State of polarization of a stochastic electromagnetic beam in an optical resonator

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2710 J. Opt. Soc. Am. A/Vol. 25, No. 11 /November 2008 Korotkova et al.

State of polarization of a stochastic electromagneticbeam in an optical resonator

Olga Korotkova,1 Min Yao,2 Yangjian Cai,3,* Halil T. Eyyuboglu,4 and Yahya Baykal4

1Department of Physics, University of Miami, Coral Gables, Florida 33146, USA2College of Science and Technology, Zhejiang Education Institute, Hangzhou, 310012, China

3Max-Planck-Research-Group, Institute of Optics, Information and Photonics, University of Erlangen,Staudtstr. 7/B2 D-91058 Erlangen, Germany

4Department of Electronic and Communication Engineering, Çankaya University, Ögretmenler Cad. 14, Yüzüncüyıl06530 Balgat Ankara, Turkey

*Corresponding author: yangjian�[email protected]

Received May 22, 2008; revised August 18, 2008; accepted August 26, 2008;posted September 4, 2008 (Doc. ID 96464); published October 15, 2008

On the basis of the unified theory of coherence and polarization, we investigate the behavior of the state ofpolarization of a stochastic electromagnetic beam in a Gaussian cavity. Formulations both in terms of Stokesparameters and in terms of polarization ellipse are given. We show that the state of polarization stabilizes,except in the case of a lossless cavity, after several passages between the mirrors, exhibiting monotonic or os-cillatory behavior depending on the parameters of the resonator. We also find that an initially (spatially) uni-formly polarized beam remains nonuniformly polarized even for a large number of passages between the mir-rors of the cavity. © 2008 Optical Society of America

OCIS codes: 030.1640, 260.5430.

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. INTRODUCTIONhe theory of beam propagation in laser resonators was

ormulated a long time ago for monochromatic scalarelds [1,2]. The transverse modes of the resonator, fre-uently called the Fox–Li modes, were found to be relatedo the physical parameters of the cavity and the charac-eristics of the light source. The generalization of theox–Li theory to partially coherent, sometimes calledandom or stochastic fields, i.e., fields with any state of co-erence, was later made [3–5]. The effects of the resona-or on the statistical properties of light, such as its spec-ral density and its state of coherence, were also reported6,7].

Only recently, with the development of the theory oflectromagnetic (vectorial) partially coherent fields (see8], it has become possible to predict behavior of polariza-ion properties of light beams on propagation in vacuumnd on interaction with various linear media and linearptical systems, which may affect light in a deterministicr random manner (see [9–13]). In [14] the theory of reso-ator modes was developed for electromagnetic partiallyoherent fields (see also [15] for an alternative theory),here the transverse electromagnetic modes have been

elated to the classic Fox–Li modes [1] of the cavity. Veryittle, however, is known about the behavior of the polar-zation properties of partially coherent, electromagneticight beams in resonators. The only publication of thisind (see [16]) addressed the evolution of the degree of po-

arization of a typical electromagnetic partially coherenteam.In this paper we will extend the analysis of [16] by cal-

ulating the complete state of polarization of a typicallectromagnetic partially coherent beam interacting with

1084-7529/08/112710-11/$15.00 © 2

he laser resonator. We restrict our attention to a broadlass of beams, namely, to the electromagnetic Gaussianchell-model [EGSM] beams [9], which has recently re-eived special attention due to its remarkable tractabilityn the one hand and due to its potential to exhibit all thehenomena of interest stemming from its electromagnetics well as its random nature on the other hand. In Sectionwe will review the general theory of interaction of theGSM beams with optical resonators and, in particular,

heir cross-spectral matrices, from which all the quanti-ies of interest may be determined in the later sections.

Since, historically, there have been known two alterna-ive descriptions of polarization properties of light, weill also consider them separately. In order to completelyescribe the polarization properties of a random beam,tokes introduced (see [17]), as early as in 1852, the set of

our independent parameters, since then known as Stokesarameters, which remain extremely popular in experi-ental optics due to the relative simplicity of their mea-

urement. We note that for deterministic (monochro-atic) beams, Stokes theory is also applicable, but in this

ase only three independent parameters are required toharacterize the polarization properties. In Section 3 weill review the general expressions for the Stokes param-ters of an electromagnetic partially coherent beam andhen the expressions for the particular case of an EGSMeam; finally, we will provide numerical examples of theehavior of the Stokes parameters of the EGSM beams inptical resonators.

The other approach for characterization of the polariza-ion properties of light beams is based on 2�2 correlationatrices [8] and originates from the fact that the correla-

ion matrix of any partially polarized beam can be locally

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Korotkova et al. Vol. 25, No. 11 /November 2008 /J. Opt. Soc. Am. A 2711

epresented as a sum of a completely polarized beam that,s a matter of fact, is indistinguishable from the mono-hromatic beam, which we will call an equivalent mono-hromatic field, and a completely unpolarized beam (see18] for such a decomposition in the space–time domainnd [19] in the space–frequency domain). Further, it cane shown that at any point within the beam, the motion ofhe equivalent monochromatic field is constrained to alosed elliptic curve. The parameters of the ellipse, usu-lly referred to as the polarization ellipse, can be deter-ined from the elements of the correlation matrix. In Sec-

ion 4 we will review the expressions for the ellipsometricarameters of the electromagnetic partially coherenteams and then will give numerical examples of the be-avior of the ellipse of EGSM beams in optical resonators.

. GENERAL THEORY OF INTERACTION OFN ELECTROMAGNETIC PARTIALLYOHERENT BEAM WITH A RESONATORn optical resonator is composed of two reflecting sur-

aces (mirrors) with the same or different sizes and cur-atures [see Fig. 1(a)]. Originally, the beam of light is sento one of the mirrors in such a way that it then travelsack and forth in the cavity. For simplicity we will as-ume that both mirrors are spherical, with radius of cur-ature R, mirror spot size �, and distance between theenters of the mirrors L. The interaction of light withuch a resonator is equivalent to its propagation through

sequence of thin spherical lenses of focal lengths fR /2 combined with filters with a Gaussian amplitude

ransmission function [6,7]. Figure 1(b) shows the equiva-ent “unfolded” version of the resonator. Depending on thealue of the stability parameter g=1−L /R, the resonatorsre classified as stable �0�g�1� or unstable �g�1� [20].Let the beam entering the cavity be produced by an

GSM source (c.f. [9]). Such a beam can be characterizedy the cross-spectral density matrix evaluated at pointsr1 ,r2�� r of the form [16]

W��r� = A�A�B�� exp�−ik

2rTM0��

−1 r�, �� = x,y;�

= x,y�, �1�

here k=2� / is the wavenumber, is the wavelength ofight, A� is the square root of the spectral density of elec-ric field component E�, B��= �B�� exp�i� is the correla-ion coefficient between Ex and Ey field components, T

ig. 1. Schematic diagram of a Gaussian cavity and its equiva-ent (unfolded) version.

tands for vector transpose, and M0��−1 is the 4�4 matrix

f the form

M0��−1 = �

1

ik� 1

2�a2 +

1

��2 I

i

k��2 I

i

k��2 I

1

ik� 1

2��2 +

1

��2 I , �2�

here I is the 2�2 identity matrix and parameters ��

nd �� are the widths of the spectral density and of thepectral degree of coherence of the beam, respectively. Inqs. (1) and (2) as well as in all the formulas below, thexplicit dependence of the cross-spectral density matrixn the oscillation frequency was omitted for simplicity.e should note, however, that for this class of model

eams all the parameters entering the formulas might, ineneral, depend on frequency. See [9,21,22] for the deri-ation of conditions to be satisfied by various parametersf the EGSM sources that produce physically realizableeamlike fields.It can be shown with the help of the ABCD matrix ap-

roach that after passing N times between the mirrors ofhe cavity, the cross-spectral density matrix of the beamt points with transverse position vectors ��1 ,�2�� isiven by the expression (see Appendix A) [16]

W�� = A�A�B��det�A + BM0��−1 ��−1/2

�exp�−ik

2�TM1��

−1 ��, �� = x,y;� = x,y�,

�3�

here det stands for the determinant of a matrix and

M1��−1 = �C + DM0��

−1 ��A + BM0��−1 �−1, �4�

nd A ,B ,C, and D are 4�4 matrices of the form

A = �A 0I

0I A*, B = �B 0I

0I − B* ,

C = �C 0I

0I − C*, D = �D 0I

0I D* , �5�

here the asterisk denotes the Hermitian operator. Forhe resonator of interest, matrices A, B, C, and D takehe form [23]

�A B

C D = �A1 B1

C1 D1N

,

�A1 B1

C1 D1 = �

I L · I

�−2

R− i

��2I �1 −2L

R− i

L

��2I , �6�

here, as before, R is the radius of curvature, � is theirror spot size of the cavity, L is the distance between

he centers of the mirrors, and N is the number of pas-ages between the mirrors. The matrix with elementsaving subscript “1” describes the single pass between the

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wo mirrors. On substituting from Eqs. (4)–(6) into Eq.3), one can determine the elements of the cross-spectralensity matrix of the EGSM beam from which all second-rder statistical properties of the beam can be found.

. EVOLUTION OF THE STOKESARAMETERS OF A BEAM IN AESONATORe will now use the analysis of the previous section for

alculation of the Stokes parameters of the beam [17] in-eracting with the optical resonator. Following [24], wexpress the Stokes parameters of the beam in terms of thelements of the cross-spectral density matrix (3) as

S0�� = Wxx�� + Wyy��,

S1�� = Wxx�� − Wyy��,

S2�� = Wxy�� + Wyx��, �

S3�� = i�Wyx�� − Wxy��. �7�

e note that the Stokes parameter S0 coincides with thepectral density of the beam.

We will now illustrate the behavior of the Stokes pa-ameters of the beam as a function of the number of pas-ages N for various values of cavity parameters � and g.or all the figures in this paper, the following parametersf the source of the beam and of the cavity will be chosenunless different values are specified in the captions): Ax1.5, Ay=1, =� /3, �Bxy�=0.3, �x=�y=�=1 cm, �yy0.225 mm, �xy=0.25 mm,L=35 cm, and =632.8 nm.Figure 2 shows the (on-axis) normalized Stokes param-

ters S1 /S0, S2 /S0, and S3 /S0 calculated from Eq. (7) withncreasing values of N for different values of cavity pa-ameter g, with �xx=0.15 mm and �=0.8 mm. From thisgure we see that after experiencing the changes formall values of N �5�N�20�, the normalized Stokes pa-ameters saturate at certain values, which may be eitherarger or smaller than initial values, determined by theavity parameters and by the initial parameters of theeam. We also note that when the resonator is stable �0
g�10�, the changes have oscillatory character while
ig. 2. The (on-axis) normalized Stokes parameters versus N for different values of cavity parameter g with �xx=0.15 mm and �0.8 mm.

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hey are monotonic for unstable resonators �g�1�, beingimilar, in this case, to changes when a beam propagatesn free space (see Fig. 3 of [24]).

Figure 3 illustrates the evolution of the (on-axis) nor-alized Stokes parameters with increasing values of N

or different values of the mirror spot size � in a Gaussianlane-parallel cavity �g=1� with �xx=0.15 mm. It can beeadily seen from the figure that with increasing values ofirror spot size �, the saturation of all the normalizedtokes parameters occurs less rapidly, requiring a largerumber of passes between the mirrors.From Figs. 2 and 3 a very general conclusion can be

rawn: The normalized Stokes parameters of the beamave qualitatively different behavior depending on thealues of stability parameter g. This might have the fol-owing physical explanation: In stable cavities, the twoompeting mechanisms of focusing and free-space diffrac-ion act on the beam during each passage between the

ig. 3. The (on-axis) normalized Stokes parameters versus Nor different values of the mirror spot size � in a Gaussian plane-arallel cavity �g=1� with � =0.15 mm.
xx
irrors. The values of the normalized Stokes parametershange in an oscillatory manner depending on whichechanism prevails for a certain point between the mir-

ors. On the other hand, for unstable resonators the effectf focusing is negligible and the behavior of the normal-zed Stokes parameters is practically monotonic, just as inhe case of free-space propagation.

. EVOLUTION OF THE POLARIZATIONLLIPSE OF A BEAM IN A RESONATORe will now discuss the behavior of the polarization el-

ipse of a beam, which is known as an alternative repre-entation of the state of polarization.

It has been recently shown in [19] how the cross-pectral density matrix can be used for finding the (spec-ral) polarization ellipse associated with the completelyolarized portion of the beam as it propagates in freepace. We will now use the same approach in order totudy the behavior of the ellipse in the resonator.

The cross-spectral density matrix (3) evaluated at coin-iding arguments can be uniquely decomposed into polar-zed and unpolarized matrices:

W�� = W�u�� + W�p��, �8�

here

W�u�� = �A�� 0

0 A��, W�p�� = � B�� D��

D*�� C��9�

ith

A�� =1

2�Wxx�� + Wyy��

+ �Wxx�� − Wyy��2 + 4�Wxy��2�,

B�� =1

2�Wxx�� − Wyy��

+ �Wxx�� − Wyy��2 + 4�Wxy��2�,

C�� =1

2�Wyy�� − Wxx��

+ �Wxx�� − Wyy��2 + 4�Wxy��2�,

D�� = Wxy. �10�

sing the fact that matrix W�p� is a singular matrix at anyoint, it was demonstrated that its elements may be writ-en as products of “equivalent monochromatic field” com-onents, say, Ex and Ey [19]. The quadratic form associ-ted with such a matrix can be shown to represent thellipse, known as a spectral polarization ellipse, of theorm

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��Ex�r�2�� − 2 Re D��Ex

�r��Ey�r�� + B��Ey

�r�2��

=�Im D��2, �11�

here Re and Im stand for real and imaginary parts, re-pectively, of complex numbers and Ex

�r��= B��cos� t�x�, Ey

�r��= C��cos� t+�y�, and �y−�x=arg�D��. Theajor and minor semiaxes of the ellipse, A1 and A2, asell as its degree of ellipticity, �, and its orientationngle, �, can be related directly to the elements of theross-spectral density matrix (3) with the help of the ex-ressions

A1,2�� =1

2� �Wxx�� − Wyy��2 + 4�Wxy��2

± �Wxx�� − Wyy��2 + 4�Re Wxy��2�1/2,

�12�

�� =A2��

A1��, �13�

�� =1

2arctan� 2 Re Wxy��

Wxx�� − Wyy�� . �14�

n Eq. (12) signs “�” and “-” between the two square rootsorrespond to A1 (major semiaxis) and A2 (minor semi-xis), respectively. On substitution from Eq. (3) into Eqs.11)–(14), we can determine the behavior of the param-ters of the polarization ellipse in the cavity.

Figures 4–11 illustrate the behavior of various param-ters of the polarization ellipse in the resonator as a func-ion of the number of passages N between the mirrors forarious values of cavity parameters. In particular, Figs. 4nd 5 show the evolution of the orientation ellipse and ofhe degree of ellipticity for several combinations of stabil-

ig. 4. Degree of ellipticity (on-axis) versus N for different val=0.8 mm: (a) � =0.15 mm, (b) � =0.15 mm, (c) g=0.5, (d) g=1
xx xx
ty parameter g and of the source correlation coefficientxx. One can see from these figures that, independently ofhe source correlations, while for stable resonators the el-ipsometric quantities exhibit an oscillatory regime firstnd then stabilize for sufficiently large values of N toome values between oscillation maxima and minima, fornstable resonators they tend monotonically to certainalues. As with propagation of the degree of polarizationcompare with corresponding figures of [16]) and of theormalized Stokes parameters, the effect of an unstableesonator resembles the effect of the free-space propaga-ion of the polarization ellipse (compare with Figs. 2 and 3f [19]).

Figures 6 and 7 explore the evolution of the degree ofllipticity and of the orientation angle of the polarizationllipse with increasing N for various values of mirror spotize � in a Gaussian plane-parallel cavity �g=1� for thexed model source (with �xx=0.15 mm). As in the case ofhe other polarization properties, the increasing values ofirror spot size � correspond to slower changes in the el-

ipsometric parameters.In Figs. 8 and 9 we illustrate the behavior of the degree

f ellipticity and the orientation angle, respectively, withgrowing number of passes N, for several fixed values of

tability parameter g, in the limiting case of the losslessavity ��→��. The parameters of the source were choseno be the same for all curves (with �xx=0.15 mm). On com-aring these figures with Figs. 4 and 5, we see that unliken a stable resonator with losses, in the lossless cavity thearameters of the polarization ellipse never stop oscillat-ng between two values.

As is seen from Figs. 4–9, the dependence of the param-ters of the polarization ellipse on the stability parameteris similar to that of the normalized Stokes parameters

compare with Figs. 2 and 3), and the physical reason forhis is the same as that given in Section 3.

For all the figures above, the evolution of the polariza-

cavity parameter g and the source correlation coefficients with
ues of.2.

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ion properties of the beam were shown only on axis. Fig-res 10 and 11 illustrate the behavior of the degree of el-

ipticity and of the orientation angle in a transverse planeor a fixed number of passes �N=30� and several fixed val-es of mirror spot size �. The cavity was chosen to be alane-parallel cavity �g=1�. It can be readily deducedrom these figures that the spatial profiles of the polariza-ion properties tend to become Gaussian for large valuesf N, even though their initial distributions are uniform.oreover, if in the center of the cavity the state of polar-

zation of the beam is determined by the model source andy the resonator, it turns out to be “more” linearly polar-zed along the x direction toward the edge of the beam. In16] a similar behavior is demonstrated for the degree ofolarization of the beam: It tends to zero at the edges ofhe beam. We note that since such profiles depend on theavity parameters, by adjusting them it is possible from a

ig. 5. Orientation angle � (on-axis) versus N for different valu0.8 mm; (a) �xx=0.15 mm, (b) �xx=0.15 mm, (c) g=0.5, (d) g=1.2

ig. 6. Degree of ellipticity (on-axis) versus N for different val-es of mirror spot size � in a Gaussian plane-parallel cavity �g1� with � =0.15 mm.
xx
iven uniformly polarized beam to obtain a beam with aariable, Gaussian-distributed, transverse polarization.n Fig. 12 the intensity distribution is shown for the sameeam-cavity scenarios as in Figs. 10 and 11.To illustrate the spatial distribution of the degree of el-

ipticity and of the orientation angle of a typical EGSMeam in greater detail, we show them in Fig. 12 as three-imensional plots versus the x and y axes of the beam af-er N=30 passes between the mirrors for two selected val-es of �. One sees that, similarly to the situation of Figs.3 and 14, in the center of the cavity the state of polar-zation of the beam is determined by the cavity param-ters and by the source parameters, but it tends to be lin-arly polarized along the x direction for points on the edgef the cavity. In Fig. 15 the three-dimensional plot illus-rates the spatial distribution of intensity for the sameeam-cavity situation as in Figs. 13 and 14.

avity parameter g and the source correlation coefficients with �

ig. 7. Orientation angle � (on-axis) versus N for different val-es of mirror spot size � in a Gaussian plane-parallel cavity �g1� with � =0.15 mm.

es of c.

xx

Fc


Fig. 8. Degree of ellipticity (on-axis) versus N for different values of g in a lossless cavity ��→�� with �xx=0.15 mm.

Fig. 9. Orientation angle � (on-axis) versus N for different values of g in a lossless cavity ��→�� with �xx=0.15 mm.

ig. 10. Degree of ellipticity versus a transverse dimension x for different values of the mirror spot size � and the source correlationoefficients in a Gaussian plane-parallel cavity �g=1� with N=30.

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. CONCLUDING REMARKSe have investigated how various polarization properties

f a typical EGSM evolve in optical resonators. In particu-ar, we focused our attention on the Stokes parameters ofhe beam, normalized by its spectral density, and on twoarameters of the polarization ellipse: the degree of ellip-icity and the orientation angle. We found that all theseuantities have a very similar qualitative dependence onhe physical parameters of the cavity and on those of theource of the beam. In particular, there is a striking dif-erence in the behavior of the polarimetric properties ofeams that interact with stable and unstable resonators.hile for stable resonators during the first several pas-

ages the polarimetric parameters may oscillate, for un-table resonators the changes are always strictly mono-onic. Moreover, all quantities that we have consideredet stabilized for a sufficiently large number N of passesetween the mirrors, except in the case of the lossless cav-ty.

It is also of interest to note that even for a sufficientlyarge number of passes N, the transverse polarizationroperties of the initially uniformly polarized beam areot, in general, uniform. The spatial distributions of allolarimetric properties that we have considered tend as-mptotically to Gaussian as N increases.

The analysis of this paper revealed a number of inter-sting phenomena relating to the interaction of initially

ig. 11. Orientation angle � versus a transverse dimension x fooefficients in a Gaussian plane-parallel cavity �g=1� with N=30

ig. 12. Spectral density S0 of the beam versus a transverse diorrelation coefficients in a Gaussian plane-parallel cavity �g=1�

niformly but arbitrarily polarized stochastic electromag-etic beams with optical resonators and may be used forhe spatial modulation of polarization properties of sucheams. Moreover, an optical resonator may be viewed as aevice capable of generating beams with prescribed polar-zation properties.

PPENDIX A: DERIVATION OF EQ. (3)he propagation of the elements of the cross-spectral den-ity matrix through a general astigmatic ABCD opticalystem can be studied with the following generalized Col-ins formula [25]:

W�� =k2

4�2�det�B��1/2��W��r�

�exp�−ik

2�rTB−1Ar − 2rTB−1� + �TDB−1��dr,

�A1�

here A, B, C, and D are expressed in the form of Eq. (5),nd A, B, C, and D are the 2�2 submatrices of the gen-ral astigmatic ABCD optical system. We note that forree-space propagation (see [25–27]), A, B, C, and D areeal quantities, implying that the “*” is not needed any-

rent values of the mirror spot size � and the source correlation

n x for different values of the mirror spot size � and the source=30.

r diffe.

mensiowith N

Fs

Fs

Fs


ig. 13. Spatial distribution of the degree of ellipticity of a typical EGSM beam for N=30 and g=1 for two different values of the mirror
pot size �.
ig. 14. Spatial distribution of the orientation angle of a typical EGSM beam for N=30 and g=1 for two different values of the mirrorpot size �.

ig. 15. Spatial distribution of spectral density S0 of a typical EGSM beam for N=30 and g=1 for two different values of the mirror spotize �.

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Es

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1

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1

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1

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1


here in Eq. (5). However, for a general optical systemith loss or gain (e.g., dispersive media, a Gaussian aper-

ure, helical gas lenses, etc.), A, B, C, and D take complexalues and “*” is then required. Note that A, B, C, and Datisfy the following well-known Luneburg relations thatescribe the symplecticity of a general astigmatic opticalystem [28]:

�B−1A�T = B−1A, �DB−1�T = DB−1,

C − DB−1A = − �B−1�T. �A2�

ubstituting Eq. (1) into Eq. (A1), we obtain (after someperation) the formula

W�� =k2A�A�B��

4�2�det�B��1/2exp�−

ik

2�TDB−1�

+ik

2�TB−1T�M0��

−1 + B−1A�−1B−1T�� exp�−

ik

2��M0��

−1 + B−1A�1/2r

− �M0��−1 + B−1A�−1/2B−1��2�dr. �A3�

hen after applying the integral formula

�−�

�

exp�− ax2�dx = �/a, �A4�

q. (A3) reduces (after vector integration) to the expres-ion

W�� =k2A�A�B��

4�2�det�B��1/2�det�M0��−1 + B−1A��1/2

�exp�−ik

2�TDB−1� +

ik

2�TB−1T

��M0��−1 + B−1A�−1B−1T� . �A5�

fter applying operations

�det�B��−1/2�det�M0��−1 + B−1A��−1/2 = �det�A + BM0��

−1 ��−1/2,

�A6�

DB−1 − B−1T�M0��−1 + B−1A�−1B−1

= �DB−1�A + BM0��−1 � − B−1��A + BM0��

−1 �−1

= �C + DM0��−1 ��A + BM0��

−1 �−1, �A7�

nd after setting

M1��−1 = �C + DM0��

−1 ��A + BM0��−1 �−1, �A8�

q. (A5) reduces to Eq. (3) in the text. In the above deri-ation, we have used the Luneburg relations [Eq. (A2)]. Ifhe EGSM beam travels once between two mirrors [seeig. 1(b)], the matrices A, B, C, and D are given by theecond term of Eq. (6) in the text (also see [7,16]).

CKNOWLEDGMENTSangjian Cai gratefully acknowledges support from thelexander von Humboldt Foundation. Olga Korotkova’sesearch is funded by the Air Force Office of Scientific Re-earch (grant FA 95500810102).

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State of polarization of a stochastic electromagnetic beam in an optical resonator

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