form! It is 413(X)eT [1 +~(A)]2e~)T —[1 —~(A)]2e~(X)T
430
(
)
-
where j3(A) (1 + 2vA)112. A proof of this important result is outlined in subsection 5.4. Substitution of eq. (4.30) into eq. (4.29) easily yields ‘2 — b ~eT C.=(—1)’~’ ‘ “ v(T+b 1)
(4.31)
-
This closed form for C1 now permits one to carry out numerical computations relatively easily. The temporal development of P(m, T) is shown in figs. 4.1 and 4.2 for moderate beam strengths ofv= 1,5. For T< 1, P(m,T) is essentially a power law distribution, eq. (4.18), as only C, is effectively nonzero. As T is made to increase successively more terms of eq. (4.8) come into play leading to the Poisson-like curves for large times. The numerical data shown in the previous graphs concerns only hypothetical situations, but the theory is capable of handling actual situations. Although data is available for short time experiments as for example Arecchi [131, the studies of Freed and Haus [31, 321 using a laser operating below 0.3
4 f’\6 0.2 8
4 6 12 6
0,1
8
20
.~
12 20
00
10
30
-
20
1-
30
00
40
NUMBER OF PHOTOELECTRONS
Fig. 4.1. F(m, T) for a Lorentz spectral line, u = 1 for T 6, 8, 12, 20 (Barakat and Glauber, 1966 unpublished).
20
80
120
I
160
I 200
I
240
NUMBER OF PHOTOELECTRONS =
2, 4,
Fig. 4.2. P(m, T) for a Lorentz spectral line, u = 5 for T = 6, 8, 12, 20, 30 (Barakat and Glauber, 1966 unpublished).
R. Barelet and J. Blake, Theory of photoelectron countingstatistics: an essay
251
threshold offer an ideal test of the theory since the time of observation, T ~ 1, is outside the short time region and the beam strength is very large, u ~ 1000. The times are especially important because they lie in the never-never land between the short time and long time approximations (to be discussed) where one is forced to employ the exact expansion of eq. (4.8). For a typical case of their data: w = 1.40 X 106 counts/sec, ‘y = 420 hertz, t = i0~ sec, yielding a beam strength v = 1060 and a normalized time T = 1.32. Both the theoretical curve computed from the exact solution and the experimental data points are compared in fig. 4.3, the agreement is excellent. The slight overshoot of the theoretical curve when the number of photons is small is probably due to the fact that the actual line shape is not a true Lorentzian. (One of us, R.B., is deeply indebted to Dr. Freed for making available the original data to him in 1965 and for several clarifying discussions on the experiments.) An unfortunate feature of the exact analysis lies in the fact that the number of eigenvalues required increases with increasing T by virtue of the constraint expressed by eq. (4.20). This is particularly serious for the Lorentz spectrum because the eigenvalues decrease so slowly. For this reason, long time asymptotic approximations are desirable. In the special case of the Lorentz spectrum, Glauber has already obtained two such approximations to the exact distribution. The first Glauber [2] approximation (in our present notation) is 1 ivT\m s,~,(fiT) m!~3
P(m,T)
—
1—
e~
(T ~=1)
~)T;
(4.32)
I. ~
id~
:~..:
::..
..
1O6
I
0
I
000
2000
I
I
3000
I
I
4000
. •. . ts.. . •.
.
I
I
5000
6000
NUMBER OF PHOTOELECTRONS Fig. 4.3. Comparison of theoretical calculation of P(m, T = 1.32 (Barakat and Glauber, 1966 unpublished).
T) for a Lorentz spectrum with experimental data of Freed and Haus: v = 1060,
R. Barakat and .1. Blake, Theory of photoelectron counting statistics: an essay
252
where 13 13(1) = (1 + 2v)112 and ~m (x) is a polynomial which can be expressed in terms of the halforder modified Bessel functions of the second kind: 2x 1/2 Sm(X)=(_) ex K,,_L (x). (4.33) Glauber’s original derivation is somewhat involved, but we can take advantage of eq. (4.30) to obtain his approximation. Let T~ 1 in eq. (4.30) then 413(A) e11 [1+13(A)]2
Q(A)
—~3(X)IT
(4.34)
The important normalization, Q(0) = 1, is preserved. Glauber’s approximate generating function follows by setting the coefficient in front of the exponential to unity, thus Q(A)
‘S-’
et’
(4.35)
13(X)IT
Upon expanding this expression in a power series about A = 1 we obtain eq. (4.32). The first few 5m (x) are: s 0(x)
s1(x)
1,
s2(x)
1+—, x
I,
s3(x)
1
33 x x
+—+---~.
(4.36)
Higher order members are obtainable from the recursion formula Sm+i(X)=_5~(X)+(l
+~)5m(X).
(4.37)
The second Glauber approximation [33]is based on the additional assumption that the beam strength is large (v ~ 1) and in the present notation reads / v ~1/2 T 1 1 I —) exp — [m — (vT)} 2} (4.38) \2irmf m 2vm Glauber actually never published the original derivation of this formula which he obtained from eq. (4.32), but we outline an alternate (unpublished) proof due to Glauber and Barakat in appendix 4.A. A discussion of these asymptotic formulae is postponed until subsection 4.9.
P(m, T)
‘~
—
4.5. Rectangular spectrum A second spectrum which admits an exact solution is the rectangular spectrum of full width z~w centered about w0 1(w) =
~
= 0.
w0 --i—
(4.39)
R. Bare let and J. Blake, Theory ofphotoelectron counting statistics: an essay
253
The normalized correlation function is g(r)
=
sinirT irT
(4.40)
where the characteristic width ‘y is now taken to be ~w/2ir. The basic integral equation can be cast into the form +1
sinc(z’—z”)
—1
lr(z—z)
,,
0,(z”) dz” = b,01(z’)
(4.41)
with c (irr/2) upon transforming to new variables z’ = (2T’ — T)/T, z” = (2T” T)/T. This is the integral equation satisfied by the angular prolate spheroidal wavefunctions S01 (c, z): —
sin c(z’—z”)
S01
(c,z”) dz” -~[R~) (c, 1)] 2 s01 (c,z’)
(4.42)
where R~)(c,z) is the corresponding radial spheroidal wavefunction. Hence, our eigenvalues are given by [Ru) (c, 1)12
(4.43)
and have been tabulated by Slepian and Sonnenblick [34]for integer values of c. Reference is made to Flammer [35] , Frieden [361 for general properties of the prolate spheroidal functions. The behavior of the rectangular eigenvalues is different than that of the Lorentz eigenvalues. Table 4.2, abstracted in part from Slepian and Sonnenblick reveals the tendency of the leading eigenvalues to cluster rapidly about unity as T is increased. From the practical point of view actual calculations indicate that the limit on the use of eq. (4.8) is T 2.546 (c 4). For larger values of time the multiple roots have to be taken into account. For small values of T (i.e., T < 1) each succeeding eigenvalue is an order of magnitude smaller than its predecessor so that only a few values of C1 are required. Under these conditions it is justifiable to employ the product representation of C1, eq. (4.7), in the numerical calculations. Unlike the Lorentz spectrum, there are no explicit asymptotic approximations. Table 4.2 First nine eigenvalues of rectangular spectrum as a function of normalized time T (from Slepian and Sonnenblick [34J)
8
/
T= 1.273
T= 2.546
T=
0 1 2 3 4 5 6 7
8.8056 (—1) 3.5564 (—1) 3.5868 (—2) 1.1522 (—3) 1.8882 (—5)
9.9586 (—1) 9.1211 (—1) 5. 1905 (—1) 1.1021 (—1) 8.8279 (—3) 3.8129 (—4) 1.0951 (—5) —
9.9990 (—1) 9.9606 (—1) 9.401,7 (—1) 6.4679 (—1) 2.0735 (—1) 2.7387 (—2) 1.9550 (—3) 9.4849 (—5)
—
—
— — — —
3.819
T= 5.082 9.9999 (—1) 9.9988 (—1) 9.9701 (—1) 9.6055 (—1) 7.4790 (—1) 3.2028 (—1) 6.0784 (—2) 6.1263 (—3) 4.1825 (—4)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
254
4.6. Numerical solution of integral equation Aside from the Lorentz and rectangular spectra, analytical solution of eq. (3.63) are possible when the spectrum is the ratio of two polynomials, Youla [37], Slepian [38]. Unfortunately, this case is of academic interest since natural spectra are not of this form, although they can probably be closely approximated by the rotating ground glass method of Martienssen and Spiller [39].In general, however, numerical techniques must be employed in order to determine the eigenvalues. We are only interested in values of T such that the eigenvalues are distinct in order to satisfy the conditions leading to eq. (4.8). Fortunately this constraint means that T must be fairly small (T < 6). Under these conditions, probably the best approach is to replace the integral equation by an equivalent system of linear equations via double application of a quadrature formula, Kopal [401. The final result is b
N n~1
H~g(IT~ —r~I)Ø(T,,)— (Tm),
m
l,2,--~,N
(4.44)
where r,, are the quadrature points for the interval (0, T) and H,, are the corresponding weight factors. This system of equations is not symmetric even though the original integral equation was. Computation of the eigenvalues is more easily performed by transforming the above system into a symmetric one. Set h,, H,~2f(Tn),then eq. (4.44) becomes (G—bI)h=0 (4.45) where Gm n =
(4 4 ~. The numerical solution of eq. (3.45) is a standard procedure, Ralston and Wilf [411, and we need not concern ourselves with the actual details since they depend upon the computer employed, special programs available, etc. In the calculations of Barakat and Glauber [18], Blake and Barakat [42], Gauss quadrature was employed. The quadrature points r,, are related to the Gauss quadrature points X~N)over the standard interval (—1, +1) by the transformation
T,,
G
nm
= ‘H Hsi~\1/2 g~. ( Tm \ m
~ [1 +X,~’)]T,
—
T,,
(n = l,2,~--,N)
(4.47)
while the transformed weight factors H,, are given by H,,
~A~f’)T,
(n
l,2,~,N)
(4.48)
where A~N)are the standard Gauss weight factors. For the theory of Gauss quadrature see Stroud and Secrest [43]. It is here that the Gauss method displays its power, because if we had employed a standard equidistant quadrature formula then we would have had to contend with a matrix of order (2N + 1) to obtain the same accuracy as N point Gauss quadrature would supply. An alternative approach to the numerical evaluation of the eigenvalues of eq. (3.63) is via Fourier series expansions, Lachs [44]. As an example, consider the Gaussian spectrum due to Doppler broadening ___
I(w) =
I (w_wo)21 exp I— I L 2cr2 ~
(4.49)
R. Bare let and J. Blake, Theory ofphotoelectron counting statistics: an essay
255
where a (variance) is the characteristic width of the spectrum. Under the condition of quasi-monochromatic radiation, the correlation function is (4.50)
g(T) = e_T2/2
where T = at, V = w/a and a is the variance of the Gauss line. It appears to be a general rule that the rate of decrease of the eigenvalues of a spectral line is directly related to the rate of decrease of the corresponding correlation function. In the case of the Gauss line, the rate of decay of g(T) is intermediate between that of the Lorentz and rectangular correlation functions. A value of N 14 was sufficiently large to permit accurate evaluation of the first ten eigenvalues of the Gauss line for values of T up to six. A typical result abstracted from Barakat and Glauber is shown in fig. 4.4. The curves are qualitatively similar to the Lorentz and rectangular curves except that the Gaussian curves are somewhat broader in the medium time region. Physical phenomena giving rise to several spectral lines are also encountered. For example, in the electrophoresis experiments of Ware and Flygare [45] each component of distinct electrophoretic mobility produces a Lorentz line of different central frequency. Another example is the Brillouin scattering from media where the velocity of sound is very small such as in the work of Katyl and Ingard [46] on surface waves along an interface between two liquids, or the work of Clark and Liao on smectic liquid crystals [47]. To this end let us consider a prototype spectrum consisting of two Lorentz lines symmetrically
1
-
I
I
I
I
I
I 1
12~~
__
20
40
60
NUMBER OF PHOTOELECTRONS
NUMBER OF PHOTOELECTRONS
P(m, T) for a Brillouin spectrum, ~ = 0.4, u = 5: A) 0, T’ 0.5, B) ~ 4, T 0.5,C) ~ = 0, T’ 2, D) ~ 4,
Fig. 4.5. Fig. 4.4.P(m,T) foraGauss spectral line, u 7.1 for T= 0.5, 1, 2, 4 (Barakat and Glauber, 1966 unpublished).
=
T= 2
(Blake and Barakat, 1972 unpublished).
R. Barekat and J. Blake, Theory of photoelectron counting statistics: an essay
256
placed about a central Lorentz line of different height and halfwidth: +1
r
I. 2+y 2J /=—I ‘[(ww1) 1 The total energy in the beam is normalized to unity I(w)
J
>
(4.51)
R.I
J(w)dw =
/
~R 1
= 1.
(4.52)
Since R1 = R1, we define a parameter a = 2R1 which represents the fraction of the total energy contained in the Brillouin lines. Thus a = 0 describes a single (Rayleigh) line and a = 1 describes a pair of (Brillouin) lines in the absence of a Rayleigh line, a is generally known as the Landau—Placzek ratio. The actual Brillouin lines are spaced (approximately) symmetrically about the Rayleigh line so that — w0 = w0 w..1. Let us define a dimensionless quantity z~ (w1 w0)/y0. Similarly, we define “Y~I’ve = ~., /‘y~which gives the ratio of the width of the Brillouin line to the Rayleigh line. As before, we measure time in units of l/y~,and measure beam strength in units of 7~by setting w/”y0. A summary of these parameters is given in table 4.3. The correlation function of the Brillouin spectrum is —
—
g(T)exp{—iw0T/’y0}[(l
+ae_~T~cos~T].
_a)e~TI
(4.53)
We can drop the exponential term outside the brackets in accordance with our previous discussion so that the kernel of eq. (4.44) is the term in brackets. A typical plot of P(m, T) for such a Brillouin spectrum is shown in fig. 4.5. Note that as z~increases the distribution shifts so as to make P(0, T) larger. This is a reflection of the heterodyning between the Brillouin and Rayleigh lines which produces fluctuations in intensity on a times scale of the order of ~ These fluctuations in intensity will increase the “clumping” of the photons, making it more 1. likely to see none at all arrive during a time T, so long as T is of the order of Z~ Table 4.3 Summary of dimensionless parameters Brillouin spectrum Symbol
Definition 2R_,
Meaning
2R
=
1
=
1
—
R0
fraction of energy in
Brillouin lines (w,
—
w0)/’y0
=
v
w/-y,
ratio of frequency shift to Rayleigh half width ratio of Brillouin to Rayleigh half widths number of photocounts in one correlation time y~’
R. Barekat and J. Blake, Theory ofphotoelectron counting statistics: an essay
257
A
.77
I
I
10
I
75
I
a
4~”~~A
~
Fig. 4.6. P(0, T) for a Brillouin spectrum as a function of e for T 1, v land ~ = 1,2,3,4,6,10 (Blake and Barakat, 1972 unpublished).
Fig. 4.7. P(0, T) for a Brillouin spectrum as a function of ~ fore 0.7, v 1,8 = land A)T~’0.5, B) T~1, C) T= 2, D) T = 4 (Blake and Barakat, 1972 unpublished).
To examine this effect more closely we have plotted P(O, T) in fig. 4.6 as a function of a, and in fig. 4.7 as a function of ~. For small ~, the individual lines are not distinct and the “clumping” of the photoelectrons increases monotonically as a increases. For ~ > 3, the lines become distinct, and if we make a> 0.7 we decrease the beating, thereby reducing P(0, T). Similarly, increasing ~ increases P(0, T) until the counting time contains several fluctuations of intensity. Past this point where ~> T’, P(0, T) levels off and displays oscillating behaviour. 4. 7. Long-time asymptotics The expression for P(m, T) given in eq. (4.8) is valid only for times T for which the eigenvalues are distinct. For longer observation times, where the first N eigenvalues are confluent (i.e., b0 = b1 =
..
.
bN > bN -
P(m,T)
~
1
>
~ Barakat has shown that P(m, T) becomes
~
m
(l±b~)
(
bi’)j~-) dX1~h[~~X)m1
H(bI’ (4.54)
The tedious details are omitted. This expression can be employed for calculations although the final
R. Baralet and J. Blake, Theory of photoelectron counting statistics: an
258
essay
results hardly justify the elaborate computations. What we really want is an asymptotic approximation. In order to obtain a simple useful approximation to P(m, T) for T 1, we assume that only the N confluent eigenvalues b0, contribute substantially to W(~l).This is tantamount to letting the ~‘
singularities of eq. (4.5) reduce to just a single pole of order N, where N is large but as yet unspecified. When the residue is evaluated by contour integration in the complex A-plane we obtain
c (ba’e~dA + A)N
1 1 W(~)—— b~f 2iri =
(4.55)
J
&2N-1
e_n/!~o,
~
b~’F(N) 0.
(4.56)
~<0.
The resultant probability distribution is then easily shown to be P(m T)
(m+N) [‘(N)
F
(m + 1)
b~ m+N (1 + b0)
.
(4.57)
However, b 0 and N are really as yet unknown parameters, but they can be expressed in terms of observable quantities. In order to demonstrate this, we first note from eq. (4.23):
11
(~)~
b~= T
(4.58)
because uT is the average counting rate (m). Hence, ~
b1 ~ Nb0 ~ (m>
(4.59)
j=0
connecting.
(4.60)
b
(4.61)
Consequently, b0 and N are expressed in terms of (m) and 12(T). When ‘2 (T) is written out explicitly it is 12
(fl
=
J
2(r’
g
—
r”)12 dt’ dt”.
(4.62)
R. Barak.at and J. Blake, Theory of photoelectron counting statistics: an essay
259
Fortunately we are only interested in large values of T for which we can prove that (T~l) I2(fl’~’6gT, where is a numerical constant dependent upon the spectrum under consideration. For the Lorentz spectrum, we have
1
T,
2T]
(T~ 1)
(4.63)
(4.64)
2(T)—~[2T— 1 + e_ so that = 1. We can also show that =
1,
=
~/W
rectangular spectrum Gauss spectrum.
Surprisingly enough 6g is unity for both the Lorentz and rectangular spectra. The line shape is still contained in the asymptotic expression through the constant As the time of observation becomes essentially infinite, p(m, T) must become independent of the actual shape of the spectrum and only depend on two parameters w and t rather than three parameters w, t, ‘y. In order to prove that eq. (4.54) passes over to the Poisson distribution, we note that a)m 0 ‘~‘~gbo
(4.65)
asT-~°°
16g b)N~ T
and rewrite eq. (4.57) in the form P(m fl~
I”(m + T/~Ti r (m) g m!F(T/~g)11(m) + T/i5g ________
m
r II
I
+
(m) yTf6g I (T/lSg)J
(466)
using these expressions. However,
r lim
(TI8g)
-+
lim
(TIög)~
(m)
Ii + I
1_(T/ög)
I
F(m+ T/ô ) r g
F(TI~g)
(4.67)
= e_
(TIiSg) .i
(m)
I _______I
1”
I.(m)+TI~gJ
(m)
(468)
so that lim (T/óg)~
P(m,T)
(m)m e~m~ m!
(4.69)
which is what we wanted to prove. Examination of figs. 4.1 .and 4.2 reveals that P(m , T) is still not in the Poisson regime even for an exceptionally long time of T = 30, since the curves are not symmetric about the mean value.
R. Bare/cat and J. Blake, Theory of photoelectron counting statistics: an essay
260
Equation (4.57) can be recast into the form P(m,T)~
F(m+N)
_______________
F(N)m!
[1 + (m)/N]N[1
(4.70)
.
+N/(m)]m
This expression was first proposed by Mandel [481,see also Mandel and Wolf [491,as an approximation to P(,n, T) in the semiclassical interpretation. He assumed that P(m, T) could be written in the form, eq. (4.70), and adjusted the parameter N (s in his notation) such that eq. (4.70) and the (unknown) exact distribution possess the same second moment. The present analysis demonstrates how this type of asymptotic formula arises naturally out of the Glauber formalism. Reference is made to Bedard, Chang and Mandel [50]for applications of eq. (4.70) to the calculation of factorial moments. Our illustrative examples are confined to the Lorentz and rectangular spectra for which we have exact results. In view of our discussion of the rate of decrease of the eigenvalues of the Lorentz spectrum in section 5 (especially with regard to table 4. 1), we should anticipate that eq. (4.57) will not yield a particularly good approximation for T < 10. The expectation is borne out by an examination of figs. 4.8 and 4.9 for T 6 and-u = 5,10 respectively. This approximation (solid triangles) underestimates the peak values. Glauber’s Bessel function approximation (solid circles) given by eq. (4.32) is certainly better. It suffers the disadvantage that the computation of the high order Bessel functions
4.0
I
I
I
I,o(
— 3.2-
£
-
I
2.C
— -
aa•
I
I
1,1
I
I
I
I
I
I
I
I
I
I -
—
a
I
0
•
-
-
a -
-
0
-
-
a
-
-
N -
6
-
-
-
—
1-1.0-
O~16-
-
a,
0.8’
0
0
-
-
-
a
0
liii
-
20
111111
40
60
BO
NUMBER OF PHOTOELECTRONS
Fig. 4.8. P(m, T) for a Lorentz line: u = 5, T= 6, = 30. The solid line is the exact value; solid circles are from Glauber’s Bessel function approximation, eq. (4.32); open circles are from Glauber’s second approximation, eq. (4.38); solid triangles are from eq. (4.57) (Barakat, 1967 unpublished)
0
oil
a
I
40
III
Il-Ill
80
120
160
NUMBER OF PHOTOELECTRONS
Fig. 4.9.P(m,T) for a Lorentz line, v = 10, T 6,
R. Bare kat and J. Blake, Theory ofphotoelectron counting statistics: en essay
261
is just about as complicated as the exact solution. Glauber’s second approximation, eq. (4.38), is dependent on 1.1 as well as T. The fit is better at u = 10 than at u = 5 in accordance with the basic assumption V ~ 1. Calculations were also performed at T = 12 for v = 1,5,10,100. Glauber’s Bessel function approximation was practically indistinguishable from the exact solution, whereas the other two approximations still tended to undershoot and overshoot the peak values respectively. These long time calculations should be put into proper perspective. Such long normalized times can only be achieved by taking t large since ‘y is basically fixed. However it is difficult to stabilize a laser operating below threshold for much longer than t = 10”~sec. In fact the largest value of T of which we are aware is T = 1.32 in the pioneering experiments of Freed and Haus [31]; this value of T hardly puts it in the long-time region. We now pass to a consideration of the rectangular spectrum. Here the situation is decidedly more favorable for the application of the asymptotic formula in view of the fact that the leading eigenvalues rapidly cluster around unity as T is increased. The largest value of T for which the eigenvalues do not coalesce is T = 2.546 (see section 5 and table 4.2). We chose this value of T to check the exact probability distribution vs the asymptotic formula. Some typical results are summarized in fig. 4.10 for v = 5, (m) = 12.73. The solid line represents the exact solution, while the solid circles represent the long time asymptotic solution, eq. (4.57). There is still a tendency to underestimate slightly the maximum value. The dotted curve is for a Lorentz line (exact solution) with T = 2.546, v = 5 and is included for comparison. 4.8. Variance of photoelectron counts The function W(~2)is easily seen to be a probability density function for 0’( T < oo, thereby implying that m (t) is always a mixed Poisson stochastic process. This fact follows directly from the condition that Q(A) is a moment generating function. Furthermore we have ~ W(~2)d~2=~ C1=l.
(4.71)
I
I
I
I
CO:1~1I52~25 NUMBER OF PHOTOELECTRONS
Fig. 4.10. P(m, T) for a rectangular spectrum; v = 5, T = 2.546, (m> = 12.73. The solid line is the exact value, solid circles are from eq. (4.57). The dotted line is for a corresponding Lorentz line and is included for comparison (Barakat, 1967 unpublished).
R. Barakat and J. Blake, Theory ofphotoelectron counting stetistics: an essay
262
The variance of the distribution of photoelectron counts is easily derived in the context of a mixed Poisson process. If the eigenvalues are distinct, we can use eq. (3.25) to prove that (~2)
=2
~
m~C/.
(4.72)
j=O
In the short time approximation only the first term in the series is important, thus 2 — 2m~= 2(m)
(4.73)
so that var(m)~i(m)[1 +(m)].
(4.74)
Note that the line shape does not enter into expression in accordance with eq. (4. 1 8). In the long time region, we can show that (~TZ2)—m~N(N+l)(m)2
+v2ö 5T
(4.75)
It then follows that var (m) = (m) [1 + iSg(m)/T1.
(4.76)
When the spectrum is Lorentzian, then this expression is identical with the result obtained by Glauber using his Bessel function approximation. Regardless of the length of observation, it is simple to prove that var&2 -412(T).
(4.77)
The usefulness of this expression is that the right hand side can be evaluated in terms of the second iterated kernel without the necessity of having to know the mixing density W(~)directly. The behavior of W(~2)as a function of time is worth an explicit statement. In the short-time region, W(~2)is a negative exponential density which decays as time increases to a bell shaped curve. This bell shaped curve goes over into a chi-squared density as time is further increased. Finally as time is made to approach infinity W(~) approaches a Dirac delta function centered at &2 = (m), lim W(~Z)= ~[~2 — (m)].
(4.78)
4.9. Superposition of coherent and multiple mode chaotic radiation* The present subsection is devoted to a natural extension of the theory to cover a particularly interesting situation: the superposition of a coherent excitation and a chaotic one. In particular we consider the superposition of a coherent signal and a noise excitation in an infinite number of modes. Such a situation is a simple modeling of a laser operating just above threshold where the “signal” is to be identified as the laser line and the chaotic background is the “noise”. The chaotic background is taken to be either a Lorentz spectrum or somewhat more realistically, a Gauss spectrum. *This subsection is based upon an unpublished report by Barakat (1968).
R. Bare kat and J. Blake, Theory of photoelectron counting statistics: an essay
263
Although the analysis about to follow actually holds independent of the time of observation, we note that in practice a laser operating just above threshold is quite unstable. Consequently, our simple signal plus noise model is only physically realistic for short counting times. If the counting time is very short, then only one mode of the background effectively interacts with the coherent mode. However, as the counting time increases additional background modes come into play which makes the subsequent mathematics somewhat complicated. In view of these remarks, we can expect that counting times sufficiently large to include three or four chaotic modes will certainly cover the practical experimental situation. If the amplitude of the coherent signal is taken to q;, then the P function for the superposed radiation is (see eq. (2.37)) P[{ctk}]
11
1
21 Iak —6 ki q’1 k
exp 1~_
k lr(nk)
L
(4.79)
‘~~k>
Only the lth mode possesses a coherent component. The subscript 1 can take on any integer value, however the case of most interest is I = 0 (the coherent component lies in the center of the chaotic background line). The single-fold generating function corresponding to eq. (4.1) is now given by Q(A) = exp{—1q 2 } ~exp{—~~(1+!~3}exp{_(q,*~, +q 11 1~3,*}H~Lk
(4.80)
where (4.81)
2. qk ~q~/{(n~)}”
An explicit evaluation of this multiple integral can be accomplished by reducing the complex quadratic form in the exponential via a unitary transformation. The final result is:
Q(X)exp{—Iq,12}
+Xbk)’ exp
H {(l
[1l1I).
(4.82)
Unlike the completely chaotic situation, we are now forced to deal explicitly with the matrix U which diagonalizes A. The elements of U have the following significance; Ukl is the kth eigenvector component corresponding to the lth eigenvalue. Thus, the eigenvectors serve to determine where the signal mode is with respect to the chaotic modes. We recall that only the eigenvalues were of significance in the completely chaotic case. Note that
Iq,I2 If
IqI2/(n~) signal/local noise.
=
we set 5k
=
(mk)”, hkl
U~
(4.83)
2,then eq. (4.5) becomes 1q1I
W(fZ)
= (fi
k
5k) exp{—lq,l2}
2iri
~ J
jj (exp k~
[hk,sk/(X + ak)] ~ e~ dA. X+Sk
(4.84)
J
The singularities of the integrand are at A = ~5k’ but they are now essential singularities and not just simple poles as in the completely chaotic case.
R. Barak.at and J. Blake, Theory of photoelectron counting statistics: an essay
264
At this stage, it is probably more enlightening to consider only one spectral mode so that eq. (4.84) reads W1(~2) exp{—Iq,12 —&2s0}
J z’
exP{&z(z + h&s0)}d
(4.85)
where z = A + s0. The integral can be expressed in a series of modified Bessel functions of the first kind, we note [51] that 2 n/2 exp frz(z + ~ ~ In [2(h 2]. (4.86) ~2z
)}
~
°°
01s0~2)”
When this series is substituted into eq. (4.85) and termwise integrated, W 2
~2s
—
W1 (~)= s0 exp{—1q1j
1 (~2)becomes
2].
(4.87)
0} I~[2(h01s0~)”
The higher modes can be treated in similar fashion. For two spectral modes, eq. (4.84) becomes exp (—Iq,12) ~
=
[(A +s 0) (A +s1)]
27rl
c
1
e~ exp( hOlsO A+s0
) exp(A+s1 )dA. h11s1
(4.88)
We first evaluate the integrand at the singular point A = —s0 and then at the other singular point A = —s1. Upon setting A0 + s0 = z, we have
2&‘Ls
—
W2(~2)= exp [—1q11
0I
r
~
2in
=
z
(4.89)
C
1 exp {—&2 (z + h A(z)
A(z)B(z) dz
——
01 s0/&2z)}
~ B(z)(s1 —s0 —z)~ expi
h11s1
(4.90) 1
I.
(4.91)
—ZJ
L~i~
A(z) is the generating function for the modified Bessel functions, see eq. (4.86), while B(z) can be
expressed in terms of Laguerre polynomials B(z)exp[hh151] C10
exp[h1151} 10
where c10 ~s1
—
{(i _±)Jexp[hh151(_~_)(l C10
Cio
(._~_.)
~ m—o C10
do
~_)]} do
(4.92)
~ 10
~ and Lm (x) is the Laguerre polynomial defined by
e-”
d~
m e~I .
(4.93)
m! dxtm [x Lm(x)= 1F1 (—m,l,x)=—
Upon substituting the appropriate expansions of A and B into eq. (4.89), we can prove that the double series expansion reduces to a single series, namely (sosi) exp
{—
q 2 +h 11 11s1
—
~iso) ~(holso)n/2
ç—
2]L-~ [I~(2 h01s0&~)”
~
(4.94)
R. Barekat end J. Blake, Theory ofphotoelectron counting statistics: an essay
265
This is the contribution to W2 (~)from A = —s0. The contribution from A = —~i can be evaluated in exactly the same manner by letting A + s~= —z. The final expression for the two mode case is given by ~ W2(~Z)= ( __) C
exp [hlisi
c10
10j +~_—
2]
~2s~ — Iq,1
—
exp
(hO1SO
~
—&~~Si—
j
Iq,12]
mn[2(hol5o~)h/2]Ln
~
n=O
[hoiso
n/2
h11s1
[hizsil C10 I 1h01s01
n/2
____
\Isosl) ~0,
____
In [2(h11s1
n = 0(~~~1)
~)1/2]
Ln
(4.95) 5~
S
where CJç/ 1~ The extension to three or more modes is obvious; the final answer for the three mode case is 2] W + h21s2 — ~ Iq,12] j ( h01s0 ) n/2 I~,[2(h01s0~2)” 3(~2) ( 505152) exp [hizsi c10 c2~ ,~= ~ —
1L (hllsl)L i~(k+ /
—
n) ~
~
k0
—
_____
I
2152) C10
1=0
expi1h01s0 -I- h~1s2— ~
c01c21, +(sosls2
~
1q11 2]
(cOl)~”(c2l)~’Lk (_
k0/0
n)
~
~
(c
k0
/0
0~)~ (d12)’Lk
01s0
)L1
( h21s2 ) ~21
C01
“
+ s0s1s2 exp [ho’so + h11s1 —~s2—Iq,~ 2] (co2ci2) c02 c12 —
n/2
=
h
°°
~
°~(k+/
C21
h11s1 2] ~-(~) mn[2(hii5i~)”
~
n
~21
°°
(h
(c10)_k (c20~
(
j =
(h2:s2 ~
0
2]
n/2
‘n
holso)L(hllsl) d02
[2(h21s2~)~ (4.96)
C12
Having determined W(f~)the evaluation of P(m,T) follows directly. When W(&2) is substituted into eq. (4.5), we encounter the function 2] Rmn
=
0
e_(1~)n ~m-n/2
j
_____
n!
(hs~~/2e’~1~1 +s) (1 +S)m+
d~2,
m,n~0
[2(hsIZ)” F (n — m, n +
hs\
1, —
1
where 1F1 is the confluent hypergeometric function.
j—~-—)
(4.97)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: en essay
266
It is then a simple matter to prove that the final result for one mode is d~2
112]
P1(m,T)
~m!
~
~—Iq~I~ 0
~
m
I~ [2(h01s0~)
1h expt 01s0 +~
1
s0(l +s0)~ =
s0 (1
m +
—
Iq112) 1F1 ~—m,l,
exp1 h 2 I 01s0 + s~— Iq11
1
s0)
h1~1s~ 1 +s01 ‘~
0
}
)
Lm (~0150
1 +s
(4.98)
0
where L~ is the Laguerre polynomial. Before commenting on this expression, we write out the explicit formulae forP2(m,T)andP3(m, T). The two mode expression is P2(m,T)
—_Jexp
=
(~ ~
+
(sosl\
+ h01s0 (1 + 1 +S0 ~
C10
j
)~rn — i
(ho5o)~L0 ~ o
n
C10
C10
d10i
X ~
(n
—
m, n + 1
h01s0
1 +S0
x
~0
L~
~Ol (~)n
n=
(
C Ibis0)
2
( sos1 s2 )exp {
q,1
\
+
C10 C20
• ~
(h0~s0)fl 1F1 (n
—
1F1 (n
• ~
hOsO
+/
—
n)
_
—
~
~
k0
j0
(c1~)~ (c20)’
(h11s1)~iFi(n_m,n+
l,—’~”~’~(k+j—n) ~ I +s1/ k0
(h21s2)~1F1 (n
) exp {
—
m, n + 1
—
h21s2 1+52
XLk(~.--~-0_i_50)LJ j ~—
____
C12)
~
/121521~(l C21 ,1
+s1) —rn—i
(Coi)~(C21)~’
j0
2 + hois —
Iq,1
0 + h11s1 + h21s2 1(1 + c02 C12 1
C21 I
n=0
1+51
C01
X Lk (_~~i5o)L.(_~~2152~+(sosis2
• ~
) ~(k
l+s~
/
n =0
\
1
(4.99)
01s0 + h us1 + 1121s21 i(l +s0) 1 +s0 C10 C20 I
m, n + i
C20
-
m, ~ + 1 h1isi) 1 +s~
(_~.L~!’) +( sosls2) exp{ —jq 12 + h01s0 + h11s1 +
h1is1~L C10 F /
Co1
—
) (1 + s1Ym
h
-
-
C0s0 + h11s1 01 1 + S1
\C01!
n0
X Lk (_
exp{_ 1q11
01
Tl1e three mode expression is P3(m,T)
2 +h
~
+
)~ (k + /
—
n) ~
~
52)_rn_i
(c02)_k (c12Y’
k=0j0
(4.100)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: en essay
267
We again remark that higher modes can be explicitly written out by induction. We return to the one mode case as expressed in eq. (4.98). Glauber [33] and Lachs [52] have already obtained a special case of this result by different procedures. In Glauber’s notation r Si r S 1 Jytm P(m T) = — exp — IL I— I (4.101) (1 +N)m+l 1 1 +NJ m ~ N(l +N)J where S represents the mean number of quanta which would be counted in the coherent field and N the mean number of quanta in the noise field. Since eq. (4.101) is only realistic for T 1, we then note -~
s
2 uT;
0 ~ (uT)’, so that S
(T~
(4.102)
1)
h01 ~ lU01q1I
IU
2uT, NvT; (T<<1). 0~q,I When the signal lies at the center of the chaotic spectrum, then 1 =
(4.103) 0
and
U 00~1,
U01~0
(1>0).
(4.104)
Thus although eqs. (4.98) and (4.101) have the same functional form, the latter is less general because it was derived under the assumption that both signal and noise are in the same mode. No such assumption was made for eq. (4.99). In fact, if the signal is not in the central mode (1 0), then eq. (4.99) reduces to the simple power law distribution m/(l +VT)m+l. (4.105) P(m,T)~(uT) Longer counting times are required to detect the signal when it is not in the center of the background line. A typical numerical result is shown in fig. 4.11; here the beam strength u and the normalized time T are fixed and the signal-to-noise ratio 1q 0 12 varied. The calculation was carried out using two chaotic modes. In fact, P(m,T) is very close to being a Poisson distribution for Iq0I~= 9, even though T is still very small. When the noise field is completely absent, the generating function for the single mode coherent field is Q(A) = e_(m~
(4.106)
which leads to a Poisson distribution (m)m e~m>
4 107
m! ) where (m> = S. Thus the Poisson distribution for the photoelectron counts arises from two different conditions: from the limit of very long counting for a multimode chaotic field (eq. (4.69)); from a single mode coherent field plus a multimode chaotic field in the limit of large signal-to-noise. P(m,T)
.
4.10. P(m, T) for partially polarized chaotic light
Thus far the analysis has been confined to chaotic light that is completely polarized. An extension of the analysis to include partially polarized chaotic light is of some interest.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an
268
I
I
I
I
I
B
3.0
essay
I
-
C I-
E
D 1.5-
C
0
-
I
I
I
Fig. 4.11. P(m, T) for a coherent line in center of a Gauss spectrum for v 1q 2 = 3, C) 1q 2 = 5, D) Iq 2 = 9, (Barakat, 1968 unpublished). 01
0(
I
20 40 60 NUMBER OF PHOTOELECTRONS =
80
2
10, T = 0.52 as a function of 1q
0
2:
=
0.5, B)
A) Iq,1
0I
In order to develop the theory, we need some background information on partially polarized Gaussian light. The standard reference is Born and Wolf [53]. The coherency matrix for plane waves is [‘~
(r)
F 12(r) (4.108)
F21
(r)
F22(r)
where F~1(r)= (~7(t)~1 (t + r))/(~7(t)~/(t))
(4.109)
and 1.5~(t)is the ith component of the field at time t. We further restrict the analysis by letting t Note that tr f(0)
= total
average intensity
(I).
=
0.
(4. 110)
J is also hermitian (I = J +) which means that it possesses real eigenvalues and can be diagonalized. The eigenvalues of J(0) are A2 trJ+~[(tr)2 ~_4detJ]~2
(4.111)
A2~~trJ_~[(trJ)2 _4detJ]1~’2. The two limiting cases of polarized light and unpolarized light are defined in terms of the eigenvalues. Light is completely unpolarized when A1 = A2. This means that the term in brackets is zero,
R. Barekat and J. Blake, Theory ofphotoelectmn counting statistics: an essay
orF11
= F22, F12
F21
269
0
10
(4.112) 01 Light is completely polarized when A1 * 0, A2 0; or when det J = 0. The intermediate situation can be handled via a linear superposition of these limiting cases. We write (I~)0 B D + =J~+J~, (4.113) 0 (I) D* C 2 = 0). The average intensity of the unpolarized u where of~u =is~‘u~ zerowhile (i.e., BC 1D1 intensity of the polarized part of the radiation part ofthe the determinant radiation is tr the — average is trJ~= (In). The degree of polarization, ~is defined as trf p trJ~+ trJ~
=
r
4 detfl
I
(trJ)2 J
I
1/2
.
(4.114)
Both det J and tr J are invariants with respect to unitary transformations so that ~ is independent of the orientation of the coordinate system. Furthermore ~ is bounded (4.115)
~
with ~ = 0 corresponding to unpolarized light and £3D be expressed in terms of
= 1
to polarized light. The eigenvalues of J can
A 1 =~(l
X2
+~)(I) (4.116)
~(l —~)(I).
Given this background information, our basic integral equation, eq. (3.63), becomes the coupled set
~ F~1(t
—
t’)Ø1 (t’) dt’ =
cb1(t)
(4.117)
with i = 1,2. If we confine ourselves to the short counting time approximation, this reduces to the algebraic equations T>~
i
F11(0)Ø1b~1
1,2.
(4.118)
The eigenvalues of this set of equations are b1 ~
(1
+.9)VT,
b2
~(1
—~)vT
(4.119)
upon employing eq. (4.116) and (I> = (m> = VT. Note that b1 + b2 = vT in accordance with eq. (4.23). Thus the mean number of photoelectrons is independent of the degree of polarization for T ~ 1.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
270
The single-fold generating function is (4.120) Q(A) = (1 + Ab1)1 (I + Ab2)1. If 0< ?< I, then b 1 ~‘b2 and we can employ eq. (4.6) to obtain the mixing density function: 2n/(i 1?)VT e_2u/(~_.,)UT] (4.121) W(~)= (~VTY~[e_ The probability of obtaining m photoelectrons in time T follows directly from eq. (3.18) and is —
1 ir (1 +~?/)VT 1P~+1 1 (1 —.~?)vT lrn+lI I —I I } .~VT~L2 + (1 +?)vTJ 12 + (I —,9)vTJ I
(4.122)
P(rn,T)—fl
in agreement with Mandel [54]. The dependence of P(m, T) on ~?is quite small as witness table 4.4. The variance of m(t) is easily obtained via eq. (4.6). We have (~2) =
~2
W(~)d~= (3
rn2,
+~2)
T ~ 1.
(4.123)
It follows that var(m)(m)[1
+~(l +~2)(m)],
T~l.
(4.124)
The variance of m(t) is a maximum for polarized light and a minimum for unpolarized light. The variance is a measure of the tendency for photons to clump together, which is why P(0, T), and P(m, T) form ~ (rn), are largest for~ 1, see table 4.4. The corresponding results for unpolarized light are obtainable in analogous fashion. The eigenvalues b 1 and b2 are now equal, so that the generating function Q(A) is 2. (4.125) Q(A)=(l +AVT/2~ We leave to the reader the task of proving that W(~) P(rn,T)
-~--
(vT)2
(4.126)
e2~~/~T
4(m + l)(vT)m
(2+VT)m+2
(4.127)
.
P(m, T) as a
Table 4.4 function of~in short counting time approximation: u =
5,
T 0.2 m
~
0
.444 (0) .296 (0) .148 (0) .658 (—1) .274 (—2) .110 (—1) .427 (—2) .163 (—2) .609 (—3) .225 (—3)
1 2 3 4 5 6 7 8 9
0
~
0.3
.449 (0) .293 (0) .146 (0) .652 (—1) .277 (—1) .114 (—1) .465 (—2) .187 (—2) .744 (—3) .295 (—3)
~
0.5
.457 (0) .287 (0) .141 (0) .643 (—1) .283 (—1) .123 (—1) .529 (—2) .227 (—2) .975 (—3) .418 (—3)
~
0.8
.479 (0) .270 (0) .132 (0) .628 (—1) .298 (—1) .141 (—1) .669 (—2) .317 (—2) .150 (—2) .711 (—3)
~
1.0
.500 (0) .250 (0) .125 (0) .500 (—1) .250 (—1) .125 (—1) .500 (—2) .250 (—2) .125 (—2) .500 (—3)
R. Barekat and J. Blake, Theory ofphotoelectron counting statistics: an essay
271
An interesting problem would be the extension of the analysis to cover longer counting times, where one would have to deal with eq. (4.116) without the simplification afforded by eq. (4.117). 4.11. P(m, T) for statistically independent fields If the incident field consists of two statistically independent fields, then the total field correlation function is the sum of the correlation functionsg1(t), g2(t) of the individual fields. Under these circumstances, the generating function of the total field is Q(X) = Qi (X)Q2(X).
(4.128)
The single-fold probability distribution can be obtained in the usual manner via eqs. (4.5) and (3.18). However, for this special case, it can also be expressed as the convolution of the individual probability distributions P1 (m, T) and P2 (m, T) P(m,T)
=
P1 (k, T)P2 (m
k0
—
k, T).
(4.129)
This expression is easily evaluated on a computer, provided P1 and P2 are in reasonably simple form. It suffers the disadvantage of any discrete convolution expression, that in order to evaluate P(m, T) for a fixed m, all integers less than m must be considered. Lachs [44] has utilized eq. (4.129) to evaluate P(m, T) for an incident field consisting of the superposition of a coherent field and a chaotic field having a Lorentz spectrum. Obviously this is not the same physical situation as discussed in subsection 4.9. Appendix 4.A: Derivation of Glauber’s second asymptotic approximation We now sketch a derivation of eq. (4.38) given by Glauber and Barakat. A somewhat more elaborate version was subsequently published independently by Lax [14]. We can write tm —1 ç Q(A)dA
a
(_l)m
____
Q(A)
—
ax
2in
X1
m+l
(4.A.l)
c (1 —A)
by virtue of Cauchy’s theorem on derivatives. Here A is a complex variable and c is a contour in the complex A-plane which encloses the singularities of the integrand. We next substitute into this expression, the approximate generating function, eq. (4.35), and change the variable of integration from A to 13(A) 13. The final result is P(m,T)—
_42V~m ‘
/
27ri
eT
( J (1
1~e~’~
dA t’ . +2v_132)m+l
Assuming that the beam strength v
~‘
1,
allows us to approximate the second term of the identity
) l+2v A2
(1
+2v_13)_m_l
=(1
(4.A.2)
~‘
+2v)_m_1(l
—
~
—in—i (4.A.3)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
272
by an exponential (i
—
\
p2
~—m—1
~‘
)
l+2v-
(4.A.4)
e(m + 1)132/(1 + 2v)
Hence eq. (4.A.2) simplifies to —4(2v)m P(m,T)
(I + 2v)m
~‘
(m + 13 exp [(I
2~i
—
1)132
13)T + (1 + 2v) ]d13.
(4.A.5)
The saddle point method [86] is appropriate since the exponential term predominates in the integrand. If we call —f3)T+(m+ 1)132/(1 +2v)
h(j3)(l
(4.A.6)
then the saddle points are located at the zeros of the derivatives of h with 13. In our case there is only one zero and it is located at 13 = 13~ where (1 + 2v)T 130 =
2(m+ 1)
(4.A.7)
.
Since we are already in the long time region (T ~ 1), we can expect that the peak value of P(m, T) versus m for a fixed T will be in the vicinity of m = (m) = vT as witness the curves in figs. 4.1 and 4.2. It follows that 1 F—(vT)2
h(j3 0)
—
I
2vi
1 + 2(vT)J .
rn
(4.A.8)
Following the usual recipe for saddle point integration, we have m13 13o) 1 jm + l\ P(m T) _______________ exp —I I ~2 ds 4(2v) 0 ehz( (1 + 2vY”~ 2ir ~ I \l + 2v1 I ~‘
‘~
—
2v)312 T (2v)m (2~)1/2 (m + 1)3/2 (1 + 2v)m+ 1 The power law distribution can be further approximated, since v is a standard procedure in statistics), thus e”~13°~ (1 +
(2v)m
1
(l+2v)m+l
(l+2v)
~‘
—~
{
. )
1, by a negative exponential (this
2v. Consequently
(vT) 2 exp{ (2iw)”2 (m + l)~”
(2irv)”2rn3”2 exp
.
(4.A.l0)
1
—
One final approximation (!), we let m + 1 large (m) anyway. Finally P(m,T)
(
e_m/2V.
We may as well also write (I + 2v) P(m,T)
4A9
(vT)2 2v rn —
____
—
11
2(vT) + mIs.
(4.A.l I)
i-I
m since the formula is only good in the vicinity of
l(vTv~Th)2J
(4.A.l2)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: en essay
273
which is Glauber’s second approximation, eq. (4.38). Lax [14] has also derived this expression by a similar method, but he obtains a further multiplicative factor on the right hand 2 [i
+~-(~+~-)]. 2 m vT
(4.A.13)
This factor is only important when m ~ vT, but the correction is very small.
5. Two-fold statistics for Gaussian light Although the determination of the one-fold statistics is interesting, nevertheless the limited information that is obtained is insufficient for many purposes. Consequently it is necessary to develop and exploit the formalism appropriate for two-fold statistics. Of particular importance is the photoelectron correlation function and its variant, the clipped photoelectron correlation function. This section is devoted to the analysis of these functions and their physical consequence. 5.1. Correlation and clipped correlation functions for photoelectrons The measurable quantity available in a photoelectron counting experiment which (in principle) is most simply related to the spectrum of the incident light is the photoelectron auto-correlation function. If rn (t) is the number of photoelectrons counted during a time interval of length T centered at time t, we may define the joint counting rate at times separated by a delay time r as C(r)=(m1(t)m2(t + r)).
(5.1)
It is customary to normalize this correlation function to be unity at infinity, and so we define the photoelectron correlation function by
(m(t) m(t + r)) 2 (5.2) c(r) = C(oo) = (m(t)) We shall use capital letters C (or G, in the case of continuous random functions) to denote joint moments, and lower case letters c (and g) to denote normalized joint moments or correlation functions. When the “window”, T, becomes much shorter than the time over which the field describing a chaotic light source fluctuates, we will see shortly that lim c(r) ~ 1 + Ig(r)12 (5.3) —
C(r)
T-’ 0
where g(r) is the normalized field autocorrelation function. Unfortunately it is frequently quite difficult to measure c(r) directly in a photoelectron counting statistics (PCS) experiment, as this measurement requires either storing the values recorded for m(t) in a vast array, or performing real time multiplications on a time scale of the order of 10—6 to 10-8 seconds. One may avoid this problem by applying the technique of “clipping” — that is by replacing a signal, before autocorrelation, by a series of ones or zeros depending upon whether the signal lies above or below a predetermined “clipping level”. From the auto-correlation function of
R. Barakat and .1. Blake, Theory of photoelectron counting statistics: an essay
274
this “clipped” signal, one can extract information about the full correlation function. This technique was originally developed to analyze the reflected microwave fields encountered in radar applications, the application to PCS experiments is due to Jakeman and Pike [55]. As the technique of clipping is rather specialized, it is useful to devote several paragraphs to a discussion of the general formulation of this problem. Given random processes X and Y which take on the values x and y at a particular instant of time, we define the unnormalized correlation function as G(x,y)~(XY)
p(x,y)xydxdy.
(5.4)
To define the clipped correlation function we replace X and Y by the f(x) and g(y) defined through f(x) g(y)
1,
ifx>~b
0,
ifx
1,
ify~b’
0,
ify
(5.5) (5.6)
where the clipping levels b, b’ are specified. The correlation function of the processes f and g is called the clipped correlation function of the processes X and Y: Gbb (x,y)—G(f,g)-~
J
dxJ
dyp(x.y).
(5.7)
The exact nature of the statistics describing the processes X and Y determines the way Gb,,’ (x,y) is related to G(x,y). For example, if X and Y are each Gaussian random processes with zero mean, Van Vleck and Middleton [56]have shown that 2 G00 (x,y)—arcsinG(x,y).
(5.8)
Generally the relation between the clipped and full correlation functions is considerably more complex. The difficulty of obtaining G(x,y) from G,,,,’ (x,y) is considerably reduced when clipping is applied to only one of the processes, X. This is commonly called single channel clipping. Let us define a singly clipped correlation function, Gb (x,y) as
J
G~(x,y)~G(f,y)-~ dxJ
dyyp(x,y).
(5.9)
In order to evaluate G~(x,y)experimentally we must perform the operation of multiplyingy by a zero or a one, depending upon whether x lies above or below the clipping level, b. This can be done by a simple gate circuit; Gb(x,y) is scarcely more difficult to evaluate than G,,,, (x,y). To apply these general ideas to the problem of evaluating spectra from PCS experiments, we replace the continuous random processes X and Y by the discrete random processes m(t) and
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
275
m(t + r). We define the (unnormalized) clipped correlation function for photocounts as
(5.10)
(mk(t) m(t + r))
Ck(r)
where, in analogy to eq. (5.5): mk(t)= 1 =
ifm(t)>k
0
(5.11)
ifm(t)
Accordingly we see that ~ m,
k
m20
a
ax2
m2P(m1,t;m2,t+r) (_l)mI
a
m1!
axr~
m,k
Q(X1X2)
x1=i~
(5.12)
We will now examine several methods for evaluating Q (A1 , A2), and then show by example how C(r) and Ck(r) can be determined. 5.2. Solution for short counting times Equation (3.62) is the integral equation whose eigenvalues yield the double generating function. Because of the two domains of definition, (0,T) and (r, r + T), it is not of the standard symmetric Fredholm type. It is only when r = 0 that the two domains coalesce and we retrieve the standard type again. We now outline a numerical technique for determining the eigenvalues of the “gap” integral equation due to Blake and Barakat [57].It is probably easier to understand the rationale behind their method by first considering the short counting time approximation where T is small compared to the time over which g(r) decays. Under this condition, eq. (3.62) can be approximated by A1Tg(t)~1(0)+ A2Tg(t
—
1~! ~ 1(t)
r)çb1(t).~-
(5.13)
with / = 1,2. This equation must be satisfied at t that X1T A2Tg(r) q~(0) ~(0)
= 0,
r simultaneously. These two conditions require
(5.14) A1 Tg(r)
A2 T
çb(r)
0(r)
where we have utilized the fact that g(r) = g(—r) and g(0) = 1. To compute the generating function we need not evaluate the m1 individually, but consider instead their product in the form (1 + m1) X (1 + m2). To form this product, we rewrite eq. (5.14) in the form 1
+X1vT
A2vTg(r)
0(0)
0(0) =
A1vTg(r)
1 +A2vT
0(r)
(1 +
m1)
.
0(r)
(5.15)
R. Barakat and .1. Blake, Theory of photoelectron counting statistics: an essay
276
Clearly the matrix has eigenvalues (1 + m1), (1 + m2), and as the determinant of a matrix is the product of the eigenvalues we have 1 +A1vT
A2vTg(r)
A1vTg(r)
1 +X2vT
Q(A1,A2)~=det
(5.16)
.
Upon evaluating the determinant we have 2 (1— lg(r)12)]’. (5.17) Q(A1,A2)= [1+(A1 +A2)vT+X1X2(vT) This expression was first obtained by Bedard [58] and Arecchi, Berné and Sona [591. We have extended the analysis to include the case of partially polarized light, see appendix 5.A. To evaluate the photoelectron correlation function defined by eq. (5.1), we differentiate the generating function as follows (rn
2 [1 + g(r)~2)]~1. 1 (t)rn2(t + r))
Q(A1,A2)~A,
~
=
(5.18)
(vT)
=A,
Thus the normalized photoelectron correlation function is c(r)
1 + Ig(r)12.
(5.19)
The clipped photoelectron correlation function can also be evaluated in closed form in the short counting time approximation (see next subsection for details): Ck (r) =
a — ~
n
k
(—l)~ a~ n! ~ Q(A
k
(VT)k+~
1, A2)
A,
=
1~X2
=0
=
{~+ I + vT Ig(r)12].
(1 + vT)k
(5.20)
Finally the joint probability P(m1, t; m2, t + r) can be obtained from the generating function via lm+m
P(rn1,t;m2,t+r)
(— )
~I
am 2
m1 .rn2 .
+01
rn
oX1
Q(X1,A2)~~
2 I
2
(5.21)
=1~
=~
1
2
Arrechi, Berné and Sona [59] show that P(m1 t;m2 t + r) where A 1 B~ 1
+ +
C—A(1
=
m’ + 022 (mi + m2)! (vTB) 2F1 (—rn1, —rn2, —m1 m1!m2! Am1+tm2~
—
m2, C)
(5.22)
2 (I —girl2) 2vT+ (vT) vT(1 Ig(r)12) —
—
g(r)12)/B
(5.23)
and 2F1 is the hypergeometric function. Note that the probability of obtaining no photoelectrons is 2 (1 Jg(r)12)]’ (5.24) P(0,t;0,t+ r) Q(l,l) [1 + 2vT+ (vT) so that a knowledge of P(0,t; 0, t + r) should enable one to obtain lg(r)12. Reference is made to Furcinitti et al. [60] for details of the experimental results. —
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
277
In the short counting time approximation the single-fold probabilities and moments contain no spectral information. This is not true of second and higher order quantities, their behavior as a function of the delay times is strongly influenced by the field correlation function g(r). 5.3. Solutions for arbitrary counting times Short counting times, as one can surmise, are the ne ultra plus of the experimentalist. However, the short counting time approximation cannot always be realized experimentally. For this reason, we now consider the extension of the analysis to cover finite counting times [57]. By employing N-point quadrature to improve the approximation to the integrals in eq. (3.62), one can construct a generalization, valid for arbitrarily long counting times. The final equation can be expressed in a form identical to eq. (5.16) except that each element of the 2 X 2 matrix is itself an N X N matrix. Applying N-point quadrature to eq. (3.62) yields N
A1 ~
T T Hjg(t ——x1)0(—-x1) + A2
N
T T 2 —-~-x1— r)0(-~-x1+ ~)_7 m1çb(t),
(5.25)
where we have transformed the variable limits of integration to the standard interval (—1, +1). The H1 are the weight factors corresponding to the quadrature points x1. A particular scheme, Gauss quadrature, is preferable for two reasons: first, N-point Gauss quadrature is exact for any polynomial of degree (2N — 1) or less; and second, because the quadrature points are not equally spaced, we are less likely to encounter difficulty if the integrand oscillates regularly. We require that eq. (5.25) be satisfied at the 2N values oft given by ~
(5.26)
This requirement yields a system of 2N homogeneous equations for the 2N unknowns 0(t1), which has nonzero solutions only for the eigenvalues m1. This system of equations can be cast into matrix form by defining the N X N matrix BN (r): h11g(r) h12g{~T(x2—x1)+r} h1Ng{~T(x~ —x1)+r} ...
BN(r)
=
h21g{~T(x1 —x2) + r}
hNl
g{~T(x1
—
xN) + r}
h22g(r)
.
.
.
h~2g{~T(x2— XN) + r} ...
h,~~,g{~-T(x~ —x2) + r} h~~g(r) (5.27)
where hq
2, and the column matrix ~N(r) is given by (H,H1)” 0(~Tx 1+r)
4~N(r)= 0(~Tx2+r) 0(~TxN+r)
.
(5.28)
R. Barekat and J. Blake, Theory of photoelectron counting statistics: an essay
278
Consequently the requirement that eq. (5.27) holds for the succinctly written as ‘N +~X 1vTBN(0)
~X2vTBN(—r)
2N
values of t given by eq. (5.26) can be
~N(0)
(5.29) ~A1 vTBN(r)
+~A2vTBN(0)
‘N
~N(T)
where ‘N is the unit matrix of order N. However, the determinant of the coefficient matrix is simply the reciprocal of the double generating function Q(A1X2): Q(A1A2)’ H (I +mk) k
2X =
det
{‘N
+ ~vT(A1 + A2)BN(0) + (~vT)
1X2 EBN(0)BN(0)
—
BN(r)BN(—r)]}.
(5.30) It can be shown that eq. (5.30) can be differentiated to provide an explicit expression for the photoelectron correlation function
2[l +~trBN(r)BN(—r)] (rn1m2)
(5.31)
(vT)
Q(X1,A2)
ax1ax2
where m 1 m(t), m2 m(t + r). See appendix details. Upon normalizing the photoelectron 2 = (vT)2,5.B wefor have correlation function by dividing by (rn) c(r) = (m 2 = 1 + ~ tr BN(r)BN (—r). (5.32) 1m2 )/(m) c(r) can be cast into integral form by examining the limit as N approaches infinity. Since
N~=
hm
trBN(r)BN(—r)=
~
H
2 1H,~~g[~(x~ —x1)T+r]}
T/2 =
T/2
f
f
—j
—T/2
[g(t’
—
t”
+ r)] 2 dt’ dt”
(5.33)
—T/2
we have c(r)
=
I +
~1
f
T/2
T/2
f
[g(t’
—
t” +
r)J 2 dt’dt”.
(5.34)
—T/2 —T/2
We can recover the short counting time approximation, eq. (5.19), by letting T approach zero. Blake and Barakat [57] have shown that the full photoelectron correlation function for a Lorentz spectrum is c(r)
=
=
1 + e2T(5i~T) 1 + T2[.~
,
e_2(T+T)
0 ~ T~r +~e2(TT)
by direct integration of eq. (5.34).
_~
e_2T
+ T—r],
0~
(5.35)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
279
To obtain the clipped photoelectron correlation function Ck (r) we employ eq. (5.12). For N = we recall that the generating function is given by eq. (5.17). In this case the differentiation and subsequent summations can be carried out directly by using the substitutions 1 +vTX2 2[l —g2(r)]X V—vT+(vT) 2.
1,
U
With Q now expressible as (U +
(5.36)
A1 V)~,we
have tm’
am, ~ Q(A
(V)
mi (U+ V)mt+~ 1,X2)~A, =1 m1!(—l) and the sum over rn 1 can be evaluated easily:
am,
(~~l~j
m1
m,k
1
ax1~
(5.37)
k
V
x,=iU’~U+V!
Upon performing the remaining differentiation, we recover
(
Ck(r)
VT)
VT(l + 1 +vT g2(r))
(5.20)
This expression was first obtained by Jakeman [20]. In order to obtain an expression for Ck(r) valid for longer counting times, we use the generating function given in eq. (5.30) with N = 2. In this case we can show that 2 —X2 —W2) (s 1+s1+s2 +s1s2(l + Y 1 +s2)Y+s1s2(2Y—XU—XW) = det 2 —X2 —U2) (s1 +s2)Y+s1s2(2Y—XU--XW) l+s1 +s2+s1s2(l+Y (5.39) where s 1
(~vT)X1and
Xg(r),
Ug(r.+L~)
Yg(i~),
Wg(r—L~)
with ~
~
(x2
—
x1 )T. We can evaluate the determinant explicitly with the result that
Q[L+Ms2+O(s~)]’
(5.41)
where (5.42)
2)s~
L = 1 + 2s1 + (1 — Y M = 2+ s 2 — U2 1 (4 — 2X Thus
01
—
W2) + s~(2—
2Y2
—
2X2
—
U2
—
W2 + 2XYU + 2XYW).
(5.43)
vTM (5.44)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
280
We must now evaluate the sum over m1: Ck(r)=
ivT\
0021
(_l)mi
°°
—
—(—I ‘ 2 ‘ m,—k -
iM\
2/ —I—I
x,=i
(5.45)
.
m1! OAT’ \L Unfortunately we have not found any way to carry out this summation in closed form. For small k we can use a trick due to Jakeman [20]. Since (—l)°’’ am, a ~
—
-
00
00
m
m~—0
1!
~Q(A1X2)~ ~2=o
~
=
= m,—O
~
022—0
m2P(m1,m2)
(m1),
(5.46)
we can rewrite the clipped correlation function as (_l)m,
Ck(r)=_~
=
vT
k—i
~~Q(A1A2)~
mi!
(rn1) +
a
0””
(_l)m,
k—i
—
______
0””
(_l)m, am, mi!
/
— 3’\I---
a
~
M~
L2/I —
x,=i
2 m, 0 m1! 0X7 Equation (5.46) is easy to apply for small k; in fact
(5.47)
.
=
C 1
(r)
=
/vT\/ M (m1 ) + ~—J~——2 \ 2 1 \ L vT2
C
2(r)=Ci(r)_(_~-) C 3 (r) C
=
4(r)C3(r)
~
2ML’
~
1 vT ~
C2 (r) + —(-—) 1 vT
M’
V
-
)
2 4 4M’L’ + 2ML” 6ML’ (~} + V L 6L”M’ 6M”L’ 18M’L’2 + 18MLL” M”
—
‘~
(
24L’M
—
L~
+
—
V
)
(5.48)
where the prime denotes differentiation with respect to s and all expressions are to be evaluated at s = ~vT. Higher order Ck (r) can be calculated via the recurrence relation (1)k_i
Ck(r)Ckl(r)+
(k— 1)!
0k—1 1 ~Q(A ~
0A~
1A2)
~
k> 1.
(5.49)
A2 = 0 It
is instructive to apply our numerical techniques to the case of a Brillouin spectrum. Reference is
again made to table 4.3 for definitions of the parameters c~,~, and ‘y which specify the Landau—
Placzek ratio, frequency shift, and halfwidths of the spectrum. A superposition of spectral lines, such as the Brillouin spectrum, will display hetrodyning and thus cause c(r) to oscillate on a dimensionless time scale of l/~.c(r) is plotted as a function of r for counting times T = 0.1, 0.2 in figs. 5.1, 5.2. We selected these counting times because T = 0. 1 is just long enough to show departure from the
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
281
0.400306091.2
__
INTERVAL TIME
Cr)
Fig. 5.1. Photoelectron correlation function c(r) for a Brilouin spectrum(~= 10,6 = 1)withT0.1, v 1: —••—,a 0; —----,a=0.4;—,a=0.6;—.—,a’~ 1.0 (Blake and Barakat [571).
INTERVAL TIME (r) Fig. 5.2. Photoelectron correlation function c(r) for a Brillouin s~~trum(~ = 10,6 = 1)withTo.2, v 1: _.._,n 0; ,n’0.4;—,n=0.6;—.—,o= 1.0 (Blake and Barakat [57]).
short counting time approximation and T = 0.2 is just long enough to cause averaging over major details of the spectrum. For T ~ l/z~,c(r) tends to a Gaussian-like shape and the oscillations vanish entirely. In making these calculations, we employed six quadrature points (N = 2 in the notation of subsection 5.3). For counting times as short as T = 0.2, the eigenvalues fall off quite rapidly and in practice only the first two make a significant numerical contribution. This is consistent with the behavior of the corresponding eigenvalues of the single-fold problem (subsection 4.6). The clipped correlation function ck(r) for a Brillouin spectrum is shown in figs. 5.3 and 5.4. As is customary, we have normalized by dividing by the product of the true and clipped mean counting rates: ck(r)—
Ck(r)
(5.50)
From the figures we note that as T becomes larger, there is a decrease in the slope of ck(r) at the origin, and an averaging over details in the spectrum. T = 0.2 is large enough to separate the curves 2.0 I
INTERVAL TIME
I
Cr)
Fig. 5.3. Clipped photoelectron correlation function c,(r) for a Brillouin spectrum (~= 10,6 = 1), with T = 0.1, v = 2.0 (uT ,a0.4;~,a0.6;_._,n1.0 (Blake and Barakat [57]).
2.0 I
INTERVAL TIME Cr) Fig. 5.4. Clipped photoelectron correlation function c,(r) for a Brillouin spectrum (~ = 10,6 = 1), with T = 0.2, v = 1.0 (uT ‘0.2);—..—,n0; (Blake and Barakat [57]).
R. Barakatand J. Blake, Theory of photoelectron counting statistics: an essay
282
of different a values from each other at r = 0. In order to give some idea of the effects of clipping at different levels, we show, in fig. 5.5, ck(r) for a Lorentz spectrum at T = 0.1. The method employed in this section can be applied directly to any other quasi-monochromatic spectrum, of course. By using larger values of N one can treat longer counting times, although for k> 2 the algebra for computing the clipped correlation function becomes considerably more cumbersome. 5.4. Q(Ai,X2) for the Lorentz spectrum Jakeman [20] has shown that Q(X1,X2) can be evaluated in closed form for Gaussian light having a Lorentz spectrum; this solution includes Q(X) as a special case. In view of the importance of this result we will outline the relevant analysis especially since Jakeman only presents the analysis in bare outline. We wish to thank Dr. Jakeman for helpful correspondence on this point. The field correlation function for the Lorentz spectrum was given in eq. (4.26). Substituting this kernel into the integral equation, eq. (3.26), we have A1 ~ e ~
~
Ø(t’) dt’ + A2
e_Itt’I ~(t’)
r~fT
(5.51)
dt’ =~ 0(t),
v
0
where we have made the intervals explicit: t1 = [0,t], t2 = Er, r + TI. Let us rewrite the left hand side so as to eliminate the absolute value signs in the exponentials: A1 ~ e_t
+
~‘
0(t’) dt’+ A1 ~ ett’ 0(t’)dt’ + ~2 ~ T ett’ Ø(t’)dt’ ~
(t),
0 ~ t ~ T (5.52)
and A1 ~et+
~‘
0(t’) dt’ + A2 ~ e_t+ t’ 0(t’) dt’ + A2
J
e~
~‘
0(t’) dt’
~ 0(t),
r~
T.
(5.53) 4 I
I
I
\
~0
0.3
0.6
0.9
1.2
INTERVAL TIME Cr) Fig.
5.5. ck(r) for a Lorentz spectrum
with T
0.1, v
0.1:
,
k = 1; —,k
=
2; --.—,k
=
3 (Blake and Barakat [57]).
R. Barakat and J.
Blake,
Theory of photoelectron counting statistics: an essay
283
Upon differentiating these equations twice with respect to time, we obtain the ordinary differential equations 0~t~T (5.54) I
—(-~—2X2)0(t)0,
~0”(t)
r~
The solutions to these equations may be written as 0(t)A cos7t+Bsin’yt, Ccos~t+Dsin~t,
0~t~T
r~t~r+ T
(5.55)
where we have defined 72
i2A1v
\
— 1)
~2
m
12X2v
\
— 1)
(5.56)
.
m
In differentiating eqs. (5.52) and (5.53) twice, we have essentially lost the boundary conditions. To reintroduce this information, we substitute the general solution given by eq. (5.54) into eqs. (5.52) and (5.53) and require validity for all values of t within the pair of domains (rangel)
0~t~T,
r
(5.57)
Thus, A1
et
+
~‘
~A cos7t’ + B sin7t’) dt’ + A1
e~
— ~‘
(A cosyt’.+ B sinyt’) dt’
r+T
+ A2
~ et
—
‘
(C cos~t’+ D sin~t’)dt’
~ (A cosyt + B cos’yt)
(5.58)
while eq. (5.52) becomes ~1
e_t+t’ (A cos7t’ +B sin7t’)dt’ + A2 ~ e~~’(Ccos6t’ +Dsinót’)dt’ r+T
+ A2
~
et_t’ (Ccos~t’+D sin~t’)dt’ ~~Ccos6t
+D sin~t).
(5.59)
Performing the integrations is straightforward. Then, collecting all terms in e ~and all terms in the result is —
e+t,
e_t{_A + ‘yB} + e+t {Ae_T(_cosoyT + ysin7T) + Be_T(_sin7T
—
‘ycos7T)
+ Ce_T[e_T~sin6(r + T) — e_Tcos6(r + T) + cos~r— 6sinór] T [_e_Tsin&(r + T) — e_T6cos~(r+ T) + sin~r+ ~cos~r] } = + De
0.
(5.60)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
284
Similarly, eq. (5.59) requires that et{A [eTcosyT + eT7sin~yT 1] + B [eTsin~yT eT7cosyT 7 (cos6T + ~sin~T) + DeT(ôcos~T sin6T)} + Ce + e0 {Ce~[~sin~(r + T) — cosö(r + T)] + DeT [—sin~(r + T) —
—
—
(5.61)
—
—
öcos~(r+ T)] } = 0.
The coefficients of the bracketed terms are independent of each other, consequently the only way for eqs. (5.60) and (5.61) to vanish is for the bracketed terms to vanish individually. This leads to four simultaneous equations, which we write in matrix form MC=0
(5.62)
where C is the column vector C +
=
(A ,B,C,D) and M is 7
—1
0
eT(—cos~yT+ysinyT)
_-e7(sinyT+ ycosyT)
—1 + eT(cosyT + ysinyT)
+eT(sinyT
— M
0
—
ycosyfl
0
0
eT(cos~r— ~sin~r)
eT(sinor + 6cos~r)
—cosb(r
—eT(cos&r + 5sin~r)
—sin~(r+ T) — lcos& (r + T) _eT(sinor — &cos6r)
bsin~(r+ 7)
—sin~(r+ T)
+ T) + ~sin~(r + T)
—
cos~(r+ T)
—
~cos6(r +T)
(5.63) For eq. (5.62) to have a nontrivial solution the determinant of M must vanish. To evaluate det M conveniently, we begin by performing several elementary row operations casting M into the following form. —1
M
y
eT(—cosyT + ysinyT)
—eT(sinyT + ycosyT)
eT(cosyT + ysinyT)
eT(sinyT
0
—
0
0
-ycosyT)
0
eT(sin~r +
~cosbr)
—cT(cos~r+ 8sin~r)
--eT(sin5r
—
~cos~r)
ôsinó(r + T)
—sin6(r +
T)
eT(cosor
—
~sinör)
—
cos6(r + T)
—
öcosä(r
+
T)1
(5.64)
It is convenient to define the functions a
sinö(r + T) + i5cosö(r + T)
/3
~sin~ (r + T) 2
F( 7,T)
—
(y
G(y,T)~(y2+
—
costS (r + T)
l)sin’yT
—
2’ycosyT
l)sin7T.
(5.65)
In terms of these definitions detM = e2TF(-y,T) {(/3tS
—
a)costSr
—
([3 + atS)sini5r}
—
e2TG(~y,T){(j3tS + a)cosbr + (/3
—
atS)sintSr}. (5.66)
The bracketed terms are now expanded using trigonometric identities, tedius manipulations then yield —
a)cos6r
—
([3 + atS)sintSr}
{(/3a + a)costSr + ([3
—
F(cS,T)
(5.67)
atS)sin6rH G(6,T).
(5.68)
=
R.
Barakat and J. Blake,
Theory ofphotoelectron counting statistics: an essay
285
Consequently, the determinant of !i~becomes (5.69) detM = e2TF(7,T)F(tS,T) — e2TG(oy,T)G(tS,T). For a specified pair of values of A 1 and A2, m will be an eigenvalue of the basic integral equation only when det M = 0, or 2TF(y,T)F(tS,T) e_2TG(~y,T)G(tS,fl 0. (5.70) e_ There are two ways to obtain the one-fold case. The first is to set A 2 = 0 in eq. (5.70); the result is 2)sinyT+2ycos7T0 (5.71) (1 —7 which is equivalent to eq. (4.28) with y. The second way is to let r = T/2 and A 1 = A2 A. A straightforward, but tedious analysis, again leads to eq. (5.71) as expected since the answers are time translational invariant. We now pass on to the problem of obtaining Q(A1 ,A2) and Q(A) in closed form. Consider the function 2TF(g,T)F(d,T) + e2TG(g,T)G(d,T)] (5.72) P(~)= —~--[e 4gd —
where we have defined =
2A 1v~—1
2 = 2A
2v~—
d
1
(5.73)
and set ~ = 1/rn. P(fl is simply detM multiplied by the factor —(4gd)~exp(—4T — 4r). If we extend ~ into the complex plane, then P(~)is an entire function of ~. Furthermore, let M(r) be the maximum of P(~)on the circle Izi = r. The order, p, of the entire functionP(~)is given by — loglogM(r) lim ~ (5.74) Pr_ooo log~ r-o~2log~ 2 As discussed in texts on complex variable theory, Ahlfors [61], Titchmarsh [62], if an entire function is of fractional order less than unity, then the order p is bounded by h ‘~p ~ h + 1 where h, an integer, is the genus of the function. Here h must be zero. Since P(~)is an entire function of genus zero and is also devoid of zeros at the origin then it can be expressed as an infmite product via the Hadamard factorization theorem in the form P(fl=C
H
(i
\
k0
——‘i
(5.75)
~k’
where C is a numerical constant and {~k} are the zeros of P(~).Obviously, mk The double generating function Q(A1,A2), can now be written as —
=
H
k0
~°
(1 +rnk)=C
II
k0\
i l~ (1 +_—)=
P(—l)
=
C
lftk.
(5.76)
.
2 withj = 1,2, then When
~ = —1,
both g and d become pure imagery. If we define (3~=
(1 +
2vA,)b’
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
286
P(—l)becomes
4131/32
—
e2T[(l + A1v)sinh/31 T+ 2/3k cosh/31 T]
4131132e
[(1
+
A2v)sinh/32T + 2/32cosh/32T1
(A1vsinh[31 T) (A2vcosh[32T).
(5.77)
Since Q(0,0) = 1, then C 1. Upon carrying out some further computations, we can express the double generating function as 1 =e_2T [~([3~ +(311)sinh/3 Q(A1,A2) 1T+ cosh(31T1 [~(/32 +j3~’)sinhj32T+coshj32T] 1)sinhj3 e_2T [~ (I3~— j3j 1 TI [~ (/32 /3~’)sinh/32TI. (5.78) —
—
This is essentially the result obtained by Jakeman [20]. Note that the two time intervals enter the expression in a completely symmetric manner. To obtain the one-fold generating function, we set A2 = 0 (i.e., /32 = I); then the result is 1)sinh/3T + cosh/3T]. (5.79) e_T [~ (/3 + f3 Simple manipulations then yield the form quoted in eq. (4.30). We should note that the result quoted in eq. (5.79) was first evaluated by Slepian [381using contour integration. Srinivasan [63] also obtained eq. (5.79) by using the Hadamard factorization approach. It should be noted that Wittke [64] was evidently the first to employ the theory of entire functions to such problems, he obtained eq. (5.79) for the closely related problem of the autocorrelation function of the output of a nonlinear angle modulator. In the limit that T becomes much shorter than the coherence time of the incident light (T ~ 1), one can derive several approximate and useful results. In this limit, we have from eq. (5.78) Q(A
1 = 1 + (vT) (A 2A 27). (5.80) 1,A2) 1 + A2) + (vT) 1 A2 (1 e To this approximation only two eigenvalues of the gap integral equation contribute to the product in eq. (5.76) —
(vT)
m 1
I
1 A2 (1 —
(A1 + A2)
1 +
2!~i_~(A1+A2)
{ 1—
—i—
rn
4A 1
—
2T)1 e2
+ A2)
.1
1/2
2~)j~2j [1_
(5.81)
4A2e~
m 3m40. As the reader can show, these results also follow from taking the limit T —~0 of eq. (5.70) and solving the resultant quadratic equation. Srinivasan et al. [651 have obtained the N-fold equivalent of the transcendental equation, eq. (5.70), for the eigenvalues. Additionally, the single-fold casehas been treated for a spectrum consisting of a superposition of Lorentz lines by Tomau and Echtermeyer [661. Both these solutions make use of a technique resembling Laplace transform, but with a finite upper limit of integration. Unfortunately these explicit formulae for the eigenvalues are extremely complicated and not very suited to numerical computations.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay 5.5.
287
Higher order photoelectron statistics
While interest has usually been focused on second order correlations measurements of the clipped and unclipped correlation functions third order experiments are both practical and useful. Davidson and Mandel [67] and Davidson [68] have measured third order correlations with a time-to-amplitude converter; and more recently, Corti and Degiorgio [69] have performed third-order measurements with a fast digital correlator. Therefore, in this subsection we demonstrate how third order statistics may be extracted from the basic integral equation for Q. Our approach is a direct generalization of the techniques outlined in subsection 5.2. Generalizing eq. (3.62) to include three counting intervals, we write following Blake and Barakat [701 —
—
~ A1
g(t
—
t’)
0(t’) dt’ ~
(5.82)
0(t).
Taking the three counting intervals, each of the same width T (where T is taken to be small compared with the time required for g(t) to change significantly), we have ~ A. g(t
—
(5.83)
t1) 0(t1) ~0(t).
Let t take the values t1, t2 and t3 in turn. For eq. (5.83) to be satisfied we must have A1
A2g(t1
—
t2)
A1g(t2
—
t1)
A2
A1g(t3
—
t1)
A2g(t3 — t2)
A3g(t1
—
t3)
0(t1)
0(t1)
A3g(t2
—
t3)
0(t2)
0(t2)
0(t3)
0(t3)
t3
.
(5.84)
Let us define the matrices 0(t1)
0~
(5.85)
0(t2) 0(t3)
A1vT+ 1 A
A2uTg(t1
A1vTg(t2
—
r1)
A2vT+ 1
A1vTg(t3
—
t1)
A2vTg(t3
—
t2)
A3vTg(t1
—
t3)
A3vTg(t2 — t3) —
t2)
A3vT+
.
(5.86)
1
Then eq. (5.84) can be rewritten as Aç~(1 +m)Ø.
(5.87)
The matrix A has eigenvalues (1 + m1) where/ = 1,2,3, and we have Q(A1,A2,A3)11(1
+m1~’ (detAY’,
(5.88)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
288
where detA 00(1 + A1vT)(l + A2vT)(l + A3vT) + 2(vT)3 A1A2X3g12g23g31 2 [(1 + A —(vT) 2vT)A2A3g~3+(1 + A2vT)A1A3g~1+ (1 + A3vT)X1X2g~2]
(5.89)
and g11 —g(t, — t1). With Q(A1 ,A2,A3) in closed form, we are in a position to evaluate joint probabilities and moments by direct differentiation. For example, the probability of obtaining no photoelectrons is P(0,t1 ; 0, t2
0,t3) =
Q(A1 ,A2,A3)~A, A2 A3 1 3 + 2(vT)3g ~I(l+ vT) 12g23g31 =
=
=
2 (I + vT)(g~ —
(vT)
1. 2+g~3+g~1)}
(5.90)
Another example of particular experimental interest is the triple photoelectron correlation function: c(r
3, 1,r2)~(m1m2m3)/(m) for delay time r 1 = t2 t1 and r2 = t3
—
t2.
From eq. (3.65)
A =A
=A
=0
—
~3rr1 ~‘.
(m1m2rn3) =
~
1’”2, 22~3
(5.91)
‘3
(5.92) 1
2
3
Upon performing the necessary manipulation we obtain 2(r 2(r 2(r c(r1,r2) 1 + 2g(r1)g(r2)g(r1 +r2)+g 1) +g 2)+g 1 +r2). Note that c(r1,r2)
(5.93)
c(r2,r1).
Equation (5.93) must reduce to the expression for c(r) given in eq. (5.19) if r2 (or alternately r1) becomes large. Since lim g(r2)
0,
(5.94)
0 00
we have 2(r
lim
c(r1,r2)= I +g
1)=c(r1).
(5.95)
The bahavior of c(r1,r2) is shown in fig. 5.6 for a Lorentz line shape as a function of r2 for fixed values of r1. The curves undergo a monotone decrease to their limiting value c(r1). The main effect of increasing ~r1is to decrease the value of c(r1, 0) relative to the background value of c(r1). c(r1, r2) for a Brillouin line is shown in fig. 5.7. We get the oscillatory behavior characteristic of hetrodyning. A particularly useful caclulation would be the evaluation of the clipped photoelectron correlation functions from Q(A1A2A3). This problem certainly deserves further work. If we limit our interest entirely to joint moments, we can avoid direct reference to the generating function, and extend counting times to longer values. We take the expression (m1 m2 m3) as an example, this permits us to derive eq. (5.93) in a very different way. It follows directly from eqs. (3.36) and (5.92) that we can write (m1m2m3)
=
(&~(t1, {ak})
~(t2, {ak}) ~T~(t3,~ak})),
(5.96)
R. Barakatand J. Blake, Theory of photoelectron counting statistics: an essay C
c
0
I 0.4
I
I
I
I
I 0.8
I
I
I
I 1.2
I
1.6
Fig. 5.6. Triple photoelectron correlation function for a Lorentz line as a function of ~2 for fixed values of i~,: = 0.1;—.—, r, = 0.4 (Blake and Barakat [70]). I
I
I
Fig. 5.7. Triple photoelectron correlation function for a Brillouin line , r, = 0;—, r, = 0.1; —.—, r, = 0.4 (Blake and Barakat [70]).
289
,
r,
=
0;
—,
I
(~= 5, ~
=
1) as a function of
~2for fixed values of r,:
where ~
{ctk})Ns
~
~%“(t’,{ak})
~(t’,
(5.97)
{ak})dt’
l-
and where (• .) indicates an average using the weight function P({ak}). When P({ak}) is given by the Gaussian distribution, eq. (3.36), S(t, fak}) is a complex Gaussian random variable, for it is simply a linear combination of the ak’s. A convenient summary of basic properties of such variables is given by Mehta [17]. The even order moments of ak can be expanded in terms of the second moments of writing a,, a,
a1)= ~
a1)(a~’~
a1>,
(5.98)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
290
where ~ stands for a sum over the m! permutations p,q, m = 3) then yields
, r of 1 ,2,
.
...
,
m. This formula (for
(I S(t’)12 16(t” + T1 )~2 6(t” + T1 + r2)12) + 2 Re (~* (t’)~(t” + r 1)) (~*(~1~ + r1 )~(t” + ~1 + r2)) (6 *(tlFf + ~1 + T2 2){~(g*(~F)~ (t” + r + (I~(t’)~ 1 ))~2 + I((*(tU + r1 )~(t” + r1 + r2 ))12 +
I(~*(tflP+ ~1
(5.99)
+ T2)~(t~))I2},
where we have used the fact that 2) = (V (t” + ri)12)= (I~(t”+ Ti + r (V (t’)1 2)j~).
(5.100)
By interchanging ensemble averaging and integration in eq. (5.96), we have 3 Km1 m2m3>
=
~ dt” (I~(t’)2 I~(t”+ r
dt’ [dt”
I~(t”+
1 )12
(Ns)
~1
+ r2 )12).
(5.101)
Substitution of eq. (5.99) into this triple integral and appropriate normalization yields the triple photoelectron correlation function c(Ti, r2): 2 + ~ g(T1,r2)
=
1 +~ +
[dt’ T
I dt’ T2J
I
0
0
—
+
dt’
~ dt” g(t” + r1 dt” g(t’
—
~dt” g(t” + r
2 2
r
—
t”)
t”)2 + 1
dt”
~dt”
t’)
---
—
r2
—
t
dt” g(t” + T 1
—
t’)g(t” + r2
—
t”)g(t’
—
—
—
t”).
(5.102)
Note that we have utilized the fact that the field correlation function g(t) is real when the radiation is quasimonochromatic. When T is short enough to permit g(t) to be treated as a constant, eq. (5.102) reduces to eq. (5.93). Similar techniques permit the evaluation of other moments in terms of the field correlation g(t). However, the joint probability functions P(rn1 ,m2, , mN) and the clipped photoelectron correlation functions can only be evaluated in terms of the N fold generating functions. Appendix 5. A: Q(A1, A2) for partially polarized light In this appendix we generalize eq. (5.16) forQ(A1,A2) in the short counting time approximation to include partially polarized light. Following the notation of subsection 4.11, we write the generalization of eq. (4.117), whose eigenvalues determine Q(A1, A2), as A1 ~
~ F11 (t
— t’)01(t’)
dt’ + A2 ~
F1~(t— t’)0,(t’)dt’
=~
0~(t).
(5.A. 1)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
291
Upon approximating the integrals as we did in eq. (5.3), we obtain 2 A
1
T
~
m
2 F11(t)0,(O)
+ A2T ~
j=1
F11(t —
T)0,(T)
(5.A.2)
I) 01(t).
f=1
The requirement that this holds for t = 0~(O) A21(—r) A2J(0)
A1J(0)
A1J(r)
0,
r can be expressed as
0~(0)
02(0) 01(r)
rn 02(0) uT 01(T)
02(r)
02(T)
(5.A.3)
where 1(r) is the polarization matrix defined in eq. (4.108). The generating function Q(A1, A2) follows by forming the product of one plus the eigenvalues, we have det
1 Q(A1,A2)
1+ A~vTJ(O) A2vTJ(—r) A 2vTJ(r) 1+ A2vTJ(0)
(5.A.4)
where I~sthe 2 X 2 unit matrix. Unlike the single-fold case where a single parameter, ~, sufficed, an evaluation of Q(A1, A2) requires a knowledge of J(r) for all delay times. Appendix 5.B: Evaluation of the photoelectron correlation function Upon employing eq. (5.31), we can write the photoelectron correlation function in the form (m(t)m(t + r)>
2
=
a Aa 1
(det Ay1
(5.B.l)
a A2
where A ~~+!?~
+ (U7’)2
(A1 + A2)E(0)
[E(0)E(0)
—E(r)E(—r)].
(5.B.2)
The matrix differentiation can be performed explicitly. Setting (det A) ~A for convenience, we have
2
a 8A 1
a~.2
r
8A aA
I.
ax.,
=I~3——A2
A’ A, =
=
0
8A2
~2A i
aA, aA2Jx,
aA aA
a2A
8A1 ~A2
aA1aA2
2———
I
=
= 0
-
x,
=
A2
=
0
(5.B.3) where we have used the fact that
AI~~0
1.
(5.B.4)
By employing the identity detA
exp[trlogA]
(5.B.5)
R. Barakat and .1. Blake, Theory of photoelectron counting statistics: an essay
292
the following two formulas can be derived
a aA1
—A
A trA~—A
(5.B.6)
aA1
—
tr(A1
tr(A1 ~A)
A =A
tr (A
A ~-_A)}
2
~A)
+ tr(A1
aA1aA2A) (5.B.7)
.
Substituting A from eq. (5.B.2), we obtain 2 A~[B(0)E(0)—B(r)B(—r)i,
(~)E(o)+(~)
~
a2 ~
vT\2
A
=
-
11,2
(5.B.8)
-
(-~—) [B(0)B(0)
—
B(r)B(—r)1.
(5.B.9)
When these expressions are substituted in eq. (5.B.3) we have (m(t)m(t + r)>
(~)2
{[trE(0)]2
—
tr [E(r)~(—r)I } = (vT)~ {l + ~tr [E(r)~(— T)1} (5.B. 10)
where we have used trE(0)~B11—2.
(5.B.ll)
6. Time interval statistics 6.1. Derivation of time interval probability densities Another useful approach to photoelectron counting statistics is through the time interval statistics. Since the underlying stochastic process is non-Poissonian, we would expect that measurements of such quantities as the distribution of time intervals elapsing between consecutive photoelectron counts would yield valuable information. The value of these techniques lies in the ease with which the distributions can be measured over an exceptionally wide range of time scales by using a time-toamplitude converter (Scarl [71], Davidson and Mandel [67], Chopra and Mandel [72], Kelly and Blake [73]). Strictly speaking, time interval probability densities constitute a problem in two and three-fold probabilities which must be expressed as functions of Q(A1, A2) and Q(AI, A2, A3). However it is possible to cast the problem into a form involving only the single-fold generating function Q(A) under certain limiting conditions. We follow an unpublished derivation of Blake and Barakat. We denote by A,B,C the following three events:
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
293
A. Arrival of one photoelectron during (t1 Z~t,t1) B. Arrival of no photoelectrons during (t1, t2) C. Arrival of one photoelectron during (t2, t2 + tSt) where t1
a aA3
P(B,C)=——Q(A1,A2,A3)
(6.1)
.
A,=o A2 = A3
=
1
Here, Q(A1,A2,A3)(exp{—A1~21—A2,Q2 —A3~23})
(6.2)
where 2 dt’,
/ = 1,2,3
(6.3)
Ns ti~ ~(t’)I and t~stands for the appropriate time interval. Similarly, the probability of time intervals elapsing between consecutive photoelectron counts is the probability that, given a photoelectron count during (t 1 — z~t,t1), then the next subsequent count occurs during (t2, t2 + tSt). This conditional probability is
P(B,CIA) =
P(A
B C)
P(A)
(6.4)
Again, using, eq. (3.64) we have P(A,B,C)=
2 a aA1aA3
Q(A 1,A2,A3)
(6.5) A,=A2=A3=1
and P(A)=~-
Q(A1,A2,A3)~
(6.6)
.
A2
=
A3
=
0
Thus, P(A)
Ns(J
L I(t’)12 dt’}).
I~(tl)l2dt’ exp (_Ns ~,
(6.7)
In order to proceed further we now assume that the intervals L~tand t5t are small in two respects: 1. They are small compared to the time scale on which (Ig(t)12> fluctuates. 2. They are short enough so that the probability of more than one photoelectron count occurring during either interval is negligible. In this case we have I(t’)12 dt’ ~ ~(t 11
—~t
2~t, 1)I
(6.8)
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
294
I~(t’)I2dt’)
J
exp {_Ns
~ 1,
(6.9)
ti-it
P(A)~NsI6(ti)I2i~t=w&.
(6.10)
In like manner P(B,C) ~ Ns (exp ~—Ns ~2 I~(t’)2 dt’) I ~ (t 2)
2 ~1(ti
P(A, B,C) ~ (Ns)
P(B,CIA)
)12
exp t_Ns
)12 exp (_Ns
~(t1
J
j
(6.11)
2~,
I I(t’)I~dt’} I1(t
I ~(t’)l~ dt’
2)12)tSt ~t.
} I t(t2
Now consider the single-fold generating function with
~2
)12)tSt.
(6.12)
(6.13)
~+ ~1
Q(A) =(exp f_Ns ~ k~(tP)I2 dt’}).
(6.14)
Since g (t) is a stationary stochastic process, then we may view either t1 or t2 as arbitrary so long as the interval between them remains t. Differentiating eq. (6.14) with respect to t and considering t1 to be fixed, we have:
A
=
=
—Ns (exp (_Ns ~2I~(tl)I2 dt’} 16(t2)12).
(6.15)
Comparison with eq. (6.11) yields P(B,C)
=
—tSt
-p-- Q(A) at
.
(6.16)
If we define the probability density V(t) V(t) ~~-P(B,C)
(6.17)
then V(t)
=
a
——Q(A) at
.
(6.18)
Since V(t) is a probability density then it must satisfy the condition V(t)dt= 1
(6.19)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
295
in addition to being non-negative. This is a useful check on calculations. A second differentiation of eq. (6.15), now viewing t2 as arbitrary and fixed yields: 2 dt’) Ii~(t (I~(ti)I2 exp (_Ns ~ I~(t’)1
(6.20)
2) 2).
Comparison with eq. (6.12) yields
2 a at
=P(A,B,C).
tSt&—~Q(A)
(6.21)
We define the conditional probability density P(t) 00
1 a2 P(B,CIA) =—-—j
1 —
wat
tSt
(6.22)
.
Q(A) A=1
In applications, we will find it easier to write V and P in terms of the normalized time T and the normalized count rate v. Consequently, probability density function of the time intervals which elapse before the occurrence of the first photoelectron count
V(T)
a
= —
—
Q(A)
(6.23) A=1
and P(T)
conditional probability density function that given a photoelectron count at T = next count occurs at time T
a2
i
.
—~Q(A)
v ~3T
0,
the
(6.24)
A1
Equations (6.23) and (6.24) for the distribution of time intervals have been given previously by Glauber [2, 3]. Though not strictly rigorous, his arguments add some intuitive insight to this discussion of time interval statistics. Recall that the generating function evaluated at A 1,
Q(A)1A
1
=P(0,T)
(6.25)
is simply the probability that no photocounts have occurred during a time interval from t = 0 to 00 T. For a stationary process, this interval may be given an arbitrary translation. Clearly the probability that at least one photocount occurred during the interval is 1 — P(0, T). If we choose, the initial instant at random, then the probability that the first photocount occurs between T and
T+tSTis - P(B,C)=P(0,T)[1
—P(0,5T)] =P(0,T)_P(0,T+6T).
Thus the probability density function V(T) reads V(T)
1
=
—P(B,C) 00 6T
a ——
~T
Q(A,T)
.
A1
(6.26)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
296
Notice that we have set P(0,T)P(0,6T)=P(0,T+
6T)
(6.27)
in deriving eq. (6.26), which would be strictly true only were the photocounts uncorrelated. To find the distributions of time intervals between consecutive photocounts we may employ the same technique: F(A,B,C) 00 [1
— P(0,~T)]P(0,T)
[1
—
P(0, 6T)],
(6.28)
where again we have made use of the stationary nature of the process. Thus P(A,B,C)=
[P(0,T + L~T+ 6T)
and
—
P(T + ST)]
2
1 a Q(A) P(T)=P(B,CIA)=___
vaT2
.
—
[P(0,T + aT)
—
P(0,fl],
(6.29)
(6.30)
A=i
Again, the argument is strictly rigorous only when the photocounts are uncorrelated. Because the distribution of time intervals is perhaps less familiar intuitively than such comparable quantities as the photoelectron correlation function, it is useful to consider some simple examples. First, suppose the photoelectrons arrive randomly (as from a perfectly coherent source or a very long counting time). Then,
Q(A,T)
(6.31)
e_~~~)T,
and V(T)ve~T
(6.32)
P(T)
(6.33)
=
v e~T.
V(T) and P(T) are identical in this case (i.e., Poisson statistics) because starting the time intervals at random or at photocounts are equivalent conditions. In other words, both V and P are negative exponential probability density functions. Notice that the photoelectron correlation function would be independent of Tin this case: c(T)
=
1.
(6.34)
As a second example, suppose that the count rate v is small enough so that vT ~ 1 for those T we are considering. Then from eq. (6.13), we have P(T)
v[l + Ig(r)121
=
vc(r).
(6.35)
Thus, for small vT, P(T) resembles the photoelectron correlation function, which is typically the way P(T) is treated in PCS experiments. Unlike c(T) which always approaches a constant background value, P(T) eventually begins to decay exponentially because for large enough T, Q(A,T) will approach the function given in eq. (6.3 1). 6.2. Short counting time, partial polarization Let us now examine the behavior of V and P when the counting time is short (so that only one mode of the field is effective) and the light is partially polarized.
R.
For 0 ~
Barakatand J.
Blake, Theory of photoelectron counting statistics: an essay
297
1, we must differentiate eq. (4.120), the final result is
V(T
=
‘
(1
(1 +,~)v/2 2(1 + b + b
(1 —.~)v/2
(6.36) 2
2)
1)
where we recall that
+
(1 + b1)(l
b2)
+
-
b 1 ~
(1
b2 ~
+~)vT,
(1 —~)vT.
Note that for completely polarized light, ~
=
(6.37) 1; this reduces to (6.38)
2. V(T,l)
V/(1 +vT)
For completely unpolarized light,.~ 0; we differentiate eq. (4.125) and obtain V(T,0)
=
vi
(1 +
~ vT)2.
(6.39)
Before discussing the consequences of these formulae, let us derive P(T,~). The corresponding formulae for P(T,~)follow by additional differentiation with respect to T and can be cast into the form 3 v/2 +~vT)4’
(6.40)
P(T,0)=
(1
P(Tc~)=
(1 +~)2v/2 + (1 + b 3(l + b 1) 2)
P(T, 1) =
(1
—~)2v/2
(1 + b1)(1 +
3
+
b2)
(1 ~2)(2 + vT)v/4 (1 + b 2(l + b 2 1) 2)
(6.41)
1 ~
(6.42)
P(T,~)is greater than V(T, ~) at fixed values of v and ~ for T ~ 0, and then decreases more rapidly than V(T, ~) with increasing time. The physical reason for this behavior is easy to see. When we require that a photon be counted at T = 0, we are giving additional weight to those cases in which the field strength has fluctuated toward large values. In those cases the subsequent count tends naturally to occur more quickly. The ratio of V to P when T 0 is given by P(O, ~) 3 +.~ V(0,~)= 2
(6.43)
so that R (1) = 2 and R (0) = Completely polarized light thus enhances this ratio as we would expect. The behavior of V and P for the limiting values ~ 0,1 is shown in figs. 6.1 and 6.2. ~.
6.3. Lorentz spectrum Since Q(A) is given in closed form, see eq. (4.30), we can differentiate it explicitly with respect to T and obtain V(T) and P(T). The final results are V(T
=
4/3(1
_/32)[(l [(1
_(3)e(i~)T +/3)e(1~)T] /3)2 e~T —(1—(1 +/3)2ePT]2
.
(644)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
298
2.C I
.5-
I
I
I
\
I-c’
\
_5~~.
>
c0
0
I
.2I
I
4I
I
.6I
I
.8
_____________________
C
0
4
T
I
-~
T
Fig. 6.2. P(T) in the presence of partial polarization, u = 1: , = 1;—, ~ 0 (Barakat and Blake, 1973 unpublished).
Fig. 6.1. V(T) in the presence of partial polarization, u = 1: , = 1;—, ? 0 (Barakat and Blake, 1973 unpublished).
and 3~T +
4/3(1
— [32)2 eT [(1 + [3)4 e
(I + [3)3(5/3_3) ePT + (I
v[(l
+(3)2
et~T_(1
j3)2
—
~3)3(5/3 +
3)
—(1
e~T
—
[3)4 e3aT]
e_~T]4
(6.45) When T is large, the asymptotic behavior of these two functions is given by V(T)-=—
4/3(1 —[3)
e(1_~T
T>> 1
(6.46)
T~’1
(6.47)
(l+/3)2
P(T) -
(1
_/3)2
v(l
+/3)2
e~’
~)T,
so that the decay is exponential in time. The larger v, the more rapid the decay. Graphs of V and P are shown in figs. 6.3 and 6.4. The probability that no photoelectrons are recorded during the time interval T is P(0,T)=
— (1 +/3)2
4/3 eT e~~T_(l
(6.48) /3)2 e_~~T~
The long time behavior is P(0,T)
413 —j
(1
+/3)2
e~’_Ø)T
(6.49)
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
299
I0~
-
101 ___________________________________
\~%\ ~
. -
\
\
~ \
01
-
N
\
N.
\
N.•
~~__ .‘—.
\ \
.
—.
--.
~
01
—S.
N..
--,~
N.
\
.5
S.
.=~ S_S5 S....
:
N S...
.
I
N
.0
N I’..
.5
2.0
62
2.5
I
0
.5
NORMALIZED TIME (T)
.0
I
1.5
~
I
2.0
2.5
NORMALIZED TIME (T)
V(T) for Lorentz spectrum: — . —, v = 1; , v = 2; v = 5;—, u = 10 (Barakat and Glauber [18]).
Fig. 6.4. P(T) for Lorentz spectrum: — . —, v 1; , v = 2; —. .—, v = 5; —, u = 10 (Barakatand Glauber [18]).
Fig. 6.3. —. .—,
S.
\~.
N \
0
-
~S
N 62
..
and is also exponential decaying. When uT 4 1, then eq. (6.45) simplifies to eq. (6.35). See fig. 6.5. A discussion of some experimental results is not without interest and we follow Kelly and Blake [73,74]. Gaussian light having a Lorentz spectrum is produced by scattering coherent laser light from a suspension of polystyrene spheres. To measure P(0, T), one records the relative frequency that no photoelectrons are counted during a time T. The results of such measurements for a wide range of average count rates, v, are shown in fig. 6.6. In addition to the exact results the short counting time
0~~0~5~0
NORMALIZED TIME (T) Fig. 6.5.P(0,T)foraLorentz line: —..
.—,
v
0.5;——,
U
1~
, or
~
v= 5;—, u= 10(Barakat and Glauber (18]).
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
300
5
/
/
/
/
4
/
Fig. 6.6. P(0,T)~ for a Lorentz spectrum: — - —, v = 1; -——, u = 1.72; values are denoted by ., ~ ., respectively (Kelly and Blake [741).
,
=
3 as computed from eq. (6.48). Experimental
approximation eq. (6.36) is also plotted. Note that the reciprocal of P(T) is plotted in order to make the short counting time approximation a linear function of T. The experimental results are in excellent agreement with the exact calculations; the short counting time approximation is reasonable only for T<0.6. 6.4. Numerical schemes for V(T) and P(T) The techniques just described hold only for special spectral profiles. We now present a numerical scheme developed by Blake and Barakat [421 for the evaluation of V(T) and P(T) when the input profile is arbitrary (but quasimonochromatic). Perhaps the most obvious way to obtain V and P is by direct numerical differentiation of the single fold generating function with respect to time after calculating numerically the first N eigenvalues of eq. (3.63) and then forming N
Q(A)
H (I +
/=0
~
(6.50)
Unfortunately this procedure becomes computationally useless because the dependence of the eigenvalues upon T turns out to be unstable when T is even moderately large. The Blake—Barakat scheme employs the representation of Q in terms of the iterated kernels of the integral equation, eq. (4.24). The advantage of this approach is that the noise generated by diagonalizing a matrix, whose every element depends on T, is eliminated. The disadvantage is that although eq. (4.24) is formally valid for all v and T, we are restricted to vT < 1 in order to apply it for computational purposes. This is not a serious restriction as most of the current experimental situations fall into this range.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
301
The practical question is reduced to determining the number of integrals of the iterated ‘r which must be retained in order to give a reasonable estimate of Q and its first two derivatives with respect to time. Fortunately we have the Lorentz spectrum results as a check, see previous section. Furthermore the ‘r functions can be evaluated in closed form for a Lorentz spectrum, thus affording us further numerical checks. The first three ~tr(T)are: 11(T)= T 2T) 12(T)-}(2T_
I
1 +e
2T]. (6.51) 3(T)~[T_ 1 +(T+ 1)e_ Extensive numerical calculations performed on a variety of spectral line shapes by Blake and Barakat (unpublished calculations) have shown that as long as vT < 1, only three 1r(T) are needed to determine Q accurately enough to permit first and second derivatives to be taken without serious loss of error. Thus, Q(A)
exp [_AvIi + (Au)2
12 —
__
13]
(6.52)
.
Furthermore, 15 point Gauss quadarture, was found to be sufficiently accurate for the numerical calculation of the ‘r (in fact departures of less than 0.004 were detected when compared to the closed form solutions of J~for the Lorentz line given above). Numerical differentiation of Q was then performed using a 5-point Lagrange formula. The mesh spacing h employed was h = 0.01 although values of h <0.05 produced the proper behavior of Q’ and Q”. Extensive checks were made against the exact values available for a Lorentz line. As an example of the application of the numerical method we again consider the Brillioun spectrum, eq. (4.51). P(T) is shown in fig. 6.7 for a count rate v = 0.1 and in fig. 6.8 for v = 1. The value a 0 corresponds to a single Lorentz line. As a increases, we see evidence of hetro1.3
000:40608
Fig. 6.7. P(T) for a Brillouin spectrum, ~ = 10, ~ = 1, v = 0.1: ,a1(Blake and Barakat (42]).
I
I
I
_
Fig. 6.8. P(T) for a Brillouin spectrum, ~ = 10,7 = 1, u = 1: —..—, ao;_._,a_0.4;____,ao.6; ,arl(Blake and Barakat [42]). -
R. Barakat and J. Blake, Theory of phott,electron counting statistics: an essay
302
dyning between the individual lines in the form of oscillations in P(T) unlike the Lorentz (and other single) spectral lines which decay to zero in a monotone fashion. In the limit of very small count rates, P(T) approaches the joint counting rate of samples taken at t = 0 and t = T, and thus becomes the full photoelectron correlation function in accordance with eq. (6.35). 7. Non-Gaussian statistics 7.1. Introduction
To this point we have treated the case of Gaussian light. This is to say that the P function is a product of Gaussian distributions. This description is accurate whenever the scattered field is produced by a large number of contributions (not necessarily independent) so that the central limit theorem in one of its many forms is operative. When there are many independent scatterers in the scattering volume, fluctuations in light intensity are typically due to motion of the scatterers on the same spatial scale as the wavelength of the incident radiation. These motions shift the phase of the various contributions and cause the intensity to fluctuate. However if there is only a small number of scatterers in the scattering volume, there will be additional fluctuations in the light intensity as scatterers drift into and out of the scattering volume. Further, the scattered light is no longer Gaussian because the central limit theorem cannot be invoked. Nevertheless useful information can be extracted from the photocount statistics, though it is necessary to set up different machinery for extracting the information. Because the light is non-Gaussian, the first and second moments are no longer sufficient to characterize the distribution. It is this fact which makes the analysis of the higher moments of W and P particularly interesting. Following Schaefer and Pusey [75], we consider the field produced when there are N(t) identical scatterers within the scattering volume. Then we have the total scattered field S(t) given by N(t)
~
exp {ikr1(t)}
(7.1)
/=1
where k is the scattering vector and the position of the /th scatterer. Since the i~are random, we can picture the sum in eq. (7.1) as a random walk in the complex plane. The step length may be constant if each particle makes an identical contribution to the light intensity. This need not be the case, and we will describe the random walk in the most general case possible — by specifying the distribution of contributions each scatterer may make to the scattered field, and by allowing the number of scatterers within the scattering volume itself to be random. 7.2. Fixed number of scatterers
The probability of obtaining m photoelectrons in a time interval T is given by eq. (3.21) where the inverse Laplace transform of the single-fold generating function Q(X). The variable 2 is now to be interpreted as the time integrated intensity of the scattered field: W(fZ) is
~
j1(t)~dt
(7.2)
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
303
here s is a sensitivity factor which reflects the quantum efficiency of the detector and the spatial coherence of the light. W(~)is a non-negative function of both ~2and T. Schaefer and Pusey [75]have shown that the probability density function of the scattered light W(~2)can be considered as a random walk problem and they treat the case where all scattering particles are identical, see: Schaefer and Pusey [761; Pusey, Schaefer and Koppel [77]. However we follow the more general analysis of Barakat and Blake [781. The probability density function W(~2)has been evaluated for short counting times by Barakat [79]under the assumption that the light is scattered by N independent particles: W(~2IN)=
~N(~’O
~
(~1/2t)
t dt
(7.3)
where ~N(t)
H n1
=
~ fan (a~)J0(tan) dan.
0
(7.4)
Here a~is the contribution of the nth particle to the amplitude of the scattered light; specifically if there is only one scatterer in the system, then 1”2. (7.5) a1 (S21) fan (an) is the probability density function of an. In the situation envisaged, all the scattering particles will have the same probability density so that eq. (7.4) reduces to
Q~N(t)
If fa(a)
[~fa(’1)Jo(ta)t1dl].
(7.6)
obeys ~
fa(”)
— (~2~ >1/2]
(7.7)
(in other words, all the particles are of the same fixed size and thus scatter light equally), then 2 t)]N. (7.8) 0((~21)” Because the observation time T is taken to be smaller than the reciprocal of the characteristic line width of the spectrum of the scattered light, the shape of the spectrum of the scattered light does not enter into the analysis and the two relevant parameters are the time T and the average count rate w. In other words we are looking at a single mode scattering problem. In an experiment, it is very difficult to keep the number of scatterers fixed in the scattering volume and it is more realistic to consider that the number of scatterers is itself a random variable. We take N to be a discrete random variable having a Poisson distribution. Thus (I’s1), the average number of independent scatterers in V, characterizes the random scatterers since (.7%!) is also the variance. Any higher moments of the Poisson distribution can be expressed in terms of (N). Since the solution for a random (Poisson) number of scatterers depends on the solution for a ØN(t)
=
[J
R. Barakat and J. Blake,
304
Theory
of photoelectron counting statistics: an essay
fixed number of scatterers, we begin with the latter case. For N = 0 (no scatterers), we have W(~I0)=1
~
j
(~1/2
t)t dt
=
b(~).
(7.9)
Consequently, P(,nIO)1,
m=0
Q,
rnzO.
(7.10)
is exactly what we expect, namely that the probability of obtaining no photoelectrons is unity. For N ~ 1, we express W(~2IN)in the form of a Fourier—Bessel series whose coefficients are sampled values of the characteristic function ~N(t) of W(~2IN).We now assume that This
a>c~ where c~is finite. Obviously
(7.11)
fa(~1)~0,
(7.12) W(~2IN)~0, ~2>N2a2. Consequently ~N(t) is a bandlimited function since its Fourier—Bessel transform W(~.2lN)vanishes
identically outside a compact region. It can be shown that (see details in Barakat [79]):
W(~IN)= ~ n1
[~i(~, N2a2[J(y)]2 1/Na)]N fl
~ \ Nct
2a2 0~~~N elsewhere,
(7.13)
where Yl,Y2,~ are the positive roots of J 0 [i.e., J0(y~,)0001. Since this is a Fourier—Bessel series, its convergence is governed by the smoothness (continuity) of W(~jN).The smoother W(~ZlN)is, the more rapid is the convergence of its series expansion. In the special case where all the particles are of the same fixed size, it is possible to obtain closed form solutions of W(fLIN) forN 1,2,3. These expressions are listed in appendix 7.A. As N becomes very large, W(~7IN)approaches a negative exponential probability density
W(~IN) ~~2) —e~~~> ~
(7.14)
so that the underlying statistics of the field amplitudes are Gaussian. The corresponding expression for P(m IN) is the Bose—Einstein distribution “-~yfl
P(mIN) (1 + (~))m +1 (7.15) The expressions in eqs. (7.14) and (7.15) are the leading terms in the asymptotic series in powers of see appendix 7.B for details. In spite of the fact that W(~IN)varies drastically for small N, the resultant photoelectron counting distributions are very tame creatures. This is simply a consequence of the smoothing action of the Poisson term in eq. (3.21). The factor ~ effectively damps out any irregular behavior of W(~ZIN) for ~ 4 1, while the negative exponential accomplishes the same result for ~Z> 1. It is for this -~
R. Barekat and J. Blake, Theory of photoelectron counting statistics: an essay
305
reason that we have confined our numerical calculations to the case of identical particles. Even though W(fZIN) will be somewhat different according to the situation considered (i.e., particles with different fixed sizes, particles with range of different sizes, etc.), the smoothing action of the integrand in eq. (3.21) is the determining factor. The results forN= 1,2,3,8 are shown in figs. 7.1 through 7.4 (see curves with open circles), in all cases (~l~ > = 1. Note that the maxima of these distributions are at m = 0, as we would expect. We have chosen to work with (~Z>,but it is also just as easy to work with the average total count rate w w
=
(7.16)
(f~)fT=N(~2,>/T.
The question naturally arises as to how large N must be in order that P(mlN) approximate the Bose—Einstein distribution. Answering this question is not a simple matter, but we can offer the 0.6
I
I
I
I
I
I
I
I
I
0.5
-
0.4-
-
0.4 I
NUMBER OF PHOTOELECTRONS Fig. 7.1. Photoelectron counting distributions: P(m Ii), open circles; P(m (1>), solid circles (Barakat and Blake [781).
~
I
I
I
I
I
I
I
I
I
I
I
I
I
NUMBER OF PHOTOELECTRONS Fig. 7.2. Photoelectron counting distributions: P(m12), open circles;P(m 1(2)), solid circles (Barakat and Blake [78]).
I
O.1~
0.3
I
O~h1
___
04
I
-
I I
I
I
I
I
I
I
I
I
I
I
I
I
I
0.12
¼\%~a~l::~il:=~=r1srr.
NUMBER OF PHOTOELECTRONS Fig. 7.3. Photoelectron counting distributions: P(m13), open circles;P(m 1(3>), solid circles (Barakat and Blake (78]).
E.E3~S.SO.~Q. NUMBER OF PHOTOELECTRONS Fig. 7.4. Photoelectron counting distributions: P(m18), open circles;P(m 1<8)), solid circles (Barakat and Blake (781).
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
306
following remarks. Based on test calculations (not reproduced here), it appears that for practical purposes N ~ 5 gives a Bose—Einstein distribution. Of course, this is not strictly true. Nevertheless an experimenter would be hard put to distinguish P(m15) from P(mj 100), say, purely on the basis of a single-fold counting experiment. The probability of obtaining no photoelectrons is a useful statistic and its behavior is depicted in fig. 7.5. 7.3. Random number of scatterers We have just examined the case of N fixed so we can interpret these results as holding for a canonical ensemble in the language of statistical mechanics. The case of random N leads to an interpretation as a grand canonical ensemble. It is a simple exercise in probability theory to prove that if N is distributed according to a Poisson distribution having a mean value (N>, then 1’T>
W(~2I(N>)=NrrO W(~IN)(N>”~ N!e~
P(mI(N>)
(7.17)
(N)N e_
N!
NO
(7.18)
As (N> is made to increase, the Poisson distribution becomes very peaked at (N> = N and acts somewhat like a Dirac delta function centered at (N> = N. Consequently W(~I(N>)~ W(fZIN),
P(mI(N))~P(mIN)
(7.19)
for large (N>. We can easily plot W as a function of (N>, but it hardly seems worthwhile to do so and we pass directly to the photoelectron statistics. Numerical values of P(mI(N>) for (N) = 1,2,3,8 are plotted in figs. 7.1 through 7.4 (see solid circles). As we would expect, the photoelectron counting distri-
I
O2~~oO~~
N, (N> Fig. 7.5. Photoelectron counting distribution:
P(OIN),
open circles;P(0I(N>), solid line (Barakat and Blake [781).
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
307
butions for the deterministic and stochastic situations are measurably different for one and two scatterers. However, even for three scatterers, the two situations are not very different. For more than three scatterers, the two situations yield practically the same result as witness N = (N> 8 (fig. 7.4). We can also derive integral representations for both W(fZI(N>) and P(mL(N>). If we substitute eq. (7.17) into eq. (7.3) and sum the series, we obtain W(fZI(N))
~
exp {—(N> [1
— ~
1
(t)] } J0 (~1/2 t)t
(7.20)
dt.
When eq. (7.20) is substituted into eq. (3.21), we can prove that 2 ‘~ Lm (t2/4) t P(mI(N))
~
exp {—(N>t 1
— ~,
dt)
dt
(t)} } e_t
eY exp {—(N>[l —~,(2y”2)]}Lm (y) dy
(7.21)
where Lm (y) is the Laguerre polynomial. In its second form, the integral can be evaluated via Gauss—Laguerre quadrature (Abramowitz and Stegun [80]) if desired. Probably the best way to distinguish these two situations under the constraint of single-fold counting is to consider the probability of obtaining no photoelectrons during the measurement. The resulting calculations are shown in fig. 7.6. It is possible to distinguish the two situations for N = (N> ~ 5; but certainly not for a larger number of scatterers. 2.1
2.2
I
I
I
111111
I
III
Ill-I
-
>._\
-
1.6
\
\
\. \
0 0
“ ~ OO~&~~ ~
LU
~
10
50
100
N,(N) Fig. 7.6. V(N,6), V(, 6) as functions of N and (N> for fixed 6; V(N, 0), open circles; V(N, 0.2), solid circles; V((N>, 0), dotted line; V((N), 0.2), solid line (Barakat and Blake [781).
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
308 7.4.
Moments of JV(~ZIN)andW(~I(N>)
In order to calculate the moments (more precisely the conditional moments about the origin), we recall the generating function introduced in eq. (3.7):
Q(XIN)
~ e~ W(~2IN)d~2
(7.22)
from which the moments can be obtained by differentiation ak —~Q(XIN)
(~kIN>(_l)
,
k=
~
(7.23)
ax Q(AIN) can be expressed in the form Q(XIN)
e21~X ~N(t)t
=
dt
(7.24)
by substituting eq. (7.3) into eq. (7.22) and integrating out the Bessel function. It is convenient to express Q as a power series in A, for then the differentiations are trivial. To this end, we express cbN(t) as power series in t2 by expanding J 0 (ta) in a power series and then integrating termwise~we have
[~
Q~N(t)00
A?lt2n]
(7.25)
where 2), A0
=
1,
A2
=
A
—~ (a
4), 3
=
(7.26)
~(a
and ~ fa(a)~”~1a
(7.27)
We now rewrite ØN(t) in the form ~N(t)
=
~
(7.28)
B~t2~.
The B coefficients can be calculated from the A coefficients via the recurrence relation, Gould [811 Bn =
1
~ 0
[k(N+
1)
—nIAkBn_k,
n~’1.
(7.29)
k1
When eq. (7.28) is substituted into eq. (7.24), the resulting integration is trivial and the final result
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
309
is that Q(AIN) is a power series in A. The resulting differentiations can now be performed and the first four moments about the origin are (~2IN>N(a2) (~2 IN) = 2N(N
— 1)
(a2>2 + N(a4)
(~VIN>6N(N_1)(N_2)(a2>3 +9N(N—1)(a2>(a4)+N(a6) (~24IIV)24N(N— l)(N—2)(N—3)(a2)4+72N(N—l)(N—2)(a2>2(a4) + l6N(N— l)(a2>(a6>+N(a8).
(7.30)
When all the particles are of fixed size then these expressions reduce to those given in Pusey, Schaefer and Koppel [77]. The limiting values of (IZ’9N), when N ~ 1, are (akIN)
,~.
k!N” (a2)”
(7.31)
characteristic of a negative exponential probability density function, as expected. Thus the rate at which the moments approach their limiting values depends not only on N (as in the case of indentical particles) but also on the spread of the probability density function f(a) as measured by the ratio of the various moments of a. Since W(~2IN)is a true probability distribution as regards our problem, then P(mIN) is the probability distribution of a mixed Poisson process, Haight [82]. It is a well known property of mixed Poisson processes that the kth factorial moment defined by l)IN),
(m~”~IN)—(m(m—l)~(m—k+
k1,2,
(7.32)
is related to the kth moment of W(~IN)by (m~”~ IN>
=
(a” IN).
(7.33)
Thus we can list the factorial moments of P(mlN) once we have the moments of W(~Tl4N). The same procedure can be employed to determine (~2”1(N)), we define the generating function Q(ARN>)
~ e~ W(~TLI(N))d~.
(7.34)
It follows that Q(AI(N)) =
exp {—(N> [1
—
0, (t)]} et2/4X
tdt.
(7.35)
In order to obtain the moments, we expand the first term in, the integrand in a power series in t; thus exp {—(N) [1 —0, (t)]}
~ n=0
cnt~
(7.36)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
310
where (N)(a2)
C 01,
C
C2=— 22 (N) (a4)
(N)
3
+
32
(7.37)
...
64
Proceeding as before, we obtain Q(XI(N)) as a power series in A. Differentiation then yields the moments about the origin, of which the first four are 2)
(&2I(N)) (N)(a 1(N)) 2(N)2(a2 >2 + (N) (a4) (&2~1(N))
6(N)3(a2)3 + 9(N)2(a2)(a4) + (N)(a6)
(~24l(N)) 24(N)4(a2)4 + 72(N)2(a2)2(a4) + 34(N)2(a2)(a6) + (N)(a8).
(7.39)
When all the particles are of same fixed size, then these expressions also reduce to those given in Pusey, Schaefer and Koppel [77]. The limiting values of (~Z”(N)) for large (N) are again given by eq. (7.3 1) with N replaced by (N). However specifying (N> instead of N permits the number of particles in the scattering volume to fluctuate. This introduces additional fluctuations in the intensity of the scattered light. Accordingly the moments for the random case will approach their limiting values more slowly than the corresponding moments in the deterministic case. P(mI(N)) is also the probability distribution of a mixed Poisson process because W(~ZI(N)) is a probability density; consequently (m~”~I(N>) = (&V’I(N)).
(7.40)
It is useful to examine the case where fa(a) is sharply peaked around its mean value because this provides an approximation to a slightly polydisperse suspension of particles. For computational simplicity, we let exp {—(a
fa(a)
—
(~,)l/2)2/2a2
}
(7.41)
where (~2~) cj~. The moments of a are ~‘
~ f~(a~”da.
(7.42)
If we form the dimensionless parameter ~ a~/(~,),then for ö 4 1, the negative contribution of the probability density function in eq. (7.41) is negligible and we can safely replace the limits (0,oo) in eq. (7.42) by (_oo,00). Consequently, we obtain (a2)(l +6)(fZ,)
(a4)
=
(1 + 66 +
(a6>
=
(1 + 156 +
(a8)
(1
362)(&2)2 4562
+ l5ö~)(&~~)~
+ 286 + 21062 + 4206~+ l0S6~)(~,)~.
(7.43)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
311
In order to utilize these explicit formulae, let us consider for example, the variance of ~7:
(7.44)
var (&~IN)= N(N — 2)(a2>2 + N(a4> var (~2I(N>)= (N)2(a2)2 + (N)(a4)
(7.45)
then V(N,6)~~~ V((N), 6)
(1
=
+~)+~(6_362+263
00 (1 +
~~s)
~
+
(6
— 362
—~~)
+ 26~—...).
(7.46)
(7.47)
The first terms in parentheses are the contributions when all the scatterers are identical. The second term represents the influence of the width of fa(a), and arises from the varying amounts of light scattered by different sized particles. The second term is the same for both situations. Some typical numerical results are shown in fig. 7.6 and are self explanatory. Other moments can be handled in much the same manner. 7.5. Coefficients of skewness and excess for W Since both W(~2IN)and W(~ZI(N))are far different from their limitingvalues for small N and (N), it is important to evaluate their global shape parameters: coefficient of skewness, and coefficient of excess. The coefficient of skewness -y, is defined as
~
—
(~))3
)/((cz —
(7.48)
(~))2 )3/2
The skewness is a normalized measure of the mode minus the mean and serves as one indicator of the length of the “tail” of the density function. If>’, is positive (negative), then the corresponding probability density function is skewed to the right (left) of the mode. The larger 17,1, the longer the resultant tail. If-y, = 0, then the density function is symmetric about the mean. In the special case where all the scatterers are of the same fixed size, the explicit formulae are 2N—4 =
(N2
—
N)”2
3
_
2 —
+ O~N2)
(7.49)
and 2(N)3 + 6(N)2 + (N> ((N)2 + (N))312
3
_
2 + ~
O((N)2).
(7.50)
Both approach the limiting value, -y, 2, characteristic of a negative exponential probability density function. Some typical curves are shown in fig. 7.7. For fixed N, the result of allowing 6 > 0 is to hasten the rate at which ~, (N) tends to two. When N is allowed to fluctuate, ‘y, ((N>) is larger than the limiting value; thereby indicating that the “tail” dies off more slowly than for the corresponding fixed case. This is exactly what we would expect. The coefficient of excess, ~Y2,is defined as (ç~>)4>— 3((~ — (~))2>2 72 = ((~~(ç~>)2>2 (7.51) —
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
312
4
I
I
IIIIII~
I
.
z •
W
•
00
I
0
0
0
CJ)
0 0
0
~
1111111
I
I
10 N,
1
IIIII
100
Fig. 7.7. Coefficient of skewness, -y
1, as a function ofNand (N) for fixed 6: ~y,(N, 0), open circles; y, (N, 0.2), solid circles; .y, (, 0) dotted line; .y, ((N>, 0.2) solid line (Barakat and Blake [781).
and is a measure of the peakedness of the corresponding density function about the mode. 72 0 for a Gaussian, and 72 = 6 for a negative exponential. Explicit formulae for 72, when all the scatterers are identical, are 3 36N2 + 45N — 15 24 6N N3—N 6———+O(N2) (7.52) _
—
and 6(N)3 + 36(N)2 + 39(N) + 1
24
3 + 2(N)2 + (N> 6 + ‘~ + O((NY2). (7.53) The limiting value 72 6 characteristic of a negative exponential is approached in both cases, note that 72 (N) < 6, while ‘ 12((N>) > 6. Some typical curves are shown in fig. 7.8. ‘~
72((N))
7.6.
(N)
=
Variance ofF
We can list the factorial moments ofP(mIN) and P(mI(N)) by virtue of eqs. (7.33) and (7.40). However we content ourselves with deriving the second factorial moment (variance). Thus 2)2 + N(a4) + N(a2) (7.54) var (mIN) = N(N — 2)(a var (mI(N)) = (N)2 (a2)2 + N(a4) + (N)(a2). (7.55) When fa(a) is the Gaussian discussed in eq. (7.41), then we can show that var (mIN)
=
[(N2
—
N)(~2
2)N2(~2,>2+ 6N(~Z,>
1)2 + N(~Z,>]+ (26 + ä var (mI(N>) 00 [((N)2 + (N>)(fZ 2 + (N>(&2,)] + (26 1)
+
(7.56)
62)((N)2 + 2(N))(&2 2 + 6N(cz,). 1)
(7.57)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
• •o o~
0 x LLi
313
O
.00
—
-
0
-
S
0 —12
I
I
1111111 -
10
Fig. 7.8. Coefficient of excess, ~z, as a function ofN and (N> for fixed 6:
I
I
11111’
100
72(N, 0), open circles; 72(N, 0.2), solid circles;
72(, 0), dotted line; .y2(, 0.2), solid line (Barakat and Blake [78]).
When all particles are identical 2
—
N)(~2,)2
+
N(~,>
(7.58)
var (mIN) = (N var (mI(N>) = ((N2) + (N>)(~2,)2+ (N)(&2,).
(7.59)
The variance for the stochastic situation is larger than that for the deterministic situation as expected, both approach the same limiting value. 7. 7. Time interval statistics
In the present subsection we evaluate the time interval photoelectron statistics for light scattered by an arbitrary number of independent particles. We follow the analysis of Blake and Barakat [831 and refer to their paper for the computational details. We confine our attention to the case where all the scatterers are of fixed size. We are interested in determining the behavior of V(T) and P( T), see subsection 7. 1, for both fixed and random number of scatterers. In accordance with the formulae of subsection 7.1, we need to evaluate the first and second time derivatives of Q(XIN)l~... 1
W(~2IN)e
d~2.
(7.60)
The time dependence in Q(AIN), which we have deliberately suppressed, enters through (~2,>.For an average counting rate w, the total number of counts recorded during a time interval T is given by (m>wTN(f~,).
(7.61)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
314
Accordingly, (~7,) wT/N, and we may write the upper limit of integration as The case N I is, of course, trivial. We have Q(AIl)e~T
N2(&2,>
=
NwT
(7.62)
characteristic of a Poisson distribution. The time interval probability densities are V(TIl) =F(TIl)
=
w
(7.63)
e~T.
V(TI 1) and F(TJ I) are identical because with only one scatterer the light is not modulated in intensity, and hence remains coherent. In this case photocounts occur at random, and to start a time interval at a photocount is equivalent to starting the interval at random. ForN= 2, we can utilize Chebyschev quadrature [801 to evaluate the generating function I
Q(X!2)I~
~
~
J
e’~ d~ ~
—
(7.64)
~)1/2~
ForN~ 3, we employ eq. (7.13) and show explicitly the Tdependence of W(&211V) by writing W(~2,~) where we temporarily suppress the N dependence. With this convention we have W(~,~) e~ d~= W(~) e~ +
W(~) e~ d~2
(7.65)
and a
d2~
~ W(~,~)e~ d&~e~~
+
~
W(~2,~)e~’ d~2
(7.66)
where d
d Nw
j.
(7.67)
With these observations, we can easily show that V(TIN)
=
—Nw
W(~,~) e~ d~
(7.68)
and P(TIN)N2w [e_t ~W(~,~)In~
+
J
~2
W(~) e~ d~}.
(7.69)
Some typical numerical results for both V(TIN) and P(TIN) are shown in figs. 7.9 and 7.10 for various values of N and fixed total average count rate. The differences are especially noticeable for P(TIN). As N is made to increase, they approach the limiting values predicted on the basis of single mode Gaussian (chaotic) light N—l’~o
V(TIN)
P(TJN)
=
=
w/(l + wT)2
(7.70)
2w/(l + wT)3.
(7.71)
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
315
This limiting behavior is reached when N = 8 for V(TIN); however a value of N = 20 is required for P( TIN) to reach its limiting behavior. We have already noted that coherent light, N 1, leads to negative exponential probability densities for V and P characteristic of a Poisson process. Now the Poisson process has the property that it is a maximum disorder process in that there is no tendency toward clumping of photocounts. For N> 1, the underlying process is no longer Poisson and we expect clumping. A crude measure of the clumping or overall correlation of the photocounts is available through the ratio R(N) =P(OIN)/V(OIN).
(7.72)
For coherent light R (1) = 1. With increasing N, R increases until assuming the value 2 at N = oo, see figs. 7.9 and 7.10. This reflects the increasingly larger fluctuations in the field intensity which become possible as N increases. Recalling that for small w, P behaves like the photoelectron correlation function (Davidson and Mandel [67]), then we can interpret this enhancement as indicating greater photocount correlation. To obtain the generating function for a random number of scatterers distributed according to a Poisson distribution, we simply integrate eq. (7.17); appropriate differentiations yield V(TI(N>)=
V(TIN)
N1
(N)~’e~I~> N!
(7.73)
and (N>” e~~> P(TI(N>)=>~P(TIN) N!
(7.74)
When (N) is large, the Poisson distribution peaks around (N) centered at (N) N. Thus
=
N and acts like a Dirac delta function
V(TI(N>) ~ V(TIN) for(N)~ 1.
(7.75)
P(TI(N)) ~ P(TIN) LC
::
1
I
I
I
I
-
I
o~:.6
Fig. 7.9. V(T LN) for light scattered from N independent scatterers, average counting rate is unity. —, N = 1; ,N=2;—..-,N=4;—..—,N8.ThecurveforN~ is indistinguishable from that for N = 8 (Blake and Barakat [84]).
2.C.
I
I
I
I
I
::0I~2~4Io6
Fig. 7.10. P(T IN) for light scattered from N independent scatterers, average counting rate is unity. —, N = 1; N’~2;—.—,N~’4;—..—,N~’ 8; ,N~o(Blakeand Barakat [83]).
,
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
316
However when (N) is large enough for these approximations to hold, then N is large enough for the limiting cases, eqs. (7.70) and (7.71) to be valid. In order to show the difference between a fixed and a random number of scatterers let N = (N) = The results are summarized in figs. 7.11 and 7.12. Note that V(TI(3>) ~ V(Tl 3), at least over the range 0 ~ T ~ 0.6. However since ~ V(TI(3)) dt
=
~ V(TI3) dT 00 1
3.
(7.76)
then V(T13) must eventually be greater than V(tI(3)). The increase in photocount correlation (clumping) over that of a pure Poisson process is even more pronounced for a random number of scatterers. For fixed N the only fluctuations in the field intensity are those that arise from interference terms; with N random there are additional contributions to the fluctuations asNitself fluctuates. Thus we would expect that P(0I(N>) would be greater than P(OIN); this is true as witness fig. 7.12. 7.8. Photoelectron correlation function
We now pass to the consideration of second and higher order statistics of the scattered light for non-Gaussian situations. Of particular importance are the full and clipped photoelectron correlation functions. The development of this and the next section lean heavily on the work of Chen, Tartaglia and Pusey [211, see also Chen and Tartaglia [841, Schaefer and Berne [85]. The full photoelectron correlation function given by eq. (5.2), can also be expressed in terms of the integrated intensities
c(r),
c(r)
=
(~,~2>/(~1)2.
(7.77)
We confine our attention to short counting times only, so that we approximate cz. by =
NsT f” (t1) g* (t1)
o:OiQi2I~4iQ6 Fig. 7.11. Comparison of V(TIN) and V(TRN>) for: , N = 3; —, (N> = 3; average counting rate is unity (Blake and Barakat [83]).
(7.78)
I~__
Fig. 7.12. Comparison ofP(TIN) and P(TI) for: N = 3; —, (N> = 3; average counting rate is unity (Blake and Barakat [83]).
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
317
where N is the number of independent scatteringparticles, and write the scattered field as ~ b1(t)exp{ik~(t)}
(7.79)
where b1(t) =
1,
if
particle j is in scattering volume v
=
0,
if
particle j is not in scattering volume u.
(7
80)
To evaluate c(r), we first calculate (~Z,~22> 4 ~ (b~(t,)b
<~1~2>
=
1(t,)b~(t2)b,(t2) exp {—ik,~(t,) + ik r~(t1)
(NsT)
— ik i~(t2)
jfk!
4 ~ (b,,b,,b~ + i1t~i~(t2)} = (NsT)
2b,2A/~A11A,~’2A~>
(7.81)
i/k!
where b,1
b,(t1);
A,1
exp {iI
(7.82)
i~(t,)}
and the sum over i, /, k, 1 runs over all the particles in the scattering volume. There are two different time scales to consider. 1. The time for a particle to move a distance comparable to the wavelength of the incident light (the time scale on which I(A,,A~2)I—g12 fluctuates), 2. The time for a particle to move a distance equal to the cube root of the scattering volume (the time scale on which (b,, b,2> fluctuates). The second time scale is typically a thousand times larger than the first. Thus, the b’s and A’s are essentially independent random variables and we can factor eq. (7.81) 4 ~ (by, b, b~ ~2,
~2>
=
(NsT)
2b,2)(A1’~AJ,A~2A,2>.
(7.83)
i/kl
Since the particles are independent, the ensemble brackets containing the A’s vanish unless i = I, k~l,ori~lj~k. Nowsince g,2 —g(t,
— t2)
=
I(A,,A72)I 00 KA,,A72>I
(7.84)
we can rewrite eq. (7.83) as 4 (~1~2>
=
(NsT)
{~ 1k
(b~,b~
}.
2>+ ~ (b,1 b,2) (b1, b,2> g~2 i/
(7.85)
In the scattering situation under investigation there are N independent particles in the containing vessel (having volume V). The scattering volume u is a small fraction of V. Since the system is spatially homogeneous, the average number of particles in the scattering volume v is (M> = N(v/V)
(7.86)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
318
The instantaneous numberM(t) can be written as M(t) 00 ~
b1(t)
(7.87)
so that (M(t)>
~
=
(b~(t))00 N(b>.
(7.88)
By comparing eqs. (7.86) and (7.88), we have (b) = v/V. Returning to eq. (7.85), we note that the difference in time scales permits us (to a good approximation) to let t, = t2 in the b’s, whenever they are multiplied by g~2.This is because b,(t) does not vary in the time scale necessary for the fast correlation in g,2 to vanish. Consequently, the second term on the right hand side of eq. (7.85) can be summed 2g~ ~
(b~(t1))(b~t2))g~2 ~
(b1(t,))(b1(t1))g~2 N(N_ 1) (b)
2.
(7.89)
j#!
The first term must include the two different times. We have N
~
j/V
(bI(t,)bk(t2))
00
i,k
(,,~b. 0’,) i
N
~
b1(t2)) 00 (M(t,)M(t2)>.
(7.90)
j
But M(t) can be written as M(t) = (M(t)> + 6M(t) where 6M0’) is the deviation of the number of particles from the average number. Since (6M(t)> 0, we can write 2) + (6M(t, )6M(t 2 (b)2 + (6M(t,)6M(t (M(t, )M(t2)) = (M 2)) = N 2)). (7.91) Consequently, eq. (7.85) becomes 2(b)2 +N(N— 1) (b)2g~ (~)2
N
~
{N
2+ (6M(t,)6M(t2))}.
To obtain the result for a fixed number of particles in the scattering volume, we let (b) (M). The last term vanishes leaving
c(r)
1 + (I
—
(7.92)
1 and
(7.93)
)g2(r)
Thus, there is a correction term of orderN’ to the Gaussian result. When the number of scatterers is random then some ansatz must be postulated in order to evaluate (6M (t, ) 6M (t2 )>. Assume that (6M(t,) 6M(t2)) 00 exp {—~It1
— t2
l} =
(7.94)
e_~T
where ,3 is a constant. The thermodynamic limit N 00, v/V (b> -÷0 while (N) remains finite is important. Chen et al. point out that this particular limiting process is required in order to be able to neglect the unknown wall effect of the scattering vessel, and also because v/V can be made as -~
R. Barakat and J. Blake,
Theory of photoelectron counting statistics: an essay
319
small as iO~.As N oo, the random variableM becomes Poisson distributed so that (M(M 2. It is a consequence of eq. (7.94) that = (M) —~
—
________
(b>) exp {—~It, — t 2
In this limit
+g2(r)
}
+ (b>.
— 1))
(7.95)
~
+ ~e_~T + 0 (7.96) 1 In the next subsection, we consider the clipped photoelectron correlation function. In order to study this function, we need to obtain higher order two-time intensity correlation functions, i.e., (~Z~ ~7~)where k is a positive integer. Chen et al. show by induction that c(r)
(M>”~’ ~ C,(k) (N1+
(~)k+1
11
(b)’~’+ 1N111 (bY
[(1
—
(b>)
e_~T +
+f(k,N,(b>)g2(r)
(b)]
} (7.97)
where N[’I—N(N—l).•~(N—l+l)
C,(k)
~ (k:a,,
,ak)
(1!)al
(7.98)
(2!)aB... (k!yzk.
The summation is over the set of a’s that simultaneously satisfy a,+2a 2+.”+kakk
a2 +a2 +
1.
+ak
(7.99)
The coefficient (k:a,, .. . , a,~)’is the number of ways of partitioning k different objects into a1 subsets containing / objects for / = 1, 2, ... , 1. It is tabulated in [80]. The function f(k, N, (b>) is as yet unknown. To obtain the intensity moments, let t ~ in eq. (7.97) and also let g(r) 0, then —~
(k)
1
-~
k
1~’1 (bY + ‘+ lJVt1 (b>’~‘1 (M>k+ 1 ~ C,(k)[N For a fixed number of particles this reduces to
(7.100)
(ç~)k
(~>k
=
N~’ ~
C,(k) [N~’~~l + lNUl]
The first four moments are /o~2
(&22>=_(2N2_N) N2
=
~~C,(k)N1hI.
(7.101)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
320 (~)3
(&23)=-~-(6N3_9N2+4N) (~)4 (~4)=
(24N4
—h--
—
72N3 + 82N2
—
33N).
(7.102)
When the number of particles is random, then
>
(czk)
k
C,(k) (MY
(7.103)
~
To obtain the unknown factor f(k,N, Kb)) we go back to eq. (7.97). Set
~,
t 2,
1(b)’ + 1 + iN111 Kb)’] + f(k,N, Kb)). (M)k +
C, (k)
1
=
(~k+ 1) (~)k + 1 —
KM>’
k ~
(7.104)
1’ + ‘
f(k, N, Kb)) can be determined for a fixed number of particles by letting Kb> random number of particles, we easily obtain f(k,N,0)
the result is
=
1 and N = (M). For a
KM>’
{~—~ + ok ~
+ 1].
(7.105)
The two-time higher order intensity correlation function given in eq. (7.97) can now be explicitly evaluated. For fixed numbers of scattering particles the first few are (&Z,~Z~) (~>3
=
(>4
=
~
1
(6
9
8
4
(18
63
(4_~+~)g2(r)
(2_~)+
4
_~+—~)+
78 33\ ~-+~._~)~2(r).
(7.106)
(7.107)
The corresponding expressions for the random situation are more involved. Chen et al. worked out the explicit expansion of ~ in a power series in 11(M), reference is made to the original paper for the actual expression (eq. (19)). 7.9. Clipped photoelectron correlation function The clipped photoelectron correlation function Ck(r) is defined in eq. (5.12). However, since m, and m 2 enter into the expression symmetrically, we will now write Ck
~
~
m,0
m,k
m1P(m1,m2)
(7.108)
where we omit the explicit time variables for typographic convenience. We are, of course, only interested in very short sampling times. The double series can be recast into a series involving the
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
321
joint higher moments of the integrated intensity ~‘> 1 (rn>’ (—(rn>)1 (cl, &2~~ Ck(rn>_(m> ~ -j~---- ~ ‘=0 /=0 j! K2>’~’~1 If the
clipping le1vel is set at k =
then
1,
(—(m))/(~2
G 0
(7.109)
1~>
(7.110)
1~2
(m>—(m> ~
~‘
/=0
The average clipped count (mk>
~
P(m)
(7.111)
mk
can also be written as (mk>
1 (~‘~~> (rn)’
1— k1 ~ l=0
(—(rn>) j!
°~
(7.112)
(f~>’~j
/=0
Consequently (—1~,n>)1 (~2~>
(m 1>
I
—
(7.113) /=0
Since (fZ’> and (&2~cz’2> have already been evaluated, it is simply a question of tedious algebra to substitute these expressions into eqs. (7.110) and (7.113). If the clipping level is set at k = 1, then to order N’, (m>
(rn,>
1 +
and
1 1+—
(m>
C 1 = (m>(rn1>
N
k~l
+
[ (m) 2(1 +
2 (r) I g + (m>
(7.114)
1)
(m>)2
[(rn)22(1++(m>(rn>)2+
g2 (r) (rn>)
— N(1 +
21 ~
(7.115)
The corresponding results for a random number of scatterers are more involved and we refer to Chen et al. for the explicit formulae. Appendix 7.A: Formulas for scattered intensity
-
ForN= 1, we have W(~2Il)=
~
J
2t)J 0(K21>”
0
1’2t)tdt 0(~2
=
6(~2— (~Z 1>)
(7.A.1)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
322
(i.e., there are no fluctuations in the light scattered by a single particle). Thus, the system is in a coherent state [2, 3] and the photoelectron counting distribution is Poisson with mean value (~)
P(rnlI)~ (~)~e~’>.
(7.A.2)
ForN 2, Rayleigh [86] has evaluated the equivalent of W(~l2),his result translated into the language of the present problem is W(~2I2) ~[~(4(~) =
— ~
1/2’
~
0,
elsewhere.
Note the square root singularities at &2 distribution is P(mI2) 00
1
4(&Z~>
corresponding photoelectron counting
~m e_n d~
~
—
irm !
0,4(~2~). The
(7.A.3)
0
[~2(4(~~>
—
(7.A.4)
.
~)]
1/2
Although this integral cannot be evaluated analytically, it is easily handled by Chebyschev quadrature which automatically takes account of the square root singularities at the end points of the integration. Finally, N = 3 can be expressed in terms of elliptic integrals, Slack [871: W(S~I3)= ~.2 ((~)1/2
00
+
[(3(~)1/2 2K(k) — p1/2) ((~)l/2
~1/2)
+
~-~,1/2 )] 1/2
‘
0~2~(~7) 1
K(l/k) 2 (~ >3/4 ~ 2ir
=0,
elsewhere (7.A.5)
where K(k) is the complete elliptic integral of the first kind with modulus k given by k
=
((~i>~/2 +
~1/2)
4(~l 314.cZ314 1) ~1/2) ((~)1/2 + ~1/2)J
[(3(~>1/2
—
1/V
(7.A.6)
The elliptic integral is logarithmically infinite at k 1 (i.e., when &2 = (fZ,)). The logarithmic singularity is very weak and P(rn13) can be evaluated numerically by a trapezoidal rule with a very close mesh.
Appendix 7.B: Asympto tics of W(&~IN) and P(m IN) The probability density function of the envelope r of the sum ofN independently distributed random sine waves approaches the Rayleigh probability density function as N becomes very large. If we apply the procedure described in Cramer [88] for corrections to the central limit theorem, we can show (details are omitted) that f(r)
2r N(a >
exp(—r2/N(a2>)
[1
+D 1 +
...]
(7.B.l)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
323
where 2/N(a2) + r4 IN2 (a2>-2) (N(a4>
—
(2 — 4r
—
2N(a2>2)
7B2
D 2 (a2>2 ( ) = counting time approximation, 4N In the1 short we can equate the integrated intensity ~2to the square of the envelope r to within a proportiona1it~onstant. Upon transforming to the new independe~tvariab1e~2,eq. (7.B. 1) yields .
W(~2IN)
e~n/[1
‘~
+
C 2
.
(7.B.3)
+...]
where 2)2 \ I 2N2 (a2 >2 — 4N(a2 >~2+ ~22 \ C 2 (a2 >2 4N2 (a2>2 (7.B.4) I N(a”> — 2N(a 2 ~ N Note that the fourth moment (a4> enters into the correction term. In the special case of identical particles (a4> (a2>2, the correction term reduces to the classical result of Pearson [89] and Rayleigh [861: C 2
N[2
(7.B.5)
(&•~>~4(~:)2].
If the particles are distributed according to the narrow Gaussian, eq. (7.41), then 00 —
(l_26_~2) 1 N [2
~
____
— (~>+ 4(~2>21~
(7.B.6)
The asymptotic expression for P(rnIN) can be obtained by appropriate integration ofeq. The final result is 4>
P(rnIN)
Bm (~TZ>)+
[
N(a
—
2N(a2>2
N2 (a2>2
(m +
] [~Bm ((&2>) —
4(~>2
1)
~2>
Bm + 1
(~~2))
1
(rn+l)(rn+2)
+
(7.B.3).
Bm
+2
((~2>)J
(7.B.7)
where Bm ((~2>)
(~l>~/(1 + (fL>)tm + 1
(7.B.8)
The leading terms of eqs. (7.B.3) and (7.B.7) are the negative exponential probability density
function, and the Bose—Einstein distribution.
8. The inversion problem 8.1. Introduction We have so far examined the way spectral properties of the light incident upon a detector determine the emissions of photoelectrons. These photocount distributions are typically described by
324
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
specifying one of three statistical quantities: (1) factorial moments of single-fold distributions; (2) the distribution of time intervals between consecutive photocounts; or (3) the full or “clipped” photoelectron correlation function. In this subsection we consider the problem of determining the spectrum of the incident light from measurements of these three quantities. Extracting spectral information from the first sort of experiment is made difficult by the great accuracy of the data required — that is, the moments depend only weakly on details of the particular spectrum. A discussion of this problem with references is given by Meltzer and Mandel [90], and in the last subsection of this section. While the second method is less sensitive to inaccuracies in the original data, evaluating the spectral parameters is a subtle problem involving the approximation of very low counting rate (Scarl [71]), an iterative solution (Davidson and Mandel [67]), or methods suitable only for particular spectra (Kelly and Blake [73], Tornau and Echtermeyer [66]). Because it yields the most straightforward analysis, and because of its efficiency in information gathering capacity (Kelly [91] ; Degiorgio and Lastovka [24]) the third method is thus in most general use. Section 5 sketched the evaluation of the full and clipped photoelectron correlation functions for quasi-monochromatic radiation of arbitrary spectral profile. Let us view this as a quantum mechanical black box in which the input is the spectrum of correlation function of the incident radiation and the output is the photoelectron correlation function of interest. We term this the direct problem. Although the direct problem is interesting and important to the theoretician, the experimentalist faces a different problem. He is given the output and wishes to determine the input, measuring a photoelectron correlation function (with the attendant noise) and needing the input spectral profile or field correlation function. In other words, he must contend with the inverse problem. Inverse problems are known to be numerically unstable [92, 93], and care must be taken to stabilize the solutions. When the normalized counting time is very small (the ideal situation from the experimentalist’s point of view), there is a simple relation between the photoelectron correlation function and the field correlation function. This aspect of the inversion problem has been successfully dealt with by several groups of investigators. However, it is not always possible to guarantee that the experiments can be conducted in these ideally very short counting times and the question is how best to extract the input data from experimental data corresponding to somewhat longer counting times while insuring against numerical instabilities in the inversion process. The purpose of this section, then, is to investigate various schemes for carrying out this program. The first section is devoted to the situation where there is no a priori knowledge of the functional form of the input field correlation function. The method of singular value decomposition is employed to effect a stable inversion. In the second section we assume that the experimenter already knows the functional form of the input and is interested in estimating one or more parameters characterizing the input. Reference is made to Blake and Barakat [94] for the full details. 8.2. In version via singular value decomposition Assume we are given measurements of the full correlation function 2 (8.1) c(r 1) = (rn(t)rn(t + r1))/(m(t)) at the discrete points r 1 (1 1,... , M) where rn (t) is the number of photoelectrons counted in the
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
interval from (t
— T/2)
(1~~(t)
to (t + TI2). We desire to reconstruct the field correlation function (t+r)>
~(—)
~.
g(T)
325
(8.2)
~.
((+) 0’) f’() (t)>
from our experimentally measured values of c(r,) The full photoelectron correlation function can be written in the form (see eq. (5.32)) c(r)
1 +~
(8.3)
trBN(T)BN(—r)
where BN is an N X N matrix of the form {BN(r)}l/ = (H,H1)”2 g [~ T (x1 x,) + TI —
(8.4)
here x1 are the Nth order Gauss quadrature points and H, the corresponding weight factors. For very short counting times, T < 0.05, we can set N = 1 in eq. (8.3) obtaining 2(r). (8.5) c(r) 1 +g Since we are dealing only with quasi-monochromatic radiation, g(r) is real and even. The values of c and g are related pointwise to each other; that is, both sides of the equation are evaluated at the same single time point ‘r = r 1. For this case, we refer the reader to the following references: Degiorgio and Lastovka [24] ; Jakeman, Pike and Swain [95] ; Kelly [91]. It is not always possible to insure experimentally that T < 0.05. This raises the question of how best to extract spectral information from experimental data corresponding to longer counting times. In order to take account of these somewhat longer counting times, we take eq. (8.3) with N = 2. In this case g(r)
g(r—6)
g(T+~)
g(T)
(8.6)
.132(T)—
where ~ = (x2
—
x, ) T/2. The corresponding correlation function is
c(r)l+~[g2(T_~)+2g2(r)+g2(T+~)].
(8.7)
Now the values of c(r) are related to those of g at times T, r ±~ and the inversion is not straightforward. In order to obtain a formal inversion we let = j~and define the matrices ê, g, A:
{e}1
c(j~)—
(8.8) (8.9)
{A}11~ 26~~ +
~
1
+~
1
(8.10)
where i,j = 1,2,... , M. The purpose of the term —i,5~,,in the definition of ê is to take account of the truncation sequence of equations / = enough 1 by using the fact g(0)a delay 1. With the assumption 2 (Ms)of=the 0 (i.e., the PCS instrumentathas channels to that provide of several correthat g lation times) we have e=f+~Ag (8.11) where is a column matrix whose elements are unity. Solution of this matrix equation yields the f
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
326
unknown matrix g in the form g 00 4A
1
(ê
—
1)
(8.12)
and thus g(r1). The matrix A given by eq. (8.10) is anM X M symmetric continuant matrix. We have been able to express the inverse of A in closed form: (A’),1=j(M—/+l), (_I)i+/(M_/+l),
i
(8.13)
i>i We derived this result by noting that the cofactors in the explicit expression for A ~‘for M X M can be reduced to the same form as A for the size (M 1) X (M — I) and employing the fact that —
detAM+l
(8.14)
for an M X M continuant matrix. 1, it is still not useful to attempt a reconstruction of Even with a simple expression forAof the problem becomes painfully evident. Input data g by using eq.such (8.12). The ill-posed nature accurate to 1% calculated by using eq. (8.3) with N 6 were employed and the formal inversion numerically evaluated. The final results are shown in fig. 8. 1; the solid line is the theoretical value ofg2 (T) while the open circles represent the values obtained via eq. (8. 12). This difficulty could also have been anticipated by examining the eigenvalues of A which are I irk 1 =2 Il + cos I I M+lJ
(8.15)
where M is the size of the matrix. The three smallest eigenvalues are listed in table 8.1 for matrices of size M = 10, 20, 40. Observe that as M increases, the higher order eigenvalues become very small. i_c
I
I
0.8—
—
0.6—
-
0.4—
-
I-
N Dl
0.2—
TIME (UNITS OF ~) Fig. 8.1. Reconstructions ofg’(r) for a Brillouin spectrum (a = 1.0,6 = 5.0, ~ = 1.0) for a count time T = 0.1: —, value; o formal inversion with 1% noise; • singular value decomposition with 5% noise (Blake and Barakat [94]).
theoretical
R. Barakar and J. Blake, Theory of photoelectron countingstatistics: an essay
Table 8.1
327
-
Three smallest eigenvalues of A
M
XM_2
~‘-M—1
XM
10 20 40
0.7341 0.2014 0.0528
0.3262 0.0895 0.0234
0.0815 0.0224 0.0058
If (ê 1) has a component along the kth eigenvector of A, then this component will be multiplied by (Ak) 1 when we operate with A ~. Thus these high order eigenvalues are extremely sensitive to the slightest error (i.e., noise) and induce large errors in the final result. We can circumvent this ill conditioning by performing a singular value decomposition of A. Any real m X n (rn ~ n) matrix A can be factored as follows [96] —
A
L/’’J~~
=
(8.16)
where V is a square matrix of size n X n; U is a square matrix of size m X m, and ~diag(a1,a2,...
,a~).
(8.17)
U is composed of the n orthonormal eigenvectors belonging to the n largest eigenvalues of AA+, while V is composed of the n orthonormal eigenvectors of A+A. The singular values a1, , a,~are the non-negative square roots of the eigenvalues of A+A. We form the pseudoinverse of A by writing ~
A°
U~
(8.18)
where ‘A
j+
+.~
—uiag1~o,,a2, a7 = -~- if a~0 UI 0 if g1 0. —
+
,a,~,
(8.20)
If A is simply an M X M nonsingular matrix, as in our case, then A ~ A and we seem to have gained nothing. By expressing A -i as in eq. (8.18) we have, however, gained the option of ignoring those eigenvectors which are greatly magnified by the application ofA’. Specifically, let us set 001/C, ifa1>c
0
ifa1<~
(8.21)
where one reasonable criterion for picking e is c/a0
~‘
noise.
(8.22)
Physically, this procedure has the effect of ignoring the very high frequency components ofg (r). In applying the singular value decomposition to our problem we used an algorithm described by Golub and Reinsch [96]. We now apply the procedure to an input spectrum consisting of the superposition of two Lorentz lines (Brillouin) placed symmetrically about a central Lorentz line (Rayleigh) of different height
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
328
and halfwidth. This spectrum can be written in the form
R1
=
~
—
~2
(8 23)
Normalizing the power in the beam to unity, and defining a parameter 2R1 which represents the fraction of total power in both Brillouin lines (the Landau—Placzek ratio), we may write the field correlation function as g(r)
=
(1
—
a) e’~ +
e_I’T
cos T~.
(8.24)
Here we measure time in dimensionless units and express the frequency shift ~ (w1 — w0) (F/y) in the reciprocal of those units. For convenience in the calculations, the linewidths of the Rayleigh and Brillouin components are taken to be equal; this is very close to the truth in many experiments. For the present purpose, let us assume that we know nothing of the input spectrum except the output full photoelectron correlation function. In order to mimic the experimental situation we added 5% noise to the photoelectron correlation function which we calculated as before. Equation (8.22) then e = 0.1, that forM = 20input we should The reconstruction 2 (r) via thisyielded approach (for so exactly the same data asset in at9 the formal000. inversion) is also shownofin gfig. 8.1, see the solid circles. The majority of reconstructed points are now very close to the true curve in spite of the fact that we included 5% noise rather than the 1% noise for the formal inversion. At present there is no corresponding analysis for the clipped correlation function. This is an interesting problem and deserves further work. 8.3. Parameter estimation When the general form of the spectrum of the incident light is known, it is useful to add this information to the specification of the problem. The most common way to do this is to describe the spectrum by specifying a set of parameters (e.g., line width, etc.) which we will denote by ~ . . . ,/3,,. It is convenient to regard the whole collection of parameters as a vector i~having components (j3~,~ . . ,f1,,). Our problem now reduces to the estimation of these parameters from the measured data of the full or clipped photoelectron correlation function. We consider both the full photoelectron correlation function c(T) and the clipped photoelectron correlation function ck(T). Let us denote the set of predicted values of either of these functions over a mesh of time arguments by the vector (8.25)
where we include ~ in the argument to remind us that the components of the predicted correlation functions depend upon the parameters. The measured values of the correlation functions will be denoted by the vector
{e}
1
c(r1)
or ck(r,)
(8.26)
with the j~omitted. We now face the problem of selecting 13 so that ê(J3) is as “close” as possible to the measured values ê. A common and useful way of formulating this problem is the method of least squares in
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
which we select that j~which minimize the quantity —ê~][ê(13)—ê]. IIê(~)—êII200 [eq3)~
329
(8.27)
If the dependence of ê(~)upon 13 is linear, one can derive an expression for ~ in closed form. More generally it is necessary to solve for 13 by iteration. The calculation is straightforward (see Wolberg [97]) and we m~relystate the result: ~ 13n 1T~(!3n)[Vf(.8n)]~f(!3n) where f3,.~ is the nth iterate. The various functions are defined by: fi~
ê—
ê(j3)
f(13)}iI
(8.28)
(8.29) {ê(1~)}
{t(Th~’
1
(8.30)
(8.31)
~[Vf(1~)]~[vf(j3)1.
To apply eq. (8.28) to the full photoelectron correlation function it is easiest to make direct use of eq. (8.11), as the derivatives ofH are easily written down. Applying eq. (8.28) to the clipped correlation function requires taking eq. (5.47) as a starting point. The problems of multiple parameter estimation for complicated spectra have been discussed by Blake and Barakat [94]. In this subsection we will therefore treat only a simple illustration involving a single parameter. For the Lorentz spectrum the field correlation function is g(t)
=
e~
=
(8.32)
eI’ITI
where we measure time in dimensionless units chosen to make F near to unity. The vector of parameters 13 now consists solely of F. Some typical numerical results obtained by using eq. (8.28) are summarized in table 8.2. The simulated experimental values for the full correlation function were calculated form eq. (5.32) with N = 6. The clipped correlation function’s simulated values were calculated from eq. (5.47). We added random noise (see column 3) to the calculated values an three experiment. 2~ to0.imitate The first entries in table 8.2 show In deriving eq. (8.11) we assumed that g(M~)I Table 8.2 Parameter estimation of r for Lorenta spectrum for full and clipped
correlation functions T
M
% Noise
F
Full (N
0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.2 0.2
20 10 5 20 10 20 20 20 20 10
5 5 5 5 5 2 10 15 5 5
1.3 1.3 1.3 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1.294 1.276 1.107 1.001 1.002 1.000 1.034 1.046 1.028 1.017
=
2)
Clipped (N = 2)
-
— — — 1.005
1.007 1.001 1.030 1.101 1.012 1.001
330
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
cases where this is not so; in fact Jg(M~)f2 0.223 whereM 20; 0.472 forM 10; 0.687 for M 5. Thus M = 5 and 10 violate this condition severely and this is reflected by the errors in the fitted values of F by 23% and 5% respectively. ForM = 20 the fitted value is correct to 0.7%. We anticipate no significant error on this account as long as g(M~)I2 <0.2 which is true for the rest of these calculations. In practice, this restriction requires that the correlation instrument has enough channels to scan at least one correlation time. Subsequent entries in table 8.2 show fitted values for counting times of 0.1 and 0.2. While the short counting time approximation, eq. (8.5), would tend to underestimate F as the counting time increases, the N 2 solution does not, and remains accurate for all counting times <0.5. The iteration scheme is quite insensitive to noise and converges in 3 or 4 iterations even in the presence of far larger amounts of noise than shown here.
9.
Influence of spatial correlations
In section 3 we made the assumption that all the atoms of the photodetector experience the same field; that is, that the detector subtends an angle negligible compared to a single coherence angle. In any practical experiment this assumption is not rigorously true, and it is therefore appropriate to examine the alterations in some of our previous results caused by the effects of partial spatial coherence. 9.1. Heuristics of spatial coherence
Before commencing with the mathematical treatment we will find it useful to examine the problem heurestically, particularly in order to point out why one would expect to see that the modifications to the simplest results required by finite detector area resemble the modifications required by finite counting time. One can determine information about a spectral profile by examining the correlation of photons in time. Similarly, one can determine information about a spatial profile by examining the correlation of photons over a region in space. In fact the first practical photo correlation experiments were of this second type, namely those made by Hanbury Brown and Twiss [98, 99] to determine stellar diameters. We begin by imagining two detectors, each of negligible size, placed at the points r 1 and r2. An experiment is performed to determine the quantity (n1n2> where n1 is the number of photocounts recorded by detector/in a time T, negligible compared with the time over which the intensity of the light fluctuates. The value of (n1n2> will, of course, depend upon the type of light incident, and in most cases, upon r1 and r2. First, let the detectors be illuminated by perfectly coherent light (strictly speaking, by light2)(r with coherence of order two or greater). Under our assumptions (n1 n2) will be proportional to G( 1t1 ,r2t2 r2 t2,r1 t1), and since the criterion for second order coherence is that 2)(rit G( 1,r2t2r2t2,r1t1)= G~’~(r1t1,r1t1)G~’~(r2t2,r2t2) (9.1) then (n1n2> will not depend upon r1 or r2 as long as the field is stationary in space. That is to say there is no Hanbury Brown—Twiss (HBT) effect. This is reasonable, for within such a field there are no fluctuations in intensity; the only fluctuations in n, arise from shot noise and the shot noise in the two detectors will not be correlated. Clearly also, if we set r1 = r2 and looked at correlations in time
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
331
for the photocounts, we would also find (n1n2) (n>2. We might explain both these results in an alternative way by observing that the fourth moment of a 6-function distribution is just the square of the second moment. For coherent light the P-function is given by P[{ak}]
1~1 6(c4—nk)
(9.2)
and therefore (n 1n2> = (n1)(n2), no matter whether r1 = r2 or ~1 t2, or both conditions held at once. Next, let the detectors be illuminated by quasimonochromatic Gaussian light described by the distribution function eq. (2.36). In this case we have 2~ (r 1~(r (r 1~ 1~ (r 2 (9.3) G~ 1t1,r2t2 r2t2,r1t1) = G~ 1t1,r1t1)G~2t2,r2t2) + IG~ 1t1,r2t2)( when r 2. This is a consequence of the 1 = r2, and t1 = t2, therefore, we will always find (n1n2) = 2(n) fact that the fourth moment of a Gaussian distribution is always twice the square of the second moment. Now as r 2. If, for 1 departs from r2, of (n1n2) will decrease until value (n> (n example, the source consists two points separated byitareaches distancethe d, limiting we would expect 1 n2) to decrease as a function of r1 — r21 in such a way that 2when the angle between r1 and r2 was largeis for light with a mean wavelength A. This to sin~(Aid), would be close to (n)[1001 which says that we would expect light acompared consequence of the Van(n1n2) Cittert—Zernike theorem striking different maxima of a diffraction pattern to be incoherent. Were the source extended in space with characteristic length d (for example, a star of diameter d) we would expect roughly the behavior shown in fig. 9.1. This is analagous to the decay of the time correlation function (n 1n2) as It1 — t2~becomes large compared with (L~w)’ where (sw) is the spectral width of the light.
2:~~-
(r1r2)/J~ Fig. 9.1. Schematic plot of the correlation ofn(r,) n(r2) as a function of (r~— r2)/ .JX~.i~Ais the “coherence area” defined for a particular geometry in eq. (9.34).
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
332
9.2. Generating function
The heuristic ideas presented in the previous section can be made more precise. We will follow in broad outline Kelly’s treatment [22] of the problem because it interleaves closely with the formalism developed thus far in the essay. However our analysis is somewhat less general than Kelly’s.
We have already made several assumptions about the nature of the Gaussian g-field: it is stationary in space and time, its spectrum is symmetric about some central frequency w0 (quasimonochromatic). To these we add another: the maximum dimensions of the source are much less than R0, where R0 is the average distance between source and detector. When these assumptions apply, we can write G~’~ in the form 1~ (r 2)g(y G~ 1t1,r2t2) = (111 1 —y2)h(r1 — r2) exp {—io.,0 (x1 x2)} (9.4) —
where g(t) —
1(w) e~w
1~ (r G~ 1 t, r2t)/ G~’~ (00,00) 2w 1r1 0 x—t—. 2R0C
h (r 1
(9.5)
tdw
J
(2)
r2)
In y~t___, c
(9.6) (9.7)
The r2). The van Cittert—Zernike theorem requires that 1~ (r shape of the source determines h (r1 G~ 1t1, r2t1) be determined by the diffraction pattern that would be produced were the source viewed as a mask through which plane monochromatic light were allowed to pass; that is, —
(9.8)
2R h(r1
—
r2)
~-
~ exp {—iw0(r1
—
r2)/cR0}d
where B is the area of the source. We have already seen in section 3 that the double generating function Q(A 1 ,A2) can be expressed in terms of the eigenvalues of an integral equation, eq. (3.59). In order to account for the fact that the detector has finite area we must modify eq. (3.59) by including explicitly the integration over the area of the detector ~-
[A1
J
dt’
J
2r’ + A d
2 ~ dt’
J
1~ (r 1t1,r1’t’)41(r’1t’)
G~
d2rIJ
.~!
41(r1t).
(9.9)
The evaluation of the eigenvalues will follow much the same technique as discussed in section 4, with the emphasis now on including the effects of finite detector area A instead of finite counting time T. We assume that T is much smaller than the time over whichg(t) varies significantly. Thus we can perform the integration over t’ trivially in eq. (9.9). The result is 2r’ + 1~(t,r)
=
~
)‘~(0,r’)h(r— r’)g(t) d
l’~(r,r’)h(r
—
r’)g(t
—
r) d2r’
(9.10)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an
essay
333
where we have defined T+ t
2/2R
exp {iw0 1r1
Y (t,r)
0C}
~
exp {—iw0t’} 4~(rat’) dt’.
(9.11)
The kernel of eq. (9.11) factors into space and time components. Therefore it is reasonable to look for eigenfunctions which factor similarly Y1(t,r)
=
Y1 (r,0)f1(r).
(9.12)
We shall require the f1 (n) to satisfy orthonormality conditions similar to those required of the eigenfunctions 41(t) 2nl=tI, (9.13) ~ f*(r?)f(rI)d and 2r’
b 1f1(r)
-~-
(9.14)
~ h(r—r’)f1(r’)d
where the b 1 are the spatial eigenvalues corresponding to the geometry of the particular detector area A and particular source which has determined h (r — r’). Upon substituting for Y,(t,r) in eq. (9.10), we find that any eigenvalue m must satisfy
(0,0)b,[A1Y1(0,0)+A2g*(r)Y(r,0)]
~-
(9.15)
and simultaneously ~
Y1(r,0)
b,[A1g(r)Y(0,0) + A2Y1(r,0)].
(9.16)
These two equations can be combined as a matrix equation .~
~
(9.17)
where A1
A2g*(r)
B
(9.18)
A1g(’r)
A2
lyo,o)
(9.19)
There will be two values of m determined by each l,~.
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
334
ln order to solve eq. (9.17) directly in terms of the product form
~k
(I +
we rewrite it in the
rnk),
(Wb,B+f)iyzr(l +rn)Y~,
(9.20)
which displays explicitly all the rn’s determined by the distinct b’s wb1B+f
0
0
0
wb2B +1
0
0
0
wb3B+f
11k
... Y2
(l
+m)
(9.21)
.
Y2
(1 + mk) is the determinant
The product wb
0
0
wb2B+I
0
1B+I
0
...
-
-
0
0
=
wb3B+1
H IIwb,B + I II = II (1 + rnk) I
...
(9.22)
k
where we have made use of the block diagonal structure in evaluating the determinant. Since wb1A1 + I wb 1A1g(r)
IIwb1A+fIl
wb,A2g*(r)
2b?[l =
—g2(r)] (9.23)
1 +(X1 +A2)wb,+XiA2w
wb1A2 + I
we have
2[l —g2(r)]}.
(9.24)
Q’(A1,A2)11 {l +(A1 +X2)wb1+X1A2(wb1)
When the detector is so small that eq. (9.24) can be satisfied only for b 1 then it simplifies to the usual result
Q’(A1,A2)
2[l
=
1, b2
b3
=
—g2(r)].
.
.
.
=
0,
(9.25)
I +(X1 +A2)w+A2X2w
9.3. Photoelectron correlation function The photoelectron correlation function is 2 +g2(r) ~ b~}.
(n 1n2)=
~-
Q(A1,A2)I~~=0
~—
_w2[(~
(9.26)
b,)
Both sums can be evaluated. The kernel of eq. (9.14) can be expanded in a bilinear series ~ b 1)7(r)f,(r’)
(9.27)
R. Barakat and 1. Blake, Theory ofphotoelectron counting statistics: an essay
335
assuming no degeneracy. Now set n = r’ and integrate over A, we have 1
=
~ b,.
(9.28)
Next, multiply both sides of eq. (9.27) by h (r — r’) and integrate twice, once with respect to r and once with respect to r’; the final result is d2r’ Ih(r—r’)I2 ~f3.
d2r
~ b~
(9.29)
Consequently, (n
2[1 +8g2(r)]. (9.30) 1n2)~w Since ~b 1 = 1, and b, >0, therefore, ~b1~ 1 where equality holds only when b0 = 1 and b1 = 0, /> 1. This means that 1~will approach 1 for a detector which subtends an area far less than a “coherence area” and will decrease significantly for larger detectors. This will have the effect of reducing the time dependent component of c(t). Clearly, for a detector subtending many coherence areas c(t) will become constant as such a detector will average over many independent fluctuations. 9.4. Clipped photoelectron correlation function The influence of spatial coherence on the clipped correlation function defmed by eq. (5.10) requires a more elaborate analysis. Equation (9.30) demonstrated that one needed only a single parameter, 13, to characterize the influence of the detector geometry on the full correlation function. This turns out not to be the case for the clipped correlation function; rather, it is necessary to evaluate all the eigenvalues of eq. (9.14). Once this is done, it becomes straightforward, albeit tedious, to differentiate Q (A1, A2) to obtain Ck (r) via the basic expression k — 1
(1)m1 m1!
am, + 1
aA7’~-aA2Q(A1,A2) x,=i
(5.47)
X~=O
m,0
where w (m1) is the average counting rate. Analytic evaluation of the eigenvalues is possible only for certain limited geometries (see subsections 4.4—4.6). One example which admits a closed form solution and which is of considerable experimental interest is a square source of side length a and a parallel square detector of side length b a distance R0 away. In this case we have from eq. (9.8) that -
h(r—r)
sinpa(x—x’) ~ia(x—x)
,
sinua(y—y’) (9.31)
P’z(YY )
where ~.t = w0/cR0. Equation (9.14) becomes bkfk(x,y) =B1F1(X)BmFm (Y) 2~lsin1Aa(x_x’)F(X~).! ~dy’ “~‘ b —b/2 ~za(x—x ) b —b/2 ..!
smn~’~ pa(y—y )
Fm(Y’).
(9.32)
336
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
We have made explicit the factorization of fk(x,y) and bk required by the form of h(x x’, y — y’). Equation (9.32) is a product of integral equations satisfied by the prolate spheroidal wave functions, see eqs. (4.42) and (4.43), provided we set B1 = (ir/2c0)A1, B~= (1T/2C0)Am where the A’s are the —
eigenvalues tabulated by Slepian and Sonnenblick [34]. Clearly bk ~BiBm consequence of the symmetry of the square. The quantity c0 is defined by c0 —pab/2.
is
degenerate, a
(9.33)
c0 is a dimensionless quantity related to the number of “coherence areas” the detector subtends. For our example the coherence area z~.Ais given by 2c2R~/w2a2 = 4ir2/~t2a2. (9.34) i.~A 4ir The number of coherence areas interrogated is thus ‘=
Nb2/I~A
c~/7r2.
(9.35)
When c 0 is small, it can be shown that b0 ~
—i
4c0
A~,
b1=b2=...~0.
(9.36)
Table 9. 1 gives values for the bk as a function of c0. In the case of other geometries, evaluation of the eigenvalues is a very complicated numerical problem. If the source and detector are discs of radii a and b respectively, the analog of eq. (9.32) is bkfk(r)
I
0
2J1 [~aIr—r’I] pair—rI
(9.37)
‘2~r’dr’.
This integral equation has not been studied. Kelly’s results for this geometry are incorrect. Once the eigenvalues have been determined and inserted into eq. (9.24), Ck(r) can be evaluated via Table 9.1 A. First nine eigenvalues bk as a function ofc0 for square geometry
k
c01
c02
c03
c04
c05
0 1 2 3 4 5
.80891 .08871 .08871 .00973 .00019 .00019
.47830 .19317 .19317 .07802 .00786 .00786
.26106 .18993 .18993 .13818 .03992 .03992
.15294 .14008 .14008 .12830 .0730 .0730
.09856 .09664 .09664 .09476 .07736 .07736
6 7
— — —
.00079
.01154
.0415
.06316
.0010 .0010
.0088 .0088
.02712 .02712
8
— —
B. Numbering scheme for eigenvalues bk kO 123456789 nOlOl 212323 mO 011122233
R. Barakatand I. Blake, Theory ofphotoelectron counting statistics: an essay
337
eq. (5.24) by appropriate differentiation of Q(A1 , A2). We will now evaluate explicitly c1 (r) and c2(r). It is convenient to define 2 [1 —g2(r)]}’ (9.38) Q1(A1,A2) {1 + (A1 + A2)wb1 + A1A2(wb1) so that
a C
Q1(A1,A2)
~—H
1(T)w+
a
C 2(r)
=
w+
(9.39)
2
(~—
—
a
8AaA) H Q1(A1,A2).
(9.40)
Tedious manipulations then yield 2(r)
C1(r) =w + 11(1 + wb1Y’
{_~
~
+g
k
,Y’ (—(1 + w) ~
C 2(r)w+H(l
(wbk)2 (l+wbk) wbk
2(r) ~ +g
Ck(r) ~
+
[( wbk
+bwk)
(wb~)2 ~
(wbk)2 ~ + ~
k 1 +wbk k ((1 +wbk)/ The normalized clipped correlation function ck(’r) is
C
(9.41)
+wb k
ck(r)
}
k
(l+wb,~) (wb,~)3 1
(1 +wbk)2i~
(9.42)
Ck(’r) =
(5.50)
.
With this normalization, we have
ck(r)
=
1 + ct~g2(r)
(9.43)
with a 1
=
{ ~k
2
(1 (wbk) + wbk))/{~1II (1 + b 1) 2
a2
~
(wbk)
k
(1 +wbk)
— 1])
wbk
(9.44)
(wbk)3
l/tw[fl (1 + wb 1)
(1 +wbk) +
k
(1 +wbk)2Jik LI
—
~ k
1 wbk ( +wbk)
—
l]}.
(9.45) Figure 9.2 displays the behavior of a1 and a2 as a function c0. All nine eigenvalues listed in table 9.1 A were used in the evaluation. Clearly a1 and a2 approach zero as c0 is made large. For c0 > 3 the eigenvalues fall off very slowly and a large number of them must be retained in order to insure reasonable accuracy. Simple asymptotic expressions for a1 and a2 would be desirable.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
338
I.e -
a 2
Fig. 9.2. Plot of aj, defined in eq. (9.43) as a function of c0.
Acknowledgments We wish to thank Professor R.J. Glauber for stimulating discussions on problems in quantum optics. Barakat’s work was supported in part by Air Force Office of Scientific Research.
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Richard BARAKAT and Julian BLAKE Division of Applied Sciences, Harvard University, Cambridge, MA 02138, US.A.
I
NORTH-HOLLAND PUBLISHING COMPANY
—
AMSTERDAM
PHYSICS REPORTS (Review Section of Physics Letters) 60, No. 5 (1980) 225 -~340. NORTH-HOLLAND PUBLISHING COMPANY
THEORY OF PHOTOELECTRON COUNTING STATISTICS: AN ESSAY R. BARAKAT* and J. BLAKE Division of Applied Sciences, Harvard University, Cambridge, MA 02138, U.S.A. Received September 1979
Contents: 1. Introduction
227
2 Summary of the Glauber theory of quantum optics 2.1. Coherent states and correlation functions 2.2. P-representation 2.3. The n-atom detector 3. Photoelectron generating functions 3.1. Single-fold photoelectron generating function 3.2. Mixed Poisson process 3.3. N-fold photoelectron generating function 3.4. Eigenvalue problem for generating functions 4. Single-fold statistics for Gaussian light 4.1. Evaluation of Q(~’.) 4.2.P(m,T) for completely chaotic case 4.3. Eigenvalue problem for Q(X) 4.4. Lorentz spectrum 4.5. Rectangular spectrum 4.6. Numerical solution of integral equation 4.7. Long-time asymptotics 4.8. Variance of photoelectron counts 4.9. Superposition of coherent and multiple mode chaotic radiation 4.10. P(m,T) for partially polarized chaotic light 4.11. P(m,T) for statistically independent fields Appendix 4.A: Derivation of Glauber’s second asymptotic approximation 5. Two-fold statistics for Gaussian light 5.1. Correlation and clipped correlation functions for photoelectrons 5.2. Solution for short counting times 5.3. Solutions for arbitrary counting times 5.4. Q(X1, X2) for Lorentz spectrum 5.5. Higher order photoelectron statistics
228 228 230 233 234 234 237 239 241 244 245 245 247 248 252 254 257 261 262 267 271 271 273 273 275 277 282 287
Appendix 5.A: Q(X
1, ?.2) for partially polarized light Appendix 5.B: Evaluation of the photoelectron correlation function 6. Time interval statistics 6.1. Derivation of time interval probability densities 6.2. Short counting time, partial polarization 6.3. Lorentz spectrum 6.4. Numerical schemes for V(I) and P(T) 7. Non-Gaussian statistics 7.1. Introduction 7.2. Fixed number of scatterers 7.3. Random number of scatterers 7.4. Moments of W(fflIV) and LV(f~I(N>) 7.5. Coefficients of skewness and excess for W 7.6. Variance ofP 7.7. Time interval statistics 7.8. Photoelectron correlation function 7.9. Clipped photoelectron correlation function Appendix 7.A: Formulas for scattered intensity Appendix 7.B: Asymptotics of W(~IN)and P(mIN) 8. The inversion problem 8.1. Introduction 8.2. Inversion via singular value decomposition 8.3. Parameter estimation 9. Influence of spatial correlations 9.1. Heuristics of spatial coherence 9.2. Generating function 9.3. Photoelectron correlation function 9.4. Clipped photoelectron correlation function Acknowledgments References
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R. Barakat and 1. Blake, Theory of photoelectron counting statistics: an essay
227
1. Introduction One of the most useful methods of studying the statistical and spectral properties of light is by means of photoelectron counting detectors. The detection process is, of course, intrinsically quantum mechanical. The simplest photoelectron detector, for example, is a single atom free to undergo photoemission after which the ejected electron is observed. An important first step was carried out by L. Mandel [1] who derived a photoelectron counting formula (that is, a relation between the intensity and the probability of photodetection). The development employed semiclassical techniques in that the radiation field is treated classically and the interaction with the electrons in the detector is treated quantum mechanically. The quantum theory of coherence phenomena (which subsums photoelectron counting) was mainly originated by R.J. Glauber in the early 1 960s. However, a detailed account of the theory did not appear until the publication of his Les Houches Lectures [2] in 1966. Further amplifications and elaborations of the theory were subsequently published [3, 4, 5]. Included in Glauber’s work is a complete quantum derivation of the photoelectron counting formula which has served as the basic “building block” for subsequent research. Although the semiclassical and quantum approaches employ similar mathematics the underlying physics is different. The contrasting sentiments of the two camps are probably best summarized in an exchange between Mandel—Woif [61 for the semiclassical and Jakeman—Pike [7] for the full quantum protagonists. There now exist several volumes devoted in part to photoelectron counting: Klauder and Sudarshan [8], Perina [9], Loudon [10], Crosignani, DiPorto and Bertolotti [11] and Saleh [12] as well as review articles of which the most pertinent are: Arecchi [131, Lax [14], Jakeman [1 5] and Pike and Jakeman [16], reference is also made to Mehta [17] for the semiclassical viewpoint. The purpose of the present essay is to provide a detailed analysis of those theoretical aspects of photoelectron counting which are capable of experimental verification. Most of our interest is in the physical phenomena themselves, while part is in the mathematical techniques. Many of the mathematical methods used in the analysis of the photoelectron counting problem are generally unfamiliar to physicists interested in the subject. For this reason we have developed the essay in such a fashion that, although primary interest is focused on the physical phenomena, we have also taken pains to carry out enough of the analysis so that the reader can follow the main details. We have chosen to present a consistently quantum mechanical version of the subject, in that we follow the Glauber theory throughout. In fact, we assume the reader to be familiar with [2]. No attempt has been made to provide an exhaustive list of articles written on the subject, rather, our references are somewhat eclectic and are chosen to reinforce the main text. Saleh’s recent text [12] contains a reference list of 521 items (!), sufficient to whet the appetite of any photoelectron counting bibliophile. Prominent among our references are the publications of the Royal Radar Establishment (Jakeman, Oliver, Pike) whose viewpoint we generally share. The plan of the essay is as follows. Section 2 is devoted to a summary of those aspects of Glauber’s theory directly pertinent to photoelectron counting. The general theory is continued in section 3, with emphasis upon the development of N-fold photoelectron generating function which appears here for the first time. Initially, the central theme, as conceived by Glauber, was the determination of the single-fold statistics of the photoelectrons for Gaussian light. Section 4 summarizes the various aspects of the problem and is based in part upon an unpublished manuscript by Barakat and Glauber [18] as well as unpublished work by Barakat and Blake. Some of this work was independently carried out and published by Jakeman and Pike [19].
8
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
Although the determination of the one-fold statistics is interesting, the information that is obtaine insufficient for many purposes. It is necessary to develop and exploit the formalism appropriate ~rtwo-fold statistics (especially correlation functions). Section 5 covers this aspect of the problem regards Gaussian light. The elegant paper by Jakeman [20] giving an exact solution to the -oblem for a Lorentz line is covered in full, especially since the analytical details are essentially nitted from the original paper. We are indebted to Dr. Jakeman for helpful correspondence. npublished work by us on the double generating function for partially polarized light is also .mmarized. Time interval statistics form the subject of section 6. Practically all the work is based on published Ld unpublished work by Glauber and Barakat, and by Blake and Barakat. To this point we have ~ated the case of Gaussian light. In many problems, such as scattering by a small number of trticles, the scattered light is non-Gaussian. Thus, the first and second moments are insufficient to Laracterize the distribution and higher moments are needed. Section 7 reexamines the various oblems discussed in sections 4, 5 and 6 for non-Gaussian light including the Chen, Tartaglia and isey [211 version of the non-Gaussian photoelectron correlation function. The problem of termining the spectrum of the incident light from measurements of the quantities discussed in evious sections is briefly discussed in section 8. Finally, the influence of spatial correlations on totoelectron counting, as outlined in Kelly [22] , is summarized in section 9. A number of topics have been omitted, such as dead time corrections, processing techniques, mpling scheme, etc., see [15, 23, 24] for such information. The thrust of this essay is on the incipal underlying physical ideas and not necessarily on applications of these to technologically eful applications. Summary of the Glauber theory of quantum optics This section is a summary of Glauber’s full quantum description of the interaction of the electroignetic field with the n-atom detector. Of particular importance, insofar as photoelectron counting itistics is concerned, are the coherent state description of the electromagnetic field and the representation of the density operator. t. Coherent states and correlation functions Let us review briefly Glauber’s coherent state description of the electromagnetic field. The umption of finite boundary conditions of a general sort permits us to describe the field as conting of a discrete set of modes, each analogous to a single harmonic oscillator. If ak is the nihilation operator for a photon in the kth mode, and its adjoint, a~ the corresponding creation erator, then these operators obey the commutation rules [ak,a~] [ak,ak’]
=
ókk
[a~i,a~J
=
0.
(2.1)
The stationary states for the kth mode are those consisting of precisely n photons. They can be istructed from the vacuum state I ~ by writing In)k
(n!)1”2 (a~)n I0)~~, n
=
1,2,
-
.
-
.
(2.2)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
229
The n-photon state is an eigenstate of the number operator (2.3)
Nk —akak
since NkIn)k =nln)k.
(2.4)
Though complete and orthonormal, these n-photon states are not a particularly useful basis when the quantum numbers describing the field are large. In this limit, a more useful basis is the set of eigenstates of the annihilation operator, ak: =o~kI{ak})
akl{ak})
(2.5)
where the ~‘~k} are arbitrary complex numbers. These states, introduced by Glauber, are the coherent states. They may be expressed as a sum of the n-photon states by writing (2.6) Ictk)=exp {—~IakI2} (n!)”~ (ak)’~In>k for the kth mode, and then forming the direct product over all the various modes:
I{ak})=~Ia)k.
(2.7)
The average number of photons in the kth mode, = =
,Tk,
is given by
({ak}INkI {o~k}) IakI exp{—IctkI2} ~ ~9~I~1çI = I°~kI.
(2.8)
Similarly the probability that a given coherent state has exactly n photons in a particular mode is a Poisson distribution
I(flkl{~})I = I(flkl Hk exp {—
=
=
(ñ~)’1e
—
2(ak)”k’ Ink’)k’I
2Iak’I}
~
(n~’Y”
(2.9)
n~
The reason for the name coherent state is that these states satisfy the classical condition for nth order coherence; we will demonstrate this shortly. Finally, we note that the coherent states form an overcomplete set, and are not orthogonal. Consequently the expansion of an arbitrary state vector is not completely straightforward. It is useful to express the electric field operator as the sum of two nonhermitian operators: E
(Pt) = ~
(Pt) + E~—~ (Pt).
(2.10)
The positive frequency part ~ contains all terms which vary as exp (—iwt), the negative frequency part ~ all those which vary as exp (iwt). Such a decomposition is useful because we may write E(+) entirely in terms of annihilation operators and E~)entirely in terms of creation operators. For example, if we employ box normalization we have ~
(Pt)
=
i ~ (~hw,~)”~ akUk(r) exp ~—iwkt}
where the mode functions, Uk(P), form a complete orthonormal set. Since
(2.11)
E(+) (Pt) is a linear
R. Barakar and J. Blake, Theory of photoelectron counting statistics: an essay
230
colnbination of the ak, we see that the coherent states are eigenstates of E(+) (Pt): E(~~(Ft)I {~k}~=~(Ft)I {ctk})
(2.12)
where ~ (Pt) is the c-number field derived from our expression for E(+) (Ft) when all tile
ak
are
replaced by ctk t (F,
t,
{ctk})
=
i ~ (~liw,,~)~ctkUk(P) exp {—iwkt}
~ c~ê,jPt).
(2.13)
In most physically realizable situations the field is not in a pure state; our knowledge of the parameters describing the field is incomplete. In making predictions of any measurements, then, we must average over an ensemble of fields prepared in a manner which makes them all consistent with whatever knowledge we do have. The most convenient way to handle this averaging is through the use of the density operator (2.14)
P~fI~3)(/3I}av
where I ~3)is one of tile possible field states. and {
}av
denotes an ensemble average. The expectation
value for any operator is given by (2.15)
{I~9I/3)}av=trp&.
We will find it useful to introduce a class of functions, called nth order correlation functions, defined through G~°~ (x1
..
.
x2~) tr [pEL) (x1)-
.
.
E~)(x0)E(~)(x~+ ~)
..
.
EW (x20)]
(2.16)
where x1 represents the /th space-time point, (F1, t1). The product of field operators is normally ordered, that is all the positive frequency factors precede the negative frequency factors when reading from right to left. Because of this normal ordering, the value of G(?1) (x1’ x20)in a coherent state becomes a product, each of whose factors depends upon only a single space-time point; .
G(°)(x1--x2~)—~ g*(x1 ~ak})’ ..g*(x0, {~k}) ~(x0~
i~~k})’’
g(x20, ~k}).
(2.17)
For fields which are not pure coherent states we will still be able to make use of this property to provide a useful alternate representation for ~ (x1’ ‘
2.2. P-representation
By using the overcompleteness of the coherent states, we may express any density operator by writing pHHI{ctk})({ctk}IpI{I3k,}>({I3k,}Id2akd2~k,.
(2.18)
For a large class of fields we may collapse this integral into a simpler expression: 2ak, (2.19) p=~P({ak})I{ak})({c~k}Ifld If a real-valued function, P({ctk}), exists so that it is possible to express pin this way, we say that the field has a P-representation. In such a case we may evaluate the average of any normally ordered
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
product of the operators E(+) and ~ becomes ~
(x1
231
easily. In particular, the nth order correlation function 2ak.
x2,~)= ~P({ak}) g*(x1 {ak})~ g*(~, {~h,c})~(Xn +
. . .
.
1~{a,~})-. . ~
{ak}) H d
(2.20) The P-representation for a pure coherent state is easily obtained. For simplicity we consider a single mode only. In this case the density operator for a pure coherent state is p1j3)( 131.
(2.21)
We now look upon eq. (2.19) as an integral equation for the unknown function P(a). The solution, by inspection, is P(ct) =
(2.22)
5(2) (a — 13)
where 6(2) (a) is the two-dimensional Dirac delta function 6(2)
(a)_ ö(Rea)6(Ima).
(2.23)
Generally, the integral equation cannot be solved by inspection and we must make use of the normally ordered characteristic function X N (s)
tr [p ~
e5*~~ 1
(2.24)
where N stands for normal ordering. Substituting the P-representation of p into this expression, we have XN (s)
=
f d2ctP(a) e(s~*
—
s*a)
(2.25)
This is a diagonalized form of the two-dimensional Fourier transform of the P-function. if the Fourier transform can be inverted, then P(ct)
=
~
~ d2s XN
(s) ~
— sa*)
(2.26)
which yields P in terms of p through the definition of XN. The conditions under which this formal inversion is possible have been described by Cahill and Glauber [25]. Fortunately this inversion is possible for the fields which interest us in photoelectron counting experiments. Perhaps the most important state from our point of view is the chaotic state, the most common state occurring in nature. Again restricting our consideration to a single mode, we define chaos to be the state with the maximum entropy S = —tr [p log p1
(2.27)
subject to the constraints of a fixed mean occupation number ñ—(n>—tr [pa~a],
(2.28)
and the normalizing condition trp= 1.
(2.29)
R. Baralcat and J. Blake, Theory of photoelectron counting statistics.’ an essay
232
Following the usual procedures of the calculus of variations we introduce two unknown Lagrange multipliers m~?72, and set the variation of S equal to zero. This requires that pexp{—r~1 —~2a~a}.
(2.30)
The Lagrange multipliers are determined from the two constraint equations. We find, then, that I
I (n)
laa
~i +(n)i
(2.31)
-
Following the procedure for determining the function P(cs), we first must evaluate XN (s). This can be shown to be 2}. (2.32) XN(s) exp{—(n)Is( Using eq. (2.26) for the inverse Fourier transform, we have P(a)
ir(n>
exp{—1a12/(n)}.
(2.33)
Thus tile P-function describing a single mode chaotic state is a Gaussian with variance proportional to (11). Thermal equilibrium states, typically generated by the usual thermal light sources (mercury arc lamps, or lasers operating below threshold) are special cases of the general chaotic state in which the mean occupation number for each mode is given by the Planck law (nk)
=
[exp (/lwkIkBT)
—
1] —1
(2.34)
The P-representation obeys a convolution property: if P 1 (a), P2 (a) describe fields which would be produced by two independent sources, then the P-function of the sum is P(a) ~P1(a1)P2(a_a’)d2a’.
(2.35)
This convolution property is useful because we can use it to generate the description of new fields. Consider, for example, N independent chaotic modes, each with mean occupation number (nk). By repeated application of the convolution integral it can be shown easily that the P-function for the total field is multivariate Gaussian 2/(nk)}. P({ak})
k~1~k>
(2.36)
exp{—IakI
We will need also the P-function for the sum of a chaotic state and a coherent state. We have ~ 62(ct’_13) ~_~.~~exp{_Ia_&I2/(n)}d2a?
ir(n)
=
~exp{_Ia_13I2/(n>}. ~r(n)
The P-function is still Gaussian, but with nonzero mean 13.
(2.37)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
233
2.3. The n-atom detector Let us consider a detector which functions by absorbing photons from the incident light. Upon absorbing a photon, the detector displays an observable change which we can use as a signal that a photon has arrived. The simplest example of such a detector is a single atom which undergoes ionization by photoabsorption. We first examine the behavior of this simplest detector, and then discuss the behavior of a collection of these atoms, a model which is a good approximation to the photocathode of the photomultiplier usually used in photon counting experiments. We assume that the atom is much smaller than the wavelength of the incident radiation. If, additionally, the field is weak, we may describe the effect of the field by writing the interaction Hamiltonian in the electric dipole approximation: H1 = —e
i~ E(ft),
(2.38)
where i~is the coordinate operator of the nth electron of the atom relative to its nucleus, and the sum is taken over the valance electrons in the atom. Glauber has shown, using first order time-dependent perturbation theory, that the probability of a photoabsorption during an observation time t is given by pO) (t) =
~-~--
JJ
G(’) (Pt’, Pt”) dt’dt”
s(w) exp{i(t” — t’)w}dw,
(2.39)
where s(w) is the sensitivity of the detector. s(w) contains the matrix element of q between initial and final atomic states, and may be frequency dependent. We shall assume that s(w) is essentially constant over the frequency range of the incident light, which is an excellent approximation in most photon counting experiments. The o.,-integral then reduces to a delta function, and eq. (2.39) becomes 1~ (Pt’, Pt’) dt’. ~(l)
(t) = s
(2.40)
G~
The counting rate, the probability per unit time of detecting a photon, is w(t)
d =
p(’~(t)
=
sG~’~ (Pt, Pt).
(2.41)
This is proportional to G~’~ (Pt, Pt), the first order correlation function, which is the average intensity of the light. In deriving eq. (2.39) from the interaction Hamiltonian, eq. (2.38), it is necessary to make one further assumption which warrants some discussion. We have pictured our detector as functioning by absorbing a photon. Actually, transitions which do not conserve energy — transitions in which a photon is emitted during ionization — are possible as well with the form of the interaction Hamiltonian given in eq. (2.38). If we replace E(Pt) with the sum of the negative and positive frequency parts, we find that the matrix elements of the negative frequency part, E(), oscillate very rapidly. Since the observation time is necessarily much longer than these oscillations, the contribution of ~ is negligible. Recalling that ~ is composed entirely of annihilation operators, we see that neglecting ~ is equivalent to ignoring those occasions on which a photon is emitted
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
234
during ionization. In deriving eq. (2.39) we thus impose conservation of energy in a way not strictly compatible with the uncertainty principle. Were we not to make this approximation, we would lose tile nornial ordering in eq. (2.39) and the problem would become untractible. We next consider a detector consisting of an array of n atoms, each atom behaving independently, and in exactly the same way we have pictured our one atom detector. If we expose the detector to tile field at t = 0, we may ask tile probability that the first photoabsorption occurs during the time interval (0, t1), the second during (0,t2). and so on. As long as the total number of photoabsorptions is much smaller than the total number of atoms, this compound probability is given by
f J
tO
tl
p(~~) (t1,’~ ‘t0)
s°
‘‘‘
G~°~ (F1 ti’’
‘Fflt~, F0t;~.‘‘Fi
t~)fl dt
(2.42)
where F1 is the position of the jth atom. Tile derivation of this result, due to Glauber, follows tile same approach used in analyzing the single atom detector. As before, we can ask for the average rate at which the photons are counted, in this case, the rate at which this compound event occurs. This is given by W(t1
- ..
t0)
=
ut1
a°~ ‘
ut0
p~°~ (t1’’
‘t0)
=
s°G~°~ (F1 t1~ F~t0,f~t0~’’ F~t~). . .
(2.43)
3. Photoelectron generating functions This section is devoted to the derivation of photoelectron generating functions using the tools developed in the previous section. 3.]. Single-fold photoelectron generating function
Let us consider a photoelectron detector which is in tile form of a plane perpendicular to the direction of propagation of a collimated beam of plane waves. We further assume that the large number, N, of atoms in the photosurface all experience the same field; that is the detector subtends less than one coherence angle. In this case all atoms are equivalent, and each contributes equally to the dynamics of the detection process. While this is far from the most general geometry, it approximates that used in most experiments. The difficult question of the effects of including more than one coherence angle will be treated in a subsequent section. Each atom of the photodetector has only two states of interest to us: the ground state, and an excited state in which the atom has absorbed a photon and emitted a photoelectron. This suggests defining an operatorz1 with the property: = =
0, if jth atom is in its ground state 1, if jth atom has emitted a photoelectron.
We shall ignore those few occasions on which an excited atom decays to its ground state during the time of observation. The expectation value of z1(t) is then a random function of time which is initially zero and jumps discontinuously to unity at some unknown later time. By using the operators we can describe the counting statistics of the n-atom detector in a much more useful way than
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
simply stating the value of the joint probability that atoms i,j, k, times t1, t1, tk,
235
have emitted photoelectrons by
~-
The photoelectrons emitted by different atoms are indistinguishable. Accordingly we can ask questions only about the total number of photoelectrons emitted during some time interval, t. Let us define the random operator Z(t)_ ~ z1(t),
(3.1)
whose expectation value is the total number of photoelectrons emitted by all the atoms in the detector. If we expose the detector to light for a fixed time period (by means either of a mechanical shutter or an electronic gate) we will find that Z(t) has taken on a particular value, and that this value will be different each time we repeat the experiment. What we must evaluate then, is the single-fold probability distribution ofZ(t). Specifying the values of z~(t)for / = 1, N uniquely determines the final state of the detector. The most complete description we can hope to calculate, however, is the probability distribution of these values, p [z1 (t), z2 (t), , ZN (t)1, which we abbreviate p(z1, , ZN, t). Once this distribution is determined, we can compute any observable function of z(t) by evaluating the ensemble average: . . -
...
~4
(f[z(t)1)
..
~ zk(t)] p(z~,. ~,zN,t).
allz1
(3.2)
.
k
Our task now is to express the distribution p(z1, ZN, t) as a function of the joint probabilities p~(t1,~ t,,) which we evaluated in the last section. Consider first the marginal distribution for the /th atom alone: .
. -
p1(Z1,
t)
p(z1,
=
,ZN,
t).
(3.3)
all zk (k * I)
1)(t) can be related to this marginal distribution by noticing
The one-atom transition probability p( that p(1)(t)—p 1(l,t)
~
(3.4)
Z1p~(Z1,t).
Zj
Similarly, the n-atom transition probability, p(fl) (t1, be expressed as p(~)(t1,~
.
, t~)5
tnt
=
Z1Z2
~
ZN p(Z1,z2
tn) evaluated
,ZN, t).
with t1
=
t2
=
=
tn
=
t
can
(3.5)
The marginal probabilities make it easy to express ensemble averages over equivalent final states of the detector: (Z(t))~~
p1(1,t)
(3.6)
and in particular, the single-fold generating function Q(X) defined by m p(m,t)((l \)m) Q(X)~ ~ (1 —X)
(3.7)
Z1(t~
j~1
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
236
becomes (1 _X)Z(t)
(3.8)
p(Z1’’.ZNt)
allzk
The operators z~(t)commute since they refer to different atoms, therefore (I
—
X)z(t)
Further, as (1
—
N
(3.9)
t)
fl (1
= j=1
—
X)ZJ(
can take only the values one or zero:
X)z/(s)
1
=
—
Xz
(3.10)
1(t).
Substituting this expression into eq. (3.8) yields Q(X)
=
~
[II
allzj~
jl
(1
—
~~)] p(z1,~
,ZN,
(3.11)
t).
We now expand the product and obtain Q(X) as a power series in X Q(X)
=
~
(—X)° ~
n0
=
~
allzk
~
Z1
‘Z1
-
p(Z1,’
“
,ZN, t)
~
(—X)°~p~”~(ti,’’~,t0)I
(3.12)
~
where .fF indicates that the sum is to be taken over all the n-fold combinations. All the atoms in the detector are exposed to the same field by our original assumptions, and so the sum over the n-fold combinations is just the number of possible ways of grouping n objects from a total collection of N: (t1,’’
.
~
=
=‘=~~=~
Since the number of excited atoms, we can write N! n!(N—n)!
n! (N— n)! p(fl) (t1,’’ ,t0)j~
ii,
‘
=...=
~
(3.13)
is far less than the total number of atoms in the detector, N,
N° n!
(3.14)
Finally, substituting eq. (2.42) which gives p(0) (t1, function, G~°~ (t1,ta), we have
.
.
.
, t~)interms of the nth order correlation
. .
Q(X) = ~
...
~
(t~’ ‘t~,t~---t)
U dt.
(3.15)
Actually the sum should terminate at N, but since N is so large we commit little error in letting N be infinite.
R. Barakat and I. Blake, Theory of photoelectron counting statistics.’ an essay
237
This sum looks almost like the exponential function series. However, ~ generally involves a product of noncommuting operators and as it is not possible to collapse the series in its present form. If the field has a P-representation, then ~ can in fact be expressed as a product of c-numbers, as we saw in section 2.2. In fact, eq. (2.20) may be generalized to the following identity (Ns)~
. -.
G(~)(t~
.
.
.
t~,t~ ..
.
t~)Hdt;
=
~P(~ak}) ~(t,
{ak})
H d2ak
(3.16)
where (Ns)
I1(t’,
{ak})I2dt’
(3.17)
is the time integrated intensity of the random field amplitudes and is, of course, itself a random process in time. Thus for fields which can be described by the P-representation, we may write the expression for the generating function in a very compact form: Q (X)
P( {ak}) exp ~—X~2(t,{ak}) } H d2 ak.
=
(3.18)
Equation (3.18) resembles a Laplace transform and we will find it advantageous to define the function W(~2) ~P({ak}) 6[~2 — ~2’(t, {ak})] H ~
(3.19)
We can now write the generating function as Q(X)
=
~ W(~2,t)e ~ dH
(3.20)
so that Q and W form a Laplace transform pair. Upon differentiating Q(X, t) in accordance with eq. (1.4) we have -
P(m,t)j
W(12)
m!
d~’Z.
(3.21)
The usefulness of this result is obvious because P(m, t) can be obtained by integration rather than by repeated differentiation of Q(X). The factorial moments are easily shown to be (m-~.(m—/+ 1))m(m(/))’~ W(~2)~’2’d~.
(3.22)
3.2. Mixed Poisson process The integral representation of P(m, t) given in eq. (3.21) is more than just a useful computational device. Suppose that W(fZ) were itself a continuous probability density function with respect to ~2
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
238
over the interval (0,oo). Then m(t) is termed a mixed Poisson process (the terms compound and generalized are also used). The interpretation is simple. The real variable ~‘1~ in eq. (3. 1 7) is nonnegative since it is confined to the interval (0,oo). The factor ~m exp (—&Z)/m! in the integrand is a Poisson distribution with respect to ,~2,the mean of the distribution. Consequently P(rn, t) can be interpreted as a probability distribution in which the mean, is a random process having first order probability density W(~).Since &Z is non-negative it must be a non-Gaussian random process, even though it is a Gaussian random process. Thus eq. (3.21) connects the statistics of the continuous random process ~2(t) to the statistics of the discrete random process m(t). Necessary and sufficient conditions on W(&2) for it to be a probability density function over the interval (0,oo) with respect &2 are that it be a real function and simultaneously satisfy ~
1. ~ W(2)d~2—1, forallt 2. W(~)~ 0,
for all ~2and t.
The first condition is automatically satisfied irrespective of the second condition by setting X = 0 in eq. (3.7) and noting that ~ in
P(m,t)
1.
(3.23)
0
The non-negative constraint imposed by condition 2 must be handled on an individual basis. The mere fact that P(m, t) is expressible in the form given in eq. (3.21) does not imply that rn(t) is a mixed Poisson process because W(~2)can be negative over a finite interval and still satisfy condition 1. Reference is made to Schell and Barakat [261 for a physical situation where W takes on negative values and thus rn(t) is not always a mixed Poisson process. We can anticipate that m(t) is mixed Poisson and a formal proof will be given shortly. Since W(f~)is thus a probability density function, then the first moments of m(t) and cZ(t) are connected by m
(rn)
d~’2W(&2)e n
=
_______=
~1(mU!
0
J
&2W(~)df~= (~2).
(3.24)
0
The average number of photoelectron counts is thus equal to the ensemble average of ~2. The relation between the variances of m and &2 can be evaluated in similar fashion and is varm
(m)+var~
(3.25)
where var ~
J
2 [~‘~ —
(~)12 W(~2)d&~=
(~72)—
(3.26)
(rn)
This expression is important because it shows that the variance of the probability distribution of the number of photoelectrons is composed of two terms: the first term on the right-hand side of eq.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
239
(3.25) is the variance of a Poisson process and would be the only term if the photons showed no tendency to cluster; the second term is the contribution of the mixing density W(~2).Since this second contribution is positive, the variance of a mixed Poisson process is greater than the variance of a simple Poisson process having the same mean. This second term, var ~ is a manifestation of the fact that photons, when in the thermal equilibrium, obey Bose statistics so that the photon arrival times are not statistically independent of one another but are correlated. 3.3. N-fold photoelectron generating function Almost all experiments designed to extract spectral information from photoelectron arrivals make use of two-fold or second order statistics: conditional probabilities, correlation functions, or clipped correlation functions. The reason for this is that the details of the spectral profile are much more strongly reflected in these two-fold distributions than in the single-fold quantities. As the generalization to three- and higher-fold statistics follows easily from the two-fold case, we will treat only the two-fold problem in order to keep the notation as simple as possible. Our interest is now in the joint probability P(m1, t1 rn2, t2) that m1 photoelectrons are counted during a time interval designated by t1, and subsequently m2 are counted during the interval r2. Notice that we are using t1 and t2 to specify an interval in time, and not necessarily that range t = (0, t1). As before, it is useful to define a two-fold generating function by m’ (1 (1 —X1)
Q(A1,X2)= in1
_A
P(m1,t1m2,t2).
2)2
(3.27)
Ut2
1 and Z2, whose expectation values are the number of Let us define a pair of random Z photoelectrons counted during operators, t 1 and t2, respectively. Each may in turn be written as a sum over the contributions of the individual atoms in the detector: Z’ = ~
4,
4
4
where
4 4
=
and
2
=
z
4
(3.28)
are random operators defined by
1
if the /th atom emits a photoelectron during time interval t
0 1 0
otherwise; if the jth atom emits a photoelectron during time interval t2 otherwise.
1 = = =
Thus, 2) Q(X1,X2)=
((1
—
=(INII \j=1
4
X1)z’ (1
— A)Z
(1 —zJXi)
N
\
II (1 —z~X
2)). k1
Following the development in subsection 3.1, let us define the probability distribution of as p(z~ Z~, t1 z~ z~,t2), and note that this probability distribution is related to the . .
(3.29)
/
4
and
R. Barakatand J. Blake, Theory of photoelectron countingstatistics: an essay
240
(rn1 + rn2)th-order correlation function through ~ ~ {z)}
{4~
(5) ~
+
~
~dt~-- Jdt’ _______
~dt~
_______J
+ 1
~dt’in,
‘‘‘
+
G(m,
in,
+
in,)
(t~
-
+ m,;
~
+ ~,
. . -
t~).
_____________
‘.—~-
tl
(3.30)
t2
The expectation value in eq. (3.29) for Q(X1,X2) can now be expressed as an explicit sum over the probability distribution of 4 and 4: Q(X1,X2)=~ ~ ,
{(_Xl)m1(_X2)m2
z~
~
rn,
.
LF,
-
~ z~’.‘z~ p(z~’ z~,ti;z.z~,t2)}(3.3l)
-z~
where,~indicates a sum over all permutations of z~ z’,~,..As all the atoms see the same field, this sum over the permutations is just N!/(m~!(N rn,)!) times any of the single terms, and as N rn~, we may replace this factor with N’°i/m~!.Thus the generating function becomes -
~‘-
m’ (—X ~
Q(X1,X2)
~
(—X1N)
in,
~ z~ -z~ p(z~-‘-z~,t1z~”’z~,t2)
2N)
‘
(ml)!(m2)!
in2
(—X~Ny~’ (_X2N)in2
=
in,
2
{z)}{Z~}
~dt’1--- ~dt~
~
m, ________
_)
______________
..—
t2
tl
(3.32) where we have employed eq. (3.30) to express p(zi,- z~,t2) in terms of the correlation function. By expanding G(m, + in,) (t~ ~fl + in~ + ~, t~)in the P-representation, one may prove the following identity (a generalization of eq. (3.16)): - -
. . -
(Ns)in,+m, Jdt~---~dt~, Jdt,,+1 ________
_)
‘—
Jdt’in,
..
..
.
+
in2
G(m1+m2)(t~
-t~,+~
tl
...t’~ ‘3
t2
2ak fP({ak})~m,
t,,.~
______________
(t1,’{ak})~’2(t2,
(3.33)
{ak})Hd k
where &2(t 1, {ak})
Ns ~ I~(t’,{ak}) 12 dt’.
(3.34)
ti
Thus,
m2
Q(X1, X2) = m,,rn,
(rn
1
~ P({ak})~in1 (t,,{ak})~
(t
2ak 2, ~ak}) H d
1)!(m2)’
= (exp {—X,~2(t1, {aAj)}
exp {—X2~’2(t2,{ak})).
(3.35)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
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In this form Q(X1, A2) is an obvious generalization of eq. (3.7). The single-fold generating function Q(X) is obtained by setting A2 = 0. For this reason, it will be more efficient in the subsequent section to treat initially the evaluation of Q(A1, A2) and thus obtain Q(A) automatically. Having computed Q(X1, A2), we can then prove by induction that the n-fold generating function is ~
(exp{—A,~(t1,{ak})}
~~n)
..
~exp{—A~&~(t~,{ak})}).
(3.36)
3.4. Eigenvalue problem for generating functions The generating function can be evaluated by solving an eigenvalue problem involving an infinite matrix whose eigenvalues and eigenvectors determine Q(A1 ,A2). However, a useful computational approach for the completely chaotic case, where only the eigenvalues are needed (more on this point shortly), is to convert the infinite matrix eigenvalue problem into an integral equation whose eigenvalues are identical to those of the original problem. The P-representation for completely chaotic field is given in eq. (2.36) where (nk> is the mean occupation number for the kth mode. From the central limit theorem, any light source which we can in principle divide into many independent parts will radiate chaotic light. The set of all (nk) display the spectral profile of the radiation, 1(w), through the expression 1(w)
= hw(nk)
(3.37)
where w is the frequency. Upon substituting eq. (2.36) into eq. (3.35) for the two-fold generating function we obtain: 2 dt’ Q(A1,A2) ~exp ~ IakI — X,Ns ~ S(t’, {ak})1 k (nk) d2a A 2 dt’ H Ic (3.38) 2Ns I e’(t’, {ak})1 ) k in the form of a multiple integral. This expression can be simplified by defining the Hermitian matrix
f—
—
Mkk~ Ns {nk
J
[(x
1~
dt’ + X2 ~ dt’) e~(rt’)ek (rt’)](nk~)h12}
(3.39)
where ek(t) is given by eq. (2.18). Also set (3.40) With these definitions, we can rewrite eq. (3.38) in the succinct form Q(A1 ,A2) = ~exp {~+ (I + fi~} H
dyk
(3.41)
here ~ the unit matrix. Since M is Hermitian, it can be diagonalized by a unitary matrix U: CT~J1~UAdiag(m1,m2,-..)
U’~=~3.
(3.42)
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242
Performing the necessary change of variables, we can perform the integration explicitly: r d213 Q(A 3(I+ A)f3} Ii J k 1,X2) = ~exp{—1 -
~ exp{—13~(l~k)13k}
d13k
11(l
(3.43)
~‘~k)
where 1~k = 1~k(A
1, A2) are the eigenvalues of M. Consequently the problem shifts to the evaluation of these eigenvaiues. As we have just remarked there are two possible approaches. The first is to consider M itself by truncating it to finite size and evaluating the eigenvalues via any standard diagonalization scheme. The second is to transform the original problem into an equivalent eigenvalue problem involving an homogeneous Fredholm integral equation of the second kind having a symmetric kernel. Actually the methods complement each other in that the matrix formulation involves tile line shape 1(w) directly whereas the integral equation formulation involves the Fourier transform of the line shape, the correlation function, as its kernel. Although the matrix formulation is perfectly valid, the disadvantage is that the eigenvalues depend in a complicated way upon A~and A2 making the ensuing computations very difficult. (Nevertheless, we actually performed several calculations using tile matrix method to check the integral equation method.) With these remarks, we now outline a derivation of the integral equation formulation of Q(A1, A2) originally due to Jakeman [19]. Following tile formalism developed in subsection 2.2, we see that the statistical independence of the different modes requires that 6k, k, (3.45) (ar, ak2) = (nk) We will not work with the functions ek(t) appearing in eq. (2.28), which are orthonormal over tile time interval (_oo, 00), but with a set of functions ~ [5] which satisfy the equation -
(xi
J dt’ + A2 J
dt’) ~$(t’)Ø,(t’)
We further require that ek (t)
=
ek(t)
= 6~.
(3.46)
be related to ~1(t) by a unitary transform
~ Sk/ø/(t)
(3.47)
where ~S*SIc!
if
=Lö
-
ki’
~
~S =6 if-
4
k/Il
Corresponding to the coefficients, ak, the projection of a1
= [x1 ~ dt’ + A2
~ dl”]
~7(t’)I(t’, {ak }) =
I (t,{ak})
~ a1S,~J.
along a particular ~~(t) is given by (3.49)
This permits us to write A1 ~2(t1, {cz~}) + A2 ~(t2, ~ak })
= Ns ~
2. 1a11
(3.50)
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243
From the various sets of ~1(t), we assume that there is one particular set for which the a1 are independent
6~
(a1a1>
= (r1) 1
(3.51)
where the Kr1> are linear combinations of the (nk) whose explicit form we momentarily leave unspecified. The P-function for {a1} is 2/(r II
—~—-
exp{—1a11
1)}
(3.52)
and follows directly from eq. (2.33) and eq. (2.35). The double generating function can now be written directly in terms of Kr1> by following the approach we employed to derive eq. (3.43): 21 H ~d2a. = H 1 Q(A 2 ~ (3.53) r Ia.1 Kr 1, A2) = exp i. —NsIa,1 1> .i / ir(r1> / 1 + Ns(r1> .
—
The eigenvalues of the matrix M are NsKrk>. To prove this statement, we consider the matrix element (S~MS)kk’.We have 2 S/k’. (3.54) ($~I~IS)kk’ Ns~A1tl dt’ + A2 t dl”) ~S~1~(n,)hi’2 e7(t’)e1(t’)(n,)”
J
f
2
I,!
Upon expressing ek(t) in terms of~1(t)via eq. (3.47) and making use of the orthonormality of the 2
S,~S,~’S
(S’~MS)11. Ns i,j,r ~ (n)”
2 1~S11~(n,>”
= Ns ~ (n
1> S~Slk’= Ns ~ (a7a1)S~SIk’.
(3.55)
Next use eq. (3.49) 6kk’.
(S~MS)kk’= Ns ~ (ct7S7~ct~S(k’> = Ns(aak’> = Ns(rk> Consequently, m 1
(3.56)
= Ns(rk> are the eigenvalues of M.
To construct an integral equation with eigenvalues Kr1> we note that from eq. (3.49) we can derive the following expression for G~’~ (t’, t): G~’~ (t’, t)
=
=
(g*(~~ {ak})g(t,
{ak}))
~ (/~‘(t’)a1/~ (t)> = ~ (r~>~7(t’)~1 (t).
(3.57)
Next multiply both sides by Ø1(t’) and operate on them by the integral operator
(x,
f dt’ + A2 ~ dr’).
(3.58)
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244
The final result is the integral equation
(x1 ~ dt’ + A2
J
dt’) G(’~(t~t) = (i~)Ø~ (t).
(3.59)
1~ (t~t) becomes G~’~ (It’
—
tI). Use of the normalized
If the process is stationary in time, then G~ correlation function g(It’
—
tI) =
G~1~ (It’— tI)/G~’)(0)
(3.60)
and the average counting rate of the detector, w. wNsG(t,t)
(3.61)
permits us to rewrite eq. (3.59) in its final form (A
J
1
dt’ + A2 ~ dt’) g(It’
—
tI)01 (t’)
~! ~ (t).
(3.62)
We have thus found an integral equation whose eigen values are precisely the same as those of the matrix M, and it is this equation which forms a basis for the development of the next few subsections. By setting A2 = 0, we obtain an integral equation A
J g(t’
—
t)4~(t’) dt’ ~
~ (t),
(3.63)
whose eigenvalues enable us to evaluate the single-fold generating function Q (A). When the generating function has been evaluated as an analytic function of A1 and A2, the joint probabilities and factorial moments may be extracted by using the definition, eq. (3.27). Specifically, it is easy to show that P(m1t,;m2t2)=
(—l)”~’+in2 m1 !m2!
~3in,
~
ain, ~
Q(A1A~~
(3.64)
and
1
((m
a”
a
1) (m,
—
1)-~-(m1 —1+ l)(rn2)(m2
--
I)-- -(m2
—
k + l)>
(—l)’~”~ U14
Q(AX) x, =o N2
U!~.2
=
0
(3.65) Generalization of eqs. (3.64) and (3.65) may be derived easily for the n-fold problem.
4. Single-fold statistics for Gaussian light The determination of the single-fold statistics of the photoelectrons, namely the probability distribution of the photoelectrons, was initially the central theme. This section is devoted to
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
245
summarizing the relevant analyses and discussing the physical consequences. Subsections 4~2—4.6 are based upon an unpublished report by Barakat and Glauber [18] , while subsections 4.9—4.10 are also based on unpublished work by Barakat and Blake. 4.]. Evaluation of Q(A) We now proceed to the evaluation of Q(A) via the integral representation of Q(A1 ,A2) in eq. (3.43) with A2 ~0: Q(A) = j’exp~_~(1+ft~)j~} H where M is given by eq. (3.39) with A2
(4.1) 0. In the case of completely chaotic light
Q(A)H(l +mk).
(4.2)
k
From eq. (3.63) we see that each of the eigenvalues mIc is directly proportional to A. This permits us to exhibit explicitly the dependence of Q(A) upon A in this singlefold case. Before doing so, however, we will rewrite eq. (3.63) in terms of dimensionless variables. The spectrum of the light will have some characteristic width, ‘-y, which will be more precisely defined for particular spectra. We take as our dimensionless unit of time 1 /y, setting T = ‘yt. In order to avoid a proliferation of symbols we will still use Tfor the counting time, but measure it in units of l/’~’.The corresponding dimensionless variable for average counting rate is v = w/’y. We may now rewrite eq. (3.63) as
—
r’)Ok(r’)
dr’
where we have defined bk
(4.3)
Ok(r)
mI/A. This allows us to write the generating function as
(1 +Ab1)’.
Q(A)=H
I
(4.4)
4.2. P(m, T)for completely chaotic case Assuming for the moment that the eigenvalues b1 are known, we can evaluate P(m, T) by first obtaining W(~2)and then utilizing eq. (3.16). W(~)can be expressed in terms of Q(A)by the usual complex inversion integral W(~)=
1
J
X0+ioo
—
2iri A0
=
1 2iri
—
Q(A) e~~’ dA
—
J
X0+ioo
No —
-
I~O
H (1 + Ab1)” ~
j=i
dA.
(4.5)
R. Bara let and J. Blake, Theory of photoelectron counting statistics: an essay
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It will be shown in the next subsection that all tile b1 are real and positive so that the poles of tile integrand are located at A, = (—b1)” and thus lie in the left hand side of the complex A plane. If in addition, the poles are distinct (i.e., b0 > b~> b2 > ) then all the poles are simple. Under these circumstances, we close the contour in tile left hand plane and a straightforward calculation of the residues results in the expression --
W(~)~
je~Ib/,
‘
~
(4.6)
where
çfi’
(1
(4.7)
_-.~-)~‘.
The prime indicates that the term 1 = / is to be omitted. Note that W(~2)is a non~negativefunction of~2. The single-fold probability distribution P(rn, T) follows directly by substituting eq. (4.6) into eq. (3.18) and integrating termwise: P(rn,T)
(b.)rn (1 +b1)m+l
j0
-
(4.8)
The probability distribution P(m, T) of the photoncounts is thus an infinite sum of power law distribution when the eigenvalues of M are distinct. However as the time of observation is increased, the simple poles will coalesce giving rise to multiple poles, in which case both eqs. (4.6) and (4.8) have to be modified (this will be done shortly). There are constraints on the C1 functions which are very useful as a check on the numerical computations. Since P(m, T) is a probability distribution, then ~
P(m,T)
1.
(4.9)
0
in=
If we substitute eq. (4.8) into this series, we find that ~ 1=
(4.10)
C~=l. 0
This result also follows from integration of the mixing density W(~)with respect to &2, i.e., ~ W(~)d~2=> ~
1.
(4.11)
Another constraint follows from (m)vT
(4.12)
where v is the average counting rate. We have ~ rn0
mP(m,T)
~
b1C1uT.
(4.13)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
247
The factorial moments of P(m,T) can be shown to be given by
~ (b1Y’C,.
(4.14)
i=’o 4.3. Eigenvalue problem for Q(A) Before we can proceed any further we must consider the evaluation of the eigenvalues b1 as a function of the time of observation and spectral characteristics of the incident radiation. The eigenvalue problem for the infinite matrix M is (4.15) which as we saw in subsection 3.4 is equivalent to solving the integral equation, eq. (3.62). In a sense, eq. (3.62) is the “Fourier transform” of eq. (4.15) because the input spectrum hw(nk) occurs in the matrix description while its Fourier transform, the correlation function, appears in the integral equation description. For the single-fold case, the matrix elements appearing in eq. (4.15) are given by exp{i(wk D[hwk(nk>]
Mkk.
1/2
[hw1’ (n1)}
1/2
—
wk’)t}
(4.16)
—
Wk’)
where D is a real constant. Unfortunately, M is a rapidly oscillating function of the counting time which makes numerical calculations of b1 useful only for fairly short times. Although some numerical calculations were carried out on finite sections of M by Barakat and Glauber [181, a more useful computational approach is the integral equation formulation of eq. (3.63). Leaving aside the actual computation of the eigenvalues for the moment, let us state two theorems on Schmidt—Hilbert integral equations which will be employed in the subsequent analysis. Since g(r) is a correlation function, then it is known to be non-negative definite in the sense of Bochner [27]. The first theorem states: If g(T) is non-negative definite, then all the eigenvalues of eq. (3.63) are positive real numbers and can be sequentially ordered in the form b0 >~b,~ b2 ~s.. >0. As an immediate application of this theorem, if T ~ 1, then b0 ~ T with the remaining eigenvalues at least an order of magnitude smaller. The product functions C1 of eq. (4.7) become .
C~0
C0~l,
(j~l)
(4.17)
and the probability function P(m, T) degenerates to m/(l + nfl” + 1 (4.18) P(m, T) ~ (vT) This is the well-known Bose—Einstein (or power law) distribution expressed in terms of its mean (m> = uT. Note that the line width, y, is absent. Thus any spectral line shape will yield the same power law distribution in the short counting time approximation. The second theorem (Mercer’s theorem) states that if g(r) is non-negative definite, then it possesses an expansion of the form [28]
g(Ir’
—
r”I)
=
~
b,0 1(r’)0,~’(r”)
/=0
(4.19)
R. Bara/cat and J. Blake, Theory of photoelectron counting statistics: an essay
248
where 01(r) are tile orthonormalized eigenfunctions of eq. (3.62). If we set r’ = r” and integrate both sides of eq. (3.62) with respect to r’ (from 0 to T), we obtain
[g(o)
dT’
=T=
b1.
(4.20)
The sum of the eigenvaiues is equal to the normalized time interval of observation. The outcome of this result is that one has to contend with successively more eigenvalues as the time of observation is lengthened. As the time of observation T is increased even more, then the larger eigenvalues tend to coalesce (b0 = b1 > b2 >-- -> 0) because g(r) begins to act like a Dirac delta function. Consequently, eq. (3.62) will possess a number of eigenfunctions having the same eigenvalue and in the limit as T tends to infinity there will be one eigenvalue corresponding to an infinite number of eigenfunctions. It is precisely this feature of the problem which makes the rigorous analysis so complicated. Nevertheless, we will turn this degeneracy to our advantage shortly. The final general result we require is the bilinear series for the iterated kernel [281 g~(T’,r”) ~
b20,(T)07(r”)
(4.21)
1=0
where g~(Tcr”) = [g(T’,y)g~_
1 (t,T”)
dy,
r~ 2
(4.22)
g, (r, T”) = g(T’,r”). Let r’ = r” in eq. (4.21) and integrate with respect to r’ over the interval (0,r). Define a function Ir(T) by the equation:
Jr(fl~J~r(T”T’)dT’
=
-..
J g(T~y1)g(T~y2) g(~y . ..
-
i)g(yr
1’
r’)Hdy,
~
~tr
[Mn.
(4.23)
The usefulness of this expression lies in tile fact that the generating function can be expressed without the necessity of evaluating the eigenvalues individually through the relation Q(A) = exp
~r~1
~
1r(T)}
(4.24)
We will utilize this result shortly in our discussion of time interval statistics. 4.4. Lorentz spectrum Generally the eigenvalues of eq. (3.62) cannot be calculated explicitly and resort must be made to numerical techniques. However there are at least two spectra for which the eigenvalues can be written
R. Bara kat and J. Blake, Theory of photoelectron counting statistics: an essay
249
out explicitly; they are the Lorentz line and the rectangular line. Furthermore these two line shapes represent the extremes in behavior; the rectangular line has an infinitely sharp cutoff, while the Lorentz line decays very slowly. In fact, the Lorentz line has the slowest possible decay. These two cases are extremely useful because they serve as checks against the long-time asymptotic expressions to be discussed in subsection 4.6, as well as being important in their own right. The Lorentz spectrum is described by I(w)=hw(nk)=
2
(4.25)
2
(w—w0) +‘-y where A, half-width at half-height, serves as the characteristic width of the spectral line. In view of remarks made in the previous section concerning narrow, symmetric spectral lines, we can simply let w0 = 0. The Fourier transform of eq. (4.25) is the correlation function and when suitably normalized is g(T)e_ITI,
(T-’yt).
(4.26)
Kac and Siegert [29, 30] have shown that the eigenvalues of eq. (3.62) with eq. (4.26) as the displacement kernel are given by b. = 2 (1 + where
(4.27)
02)_I
are the positive roots of the transcendental equation
(I _0~)tan(T01)_2ij,1,
/=0,1,~.
(4.28)
See subsection 5.4 for details. Only the first eigenvalue b0 ~ T is of any importance in the short time region (T ‘~ 1). However, the eigenvalues decrease very slowly when T is outside this restricted range. In order to obtain a general idea as to the behavior of the Lorentz eigenvalues, see table 4. 1. Note the slowness with which the eigenvalues decay as T is increased. The practical outcome of this turn of events renders the direct evaluation of C1 almost impossible if we were so foolhardy as to evaluate the infinite product for T> 4. Table 4.1 First eight eigenvalues of Lorentz spectrum as a function of normalized time T (from Barakat and Glauber, unpublished report of 1966)
/
T0.5
T=4
T=8
0 1 2 3 4
0.42664 0.04159 0.01199 0.00549 0.00312
1.55049 0.86588 0.46308 0.26776 0.16923
1.81827 1.41548 1.01634 0.71643 0.51257
5
0.00201
0.11503
0.37674
6
0.00140 0.00103
0.08271 0.06210
0.28502 0.22149
7
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
250
This apparent difficulty can be circumvented by utilizing the identity C= /
b. ‘ 1 (A)/dAI~ dQ
=
(4~)9)
-bJ’
In the special case of a Lorentz line, Q(A) is known in closed form! It is 413(X)eT [1 +~(A)]2e~)T —[1 —~(A)]2e~(X)T
430
(
)
-
where j3(A) (1 + 2vA)112. A proof of this important result is outlined in subsection 5.4. Substitution of eq. (4.30) into eq. (4.29) easily yields ‘2 — b ~eT C.=(—1)’~’ ‘ “ v(T+b 1)
(4.31)
-
This closed form for C1 now permits one to carry out numerical computations relatively easily. The temporal development of P(m, T) is shown in figs. 4.1 and 4.2 for moderate beam strengths ofv= 1,5. For T< 1, P(m,T) is essentially a power law distribution, eq. (4.18), as only C, is effectively nonzero. As T is made to increase successively more terms of eq. (4.8) come into play leading to the Poisson-like curves for large times. The numerical data shown in the previous graphs concerns only hypothetical situations, but the theory is capable of handling actual situations. Although data is available for short time experiments as for example Arecchi [131, the studies of Freed and Haus [31, 321 using a laser operating below 0.3
4 f’\6 0.2 8
4 6 12 6
0,1
8
20
.~
12 20
00
10
30
-
20
1-
30
00
40
NUMBER OF PHOTOELECTRONS
Fig. 4.1. F(m, T) for a Lorentz spectral line, u = 1 for T 6, 8, 12, 20 (Barakat and Glauber, 1966 unpublished).
20
80
120
I
160
I 200
I
240
NUMBER OF PHOTOELECTRONS =
2, 4,
Fig. 4.2. P(m, T) for a Lorentz spectral line, u = 5 for T = 6, 8, 12, 20, 30 (Barakat and Glauber, 1966 unpublished).
R. Barelet and J. Blake, Theory of photoelectron countingstatistics: an essay
251
threshold offer an ideal test of the theory since the time of observation, T ~ 1, is outside the short time region and the beam strength is very large, u ~ 1000. The times are especially important because they lie in the never-never land between the short time and long time approximations (to be discussed) where one is forced to employ the exact expansion of eq. (4.8). For a typical case of their data: w = 1.40 X 106 counts/sec, ‘y = 420 hertz, t = i0~ sec, yielding a beam strength v = 1060 and a normalized time T = 1.32. Both the theoretical curve computed from the exact solution and the experimental data points are compared in fig. 4.3, the agreement is excellent. The slight overshoot of the theoretical curve when the number of photons is small is probably due to the fact that the actual line shape is not a true Lorentzian. (One of us, R.B., is deeply indebted to Dr. Freed for making available the original data to him in 1965 and for several clarifying discussions on the experiments.) An unfortunate feature of the exact analysis lies in the fact that the number of eigenvalues required increases with increasing T by virtue of the constraint expressed by eq. (4.20). This is particularly serious for the Lorentz spectrum because the eigenvalues decrease so slowly. For this reason, long time asymptotic approximations are desirable. In the special case of the Lorentz spectrum, Glauber has already obtained two such approximations to the exact distribution. The first Glauber [2] approximation (in our present notation) is 1 ivT\m s,~,(fiT) m!~3
P(m,T)
—
1—
e~
(T ~=1)
~)T;
(4.32)
I. ~
id~
:~..:
::..
..
1O6
I
0
I
000
2000
I
I
3000
I
I
4000
. •. . ts.. . •.
.
I
I
5000
6000
NUMBER OF PHOTOELECTRONS Fig. 4.3. Comparison of theoretical calculation of P(m, T = 1.32 (Barakat and Glauber, 1966 unpublished).
T) for a Lorentz spectrum with experimental data of Freed and Haus: v = 1060,
R. Barakat and .1. Blake, Theory of photoelectron counting statistics: an essay
252
where 13 13(1) = (1 + 2v)112 and ~m (x) is a polynomial which can be expressed in terms of the halforder modified Bessel functions of the second kind: 2x 1/2 Sm(X)=(_) ex K,,_L (x). (4.33) Glauber’s original derivation is somewhat involved, but we can take advantage of eq. (4.30) to obtain his approximation. Let T~ 1 in eq. (4.30) then 413(A) e11 [1+13(A)]2
Q(A)
—~3(X)IT
(4.34)
The important normalization, Q(0) = 1, is preserved. Glauber’s approximate generating function follows by setting the coefficient in front of the exponential to unity, thus Q(A)
‘S-’
et’
(4.35)
13(X)IT
Upon expanding this expression in a power series about A = 1 we obtain eq. (4.32). The first few 5m (x) are: s 0(x)
s1(x)
1,
s2(x)
1+—, x
I,
s3(x)
1
33 x x
+—+---~.
(4.36)
Higher order members are obtainable from the recursion formula Sm+i(X)=_5~(X)+(l
+~)5m(X).
(4.37)
The second Glauber approximation [33]is based on the additional assumption that the beam strength is large (v ~ 1) and in the present notation reads / v ~1/2 T 1 1 I —) exp — [m — (vT)} 2} (4.38) \2irmf m 2vm Glauber actually never published the original derivation of this formula which he obtained from eq. (4.32), but we outline an alternate (unpublished) proof due to Glauber and Barakat in appendix 4.A. A discussion of these asymptotic formulae is postponed until subsection 4.9.
P(m, T)
‘~
—
4.5. Rectangular spectrum A second spectrum which admits an exact solution is the rectangular spectrum of full width z~w centered about w0 1(w) =
~
= 0.
w0 --i—
(4.39)
R. Bare let and J. Blake, Theory ofphotoelectron counting statistics: an essay
253
The normalized correlation function is g(r)
=
sinirT irT
(4.40)
where the characteristic width ‘y is now taken to be ~w/2ir. The basic integral equation can be cast into the form +1
sinc(z’—z”)
—1
lr(z—z)
,,
0,(z”) dz” = b,01(z’)
(4.41)
with c (irr/2) upon transforming to new variables z’ = (2T’ — T)/T, z” = (2T” T)/T. This is the integral equation satisfied by the angular prolate spheroidal wavefunctions S01 (c, z): —
sin c(z’—z”)
S01
(c,z”) dz” -~[R~) (c, 1)] 2 s01 (c,z’)
(4.42)
where R~)(c,z) is the corresponding radial spheroidal wavefunction. Hence, our eigenvalues are given by [Ru) (c, 1)12
(4.43)
and have been tabulated by Slepian and Sonnenblick [34]for integer values of c. Reference is made to Flammer [35] , Frieden [361 for general properties of the prolate spheroidal functions. The behavior of the rectangular eigenvalues is different than that of the Lorentz eigenvalues. Table 4.2, abstracted in part from Slepian and Sonnenblick reveals the tendency of the leading eigenvalues to cluster rapidly about unity as T is increased. From the practical point of view actual calculations indicate that the limit on the use of eq. (4.8) is T 2.546 (c 4). For larger values of time the multiple roots have to be taken into account. For small values of T (i.e., T < 1) each succeeding eigenvalue is an order of magnitude smaller than its predecessor so that only a few values of C1 are required. Under these conditions it is justifiable to employ the product representation of C1, eq. (4.7), in the numerical calculations. Unlike the Lorentz spectrum, there are no explicit asymptotic approximations. Table 4.2 First nine eigenvalues of rectangular spectrum as a function of normalized time T (from Slepian and Sonnenblick [34J)
8
/
T= 1.273
T= 2.546
T=
0 1 2 3 4 5 6 7
8.8056 (—1) 3.5564 (—1) 3.5868 (—2) 1.1522 (—3) 1.8882 (—5)
9.9586 (—1) 9.1211 (—1) 5. 1905 (—1) 1.1021 (—1) 8.8279 (—3) 3.8129 (—4) 1.0951 (—5) —
9.9990 (—1) 9.9606 (—1) 9.401,7 (—1) 6.4679 (—1) 2.0735 (—1) 2.7387 (—2) 1.9550 (—3) 9.4849 (—5)
—
—
— — — —
3.819
T= 5.082 9.9999 (—1) 9.9988 (—1) 9.9701 (—1) 9.6055 (—1) 7.4790 (—1) 3.2028 (—1) 6.0784 (—2) 6.1263 (—3) 4.1825 (—4)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
254
4.6. Numerical solution of integral equation Aside from the Lorentz and rectangular spectra, analytical solution of eq. (3.63) are possible when the spectrum is the ratio of two polynomials, Youla [37], Slepian [38]. Unfortunately, this case is of academic interest since natural spectra are not of this form, although they can probably be closely approximated by the rotating ground glass method of Martienssen and Spiller [39].In general, however, numerical techniques must be employed in order to determine the eigenvalues. We are only interested in values of T such that the eigenvalues are distinct in order to satisfy the conditions leading to eq. (4.8). Fortunately this constraint means that T must be fairly small (T < 6). Under these conditions, probably the best approach is to replace the integral equation by an equivalent system of linear equations via double application of a quadrature formula, Kopal [401. The final result is b
N n~1
H~g(IT~ —r~I)Ø(T,,)— (Tm),
m
l,2,--~,N
(4.44)
where r,, are the quadrature points for the interval (0, T) and H,, are the corresponding weight factors. This system of equations is not symmetric even though the original integral equation was. Computation of the eigenvalues is more easily performed by transforming the above system into a symmetric one. Set h,, H,~2f(Tn),then eq. (4.44) becomes (G—bI)h=0 (4.45) where Gm n =
(4 4 ~. The numerical solution of eq. (3.45) is a standard procedure, Ralston and Wilf [411, and we need not concern ourselves with the actual details since they depend upon the computer employed, special programs available, etc. In the calculations of Barakat and Glauber [18], Blake and Barakat [42], Gauss quadrature was employed. The quadrature points r,, are related to the Gauss quadrature points X~N)over the standard interval (—1, +1) by the transformation
T,,
G
nm
= ‘H Hsi~\1/2 g~. ( Tm \ m
~ [1 +X,~’)]T,
—
T,,
(n = l,2,~--,N)
(4.47)
while the transformed weight factors H,, are given by H,,
~A~f’)T,
(n
l,2,~,N)
(4.48)
where A~N)are the standard Gauss weight factors. For the theory of Gauss quadrature see Stroud and Secrest [43]. It is here that the Gauss method displays its power, because if we had employed a standard equidistant quadrature formula then we would have had to contend with a matrix of order (2N + 1) to obtain the same accuracy as N point Gauss quadrature would supply. An alternative approach to the numerical evaluation of the eigenvalues of eq. (3.63) is via Fourier series expansions, Lachs [44]. As an example, consider the Gaussian spectrum due to Doppler broadening ___
I(w) =
I (w_wo)21 exp I— I L 2cr2 ~
(4.49)
R. Bare let and J. Blake, Theory ofphotoelectron counting statistics: an essay
255
where a (variance) is the characteristic width of the spectrum. Under the condition of quasi-monochromatic radiation, the correlation function is (4.50)
g(T) = e_T2/2
where T = at, V = w/a and a is the variance of the Gauss line. It appears to be a general rule that the rate of decrease of the eigenvalues of a spectral line is directly related to the rate of decrease of the corresponding correlation function. In the case of the Gauss line, the rate of decay of g(T) is intermediate between that of the Lorentz and rectangular correlation functions. A value of N 14 was sufficiently large to permit accurate evaluation of the first ten eigenvalues of the Gauss line for values of T up to six. A typical result abstracted from Barakat and Glauber is shown in fig. 4.4. The curves are qualitatively similar to the Lorentz and rectangular curves except that the Gaussian curves are somewhat broader in the medium time region. Physical phenomena giving rise to several spectral lines are also encountered. For example, in the electrophoresis experiments of Ware and Flygare [45] each component of distinct electrophoretic mobility produces a Lorentz line of different central frequency. Another example is the Brillouin scattering from media where the velocity of sound is very small such as in the work of Katyl and Ingard [46] on surface waves along an interface between two liquids, or the work of Clark and Liao on smectic liquid crystals [47]. To this end let us consider a prototype spectrum consisting of two Lorentz lines symmetrically
1
-
I
I
I
I
I
I 1
12~~
__
20
40
60
NUMBER OF PHOTOELECTRONS
NUMBER OF PHOTOELECTRONS
P(m, T) for a Brillouin spectrum, ~ = 0.4, u = 5: A) 0, T’ 0.5, B) ~ 4, T 0.5,C) ~ = 0, T’ 2, D) ~ 4,
Fig. 4.5. Fig. 4.4.P(m,T) foraGauss spectral line, u 7.1 for T= 0.5, 1, 2, 4 (Barakat and Glauber, 1966 unpublished).
=
T= 2
(Blake and Barakat, 1972 unpublished).
R. Barekat and J. Blake, Theory of photoelectron counting statistics: an essay
256
placed about a central Lorentz line of different height and halfwidth: +1
r
I. 2+y 2J /=—I ‘[(ww1) 1 The total energy in the beam is normalized to unity I(w)
J
>
(4.51)
R.I
J(w)dw =
/
~R 1
= 1.
(4.52)
Since R1 = R1, we define a parameter a = 2R1 which represents the fraction of the total energy contained in the Brillouin lines. Thus a = 0 describes a single (Rayleigh) line and a = 1 describes a pair of (Brillouin) lines in the absence of a Rayleigh line, a is generally known as the Landau—Placzek ratio. The actual Brillouin lines are spaced (approximately) symmetrically about the Rayleigh line so that — w0 = w0 w..1. Let us define a dimensionless quantity z~ (w1 w0)/y0. Similarly, we define “Y~I’ve = ~., /‘y~which gives the ratio of the width of the Brillouin line to the Rayleigh line. As before, we measure time in units of l/y~,and measure beam strength in units of 7~by setting w/”y0. A summary of these parameters is given in table 4.3. The correlation function of the Brillouin spectrum is —
—
g(T)exp{—iw0T/’y0}[(l
+ae_~T~cos~T].
_a)e~TI
(4.53)
We can drop the exponential term outside the brackets in accordance with our previous discussion so that the kernel of eq. (4.44) is the term in brackets. A typical plot of P(m, T) for such a Brillouin spectrum is shown in fig. 4.5. Note that as z~increases the distribution shifts so as to make P(0, T) larger. This is a reflection of the heterodyning between the Brillouin and Rayleigh lines which produces fluctuations in intensity on a times scale of the order of ~ These fluctuations in intensity will increase the “clumping” of the photons, making it more 1. likely to see none at all arrive during a time T, so long as T is of the order of Z~ Table 4.3 Summary of dimensionless parameters Brillouin spectrum Symbol
Definition 2R_,
Meaning
2R
=
1
=
1
—
R0
fraction of energy in
Brillouin lines (w,
—
w0)/’y0
=
v
w/-y,
ratio of frequency shift to Rayleigh half width ratio of Brillouin to Rayleigh half widths number of photocounts in one correlation time y~’
R. Barekat and J. Blake, Theory ofphotoelectron counting statistics: an essay
257
A
.77
I
I
10
I
75
I
a
4~”~~A
~
Fig. 4.6. P(0, T) for a Brillouin spectrum as a function of e for T 1, v land ~ = 1,2,3,4,6,10 (Blake and Barakat, 1972 unpublished).
Fig. 4.7. P(0, T) for a Brillouin spectrum as a function of ~ fore 0.7, v 1,8 = land A)T~’0.5, B) T~1, C) T= 2, D) T = 4 (Blake and Barakat, 1972 unpublished).
To examine this effect more closely we have plotted P(O, T) in fig. 4.6 as a function of a, and in fig. 4.7 as a function of ~. For small ~, the individual lines are not distinct and the “clumping” of the photoelectrons increases monotonically as a increases. For ~ > 3, the lines become distinct, and if we make a> 0.7 we decrease the beating, thereby reducing P(0, T). Similarly, increasing ~ increases P(0, T) until the counting time contains several fluctuations of intensity. Past this point where ~> T’, P(0, T) levels off and displays oscillating behaviour. 4. 7. Long-time asymptotics The expression for P(m, T) given in eq. (4.8) is valid only for times T for which the eigenvalues are distinct. For longer observation times, where the first N eigenvalues are confluent (i.e., b0 = b1 =
..
.
bN > bN -
P(m,T)
~
1
>
~ Barakat has shown that P(m, T) becomes
~
m
(l±b~)
(
bi’)j~-) dX1~h[~~X)m1
H(bI’ (4.54)
The tedious details are omitted. This expression can be employed for calculations although the final
R. Baralet and J. Blake, Theory of photoelectron counting statistics: an
258
essay
results hardly justify the elaborate computations. What we really want is an asymptotic approximation. In order to obtain a simple useful approximation to P(m, T) for T 1, we assume that only the N confluent eigenvalues b0, contribute substantially to W(~l).This is tantamount to letting the ~‘
singularities of eq. (4.5) reduce to just a single pole of order N, where N is large but as yet unspecified. When the residue is evaluated by contour integration in the complex A-plane we obtain
c (ba’e~dA + A)N
1 1 W(~)—— b~f 2iri =
(4.55)
J
&2N-1
e_n/!~o,
~
b~’F(N) 0.
(4.56)
~<0.
The resultant probability distribution is then easily shown to be P(m T)
(m+N) [‘(N)
F
(m + 1)
b~ m+N (1 + b0)
.
(4.57)
However, b 0 and N are really as yet unknown parameters, but they can be expressed in terms of observable quantities. In order to demonstrate this, we first note from eq. (4.23):
11
(~)~
b~= T
(4.58)
because uT is the average counting rate (m). Hence, ~
b1 ~ Nb0 ~ (m>
(4.59)
j=0
connecting
(4.60)
b
(4.61)
Consequently, b0 and N are expressed in terms of (m) and 12(T). When ‘2 (T) is written out explicitly it is 12
(fl
=
J
2(r’
g
—
r”)12 dt’ dt”.
(4.62)
R. Barak.at and J. Blake, Theory of photoelectron counting statistics: an essay
259
Fortunately we are only interested in large values of T for which we can prove that (T~l) I2(fl’~’6gT, where is a numerical constant dependent upon the spectrum under consideration. For the Lorentz spectrum, we have
1
T,
2T]
(T~ 1)
(4.63)
(4.64)
2(T)—~[2T— 1 + e_ so that = 1. We can also show that =
1,
=
~/W
rectangular spectrum Gauss spectrum.
Surprisingly enough 6g is unity for both the Lorentz and rectangular spectra. The line shape is still contained in the asymptotic expression through the constant As the time of observation becomes essentially infinite, p(m, T) must become independent of the actual shape of the spectrum and only depend on two parameters w and t rather than three parameters w, t, ‘y. In order to prove that eq. (4.54) passes over to the Poisson distribution, we note that a)m 0 ‘~‘~gbo
(4.65)
asT-~°°
16g b)N~ T
and rewrite eq. (4.57) in the form P(m fl~
I”(m + T/~Ti r (m) g m!F(T/~g)11(m) + T/i5g ________
m
r II
I
+
(m) yTf6g I (T/lSg)J
(466)
using these expressions. However,
r lim
(TI8g)
-+
lim
(TIög)~
(m)
Ii + I
1_(T/ög)
I
F(m+ T/ô ) r g
F(TI~g)
(4.67)
= e_
(TIiSg) .i
(m)
I _______I
1”
I.(m)+TI~gJ
(m)
(468)
so that lim (T/óg)~
P(m,T)
(m)m e~m~ m!
(4.69)
which is what we wanted to prove. Examination of figs. 4.1 .and 4.2 reveals that P(m , T) is still not in the Poisson regime even for an exceptionally long time of T = 30, since the curves are not symmetric about the mean value.
R. Bare/cat and J. Blake, Theory of photoelectron counting statistics: an essay
260
Equation (4.57) can be recast into the form P(m,T)~
F(m+N)
_______________
F(N)m!
[1 + (m)/N]N[1
(4.70)
.
+N/(m)]m
This expression was first proposed by Mandel [481,see also Mandel and Wolf [491,as an approximation to P(,n, T) in the semiclassical interpretation. He assumed that P(m, T) could be written in the form, eq. (4.70), and adjusted the parameter N (s in his notation) such that eq. (4.70) and the (unknown) exact distribution possess the same second moment. The present analysis demonstrates how this type of asymptotic formula arises naturally out of the Glauber formalism. Reference is made to Bedard, Chang and Mandel [50]for applications of eq. (4.70) to the calculation of factorial moments. Our illustrative examples are confined to the Lorentz and rectangular spectra for which we have exact results. In view of our discussion of the rate of decrease of the eigenvalues of the Lorentz spectrum in section 5 (especially with regard to table 4. 1), we should anticipate that eq. (4.57) will not yield a particularly good approximation for T < 10. The expectation is borne out by an examination of figs. 4.8 and 4.9 for T 6 and-u = 5,10 respectively. This approximation (solid triangles) underestimates the peak values. Glauber’s Bessel function approximation (solid circles) given by eq. (4.32) is certainly better. It suffers the disadvantage that the computation of the high order Bessel functions
4.0
I
I
I
I,o(
— 3.2-
£
-
I
2.C
— -
aa•
I
I
1,1
I
I
I
I
I
I
I
I
I
I -
—
a
I
0
•
-
-
a -
-
0
-
-
a
-
-
N -
6
-
-
-
—
1-1.0-
O~16-
-
a,
0.8’
0
0
-
-
-
a
0
liii
-
20
111111
40
60
BO
NUMBER OF PHOTOELECTRONS
Fig. 4.8. P(m, T) for a Lorentz line: u = 5, T= 6,
0
oil
a
I
40
III
Il-Ill
80
120
160
NUMBER OF PHOTOELECTRONS
Fig. 4.9.P(m,T) for a Lorentz line, v = 10, T 6,
R. Bare kat and J. Blake, Theory ofphotoelectron counting statistics: en essay
261
is just about as complicated as the exact solution. Glauber’s second approximation, eq. (4.38), is dependent on 1.1 as well as T. The fit is better at u = 10 than at u = 5 in accordance with the basic assumption V ~ 1. Calculations were also performed at T = 12 for v = 1,5,10,100. Glauber’s Bessel function approximation was practically indistinguishable from the exact solution, whereas the other two approximations still tended to undershoot and overshoot the peak values respectively. These long time calculations should be put into proper perspective. Such long normalized times can only be achieved by taking t large since ‘y is basically fixed. However it is difficult to stabilize a laser operating below threshold for much longer than t = 10”~sec. In fact the largest value of T of which we are aware is T = 1.32 in the pioneering experiments of Freed and Haus [31]; this value of T hardly puts it in the long-time region. We now pass to a consideration of the rectangular spectrum. Here the situation is decidedly more favorable for the application of the asymptotic formula in view of the fact that the leading eigenvalues rapidly cluster around unity as T is increased. The largest value of T for which the eigenvalues do not coalesce is T = 2.546 (see section 5 and table 4.2). We chose this value of T to check the exact probability distribution vs the asymptotic formula. Some typical results are summarized in fig. 4.10 for v = 5, (m) = 12.73. The solid line represents the exact solution, while the solid circles represent the long time asymptotic solution, eq. (4.57). There is still a tendency to underestimate slightly the maximum value. The dotted curve is for a Lorentz line (exact solution) with T = 2.546, v = 5 and is included for comparison. 4.8. Variance of photoelectron counts The function W(~2)is easily seen to be a probability density function for 0’( T < oo, thereby implying that m (t) is always a mixed Poisson stochastic process. This fact follows directly from the condition that Q(A) is a moment generating function. Furthermore we have ~ W(~2)d~2=~ C1=l.
(4.71)
I
I
I
I
CO:1~1I52~25 NUMBER OF PHOTOELECTRONS
Fig. 4.10. P(m, T) for a rectangular spectrum; v = 5, T = 2.546, (m> = 12.73. The solid line is the exact value, solid circles are from eq. (4.57). The dotted line is for a corresponding Lorentz line and is included for comparison (Barakat, 1967 unpublished).
R. Barakat and J. Blake, Theory ofphotoelectron counting stetistics: an essay
262
The variance of the distribution of photoelectron counts is easily derived in the context of a mixed Poisson process. If the eigenvalues are distinct, we can use eq. (3.25) to prove that (~2)
=2
~
m~C/.
(4.72)
j=O
In the short time approximation only the first term in the series is important, thus 2 — 2m~= 2(m)
(4.73)
so that var(m)~i(m)[1 +(m)].
(4.74)
Note that the line shape does not enter into expression in accordance with eq. (4. 1 8). In the long time region, we can show that (~TZ2)—m~N(N+l)(m)2
+v2ö 5T
(4.75)
It then follows that var (m) = (m) [1 + iSg(m)/T1.
(4.76)
When the spectrum is Lorentzian, then this expression is identical with the result obtained by Glauber using his Bessel function approximation. Regardless of the length of observation, it is simple to prove that var&2 -412(T).
(4.77)
The usefulness of this expression is that the right hand side can be evaluated in terms of the second iterated kernel without the necessity of having to know the mixing density W(~)directly. The behavior of W(~2)as a function of time is worth an explicit statement. In the short-time region, W(~2)is a negative exponential density which decays as time increases to a bell shaped curve. This bell shaped curve goes over into a chi-squared density as time is further increased. Finally as time is made to approach infinity W(~) approaches a Dirac delta function centered at &2 = (m), lim W(~Z)= ~[~2 — (m)].
(4.78)
4.9. Superposition of coherent and multiple mode chaotic radiation* The present subsection is devoted to a natural extension of the theory to cover a particularly interesting situation: the superposition of a coherent excitation and a chaotic one. In particular we consider the superposition of a coherent signal and a noise excitation in an infinite number of modes. Such a situation is a simple modeling of a laser operating just above threshold where the “signal” is to be identified as the laser line and the chaotic background is the “noise”. The chaotic background is taken to be either a Lorentz spectrum or somewhat more realistically, a Gauss spectrum. *This subsection is based upon an unpublished report by Barakat (1968).
R. Bare kat and J. Blake, Theory of photoelectron counting statistics: an essay
263
Although the analysis about to follow actually holds independent of the time of observation, we note that in practice a laser operating just above threshold is quite unstable. Consequently, our simple signal plus noise model is only physically realistic for short counting times. If the counting time is very short, then only one mode of the background effectively interacts with the coherent mode. However, as the counting time increases additional background modes come into play which makes the subsequent mathematics somewhat complicated. In view of these remarks, we can expect that counting times sufficiently large to include three or four chaotic modes will certainly cover the practical experimental situation. If the amplitude of the coherent signal is taken to q;, then the P function for the superposed radiation is (see eq. (2.37)) P[{ctk}]
11
1
21 Iak —6 ki q’1 k
exp 1~_
k lr(nk)
L
(4.79)
‘~~k>
Only the lth mode possesses a coherent component. The subscript 1 can take on any integer value, however the case of most interest is I = 0 (the coherent component lies in the center of the chaotic background line). The single-fold generating function corresponding to eq. (4.1) is now given by Q(A) = exp{—1q 2 } ~exp{—~~(1+!~3}exp{_(q,*~, +q 11 1~3,*}H~Lk
(4.80)
where (4.81)
2. qk ~q~/{(n~)}”
An explicit evaluation of this multiple integral can be accomplished by reducing the complex quadratic form in the exponential via a unitary transformation. The final result is:
Q(X)exp{—Iq,12}
+Xbk)’ exp
H {(l
[1l1I).
(4.82)
Unlike the completely chaotic situation, we are now forced to deal explicitly with the matrix U which diagonalizes A. The elements of U have the following significance; Ukl is the kth eigenvector component corresponding to the lth eigenvalue. Thus, the eigenvectors serve to determine where the signal mode is with respect to the chaotic modes. We recall that only the eigenvalues were of significance in the completely chaotic case. Note that
Iq,I2 If
IqI2/(n~) signal/local noise.
=
we set 5k
=
(mk)”, hkl
U~
(4.83)
2,then eq. (4.5) becomes 1q1I
W(fZ)
= (fi
k
5k) exp{—lq,l2}
2iri
~ J
jj (exp k~
[hk,sk/(X + ak)] ~ e~ dA. X+Sk
(4.84)
J
The singularities of the integrand are at A = ~5k’ but they are now essential singularities and not just simple poles as in the completely chaotic case.
R. Barak.at and J. Blake, Theory of photoelectron counting statistics: an essay
264
At this stage, it is probably more enlightening to consider only one spectral mode so that eq. (4.84) reads W1(~2) exp{—Iq,12 —&2s0}
J z’
exP{&z(z + h&s0)}d
(4.85)
where z = A + s0. The integral can be expressed in a series of modified Bessel functions of the first kind, we note [51] that 2 n/2 exp frz(z + ~ ~ In [2(h 2]. (4.86) ~2z
)}
~
°°
01s0~2)”
When this series is substituted into eq. (4.85) and termwise integrated, W 2
~2s
—
W1 (~)= s0 exp{—1q1j
1 (~2)becomes
2].
(4.87)
0} I~[2(h01s0~)”
The higher modes can be treated in similar fashion. For two spectral modes, eq. (4.84) becomes exp (—Iq,12) ~
=
[(A +s 0) (A +s1)]
27rl
c
1
e~ exp( hOlsO A+s0
) exp(A+s1 )dA. h11s1
(4.88)
We first evaluate the integrand at the singular point A = —s0 and then at the other singular point A = —s1. Upon setting A0 + s0 = z, we have
2&‘Ls
—
W2(~2)= exp [—1q11
0I
r
~
2in
=
z
(4.89)
C
1 exp {—&2 (z + h A(z)
A(z)B(z) dz
——
01 s0/&2z)}
~ B(z)(s1 —s0 —z)~ expi
h11s1
(4.90) 1
I.
(4.91)
—ZJ
L~i~
A(z) is the generating function for the modified Bessel functions, see eq. (4.86), while B(z) can be
expressed in terms of Laguerre polynomials B(z)exp[hh151] C10
exp[h1151} 10
where c10 ~s1
—
{(i _±)Jexp[hh151(_~_)(l C10
Cio
(._~_.)
~ m—o C10
do
~_)]} do
(4.92)
~ 10
~ and Lm (x) is the Laguerre polynomial defined by
e-”
d~
m e~I .
(4.93)
m! dxtm [x Lm(x)= 1F1 (—m,l,x)=—
Upon substituting the appropriate expansions of A and B into eq. (4.89), we can prove that the double series expansion reduces to a single series, namely (sosi) exp
{—
q 2 +h 11 11s1
—
~iso) ~(holso)n/2
ç—
2]L-~ [I~(2 h01s0&~)”
~
(4.94)
R. Barekat end J. Blake, Theory ofphotoelectron counting statistics: an essay
265
This is the contribution to W2 (~)from A = —s0. The contribution from A = —~i can be evaluated in exactly the same manner by letting A + s~= —z. The final expression for the two mode case is given by ~ W2(~Z)= ( __) C
exp [hlisi
c10
10j +~_—
2]
~2s~ — Iq,1
—
exp
(hO1SO
~
—&~~Si—
j
Iq,12]
mn[2(hol5o~)h/2]Ln
~
n=O
[hoiso
n/2
h11s1
[hizsil C10 I 1h01s01
n/2
____
\Isosl) ~0,
____
In [2(h11s1
n = 0(~~~1)
~)1/2]
Ln
(4.95) 5~
S
where CJç/ 1~ The extension to three or more modes is obvious; the final answer for the three mode case is 2] W + h21s2 — ~ Iq,12] j ( h01s0 ) n/2 I~,[2(h01s0~2)” 3(~2) ( 505152) exp [hizsi c10 c2~ ,~= ~ —
1L (hllsl)L i~(k+ /
—
n) ~
~
k0
—
_____
I
2152) C10
1=0
expi1h01s0 -I- h~1s2— ~
c01c21, +(sosls2
~
1q11 2]
(cOl)~”(c2l)~’Lk (_
k0/0
n)
~
~
(c
k0
/0
0~)~ (d12)’Lk
01s0
)L1
( h21s2 ) ~21
C01
“
+ s0s1s2 exp [ho’so + h11s1 —~s2—Iq,~ 2] (co2ci2) c02 c12 —
n/2
=
h
°°
~
°~(k+/
C21
h11s1 2] ~-(~) mn[2(hii5i~)”
~
n
~21
°°
(h
(c10)_k (c20~
(
j =
(h2:s2 ~
0
2]
n/2
‘n
holso)L(hllsl) d02
[2(h21s2~)~ (4.96)
C12
Having determined W(f~)the evaluation of P(m,T) follows directly. When W(&2) is substituted into eq. (4.5), we encounter the function 2] Rmn
=
0
e_(1~)n ~m-n/2
j
_____
n!
(hs~~/2e’~1~1 +s) (1 +S)m+
d~2,
m,n~0
[2(hsIZ)” F (n — m, n +
hs\
1, —
1
where 1F1 is the confluent hypergeometric function.
j—~-—)
(4.97)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: en essay
266
It is then a simple matter to prove that the final result for one mode is d~2
112]
P1(m,T)
~m!
~
~—Iq~I~ 0
~
m
I~ [2(h01s0~)
1h expt 01s0 +~
1
s0(l +s0)~ =
s0 (1
m +
—
Iq112) 1F1 ~—m,l,
exp1 h 2 I 01s0 + s~— Iq11
1
s0)
h1~1s~ 1 +s01 ‘~
0
}
)
Lm (~0150
1 +s
(4.98)
0
where L~ is the Laguerre polynomial. Before commenting on this expression, we write out the explicit formulae forP2(m,T)andP3(m, T). The two mode expression is P2(m,T)
—_Jexp
=
(~ ~
+
(sosl\
+ h01s0 (1 + 1 +S0 ~
C10
j
)~rn — i
(ho5o)~L0 ~ o
n
C10
C10
d10i
X ~
(n
—
m, n + 1
h01s0
1 +S0
x
~0
L~
~Ol (~)n
n=
(
C Ibis0)
2
( sos1 s2 )exp {
q,1
\
+
C10 C20
• ~
(h0~s0)fl 1F1 (n
—
1F1 (n
• ~
hOsO
+/
—
n)
_
—
~
~
k0
j0
(c1~)~ (c20)’
(h11s1)~iFi(n_m,n+
l,—’~”~’~(k+j—n) ~ I +s1/ k0
(h21s2)~1F1 (n
) exp {
—
m, n + 1
—
h21s2 1+52
XLk(~.--~-0_i_50)LJ j ~—
____
C12)
~
/121521~(l C21 ,1
+s1) —rn—i
(Coi)~(C21)~’
j0
2 + hois —
Iq,1
0 + h11s1 + h21s2 1(1 + c02 C12 1
C21 I
n=0
1+51
C01
X Lk (_~~i5o)L.(_~~2152~+(sosis2
• ~
) ~(k
l+s~
/
n =0
\
1
(4.99)
01s0 + h us1 + 1121s21 i(l +s0) 1 +s0 C10 C20 I
m, n + i
C20
-
m, ~ + 1 h1isi) 1 +s~
(_~.L~!’) +( sosls2) exp{ —jq 12 + h01s0 + h11s1 +
h1is1~L C10 F /
Co1
—
) (1 + s1Ym
h
-
-
C0s0 + h11s1 01 1 + S1
\C01!
n0
X Lk (_
exp{_ 1q11
01
Tl1e three mode expression is P3(m,T)
2 +h
~
+
)~ (k + /
—
n) ~
~
52)_rn_i
(c02)_k (c12Y’
k=0j0
(4.100)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: en essay
267
We again remark that higher modes can be explicitly written out by induction. We return to the one mode case as expressed in eq. (4.98). Glauber [33] and Lachs [52] have already obtained a special case of this result by different procedures. In Glauber’s notation r Si r S 1 Jytm P(m T) = — exp — IL I— I (4.101) (1 +N)m+l 1 1 +NJ m ~ N(l +N)J where S represents the mean number of quanta which would be counted in the coherent field and N the mean number of quanta in the noise field. Since eq. (4.101) is only realistic for T 1, we then note -~
s
2 uT;
0 ~ (uT)’, so that S
(T~
(4.102)
1)
h01 ~ lU01q1I
IU
2uT, NvT; (T<<1). 0~q,I When the signal lies at the center of the chaotic spectrum, then 1 =
(4.103) 0
and
U 00~1,
U01~0
(1>0).
(4.104)
Thus although eqs. (4.98) and (4.101) have the same functional form, the latter is less general because it was derived under the assumption that both signal and noise are in the same mode. No such assumption was made for eq. (4.99). In fact, if the signal is not in the central mode (1 0), then eq. (4.99) reduces to the simple power law distribution m/(l +VT)m+l. (4.105) P(m,T)~(uT) Longer counting times are required to detect the signal when it is not in the center of the background line. A typical numerical result is shown in fig. 4.11; here the beam strength u and the normalized time T are fixed and the signal-to-noise ratio 1q 0 12 varied. The calculation was carried out using two chaotic modes. In fact, P(m,T) is very close to being a Poisson distribution for Iq0I~= 9, even though T is still very small. When the noise field is completely absent, the generating function for the single mode coherent field is Q(A) = e_(m~
(4.106)
which leads to a Poisson distribution (m)m e~m>
4 107
m! ) where (m> = S. Thus the Poisson distribution for the photoelectron counts arises from two different conditions: from the limit of very long counting for a multimode chaotic field (eq. (4.69)); from a single mode coherent field plus a multimode chaotic field in the limit of large signal-to-noise. P(m,T)
.
4.10. P(m, T) for partially polarized chaotic light
Thus far the analysis has been confined to chaotic light that is completely polarized. An extension of the analysis to include partially polarized chaotic light is of some interest.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an
268
I
I
I
I
I
B
3.0
essay
I
-
C I-
E
D 1.5-
C
0
-
I
I
I
Fig. 4.11. P(m, T) for a coherent line in center of a Gauss spectrum for v 1q 2 = 3, C) 1q 2 = 5, D) Iq 2 = 9, (Barakat, 1968 unpublished). 01
0(
I
20 40 60 NUMBER OF PHOTOELECTRONS =
80
2
10, T = 0.52 as a function of 1q
0
2:
=
0.5, B)
A) Iq,1
0I
In order to develop the theory, we need some background information on partially polarized Gaussian light. The standard reference is Born and Wolf [53]. The coherency matrix for plane waves is [‘~
(r)
F 12(r) (4.108)
F21
(r)
F22(r)
where F~1(r)= (~7(t)~1 (t + r))/(~7(t)~/(t))
(4.109)
and 1.5~(t)is the ith component of the field at time t. We further restrict the analysis by letting t Note that tr f(0)
= total
average intensity
(I).
=
0.
(4. 110)
J is also hermitian (I = J +) which means that it possesses real eigenvalues and can be diagonalized. The eigenvalues of J(0) are A2 trJ+~[(tr)2 ~_4detJ]~2
(4.111)
A2~~trJ_~[(trJ)2 _4detJ]1~’2. The two limiting cases of polarized light and unpolarized light are defined in terms of the eigenvalues. Light is completely unpolarized when A1 = A2. This means that the term in brackets is zero,
R. Barekat and J. Blake, Theory ofphotoelectmn counting statistics: an essay
orF11
= F22, F12
F21
269
0
10
(4.112) 01 Light is completely polarized when A1 * 0, A2 0; or when det J = 0. The intermediate situation can be handled via a linear superposition of these limiting cases. We write (I~)0 B D + =J~+J~, (4.113) 0 (I) D* C 2 = 0). The average intensity of the unpolarized u where of~u =is~‘u~ zerowhile (i.e., BC 1D1 intensity of the polarized part of the radiation part ofthe the determinant radiation is tr the — average is trJ~= (In). The degree of polarization, ~is defined as trf p trJ~+ trJ~
=
r
4 detfl
I
(trJ)2 J
I
1/2
.
(4.114)
Both det J and tr J are invariants with respect to unitary transformations so that ~ is independent of the orientation of the coordinate system. Furthermore ~ is bounded (4.115)
~
with ~ = 0 corresponding to unpolarized light and £3D be expressed in terms of
= 1
to polarized light. The eigenvalues of J can
A 1 =~(l
X2
+~)(I) (4.116)
~(l —~)(I).
Given this background information, our basic integral equation, eq. (3.63), becomes the coupled set
~ F~1(t
—
t’)Ø1 (t’) dt’ =
cb1(t)
(4.117)
with i = 1,2. If we confine ourselves to the short counting time approximation, this reduces to the algebraic equations T>~
i
F11(0)Ø1b~1
1,2.
(4.118)
The eigenvalues of this set of equations are b1 ~
(1
+.9)VT,
b2
~(1
—~)vT
(4.119)
upon employing eq. (4.116) and (I> = (m> = VT. Note that b1 + b2 = vT in accordance with eq. (4.23). Thus the mean number of photoelectrons is independent of the degree of polarization for T ~ 1.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
270
The single-fold generating function is (4.120) Q(A) = (1 + Ab1)1 (I + Ab2)1. If 0< ?< I, then b 1 ~‘b2 and we can employ eq. (4.6) to obtain the mixing density function: 2n/(i 1?)VT e_2u/(~_.,)UT] (4.121) W(~)= (~VTY~[e_ The probability of obtaining m photoelectrons in time T follows directly from eq. (3.18) and is —
1 ir (1 +~?/)VT 1P~+1 1 (1 —.~?)vT lrn+lI I —I I } .~VT~L2 + (1 +?)vTJ 12 + (I —,9)vTJ I
(4.122)
P(rn,T)—fl
in agreement with Mandel [54]. The dependence of P(m, T) on ~?is quite small as witness table 4.4. The variance of m(t) is easily obtained via eq. (4.6). We have (~2) =
~2
W(~)d~= (3
rn2,
+~2)
T ~ 1.
(4.123)
It follows that var(m)(m)[1
+~(l +~2)(m)],
T~l.
(4.124)
The variance of m(t) is a maximum for polarized light and a minimum for unpolarized light. The variance is a measure of the tendency for photons to clump together, which is why P(0, T), and P(m, T) form ~ (rn), are largest for~ 1, see table 4.4. The corresponding results for unpolarized light are obtainable in analogous fashion. The eigenvalues b 1 and b2 are now equal, so that the generating function Q(A) is 2. (4.125) Q(A)=(l +AVT/2~ We leave to the reader the task of proving that W(~) P(rn,T)
-~--
(vT)2
(4.126)
e2~~/~T
4(m + l)(vT)m
(2+VT)m+2
(4.127)
.
P(m, T) as a
Table 4.4 function of~in short counting time approximation: u =
5,
T 0.2 m
~
0
.444 (0) .296 (0) .148 (0) .658 (—1) .274 (—2) .110 (—1) .427 (—2) .163 (—2) .609 (—3) .225 (—3)
1 2 3 4 5 6 7 8 9
0
~
0.3
.449 (0) .293 (0) .146 (0) .652 (—1) .277 (—1) .114 (—1) .465 (—2) .187 (—2) .744 (—3) .295 (—3)
~
0.5
.457 (0) .287 (0) .141 (0) .643 (—1) .283 (—1) .123 (—1) .529 (—2) .227 (—2) .975 (—3) .418 (—3)
~
0.8
.479 (0) .270 (0) .132 (0) .628 (—1) .298 (—1) .141 (—1) .669 (—2) .317 (—2) .150 (—2) .711 (—3)
~
1.0
.500 (0) .250 (0) .125 (0) .500 (—1) .250 (—1) .125 (—1) .500 (—2) .250 (—2) .125 (—2) .500 (—3)
R. Barekat and J. Blake, Theory ofphotoelectron counting statistics: an essay
271
An interesting problem would be the extension of the analysis to cover longer counting times, where one would have to deal with eq. (4.116) without the simplification afforded by eq. (4.117). 4.11. P(m, T) for statistically independent fields If the incident field consists of two statistically independent fields, then the total field correlation function is the sum of the correlation functionsg1(t), g2(t) of the individual fields. Under these circumstances, the generating function of the total field is Q(X) = Qi (X)Q2(X).
(4.128)
The single-fold probability distribution can be obtained in the usual manner via eqs. (4.5) and (3.18). However, for this special case, it can also be expressed as the convolution of the individual probability distributions P1 (m, T) and P2 (m, T) P(m,T)
=
P1 (k, T)P2 (m
k0
—
k, T).
(4.129)
This expression is easily evaluated on a computer, provided P1 and P2 are in reasonably simple form. It suffers the disadvantage of any discrete convolution expression, that in order to evaluate P(m, T) for a fixed m, all integers less than m must be considered. Lachs [44] has utilized eq. (4.129) to evaluate P(m, T) for an incident field consisting of the superposition of a coherent field and a chaotic field having a Lorentz spectrum. Obviously this is not the same physical situation as discussed in subsection 4.9. Appendix 4.A: Derivation of Glauber’s second asymptotic approximation We now sketch a derivation of eq. (4.38) given by Glauber and Barakat. A somewhat more elaborate version was subsequently published independently by Lax [14]. We can write tm —1 ç Q(A)dA
a
(_l)m
____
Q(A)
—
ax
2in
X1
m+l
(4.A.l)
c (1 —A)
by virtue of Cauchy’s theorem on derivatives. Here A is a complex variable and c is a contour in the complex A-plane which encloses the singularities of the integrand. We next substitute into this expression, the approximate generating function, eq. (4.35), and change the variable of integration from A to 13(A) 13. The final result is P(m,T)—
_42V~m ‘
/
27ri
eT
( J (1
1~e~’~
dA t’ . +2v_132)m+l
Assuming that the beam strength v
~‘
1,
allows us to approximate the second term of the identity
) l+2v A2
(1
+2v_13)_m_l
=(1
(4.A.2)
~‘
+2v)_m_1(l
—
~
—in—i (4.A.3)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
272
by an exponential (i
—
\
p2
~—m—1
~‘
)
l+2v-
(4.A.4)
e(m + 1)132/(1 + 2v)
Hence eq. (4.A.2) simplifies to —4(2v)m P(m,T)
(I + 2v)m
~‘
(m + 13 exp [(I
2~i
—
1)132
13)T + (1 + 2v) ]d13.
(4.A.5)
The saddle point method [86] is appropriate since the exponential term predominates in the integrand. If we call —f3)T+(m+ 1)132/(1 +2v)
h(j3)(l
(4.A.6)
then the saddle points are located at the zeros of the derivatives of h with 13. In our case there is only one zero and it is located at 13 = 13~ where (1 + 2v)T 130 =
2(m+ 1)
(4.A.7)
.
Since we are already in the long time region (T ~ 1), we can expect that the peak value of P(m, T) versus m for a fixed T will be in the vicinity of m = (m) = vT as witness the curves in figs. 4.1 and 4.2. It follows that 1 F—(vT)2
h(j3 0)
—
I
2vi
1 + 2(vT)J .
rn
(4.A.8)
Following the usual recipe for saddle point integration, we have m13 13o) 1 jm + l\ P(m T) _______________ exp —I I ~2 ds 4(2v) 0 ehz( (1 + 2vY”~ 2ir ~ I \l + 2v1 I ~‘
‘~
—
2v)312 T (2v)m (2~)1/2 (m + 1)3/2 (1 + 2v)m+ 1 The power law distribution can be further approximated, since v is a standard procedure in statistics), thus e”~13°~ (1 +
(2v)m
1
(l+2v)m+l
(l+2v)
~‘
—~
{
. )
1, by a negative exponential (this
2v. Consequently
(vT) 2 exp{ (2iw)”2 (m + l)~”
(2irv)”2rn3”2 exp
.
(4.A.l0)
1
—
One final approximation (!), we let m + 1 large (m) anyway. Finally P(m,T)
(
e_m/2V.
We may as well also write (I + 2v) P(m,T)
4A9
(vT)2 2v rn —
____
—
11
2(vT) + mIs.
(4.A.l I)
i-I
m since the formula is only good in the vicinity of
l(vTv~Th)2J
(4.A.l2)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: en essay
273
which is Glauber’s second approximation, eq. (4.38). Lax [14] has also derived this expression by a similar method, but he obtains a further multiplicative factor on the right hand 2 [i
+~-(~+~-)]. 2 m vT
(4.A.13)
This factor is only important when m ~ vT, but the correction is very small.
5. Two-fold statistics for Gaussian light Although the determination of the one-fold statistics is interesting, nevertheless the limited information that is obtained is insufficient for many purposes. Consequently it is necessary to develop and exploit the formalism appropriate for two-fold statistics. Of particular importance is the photoelectron correlation function and its variant, the clipped photoelectron correlation function. This section is devoted to the analysis of these functions and their physical consequence. 5.1. Correlation and clipped correlation functions for photoelectrons The measurable quantity available in a photoelectron counting experiment which (in principle) is most simply related to the spectrum of the incident light is the photoelectron auto-correlation function. If rn (t) is the number of photoelectrons counted during a time interval of length T centered at time t, we may define the joint counting rate at times separated by a delay time r as C(r)=(m1(t)m2(t + r)).
(5.1)
It is customary to normalize this correlation function to be unity at infinity, and so we define the photoelectron correlation function by
(m(t) m(t + r)) 2 (5.2) c(r) = C(oo) = (m(t)) We shall use capital letters C (or G, in the case of continuous random functions) to denote joint moments, and lower case letters c (and g) to denote normalized joint moments or correlation functions. When the “window”, T, becomes much shorter than the time over which the field describing a chaotic light source fluctuates, we will see shortly that lim c(r) ~ 1 + Ig(r)12 (5.3) —
C(r)
T-’ 0
where g(r) is the normalized field autocorrelation function. Unfortunately it is frequently quite difficult to measure c(r) directly in a photoelectron counting statistics (PCS) experiment, as this measurement requires either storing the values recorded for m(t) in a vast array, or performing real time multiplications on a time scale of the order of 10—6 to 10-8 seconds. One may avoid this problem by applying the technique of “clipping” — that is by replacing a signal, before autocorrelation, by a series of ones or zeros depending upon whether the signal lies above or below a predetermined “clipping level”. From the auto-correlation function of
R. Barakat and .1. Blake, Theory of photoelectron counting statistics: an essay
274
this “clipped” signal, one can extract information about the full correlation function. This technique was originally developed to analyze the reflected microwave fields encountered in radar applications, the application to PCS experiments is due to Jakeman and Pike [55]. As the technique of clipping is rather specialized, it is useful to devote several paragraphs to a discussion of the general formulation of this problem. Given random processes X and Y which take on the values x and y at a particular instant of time, we define the unnormalized correlation function as G(x,y)~(XY)
p(x,y)xydxdy.
(5.4)
To define the clipped correlation function we replace X and Y by the f(x) and g(y) defined through f(x) g(y)
1,
ifx>~b
0,
ifx
1,
ify~b’
0,
ify
(5.5) (5.6)
where the clipping levels b, b’ are specified. The correlation function of the processes f and g is called the clipped correlation function of the processes X and Y: Gbb (x,y)—G(f,g)-~
J
dxJ
dyp(x.y).
(5.7)
The exact nature of the statistics describing the processes X and Y determines the way Gb,,’ (x,y) is related to G(x,y). For example, if X and Y are each Gaussian random processes with zero mean, Van Vleck and Middleton [56]have shown that 2 G00 (x,y)—arcsinG(x,y).
(5.8)
Generally the relation between the clipped and full correlation functions is considerably more complex. The difficulty of obtaining G(x,y) from G,,,,’ (x,y) is considerably reduced when clipping is applied to only one of the processes, X. This is commonly called single channel clipping. Let us define a singly clipped correlation function, Gb (x,y) as
J
G~(x,y)~G(f,y)-~ dxJ
dyyp(x,y).
(5.9)
In order to evaluate G~(x,y)experimentally we must perform the operation of multiplyingy by a zero or a one, depending upon whether x lies above or below the clipping level, b. This can be done by a simple gate circuit; Gb(x,y) is scarcely more difficult to evaluate than G,,,, (x,y). To apply these general ideas to the problem of evaluating spectra from PCS experiments, we replace the continuous random processes X and Y by the discrete random processes m(t) and
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
275
m(t + r). We define the (unnormalized) clipped correlation function for photocounts as
(5.10)
(mk(t) m(t + r))
Ck(r)
where, in analogy to eq. (5.5): mk(t)= 1 =
ifm(t)>k
0
(5.11)
ifm(t)
Accordingly we see that ~ m,
k
m20
a
ax2
m2P(m1,t;m2,t+r) (_l)mI
a
m1!
axr~
m,k
Q(X1X2)
x1=i~
(5.12)
We will now examine several methods for evaluating Q (A1 , A2), and then show by example how C(r) and Ck(r) can be determined. 5.2. Solution for short counting times Equation (3.62) is the integral equation whose eigenvalues yield the double generating function. Because of the two domains of definition, (0,T) and (r, r + T), it is not of the standard symmetric Fredholm type. It is only when r = 0 that the two domains coalesce and we retrieve the standard type again. We now outline a numerical technique for determining the eigenvalues of the “gap” integral equation due to Blake and Barakat [57].It is probably easier to understand the rationale behind their method by first considering the short counting time approximation where T is small compared to the time over which g(r) decays. Under this condition, eq. (3.62) can be approximated by A1Tg(t)~1(0)+ A2Tg(t
—
1~! ~ 1(t)
r)çb1(t).~-
(5.13)
with / = 1,2. This equation must be satisfied at t that X1T A2Tg(r) q~(0) ~(0)
= 0,
r simultaneously. These two conditions require
(5.14) A1 Tg(r)
A2 T
çb(r)
0(r)
where we have utilized the fact that g(r) = g(—r) and g(0) = 1. To compute the generating function we need not evaluate the m1 individually, but consider instead their product in the form (1 + m1) X (1 + m2). To form this product, we rewrite eq. (5.14) in the form 1
+X1vT
A2vTg(r)
0(0)
0(0) =
A1vTg(r)
1 +A2vT
0(r)
(1 +
m1)
.
0(r)
(5.15)
R. Barakat and .1. Blake, Theory of photoelectron counting statistics: an essay
276
Clearly the matrix has eigenvalues (1 + m1), (1 + m2), and as the determinant of a matrix is the product of the eigenvalues we have 1 +A1vT
A2vTg(r)
A1vTg(r)
1 +X2vT
Q(A1,A2)~=det
(5.16)
.
Upon evaluating the determinant we have 2 (1— lg(r)12)]’. (5.17) Q(A1,A2)= [1+(A1 +A2)vT+X1X2(vT) This expression was first obtained by Bedard [58] and Arecchi, Berné and Sona [591. We have extended the analysis to include the case of partially polarized light, see appendix 5.A. To evaluate the photoelectron correlation function defined by eq. (5.1), we differentiate the generating function as follows (rn
2 [1 + g(r)~2)]~1. 1 (t)rn2(t + r))
Q(A1,A2)~A,
~
=
(5.18)
(vT)
=A,
Thus the normalized photoelectron correlation function is c(r)
1 + Ig(r)12.
(5.19)
The clipped photoelectron correlation function can also be evaluated in closed form in the short counting time approximation (see next subsection for details): Ck (r) =
a — ~
n
k
(—l)~ a~ n! ~ Q(A
k
(VT)k+~
1, A2)
A,
=
1~X2
=0
=
{~+ I + vT Ig(r)12].
(1 + vT)k
(5.20)
Finally the joint probability P(m1, t; m2, t + r) can be obtained from the generating function via lm+m
P(rn1,t;m2,t+r)
(— )
~I
am 2
m1 .rn2 .
+01
rn
oX1
Q(X1,A2)~~
2 I
2
(5.21)
=1~
=~
1
2
Arrechi, Berné and Sona [59] show that P(m1 t;m2 t + r) where A 1 B~ 1
+ +
C—A(1
=
m’ + 022 (mi + m2)! (vTB) 2F1 (—rn1, —rn2, —m1 m1!m2! Am1+tm2~
—
m2, C)
(5.22)
2 (I —girl2) 2vT+ (vT) vT(1 Ig(r)12) —
—
g(r)12)/B
(5.23)
and 2F1 is the hypergeometric function. Note that the probability of obtaining no photoelectrons is 2 (1 Jg(r)12)]’ (5.24) P(0,t;0,t+ r) Q(l,l) [1 + 2vT+ (vT) so that a knowledge of P(0,t; 0, t + r) should enable one to obtain lg(r)12. Reference is made to Furcinitti et al. [60] for details of the experimental results. —
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
277
In the short counting time approximation the single-fold probabilities and moments contain no spectral information. This is not true of second and higher order quantities, their behavior as a function of the delay times is strongly influenced by the field correlation function g(r). 5.3. Solutions for arbitrary counting times Short counting times, as one can surmise, are the ne ultra plus of the experimentalist. However, the short counting time approximation cannot always be realized experimentally. For this reason, we now consider the extension of the analysis to cover finite counting times [57]. By employing N-point quadrature to improve the approximation to the integrals in eq. (3.62), one can construct a generalization, valid for arbitrarily long counting times. The final equation can be expressed in a form identical to eq. (5.16) except that each element of the 2 X 2 matrix is itself an N X N matrix. Applying N-point quadrature to eq. (3.62) yields N
A1 ~
T T Hjg(t ——x1)0(—-x1) + A2
N
T T 2 —-~-x1— r)0(-~-x1+ ~)_7 m1çb(t),
(5.25)
where we have transformed the variable limits of integration to the standard interval (—1, +1). The H1 are the weight factors corresponding to the quadrature points x1. A particular scheme, Gauss quadrature, is preferable for two reasons: first, N-point Gauss quadrature is exact for any polynomial of degree (2N — 1) or less; and second, because the quadrature points are not equally spaced, we are less likely to encounter difficulty if the integrand oscillates regularly. We require that eq. (5.25) be satisfied at the 2N values oft given by ~
(5.26)
This requirement yields a system of 2N homogeneous equations for the 2N unknowns 0(t1), which has nonzero solutions only for the eigenvalues m1. This system of equations can be cast into matrix form by defining the N X N matrix BN (r): h11g(r) h12g{~T(x2—x1)+r} h1Ng{~T(x~ —x1)+r} ...
BN(r)
=
h21g{~T(x1 —x2) + r}
hNl
g{~T(x1
—
xN) + r}
h22g(r)
.
.
.
h~2g{~T(x2— XN) + r} ...
h,~~,g{~-T(x~ —x2) + r} h~~g(r) (5.27)
where hq
2, and the column matrix ~N(r) is given by (H,H1)” 0(~Tx 1+r)
4~N(r)= 0(~Tx2+r) 0(~TxN+r)
.
(5.28)
R. Barekat and J. Blake, Theory of photoelectron counting statistics: an essay
278
Consequently the requirement that eq. (5.27) holds for the succinctly written as ‘N +~X 1vTBN(0)
~X2vTBN(—r)
2N
values of t given by eq. (5.26) can be
~N(0)
(5.29) ~A1 vTBN(r)
+~A2vTBN(0)
‘N
~N(T)
where ‘N is the unit matrix of order N. However, the determinant of the coefficient matrix is simply the reciprocal of the double generating function Q(A1X2): Q(A1A2)’ H (I +mk) k
2X =
det
{‘N
+ ~vT(A1 + A2)BN(0) + (~vT)
1X2 EBN(0)BN(0)
—
BN(r)BN(—r)]}.
(5.30) It can be shown that eq. (5.30) can be differentiated to provide an explicit expression for the photoelectron correlation function
2[l +~trBN(r)BN(—r)] (rn1m2)
(5.31)
(vT)
Q(X1,A2)
ax1ax2
where m 1 m(t), m2 m(t + r). See appendix details. Upon normalizing the photoelectron 2 = (vT)2,5.B wefor have correlation function by dividing by (rn) c(r) = (m 2 = 1 + ~ tr BN(r)BN (—r). (5.32) 1m2 )/(m) c(r) can be cast into integral form by examining the limit as N approaches infinity. Since
N~=
hm
trBN(r)BN(—r)=
~
H
2 1H,~~g[~(x~ —x1)T+r]}
T/2 =
T/2
f
f
—j
—T/2
[g(t’
—
t”
+ r)] 2 dt’ dt”
(5.33)
—T/2
we have c(r)
=
I +
~1
f
T/2
T/2
f
[g(t’
—
t” +
r)J 2 dt’dt”.
(5.34)
—T/2 —T/2
We can recover the short counting time approximation, eq. (5.19), by letting T approach zero. Blake and Barakat [57] have shown that the full photoelectron correlation function for a Lorentz spectrum is c(r)
=
=
1 + e2T(5i~T) 1 + T2[.~
,
e_2(T+T)
0 ~ T~r +~e2(TT)
by direct integration of eq. (5.34).
_~
e_2T
+ T—r],
0~
(5.35)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
279
To obtain the clipped photoelectron correlation function Ck (r) we employ eq. (5.12). For N = we recall that the generating function is given by eq. (5.17). In this case the differentiation and subsequent summations can be carried out directly by using the substitutions 1 +vTX2 2[l —g2(r)]X V—vT+(vT) 2.
1,
U
With Q now expressible as (U +
(5.36)
A1 V)~,we
have tm’
am, ~ Q(A
(V)
mi (U+ V)mt+~ 1,X2)~A, =1 m1!(—l) and the sum over rn 1 can be evaluated easily:
am,
(~~l~j
m1
m,k
1
ax1~
(5.37)
k
V
x,=iU’~U+V!
Upon performing the remaining differentiation, we recover
(
Ck(r)
VT)
VT(l + 1 +vT g2(r))
(5.20)
This expression was first obtained by Jakeman [20]. In order to obtain an expression for Ck(r) valid for longer counting times, we use the generating function given in eq. (5.30) with N = 2. In this case we can show that 2 —X2 —W2) (s 1+s1+s2 +s1s2(l + Y 1 +s2)Y+s1s2(2Y—XU—XW) = det 2 —X2 —U2) (s1 +s2)Y+s1s2(2Y—XU--XW) l+s1 +s2+s1s2(l+Y (5.39) where s 1
(~vT)X1and
Xg(r),
Ug(r.+L~)
Yg(i~),
Wg(r—L~)
with ~
~
(x2
—
x1 )T. We can evaluate the determinant explicitly with the result that
Q[L+Ms2+O(s~)]’
(5.41)
where (5.42)
2)s~
L = 1 + 2s1 + (1 — Y M = 2+ s 2 — U2 1 (4 — 2X Thus
01
—
W2) + s~(2—
2Y2
—
2X2
—
U2
—
W2 + 2XYU + 2XYW).
(5.43)
vTM (5.44)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
280
We must now evaluate the sum over m1: Ck(r)=
ivT\
0021
(_l)mi
°°
—
—(—I ‘ 2 ‘ m,—k -
iM\
2/ —I—I
x,=i
(5.45)
.
m1! OAT’ \L Unfortunately we have not found any way to carry out this summation in closed form. For small k we can use a trick due to Jakeman [20]. Since (—l)°’’ am, a ~
—
-
00
00
m
m~—0
1!
~Q(A1X2)~ ~2=o
~
=
= m,—O
~
022—0
m2P(m1,m2)
(m1),
(5.46)
we can rewrite the clipped correlation function as (_l)m,
Ck(r)=_~
=
vT
k—i
~~Q(A1A2)~
mi!
(rn1) +
a
0””
(_l)m,
k—i
—
______
0””
(_l)m, am, mi!
/
— 3’\I---
a
~
M~
L2/I —
x,=i
2 m, 0 m1! 0X7 Equation (5.46) is easy to apply for small k; in fact
(5.47)
.
=
C 1
(r)
=
/vT\/ M (m1 ) + ~—J~——2 \ 2 1 \ L vT2
C
2(r)=Ci(r)_(_~-) C 3 (r) C
=
4(r)C3(r)
~
2ML’
~
1 vT ~
C2 (r) + —(-—) 1 vT
M’
V
-
)
2 4 4M’L’ + 2ML” 6ML’ (~} + V L 6L”M’ 6M”L’ 18M’L’2 + 18MLL” M”
—
‘~
(
24L’M
—
L~
+
—
V
)
(5.48)
where the prime denotes differentiation with respect to s and all expressions are to be evaluated at s = ~vT. Higher order Ck (r) can be calculated via the recurrence relation (1)k_i
Ck(r)Ckl(r)+
(k— 1)!
0k—1 1 ~Q(A ~
0A~
1A2)
~
k> 1.
(5.49)
A2 = 0 It
is instructive to apply our numerical techniques to the case of a Brillouin spectrum. Reference is
again made to table 4.3 for definitions of the parameters c~,~, and ‘y which specify the Landau—
Placzek ratio, frequency shift, and halfwidths of the spectrum. A superposition of spectral lines, such as the Brillouin spectrum, will display hetrodyning and thus cause c(r) to oscillate on a dimensionless time scale of l/~.c(r) is plotted as a function of r for counting times T = 0.1, 0.2 in figs. 5.1, 5.2. We selected these counting times because T = 0. 1 is just long enough to show departure from the
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
281
0.400306091.2
__
INTERVAL TIME
Cr)
Fig. 5.1. Photoelectron correlation function c(r) for a Brilouin spectrum(~= 10,6 = 1)withT0.1, v 1: —••—,a 0; —----,a=0.4;—,a=0.6;—.—,a’~ 1.0 (Blake and Barakat [571).
INTERVAL TIME (r) Fig. 5.2. Photoelectron correlation function c(r) for a Brillouin s~~trum(~ = 10,6 = 1)withTo.2, v 1: _.._,n 0; ,n’0.4;—,n=0.6;—.—,o= 1.0 (Blake and Barakat [57]).
short counting time approximation and T = 0.2 is just long enough to cause averaging over major details of the spectrum. For T ~ l/z~,c(r) tends to a Gaussian-like shape and the oscillations vanish entirely. In making these calculations, we employed six quadrature points (N = 2 in the notation of subsection 5.3). For counting times as short as T = 0.2, the eigenvalues fall off quite rapidly and in practice only the first two make a significant numerical contribution. This is consistent with the behavior of the corresponding eigenvalues of the single-fold problem (subsection 4.6). The clipped correlation function ck(r) for a Brillouin spectrum is shown in figs. 5.3 and 5.4. As is customary, we have normalized by dividing by the product of the true and clipped mean counting rates: ck(r)—
Ck(r)
(5.50)
From the figures we note that as T becomes larger, there is a decrease in the slope of ck(r) at the origin, and an averaging over details in the spectrum. T = 0.2 is large enough to separate the curves 2.0 I
INTERVAL TIME
I
Cr)
Fig. 5.3. Clipped photoelectron correlation function c,(r) for a Brillouin spectrum (~= 10,6 = 1), with T = 0.1, v = 2.0 (uT ,a0.4;~,a0.6;_._,n1.0 (Blake and Barakat [57]).
2.0 I
INTERVAL TIME Cr) Fig. 5.4. Clipped photoelectron correlation function c,(r) for a Brillouin spectrum (~ = 10,6 = 1), with T = 0.2, v = 1.0 (uT ‘0.2);—..—,n0; (Blake and Barakat [57]).
R. Barakatand J. Blake, Theory of photoelectron counting statistics: an essay
282
of different a values from each other at r = 0. In order to give some idea of the effects of clipping at different levels, we show, in fig. 5.5, ck(r) for a Lorentz spectrum at T = 0.1. The method employed in this section can be applied directly to any other quasi-monochromatic spectrum, of course. By using larger values of N one can treat longer counting times, although for k> 2 the algebra for computing the clipped correlation function becomes considerably more cumbersome. 5.4. Q(Ai,X2) for the Lorentz spectrum Jakeman [20] has shown that Q(X1,X2) can be evaluated in closed form for Gaussian light having a Lorentz spectrum; this solution includes Q(X) as a special case. In view of the importance of this result we will outline the relevant analysis especially since Jakeman only presents the analysis in bare outline. We wish to thank Dr. Jakeman for helpful correspondence on this point. The field correlation function for the Lorentz spectrum was given in eq. (4.26). Substituting this kernel into the integral equation, eq. (3.26), we have A1 ~ e ~
~
Ø(t’) dt’ + A2
e_Itt’I ~(t’)
r~fT
(5.51)
dt’ =~ 0(t),
v
0
where we have made the intervals explicit: t1 = [0,t], t2 = Er, r + TI. Let us rewrite the left hand side so as to eliminate the absolute value signs in the exponentials: A1 ~ e_t
+
~‘
0(t’) dt’+ A1 ~ ett’ 0(t’)dt’ + ~2 ~ T ett’ Ø(t’)dt’ ~
(t),
0 ~ t ~ T (5.52)
and A1 ~et+
~‘
0(t’) dt’ + A2 ~ e_t+ t’ 0(t’) dt’ + A2
J
e~
~‘
0(t’) dt’
~ 0(t),
r~
T.
(5.53) 4 I
I
I
\
~0
0.3
0.6
0.9
1.2
INTERVAL TIME Cr) Fig.
5.5. ck(r) for a Lorentz spectrum
with T
0.1, v
0.1:
,
k = 1; —,k
=
2; --.—,k
=
3 (Blake and Barakat [57]).
R. Barakat and J.
Blake,
Theory of photoelectron counting statistics: an essay
283
Upon differentiating these equations twice with respect to time, we obtain the ordinary differential equations 0~t~T (5.54) I
—(-~—2X2)0(t)0,
~0”(t)
r~
The solutions to these equations may be written as 0(t)A cos7t+Bsin’yt, Ccos~t+Dsin~t,
0~t~T
r~t~r+ T
(5.55)
where we have defined 72
i2A1v
\
— 1)
~2
m
12X2v
\
— 1)
(5.56)
.
m
In differentiating eqs. (5.52) and (5.53) twice, we have essentially lost the boundary conditions. To reintroduce this information, we substitute the general solution given by eq. (5.54) into eqs. (5.52) and (5.53) and require validity for all values of t within the pair of domains (rangel)
0~t~T,
r
(5.57)
Thus, A1
et
+
~‘
~A cos7t’ + B sin7t’) dt’ + A1
e~
— ~‘
(A cosyt’.+ B sinyt’) dt’
r+T
+ A2
~ et
—
‘
(C cos~t’+ D sin~t’)dt’
~ (A cosyt + B cos’yt)
(5.58)
while eq. (5.52) becomes ~1
e_t+t’ (A cos7t’ +B sin7t’)dt’ + A2 ~ e~~’(Ccos6t’ +Dsinót’)dt’ r+T
+ A2
~
et_t’ (Ccos~t’+D sin~t’)dt’ ~~Ccos6t
+D sin~t).
(5.59)
Performing the integrations is straightforward. Then, collecting all terms in e ~and all terms in the result is —
e+t,
e_t{_A + ‘yB} + e+t {Ae_T(_cosoyT + ysin7T) + Be_T(_sin7T
—
‘ycos7T)
+ Ce_T[e_T~sin6(r + T) — e_Tcos6(r + T) + cos~r— 6sinór] T [_e_Tsin&(r + T) — e_T6cos~(r+ T) + sin~r+ ~cos~r] } = + De
0.
(5.60)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
284
Similarly, eq. (5.59) requires that et{A [eTcosyT + eT7sin~yT 1] + B [eTsin~yT eT7cosyT 7 (cos6T + ~sin~T) + DeT(ôcos~T sin6T)} + Ce + e0 {Ce~[~sin~(r + T) — cosö(r + T)] + DeT [—sin~(r + T) —
—
—
(5.61)
—
—
öcos~(r+ T)] } = 0.
The coefficients of the bracketed terms are independent of each other, consequently the only way for eqs. (5.60) and (5.61) to vanish is for the bracketed terms to vanish individually. This leads to four simultaneous equations, which we write in matrix form MC=0
(5.62)
where C is the column vector C +
=
(A ,B,C,D) and M is 7
—1
0
eT(—cos~yT+ysinyT)
_-e7(sinyT+ ycosyT)
—1 + eT(cosyT + ysinyT)
+eT(sinyT
— M
0
—
ycosyfl
0
0
eT(cos~r— ~sin~r)
eT(sinor + 6cos~r)
—cosb(r
—eT(cos&r + 5sin~r)
—sin~(r+ T) — lcos& (r + T) _eT(sinor — &cos6r)
bsin~(r+ 7)
—sin~(r+ T)
+ T) + ~sin~(r + T)
—
cos~(r+ T)
—
~cos6(r +T)
(5.63) For eq. (5.62) to have a nontrivial solution the determinant of M must vanish. To evaluate det M conveniently, we begin by performing several elementary row operations casting M into the following form. —1
M
y
eT(—cosyT + ysinyT)
—eT(sinyT + ycosyT)
eT(cosyT + ysinyT)
eT(sinyT
0
—
0
0
-ycosyT)
0
eT(sin~r +
~cosbr)
—cT(cos~r+ 8sin~r)
--eT(sin5r
—
~cos~r)
ôsinó(r + T)
—sin6(r +
T)
eT(cosor
—
~sinör)
—
cos6(r + T)
—
öcosä(r
+
T)1
(5.64)
It is convenient to define the functions a
sinö(r + T) + i5cosö(r + T)
/3
~sin~ (r + T) 2
F( 7,T)
—
(y
G(y,T)~(y2+
—
costS (r + T)
l)sin’yT
—
2’ycosyT
l)sin7T.
(5.65)
In terms of these definitions detM = e2TF(-y,T) {(/3tS
—
a)costSr
—
([3 + atS)sini5r}
—
e2TG(~y,T){(j3tS + a)cosbr + (/3
—
atS)sintSr}. (5.66)
The bracketed terms are now expanded using trigonometric identities, tedius manipulations then yield —
a)cos6r
—
([3 + atS)sintSr}
{(/3a + a)costSr + ([3
—
F(cS,T)
(5.67)
atS)sin6rH G(6,T).
(5.68)
=
R.
Barakat and J. Blake,
Theory ofphotoelectron counting statistics: an essay
285
Consequently, the determinant of !i~becomes (5.69) detM = e2TF(7,T)F(tS,T) — e2TG(oy,T)G(tS,T). For a specified pair of values of A 1 and A2, m will be an eigenvalue of the basic integral equation only when det M = 0, or 2TF(y,T)F(tS,T) e_2TG(~y,T)G(tS,fl 0. (5.70) e_ There are two ways to obtain the one-fold case. The first is to set A 2 = 0 in eq. (5.70); the result is 2)sinyT+2ycos7T0 (5.71) (1 —7 which is equivalent to eq. (4.28) with y. The second way is to let r = T/2 and A 1 = A2 A. A straightforward, but tedious analysis, again leads to eq. (5.71) as expected since the answers are time translational invariant. We now pass on to the problem of obtaining Q(A1 ,A2) and Q(A) in closed form. Consider the function 2TF(g,T)F(d,T) + e2TG(g,T)G(d,T)] (5.72) P(~)= —~--[e 4gd —
where we have defined =
2A 1v~—1
2 = 2A
2v~—
d
1
(5.73)
and set ~ = 1/rn. P(fl is simply detM multiplied by the factor —(4gd)~exp(—4T — 4r). If we extend ~ into the complex plane, then P(~)is an entire function of ~. Furthermore, let M(r) be the maximum of P(~)on the circle Izi = r. The order, p, of the entire functionP(~)is given by — loglogM(r) lim ~ (5.74) Pr_ooo log~ r-o~2log~ 2 As discussed in texts on complex variable theory, Ahlfors [61], Titchmarsh [62], if an entire function is of fractional order less than unity, then the order p is bounded by h ‘~p ~ h + 1 where h, an integer, is the genus of the function. Here h must be zero. Since P(~)is an entire function of genus zero and is also devoid of zeros at the origin then it can be expressed as an infmite product via the Hadamard factorization theorem in the form P(fl=C
H
(i
\
k0
——‘i
(5.75)
~k’
where C is a numerical constant and {~k} are the zeros of P(~).Obviously, mk The double generating function Q(A1,A2), can now be written as —
=
H
k0
~°
(1 +rnk)=C
II
k0\
i l~ (1 +_—)=
P(—l)
=
C
lftk.
(5.76)
.
2 withj = 1,2, then When
~ = —1,
both g and d become pure imagery. If we define (3~=
(1 +
2vA,)b’
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
286
P(—l)becomes
4131/32
—
e2T[(l + A1v)sinh/31 T+ 2/3k cosh/31 T]
4131132e
[(1
+
A2v)sinh/32T + 2/32cosh/32T1
(A1vsinh[31 T) (A2vcosh[32T).
(5.77)
Since Q(0,0) = 1, then C 1. Upon carrying out some further computations, we can express the double generating function as 1 =e_2T [~([3~ +(311)sinh/3 Q(A1,A2) 1T+ cosh(31T1 [~(/32 +j3~’)sinhj32T+coshj32T] 1)sinhj3 e_2T [~ (I3~— j3j 1 TI [~ (/32 /3~’)sinh/32TI. (5.78) —
—
This is essentially the result obtained by Jakeman [20]. Note that the two time intervals enter the expression in a completely symmetric manner. To obtain the one-fold generating function, we set A2 = 0 (i.e., /32 = I); then the result is 1)sinh/3T + cosh/3T]. (5.79) e_T [~ (/3 + f3 Simple manipulations then yield the form quoted in eq. (4.30). We should note that the result quoted in eq. (5.79) was first evaluated by Slepian [381using contour integration. Srinivasan [63] also obtained eq. (5.79) by using the Hadamard factorization approach. It should be noted that Wittke [64] was evidently the first to employ the theory of entire functions to such problems, he obtained eq. (5.79) for the closely related problem of the autocorrelation function of the output of a nonlinear angle modulator. In the limit that T becomes much shorter than the coherence time of the incident light (T ~ 1), one can derive several approximate and useful results. In this limit, we have from eq. (5.78) Q(A
1 = 1 + (vT) (A 2A 27). (5.80) 1,A2) 1 + A2) + (vT) 1 A2 (1 e To this approximation only two eigenvalues of the gap integral equation contribute to the product in eq. (5.76) —
(vT)
m 1
I
1 A2 (1 —
(A1 + A2)
1 +
2!~i_~(A1+A2)
{ 1—
—i—
rn
4A 1
—
2T)1 e2
+ A2)
.1
1/2
2~)j~2j [1_
(5.81)
4A2e~
m 3m40. As the reader can show, these results also follow from taking the limit T —~0 of eq. (5.70) and solving the resultant quadratic equation. Srinivasan et al. [651 have obtained the N-fold equivalent of the transcendental equation, eq. (5.70), for the eigenvalues. Additionally, the single-fold casehas been treated for a spectrum consisting of a superposition of Lorentz lines by Tomau and Echtermeyer [661. Both these solutions make use of a technique resembling Laplace transform, but with a finite upper limit of integration. Unfortunately these explicit formulae for the eigenvalues are extremely complicated and not very suited to numerical computations.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay 5.5.
287
Higher order photoelectron statistics
While interest has usually been focused on second order correlations measurements of the clipped and unclipped correlation functions third order experiments are both practical and useful. Davidson and Mandel [67] and Davidson [68] have measured third order correlations with a time-to-amplitude converter; and more recently, Corti and Degiorgio [69] have performed third-order measurements with a fast digital correlator. Therefore, in this subsection we demonstrate how third order statistics may be extracted from the basic integral equation for Q. Our approach is a direct generalization of the techniques outlined in subsection 5.2. Generalizing eq. (3.62) to include three counting intervals, we write following Blake and Barakat [701 —
—
~ A1
g(t
—
t’)
0(t’) dt’ ~
(5.82)
0(t).
Taking the three counting intervals, each of the same width T (where T is taken to be small compared with the time required for g(t) to change significantly), we have ~ A. g(t
—
(5.83)
t1) 0(t1) ~0(t).
Let t take the values t1, t2 and t3 in turn. For eq. (5.83) to be satisfied we must have A1
A2g(t1
—
t2)
A1g(t2
—
t1)
A2
A1g(t3
—
t1)
A2g(t3 — t2)
A3g(t1
—
t3)
0(t1)
0(t1)
A3g(t2
—
t3)
0(t2)
0(t2)
0(t3)
0(t3)
t3
.
(5.84)
Let us define the matrices 0(t1)
0~
(5.85)
0(t2) 0(t3)
A1vT+ 1 A
A2uTg(t1
A1vTg(t2
—
r1)
A2vT+ 1
A1vTg(t3
—
t1)
A2vTg(t3
—
t2)
A3vTg(t1
—
t3)
A3vTg(t2 — t3) —
t2)
A3vT+
.
(5.86)
1
Then eq. (5.84) can be rewritten as Aç~(1 +m)Ø.
(5.87)
The matrix A has eigenvalues (1 + m1) where/ = 1,2,3, and we have Q(A1,A2,A3)11(1
+m1~’ (detAY’,
(5.88)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
288
where detA 00(1 + A1vT)(l + A2vT)(l + A3vT) + 2(vT)3 A1A2X3g12g23g31 2 [(1 + A —(vT) 2vT)A2A3g~3+(1 + A2vT)A1A3g~1+ (1 + A3vT)X1X2g~2]
(5.89)
and g11 —g(t, — t1). With Q(A1 ,A2,A3) in closed form, we are in a position to evaluate joint probabilities and moments by direct differentiation. For example, the probability of obtaining no photoelectrons is P(0,t1 ; 0, t2
0,t3) =
Q(A1 ,A2,A3)~A, A2 A3 1 3 + 2(vT)3g ~I(l+ vT) 12g23g31 =
=
=
2 (I + vT)(g~ —
(vT)
1. 2+g~3+g~1)}
(5.90)
Another example of particular experimental interest is the triple photoelectron correlation function: c(r
3, 1,r2)~(m1m2m3)/(m) for delay time r 1 = t2 t1 and r2 = t3
—
t2.
From eq. (3.65)
A =A
=A
=0
—
~3rr1 ~‘.
(m1m2rn3) =
~
1’”2, 22~3
(5.91)
‘3
(5.92) 1
2
3
Upon performing the necessary manipulation we obtain 2(r 2(r 2(r c(r1,r2) 1 + 2g(r1)g(r2)g(r1 +r2)+g 1) +g 2)+g 1 +r2). Note that c(r1,r2)
(5.93)
c(r2,r1).
Equation (5.93) must reduce to the expression for c(r) given in eq. (5.19) if r2 (or alternately r1) becomes large. Since lim g(r2)
0,
(5.94)
0 00
we have 2(r
lim
c(r1,r2)= I +g
1)=c(r1).
(5.95)
The bahavior of c(r1,r2) is shown in fig. 5.6 for a Lorentz line shape as a function of r2 for fixed values of r1. The curves undergo a monotone decrease to their limiting value c(r1). The main effect of increasing ~r1is to decrease the value of c(r1, 0) relative to the background value of c(r1). c(r1, r2) for a Brillouin line is shown in fig. 5.7. We get the oscillatory behavior characteristic of hetrodyning. A particularly useful caclulation would be the evaluation of the clipped photoelectron correlation functions from Q(A1A2A3). This problem certainly deserves further work. If we limit our interest entirely to joint moments, we can avoid direct reference to the generating function, and extend counting times to longer values. We take the expression (m1 m2 m3) as an example, this permits us to derive eq. (5.93) in a very different way. It follows directly from eqs. (3.36) and (5.92) that we can write (m1m2m3)
=
(&~(t1, {ak})
~(t2, {ak}) ~T~(t3,~ak})),
(5.96)
R. Barakatand J. Blake, Theory of photoelectron counting statistics: an essay C
c
0
I 0.4
I
I
I
I
I 0.8
I
I
I
I 1.2
I
1.6
Fig. 5.6. Triple photoelectron correlation function for a Lorentz line as a function of ~2 for fixed values of i~,: = 0.1;—.—, r, = 0.4 (Blake and Barakat [70]). I
I
I
Fig. 5.7. Triple photoelectron correlation function for a Brillouin line , r, = 0;—, r, = 0.1; —.—, r, = 0.4 (Blake and Barakat [70]).
289
,
r,
=
0;
—,
I
(~= 5, ~
=
1) as a function of
~2for fixed values of r,:
where ~
{ctk})Ns
~
~%“(t’,{ak})
~(t’,
(5.97)
{ak})dt’
l-
and where (• .) indicates an average using the weight function P({ak}). When P({ak}) is given by the Gaussian distribution, eq. (3.36), S(t, fak}) is a complex Gaussian random variable, for it is simply a linear combination of the ak’s. A convenient summary of basic properties of such variables is given by Mehta [17]. The even order moments of ak can be expanded in terms of the second moments of writing a,, a,
a1)= ~
a1)(a~’~
a1>,
(5.98)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
290
where ~ stands for a sum over the m! permutations p,q, m = 3) then yields
, r of 1 ,2,
.
...
,
m. This formula (for
(I S(t’)12 16(t” + T1 )~2 6(t” + T1 + r2)12) + 2 Re (~* (t’)~(t” + r 1)) (~*(~1~ + r1 )~(t” + ~1 + r2)) (6 *(tlFf + ~1 + T2 2){~(g*(~F)~ (t” + r + (I~(t’)~ 1 ))~2 + I((*(tU + r1 )~(t” + r1 + r2 ))12 +
I(~*(tflP+ ~1
(5.99)
+ T2)~(t~))I2},
where we have used the fact that 2) = (V (t” + ri)12)= (I~(t”+ Ti + r (V (t’)1 2)j~).
(5.100)
By interchanging ensemble averaging and integration in eq. (5.96), we have 3 Km1 m2m3>
=
~ dt” (I~(t’)2 I~(t”+ r
dt’ [dt”
I~(t”+
1 )12
(Ns)
~1
+ r2 )12).
(5.101)
Substitution of eq. (5.99) into this triple integral and appropriate normalization yields the triple photoelectron correlation function c(Ti, r2): 2 + ~ g(T1,r2)
=
1 +~ +
[dt’ T
I dt’ T2J
I
0
0
—
+
dt’
~ dt” g(t” + r1 dt” g(t’
—
~dt” g(t” + r
2 2
r
—
t”)
t”)2 + 1
dt”
~dt”
t’)
---
—
r2
—
t
dt” g(t” + T 1
—
t’)g(t” + r2
—
t”)g(t’
—
—
—
t”).
(5.102)
Note that we have utilized the fact that the field correlation function g(t) is real when the radiation is quasimonochromatic. When T is short enough to permit g(t) to be treated as a constant, eq. (5.102) reduces to eq. (5.93). Similar techniques permit the evaluation of other moments in terms of the field correlation g(t). However, the joint probability functions P(rn1 ,m2, , mN) and the clipped photoelectron correlation functions can only be evaluated in terms of the N fold generating functions. Appendix 5. A: Q(A1, A2) for partially polarized light In this appendix we generalize eq. (5.16) forQ(A1,A2) in the short counting time approximation to include partially polarized light. Following the notation of subsection 4.11, we write the generalization of eq. (4.117), whose eigenvalues determine Q(A1, A2), as A1 ~
~ F11 (t
— t’)01(t’)
dt’ + A2 ~
F1~(t— t’)0,(t’)dt’
=~
0~(t).
(5.A. 1)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
291
Upon approximating the integrals as we did in eq. (5.3), we obtain 2 A
1
T
~
m
2 F11(t)0,(O)
+ A2T ~
j=1
F11(t —
T)0,(T)
(5.A.2)
I) 01(t).
f=1
The requirement that this holds for t = 0~(O) A21(—r) A2J(0)
A1J(0)
A1J(r)
0,
r can be expressed as
0~(0)
02(0) 01(r)
rn 02(0) uT 01(T)
02(r)
02(T)
(5.A.3)
where 1(r) is the polarization matrix defined in eq. (4.108). The generating function Q(A1, A2) follows by forming the product of one plus the eigenvalues, we have det
1 Q(A1,A2)
1+ A~vTJ(O) A2vTJ(—r) A 2vTJ(r) 1+ A2vTJ(0)
(5.A.4)
where I~sthe 2 X 2 unit matrix. Unlike the single-fold case where a single parameter, ~, sufficed, an evaluation of Q(A1, A2) requires a knowledge of J(r) for all delay times. Appendix 5.B: Evaluation of the photoelectron correlation function Upon employing eq. (5.31), we can write the photoelectron correlation function in the form (m(t)m(t + r)>
2
=
a Aa 1
(det Ay1
(5.B.l)
a A2
where A ~~+!?~
+ (U7’)2
(A1 + A2)E(0)
[E(0)E(0)
—E(r)E(—r)].
(5.B.2)
The matrix differentiation can be performed explicitly. Setting (det A) ~A for convenience, we have
2
a 8A 1
a~.2
r
8A aA
I.
ax.,
=I~3——A2
A’ A, =
=
0
8A2
~2A i
aA, aA2Jx,
aA aA
a2A
8A1 ~A2
aA1aA2
2———
I
=
= 0
-
x,
=
A2
=
0
(5.B.3) where we have used the fact that
AI~~0
1.
(5.B.4)
By employing the identity detA
exp[trlogA]
(5.B.5)
R. Barakat and .1. Blake, Theory of photoelectron counting statistics: an essay
292
the following two formulas can be derived
a aA1
—A
A trA~—A
(5.B.6)
aA1
—
tr(A1
tr(A1 ~A)
A =A
tr (A
A ~-_A)}
2
~A)
+ tr(A1
aA1aA2A) (5.B.7)
.
Substituting A from eq. (5.B.2), we obtain 2 A~[B(0)E(0)—B(r)B(—r)i,
(~)E(o)+(~)
~
a2 ~
vT\2
A
=
-
11,2
(5.B.8)
-
(-~—) [B(0)B(0)
—
B(r)B(—r)1.
(5.B.9)
When these expressions are substituted in eq. (5.B.3) we have (m(t)m(t + r)>
(~)2
{[trE(0)]2
—
tr [E(r)~(—r)I } = (vT)~ {l + ~tr [E(r)~(— T)1} (5.B. 10)
where we have used trE(0)~B11—2.
(5.B.ll)
6. Time interval statistics 6.1. Derivation of time interval probability densities Another useful approach to photoelectron counting statistics is through the time interval statistics. Since the underlying stochastic process is non-Poissonian, we would expect that measurements of such quantities as the distribution of time intervals elapsing between consecutive photoelectron counts would yield valuable information. The value of these techniques lies in the ease with which the distributions can be measured over an exceptionally wide range of time scales by using a time-toamplitude converter (Scarl [71], Davidson and Mandel [67], Chopra and Mandel [72], Kelly and Blake [73]). Strictly speaking, time interval probability densities constitute a problem in two and three-fold probabilities which must be expressed as functions of Q(A1, A2) and Q(AI, A2, A3). However it is possible to cast the problem into a form involving only the single-fold generating function Q(A) under certain limiting conditions. We follow an unpublished derivation of Blake and Barakat. We denote by A,B,C the following three events:
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
293
A. Arrival of one photoelectron during (t1 Z~t,t1) B. Arrival of no photoelectrons during (t1, t2) C. Arrival of one photoelectron during (t2, t2 + tSt) where t1
a aA3
P(B,C)=——Q(A1,A2,A3)
(6.1)
.
A,=o A2 = A3
=
1
Here, Q(A1,A2,A3)(exp{—A1~21—A2,Q2 —A3~23})
(6.2)
where 2 dt’,
/ = 1,2,3
(6.3)
Ns ti~ ~(t’)I and t~stands for the appropriate time interval. Similarly, the probability of time intervals elapsing between consecutive photoelectron counts is the probability that, given a photoelectron count during (t 1 — z~t,t1), then the next subsequent count occurs during (t2, t2 + tSt). This conditional probability is
P(B,CIA) =
P(A
B C)
P(A)
(6.4)
Again, using, eq. (3.64) we have P(A,B,C)=
2 a aA1aA3
Q(A 1,A2,A3)
(6.5) A,=A2=A3=1
and P(A)=~-
Q(A1,A2,A3)~
(6.6)
.
A2
=
A3
=
0
Thus, P(A)
Ns(J
L I(t’)12 dt’}).
I~(tl)l2dt’ exp (_Ns ~,
(6.7)
In order to proceed further we now assume that the intervals L~tand t5t are small in two respects: 1. They are small compared to the time scale on which (Ig(t)12> fluctuates. 2. They are short enough so that the probability of more than one photoelectron count occurring during either interval is negligible. In this case we have I(t’)12 dt’ ~ ~(t 11
—~t
2~t, 1)I
(6.8)
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
294
I~(t’)I2dt’)
J
exp {_Ns
~ 1,
(6.9)
ti-it
P(A)~NsI6(ti)I2i~t=w&.
(6.10)
In like manner P(B,C) ~ Ns (exp ~—Ns ~2 I~(t’)2 dt’) I ~ (t 2)
2 ~1(ti
P(A, B,C) ~ (Ns)
P(B,CIA)
)12
exp t_Ns
)12 exp (_Ns
~(t1
J
j
(6.11)
2~,
I I(t’)I~dt’} I1(t
I ~(t’)l~ dt’
2)12)tSt ~t.
} I t(t2
Now consider the single-fold generating function with
~2
)12)tSt.
(6.12)
(6.13)
~+ ~1
Q(A) =(exp f_Ns ~ k~(tP)I2 dt’}).
(6.14)
Since g (t) is a stationary stochastic process, then we may view either t1 or t2 as arbitrary so long as the interval between them remains t. Differentiating eq. (6.14) with respect to t and considering t1 to be fixed, we have:
A
=
=
—Ns (exp (_Ns ~2I~(tl)I2 dt’} 16(t2)12).
(6.15)
Comparison with eq. (6.11) yields P(B,C)
=
—tSt
-p-- Q(A) at
.
(6.16)
If we define the probability density V(t) V(t) ~~-P(B,C)
(6.17)
then V(t)
=
a
——Q(A) at
.
(6.18)
Since V(t) is a probability density then it must satisfy the condition V(t)dt= 1
(6.19)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
295
in addition to being non-negative. This is a useful check on calculations. A second differentiation of eq. (6.15), now viewing t2 as arbitrary and fixed yields: 2 dt’) Ii~(t (I~(ti)I2 exp (_Ns ~ I~(t’)1
(6.20)
2) 2).
Comparison with eq. (6.12) yields
2 a at
=P(A,B,C).
tSt&—~Q(A)
(6.21)
We define the conditional probability density P(t) 00
1 a2 P(B,CIA) =—-—j
1 —
wat
tSt
(6.22)
.
Q(A) A=1
In applications, we will find it easier to write V and P in terms of the normalized time T and the normalized count rate v. Consequently, probability density function of the time intervals which elapse before the occurrence of the first photoelectron count
V(T)
a
= —
—
Q(A)
(6.23) A=1
and P(T)
conditional probability density function that given a photoelectron count at T = next count occurs at time T
a2
i
.
—~Q(A)
v ~3T
0,
the
(6.24)
A1
Equations (6.23) and (6.24) for the distribution of time intervals have been given previously by Glauber [2, 3]. Though not strictly rigorous, his arguments add some intuitive insight to this discussion of time interval statistics. Recall that the generating function evaluated at A 1,
Q(A)1A
1
=P(0,T)
(6.25)
is simply the probability that no photocounts have occurred during a time interval from t = 0 to 00 T. For a stationary process, this interval may be given an arbitrary translation. Clearly the probability that at least one photocount occurred during the interval is 1 — P(0, T). If we choose, the initial instant at random, then the probability that the first photocount occurs between T and
T+tSTis - P(B,C)=P(0,T)[1
—P(0,5T)] =P(0,T)_P(0,T+6T).
Thus the probability density function V(T) reads V(T)
1
=
—P(B,C) 00 6T
a ——
~T
Q(A,T)
.
A1
(6.26)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
296
Notice that we have set P(0,T)P(0,6T)=P(0,T+
6T)
(6.27)
in deriving eq. (6.26), which would be strictly true only were the photocounts uncorrelated. To find the distributions of time intervals between consecutive photocounts we may employ the same technique: F(A,B,C) 00 [1
— P(0,~T)]P(0,T)
[1
—
P(0, 6T)],
(6.28)
where again we have made use of the stationary nature of the process. Thus P(A,B,C)=
[P(0,T + L~T+ 6T)
and
—
P(T + ST)]
2
1 a Q(A) P(T)=P(B,CIA)=___
vaT2
.
—
[P(0,T + aT)
—
P(0,fl],
(6.29)
(6.30)
A=i
Again, the argument is strictly rigorous only when the photocounts are uncorrelated. Because the distribution of time intervals is perhaps less familiar intuitively than such comparable quantities as the photoelectron correlation function, it is useful to consider some simple examples. First, suppose the photoelectrons arrive randomly (as from a perfectly coherent source or a very long counting time). Then,
Q(A,T)
(6.31)
e_~~~)T,
and V(T)ve~T
(6.32)
P(T)
(6.33)
=
v e~T.
V(T) and P(T) are identical in this case (i.e., Poisson statistics) because starting the time intervals at random or at photocounts are equivalent conditions. In other words, both V and P are negative exponential probability density functions. Notice that the photoelectron correlation function would be independent of Tin this case: c(T)
=
1.
(6.34)
As a second example, suppose that the count rate v is small enough so that vT ~ 1 for those T we are considering. Then from eq. (6.13), we have P(T)
v[l + Ig(r)121
=
vc(r).
(6.35)
Thus, for small vT, P(T) resembles the photoelectron correlation function, which is typically the way P(T) is treated in PCS experiments. Unlike c(T) which always approaches a constant background value, P(T) eventually begins to decay exponentially because for large enough T, Q(A,T) will approach the function given in eq. (6.3 1). 6.2. Short counting time, partial polarization Let us now examine the behavior of V and P when the counting time is short (so that only one mode of the field is effective) and the light is partially polarized.
R.
For 0 ~
Barakatand J.
Blake, Theory of photoelectron counting statistics: an essay
297
1, we must differentiate eq. (4.120), the final result is
V(T
=
‘
(1
(1 +,~)v/2 2(1 + b + b
(1 —.~)v/2
(6.36) 2
2)
1)
where we recall that
+
(1 + b1)(l
b2)
+
-
b 1 ~
(1
b2 ~
+~)vT,
(1 —~)vT.
Note that for completely polarized light, ~
=
(6.37) 1; this reduces to (6.38)
2. V(T,l)
V/(1 +vT)
For completely unpolarized light,.~ 0; we differentiate eq. (4.125) and obtain V(T,0)
=
vi
(1 +
~ vT)2.
(6.39)
Before discussing the consequences of these formulae, let us derive P(T,~). The corresponding formulae for P(T,~)follow by additional differentiation with respect to T and can be cast into the form 3 v/2 +~vT)4’
(6.40)
P(T,0)=
(1
P(Tc~)=
(1 +~)2v/2 + (1 + b 3(l + b 1) 2)
P(T, 1) =
(1
—~)2v/2
(1 + b1)(1 +
3
+
b2)
(1 ~2)(2 + vT)v/4 (1 + b 2(l + b 2 1) 2)
(6.41)
1 ~
(6.42)
P(T,~)is greater than V(T, ~) at fixed values of v and ~ for T ~ 0, and then decreases more rapidly than V(T, ~) with increasing time. The physical reason for this behavior is easy to see. When we require that a photon be counted at T = 0, we are giving additional weight to those cases in which the field strength has fluctuated toward large values. In those cases the subsequent count tends naturally to occur more quickly. The ratio of V to P when T 0 is given by P(O, ~) 3 +.~ V(0,~)= 2
(6.43)
so that R (1) = 2 and R (0) = Completely polarized light thus enhances this ratio as we would expect. The behavior of V and P for the limiting values ~ 0,1 is shown in figs. 6.1 and 6.2. ~.
6.3. Lorentz spectrum Since Q(A) is given in closed form, see eq. (4.30), we can differentiate it explicitly with respect to T and obtain V(T) and P(T). The final results are V(T
=
4/3(1
_/32)[(l [(1
_(3)e(i~)T +/3)e(1~)T] /3)2 e~T —(1—(1 +/3)2ePT]2
.
(644)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
298
2.C I
.5-
I
I
I
\
I-c’
\
_5~~.
>
c0
0
I
.2I
I
4I
I
.6I
I
.8
_____________________
C
0
4
T
I
-~
T
Fig. 6.2. P(T) in the presence of partial polarization, u = 1: , = 1;—, ~ 0 (Barakat and Blake, 1973 unpublished).
Fig. 6.1. V(T) in the presence of partial polarization, u = 1: , = 1;—, ? 0 (Barakat and Blake, 1973 unpublished).
and 3~T +
4/3(1
— [32)2 eT [(1 + [3)4 e
(I + [3)3(5/3_3) ePT + (I
v[(l
+(3)2
et~T_(1
j3)2
—
~3)3(5/3 +
3)
—(1
e~T
—
[3)4 e3aT]
e_~T]4
(6.45) When T is large, the asymptotic behavior of these two functions is given by V(T)-=—
4/3(1 —[3)
e(1_~T
T>> 1
(6.46)
T~’1
(6.47)
(l+/3)2
P(T) -
(1
_/3)2
v(l
+/3)2
e~’
~)T,
so that the decay is exponential in time. The larger v, the more rapid the decay. Graphs of V and P are shown in figs. 6.3 and 6.4. The probability that no photoelectrons are recorded during the time interval T is P(0,T)=
— (1 +/3)2
4/3 eT e~~T_(l
(6.48) /3)2 e_~~T~
The long time behavior is P(0,T)
413 —j
(1
+/3)2
e~’_Ø)T
(6.49)
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
299
I0~
-
101 ___________________________________
\~%\ ~
. -
\
\
~ \
01
-
N
\
N.
\
N.•
~~__ .‘—.
\ \
.
—.
--.
~
01
—S.
N..
--,~
N.
\
.5
S.
.=~ S_S5 S....
:
N S...
.
I
N
.0
N I’..
.5
2.0
62
2.5
I
0
.5
NORMALIZED TIME (T)
.0
I
1.5
~
I
2.0
2.5
NORMALIZED TIME (T)
V(T) for Lorentz spectrum: — . —, v = 1; , v = 2; v = 5;—, u = 10 (Barakat and Glauber [18]).
Fig. 6.4. P(T) for Lorentz spectrum: — . —, v 1; , v = 2; —. .—, v = 5; —, u = 10 (Barakatand Glauber [18]).
Fig. 6.3. —. .—,
S.
\~.
N \
0
-
~S
N 62
..
and is also exponential decaying. When uT 4 1, then eq. (6.45) simplifies to eq. (6.35). See fig. 6.5. A discussion of some experimental results is not without interest and we follow Kelly and Blake [73,74]. Gaussian light having a Lorentz spectrum is produced by scattering coherent laser light from a suspension of polystyrene spheres. To measure P(0, T), one records the relative frequency that no photoelectrons are counted during a time T. The results of such measurements for a wide range of average count rates, v, are shown in fig. 6.6. In addition to the exact results the short counting time
0~~0~5~0
NORMALIZED TIME (T) Fig. 6.5.P(0,T)foraLorentz line: —..
.—,
v
0.5;——,
U
1~
, or
~
v= 5;—, u= 10(Barakat and Glauber (18]).
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
300
5
/
/
/
/
4
/
Fig. 6.6. P(0,T)~ for a Lorentz spectrum: — - —, v = 1; -——, u = 1.72; values are denoted by ., ~ ., respectively (Kelly and Blake [741).
,
=
3 as computed from eq. (6.48). Experimental
approximation eq. (6.36) is also plotted. Note that the reciprocal of P(T) is plotted in order to make the short counting time approximation a linear function of T. The experimental results are in excellent agreement with the exact calculations; the short counting time approximation is reasonable only for T<0.6. 6.4. Numerical schemes for V(T) and P(T) The techniques just described hold only for special spectral profiles. We now present a numerical scheme developed by Blake and Barakat [421 for the evaluation of V(T) and P(T) when the input profile is arbitrary (but quasimonochromatic). Perhaps the most obvious way to obtain V and P is by direct numerical differentiation of the single fold generating function with respect to time after calculating numerically the first N eigenvalues of eq. (3.63) and then forming N
Q(A)
H (I +
/=0
~
(6.50)
Unfortunately this procedure becomes computationally useless because the dependence of the eigenvalues upon T turns out to be unstable when T is even moderately large. The Blake—Barakat scheme employs the representation of Q in terms of the iterated kernels of the integral equation, eq. (4.24). The advantage of this approach is that the noise generated by diagonalizing a matrix, whose every element depends on T, is eliminated. The disadvantage is that although eq. (4.24) is formally valid for all v and T, we are restricted to vT < 1 in order to apply it for computational purposes. This is not a serious restriction as most of the current experimental situations fall into this range.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
301
The practical question is reduced to determining the number of integrals of the iterated ‘r which must be retained in order to give a reasonable estimate of Q and its first two derivatives with respect to time. Fortunately we have the Lorentz spectrum results as a check, see previous section. Furthermore the ‘r functions can be evaluated in closed form for a Lorentz spectrum, thus affording us further numerical checks. The first three ~tr(T)are: 11(T)= T 2T) 12(T)-}(2T_
I
1 +e
2T]. (6.51) 3(T)~[T_ 1 +(T+ 1)e_ Extensive numerical calculations performed on a variety of spectral line shapes by Blake and Barakat (unpublished calculations) have shown that as long as vT < 1, only three 1r(T) are needed to determine Q accurately enough to permit first and second derivatives to be taken without serious loss of error. Thus, Q(A)
exp [_AvIi + (Au)2
12 —
__
13]
(6.52)
.
Furthermore, 15 point Gauss quadarture, was found to be sufficiently accurate for the numerical calculation of the ‘r (in fact departures of less than 0.004 were detected when compared to the closed form solutions of J~for the Lorentz line given above). Numerical differentiation of Q was then performed using a 5-point Lagrange formula. The mesh spacing h employed was h = 0.01 although values of h <0.05 produced the proper behavior of Q’ and Q”. Extensive checks were made against the exact values available for a Lorentz line. As an example of the application of the numerical method we again consider the Brillioun spectrum, eq. (4.51). P(T) is shown in fig. 6.7 for a count rate v = 0.1 and in fig. 6.8 for v = 1. The value a 0 corresponds to a single Lorentz line. As a increases, we see evidence of hetro1.3
000:40608
Fig. 6.7. P(T) for a Brillouin spectrum, ~ = 10, ~ = 1, v = 0.1: ,a1(Blake and Barakat (42]).
I
I
I
_
Fig. 6.8. P(T) for a Brillouin spectrum, ~ = 10,7 = 1, u = 1: —..—, ao;_._,a_0.4;____,ao.6; ,arl(Blake and Barakat [42]). -
R. Barakat and J. Blake, Theory of phott,electron counting statistics: an essay
302
dyning between the individual lines in the form of oscillations in P(T) unlike the Lorentz (and other single) spectral lines which decay to zero in a monotone fashion. In the limit of very small count rates, P(T) approaches the joint counting rate of samples taken at t = 0 and t = T, and thus becomes the full photoelectron correlation function in accordance with eq. (6.35). 7. Non-Gaussian statistics 7.1. Introduction
To this point we have treated the case of Gaussian light. This is to say that the P function is a product of Gaussian distributions. This description is accurate whenever the scattered field is produced by a large number of contributions (not necessarily independent) so that the central limit theorem in one of its many forms is operative. When there are many independent scatterers in the scattering volume, fluctuations in light intensity are typically due to motion of the scatterers on the same spatial scale as the wavelength of the incident radiation. These motions shift the phase of the various contributions and cause the intensity to fluctuate. However if there is only a small number of scatterers in the scattering volume, there will be additional fluctuations in the light intensity as scatterers drift into and out of the scattering volume. Further, the scattered light is no longer Gaussian because the central limit theorem cannot be invoked. Nevertheless useful information can be extracted from the photocount statistics, though it is necessary to set up different machinery for extracting the information. Because the light is non-Gaussian, the first and second moments are no longer sufficient to characterize the distribution. It is this fact which makes the analysis of the higher moments of W and P particularly interesting. Following Schaefer and Pusey [75], we consider the field produced when there are N(t) identical scatterers within the scattering volume. Then we have the total scattered field S(t) given by N(t)
~
exp {ikr1(t)}
(7.1)
/=1
where k is the scattering vector and the position of the /th scatterer. Since the i~are random, we can picture the sum in eq. (7.1) as a random walk in the complex plane. The step length may be constant if each particle makes an identical contribution to the light intensity. This need not be the case, and we will describe the random walk in the most general case possible — by specifying the distribution of contributions each scatterer may make to the scattered field, and by allowing the number of scatterers within the scattering volume itself to be random. 7.2. Fixed number of scatterers
The probability of obtaining m photoelectrons in a time interval T is given by eq. (3.21) where the inverse Laplace transform of the single-fold generating function Q(X). The variable 2 is now to be interpreted as the time integrated intensity of the scattered field: W(fZ) is
~
j1(t)~dt
(7.2)
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
303
here s is a sensitivity factor which reflects the quantum efficiency of the detector and the spatial coherence of the light. W(~)is a non-negative function of both ~2and T. Schaefer and Pusey [75]have shown that the probability density function of the scattered light W(~2)can be considered as a random walk problem and they treat the case where all scattering particles are identical, see: Schaefer and Pusey [761; Pusey, Schaefer and Koppel [77]. However we follow the more general analysis of Barakat and Blake [781. The probability density function W(~2)has been evaluated for short counting times by Barakat [79]under the assumption that the light is scattered by N independent particles: W(~2IN)=
~N(~’O
~
(~1/2t)
t dt
(7.3)
where ~N(t)
H n1
=
~ fan (a~)J0(tan) dan.
0
(7.4)
Here a~is the contribution of the nth particle to the amplitude of the scattered light; specifically if there is only one scatterer in the system, then 1”2. (7.5) a1 (S21) fan (an) is the probability density function of an. In the situation envisaged, all the scattering particles will have the same probability density so that eq. (7.4) reduces to
Q~N(t)
If fa(a)
[~fa(’1)Jo(ta)t1dl].
(7.6)
obeys ~
fa(”)
— (~2~ >1/2]
(7.7)
(in other words, all the particles are of the same fixed size and thus scatter light equally), then 2 t)]N. (7.8) 0((~21)” Because the observation time T is taken to be smaller than the reciprocal of the characteristic line width of the spectrum of the scattered light, the shape of the spectrum of the scattered light does not enter into the analysis and the two relevant parameters are the time T and the average count rate w. In other words we are looking at a single mode scattering problem. In an experiment, it is very difficult to keep the number of scatterers fixed in the scattering volume and it is more realistic to consider that the number of scatterers is itself a random variable. We take N to be a discrete random variable having a Poisson distribution. Thus (I’s1), the average number of independent scatterers in V, characterizes the random scatterers since (.7%!) is also the variance. Any higher moments of the Poisson distribution can be expressed in terms of (N). Since the solution for a random (Poisson) number of scatterers depends on the solution for a ØN(t)
=
[J
R. Barakat and J. Blake,
304
Theory
of photoelectron counting statistics: an essay
fixed number of scatterers, we begin with the latter case. For N = 0 (no scatterers), we have W(~I0)=1
~
j
(~1/2
t)t dt
=
b(~).
(7.9)
Consequently, P(,nIO)1,
m=0
Q,
rnzO.
(7.10)
is exactly what we expect, namely that the probability of obtaining no photoelectrons is unity. For N ~ 1, we express W(~2IN)in the form of a Fourier—Bessel series whose coefficients are sampled values of the characteristic function ~N(t) of W(~2IN).We now assume that This
a>c~ where c~is finite. Obviously
(7.11)
fa(~1)~0,
(7.12) W(~2IN)~0, ~2>N2a2. Consequently ~N(t) is a bandlimited function since its Fourier—Bessel transform W(~.2lN)vanishes
identically outside a compact region. It can be shown that (see details in Barakat [79]):
W(~IN)= ~ n1
[~i(~, N2a2[J(y)]2 1/Na)]N fl
~ \ Nct
2a2 0~~~N elsewhere,
(7.13)
where Yl,Y2,~ are the positive roots of J 0 [i.e., J0(y~,)0001. Since this is a Fourier—Bessel series, its convergence is governed by the smoothness (continuity) of W(~jN).The smoother W(~ZlN)is, the more rapid is the convergence of its series expansion. In the special case where all the particles are of the same fixed size, it is possible to obtain closed form solutions of W(fLIN) forN 1,2,3. These expressions are listed in appendix 7.A. As N becomes very large, W(~7IN)approaches a negative exponential probability density
W(~IN) ~~2) —e~~~> ~
(7.14)
so that the underlying statistics of the field amplitudes are Gaussian. The corresponding expression for P(m IN) is the Bose—Einstein distribution “-~yfl
P(mIN) (1 + (~))m +1 (7.15) The expressions in eqs. (7.14) and (7.15) are the leading terms in the asymptotic series in powers of see appendix 7.B for details. In spite of the fact that W(~IN)varies drastically for small N, the resultant photoelectron counting distributions are very tame creatures. This is simply a consequence of the smoothing action of the Poisson term in eq. (3.21). The factor ~ effectively damps out any irregular behavior of W(~ZIN) for ~ 4 1, while the negative exponential accomplishes the same result for ~Z> 1. It is for this -~
R. Barekat and J. Blake, Theory of photoelectron counting statistics: an essay
305
reason that we have confined our numerical calculations to the case of identical particles. Even though W(fZIN) will be somewhat different according to the situation considered (i.e., particles with different fixed sizes, particles with range of different sizes, etc.), the smoothing action of the integrand in eq. (3.21) is the determining factor. The results forN= 1,2,3,8 are shown in figs. 7.1 through 7.4 (see curves with open circles), in all cases (~l~ > = 1. Note that the maxima of these distributions are at m = 0, as we would expect. We have chosen to work with (~Z>,but it is also just as easy to work with the average total count rate w w
=
(7.16)
(f~)fT=N(~2,>/T.
The question naturally arises as to how large N must be in order that P(mlN) approximate the Bose—Einstein distribution. Answering this question is not a simple matter, but we can offer the 0.6
I
I
I
I
I
I
I
I
I
0.5
-
0.4-
-
0.4 I
NUMBER OF PHOTOELECTRONS Fig. 7.1. Photoelectron counting distributions: P(m Ii), open circles; P(m (1>), solid circles (Barakat and Blake [781).
~
I
I
I
I
I
I
I
I
I
I
I
I
I
NUMBER OF PHOTOELECTRONS Fig. 7.2. Photoelectron counting distributions: P(m12), open circles;P(m 1(2)), solid circles (Barakat and Blake [78]).
I
O.1~
0.3
I
O~h1
___
04
I
-
I I
I
I
I
I
I
I
I
I
I
I
I
I
I
0.12
¼\%~a~l::~il:=~=r1srr.
NUMBER OF PHOTOELECTRONS Fig. 7.3. Photoelectron counting distributions: P(m13), open circles;P(m 1(3>), solid circles (Barakat and Blake (78]).
E.E3~S.SO.~Q. NUMBER OF PHOTOELECTRONS Fig. 7.4. Photoelectron counting distributions: P(m18), open circles;P(m 1<8)), solid circles (Barakat and Blake (781).
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
306
following remarks. Based on test calculations (not reproduced here), it appears that for practical purposes N ~ 5 gives a Bose—Einstein distribution. Of course, this is not strictly true. Nevertheless an experimenter would be hard put to distinguish P(m15) from P(mj 100), say, purely on the basis of a single-fold counting experiment. The probability of obtaining no photoelectrons is a useful statistic and its behavior is depicted in fig. 7.5. 7.3. Random number of scatterers We have just examined the case of N fixed so we can interpret these results as holding for a canonical ensemble in the language of statistical mechanics. The case of random N leads to an interpretation as a grand canonical ensemble. It is a simple exercise in probability theory to prove that if N is distributed according to a Poisson distribution having a mean value (N>, then 1’T>
W(~2I(N>)=NrrO W(~IN)(N>”~ N!e~
P(mI(N>)
(7.17)
(N)N e_
N!
NO
(7.18)
As (N> is made to increase, the Poisson distribution becomes very peaked at (N> = N and acts somewhat like a Dirac delta function centered at (N> = N. Consequently W(~I(N>)~ W(fZIN),
P(mI(N))~P(mIN)
(7.19)
for large (N>. We can easily plot W as a function of (N>, but it hardly seems worthwhile to do so and we pass directly to the photoelectron statistics. Numerical values of P(mI(N>) for (N) = 1,2,3,8 are plotted in figs. 7.1 through 7.4 (see solid circles). As we would expect, the photoelectron counting distri-
I
O2~~oO~~
N, (N> Fig. 7.5. Photoelectron counting distribution:
P(OIN),
open circles;P(0I(N>), solid line (Barakat and Blake [781).
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
307
butions for the deterministic and stochastic situations are measurably different for one and two scatterers. However, even for three scatterers, the two situations are not very different. For more than three scatterers, the two situations yield practically the same result as witness N = (N> 8 (fig. 7.4). We can also derive integral representations for both W(fZI(N>) and P(mL(N>). If we substitute eq. (7.17) into eq. (7.3) and sum the series, we obtain W(fZI(N))
~
exp {—(N> [1
— ~
1
(t)] } J0 (~1/2 t)t
(7.20)
dt.
When eq. (7.20) is substituted into eq. (3.21), we can prove that 2 ‘~ Lm (t2/4) t P(mI(N))
~
exp {—(N>t 1
— ~,
dt)
dt
(t)} } e_t
eY exp {—(N>[l —~,(2y”2)]}Lm (y) dy
(7.21)
where Lm (y) is the Laguerre polynomial. In its second form, the integral can be evaluated via Gauss—Laguerre quadrature (Abramowitz and Stegun [80]) if desired. Probably the best way to distinguish these two situations under the constraint of single-fold counting is to consider the probability of obtaining no photoelectrons during the measurement. The resulting calculations are shown in fig. 7.6. It is possible to distinguish the two situations for N = (N> ~ 5; but certainly not for a larger number of scatterers. 2.1
2.2
I
I
I
111111
I
III
Ill-I
-
>._\
-
1.6
\
\
\. \
0 0
“ ~ OO~&~~ ~
LU
~
10
50
100
N,(N) Fig. 7.6. V(N,6), V(
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
308 7.4.
Moments of JV(~ZIN)andW(~I(N>)
In order to calculate the moments (more precisely the conditional moments about the origin), we recall the generating function introduced in eq. (3.7):
Q(XIN)
~ e~ W(~2IN)d~2
(7.22)
from which the moments can be obtained by differentiation ak —~Q(XIN)
(~kIN>(_l)
,
k=
~
(7.23)
ax Q(AIN) can be expressed in the form Q(XIN)
e21~X ~N(t)t
=
dt
(7.24)
by substituting eq. (7.3) into eq. (7.22) and integrating out the Bessel function. It is convenient to express Q as a power series in A, for then the differentiations are trivial. To this end, we express cbN(t) as power series in t2 by expanding J 0 (ta) in a power series and then integrating termwise~we have
[~
Q~N(t)00
A?lt2n]
(7.25)
where 2), A0
=
1,
A2
=
A
—~ (a
4), 3
=
(7.26)
~(a
and ~ fa(a)~”~1a
(7.27)
We now rewrite ØN(t) in the form ~N(t)
=
~
(7.28)
B~t2~.
The B coefficients can be calculated from the A coefficients via the recurrence relation, Gould [811 Bn =
1
~ 0
[k(N+
1)
—nIAkBn_k,
n~’1.
(7.29)
k1
When eq. (7.28) is substituted into eq. (7.24), the resulting integration is trivial and the final result
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
309
is that Q(AIN) is a power series in A. The resulting differentiations can now be performed and the first four moments about the origin are (~2IN>N(a2) (~2 IN) = 2N(N
— 1)
(a2>2 + N(a4)
(~VIN>6N(N_1)(N_2)(a2>3 +9N(N—1)(a2>(a4)+N(a6) (~24IIV)24N(N— l)(N—2)(N—3)(a2)4+72N(N—l)(N—2)(a2>2(a4) + l6N(N— l)(a2>(a6>+N(a8).
(7.30)
When all the particles are of fixed size then these expressions reduce to those given in Pusey, Schaefer and Koppel [77]. The limiting values of (IZ’9N), when N ~ 1, are (akIN)
,~.
k!N” (a2)”
(7.31)
characteristic of a negative exponential probability density function, as expected. Thus the rate at which the moments approach their limiting values depends not only on N (as in the case of indentical particles) but also on the spread of the probability density function f(a) as measured by the ratio of the various moments of a. Since W(~2IN)is a true probability distribution as regards our problem, then P(mIN) is the probability distribution of a mixed Poisson process, Haight [82]. It is a well known property of mixed Poisson processes that the kth factorial moment defined by l)IN),
(m~”~IN)—(m(m—l)~(m—k+
k1,2,
(7.32)
is related to the kth moment of W(~IN)by (m~”~ IN>
=
(a” IN).
(7.33)
Thus we can list the factorial moments of P(mlN) once we have the moments of W(~Tl4N). The same procedure can be employed to determine (~2”1(N)), we define the generating function Q(ARN>)
~ e~ W(~TLI(N))d~.
(7.34)
It follows that Q(AI(N)) =
exp {—(N> [1
—
0, (t)]} et2/4X
tdt.
(7.35)
In order to obtain the moments, we expand the first term in, the integrand in a power series in t; thus exp {—(N) [1 —0, (t)]}
~ n=0
cnt~
(7.36)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
310
where (N)(a2)
C 01,
C
C2=— 2
(N)
3
+
32
(7.37)
...
64
Proceeding as before, we obtain Q(XI(N)) as a power series in A. Differentiation then yields the moments about the origin, of which the first four are 2)
(&2I(N)) (N)(a 1(N)) 2(N)2(a2 >2 + (N) (a4) (&2~1(N))
6(N)3(a2)3 + 9(N)2(a2)(a4) + (N)(a6)
(~24l(N)) 24(N)4(a2)4 + 72(N)2(a2)2(a4) + 34(N)2(a2)(a6) + (N)(a8).
(7.39)
When all the particles are of same fixed size, then these expressions also reduce to those given in Pusey, Schaefer and Koppel [77]. The limiting values of (~Z”(N)) for large (N) are again given by eq. (7.3 1) with N replaced by (N). However specifying (N> instead of N permits the number of particles in the scattering volume to fluctuate. This introduces additional fluctuations in the intensity of the scattered light. Accordingly the moments for the random case will approach their limiting values more slowly than the corresponding moments in the deterministic case. P(mI(N)) is also the probability distribution of a mixed Poisson process because W(~ZI(N)) is a probability density; consequently (m~”~I(N>) = (&V’I(N)).
(7.40)
It is useful to examine the case where fa(a) is sharply peaked around its mean value because this provides an approximation to a slightly polydisperse suspension of particles. For computational simplicity, we let exp {—(a
fa(a)
—
(~,)l/2)2/2a2
}
(7.41)
where (~2~) cj~. The moments of a are ~‘
~ f~(a~”da.
(7.42)
If we form the dimensionless parameter ~ a~/(~,),then for ö 4 1, the negative contribution of the probability density function in eq. (7.41) is negligible and we can safely replace the limits (0,oo) in eq. (7.42) by (_oo,00). Consequently, we obtain (a2)(l +6)(fZ,)
(a4)
=
(1 + 66 +
(a6>
=
(1 + 156 +
(a8)
(1
362)(&2)2 4562
+ l5ö~)(&~~)~
+ 286 + 21062 + 4206~+ l0S6~)(~,)~.
(7.43)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
311
In order to utilize these explicit formulae, let us consider for example, the variance of ~7:
(7.44)
var (&~IN)= N(N — 2)(a2>2 + N(a4> var (~2I(N>)= (N)2(a2)2 + (N)(a4)
(7.45)
then V(N,6)~~~ V((N), 6)
(1
=
+~)+~(6_362+263
00 (1 +
~~s)
~
+
(6
— 362
—~~)
+ 26~—...).
(7.46)
(7.47)
The first terms in parentheses are the contributions when all the scatterers are identical. The second term represents the influence of the width of fa(a), and arises from the varying amounts of light scattered by different sized particles. The second term is the same for both situations. Some typical numerical results are shown in fig. 7.6 and are self explanatory. Other moments can be handled in much the same manner. 7.5. Coefficients of skewness and excess for W Since both W(~2IN)and W(~ZI(N))are far different from their limitingvalues for small N and (N), it is important to evaluate their global shape parameters: coefficient of skewness, and coefficient of excess. The coefficient of skewness -y, is defined as
~
—
(~))3
)/((cz —
(7.48)
(~))2 )3/2
The skewness is a normalized measure of the mode minus the mean and serves as one indicator of the length of the “tail” of the density function. If>’, is positive (negative), then the corresponding probability density function is skewed to the right (left) of the mode. The larger 17,1, the longer the resultant tail. If-y, = 0, then the density function is symmetric about the mean. In the special case where all the scatterers are of the same fixed size, the explicit formulae are 2N—4 =
(N2
—
N)”2
3
_
2 —
+ O~N2)
(7.49)
and 2(N)3 + 6(N)2 + (N> ((N)2 + (N))312
3
_
2 + ~
O((N)2).
(7.50)
Both approach the limiting value, -y, 2, characteristic of a negative exponential probability density function. Some typical curves are shown in fig. 7.7. For fixed N, the result of allowing 6 > 0 is to hasten the rate at which ~, (N) tends to two. When N is allowed to fluctuate, ‘y, ((N>) is larger than the limiting value; thereby indicating that the “tail” dies off more slowly than for the corresponding fixed case. This is exactly what we would expect. The coefficient of excess, ~Y2,is defined as (ç~>)4>— 3((~ — (~))2>2 72 = ((~~(ç~>)2>2 (7.51) —
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
312
4
I
I
IIIIII~
I
.
z •
W
•
00
I
0
0
0
CJ)
0 0
0
~
1111111
I
I
10 N,
1
IIIII
100
Fig. 7.7. Coefficient of skewness, -y
1, as a function ofNand (N) for fixed 6: ~y,(N, 0), open circles; y, (N, 0.2), solid circles; .y, (
and is a measure of the peakedness of the corresponding density function about the mode. 72 0 for a Gaussian, and 72 = 6 for a negative exponential. Explicit formulae for 72, when all the scatterers are identical, are 3 36N2 + 45N — 15 24 6N N3—N 6———+O(N2) (7.52) _
—
and 6(N)3 + 36(N)2 + 39(N) + 1
24
3 + 2(N)2 + (N> 6 + ‘~ + O((NY2). (7.53) The limiting value 72 6 characteristic of a negative exponential is approached in both cases, note that 72 (N) < 6, while ‘ 12((N>) > 6. Some typical curves are shown in fig. 7.8. ‘~
72((N))
7.6.
(N)
=
Variance ofF
We can list the factorial moments ofP(mIN) and P(mI(N)) by virtue of eqs. (7.33) and (7.40). However we content ourselves with deriving the second factorial moment (variance). Thus 2)2 + N(a4) + N(a2) (7.54) var (mIN) = N(N — 2)(a var (mI(N)) = (N)2 (a2)2 + N(a4) + (N)(a2). (7.55) When fa(a) is the Gaussian discussed in eq. (7.41), then we can show that var (mIN)
=
[(N2
—
N)(~2
2)N2(~2,>2+ 6N(~Z,>
1)2 + N(~Z,>]+ (26 + ä var (mI(N>) 00 [((N)2 + (N>)(fZ 2 + (N>(&2,)] + (26 1)
+
(7.56)
62)((N)2 + 2(N))(&2 2 + 6N(cz,). 1)
(7.57)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
• •o o~
0 x LLi
313
O
.00
—
-
0
-
S
0 —12
I
I
1111111 -
10
Fig. 7.8. Coefficient of excess, ~z, as a function ofN and (N> for fixed 6:
I
I
11111’
100
72(N, 0), open circles; 72(N, 0.2), solid circles;
72(
When all particles are identical 2
—
N)(~2,)2
+
N(~,>
(7.58)
var (mIN) = (N var (mI(N>) = ((N2) + (N>)(~2,)2+ (N)(&2,).
(7.59)
The variance for the stochastic situation is larger than that for the deterministic situation as expected, both approach the same limiting value. 7. 7. Time interval statistics
In the present subsection we evaluate the time interval photoelectron statistics for light scattered by an arbitrary number of independent particles. We follow the analysis of Blake and Barakat [831 and refer to their paper for the computational details. We confine our attention to the case where all the scatterers are of fixed size. We are interested in determining the behavior of V(T) and P( T), see subsection 7. 1, for both fixed and random number of scatterers. In accordance with the formulae of subsection 7.1, we need to evaluate the first and second time derivatives of Q(XIN)l~... 1
W(~2IN)e
d~2.
(7.60)
The time dependence in Q(AIN), which we have deliberately suppressed, enters through (~2,>.For an average counting rate w, the total number of counts recorded during a time interval T is given by (m>wTN(f~,).
(7.61)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
314
Accordingly, (~7,) wT/N, and we may write the upper limit of integration as The case N I is, of course, trivial. We have Q(AIl)e~T
N2(&2,>
=
NwT
(7.62)
characteristic of a Poisson distribution. The time interval probability densities are V(TIl) =F(TIl)
=
w
(7.63)
e~T.
V(TI 1) and F(TJ I) are identical because with only one scatterer the light is not modulated in intensity, and hence remains coherent. In this case photocounts occur at random, and to start a time interval at a photocount is equivalent to starting the interval at random. ForN= 2, we can utilize Chebyschev quadrature [801 to evaluate the generating function I
Q(X!2)I~
~
~
J
e’~ d~ ~
—
(7.64)
~)1/2~
ForN~ 3, we employ eq. (7.13) and show explicitly the Tdependence of W(&211V) by writing W(~2,~) where we temporarily suppress the N dependence. With this convention we have W(~,~) e~ d~= W(~) e~ +
W(~) e~ d~2
(7.65)
and a
d2~
~ W(~,~)e~ d&~e~~
+
~
W(~2,~)e~’ d~2
(7.66)
where d
d Nw
j.
(7.67)
With these observations, we can easily show that V(TIN)
=
—Nw
W(~,~) e~ d~
(7.68)
and P(TIN)N2w [e_t ~W(~,~)In~
+
J
~2
W(~) e~ d~}.
(7.69)
Some typical numerical results for both V(TIN) and P(TIN) are shown in figs. 7.9 and 7.10 for various values of N and fixed total average count rate. The differences are especially noticeable for P(TIN). As N is made to increase, they approach the limiting values predicted on the basis of single mode Gaussian (chaotic) light N—l’~o
V(TIN)
P(TJN)
=
=
w/(l + wT)2
(7.70)
2w/(l + wT)3.
(7.71)
R. Barakat and I. Blake, Theory of photoelectron counting statistics: an essay
315
This limiting behavior is reached when N = 8 for V(TIN); however a value of N = 20 is required for P( TIN) to reach its limiting behavior. We have already noted that coherent light, N 1, leads to negative exponential probability densities for V and P characteristic of a Poisson process. Now the Poisson process has the property that it is a maximum disorder process in that there is no tendency toward clumping of photocounts. For N> 1, the underlying process is no longer Poisson and we expect clumping. A crude measure of the clumping or overall correlation of the photocounts is available through the ratio R(N) =P(OIN)/V(OIN).
(7.72)
For coherent light R (1) = 1. With increasing N, R increases until assuming the value 2 at N = oo, see figs. 7.9 and 7.10. This reflects the increasingly larger fluctuations in the field intensity which become possible as N increases. Recalling that for small w, P behaves like the photoelectron correlation function (Davidson and Mandel [67]), then we can interpret this enhancement as indicating greater photocount correlation. To obtain the generating function for a random number of scatterers distributed according to a Poisson distribution, we simply integrate eq. (7.17); appropriate differentiations yield V(TI(N>)=
V(TIN)
N1
(N)~’e~I~> N!
(7.73)
and (N>” e~~> P(TI(N>)=>~P(TIN) N!
(7.74)
When (N) is large, the Poisson distribution peaks around (N) centered at (N) N. Thus
=
N and acts like a Dirac delta function
V(TI(N>) ~ V(TIN) for(N)~ 1.
(7.75)
P(TI(N)) ~ P(TIN) LC
::
1
I
I
I
I
-
I
o~:.6
Fig. 7.9. V(T LN) for light scattered from N independent scatterers, average counting rate is unity. —, N = 1; ,N=2;—..-,N=4;—..—,N8.ThecurveforN~ is indistinguishable from that for N = 8 (Blake and Barakat [84]).
2.C.
I
I
I
I
I
::0I~2~4Io6
Fig. 7.10. P(T IN) for light scattered from N independent scatterers, average counting rate is unity. —, N = 1; N’~2;—.—,N~’4;—..—,N~’ 8; ,N~o(Blakeand Barakat [83]).
,
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
316
However when (N) is large enough for these approximations to hold, then N is large enough for the limiting cases, eqs. (7.70) and (7.71) to be valid. In order to show the difference between a fixed and a random number of scatterers let N = (N) = The results are summarized in figs. 7.11 and 7.12. Note that V(TI(3>) ~ V(Tl 3), at least over the range 0 ~ T ~ 0.6. However since ~ V(TI(3)) dt
=
~ V(TI3) dT 00 1
3.
(7.76)
then V(T13) must eventually be greater than V(tI(3)). The increase in photocount correlation (clumping) over that of a pure Poisson process is even more pronounced for a random number of scatterers. For fixed N the only fluctuations in the field intensity are those that arise from interference terms; with N random there are additional contributions to the fluctuations asNitself fluctuates. Thus we would expect that P(0I(N>) would be greater than P(OIN); this is true as witness fig. 7.12. 7.8. Photoelectron correlation function
We now pass to the consideration of second and higher order statistics of the scattered light for non-Gaussian situations. Of particular importance are the full and clipped photoelectron correlation functions. The development of this and the next section lean heavily on the work of Chen, Tartaglia and Pusey [211, see also Chen and Tartaglia [841, Schaefer and Berne [85]. The full photoelectron correlation function given by eq. (5.2), can also be expressed in terms of the integrated intensities
c(r),
c(r)
=
(~,~2>/(~1)2.
(7.77)
We confine our attention to short counting times only, so that we approximate cz. by =
NsT f” (t1) g* (t1)
o:OiQi2I~4iQ6 Fig. 7.11. Comparison of V(TIN) and V(TRN>) for: , N = 3; —, (N> = 3; average counting rate is unity (Blake and Barakat [83]).
(7.78)
I~__
Fig. 7.12. Comparison ofP(TIN) and P(TI
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
317
where N is the number of independent scatteringparticles, and write the scattered field as ~ b1(t)exp{ik~(t)}
(7.79)
where b1(t) =
1,
if
particle j is in scattering volume v
=
0,
if
particle j is not in scattering volume u.
(7
80)
To evaluate c(r), we first calculate (~Z,~22> 4 ~ (b~(t,)b
<~1~2>
=
1(t,)b~(t2)b,(t2) exp {—ik,~(t,) + ik r~(t1)
(NsT)
— ik i~(t2)
jfk!
4 ~ (b,,b,,b~ + i1t~i~(t2)} = (NsT)
2b,2A/~A11A,~’2A~>
(7.81)
i/k!
where b,1
b,(t1);
A,1
exp {iI
(7.82)
i~(t,)}
and the sum over i, /, k, 1 runs over all the particles in the scattering volume. There are two different time scales to consider. 1. The time for a particle to move a distance comparable to the wavelength of the incident light (the time scale on which I(A,,A~2)I—g12 fluctuates), 2. The time for a particle to move a distance equal to the cube root of the scattering volume (the time scale on which (b,, b,2> fluctuates). The second time scale is typically a thousand times larger than the first. Thus, the b’s and A’s are essentially independent random variables and we can factor eq. (7.81) 4 ~ (by, b, b~ ~2,
~2>
=
(NsT)
2b,2)(A1’~AJ,A~2A,2>.
(7.83)
i/kl
Since the particles are independent, the ensemble brackets containing the A’s vanish unless i = I, k~l,ori~lj~k. Nowsince g,2 —g(t,
— t2)
=
I(A,,A72)I 00 KA,,A72>I
(7.84)
we can rewrite eq. (7.83) as 4 (~1~2>
=
(NsT)
{~ 1k
(b~,b~
}.
2>+ ~ (b,1 b,2) (b1, b,2> g~2 i/
(7.85)
In the scattering situation under investigation there are N independent particles in the containing vessel (having volume V). The scattering volume u is a small fraction of V. Since the system is spatially homogeneous, the average number of particles in the scattering volume v is (M> = N(v/V)
(7.86)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
318
The instantaneous numberM(t) can be written as M(t) 00 ~
b1(t)
(7.87)
so that (M(t)>
~
=
(b~(t))00 N(b>.
(7.88)
By comparing eqs. (7.86) and (7.88), we have (b) = v/V. Returning to eq. (7.85), we note that the difference in time scales permits us (to a good approximation) to let t, = t2 in the b’s, whenever they are multiplied by g~2.This is because b,(t) does not vary in the time scale necessary for the fast correlation in g,2 to vanish. Consequently, the second term on the right hand side of eq. (7.85) can be summed 2g~ ~
(b~(t1))(b~t2))g~2 ~
(b1(t,))(b1(t1))g~2 N(N_ 1) (b)
2.
(7.89)
j#!
The first term must include the two different times. We have N
~
j/V
(bI(t,)bk(t2))
00
i,k
(,,~b. 0’,) i
N
~
b1(t2)) 00 (M(t,)M(t2)>.
(7.90)
j
But M(t) can be written as M(t) = (M(t)> + 6M(t) where 6M0’) is the deviation of the number of particles from the average number. Since (6M(t)> 0, we can write 2) + (6M(t, )6M(t 2 (b)2 + (6M(t,)6M(t (M(t, )M(t2)) = (M 2)) = N 2)). (7.91) Consequently, eq. (7.85) becomes 2(b)2 +N(N— 1) (b)2g~ (~)2
N
~
{N
2+ (6M(t,)6M(t2))}.
To obtain the result for a fixed number of particles in the scattering volume, we let (b) (M). The last term vanishes leaving
c(r)
1 + (I
—
(7.92)
1 and
(7.93)
)g2(r)
Thus, there is a correction term of orderN’ to the Gaussian result. When the number of scatterers is random then some ansatz must be postulated in order to evaluate (6M (t, ) 6M (t2 )>. Assume that (6M(t,) 6M(t2)) 00 exp {—~It1
— t2
l} =
(7.94)
e_~T
where ,3 is a constant. The thermodynamic limit N 00, v/V (b> -÷0 while (N) remains finite is important. Chen et al. point out that this particular limiting process is required in order to be able to neglect the unknown wall effect of the scattering vessel, and also because v/V can be made as -~
R. Barakat and J. Blake,
Theory of photoelectron counting statistics: an essay
319
small as iO~.As N oo, the random variableM becomes Poisson distributed so that (M(M 2. It is a consequence of eq. (7.94) that = (M) —~
—
________
(b>) exp {—~It, — t 2
In this limit
+g2(r)
}
+ (b>.
— 1))
(7.95)
~
+ ~e_~T + 0 (7.96) 1 In the next subsection, we consider the clipped photoelectron correlation function. In order to study this function, we need to obtain higher order two-time intensity correlation functions, i.e., (~Z~ ~7~)where k is a positive integer. Chen et al. show by induction that c(r)
(M>”~’ ~ C,(k) (N1+
(~)k+1
11
(b)’~’+ 1N111 (bY
[(1
—
(b>)
e_~T +
+f(k,N,(b>)g2(r)
(b)]
} (7.97)
where N[’I—N(N—l).•~(N—l+l)
C,(k)
~ (k:a,,
,ak)
(1!)al
(7.98)
(2!)aB... (k!yzk.
The summation is over the set of a’s that simultaneously satisfy a,+2a 2+.”+kakk
a2 +a2 +
1.
+ak
(7.99)
The coefficient (k:a,, .. . , a,~)’is the number of ways of partitioning k different objects into a1 subsets containing / objects for / = 1, 2, ... , 1. It is tabulated in [80]. The function f(k, N, (b>) is as yet unknown. To obtain the intensity moments, let t ~ in eq. (7.97) and also let g(r) 0, then —~
(k)
1
-~
k
1~’1 (bY + ‘+ lJVt1 (b>’~‘1 (M>k+ 1 ~ C,(k)[N For a fixed number of particles this reduces to
(7.100)
(ç~)k
(~>k
=
N~’ ~
C,(k) [N~’~~l + lNUl]
The first four moments are /o~2
(&22>=_(2N2_N) N2
=
~~C,(k)N1hI.
(7.101)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
320 (~)3
(&23)=-~-(6N3_9N2+4N) (~)4 (~4)=
(24N4
—h--
—
72N3 + 82N2
—
33N).
(7.102)
When the number of particles is random, then
>
(czk)
k
C,(k) (MY
(7.103)
~
To obtain the unknown factor f(k,N, Kb)) we go back to eq. (7.97). Set
~,
t 2,
1(b)’ + 1 + iN111 Kb)’] + f(k,N, Kb)). (M)k +
C, (k)
1
=
(~k+ 1) (~)k + 1 —
KM>’
k ~
(7.104)
1’ + ‘
f(k, N, Kb)) can be determined for a fixed number of particles by letting Kb> random number of particles, we easily obtain f(k,N,0)
the result is
=
1 and N = (M). For a
KM>’
{~—~ + ok ~
+ 1].
(7.105)
The two-time higher order intensity correlation function given in eq. (7.97) can now be explicitly evaluated. For fixed numbers of scattering particles the first few are (&Z,~Z~) (~>3
=
(>4
=
~
1
(6
9
8
4
(18
63
(4_~+~)g2(r)
(2_~)+
4
_~+—~)+
78 33\ ~-+~._~)~2(r).
(7.106)
(7.107)
The corresponding expressions for the random situation are more involved. Chen et al. worked out the explicit expansion of ~ in a power series in 11(M), reference is made to the original paper for the actual expression (eq. (19)). 7.9. Clipped photoelectron correlation function The clipped photoelectron correlation function Ck(r) is defined in eq. (5.12). However, since m, and m 2 enter into the expression symmetrically, we will now write Ck
~
~
m,0
m,k
m1P(m1,m2)
(7.108)
where we omit the explicit time variables for typographic convenience. We are, of course, only interested in very short sampling times. The double series can be recast into a series involving the
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
321
joint higher moments of the integrated intensity ~‘> 1 (rn>’ (—(rn>)1 (cl, &2~~ Ck(rn>_(m> ~ -j~---- ~ ‘=0 /=0 j! K2>’~’~1 If the
clipping le1vel is set at k =
then
1,
(—(m))/(~2
G 0
(7.109)
1~>
(7.110)
1~2
(m>—(m> ~
~‘
/=0
The average clipped count (mk>
~
P(m)
(7.111)
mk
can also be written as (mk>
1 (~‘~~> (rn)’
1— k1 ~ l=0
(—(rn>) j!
°~
(7.112)
(f~>’~j
/=0
Consequently (—1~,n>)1 (~2~>
(m 1>
I
—
(7.113) /=0
Since (fZ’> and (&2~cz’2> have already been evaluated, it is simply a question of tedious algebra to substitute these expressions into eqs. (7.110) and (7.113). If the clipping level is set at k = 1, then to order N’, (m>
(rn,>
1 +
and
1 1+—
(m>
C 1 = (m>(rn1>
N
k~l
+
[ (m) 2(1 +
2 (r) I g + (m>
(7.114)
1)
(m>)2
[(rn)22(1++(m>(rn>)2+
g2 (r) (rn>)
— N(1 +
21 ~
(7.115)
The corresponding results for a random number of scatterers are more involved and we refer to Chen et al. for the explicit formulae. Appendix 7.A: Formulas for scattered intensity
-
ForN= 1, we have W(~2Il)=
~
J
2t)J 0(K21>”
0
1’2t)tdt 0(~2
=
6(~2— (~Z 1>)
(7.A.1)
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
322
(i.e., there are no fluctuations in the light scattered by a single particle). Thus, the system is in a coherent state [2, 3] and the photoelectron counting distribution is Poisson with mean value (~)
P(rnlI)~ (~)~e~’>.
(7.A.2)
ForN 2, Rayleigh [86] has evaluated the equivalent of W(~l2),his result translated into the language of the present problem is W(~2I2) ~[~(4(~) =
— ~
1/2’
~
0,
elsewhere.
Note the square root singularities at &2 distribution is P(mI2) 00
1
4(&Z~>
corresponding photoelectron counting
~m e_n d~
~
—
irm !
0,4(~2~). The
(7.A.3)
0
[~2(4(~~>
—
(7.A.4)
.
~)]
1/2
Although this integral cannot be evaluated analytically, it is easily handled by Chebyschev quadrature which automatically takes account of the square root singularities at the end points of the integration. Finally, N = 3 can be expressed in terms of elliptic integrals, Slack [871: W(S~I3)= ~.2 ((~)1/2
00
+
[(3(~)1/2 2K(k) — p1/2) ((~)l/2
~1/2)
+
~-~,1/2 )] 1/2
‘
0~2~(~7) 1
K(l/k) 2 (~ >3/4 ~ 2ir
=0,
elsewhere (7.A.5)
where K(k) is the complete elliptic integral of the first kind with modulus k given by k
=
((~i>~/2 +
~1/2)
4(~l 314.cZ314 1) ~1/2) ((~)1/2 + ~1/2)J
[(3(~>1/2
—
1/V
(7.A.6)
The elliptic integral is logarithmically infinite at k 1 (i.e., when &2 = (fZ,)). The logarithmic singularity is very weak and P(rn13) can be evaluated numerically by a trapezoidal rule with a very close mesh.
Appendix 7.B: Asympto tics of W(&~IN) and P(m IN) The probability density function of the envelope r of the sum ofN independently distributed random sine waves approaches the Rayleigh probability density function as N becomes very large. If we apply the procedure described in Cramer [88] for corrections to the central limit theorem, we can show (details are omitted) that f(r)
2r N(a >
exp(—r2/N(a2>)
[1
+D 1 +
...]
(7.B.l)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
323
where 2/N(a2) + r4 IN2 (a2>-2) (N(a4>
—
(2 — 4r
—
2N(a2>2)
7B2
D 2 (a2>2 ( ) = counting time approximation, 4N In the1 short we can equate the integrated intensity ~2to the square of the envelope r to within a proportiona1it~onstant. Upon transforming to the new independe~tvariab1e~2,eq. (7.B. 1) yields .
W(~2IN)
e~n/
‘~
+
C 2
.
(7.B.3)
+...]
where 2)2 \ I 2N2 (a2 >2 — 4N(a2 >~2+ ~22 \ C 2 (a2 >2 4N2 (a2>2 (7.B.4) I N(a”> — 2N(a 2 ~ N Note that the fourth moment (a4> enters into the correction term. In the special case of identical particles (a4> (a2>2, the correction term reduces to the classical result of Pearson [89] and Rayleigh [861: C 2
N[2
(7.B.5)
(&•~>~4(~:)2].
If the particles are distributed according to the narrow Gaussian, eq. (7.41), then 00 —
(l_26_~2) 1 N [2
~
____
— (~>+ 4(~2>21~
(7.B.6)
The asymptotic expression for P(rnIN) can be obtained by appropriate integration ofeq. The final result is 4>
P(rnIN)
Bm (~TZ>)+
[
N(a
—
2N(a2>2
N2 (a2>2
(m +
] [~Bm ((&2>) —
4(~>2
1)
~2>
Bm + 1
(~~2))
1
(rn+l)(rn+2)
+
(7.B.3).
Bm
+2
((~2>)J
(7.B.7)
where Bm ((~2>)
(~l>~/(1 + (fL>)tm + 1
(7.B.8)
The leading terms of eqs. (7.B.3) and (7.B.7) are the negative exponential probability density
function, and the Bose—Einstein distribution.
8. The inversion problem 8.1. Introduction We have so far examined the way spectral properties of the light incident upon a detector determine the emissions of photoelectrons. These photocount distributions are typically described by
324
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
specifying one of three statistical quantities: (1) factorial moments of single-fold distributions; (2) the distribution of time intervals between consecutive photocounts; or (3) the full or “clipped” photoelectron correlation function. In this subsection we consider the problem of determining the spectrum of the incident light from measurements of these three quantities. Extracting spectral information from the first sort of experiment is made difficult by the great accuracy of the data required — that is, the moments depend only weakly on details of the particular spectrum. A discussion of this problem with references is given by Meltzer and Mandel [90], and in the last subsection of this section. While the second method is less sensitive to inaccuracies in the original data, evaluating the spectral parameters is a subtle problem involving the approximation of very low counting rate (Scarl [71]), an iterative solution (Davidson and Mandel [67]), or methods suitable only for particular spectra (Kelly and Blake [73], Tornau and Echtermeyer [66]). Because it yields the most straightforward analysis, and because of its efficiency in information gathering capacity (Kelly [91] ; Degiorgio and Lastovka [24]) the third method is thus in most general use. Section 5 sketched the evaluation of the full and clipped photoelectron correlation functions for quasi-monochromatic radiation of arbitrary spectral profile. Let us view this as a quantum mechanical black box in which the input is the spectrum of correlation function of the incident radiation and the output is the photoelectron correlation function of interest. We term this the direct problem. Although the direct problem is interesting and important to the theoretician, the experimentalist faces a different problem. He is given the output and wishes to determine the input, measuring a photoelectron correlation function (with the attendant noise) and needing the input spectral profile or field correlation function. In other words, he must contend with the inverse problem. Inverse problems are known to be numerically unstable [92, 93], and care must be taken to stabilize the solutions. When the normalized counting time is very small (the ideal situation from the experimentalist’s point of view), there is a simple relation between the photoelectron correlation function and the field correlation function. This aspect of the inversion problem has been successfully dealt with by several groups of investigators. However, it is not always possible to guarantee that the experiments can be conducted in these ideally very short counting times and the question is how best to extract the input data from experimental data corresponding to somewhat longer counting times while insuring against numerical instabilities in the inversion process. The purpose of this section, then, is to investigate various schemes for carrying out this program. The first section is devoted to the situation where there is no a priori knowledge of the functional form of the input field correlation function. The method of singular value decomposition is employed to effect a stable inversion. In the second section we assume that the experimenter already knows the functional form of the input and is interested in estimating one or more parameters characterizing the input. Reference is made to Blake and Barakat [94] for the full details. 8.2. In version via singular value decomposition Assume we are given measurements of the full correlation function 2 (8.1) c(r 1) = (rn(t)rn(t + r1))/(m(t)) at the discrete points r 1 (1 1,... , M) where rn (t) is the number of photoelectrons counted in the
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
interval from (t
— T/2)
(1~~(t)
to (t + TI2). We desire to reconstruct the field correlation function (t+r)>
~(—)
~.
g(T)
325
(8.2)
~.
((+) 0’) f’() (t)>
from our experimentally measured values of c(r,) The full photoelectron correlation function can be written in the form (see eq. (5.32)) c(r)
1 +~
(8.3)
trBN(T)BN(—r)
where BN is an N X N matrix of the form {BN(r)}l/ = (H,H1)”2 g [~ T (x1 x,) + TI —
(8.4)
here x1 are the Nth order Gauss quadrature points and H, the corresponding weight factors. For very short counting times, T < 0.05, we can set N = 1 in eq. (8.3) obtaining 2(r). (8.5) c(r) 1 +g Since we are dealing only with quasi-monochromatic radiation, g(r) is real and even. The values of c and g are related pointwise to each other; that is, both sides of the equation are evaluated at the same single time point ‘r = r 1. For this case, we refer the reader to the following references: Degiorgio and Lastovka [24] ; Jakeman, Pike and Swain [95] ; Kelly [91]. It is not always possible to insure experimentally that T < 0.05. This raises the question of how best to extract spectral information from experimental data corresponding to longer counting times. In order to take account of these somewhat longer counting times, we take eq. (8.3) with N = 2. In this case g(r)
g(r—6)
g(T+~)
g(T)
(8.6)
.132(T)—
where ~ = (x2
—
x, ) T/2. The corresponding correlation function is
c(r)l+~[g2(T_~)+2g2(r)+g2(T+~)].
(8.7)
Now the values of c(r) are related to those of g at times T, r ±~ and the inversion is not straightforward. In order to obtain a formal inversion we let = j~and define the matrices ê, g, A:
{e}1
c(j~)—
(8.8) (8.9)
{A}11~ 26~~ +
~
1
+~
1
(8.10)
where i,j = 1,2,... , M. The purpose of the term —i,5~,,in the definition of ê is to take account of the truncation sequence of equations / = enough 1 by using the fact g(0)a delay 1. With the assumption 2 (Ms)of=the 0 (i.e., the PCS instrumentathas channels to that provide of several correthat g lation times) we have e=f+~Ag (8.11) where is a column matrix whose elements are unity. Solution of this matrix equation yields the f
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
326
unknown matrix g in the form g 00 4A
1
(ê
—
1)
(8.12)
and thus g(r1). The matrix A given by eq. (8.10) is anM X M symmetric continuant matrix. We have been able to express the inverse of A in closed form: (A’),1=j(M—/+l), (_I)i+/(M_/+l),
i
(8.13)
i>i We derived this result by noting that the cofactors in the explicit expression for A ~‘for M X M can be reduced to the same form as A for the size (M 1) X (M — I) and employing the fact that —
detAM+l
(8.14)
for an M X M continuant matrix. 1, it is still not useful to attempt a reconstruction of Even with a simple expression forAof the problem becomes painfully evident. Input data g by using eq.such (8.12). The ill-posed nature accurate to 1% calculated by using eq. (8.3) with N 6 were employed and the formal inversion numerically evaluated. The final results are shown in fig. 8. 1; the solid line is the theoretical value ofg2 (T) while the open circles represent the values obtained via eq. (8. 12). This difficulty could also have been anticipated by examining the eigenvalues of A which are I irk 1 =2 Il + cos I I M+lJ
(8.15)
where M is the size of the matrix. The three smallest eigenvalues are listed in table 8.1 for matrices of size M = 10, 20, 40. Observe that as M increases, the higher order eigenvalues become very small. i_c
I
I
0.8—
—
0.6—
-
0.4—
-
I-
N Dl
0.2—
TIME (UNITS OF ~) Fig. 8.1. Reconstructions ofg’(r) for a Brillouin spectrum (a = 1.0,6 = 5.0, ~ = 1.0) for a count time T = 0.1: —, value; o formal inversion with 1% noise; • singular value decomposition with 5% noise (Blake and Barakat [94]).
theoretical
R. Barakar and J. Blake, Theory of photoelectron countingstatistics: an essay
Table 8.1
327
-
Three smallest eigenvalues of A
M
XM_2
~‘-M—1
XM
10 20 40
0.7341 0.2014 0.0528
0.3262 0.0895 0.0234
0.0815 0.0224 0.0058
If (ê 1) has a component along the kth eigenvector of A, then this component will be multiplied by (Ak) 1 when we operate with A ~. Thus these high order eigenvalues are extremely sensitive to the slightest error (i.e., noise) and induce large errors in the final result. We can circumvent this ill conditioning by performing a singular value decomposition of A. Any real m X n (rn ~ n) matrix A can be factored as follows [96] —
A
L/’’J~~
=
(8.16)
where V is a square matrix of size n X n; U is a square matrix of size m X m, and ~diag(a1,a2,...
,a~).
(8.17)
U is composed of the n orthonormal eigenvectors belonging to the n largest eigenvalues of AA+, while V is composed of the n orthonormal eigenvectors of A+A. The singular values a1, , a,~are the non-negative square roots of the eigenvalues of A+A. We form the pseudoinverse of A by writing ~
A°
U~
(8.18)
where ‘A
j+
+.~
—uiag1~o,,a2, a7 = -~- if a~0 UI 0 if g1 0. —
+
,a,~,
(8.20)
If A is simply an M X M nonsingular matrix, as in our case, then A ~ A and we seem to have gained nothing. By expressing A -i as in eq. (8.18) we have, however, gained the option of ignoring those eigenvectors which are greatly magnified by the application ofA’. Specifically, let us set 001/C, ifa1>c
0
ifa1<~
(8.21)
where one reasonable criterion for picking e is c/a0
~‘
noise.
(8.22)
Physically, this procedure has the effect of ignoring the very high frequency components ofg (r). In applying the singular value decomposition to our problem we used an algorithm described by Golub and Reinsch [96]. We now apply the procedure to an input spectrum consisting of the superposition of two Lorentz lines (Brillouin) placed symmetrically about a central Lorentz line (Rayleigh) of different height
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
328
and halfwidth. This spectrum can be written in the form
R1
=
~
—
~2
(8 23)
Normalizing the power in the beam to unity, and defining a parameter 2R1 which represents the fraction of total power in both Brillouin lines (the Landau—Placzek ratio), we may write the field correlation function as g(r)
=
(1
—
a) e’~ +
e_I’T
cos T~.
(8.24)
Here we measure time in dimensionless units and express the frequency shift ~ (w1 — w0) (F/y) in the reciprocal of those units. For convenience in the calculations, the linewidths of the Rayleigh and Brillouin components are taken to be equal; this is very close to the truth in many experiments. For the present purpose, let us assume that we know nothing of the input spectrum except the output full photoelectron correlation function. In order to mimic the experimental situation we added 5% noise to the photoelectron correlation function which we calculated as before. Equation (8.22) then e = 0.1, that forM = 20input we should The reconstruction 2 (r) via thisyielded approach (for so exactly the same data asset in at9 the formal000. inversion) is also shownofin gfig. 8.1, see the solid circles. The majority of reconstructed points are now very close to the true curve in spite of the fact that we included 5% noise rather than the 1% noise for the formal inversion. At present there is no corresponding analysis for the clipped correlation function. This is an interesting problem and deserves further work. 8.3. Parameter estimation When the general form of the spectrum of the incident light is known, it is useful to add this information to the specification of the problem. The most common way to do this is to describe the spectrum by specifying a set of parameters (e.g., line width, etc.) which we will denote by ~ . . . ,/3,,. It is convenient to regard the whole collection of parameters as a vector i~having components (j3~,~ . . ,f1,,). Our problem now reduces to the estimation of these parameters from the measured data of the full or clipped photoelectron correlation function. We consider both the full photoelectron correlation function c(T) and the clipped photoelectron correlation function ck(T). Let us denote the set of predicted values of either of these functions over a mesh of time arguments by the vector (8.25)
where we include ~ in the argument to remind us that the components of the predicted correlation functions depend upon the parameters. The measured values of the correlation functions will be denoted by the vector
{e}
1
c(r1)
or ck(r,)
(8.26)
with the j~omitted. We now face the problem of selecting 13 so that ê(J3) is as “close” as possible to the measured values ê. A common and useful way of formulating this problem is the method of least squares in
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
which we select that j~which minimize the quantity —ê~][ê(13)—ê]. IIê(~)—êII200 [eq3)~
329
(8.27)
If the dependence of ê(~)upon 13 is linear, one can derive an expression for ~ in closed form. More generally it is necessary to solve for 13 by iteration. The calculation is straightforward (see Wolberg [97]) and we m~relystate the result: ~ 13n 1T~(!3n)[Vf(.8n)]~f(!3n) where f3,.~ is the nth iterate. The various functions are defined by: fi~
ê—
ê(j3)
f(13)}iI
(8.28)
(8.29) {ê(1~)}
{t(Th~’
1
(8.30)
(8.31)
~[Vf(1~)]~[vf(j3)1.
To apply eq. (8.28) to the full photoelectron correlation function it is easiest to make direct use of eq. (8.11), as the derivatives ofH are easily written down. Applying eq. (8.28) to the clipped correlation function requires taking eq. (5.47) as a starting point. The problems of multiple parameter estimation for complicated spectra have been discussed by Blake and Barakat [94]. In this subsection we will therefore treat only a simple illustration involving a single parameter. For the Lorentz spectrum the field correlation function is g(t)
=
e~
=
(8.32)
eI’ITI
where we measure time in dimensionless units chosen to make F near to unity. The vector of parameters 13 now consists solely of F. Some typical numerical results obtained by using eq. (8.28) are summarized in table 8.2. The simulated experimental values for the full correlation function were calculated form eq. (5.32) with N = 6. The clipped correlation function’s simulated values were calculated from eq. (5.47). We added random noise (see column 3) to the calculated values an three experiment. 2~ to0.imitate The first entries in table 8.2 show In deriving eq. (8.11) we assumed that g(M~)I Table 8.2 Parameter estimation of r for Lorenta spectrum for full and clipped
correlation functions T
M
% Noise
F
Full (N
0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.1 0.2 0.2
20 10 5 20 10 20 20 20 20 10
5 5 5 5 5 2 10 15 5 5
1.3 1.3 1.3 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1.294 1.276 1.107 1.001 1.002 1.000 1.034 1.046 1.028 1.017
=
2)
Clipped (N = 2)
-
— — — 1.005
1.007 1.001 1.030 1.101 1.012 1.001
330
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
cases where this is not so; in fact Jg(M~)f2 0.223 whereM 20; 0.472 forM 10; 0.687 for M 5. Thus M = 5 and 10 violate this condition severely and this is reflected by the errors in the fitted values of F by 23% and 5% respectively. ForM = 20 the fitted value is correct to 0.7%. We anticipate no significant error on this account as long as g(M~)I2 <0.2 which is true for the rest of these calculations. In practice, this restriction requires that the correlation instrument has enough channels to scan at least one correlation time. Subsequent entries in table 8.2 show fitted values for counting times of 0.1 and 0.2. While the short counting time approximation, eq. (8.5), would tend to underestimate F as the counting time increases, the N 2 solution does not, and remains accurate for all counting times <0.5. The iteration scheme is quite insensitive to noise and converges in 3 or 4 iterations even in the presence of far larger amounts of noise than shown here.
9.
Influence of spatial correlations
In section 3 we made the assumption that all the atoms of the photodetector experience the same field; that is, that the detector subtends an angle negligible compared to a single coherence angle. In any practical experiment this assumption is not rigorously true, and it is therefore appropriate to examine the alterations in some of our previous results caused by the effects of partial spatial coherence. 9.1. Heuristics of spatial coherence
Before commencing with the mathematical treatment we will find it useful to examine the problem heurestically, particularly in order to point out why one would expect to see that the modifications to the simplest results required by finite detector area resemble the modifications required by finite counting time. One can determine information about a spectral profile by examining the correlation of photons in time. Similarly, one can determine information about a spatial profile by examining the correlation of photons over a region in space. In fact the first practical photo correlation experiments were of this second type, namely those made by Hanbury Brown and Twiss [98, 99] to determine stellar diameters. We begin by imagining two detectors, each of negligible size, placed at the points r 1 and r2. An experiment is performed to determine the quantity (n1n2> where n1 is the number of photocounts recorded by detector/in a time T, negligible compared with the time over which the intensity of the light fluctuates. The value of (n1n2> will, of course, depend upon the type of light incident, and in most cases, upon r1 and r2. First, let the detectors be illuminated by perfectly coherent light (strictly speaking, by light2)(r with coherence of order two or greater). Under our assumptions (n1 n2) will be proportional to G( 1t1 ,r2t2 r2 t2,r1 t1), and since the criterion for second order coherence is that 2)(rit G( 1,r2t2r2t2,r1t1)= G~’~(r1t1,r1t1)G~’~(r2t2,r2t2) (9.1) then (n1n2> will not depend upon r1 or r2 as long as the field is stationary in space. That is to say there is no Hanbury Brown—Twiss (HBT) effect. This is reasonable, for within such a field there are no fluctuations in intensity; the only fluctuations in n, arise from shot noise and the shot noise in the two detectors will not be correlated. Clearly also, if we set r1 = r2 and looked at correlations in time
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
331
for the photocounts, we would also find (n1n2) (n>2. We might explain both these results in an alternative way by observing that the fourth moment of a 6-function distribution is just the square of the second moment. For coherent light the P-function is given by P[{ak}]
1~1 6(c4—nk)
(9.2)
and therefore (n 1n2> = (n1)(n2), no matter whether r1 = r2 or ~1 t2, or both conditions held at once. Next, let the detectors be illuminated by quasimonochromatic Gaussian light described by the distribution function eq. (2.36). In this case we have 2~ (r 1~(r (r 1~ 1~ (r 2 (9.3) G~ 1t1,r2t2 r2t2,r1t1) = G~ 1t1,r1t1)G~2t2,r2t2) + IG~ 1t1,r2t2)( when r 2. This is a consequence of the 1 = r2, and t1 = t2, therefore, we will always find (n1n2) = 2(n) fact that the fourth moment of a Gaussian distribution is always twice the square of the second moment. Now as r 2. If, for 1 departs from r2, of (n1n2) will decrease until value (n> (n example, the source consists two points separated byitareaches distancethe d, limiting we would expect 1 n2) to decrease as a function of r1 — r21 in such a way that 2when the angle between r1 and r2 was largeis for light with a mean wavelength A. This to sin~(Aid), would be close to (n)[1001 which says that we would expect light acompared consequence of the Van(n1n2) Cittert—Zernike theorem striking different maxima of a diffraction pattern to be incoherent. Were the source extended in space with characteristic length d (for example, a star of diameter d) we would expect roughly the behavior shown in fig. 9.1. This is analagous to the decay of the time correlation function (n 1n2) as It1 — t2~becomes large compared with (L~w)’ where (sw) is the spectral width of the light.
2:~~-
(r1r2)/J~ Fig. 9.1. Schematic plot of the correlation ofn(r,) n(r2) as a function of (r~— r2)/ .JX~.i~Ais the “coherence area” defined for a particular geometry in eq. (9.34).
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
332
9.2. Generating function
The heuristic ideas presented in the previous section can be made more precise. We will follow in broad outline Kelly’s treatment [22] of the problem because it interleaves closely with the formalism developed thus far in the essay. However our analysis is somewhat less general than Kelly’s.
We have already made several assumptions about the nature of the Gaussian g-field: it is stationary in space and time, its spectrum is symmetric about some central frequency w0 (quasimonochromatic). To these we add another: the maximum dimensions of the source are much less than R0, where R0 is the average distance between source and detector. When these assumptions apply, we can write G~’~ in the form 1~ (r 2)g(y G~ 1t1,r2t2) = (111 1 —y2)h(r1 — r2) exp {—io.,0 (x1 x2)} (9.4) —
where g(t) —
1(w) e~w
1~ (r G~ 1 t, r2t)/ G~’~ (00,00) 2w 1r1 0 x—t—. 2R0C
h (r 1
(9.5)
tdw
J
(2)
r2)
In y~t___, c
(9.6) (9.7)
The r2). The van Cittert—Zernike theorem requires that 1~ (r shape of the source determines h (r1 G~ 1t1, r2t1) be determined by the diffraction pattern that would be produced were the source viewed as a mask through which plane monochromatic light were allowed to pass; that is, —
(9.8)
2R h(r1
—
r2)
~-
~ exp {—iw0(r1
—
r2)/cR0}d
where B is the area of the source. We have already seen in section 3 that the double generating function Q(A 1 ,A2) can be expressed in terms of the eigenvalues of an integral equation, eq. (3.59). In order to account for the fact that the detector has finite area we must modify eq. (3.59) by including explicitly the integration over the area of the detector ~-
[A1
J
dt’
J
2r’ + A d
2 ~ dt’
J
1~ (r 1t1,r1’t’)41(r’1t’)
G~
d2rIJ
.~!
41(r1t).
(9.9)
The evaluation of the eigenvalues will follow much the same technique as discussed in section 4, with the emphasis now on including the effects of finite detector area A instead of finite counting time T. We assume that T is much smaller than the time over whichg(t) varies significantly. Thus we can perform the integration over t’ trivially in eq. (9.9). The result is 2r’ + 1~(t,r)
=
~
)‘~(0,r’)h(r— r’)g(t) d
l’~(r,r’)h(r
—
r’)g(t
—
r) d2r’
(9.10)
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an
essay
333
where we have defined T+ t
2/2R
exp {iw0 1r1
Y (t,r)
0C}
~
exp {—iw0t’} 4~(rat’) dt’.
(9.11)
The kernel of eq. (9.11) factors into space and time components. Therefore it is reasonable to look for eigenfunctions which factor similarly Y1(t,r)
=
Y1 (r,0)f1(r).
(9.12)
We shall require the f1 (n) to satisfy orthonormality conditions similar to those required of the eigenfunctions 41(t) 2nl=tI, (9.13) ~ f*(r?)f(rI)d and 2r’
b 1f1(r)
-~-
(9.14)
~ h(r—r’)f1(r’)d
where the b 1 are the spatial eigenvalues corresponding to the geometry of the particular detector area A and particular source which has determined h (r — r’). Upon substituting for Y,(t,r) in eq. (9.10), we find that any eigenvalue m must satisfy
(0,0)b,[A1Y1(0,0)+A2g*(r)Y(r,0)]
~-
(9.15)
and simultaneously ~
Y1(r,0)
b,[A1g(r)Y(0,0) + A2Y1(r,0)].
(9.16)
These two equations can be combined as a matrix equation .~
~
(9.17)
where A1
A2g*(r)
B
(9.18)
A1g(’r)
A2
lyo,o)
(9.19)
There will be two values of m determined by each l,~.
R. Barakat and J. Blake, Theory ofphotoelectron counting statistics: an essay
334
ln order to solve eq. (9.17) directly in terms of the product form
~k
(I +
we rewrite it in the
rnk),
(Wb,B+f)iyzr(l +rn)Y~,
(9.20)
which displays explicitly all the rn’s determined by the distinct b’s wb1B+f
0
0
0
wb2B +1
0
0
0
wb3B+f
11k
... Y2
(l
+m)
(9.21)
.
Y2
(1 + mk) is the determinant
The product wb
0
0
wb2B+I
0
1B+I
0
...
-
-
0
0
=
wb3B+1
H IIwb,B + I II = II (1 + rnk) I
...
(9.22)
k
where we have made use of the block diagonal structure in evaluating the determinant. Since wb1A1 + I wb 1A1g(r)
IIwb1A+fIl
wb,A2g*(r)
2b?[l =
—g2(r)] (9.23)
1 +(X1 +A2)wb,+XiA2w
wb1A2 + I
we have
2[l —g2(r)]}.
(9.24)
Q’(A1,A2)11 {l +(A1 +X2)wb1+X1A2(wb1)
When the detector is so small that eq. (9.24) can be satisfied only for b 1 then it simplifies to the usual result
Q’(A1,A2)
2[l
=
1, b2
b3
=
—g2(r)].
.
.
.
=
0,
(9.25)
I +(X1 +A2)w+A2X2w
9.3. Photoelectron correlation function The photoelectron correlation function is 2 +g2(r) ~ b~}.
(n 1n2)=
~-
Q(A1,A2)I~~=0
~—
_w2[(~
(9.26)
b,)
Both sums can be evaluated. The kernel of eq. (9.14) can be expanded in a bilinear series ~ b 1)7(r)f,(r’)
(9.27)
R. Barakat and 1. Blake, Theory ofphotoelectron counting statistics: an essay
335
assuming no degeneracy. Now set n = r’ and integrate over A, we have 1
=
~ b,.
(9.28)
Next, multiply both sides of eq. (9.27) by h (r — r’) and integrate twice, once with respect to r and once with respect to r’; the final result is d2r’ Ih(r—r’)I2 ~f3.
d2r
~ b~
(9.29)
Consequently, (n
2[1 +8g2(r)]. (9.30) 1n2)~w Since ~b 1 = 1, and b, >0, therefore, ~b1~ 1 where equality holds only when b0 = 1 and b1 = 0, /> 1. This means that 1~will approach 1 for a detector which subtends an area far less than a “coherence area” and will decrease significantly for larger detectors. This will have the effect of reducing the time dependent component of c(t). Clearly, for a detector subtending many coherence areas c(t) will become constant as such a detector will average over many independent fluctuations. 9.4. Clipped photoelectron correlation function The influence of spatial coherence on the clipped correlation function defmed by eq. (5.10) requires a more elaborate analysis. Equation (9.30) demonstrated that one needed only a single parameter, 13, to characterize the influence of the detector geometry on the full correlation function. This turns out not to be the case for the clipped correlation function; rather, it is necessary to evaluate all the eigenvalues of eq. (9.14). Once this is done, it becomes straightforward, albeit tedious, to differentiate Q (A1, A2) to obtain Ck (r) via the basic expression k — 1
(1)m1 m1!
am, + 1
aA7’~-aA2Q(A1,A2) x,=i
(5.47)
X~=O
m,0
where w (m1) is the average counting rate. Analytic evaluation of the eigenvalues is possible only for certain limited geometries (see subsections 4.4—4.6). One example which admits a closed form solution and which is of considerable experimental interest is a square source of side length a and a parallel square detector of side length b a distance R0 away. In this case we have from eq. (9.8) that -
h(r—r)
sinpa(x—x’) ~ia(x—x)
,
sinua(y—y’) (9.31)
P’z(YY )
where ~.t = w0/cR0. Equation (9.14) becomes bkfk(x,y) =B1F1(X)BmFm (Y) 2~lsin1Aa(x_x’)F(X~).! ~dy’ “~‘ b —b/2 ~za(x—x ) b —b/2 ..!
smn~’~ pa(y—y )
Fm(Y’).
(9.32)
336
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
We have made explicit the factorization of fk(x,y) and bk required by the form of h(x x’, y — y’). Equation (9.32) is a product of integral equations satisfied by the prolate spheroidal wave functions, see eqs. (4.42) and (4.43), provided we set B1 = (ir/2c0)A1, B~= (1T/2C0)Am where the A’s are the —
eigenvalues tabulated by Slepian and Sonnenblick [34]. Clearly bk ~BiBm consequence of the symmetry of the square. The quantity c0 is defined by c0 —pab/2.
is
degenerate, a
(9.33)
c0 is a dimensionless quantity related to the number of “coherence areas” the detector subtends. For our example the coherence area z~.Ais given by 2c2R~/w2a2 = 4ir2/~t2a2. (9.34) i.~A 4ir The number of coherence areas interrogated is thus ‘=
Nb2/I~A
c~/7r2.
(9.35)
When c 0 is small, it can be shown that b0 ~
—i
4c0
A~,
b1=b2=...~0.
(9.36)
Table 9. 1 gives values for the bk as a function of c0. In the case of other geometries, evaluation of the eigenvalues is a very complicated numerical problem. If the source and detector are discs of radii a and b respectively, the analog of eq. (9.32) is bkfk(r)
I
0
2J1 [~aIr—r’I] pair—rI
(9.37)
‘2~r’dr’.
This integral equation has not been studied. Kelly’s results for this geometry are incorrect. Once the eigenvalues have been determined and inserted into eq. (9.24), Ck(r) can be evaluated via Table 9.1 A. First nine eigenvalues bk as a function ofc0 for square geometry
k
c01
c02
c03
c04
c05
0 1 2 3 4 5
.80891 .08871 .08871 .00973 .00019 .00019
.47830 .19317 .19317 .07802 .00786 .00786
.26106 .18993 .18993 .13818 .03992 .03992
.15294 .14008 .14008 .12830 .0730 .0730
.09856 .09664 .09664 .09476 .07736 .07736
6 7
— — —
.00079
.01154
.0415
.06316
.0010 .0010
.0088 .0088
.02712 .02712
8
— —
B. Numbering scheme for eigenvalues bk kO 123456789 nOlOl 212323 mO 011122233
R. Barakatand I. Blake, Theory ofphotoelectron counting statistics: an essay
337
eq. (5.24) by appropriate differentiation of Q(A1 , A2). We will now evaluate explicitly c1 (r) and c2(r). It is convenient to define 2 [1 —g2(r)]}’ (9.38) Q1(A1,A2) {1 + (A1 + A2)wb1 + A1A2(wb1) so that
a C
Q1(A1,A2)
~—H
1(T)w+
a
C 2(r)
=
w+
(9.39)
2
(~—
—
a
8AaA) H Q1(A1,A2).
(9.40)
Tedious manipulations then yield 2(r)
C1(r) =w + 11(1 + wb1Y’
{_~
~
+g
k
,Y’ (—(1 + w) ~
C 2(r)w+H(l
(wbk)2 (l+wbk) wbk
2(r) ~ +g
Ck(r) ~
+
[( wbk
+bwk)
(wb~)2 ~
(wbk)2 ~ + ~
k 1 +wbk k ((1 +wbk)/ The normalized clipped correlation function ck(’r) is
C
(9.41)
+wb k
ck(r)
}
k
(l+wb,~) (wb,~)3 1
(1 +wbk)2i~
(9.42)
Ck(’r) =
(5.50)
.
With this normalization, we have
ck(r)
=
1 + ct~g2(r)
(9.43)
with a 1
=
{ ~k
2
(1 (wbk) + wbk))/{~1II (1 + b 1) 2
a2
~
(wbk)
k
(1 +wbk)
— 1])
wbk
(9.44)
(wbk)3
l/tw[fl (1 + wb 1)
(1 +wbk) +
k
(1 +wbk)2Jik LI
—
~ k
1 wbk ( +wbk)
—
l]}.
(9.45) Figure 9.2 displays the behavior of a1 and a2 as a function c0. All nine eigenvalues listed in table 9.1 A were used in the evaluation. Clearly a1 and a2 approach zero as c0 is made large. For c0 > 3 the eigenvalues fall off very slowly and a large number of them must be retained in order to insure reasonable accuracy. Simple asymptotic expressions for a1 and a2 would be desirable.
R. Barakat and J. Blake, Theory of photoelectron counting statistics: an essay
338
I.e -
a 2
Fig. 9.2. Plot of aj, defined in eq. (9.43) as a function of c0.
Acknowledgments We wish to thank Professor R.J. Glauber for stimulating discussions on problems in quantum optics. Barakat’s work was supported in part by Air Force Office of Scientific Research.
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