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Frequency dependence of the polarization of radiation emitted by polarized atoms
Physica 81C (1976) 381-391 o North-Holland Publishing Company
FREQUENCY
DEPENDENCE OF THE POLARIZATION
OF RADIATION
EMITTED BY POLARIZED
ATOMS G. NIENHUIS @s&h
Laboratonum,
Rilksumverslteit
Utrecht, Utrecht, Nederland
Received 10 September 1975
We study the possible sources of frequency-dependent polarrzatron of the radiation emitted by polarized excited atoms immersed in a perturbing gas. For an isolated line we conclude that a frequency dependence may be due to correlation between Doppler and collisional broadening, to anisotropy in the translational states or in the internal states of the perturbers, or to an external field. For an atomic multiplet the effect of lme coupling may cause transfer of radiation emitted by atoms m a certam excited state to lines attributed to other excited states. Thrs would cause a frequencydependent polarizatron. Further the simple proportionality of the intensity of a broadened multrplet line to the population of the corresponding upper state would be destroyed.
in experiments determining the collisional depolarization of resonance fluorescence. The collisions of an excited atom with the perturbing particles become manifest in a twofold way. First, these collisions modify the steady-state population of the excited atoms. Second, the collisions affect the spectral distribution of the fluorescent light with different polarizations. The first effect is the subject of many investigations concerning relaxation and transfer of atomic excited-state multipole moments [7-lo] , and we shall not discuss it further in this paper. The second effect may be looked upon as polarizationdependent line broadening. Practically all theories on spectral-line broadening consider the situation of unpolarized radiating atoms, where no polarrzation of the emitted radiation is present. The frequency dependence of the polarization of the emitted radiation has recently begun to receive attention. Schuller and Behmenburg [ 111 consider the fluorescence radiation emitted by excited atoms in a state with angular momentum Ju = 1, which is populated by absorption of linearly polarized light. The polarization of the integrated intensity is determmed by the relaxation rate of the alignment of the atomic state, and is in general different from zero. The authors claim to show that complete depolariza-
1. Introduction Probably the simplest and best-known way of detecting and analyzing atomic excited states is the observance of emitted resonance radiation. If the atoms are excited by collisions, the intensity and the polarization of the radiation reflect the characteristics of the exciting collisions [l] . If the excitation is due to absorption of radiation, the observed fluorescence is mainly used to obtain information on the collisions which an excited atom has suffered during its lifetime. Deexciting and intramultiplet mixing collisions have been investigated by observing the intensity of different fluorescence lines [2-41. Depolarizing collisions are conveniently studied by measuring the broadening of Hanle-effect signals [5,6]. These experiments have the common feature that the integrated intensity of a certain emission line in a certain polarization is measured, and that one ignores the spectral profile of the line. In the situation of a steady-state population of polarized excited atoms immersed in a bath of perturbing particles, both the frequency distribution and the polarization of the emitted radiation contain information on the interaction of a radiating atom with the surrounding medium. This situation occurs 381
382
G Nlenhuls/Frequency
tron occurs in the quasistatic wmgs of the line, from which they conclude that the polarizatron must depend on the frequency. In the present paper we study the frequency dependence of the polarization of the radiation emitted by a polarized steady-state population of excited atoms immersed in a bath of perturbing particles. Both the case of a singlet excited state and of an excited-state multiplet are considered. We analyze the information that can be extracted from the observed frequency dependence of the polarrzation (or, equivalently, the polarization dependence of the spectral profilej. The analysis 1s to a large extent independent of the details of the line-broadening mechanism, and is equally valid for the impact limit and for the quasistatrc limit. First we consider the case of spherically symmetric line broadening. In the case of a radiative transition between single states we find that the polarrzation is independent of the frequency. Thts contradicts the conclusion of ref. 11. In the case of a transition between multrplet states a frequency dependence of the polarrzatron may occur, even within one of the multiplet components. Thrs arises from the fact that radiation emitted by an atom m one of the states in the excited multiplet may contribute to the intensity of a line corresponding to decay from another excited state, due to the coupling of the lines. This may be important m experimental situatrons where the observed intensity of a multiplet component is used to derive the population of the corresponding state of the excited multiplet. Correlations between Doppler broadening and collisional broadening lift the spherical symmetry of the line broadening. We find that even the singlet state radiation may have a frequency-dependent polarization due to this correlation. This provides a possible means of investigating this correlation directly. We pay some attention to the situation of axial symmetry, which may be due either to the velocity distribution of the radiators, or to an external magnetic field. These situations will be able to produce a frequency-dependent polarization.
dependence
of polaruat+ion
n. The derivation of the equation for the spectral enussivity in thermodynamic equilibrium [ 12, 131 can be directly generalized to include the non-equilibrium case. The resulting equation for the radiant power with polarization vector e, and with frequency between w and w + dw, emitted by a unit volume of the system into the solid-angle element dL! is P(wle) dw da,
(2.1)
where e 1s the unit polarization vector of the detected radiation, and where P(ole) is the polarizationdependent spectral emissivrty, which may be written as
P(h)
= nA
g32: j
dt eeiwr e* * C(r)
l
e.
(2.2)
-cc
The matrix element Cii of the 3 X 3 Cartesian matrix C is defined as a correlation function pertaining to the system consisting of one single radiating atom in a bath of perturbers with density n.
cil(r)
=
Tr~ie-~‘Rpe(llli)Hr~~eelk’Re-(iP)Hr,
(2.3)
where i, j = x, y, z, and I is the ith component of the electric-dipole moment operator of the radiating atom, p is the hermitran density matrix and Hthe hamrltonian of the entire system, R is the position operator of the radiating atom and k is the wave vector of the radiatron in the direction of dSL The derivation of (2.1)-(2.3) is valid provided that p commutes with H, whrch we shall henceforth assume. The matrix C obeys the relation
c-t(t) = cc-0, so that we may substitute
1
in (2.2)
s -
SF
(2.4)
dt e-lWr e* * C(f) * e
_m
2. The polarization-dependent
profile
We consider a density nA of radiating atoms immersed in a gas of perturbing particles with density
=
$ Re r 0
dt e-iwte*
- C(t) - e.
(2.5)
G. Nlenhurs/Frequency
The contribution to the emitted radiation from the transition from a certain upper state multiplet denoted as 1~) to a lower state multrplet 12)is found by taking in (2.3) the matrix elements of p between states of the system with the radiator in lu), by taking matrix elements of I-(~between iZ1 and lu), and of pj between (~1 and II). Now we suppose that the internal states of the excited radiating atoms are uncorrelated to the translatronal states of the system (which we define to include the possible internal states of the perturber). Then we may write for the density matrix in (2.3) (2.6)
P = P” Pt,,
where pU IS the density matrix of the internal states of the atom in the upper state multiplet, and pt, is the density matrix for the translational states of the system. By this assumption we neglect the transfer of the anisotropy of the exerted atoms to the translational motion and to the internal states of the perturbers. This assumption is expected to be generally valid, except maybe for perturbers which are very close to the radiator, and which contribute to the far wings of the lines where the quasistatic theory is valid. In the quasistatic theory of multiplet spectra [ 141 this assumption means that the difference m the Boltzmann factors for the various magnetic substates would be neglected. This is not a serious restriction, since the Boltzmann factor is not very important in most investigated cases. Since the dipole moment p operates only on the internal states of the radiator, we can separate the trace in (2.3) in the trace Tr,, over the translational states of the system and the trace Tri,t over the internal states of the radrator, and we find from (2.3) and (2.6) C(r) = Trint ~1~ pu (k(r) CC~). Here k(t) is a tetradic operator, tion on pUl R(t)p,l
= Tr,,p,,
(2.7) defined by its opera-
e(i/fi)Ht~‘u, eik*Re-(il%%e-ik*R. (2.8)
Note that i?(t)r,l is an operator between the upper states (U I and the lower states II). If the translational
dependence
383
of polaruation
states are in thermodynamic equilibrmm in the absence of external fields, the density matrix pr, is rotationally mvanant. This does not necessarily mean that k is rotationally invariant, since R depends on the wave vector k, whrch defines a direction m space. If Doppler broadening is neglected we may leave out the terms exp(k Ik - R) m (2.8), and k becomes the rotationally invariant tetradic R, defined by (2.9) which describes collisional broadening only. However, m order to restore the rotational invariance it is not necessary to neglect Doppler broadening completely. For if we assume that collisional broadening and Doppler broadening are not correlated, we can write [15] X(r) = K&)
&(r),
where K,(t)
is the correlation
KD(r) = (eik*R(tJe-rk.R)av.
(2.10) function (2.11)
In (2.10) (. . .jav denotes an average over the initial conditions. With this assumption the spectrum is the convolution of a Doppler profile and a collisional profile. For a rotationally invariant density matrix pt,, the function KD(t) is independent of the direction of k, so that Z? given by (2.10) is rotationally invariant. If the collisronal broadening and the Doppler broadening are correlated, the factorization (2.10) is invalid and k is invariant only for rotations about k and the spherical symmetry is reduced to an axial symmetry. The reduction to axial symmetry can be due to other reasons as well. For example, an external magnetic field lifts the rotational invariance of the hamiltonian H, and makes k axially symmetric. Further if the excited states are prepared by light absorption from a laser beam with a narrow spectral distribution, it is possible to select the velocities of excited atoms to have a futed component in the direction of the incident beam, provided that Doppler broadening is not dominated by collisional broadening. The resulting axial symmetry of ptr is reflected in the symmetry properties of & In order to take full advantage of the spherical or axial symmetry, we employ as usual normalized
384
G. NtenhutslFrequency
dependence
irreducible tensorial sets. The tensor T’(JuJ1) 1s defined to have non-vamshing matrix elements only between the upper states wrth angular momentum J, and the lower states with angular momentum J1. The matrix elements are [ 161
of polarlzatlon
where .4k(JU’J1’, JuJ1) 1sa reduced matrix element defined by Ak(Ju’Jl’;JuJI)
= Tr Tl(.l,‘J,‘)t
[aT$(J,J,)]
. (2.17)
An axially symmetric tetradic & couples tensors T,“(JuJ1) with different values of k, and wrth the same values of 4, provided that the symmetry axes 1s taken to be the quantization axis [ 141. where U,M, IJrMr ; kq) 1sa Clebsch-Gordan coefficrent , and M, and M are the magnetrc quantum numbers. The tensors T,2 obey the orthonormality relation Tr,nt T,k(J,Jl>t
@‘(J,‘J,‘)
= 6kkf hqql 6.r~;
and they transform under rotations as the spherrcal harmonics Y[. Further we shall use the three spherical vectors u,, u = -1, 0, 1, which are defined by then Cartesian coordinates = +(I,
fl,
O),
240 = (O,O,
2
T;‘(‘(J,‘+‘) Bq(k’JU’J1’, k J,JJ,
k’J,‘Jl’
(2.18)
~.QJ,
$1
B Tqk(J&) =
(2.14)
1).
These vectors are convenient in finding the spherical components of vector operators. From the WignerEckart theorem [16] we know
where Bq(k’Ju’J{ ; k .I,/,) = Tr T,k’(Ju’Jl’)t [bT~(JuJ1)] . (2.19) In the following sections we shall investigate the spectral drstributron and the polarization properties of the emitted radiation. A complete description of the emrssion is expressed by eqs. (2.2) and (2.7). We shall consider the cases that I? 1s spherically symmetrrt or axially symmetric.
3. Spherical symmetry
= 1
U,ll~llJ~)
(2.15)
T,(J,J,)/~3,
JllJl
where (JUll~l]J1) denotes a reduced matrrx element, which determmes the strength of the electrrc-dipole transition .I, -f Jr. A spherically symmetric tetradrc A operating on the tensors Tt(JuJ1) couples only the tensors with the same value of k and 4, and the coupling IS independent of 4 17, 171
A^Tt(J&)
= 2 J,‘Jl
Tqk(.&‘.$) A&f&‘;
J&),
(2.16)
In thrs section we consider the case of spherically symmetric line broadenmg. This corresponds to the situation that the translational states are m thermodynamic equilibrium in the absence of external fields, and that Doppler broadening is uncorrelated to colhsional broadening. 3.1. Isolated line First we consider a transition from a single upper state with angular momentumJ, to a single lower state with angular momentum J1. The emitted radiation integrated over the frequent: is unaffected by the line broadenmg process. The integrated emissivrty for radiation with polarrzation e is according to (2.2) and (2.7)
G. NienhuislFrequency PC&)
dw = nA -e** 2nC3
C(O)*e,
(3.1)
where
It is convenient
2
of polarization
c(o) = 2
ckq
385
S4”t,
(3.6)
where
C(0) = TrrrrtCclUP”P~.
P” =
dependence
to expand pu in irreducible
Pkq q(J,Ju)t,
(3.2) tensors [7]
(3.3)
kq where Pkq = Tr Pu T,“&&).
ckq = Tr C(0) * Sz.
(3.7)
The coefficient ckq may be considered as the components of the multrpole moments of the emitted radiation. It is interesting to note that the multipole moments pkq of p,, are directly related to the multipole moments ckq of C(0). From standard recoupling considerations one derives [ 161
(3.4)
The complex numbers pkq are called the components of the multipole moments of the rank k of the excited atoms. The monopole moment poo is (2J,, + I)-$ times the fraction of the radiators in the upper state. The hrgher multipole moments describe the distribution of the excited radiating atoms over the magnetic substates. The dipole moment is usually referred to as the orientation, the quadrupole moment as the alignment of the state. The relation between the multipole moments of the excited atoms and the characteristics of the emitted radiation have been discussed by D’yakonov [ 181 and by Fano and Macek [ 11. We give here some equations for thrs relation in a slightly different form, that is suitable for our purposes. We first wish to draw attention to the fact that the 3 X 3 matrix C(0) describes the integrated intensity of the emitted radiation in every direction and with any polarizatron. This integrated intensity depends upon the polarization vector e, but it is independent of the direction of emission specified by k.This is typical for dipole radiation. For convenience we introduce a complete orthonormal set of 3 X 3 matrices Sg, which are built by recoupling the spherical vectors u0
X [(2kl + 1) (2k, + l)]:
(KQlk2q2; k,ql)
(3.8)
in terms of a Wigner 6j-symbol. we derive
From (2.15) and (3.8)
Tr PI,, T,k(JuJu) I&I = Mk (J,J,)
Sk4’
where ikf&&)
= (-1
Mk(J,/l)
(3.9)
is grven by
J,,+Jl+kcl
l(Jull/-dlJ1)12
(3.10) From (3.2), (3.3), (3.6) and (3.9) it follows that ckq = Mk CJuJd &q 3
(3.11)
and (3.5) c(o) = 2
for k = 0, 1, 2 and -k < q d k. Explicit expression0 for the matrices St are given by Carrmgton [ 191. We can expand C(0) in terms of the matrices S$
Pkq ‘& (JuJd s$t
(3.12)
ks We recover the result that only the monopole, the dipole and the quadrupole moment of p,, contribute
G. Ntenhuls/Frequency
386
to the integrated intensity. The multrpole moments of C(O), which are proportional to the multipole moments of pU, have a direct physical mterpretatron. The monopole moment co0 determmes the intensity averaged over polarrzations and drrections. The dipole moment clq determines the degree of circular polarization, and the quadrupole moment c2q determines both the linear polarization and the anisotropy of the unpolarized intensity [18, 19, 11. Now we turn to the frequency dependence of the emitted radiation. Since we consider an isolated line, JU and Jf assume only one value. Due to the assumed rotatronal invariance of rz‘, we can apply eq. (2.16) to l?
dependence
of polarzatlon
a minor fraction of all radiators [23]. Thus the argument of these authors indicates that our assumption (2.6) might not be adequate to describe the far wings of the line. 3.2. Multiplet of lines Now we consider a multiplet of upper states wrth angular momenta JU, J,‘, . . ., and a multiplet of lower states with angular momenta Jl, J1’, . . . . We assume that pu 1s diagonal m J,,, in accordance with the assumption that the density matrix p commutes with the hamiltonian H. The multipole expansion of pa 1s then
PU= 2 2 Pkq(Ju) T,k(JuJ,,i. From (2.15) and (2.7) we then find the simple result
C(t) = KI(t)
C(O) = I$(0
2 P~qJf&,J1) kq
S$tO.W
(3.15)
JU kq We use the rotational (2.15) and (2.16)
invariance
of g(t), and apply
(3.14)
&MJu’J1’) = 2 lu KI(JUJ1t J,‘J+> which IS the product of a function of time and a time-independent matrix. Combined wrth (2.2), this equality shows that the spectral profile is equal to the Fourier transform of Kl(t), for every polarization. In other words, the polarization properties of the emitted radration at a certain frequency are the same as the polarization properties of the radiation integrated over the frequency. We conclude that for an isolated spectral line in the case of spherically symmetric lure broadenmg the polarizatron IS independent of the frequency. Schuller and Behmenburg [ 1 l] obtain the contrasting conclusron that the far wmgs of the line are completely depolarized. This conclusion is arrived at by the intuitive argument that the far wings are mainly due to strong adiabatically reorienting collisions, which on the average fully randomrze the orientation of the radiators. Followmg thrs argument these authors propose that the density matrix of the radiating atoms involved m a strong interaction is proportronal to the unit matrix, and therefore these atoms produce unpolarized far wings. One should notice that this unit density matrix refers exclusively to strongly interacting radiators, which constitute only
JuJl x (J,‘IIfillJ1’) (3.16)
(Jull~llJ~l> ’
An expression for C(t) is found from (2.7), (2.15), (3.9) (3.15) and (3.16).
(J,‘ll~.(llJl’) KdJUJi;JU’J+)
(J
U
llpjlJ,)
I
(3.17)
From (3.17) we see that the atoms in the state IJ,) decaying to the state IJ1) contribute to C(t) a term of the type (3.12), multiplied by a time-dependent factor. Only the multipole moments of pU with rank k = 0, 1 and 2 contribute to C(t). The frequency distribution of the radiation emrtted by the atoms decaying from IJ,> to IJ1>is determined by the Fourier transform of the time dependent factor
G. Nienhm/Frequency
between square brackets in (3.17). This factor is independent of the multipole moments of pU. If the coupling between different transitions is negligible, K, is diagonal in (JUJI), and the spectrum is the sum of spectra of the sort described by (3.14) with polarizatron properties dependent on J, and J1. A non-negligible line coupling does not alter the polarization properties of the radiation due to a certain transition Ju + J1. But line coupling does redistribute this radiation as a function of frequency, so that the intensity is partly transferred to the lures corresponding to other transitions. This means that even rf only one component IJ,) of the upper state multiplet is populated, the emitted radiation may show lines which correspond to transitions from a different component IJu’). For example, in the classical impact lirmt for a non-moving radiator, the tetradic k(t) is given by [ 14, 17, 201 J?(t) = exp [- nta + (i/fi)t;t]
for t > 0,
where the tetradic h is the commutator internal hamiltonian h of the radiator
(3.18)
with the
dependence
387
of polarzation
is of the order of J-mixing cross sections. If the frequency splittmgs of the multiplet are much larger than nuu,,, , the off-dragonal elements are negligible and line coupling is not important. In general, if the intensity of a certain line is used as a measure of the population of the corresponding upper state, the effect of line coupling may cause an error of the order of nuo,/Aw, where Aw is a typical value of the frequency sphttmg of the multrplet. Fortunately, m several important cases line couplmg 1s absent for symmetry reasons. In particular, if the semiclassical scattering matrix S is diagonal in J1 and Ml, one finds from (3.20) that Ak(JuJ1; Ju’J1’) vanishes unless J, = Ju’ and J1 = J1’, and that the diagonal elements are independent of k. Thts is true if the lower state has only one component with angular momentum J, = 1, provided that the perturbers have zero angular momentum [21, 141. Resonance lines of alkali atoms perturbed by inert gases are an important example of the case that line coupling is zero for symmetry reasons. But m general one should be careful in utilizing the intensity of multiplet lines broadened by collisions as a measure of the population of the components of a multiplet.
(3.19)
h^/.&I = ]h, Pull and where a is a tetradic with reduced matrix elements
4. Axial symmetry In this section we derive expressrons for the polarization and the frequency drstributron in the case of axially symmetric line broadening. For simplicity we consider a transition between two single states with angular momenta J, and J1. The integrated intensity is determined by C(O), which is given by (3.12). In order to find the frequency distribution of the intensity, we must consider the variation of k(t) with time. If we choose the symmetry axis to be the z-axis, we may write according to (2.18) and (2.15)
X (J,‘M,‘ISIJ,M,)*
(J+14&91JIMI)
.I
.
(3.20)
In eq. (3.20) Pm is the Maxwell velocity distribution for the perturbers with mass m, and S is the semiclassical scattering matrix for a collision with initial relative velocity u and impact parameter b. The offdiagonal elements of K, are due to the off-diagonal elements of AI, which may be of the order of uun,, where u is a typical velocity of a perturber, and cr,
X (JulI/41J1)/d3.
(4.1)
It is convenient to express the matrix C(t) by its spherical matrix elements U, * C(t) * u,~. We apply l
388
G Nienhurs/Frequency
(2.7)
(2.15), (3.3), (3.8) and (4.1) and we find
dependence
of polarzatlon
properties of i(
X
lqJl/.41J,>12.
(4.2)
(4.3)
and Kq(k
From (4.2) we see that the multipole moments pkq of pU with rank k > 2 may contribute to the frequency distribution of the emitted radiation in the case of axial symmetry. In general eq. (4.2) cannot be separated in an overall time-dependent factor independent of the polarizations u and CJ’and a polanzation-dependent factor which does not depend on time. Hence we conclude that in the case of axial symmetry a frequency-dependent polarizatron of the radiation is to be expected. In section 2 we mentioned three possible reasons why the spherical symmetry would be reduced to an axial symmetry. (I) correlatron between collisional broadening and Doppler broadening, (ii) an axially symmetric velocity drstribution of the radiators, (iii) an external magnetic field. Case (i) is different from the other two in that the symmetry axis in this case is given by the observation direction of the emitted light. Hence only the values u, u’ = +l are physically relevant m this case, since the polarization vectors must be orthogonal to k. Only the 2 X 2 submatrix in the xy-plane of C(t) is observable. The radiation in some other direction is described by substituting in (4.2) atomic multipole moments piq wrth respect to the new symmetry axis. The 2 X 2 submatrix of the resulting matrix C in the plane orthogonal to this new axis determines the radiation in this other direction. In the two cases (ii) and (iii) the symmetry axis has a fmed direction in space, and every polarizatron wrth respect to this symmetry axis is observable. Further it is important to note that the timedependent functions K,,~(K; 1 I t) in (4.2) may not be independent. We may analyse the symmetry
; k JuJl I t)
JUJ1, k’J&‘(
r) = (-)k-
k’K_q(k.IU.Jl;
k’J,‘$J
r).
(4.4) In case (iii) of an external magnetic field the symmetry is slightly different, since the magnetic field 1s a pseudo vector and remains invariant under space reflection, whereas the wave vector and the velocrty are reversed. For case (iii) eq. (4.4) 1s no longer valid, but the symmetry relation (4.3) still holds. We conclude from (4.4) for the situation of a single transition that the time-dependent factors in (4.2) satisfy the equality
&(K; lit) = (-)“-k_t(~;
IIt)
(4.5)
in the cases (i) and (ii), but not in case (in). First we consider the case (i) of correlated Doppler and collisional broadening. Then we have to consider only the values u, u’ = +l. Due to eq. (4.5) the time-dependent factor K+ 1(~; 1 It) in (4.2) may be factored out for every value of K. Hence the polarization becomes independent of the frequency if only one value of K contributes to (4.2). This is true in two different cases. The first case is that the atomic upper state is unpolarized so that only the monopole moment poo differs from zero. Due to the Clebsch-Gordan coefficient in (4.2) we then only have to consider the value K = 1. For the relevant 2 X 2 submatrrx we then find
x lcJ,ll/41J,~12,
(4.6)
G NienhuisjFrequency
for o, (3’= f 1, where we used the explicit expression for a Gj-symbol where one of the j’s is zero [16] K K
0 = (-@
( Jl
Jl
+J2+ K[(24
+ 1) (2K + I)] +.
J2 I (4.7)
In this case the radration 1s unpolarized and isotropic for every frequency, and the spectral profile is given by the Fourier transform of K, (1; 1 I t). The second case that the polarizatron is independent of the frequency occurs when one of the angular momenta J, and J1 is zero. If J, = 0, k and 4 can attain only the value zero, and eq. (4.6) is valid. If J1 = 0, we must have J, = 1 for a drpole transition. For J, = 1 and J1 = 0 we must have K = 1, since K is found by recoupling J, and J1. From (4.2) we then find for the 2 X 2 submatrrx
dependence
measured frequency-dependent 2 X 2 polarization matrix in this direction may provide information on the multipole moments pkq with k > 2, and on the Fourier transform of K1(~; 11 t) for different values OfK.
Finally we consider the cases (ii) and (iii), where the symmetry axis has a fixed direction m space. The frequency-dependent polarization properties of the radration in every direction are then determmed by the time-dependent Cartesian 3 X 3 matrix C(t), which is given by eq. (4.2). This matrix may be determined experimentally by measuring the frequency-dependent 2 X 2 polarizatron matrix of the radration in three independent directions. This contrasts with case (i), where C depends on the direction of the detected radration. Even if the atoms are unpolarized so that only the multipole component poo differs from zero, the emitted radiation may be polarized. We find from (4.2) for thrs case
“,: * C(t)y,~ =K,(l;
Ilt)
2 pkqMk(’ kq
O)$S$+
=6,,,K,~(l;
lit)
PO0 3(2J, + 1);
lu,,~. (4.8)
So in the special case that Ju = 1 and J1 = 0, the spectral profile is independent of the atomic multipoles, and the polarization is independent of the frequency. In other cases when J1 f 0, and when the excited atoms are polarized, we may expect a frequency-dependent polarrzation. The mUltipOle moments pkq with k = 0, 1, 2 are determined experimentally by measuring the polarization properties of the integrated intensity in three independent directions. The 2 X 2 submatrix of C(t) determining the frequency-dependent polarization properties of the radiation in a certain z-direction is a linear combination of the 2 X 2 matrices m the xyplane
X l~Jull/dlJ1~12.
(4.9)
It is advantageous to expand C(t) given by (4.9) in the matrices Sg. If we use the definition (3..5), or the explicit expressions given by Carrington [ 191, we find
C(t) =
X
PO0 t I
&[K1(l;
llt) + K,(l;
llt) + K_, (1; llt)] S;+
(
+$
[K1(l,
llt) - K_1(1;
-LIK1(l;llt)+K_l(l;
+& [(-)K--]lt(l+d) 2 u,u’=* 1
389
of polarization
llt)l S;+ lIt)--o(l;llt)J
U, (KU’llU;k~)#;‘,
where the sign accounts for the symmetry relation (4.5). The coefficients of these matrices are determined theoretically by eq. (4.2). Comparison with the
x s;+
1
.
(4.10)
In case (ii) eq. (4.5) is valid, and the coefficient of SA vanishes. This means that for unpolarized atoms an axially symmetric velocity distribution can induce a
390
G. Nlenhuzs/Frequency
linear polarrzation, but no circular polarizatron. The radiation with linear polarization vector m the direction of the symmetry axis has a spectral profile which is the Fourier transform of K,(l) 1It). The radiation wrth polarizatron orthogonal to this axis has a spectral profile which is the Fourier transform of K 1(1; 1(t) = K-,(1; llt). In case (iii) eq. (4.5) is mvahd, and the radiation emitted by unpolarrzed atoms will drsplay both circular and linear polarization, which 1s what we expect for a Zeeman spectrum. The integrated intensity in both cases 1s unpolarized and isotroprc.
5. Conclusions This paper 1s devoted to a study of the frequency dependence of the polarization of the emission by polarized atoms, broadened by the mteraction with a perturbing gas. This study is relevant to investigations concerning the Hanle effect and the depolarization of resonance radiation, and may be considered as a theory of polarization-dependent line broadening. We distinguish the case of a transition between two single states, and a transition between two multiplet terms. For a single transition, the polarrzation of the integrated intensity in every directron is described by a 3 X 3 cartesran matrix C(O), eq. (3.2). By measuring the polarization of the integrated intensity in a certain direction, one determines a 2 X 2 submatrix of C(0) m a plane orthogonal to this direction. This 2 X 2 matrix is m fact the polarizatron matrix of the radiation in thrs direction [22]. The multipole moments of C(O) are proportional to the multrpole moments of the excited atoms. The monopole moment coo determines the integrated intensity averaged over polarrzations and directions. The components of the dipole moment clq determine a vector c1 which is proportional to the orientation of the excited atoms. The circular polarization of the radiation emitted in a certain directron is proportional to the component of the vector cl in that direction. The quadrupole moment of C(0) is proportional to the alignment of the excited atoms, and determines the linear polarization and the anisotropy of the emission. Multipole moments of the excited atoms with rank k > 2 do not show up in the integrated radiation.
dependence
o$ polarzatrw
In the case of a single transition we further drstinguish the case of spherically symmetric and axially symmetric line broadening. For spherical symmetry the line broadening is independent of the polarization, and hence the polarrzation does not depend on the frequency. If the velocity distributions are maxwellian and if no external fields are present, spherical symmetry holds when the Doppler broadenmg and the collisional broadening are uncorrelated. This means that a frequency-dependent polarization must be due to correlations between Doppler and collisional broadening. This provides a way of detecting and investigating this correlation directly, by measuring as a function of the frequency the polarrzation of the emission of polanzed atoms. This conclusion 1s valid irrespective of the details of the line-broadening mechanism. We only assumed that the transfer of the anisotropy of the excited atoms to the perturber system is negligible. If the spherical symmetry is reduced to an axial symmetry, be it by an axially symmetric radiator velocity distnbutron, or by an external magnetic field, the multipole moments of the excited atoms with rank k > 2 may contribute to the frequency distribution of the polarized radration. The frequency-dependent polarrzation properties of the emission in every direction 1s described by the Fourier transform of a time-dependent 3 X 3 matrix C(t). The multipole expansion of C(t) is found by coupling the multipole expansion of the excited atoms to the anisotropy of the line broadening process. The polarization-dependent profiles contain more mformation on the line broadening mechanism than the line profile emrtted by unpolarized atoms. The case of an atomic multiplet of lines is considered only for spherically symmetric hne broadening. Every transition Ju + J1 emits radiation with polarizatron propertres that are unaffected by the other transitions. But the frequency distribution of this radiation may be modified by coupling to other transitions Jn’ + Jr’. Even if only one upper state is populated, the emrssion may show lines which correspond to transitions from different excited states. This line coupling effect may cause systematic errors m experiments where the intensity of broadened multiplet lines is measured m order to find the population of a component of an excited state multiplet.
G NtenhuisfFrequeney
Summarizing we conclude that a frequencydependent polarization may be due to correlations between Doppler and collisional broadening, to anisotropy of the translational states of the system, to an external field, or to overlap or coupling with nearby spectral lines. Srmilar conclusions are possrble for the absorption of polarized light by polarized atoms.
References [l] U. Fano and J. H. Macek, Rev. mod. Phys. 45 (1973) 553. [ 21 R. Serwert in Springer Tracts in Modern Physics, G. Hohler, ed., vol. 47 (Springer Verlag, Berlin, 1968) p. 143. [ 31 M. Stupavsky and L. Krause, Can. J. Phys. 46 (1969) 2127. [4] P. L. Lrmse, P. J. Th. Zeegers and C. Th. J. Alkemade, J. quant. Spectrosc. radiative Transfer 13 (1973) 1033. [5] J. P. Barrat, D. Casalta, J. L. CoJan and J. Hamel, J. Physique 27 (1966) 608. [6]A.GallagherandE.L.Lewis,Phys.Rev.A10(1974) 231.
dependence
of polarization
391
[ 71 A. Omont, J. Physique 26 (1965) 26. [ 81 M. 1. D’yakonov and V. 1. Perel’, Sov. Phys.-JETP 21 (1965) 227. [9] C. G. Carrmgton, D. N. Stacey and J. Cooper, J. Phys. B: Atom. molec. Phys. 6 (1973) 417. [lo] V. N. Rebane and A. G. Petrashen, Opt. Spectrosc. 37 (1974) 472. [ 111 F. Schuller and W. Behmenburg, Z. Naturforsch. 30a (1975) 442. [ 121 E. W. Smith, J. Cooper, W. R. Chappell and T. Drllon, J. quant. Spectrosc. radiative Transfer 11 (1971) 1547. [ 131 G. Nienhuis, Physica 66 (1973) 245. [14] G. Nienhuis, Physrca 74 (1974) 157. [ 151 S. G. Rautian and I. I. Sobel’man, Sov. Phys.-USPEKHI 9 (1967) 701. [ 161 U. Fano and G. Racah, Irreducible Tensorial Sets (Academic Press, New York, 1959). [ 171 M. Baranger, Phys. Rev. 111 (1958) 494. 1181 M. I. D’yakonov, Sov. Phys.-JETP 20 (1965) 1484. [ 191 C. G. Carrington, J. Phys. B: Atom. molec. Phys. 4 (1971) 1222. [20] E. W. Smith, J. Cooper, W. R. Chappell and T. Dillon, J. quant. Spectrosc. radiative Transfer 11 (1971) 1567. [21] A. Omont, J. Physique 34 (1973) 179. [ 221 M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959). [ 231 F. Schuller and W. Behmenburg, private communication.
FREQUENCY
DEPENDENCE OF THE POLARIZATION
OF RADIATION
EMITTED BY POLARIZED
ATOMS G. NIENHUIS @s&h
Laboratonum,
Rilksumverslteit
Utrecht, Utrecht, Nederland
Received 10 September 1975
We study the possible sources of frequency-dependent polarrzatron of the radiation emitted by polarized excited atoms immersed in a perturbing gas. For an isolated line we conclude that a frequency dependence may be due to correlation between Doppler and collisional broadening, to anisotropy in the translational states or in the internal states of the perturbers, or to an external field. For an atomic multiplet the effect of lme coupling may cause transfer of radiation emitted by atoms m a certam excited state to lines attributed to other excited states. Thrs would cause a frequencydependent polarizatron. Further the simple proportionality of the intensity of a broadened multrplet line to the population of the corresponding upper state would be destroyed.
in experiments determining the collisional depolarization of resonance fluorescence. The collisions of an excited atom with the perturbing particles become manifest in a twofold way. First, these collisions modify the steady-state population of the excited atoms. Second, the collisions affect the spectral distribution of the fluorescent light with different polarizations. The first effect is the subject of many investigations concerning relaxation and transfer of atomic excited-state multipole moments [7-lo] , and we shall not discuss it further in this paper. The second effect may be looked upon as polarizationdependent line broadening. Practically all theories on spectral-line broadening consider the situation of unpolarized radiating atoms, where no polarrzation of the emitted radiation is present. The frequency dependence of the polarization of the emitted radiation has recently begun to receive attention. Schuller and Behmenburg [ 111 consider the fluorescence radiation emitted by excited atoms in a state with angular momentum Ju = 1, which is populated by absorption of linearly polarized light. The polarization of the integrated intensity is determmed by the relaxation rate of the alignment of the atomic state, and is in general different from zero. The authors claim to show that complete depolariza-
1. Introduction Probably the simplest and best-known way of detecting and analyzing atomic excited states is the observance of emitted resonance radiation. If the atoms are excited by collisions, the intensity and the polarization of the radiation reflect the characteristics of the exciting collisions [l] . If the excitation is due to absorption of radiation, the observed fluorescence is mainly used to obtain information on the collisions which an excited atom has suffered during its lifetime. Deexciting and intramultiplet mixing collisions have been investigated by observing the intensity of different fluorescence lines [2-41. Depolarizing collisions are conveniently studied by measuring the broadening of Hanle-effect signals [5,6]. These experiments have the common feature that the integrated intensity of a certain emission line in a certain polarization is measured, and that one ignores the spectral profile of the line. In the situation of a steady-state population of polarized excited atoms immersed in a bath of perturbing particles, both the frequency distribution and the polarization of the emitted radiation contain information on the interaction of a radiating atom with the surrounding medium. This situation occurs 381
382
G Nlenhuls/Frequency
tron occurs in the quasistatic wmgs of the line, from which they conclude that the polarizatron must depend on the frequency. In the present paper we study the frequency dependence of the polarization of the radiation emitted by a polarized steady-state population of excited atoms immersed in a bath of perturbing particles. Both the case of a singlet excited state and of an excited-state multiplet are considered. We analyze the information that can be extracted from the observed frequency dependence of the polarrzation (or, equivalently, the polarization dependence of the spectral profilej. The analysis 1s to a large extent independent of the details of the line-broadening mechanism, and is equally valid for the impact limit and for the quasistatrc limit. First we consider the case of spherically symmetric line broadening. In the case of a radiative transition between single states we find that the polarrzation is independent of the frequency. Thts contradicts the conclusion of ref. 11. In the case of a transition between multrplet states a frequency dependence of the polarrzatron may occur, even within one of the multiplet components. Thrs arises from the fact that radiation emitted by an atom m one of the states in the excited multiplet may contribute to the intensity of a line corresponding to decay from another excited state, due to the coupling of the lines. This may be important m experimental situatrons where the observed intensity of a multiplet component is used to derive the population of the corresponding state of the excited multiplet. Correlations between Doppler broadening and collisional broadening lift the spherical symmetry of the line broadening. We find that even the singlet state radiation may have a frequency-dependent polarization due to this correlation. This provides a possible means of investigating this correlation directly. We pay some attention to the situation of axial symmetry, which may be due either to the velocity distribution of the radiators, or to an external magnetic field. These situations will be able to produce a frequency-dependent polarization.
dependence
of polaruat+ion
n. The derivation of the equation for the spectral enussivity in thermodynamic equilibrium [ 12, 131 can be directly generalized to include the non-equilibrium case. The resulting equation for the radiant power with polarization vector e, and with frequency between w and w + dw, emitted by a unit volume of the system into the solid-angle element dL! is P(wle) dw da,
(2.1)
where e 1s the unit polarization vector of the detected radiation, and where P(ole) is the polarizationdependent spectral emissivrty, which may be written as
P(h)
= nA
g32: j
dt eeiwr e* * C(r)
l
e.
(2.2)
-cc
The matrix element Cii of the 3 X 3 Cartesian matrix C is defined as a correlation function pertaining to the system consisting of one single radiating atom in a bath of perturbers with density n.
cil(r)
=
Tr~ie-~‘Rpe(llli)Hr~~eelk’Re-(iP)Hr,
(2.3)
where i, j = x, y, z, and I is the ith component of the electric-dipole moment operator of the radiating atom, p is the hermitran density matrix and Hthe hamrltonian of the entire system, R is the position operator of the radiating atom and k is the wave vector of the radiatron in the direction of dSL The derivation of (2.1)-(2.3) is valid provided that p commutes with H, whrch we shall henceforth assume. The matrix C obeys the relation
c-t(t) = cc-0, so that we may substitute
1
in (2.2)
s -
SF
(2.4)
dt e-lWr e* * C(f) * e
_m
2. The polarization-dependent
profile
We consider a density nA of radiating atoms immersed in a gas of perturbing particles with density
=
$ Re r 0
dt e-iwte*
- C(t) - e.
(2.5)
G. Nlenhurs/Frequency
The contribution to the emitted radiation from the transition from a certain upper state multiplet denoted as 1~) to a lower state multrplet 12)is found by taking in (2.3) the matrix elements of p between states of the system with the radiator in lu), by taking matrix elements of I-(~between iZ1 and lu), and of pj between (~1 and II). Now we suppose that the internal states of the excited radiating atoms are uncorrelated to the translatronal states of the system (which we define to include the possible internal states of the perturber). Then we may write for the density matrix in (2.3) (2.6)
P = P” Pt,,
where pU IS the density matrix of the internal states of the atom in the upper state multiplet, and pt, is the density matrix for the translational states of the system. By this assumption we neglect the transfer of the anisotropy of the exerted atoms to the translational motion and to the internal states of the perturbers. This assumption is expected to be generally valid, except maybe for perturbers which are very close to the radiator, and which contribute to the far wings of the lines where the quasistatic theory is valid. In the quasistatic theory of multiplet spectra [ 141 this assumption means that the difference m the Boltzmann factors for the various magnetic substates would be neglected. This is not a serious restriction, since the Boltzmann factor is not very important in most investigated cases. Since the dipole moment p operates only on the internal states of the radiator, we can separate the trace in (2.3) in the trace Tr,, over the translational states of the system and the trace Tri,t over the internal states of the radrator, and we find from (2.3) and (2.6) C(r) = Trint ~1~ pu (k(r) CC~). Here k(t) is a tetradic operator, tion on pUl R(t)p,l
= Tr,,p,,
(2.7) defined by its opera-
e(i/fi)Ht~‘u, eik*Re-(il%%e-ik*R. (2.8)
Note that i?(t)r,l is an operator between the upper states (U I and the lower states II). If the translational
dependence
383
of polaruation
states are in thermodynamic equilibrmm in the absence of external fields, the density matrix pr, is rotationally mvanant. This does not necessarily mean that k is rotationally invariant, since R depends on the wave vector k, whrch defines a direction m space. If Doppler broadening is neglected we may leave out the terms exp(k Ik - R) m (2.8), and k becomes the rotationally invariant tetradic R, defined by (2.9) which describes collisional broadening only. However, m order to restore the rotational invariance it is not necessary to neglect Doppler broadening completely. For if we assume that collisional broadening and Doppler broadening are not correlated, we can write [15] X(r) = K&)
&(r),
where K,(t)
is the correlation
KD(r) = (eik*R(tJe-rk.R)av.
(2.10) function (2.11)
In (2.10) (. . .jav denotes an average over the initial conditions. With this assumption the spectrum is the convolution of a Doppler profile and a collisional profile. For a rotationally invariant density matrix pt,, the function KD(t) is independent of the direction of k, so that Z? given by (2.10) is rotationally invariant. If the collisronal broadening and the Doppler broadening are correlated, the factorization (2.10) is invalid and k is invariant only for rotations about k and the spherical symmetry is reduced to an axial symmetry. The reduction to axial symmetry can be due to other reasons as well. For example, an external magnetic field lifts the rotational invariance of the hamiltonian H, and makes k axially symmetric. Further if the excited states are prepared by light absorption from a laser beam with a narrow spectral distribution, it is possible to select the velocities of excited atoms to have a futed component in the direction of the incident beam, provided that Doppler broadening is not dominated by collisional broadening. The resulting axial symmetry of ptr is reflected in the symmetry properties of & In order to take full advantage of the spherical or axial symmetry, we employ as usual normalized
384
G. NtenhutslFrequency
dependence
irreducible tensorial sets. The tensor T’(JuJ1) 1s defined to have non-vamshing matrix elements only between the upper states wrth angular momentum J, and the lower states with angular momentum J1. The matrix elements are [ 161
of polarlzatlon
where .4k(JU’J1’, JuJ1) 1sa reduced matrix element defined by Ak(Ju’Jl’;JuJI)
= Tr Tl(.l,‘J,‘)t
[aT$(J,J,)]
. (2.17)
An axially symmetric tetradic & couples tensors T,“(JuJ1) with different values of k, and wrth the same values of 4, provided that the symmetry axes 1s taken to be the quantization axis [ 141. where U,M, IJrMr ; kq) 1sa Clebsch-Gordan coefficrent , and M, and M are the magnetrc quantum numbers. The tensors T,2 obey the orthonormality relation Tr,nt T,k(J,Jl>t
@‘(J,‘J,‘)
= 6kkf hqql 6.r~;
and they transform under rotations as the spherrcal harmonics Y[. Further we shall use the three spherical vectors u,, u = -1, 0, 1, which are defined by then Cartesian coordinates = +(I,
fl,
O),
240 = (O,O,
2
T;‘(‘(J,‘+‘) Bq(k’JU’J1’, k J,JJ,
k’J,‘Jl’
(2.18)
~.QJ,
$1
B Tqk(J&) =
(2.14)
1).
These vectors are convenient in finding the spherical components of vector operators. From the WignerEckart theorem [16] we know
where Bq(k’Ju’J{ ; k .I,/,) = Tr T,k’(Ju’Jl’)t [bT~(JuJ1)] . (2.19) In the following sections we shall investigate the spectral drstributron and the polarization properties of the emitted radiation. A complete description of the emrssion is expressed by eqs. (2.2) and (2.7). We shall consider the cases that I? 1s spherically symmetrrt or axially symmetric.
3. Spherical symmetry
= 1
U,ll~llJ~)
(2.15)
T,(J,J,)/~3,
JllJl
where (JUll~l]J1) denotes a reduced matrrx element, which determmes the strength of the electrrc-dipole transition .I, -f Jr. A spherically symmetric tetradrc A operating on the tensors Tt(JuJ1) couples only the tensors with the same value of k and 4, and the coupling IS independent of 4 17, 171
A^Tt(J&)
= 2 J,‘Jl
Tqk(.&‘.$) A&f&‘;
J&),
(2.16)
In thrs section we consider the case of spherically symmetric line broadenmg. This corresponds to the situation that the translational states are m thermodynamic equilibrium in the absence of external fields, and that Doppler broadening is uncorrelated to colhsional broadening. 3.1. Isolated line First we consider a transition from a single upper state with angular momentumJ, to a single lower state with angular momentum J1. The emitted radiation integrated over the frequent: is unaffected by the line broadenmg process. The integrated emissivrty for radiation with polarrzation e is according to (2.2) and (2.7)
G. NienhuislFrequency PC&)
dw = nA -e** 2nC3
C(O)*e,
(3.1)
where
It is convenient
2
of polarization
c(o) = 2
ckq
385
S4”t,
(3.6)
where
C(0) = TrrrrtCclUP”P~.
P” =
dependence
to expand pu in irreducible
Pkq q(J,Ju)t,
(3.2) tensors [7]
(3.3)
kq where Pkq = Tr Pu T,“&&).
ckq = Tr C(0) * Sz.
(3.7)
The coefficient ckq may be considered as the components of the multrpole moments of the emitted radiation. It is interesting to note that the multipole moments pkq of p,, are directly related to the multipole moments ckq of C(0). From standard recoupling considerations one derives [ 161
(3.4)
The complex numbers pkq are called the components of the multipole moments of the rank k of the excited atoms. The monopole moment poo is (2J,, + I)-$ times the fraction of the radiators in the upper state. The hrgher multipole moments describe the distribution of the excited radiating atoms over the magnetic substates. The dipole moment is usually referred to as the orientation, the quadrupole moment as the alignment of the state. The relation between the multipole moments of the excited atoms and the characteristics of the emitted radiation have been discussed by D’yakonov [ 181 and by Fano and Macek [ 11. We give here some equations for thrs relation in a slightly different form, that is suitable for our purposes. We first wish to draw attention to the fact that the 3 X 3 matrix C(0) describes the integrated intensity of the emitted radiation in every direction and with any polarizatron. This integrated intensity depends upon the polarization vector e, but it is independent of the direction of emission specified by k.This is typical for dipole radiation. For convenience we introduce a complete orthonormal set of 3 X 3 matrices Sg, which are built by recoupling the spherical vectors u0
X [(2kl + 1) (2k, + l)]:
(KQlk2q2; k,ql)
(3.8)
in terms of a Wigner 6j-symbol. we derive
From (2.15) and (3.8)
Tr PI,, T,k(JuJu) I&I = Mk (J,J,)
Sk4’
where ikf&&)
= (-1
Mk(J,/l)
(3.9)
is grven by
J,,+Jl+kcl
l(Jull/-dlJ1)12
(3.10) From (3.2), (3.3), (3.6) and (3.9) it follows that ckq = Mk CJuJd &q 3
(3.11)
and (3.5) c(o) = 2
for k = 0, 1, 2 and -k < q d k. Explicit expression0 for the matrices St are given by Carrmgton [ 191. We can expand C(0) in terms of the matrices S$
Pkq ‘& (JuJd s$t
(3.12)
ks We recover the result that only the monopole, the dipole and the quadrupole moment of p,, contribute
G. Ntenhuls/Frequency
386
to the integrated intensity. The multrpole moments of C(O), which are proportional to the multipole moments of pU, have a direct physical mterpretatron. The monopole moment co0 determmes the intensity averaged over polarrzations and drrections. The dipole moment clq determines the degree of circular polarization, and the quadrupole moment c2q determines both the linear polarization and the anisotropy of the unpolarized intensity [18, 19, 11. Now we turn to the frequency dependence of the emitted radiation. Since we consider an isolated line, JU and Jf assume only one value. Due to the assumed rotatronal invariance of rz‘, we can apply eq. (2.16) to l?
dependence
of polarzatlon
a minor fraction of all radiators [23]. Thus the argument of these authors indicates that our assumption (2.6) might not be adequate to describe the far wings of the line. 3.2. Multiplet of lines Now we consider a multiplet of upper states wrth angular momenta JU, J,‘, . . ., and a multiplet of lower states with angular momenta Jl, J1’, . . . . We assume that pu 1s diagonal m J,,, in accordance with the assumption that the density matrix p commutes with the hamiltonian H. The multipole expansion of pa 1s then
PU= 2 2 Pkq(Ju) T,k(JuJ,,i. From (2.15) and (2.7) we then find the simple result
C(t) = KI(t)
C(O) = I$(0
2 P~qJf&,J1) kq
S$tO.W
(3.15)
JU kq We use the rotational (2.15) and (2.16)
invariance
of g(t), and apply
(3.14)
&MJu’J1’) = 2 lu
JuJl x (J,‘IIfillJ1’) (3.16)
(Jull~llJ~l> ’
An expression for C(t) is found from (2.7), (2.15), (3.9) (3.15) and (3.16).
(J,‘ll~.(llJl’) KdJUJi;JU’J+)
(J
U
llpjlJ,)
I
(3.17)
From (3.17) we see that the atoms in the state IJ,) decaying to the state IJ1) contribute to C(t) a term of the type (3.12), multiplied by a time-dependent factor. Only the multipole moments of pU with rank k = 0, 1 and 2 contribute to C(t). The frequency distribution of the radiation emrtted by the atoms decaying from IJ,> to IJ1>is determined by the Fourier transform of the time dependent factor
G. Nienhm/Frequency
between square brackets in (3.17). This factor is independent of the multipole moments of pU. If the coupling between different transitions is negligible, K, is diagonal in (JUJI), and the spectrum is the sum of spectra of the sort described by (3.14) with polarizatron properties dependent on J, and J1. A non-negligible line coupling does not alter the polarization properties of the radiation due to a certain transition Ju + J1. But line coupling does redistribute this radiation as a function of frequency, so that the intensity is partly transferred to the lures corresponding to other transitions. This means that even rf only one component IJ,) of the upper state multiplet is populated, the emitted radiation may show lines which correspond to transitions from a different component IJu’). For example, in the classical impact lirmt for a non-moving radiator, the tetradic k(t) is given by [ 14, 17, 201 J?(t) = exp [- nta + (i/fi)t;t]
for t > 0,
where the tetradic h is the commutator internal hamiltonian h of the radiator
(3.18)
with the
dependence
387
of polarzation
is of the order of J-mixing cross sections. If the frequency splittmgs of the multiplet are much larger than nuu,,, , the off-dragonal elements are negligible and line coupling is not important. In general, if the intensity of a certain line is used as a measure of the population of the corresponding upper state, the effect of line coupling may cause an error of the order of nuo,/Aw, where Aw is a typical value of the frequency sphttmg of the multrplet. Fortunately, m several important cases line couplmg 1s absent for symmetry reasons. In particular, if the semiclassical scattering matrix S is diagonal in J1 and Ml, one finds from (3.20) that Ak(JuJ1; Ju’J1’) vanishes unless J, = Ju’ and J1 = J1’, and that the diagonal elements are independent of k. Thts is true if the lower state has only one component with angular momentum J, = 1, provided that the perturbers have zero angular momentum [21, 141. Resonance lines of alkali atoms perturbed by inert gases are an important example of the case that line coupling is zero for symmetry reasons. But m general one should be careful in utilizing the intensity of multiplet lines broadened by collisions as a measure of the population of the components of a multiplet.
(3.19)
h^/.&I = ]h, Pull and where a is a tetradic with reduced matrix elements
4. Axial symmetry In this section we derive expressrons for the polarization and the frequency drstributron in the case of axially symmetric line broadening. For simplicity we consider a transition between two single states with angular momenta J, and J1. The integrated intensity is determined by C(O), which is given by (3.12). In order to find the frequency distribution of the intensity, we must consider the variation of k(t) with time. If we choose the symmetry axis to be the z-axis, we may write according to (2.18) and (2.15)
X (J,‘M,‘ISIJ,M,)*
(J+14&91JIMI)
.I
.
(3.20)
In eq. (3.20) Pm is the Maxwell velocity distribution for the perturbers with mass m, and S is the semiclassical scattering matrix for a collision with initial relative velocity u and impact parameter b. The offdiagonal elements of K, are due to the off-diagonal elements of AI, which may be of the order of uun,, where u is a typical velocity of a perturber, and cr,
X (JulI/41J1)/d3.
(4.1)
It is convenient to express the matrix C(t) by its spherical matrix elements U, * C(t) * u,~. We apply l
388
G Nienhurs/Frequency
(2.7)
(2.15), (3.3), (3.8) and (4.1) and we find
dependence
of polarzatlon
properties of i(
X
lqJl/.41J,>12.
(4.2)
(4.3)
and Kq(k
From (4.2) we see that the multipole moments pkq of pU with rank k > 2 may contribute to the frequency distribution of the emitted radiation in the case of axial symmetry. In general eq. (4.2) cannot be separated in an overall time-dependent factor independent of the polarizations u and CJ’and a polanzation-dependent factor which does not depend on time. Hence we conclude that in the case of axial symmetry a frequency-dependent polarizatron of the radiation is to be expected. In section 2 we mentioned three possible reasons why the spherical symmetry would be reduced to an axial symmetry. (I) correlatron between collisional broadening and Doppler broadening, (ii) an axially symmetric velocity drstribution of the radiators, (iii) an external magnetic field. Case (i) is different from the other two in that the symmetry axis in this case is given by the observation direction of the emitted light. Hence only the values u, u’ = +l are physically relevant m this case, since the polarization vectors must be orthogonal to k. Only the 2 X 2 submatrix in the xy-plane of C(t) is observable. The radiation in some other direction is described by substituting in (4.2) atomic multipole moments piq wrth respect to the new symmetry axis. The 2 X 2 submatrix of the resulting matrix C in the plane orthogonal to this new axis determines the radiation in this other direction. In the two cases (ii) and (iii) the symmetry axis has a fmed direction in space, and every polarizatron wrth respect to this symmetry axis is observable. Further it is important to note that the timedependent functions K,,~(K; 1 I t) in (4.2) may not be independent. We may analyse the symmetry
; k JuJl I t)
JUJ1, k’J&‘(
r) = (-)k-
k’K_q(k.IU.Jl;
k’J,‘$J
r).
(4.4) In case (iii) of an external magnetic field the symmetry is slightly different, since the magnetic field 1s a pseudo vector and remains invariant under space reflection, whereas the wave vector and the velocrty are reversed. For case (iii) eq. (4.4) 1s no longer valid, but the symmetry relation (4.3) still holds. We conclude from (4.4) for the situation of a single transition that the time-dependent factors in (4.2) satisfy the equality
&(K; lit) = (-)“-k_t(~;
IIt)
(4.5)
in the cases (i) and (ii), but not in case (in). First we consider the case (i) of correlated Doppler and collisional broadening. Then we have to consider only the values u, u’ = +l. Due to eq. (4.5) the time-dependent factor K+ 1(~; 1 It) in (4.2) may be factored out for every value of K. Hence the polarization becomes independent of the frequency if only one value of K contributes to (4.2). This is true in two different cases. The first case is that the atomic upper state is unpolarized so that only the monopole moment poo differs from zero. Due to the Clebsch-Gordan coefficient in (4.2) we then only have to consider the value K = 1. For the relevant 2 X 2 submatrrx we then find
x lcJ,ll/41J,~12,
(4.6)
G NienhuisjFrequency
for o, (3’= f 1, where we used the explicit expression for a Gj-symbol where one of the j’s is zero [16] K K
0 = (-@
( Jl
Jl
+J2+ K[(24
+ 1) (2K + I)] +.
J2 I (4.7)
In this case the radration 1s unpolarized and isotropic for every frequency, and the spectral profile is given by the Fourier transform of K, (1; 1 I t). The second case that the polarizatron is independent of the frequency occurs when one of the angular momenta J, and J1 is zero. If J, = 0, k and 4 can attain only the value zero, and eq. (4.6) is valid. If J1 = 0, we must have J, = 1 for a drpole transition. For J, = 1 and J1 = 0 we must have K = 1, since K is found by recoupling J, and J1. From (4.2) we then find for the 2 X 2 submatrrx
dependence
measured frequency-dependent 2 X 2 polarization matrix in this direction may provide information on the multipole moments pkq with k > 2, and on the Fourier transform of K1(~; 11 t) for different values OfK.
Finally we consider the cases (ii) and (iii), where the symmetry axis has a fixed direction m space. The frequency-dependent polarization properties of the radration in every direction are then determmed by the time-dependent Cartesian 3 X 3 matrix C(t), which is given by eq. (4.2). This matrix may be determined experimentally by measuring the frequency-dependent 2 X 2 polarizatron matrix of the radration in three independent directions. This contrasts with case (i), where C depends on the direction of the detected radration. Even if the atoms are unpolarized so that only the multipole component poo differs from zero, the emitted radiation may be polarized. We find from (4.2) for thrs case
“,: * C(t)y,~ =K,(l;
Ilt)
2 pkqMk(’ kq
O)$S$+
=6,,,K,~(l;
lit)
PO0 3(2J, + 1);
lu,,~. (4.8)
So in the special case that Ju = 1 and J1 = 0, the spectral profile is independent of the atomic multipoles, and the polarization is independent of the frequency. In other cases when J1 f 0, and when the excited atoms are polarized, we may expect a frequency-dependent polarrzation. The mUltipOle moments pkq with k = 0, 1, 2 are determined experimentally by measuring the polarization properties of the integrated intensity in three independent directions. The 2 X 2 submatrix of C(t) determining the frequency-dependent polarization properties of the radiation in a certain z-direction is a linear combination of the 2 X 2 matrices m the xyplane
X l~Jull/dlJ1~12.
(4.9)
It is advantageous to expand C(t) given by (4.9) in the matrices Sg. If we use the definition (3..5), or the explicit expressions given by Carrington [ 191, we find
C(t) =
X
PO0 t I
&[K1(l;
llt) + K,(l;
llt) + K_, (1; llt)] S;+
(
+$
[K1(l,
llt) - K_1(1;
-LIK1(l;llt)+K_l(l;
+& [(-)K--]lt(l+d) 2 u,u’=* 1
389
of polarization
llt)l S;+ lIt)--o(l;llt)J
U, (KU’llU;k~)#;‘,
where the sign accounts for the symmetry relation (4.5). The coefficients of these matrices are determined theoretically by eq. (4.2). Comparison with the
x s;+
1
.
(4.10)
In case (ii) eq. (4.5) is valid, and the coefficient of SA vanishes. This means that for unpolarized atoms an axially symmetric velocity distribution can induce a
390
G. Nlenhuzs/Frequency
linear polarrzation, but no circular polarizatron. The radiation with linear polarization vector m the direction of the symmetry axis has a spectral profile which is the Fourier transform of K,(l) 1It). The radiation wrth polarizatron orthogonal to this axis has a spectral profile which is the Fourier transform of K 1(1; 1(t) = K-,(1; llt). In case (iii) eq. (4.5) is mvahd, and the radiation emitted by unpolarrzed atoms will drsplay both circular and linear polarization, which 1s what we expect for a Zeeman spectrum. The integrated intensity in both cases 1s unpolarized and isotroprc.
5. Conclusions This paper 1s devoted to a study of the frequency dependence of the polarization of the emission by polarized atoms, broadened by the mteraction with a perturbing gas. This study is relevant to investigations concerning the Hanle effect and the depolarization of resonance radiation, and may be considered as a theory of polarization-dependent line broadening. We distinguish the case of a transition between two single states, and a transition between two multiplet terms. For a single transition, the polarrzation of the integrated intensity in every directron is described by a 3 X 3 cartesran matrix C(O), eq. (3.2). By measuring the polarization of the integrated intensity in a certain direction, one determines a 2 X 2 submatrix of C(0) m a plane orthogonal to this direction. This 2 X 2 matrix is m fact the polarizatron matrix of the radiation in thrs direction [22]. The multipole moments of C(O) are proportional to the multrpole moments of the excited atoms. The monopole moment coo determines the integrated intensity averaged over polarrzations and directions. The components of the dipole moment clq determine a vector c1 which is proportional to the orientation of the excited atoms. The circular polarization of the radiation emitted in a certain directron is proportional to the component of the vector cl in that direction. The quadrupole moment of C(0) is proportional to the alignment of the excited atoms, and determines the linear polarization and the anisotropy of the emission. Multipole moments of the excited atoms with rank k > 2 do not show up in the integrated radiation.
dependence
o$ polarzatrw
In the case of a single transition we further drstinguish the case of spherically symmetric and axially symmetric line broadening. For spherical symmetry the line broadening is independent of the polarization, and hence the polarrzation does not depend on the frequency. If the velocity distributions are maxwellian and if no external fields are present, spherical symmetry holds when the Doppler broadenmg and the collisional broadening are uncorrelated. This means that a frequency-dependent polarization must be due to correlations between Doppler and collisional broadening. This provides a way of detecting and investigating this correlation directly, by measuring as a function of the frequency the polarrzation of the emission of polanzed atoms. This conclusion 1s valid irrespective of the details of the line-broadening mechanism. We only assumed that the transfer of the anisotropy of the excited atoms to the perturber system is negligible. If the spherical symmetry is reduced to an axial symmetry, be it by an axially symmetric radiator velocity distnbutron, or by an external magnetic field, the multipole moments of the excited atoms with rank k > 2 may contribute to the frequency distribution of the polarized radration. The frequency-dependent polarrzation properties of the emission in every direction 1s described by the Fourier transform of a time-dependent 3 X 3 matrix C(t). The multipole expansion of C(t) is found by coupling the multipole expansion of the excited atoms to the anisotropy of the line broadening process. The polarization-dependent profiles contain more mformation on the line broadening mechanism than the line profile emrtted by unpolarized atoms. The case of an atomic multiplet of lines is considered only for spherically symmetric hne broadening. Every transition Ju + J1 emits radiation with polarizatron propertres that are unaffected by the other transitions. But the frequency distribution of this radiation may be modified by coupling to other transitions Jn’ + Jr’. Even if only one upper state is populated, the emrssion may show lines which correspond to transitions from different excited states. This line coupling effect may cause systematic errors m experiments where the intensity of broadened multiplet lines is measured m order to find the population of a component of an excited state multiplet.
G NtenhuisfFrequeney
Summarizing we conclude that a frequencydependent polarization may be due to correlations between Doppler and collisional broadening, to anisotropy of the translational states of the system, to an external field, or to overlap or coupling with nearby spectral lines. Srmilar conclusions are possrble for the absorption of polarized light by polarized atoms.
References [l] U. Fano and J. H. Macek, Rev. mod. Phys. 45 (1973) 553. [ 21 R. Serwert in Springer Tracts in Modern Physics, G. Hohler, ed., vol. 47 (Springer Verlag, Berlin, 1968) p. 143. [ 31 M. Stupavsky and L. Krause, Can. J. Phys. 46 (1969) 2127. [4] P. L. Lrmse, P. J. Th. Zeegers and C. Th. J. Alkemade, J. quant. Spectrosc. radiative Transfer 13 (1973) 1033. [5] J. P. Barrat, D. Casalta, J. L. CoJan and J. Hamel, J. Physique 27 (1966) 608. [6]A.GallagherandE.L.Lewis,Phys.Rev.A10(1974) 231.
dependence
of polarization
391
[ 71 A. Omont, J. Physique 26 (1965) 26. [ 81 M. 1. D’yakonov and V. 1. Perel’, Sov. Phys.-JETP 21 (1965) 227. [9] C. G. Carrmgton, D. N. Stacey and J. Cooper, J. Phys. B: Atom. molec. Phys. 6 (1973) 417. [lo] V. N. Rebane and A. G. Petrashen, Opt. Spectrosc. 37 (1974) 472. [ 111 F. Schuller and W. Behmenburg, Z. Naturforsch. 30a (1975) 442. [ 121 E. W. Smith, J. Cooper, W. R. Chappell and T. Drllon, J. quant. Spectrosc. radiative Transfer 11 (1971) 1547. [ 131 G. Nienhuis, Physica 66 (1973) 245. [14] G. Nienhuis, Physrca 74 (1974) 157. [ 151 S. G. Rautian and I. I. Sobel’man, Sov. Phys.-USPEKHI 9 (1967) 701. [ 161 U. Fano and G. Racah, Irreducible Tensorial Sets (Academic Press, New York, 1959). [ 171 M. Baranger, Phys. Rev. 111 (1958) 494. 1181 M. I. D’yakonov, Sov. Phys.-JETP 20 (1965) 1484. [ 191 C. G. Carrington, J. Phys. B: Atom. molec. Phys. 4 (1971) 1222. [20] E. W. Smith, J. Cooper, W. R. Chappell and T. Dillon, J. quant. Spectrosc. radiative Transfer 11 (1971) 1567. [21] A. Omont, J. Physique 34 (1973) 179. [ 221 M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959). [ 231 F. Schuller and W. Behmenburg, private communication.