Optics Communications 233 (2004) 225–230

www.elsevier.com/locate/optcom

The far-zone behavior of the degree of polarizationof electromagnetic beams propagating through

atmospheric turbulence

Olga Korotkova *, Mohamed Salem, Emil Wolf 1

School of Optics/CREOL, University of Central Florida, 4000 Central Florida Blvd, Orlando, FL 32816-2700, USA

Received 17 December 2003; received in revised form 17 December 2003; accepted 8 January 2004

Abstract

It is shown analytically that the degree of polarization of a beam generated by an electromagnetic Gaussian Schell-

model source which propagates through atmospheric turbulence tends to its value at the source plane with increasing

distance of propagation. This result is independent of the spectral degrees of correlation of the source and of the

strength of atmospheric turbulence. These conclusions are illustrated by a numerical example.

� 2004 Elsevier B.V. All rights reserved.

PACS: 03.40Kf; 41.20.Jb; 42.25.Bs; 42.25.Kb; 92.60.Ta; 42.68.)w

Keywords: Partially coherent beams; Atmospheric turbulence; Polarization; Gaussian–Schell model beams

1. Introduction

In recent years several investigations demon-

strated that the degree of polarization of a par-

tially coherent beam which propagates in free

space changes as the beam propagates [1,2]. In thisnote we apply the recently developed unified the-

ory of coherence and polarization [3–5] to derive

* Corresponding author. Tel.: +1-407-8236864; fax: +1-407-

8236880.

E-mail address: korotkov@mail.ucf.edu (O. Korotkova).1 Present address: Department of Physics and Astronomy,

University of Rochester, Rochester, NY 14627 and The

Institute of Optics, University of Rochester, Rochester, NY

14627, USA.

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.01.005

an expression, valid at an arbitrary distance from

the source, for the degree of polarization of a beam

propagating through a turbulent atmosphere. The

source is assumed to generate an electromagnetic

generalization of the so-called Gaussian Schell-

model beam, which is a well-known model ofpartially coherent beams used in many investiga-

tions. The usual model is based on scalar theory

and it includes, as special case, some well-known

laser modes. We show that after propagating a

sufficiently long distance the degree of polariza-

tion, whilst changing with the distance of propa-

gation, returns to its initial value (its value in the

source plane), irrespective of the atmosphericconditions. Special case of this result has already

been reported in the literature [6].

ed.

226 O. Korotkova et al. / Optics Communications 233 (2004) 225–230

2. The 2� 2 cross-spectral density matrix of random

Gaussian Schell-model source

The second-order coherence and polarization

properties of a random, statistically stationaryelectromagnetic beam may be characterized by a

2� 2 electric cross-spectral density matrix [3]

W$ðr1; r2;xÞ � Wijðr1; r2;xÞ

¼ hE�i ðr1;xÞEjðr2;xÞi

ði ¼ x; y; j ¼ x; yÞ; ð2:1Þwhere E ¼ ðEx;EyÞ are members of a statistical

ensemble representing a fluctuating electric field at

a point r, at frequency x and the angular brackets

denote the average taken over the ensemble of

realizations of the electric field in the sense of the

coherence theory in the space-frequency domain(cf. [7], Section 4.7.1). The Ex and Ey components

are taken along two mutually orthogonal direc-

tions at right angle to the direction of propagation

of the beam (the z-direction).Since we are interested in propagation close to

the z-direction it is convenient to set r � ðq; zÞ,where q is a two-dimensional vector perpendicular

to the beam axis and z is the distance from thesource plane (see Fig. 1).

The elements of the cross-spectral density ma-

trix in the source plane may be expressed in the

form

W ð0Þij ðq0

1;q02;xÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSð0Þi ðq0

1;xÞq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sð0Þj ðq0

2;xÞq

�gð0Þij ðq02�q0

1;xÞ ði¼ x;y; j¼ x;yÞ;ð2:2Þ

where Sð0Þi represents the spectral density of the

component Ei of the electric field and gð0Þij is the

Fig. 1. Illustrating the notation relating to propagation of an

electromagnetic beam through a turbulent atmosphere.

spectral degree of correlation between the com-

ponents Ei and Ej in the plane z ¼ 0. These

quantities may be determined experimentally [5].

In Appendix A we show that for all values of their

argument and all pairs of the indexes i; j thespectral degrees of correlation satisfy the inequal-

ity jgð0Þij j6 1.

We will consider random electromagnetic

beams generated by Gaussian Shell-model sources.

For such sources 2

Sð0Þi ðq0;xÞ ¼ A2

i expð�q02=2r2i Þ ði ¼ x; yÞ; ð2:3Þ

gð0Þij ðq02 � q0

1;xÞ ¼ Bij exp

"� ðq0

2 � q01Þ

2

2d2ij

#

ði ¼ x; y; j ¼ x; yÞ: ð2:4ÞIn these expressions the factors Ai and Bij are in-

dependent of position but may depend on fre-

quency. The same is true about the variances ri

and dij. Moreover, the factor Bij has the following

properties:

Bij � 1 when i ¼ j; ð2:5aÞ

jBijj6 1 when i 6¼ j; ð2:5bÞ

Bji ¼ B�ij; ð2:5cÞ

asterisk denoting the complex conjugate. Eq.

(2.5a) follows from the fact that when j ¼ i gð0Þij is

just the usual spectral degree of coherence of the

scalar theory (cf. [7], Section 4.3.2), which is well

known to have the value unity when its two spatial

arguments coincide. The inequality (2.5b) followsfrom the fact that jgð0Þij j6 1 as we already noted.

The relation (2.5c) follows from Eq. (2.4) and the

relation Wjiðr1; r2;xÞ ¼ W �ij ðr2; r1;xÞ that is an im-

mediate consequence of the definition of the ma-

trix W$

. We will now derive an expression for the

degree of polarization of a Gaussian–Schell model

beam propagating through the turbulent atmo-

sphere.

2 Properties of beams generated by such sources have been

extensively studied by Gori et al. [2].

O. Korotkova et al. / Optics Communications 233 (2004) 225–230 227

3. Degree of polarization of the Gaussian Schell-

model beam in a turbulent atmosphere

The spectral degree of polarizationof the beamat

a point r is given by the expression (cf. [3], Eq. (11))

Pðr;xÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4DetW

$ðr; r;xÞ

TrW$ðr; r;xÞ

h i2vuuut ; ð3:1Þ

where DetW$

and TrW$

denote the determinant and

the trace, respectively, of the matrix W$. More ex-

plicitly, in terms of the elements of the W$-matrix

one readily finds that the degree of polarization is

given by the formula

Pðr;xÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðWxx � WyyÞ2 þ 4WxyWyx

qWxx þ Wyy

; ð3:2Þ

where the arguments of all the elements of the W$-

matrix are, of course, ðr; r;xÞ.To determine the degree of polarization across

an electromagnetic Gaussian Schell-model source

we must first determine the elements of the matrix

Wijðq1; q2;xÞ. From Eqs. (2.2) and (2.4) it follows

at once that they are given by the expression

W ð0Þij ðq0

1; q02;xÞ ¼ AiAjBij exp

"� q02

1

4r2i

þ q02

2

4r2j

!#

� exp

"� ðq0

2 � q01Þ

2

d2ij

#

ði ¼ x; y; j ¼ x; yÞ: ð3:3ÞTo simplify the analysis we will assume that

ri ¼ rj � r: ð3:4ÞThe formula (3.3) then reduces to

W ð0Þij ðq0; q0;xÞ ¼ AiAjBij exp

�� q02

2r2

� ��ði ¼ x; y; j ¼ x; yÞ: ð3:5ÞOn substituting from Eq. (3.5) into the formula

(3.2) with r � q and using Eq. (2.5c) we obtain for

the degree of polarization across the source plane

the expression

Pð0Þðq0;xÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA2

x � A2yÞ

2 þ 4A2xA2

y jBxy j2q

A2 þ A2: ð3:6Þ

x y

It is to be noted that as a consequence of the

simplifying assumption (3.4) the degree of polari-

zation across the source plane is independent of

the position vector q, i.e., it is uniform across it. It

is also of interest to note that apart from thecorrelation coefficient Bxy , the degree of polariza-

tion (3.6) in the source plane depends only on the

ratio Ax=Ay , not on the actual values of Ax and Ay .

In a recent paper [6] an expression was derived

for a cross-spectral density matrix of a electro-

magnetic Gaussian Schell-model beam at any

point r � ðq; zÞ in the half space z > 0 containing

turbulent atmosphere. The Gaussian Schell-modelsource which generated the beam was somewhat

restricted in that it was assumed that the off-di-

agonal elements W ð0Þxy ðq0;xÞ ¼ W ð0Þ

yx ðq0;xÞ ¼ 0, i.e.,

that the x and y components of the electric field

were uncorrelated at each source point. However,

it is not difficult to generalize the analysis by re-

moving this assumption and one then readily finds

by a similar argument as used in [6] that

Wijðq; q; z;xÞ ¼AiAjBij

D2ijðzÞ

exp

"� q2

2r2D2ijðzÞ

#

ði ¼ x; y; j ¼ x; yÞ; ð3:7Þwhere D2

ijðzÞ is the so-called effective beam spread.

This quantity depends on the model which one

uses for the atmospheric turbulence but, with the

commonly used models, it has the form

D2ijðzÞ ¼ 1þ aijz2 þ Tzm ði ¼ x; y; j ¼ x; yÞ;

ð3:8Þwhere

aij ¼1

ðkrÞ21

4r2

þ 1

d2ij

!: ð3:9Þ

In Eq. (3.8) T and m are parameters depending on

the atmospheric model. The formula (3.8) shows

that the spreading of the beam depends on both the

source parameters ðr; dijÞ and on the parameters

describing the model used for the turbulence ðT ;mÞ.Two expressions are currently available for the

beam spread. One is based on Tatarskii model ofthe spectrum of atmospheric fluctuations and the

other on the Kolmogorov spectrum (cf. [8], Section

3.3). For the former model (Tatarskii)

228 O. Korotkova et al. / Optics Communications 233 (2004) 225–230

T ¼ 1:093C2nl

�1=30 r�2; m ¼ 3; ð3:10Þ

and for the other model (Kolmogorov)

T ¼ 0:98ðC2nÞ

6=5k2=5r�2; m ¼ 16

5: ð3:11Þ

In these expressions k ¼ x=c is the wave number

of the light beam, C2n is the refractive index struc-

ture parameter and l0 is the inner scale of turbu-

lence.

Expression for the beam spread D2ijðzÞ based on

these two formulas has been derived in [9,10], us-ing, however, the scalar theory rather than elec-

tromagnetic theory.

Based on the definition (3.1) for the degree of

polarization, together with Eq. (3.7), we derive a

general expression for the degree of polarization

Pðq; z;xÞ of a Gaussian Schell-model beam

propagating in atmospheric turbulence. We find

that

Pðq; z;xÞ ¼ ½F ðzÞ�1=2

GðzÞ ; ð3:12Þ

where

F ðzÞ ¼ A2x

D2xxðzÞ

exp

" � q2

2rD2xxðzÞ

#

�A2y

D2yyðzÞ

exp

"� q2

2rD2yyðzÞ

#!2

þ4A2

xA2y jBxy j2

D2xyðzÞ

exp

"� q2

rD2xyðzÞ

#; ð3:13Þ

GðzÞ ¼ A2x

D2xxðzÞ

exp

"� q2

2rD2xxðzÞ

#

þA2y

D2yyðzÞ

exp

"� q2

2rD2yyðzÞ

#: ð3:14Þ

4. Far-zone behavior of the degree of polarization of

a beam propagating in a turbulent atmosphere

Based on the Tatarskii model one readily finds

that after sufficiently long propagation distance

the elements of cross-spectral density matrix of the

Gaussian Schell-model beam are given by the

expression

Wijðq; z;xÞ �AiAjBij

Tz�3 � AiAjBijaij

T 2z�4

þOðz�5Þ ð4:1Þ

as kz ! 1, O denoting the order of magnitude

symbol.On substituting from Eq. (4.1) into the expres-

sion (3.12) for the degree of polarization one finds

that as kz ! 1

Pðq; z;xÞ ! ½F ðzÞ�1=2

GðzÞ ; ð4:2Þ

where

F ðzÞ ¼A2x � A2

y

Tz�3

"�A2xaxx � A2

yayyT 2

z�4 þOðz�5Þ#2

þ 4A2xA

2yBxy

Tz�3

"�A2xA

2yBxy

T 2z�4 þOðz�5Þ

#2;

ð4:3Þ

GðzÞ ¼A2x þ A2

y

Tz�3 �

A2xaxx þ A2

yayyT 2

z�4 þOðz�5Þ:

ð4:4ÞIt is clear from the formula (4.2), together with

Eqs. (4.3) and (4.4), that the leading term (the term

in the power of z�3) does not depend on the co-

efficients for the effective beam spread aij, definedby Eq. (3.9). Moreover the ‘‘atmospheric coeffi-

cient’’ T cancels out in the expression (4.2) in the

limit as kz ! 1. Hence the degree of polarization,given by the formula (4.2), becomes

Pðq;z;xÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðA2

x �A2yÞ

2þ4A2xA

2y jBxy j2

qA2x þA2

y

as kz!1:

ð4:5ÞOn comparing Eq. (4.5) with the expression (3.6)

we see that

Pðq; z;xÞ � P0ðq;xÞ as kz ! 1 ð4:6Þ

the expressions on both sides of this formula being

actually independent of the transverse variables (q

and q0). This result was obtained on the basis of

the Tatarskii model of turbulence. If instead one

considers the Kolmogorov model one finds in

O. Korotkova et al. / Optics Communications 233 (2004) 225–230 229

place of Eq. (4.1) that the elements of the cross

spectral density matrix are

Wijðq; z;xÞ

¼ AiAjBij

Tz�16=5 � AiAjBijaij

T 2z�22=5 þOðz�28=5Þ

ði ¼ x; y; j ¼ x; yÞ: ð4:7ÞBy similar arguments as based on Eq. (4.1) one

finds that the conclusion expressed by Eq. (4.6)again holds.

In free space the expression for the degree of

polarization of a beam sufficiently far away from

the source plane can be derived from the general

formula (3.12) by setting for the beam spread Dij

the free-space value

Pðq; z;xÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2xya

2yyA

2x � 2axxayya2xyAxAy þ a2xya

2xxA

2y þ 4a2xxa

2yyA

2xA

2y jBxy j2

qðA2

xayy þ A2yaxxÞaxy

as kz ! 1; ð4:8Þ

where the aij are given by the formula (3.9). In this

case Pðq; z ! 1;xÞ 6¼ Pðq; z ! 0;xÞ and in gen-

eral: the far-zone value of the degree of polariza-

tion depends on all the parameters of the source.

Fig. 2. The change of the degree of polarization of a Gaussian–

Schell model beam propagating in a turbulent atmosphere,

calculated from Eq. (3.2). The parameters characterizing the

source are x ¼ 3� 1015 rad/sec (k ¼ 0:628 lm), A2x ¼ A2

y ¼ 0:5,

Bxy ¼ 0:2, rx ¼ ry ¼ r ¼ 5 cm, dxx ¼ dyy ¼ 0:1 mm. The pa-

rameters characterizing the atmosphere were chosen to be

C2n ¼ 10�13m2=3, l0 ¼ 5 mm.

The result expressed by Eq. (4.6) is the main

conclusion of our analysis. It shows that after a

sufficiently long distance of propagation through

the atmosphere the degree of polarization of the

beam returns to its initial value (its value in the

source plane). Moreover, since the expression (4.5)depends on the A and B coefficients, the degree of

polarization of the beam after it traveled over a

sufficiently long distance depends neither on the

spectral degrees of correlation gij of the Gaussian–

Schell model source nor on the atmospheric tur-

bulence. This conclusion is in agreement with a

result derived in [6] for a more restricted class of

beams.

In Fig. 2 we illustrate the behavior of the degreeof polarization of electromagnetic Gaussian–

Schell model beam both for propagation in tur-

bulence and in free space with increasing distance

from the source plane, for selected values of the

parameters. The figure shows that for propagation

in free space the degree of polarization acquires a

particular value after propagating at a certain

distance (z ¼ z0 say) and it retains this value as thebeam propagates further, i.e., for z > z0. On the

other hand in turbulent atmosphere the degree of

polarization returns to its initial value (the value it

has in the source plane) after it propagates over a

sufficiently long distance.

Acknowledgements

The research was supported by the US Air

Force Office of Scientific Research under Grant

No. F49260-03-1-0138, by the Engineering Re-

search Program of the Office of Basic Energy

Sciences at the US Department of Energy under

Grant No. DE-FG02-2ER45992, and by the De-

fense Advance Research Project Agency underGrant MDA 972011043.

230 O. Korotkova et al. / Optics Communications 233 (2004) 225–230

Appendix A. Proof of the inequality |gij(r1; r2;x)|< 1

We have the obvious inequality

ja1Eiðr1;xÞD

þ a2Ejðr2;xÞj2EP 0; ðA:1Þ

where a1 and a2 are arbitrary constants. This in-

equality implies that

a�1a1hE�i ðr1;xÞEiðr1;xÞi

þ a�2a2 E�j ðr1;xÞEjðr2;xÞ

D Eþ a�1a2 E�

i ðr1;xÞEjðr2;xÞ� �

þ a1a�2 E�j ðr2;xÞEiðr2;xÞ

D EP 0: ðA:2Þ

We may express (A.2) in the form

a�1a1Siðr1;xÞ þ a�2a2Sjðr2;xÞ þ a�1a2Wijðr1; r2;xÞþ a1a�2Wjiðr1; r2;xÞP 0: ðA:3Þ

It follows from the definition of the off-diagonalelements of the cross-spectral density matrix that

Wjiðr1; r2;xÞ ¼ W �ij ðr2; r1;xÞ. Using this fact and a

well-known property of non-negative definite

quadratic forms [11] it follows that the determinant

Siðr1;xÞ Wijðr1; r2;xÞW �

ij ðr1; r2;xÞ Sjðr2;xÞ

��������P 0 ðA:4Þ

implying that

jgijðr1; r2;xÞj �jWijðr1; r2;xÞjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Siðr1;xÞp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sjðr2;xÞp 6 1 ðA:5Þ

for all values of the arguments and of the suffixes.

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