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Three-fold photoelectron counting statistics for gaussian light
Volume
16, number
3
OPTICS COMMUNICATIONS
THREE-FOLD PHOTOELECTRON
March 1976
COUNTING STATISTICS FOR GAUSSIAN LIGHT
Julian BLAKE Physics Department,
Harvard University, Cambridge, Massachusetts 02138,
USA
and Richard BARAKAT Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts 02138, USA and Bolt Beranek and h’ewman Inc., Cambridge, Massachusetts 02138, Received
3 July 1975, revised manuscript
received
15 December
USA
1975
The three-fold generating function Q(hl, h2, h3) for photocount statistics is evaluated directly from the gap integral equation in the short counting time regime. The triple joint probabilities and the triple photoelectron correlation function are obtained via Q(hl, h2, h3) in terms of the field correlation function of the incident quasimonochromatic radiation. Three-fold moments are also evaluated via another approach which permits an extension to arbitrary counting times.
While interest has usually been focused on second-order correlations - the full and clipped correlation functions, for example - third-order experiments are both practical and useful. Davidson [ 11, Chopra and Mandel [I], and Davidson and Mandel [3] have used a time-to-amplitude converter to measure the third order photoelectron correlation function for a laser operating near threshold, verifying the calculations of Cantrell and Smith [4]. Recently, Corti and Degiorgio [S] have performed third-order measurements with a fast digital correlator. In this paper we demonstrate how third-order quantities can be extracted from the basic integral equation for Q(hl, h2, X3) which holds in the case of gaussian light. The general approach is a direct extension of a technique outlined previously [6] and accordingly the results hold only for short counting times. The concluding section deals with a simpler problem: the evaluation of product moments. Here arbitrary counting times can be accommodated by integrations over the field correlation functions. Generalizing Jakeman’s [7] integral equation for the double generating function to the triple generating function case, we have
c Ai lg(t t.
i=I
- t’) c$(t’)dt' =;
(1)
@(t) .
I
Here g(t) is the field correlation function for the incident quasimonochromatic radiation and u is the beam strength. The three X are the arguments of the triple generating function Q(xl, h2, h3), and m is the eigenvalue. The tj denote the successive time intervals over which photocounts are accumulating. Taking the three counting intervals to be of the same width T(where T is small compared with the time required for g(t) to change significantly), we have 3 C j=l
+ g(t-
fj>
@Ctj)
=z
#Cr>
.
(2) 303
I
.’
I
where Q is the column matrix in eq. (3). Clearly A has eigenvalues ( 1 + m) and thcrcl’orc
I-or details of this approach as applied to the double generating det A = (1 +hluT)(l
+X2uT)(l
+X,vT)
function
Q(x,. h, ). set’ (h]. I lowem.
+ 3(vT)3hth2X3.~t2~32.7R?t
+ ( I t X,U7YXl h3<& + ( 1 +hJ”T)X, _
~(uT)21(1+X,uT)X2X,RS3
(7)
XRf? I
closed form when T is small. If T IS not s11Iall nt’ cat1 emplo!, the nlatl.i\ Thus we have obtained Q(Al. X2, X3) 111 methods developed by the authors 161. With Q(ht, XI, X2) in closed form, we al-e in a position to evaluate joint probnbilitlcx atlcl IlI~~IlleIItS tl>/ dlI?Ct : differentiation. The joint probability of obtaining 1~~ photoelectrons at ft. m2 photoelectrcms in t ,, anil 111 photoelectrons in t3 is
For example. the probability
of obtaining
P(0.tl:0.t2;0.t3)=Q(l.
I. I)= [(1+tl7’?
Another example of particular
experimental
(‘(Tt,72) =(nI,n12nli)/c’?l)3 t‘or
delay times r1 = r2 hz,tnp~)
=
ml photoelectrc~ns is
luterest is the tl-iple photoeiectron
correlation
+&
+g$ll
‘.(‘)I
function
t2. Since
a3Q0,.X2.W
ax,ax2 ah
(uT)‘(I+PTH&
f IO)
.
f{. 72 = r7
+7(vT)3g12g23~31
I
III) h,zh2=Xizl,
we obtain II‘!
304
Volume
16. number
OPTICS COMMUNICATIONS
3
Fig. 1. Triple photoelectron correlation function for a Lorentz line as a function of 72 for fixed values of 71: - - - 71 = 0; ~ 7, = 0.1, - - 7, = 0.4.
March 1976
Fig. 2. Triple photoelectron correlation function for a Brillouin line (A = 5, 01= 1) as a function of 72 for fixed values of 71: - ~ - 71 = 0; -~~ 71 = 0.1. -. - 71= 0.4.
Note that ~(71, ~2) = c(r2, rI). Since lim g(r2) = 0, T2--= we have lim c(71,72) 72-‘m
= 1 +g2(rI)
(14)
= ~(7~) ,
where ~(71) is the double photoelectron correlation function [4,5]. The behavior of c(rl, r2) is shown in fig. 1 for a Lorentz line shape as a function of 72 for fixed values of 7, . The curves undergo a monotone decrease to their limiting value ~(71). The main effect of increasing 71 is to decrease the value of c(rI, 0) relative to the background value of c(r1). c(rl, 72) for a Brillouin line [6,8] is shown in fig. 2. We get the oscillating behavior characteristic of hetrodyning. A particularly useful (but complicated!) calculation would be the evaluation of the clipped photoelectron correlation functions from Q(hl, X2, h3). 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 T to larger values. We take (mlm2m3) as an example. It follows directly from Glauber [9] and Jakeman [7] (mlm2m3)
= (fi(tI,
{ok)) a(t2,
{ok)) Wt3,
{q)))
,
(15)
where R($, {o(k)) = s j &*(t’, {ok}) b(r’, {ok)) dr’ ,
(16)
‘i and where ( * ) indicates an average using the weight function P({ok}). When P({czk}) is given by a gaussian distribution characteristic of chaotic light, then &(t, {ok}) is a complex-valued gaussian random variable as it is simply a linear combination of the CY~.A convenient summary of the properties of complex-valued gaussian processes is given in [lo]. By using the properties of these processes, we can show that 305
Volume
16, number
, llarch I976
OPTIC‘S COMMUNICATIONS
3
(I&(f’)I* I&(T”+q)l* I&(t”‘+T, +T*)/2)=mw'H2)3 t
ZRe (&*(t’)
+(i&(t’)l*) By interchanging
h,“‘y?‘j~
Substitution
c(r,.
~(f”+~,))(~*(t”+~,) [I(&*(r’)
ensemble
averaging
of eq. (17)
into
eq. (1X)
g(r”‘t7,
+T>) E(t’))
t77))12
t 1(8:‘(/“‘f~,
t”‘(i&(~‘)l*
appropl-iate
to permit permit
[&(f”tT,)i’
I&(t”‘tTJ
normalization
yields ca(Tt. 72)
of other joint
photoelectron
g(f) is real when the radiation
as a constant.
then eq. ( 19) reduces
moments.
cot-relation
functions
Howevel-. the joint
IS cl~~asilnvnocht-onlatI~.
to eq. (I 2). probability
can only he evaluated
was supported with Bolt Beranek
by Air Force Office and Newman
of Scientific
Research
(AFSC)
Inc.
References 1 I 1 1:. Davidson, Phys. Rev. 185 (1969) 446. [2] S. Chopra and L. Mandel, Phys. Rev. Lett. 30 (1973) 60. [3] F. Davidson and L. Mandel, Phys. Lett. 27A (I 968) 579. (41 C.D. Cantrell and WA. Smith, Phys. Lctt. 37A (1971) 167. [S] M. Corti and V. Degiorgio, Opt. Comm. 11 (1974) 1. [61 J. Blake and R. Barakat, J. Phys. A6 (1973) 1196. [71 I,.. Jakeman, J. Phys. .43 (1970) 201. 181 J. Blake and R. Barakat. Opt. Comm. 6 (1972) 27X. 191 R. Glaubcr, ed., in: Quantum Optics (Academic Press. Ne\v York. 1969). 1101 C.L. Mehta, in: Progress in Optics. vol. 8, cd. I;. Wolf (North-Holland. Amsterdam.
306
(17)
(IX)
+77)1’:.
functions
in terms of the
functions.
research
F44620-72-C-0063
function
g(f) to be treated
the evaluation
n12, .,., rn,,) and the clipped
AT-fold generating
$(t’))i2].
T”)
the fact that the field correlation
Similar techniques
-t7$
in eq. ( 15). WC have
72) = I +
We have utilized
Barakat’s
and
+T-,))(&‘:(~“‘+T~
t l(G”(t”+~,)
and integration
= (NY)3 \[’ dr’ _ii d/[‘d
When Tis short enough P(m,,
E(t”t7,))12
&(f”‘+T,
1970)
under Contl-acl
16, number
3
OPTICS COMMUNICATIONS
THREE-FOLD PHOTOELECTRON
March 1976
COUNTING STATISTICS FOR GAUSSIAN LIGHT
Julian BLAKE Physics Department,
Harvard University, Cambridge, Massachusetts 02138,
USA
and Richard BARAKAT Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts 02138, USA and Bolt Beranek and h’ewman Inc., Cambridge, Massachusetts 02138, Received
3 July 1975, revised manuscript
received
15 December
USA
1975
The three-fold generating function Q(hl, h2, h3) for photocount statistics is evaluated directly from the gap integral equation in the short counting time regime. The triple joint probabilities and the triple photoelectron correlation function are obtained via Q(hl, h2, h3) in terms of the field correlation function of the incident quasimonochromatic radiation. Three-fold moments are also evaluated via another approach which permits an extension to arbitrary counting times.
While interest has usually been focused on second-order correlations - the full and clipped correlation functions, for example - third-order experiments are both practical and useful. Davidson [ 11, Chopra and Mandel [I], and Davidson and Mandel [3] have used a time-to-amplitude converter to measure the third order photoelectron correlation function for a laser operating near threshold, verifying the calculations of Cantrell and Smith [4]. Recently, Corti and Degiorgio [S] have performed third-order measurements with a fast digital correlator. In this paper we demonstrate how third-order quantities can be extracted from the basic integral equation for Q(hl, h2, X3) which holds in the case of gaussian light. The general approach is a direct extension of a technique outlined previously [6] and accordingly the results hold only for short counting times. The concluding section deals with a simpler problem: the evaluation of product moments. Here arbitrary counting times can be accommodated by integrations over the field correlation functions. Generalizing Jakeman’s [7] integral equation for the double generating function to the triple generating function case, we have
c Ai lg(t t.
i=I
- t’) c$(t’)dt' =;
(1)
@(t) .
I
Here g(t) is the field correlation function for the incident quasimonochromatic radiation and u is the beam strength. The three X are the arguments of the triple generating function Q(xl, h2, h3), and m is the eigenvalue. The tj denote the successive time intervals over which photocounts are accumulating. Taking the three counting intervals to be of the same width T(where T is small compared with the time required for g(t) to change significantly), we have 3 C j=l
+ g(t-
fj>
@Ctj)
=z
#Cr>
.
(2) 303
I
.’
I
where Q is the column matrix in eq. (3). Clearly A has eigenvalues ( 1 + m) and thcrcl’orc
I-or details of this approach as applied to the double generating det A = (1 +hluT)(l
+X2uT)(l
+X,vT)
function
Q(x,. h, ). set’ (h]. I lowem.
+ 3(vT)3hth2X3.~t2~32.7R?t
+ ( I t X,U7YXl h3<& + ( 1 +hJ”T)X, _
~(uT)21(1+X,uT)X2X,RS3
(7)
XRf? I
closed form when T is small. If T IS not s11Iall nt’ cat1 emplo!, the nlatl.i\ Thus we have obtained Q(Al. X2, X3) 111 methods developed by the authors 161. With Q(ht, XI, X2) in closed form, we al-e in a position to evaluate joint probnbilitlcx atlcl IlI~~IlleIItS tl>/ dlI?Ct : differentiation. The joint probability of obtaining 1~~ photoelectrons at ft. m2 photoelectrcms in t ,, anil 111 photoelectrons in t3 is
For example. the probability
of obtaining
P(0.tl:0.t2;0.t3)=Q(l.
I. I)= [(1+tl7’?
Another example of particular
experimental
(‘(Tt,72) =(nI,n12nli)/c’?l)3 t‘or
delay times r1 = r2 hz,tnp~)
=
ml photoelectrc~ns is
luterest is the tl-iple photoeiectron
correlation
+&
+g$ll
‘.(‘)I
function
t2. Since
a3Q0,.X2.W
ax,ax2 ah
(uT)‘(I+PTH&
f IO)
.
f{. 72 = r7
+7(vT)3g12g23~31
I
III) h,zh2=Xizl,
we obtain II‘!
304
Volume
16. number
OPTICS COMMUNICATIONS
3
Fig. 1. Triple photoelectron correlation function for a Lorentz line as a function of 72 for fixed values of 71: - - - 71 = 0; ~ 7, = 0.1, - - 7, = 0.4.
March 1976
Fig. 2. Triple photoelectron correlation function for a Brillouin line (A = 5, 01= 1) as a function of 72 for fixed values of 71: - ~ - 71 = 0; -~~ 71 = 0.1. -. - 71= 0.4.
Note that ~(71, ~2) = c(r2, rI). Since lim g(r2) = 0, T2--= we have lim c(71,72) 72-‘m
= 1 +g2(rI)
(14)
= ~(7~) ,
where ~(71) is the double photoelectron correlation function [4,5]. The behavior of c(rl, r2) is shown in fig. 1 for a Lorentz line shape as a function of 72 for fixed values of 7, . The curves undergo a monotone decrease to their limiting value ~(71). The main effect of increasing 71 is to decrease the value of c(rI, 0) relative to the background value of c(r1). c(rl, 72) for a Brillouin line [6,8] is shown in fig. 2. We get the oscillating behavior characteristic of hetrodyning. A particularly useful (but complicated!) calculation would be the evaluation of the clipped photoelectron correlation functions from Q(hl, X2, h3). 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 T to larger values. We take (mlm2m3) as an example. It follows directly from Glauber [9] and Jakeman [7] (mlm2m3)
= (fi(tI,
{ok)) a(t2,
{ok)) Wt3,
{q)))
,
(15)
where R($, {o(k)) = s j &*(t’, {ok}) b(r’, {ok)) dr’ ,
(16)
‘i and where ( * ) indicates an average using the weight function P({ok}). When P({czk}) is given by a gaussian distribution characteristic of chaotic light, then &(t, {ok}) is a complex-valued gaussian random variable as it is simply a linear combination of the CY~.A convenient summary of the properties of complex-valued gaussian processes is given in [lo]. By using the properties of these processes, we can show that 305
Volume
16, number
, llarch I976
OPTIC‘S COMMUNICATIONS
3
(I&(f’)I* I&(T”+q)l* I&(t”‘+T, +T*)/2)=mw'H2)3 t
ZRe (&*(t’)
+(i&(t’)l*) By interchanging
h,“‘y?‘j~
Substitution
c(r,.
~(f”+~,))(~*(t”+~,) [I(&*(r’)
ensemble
averaging
of eq. (17)
into
eq. (1X)
g(r”‘t7,
+T>) E(t’))
t77))12
t 1(8:‘(/“‘f~,
t”‘(i&(~‘)l*
appropl-iate
to permit permit
[&(f”tT,)i’
I&(t”‘tTJ
normalization
yields ca(Tt. 72)
of other joint
photoelectron
g(f) is real when the radiation
as a constant.
then eq. ( 19) reduces
moments.
cot-relation
functions
Howevel-. the joint
IS cl~~asilnvnocht-onlatI~.
to eq. (I 2). probability
can only he evaluated
was supported with Bolt Beranek
by Air Force Office and Newman
of Scientific
Research
(AFSC)
Inc.
References 1 I 1 1:. Davidson, Phys. Rev. 185 (1969) 446. [2] S. Chopra and L. Mandel, Phys. Rev. Lett. 30 (1973) 60. [3] F. Davidson and L. Mandel, Phys. Lett. 27A (I 968) 579. (41 C.D. Cantrell and WA. Smith, Phys. Lctt. 37A (1971) 167. [S] M. Corti and V. Degiorgio, Opt. Comm. 11 (1974) 1. [61 J. Blake and R. Barakat, J. Phys. A6 (1973) 1196. [71 I,.. Jakeman, J. Phys. .43 (1970) 201. 181 J. Blake and R. Barakat. Opt. Comm. 6 (1972) 27X. 191 R. Glaubcr, ed., in: Quantum Optics (Academic Press. Ne\v York. 1969). 1101 C.L. Mehta, in: Progress in Optics. vol. 8, cd. I;. Wolf (North-Holland. Amsterdam.
306
(17)
(IX)
+77)1’:.
functions
in terms of the
functions.
research
F44620-72-C-0063
function
g(f) to be treated
the evaluation
n12, .,., rn,,) and the clipped
AT-fold generating
$(t’))i2].
T”)
the fact that the field correlation
Similar techniques
-t7$
in eq. ( 15). WC have
72) = I +
We have utilized
Barakat’s
and
+T-,))(&‘:(~“‘+T~
t l(G”(t”+~,)
and integration
= (NY)3 \[’ dr’ _ii d/[‘d
When Tis short enough P(m,,
E(t”t7,))12
&(f”‘+T,
1970)
under Contl-acl