effect of polarization on the statistics of scattered waves from linear random media

Physics Letters A 163 (1992) 32-36 PHYSICS LETTERS A North-Holland

Effect of polarization on the statistics of scattered waves from linear random media

Chris t ian Brosseau Laboratoire de Spectrometric Physique ~ (CERMO), Universitg" Joseph Fourier. B.P. 87, 38402 Saint-Martin-d'Hbres Cedex, France

Received 15 October 1991; accepted for publication 9 January 1992 Communicated by J.P. Vigier

The probability density functions and moments which describe the temporal fluctuations of the amplitudes and phases for a Gaussian plane wave vectorial field are obtained as functions of the degree of polarization of the field. These results are then specialized to characterize the polarization properties of scattered Gaussian waves from a linear temporally random medium.

There has been intense activity recently in deter- mining correlations in speckle patterns by examining the influence of polarization on the statistics gov- erning the optical fields as evinced by the articles of Freund [ 1 ], Cohen et al. [ 2 ], Berkovits and Kaveh [3] to mention just a few. Interest in this topic is stimulated by the considerable effort being ex- panded on the study of speckle-pattern correlations to retrieve useful information about the properties of a scattering medium. Theoretical techniques devel- oped for the study of conductance fluctuation effects in mesoscopic conductors at low temperatures have been also applied to the transmission of light through a disordered scattering medium. The connection between these two sets of problems has been amplified in many places (see for instance ref. [4] ) and led to remarkable phenomena (e.g. weak Anderson optical localization effects, scattering enhancement around the backscattering direction of light from a rough surface). Similar problems appear in several physi- cal areas where radiation propagates through a medium containing randomly refractive index inho- mogeneities (e.g. ocean, atmosphere) and have been treated with specific techniques [ 5].

This work deals with the statistical description of temporal fluctuations of a plane wave field for ar- bitary degree of polarization. The basic objectives of

Laboratoire associ~ au C,N.R.S.

this Letter are twofold'. The first is to derive the statistical characteristics of a fluctuating Gaussian field which is partially polarized. A second aim is to ana- lyze how these results are modified by interaction with a random medium. Throughout the paper we will make use of two main assumptions: (i) the distribution of the electric field components is Gauss- ian and (ii) the random scattering medium is linear. The former can be viewed as an instance of the central limit theorem and reflects the fact that the field is a sum of a large number of statistically independent random contributions. Concerning the latter, we shall exploit the invariance of a Gaussian distribution under any linear transformation.

Following the usual complex analytic signal [6] representation of the fluctuating electric field strength of light, we shall assume moreover that the field is statistically homogeneous and temporally stationary at least in the wide sense ~. Let us assume that the approximation of a narrow band about a central frequency may be adopted (quasimonochromatic spec- trum). Our starting point is the Gaussian joint bi- variate complex probability density function (of zero mean) for the two fluctuating orthogonal compo-

~ If the field fluctuations are ergodic (this case is the one treated), ensemble averages are equal to temporal averages. For details and discussions of stationarity and ergodicity, see ref. [ 7 ]. For an account of the space-frequency representation of stationary random fields, see ref. [ 8 ].

32 0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

Volume 163, number 1,2 PHYSICS LETTERS A 9 March 1992

nents Ej(t), j = 1, 2, of the transverse electric field E at a point (with respect to a prescribed coordinate system ):

de. , ( ) p(E1,E2)= n2 exp -- E~AijEj , (1)

i , j = l

where the Hermitian and positive definite 2 X 2 matrix A has the dimension of the inverse of an intensity. Note that the equations written in the time do- main may be also expressed in the frequency space by a Fourier transform; however, since the field is stationary it is not square integrable and hence does not necessarily possess a Fourier representation: in that case one resorts to truncated E(t; T) functions over the interval [ - T, T] and proceeds to the limit as T-,oo (see footnote 1 ). Physically, the matrix A is simply related to the temporal coherency matrix q~ whose elements are given by

~o( r ) = ( A - I )o = (Ei(t+z)E~(t))

T

1 = lim j E~(t+z; T)E'~(t; T) dt (2)

r~oo 2T - - T

where the angular brackets denote an average over the ensemble that characterizes the correlations of the complex field at separated times. Here AII=(IE212)D -1, A22=(IE112)D -l and A12= - ( E I E $ ) D -1 with P being the degree of polarization, (So) the total intensity of the field (both are easily measurable parameters) and

D = l(l--p2) (So)2.

Second-order correlations of the wave field are characterized via the Wolf coherency matrix • and following standard practice [ 9,10 ] we will employ the cross-spectral purity condition, then the coherency matrix is simply the product of a Hermitian matrix q~o(0) independent of z and a scalar function of z. This condition implies that the power spectral distribution is the same for components of linear polarization in any direction.

Now instead of computing the different moments of the qOij, we may write Ej=ajexp(iOj) where aj= IEjl is the field amplitude and consider the joint distribution of the aj and 0j as well as their marginal probability density functions. We shall write A12 = [Ai2[ exp(i012). Given these background re-

marks, the joint probability density function (al, a2, G, 02 are taken at the same time) that can be re- written from eq. (1) (using the Jacobian transformation) is

p(al, a2, 01,02) ( 2

ala2 det(A) exp - ~] E*AoE j . (3) - - ~ 2 i , j = 1 /

The joint distribution of amplitudes is given by phase integration of eq. (3). We obtain

p(al, a2) =ala2 det(A) exp[ - (AIj aZl +A2za~) ] 2~x 2x

x f fexp[-2lAl2lala2 0 0

X cos(01 - 0 2 - 0 1 2 ) ] d01d02. (4)

The integration is straightforward and from results of ref. [ 11 ], eq. (4) can be put in the following form,

16ala2 ( 2 ( a~+az ~) p(al,az)= (So)Z( l_p2) exp_- ( S o ) ( l _ p 2 ) ]

. { 4Pal az "~ ,o~(So~i--~-_p2) ) , al,a2>~O, (5)

where Io is the zero-order modified Bessel function of the first kind. Note that in the general situation p(al, a2) cannot be written as a product of a function depending on al and a function depending on a2. We may also verify immediately from (5) that the marginal distributions of al and a2 are Rayleigh distributed.

Now from ( 5 ) and provided that Re ( a ) > - 2 and Re (#) > - 2 ( Re (z) stands for the real part of z), we may compute the moments of order a + fl:

oooo

(a~a~)= f fac(a~p(al,az)dalda2. (6) 0 0

Expanding the Bessel function Io contained in p(al, a2), it is straightforward albeit tedious to prove that

( a? ag ) = F( 1 + ½or)F( 1 + ½fl) det(A) A I + o t / 2 4 1 + f l / 2

I1 - 'a22

× 2 G ( l + ½ a , l+½fl, 1;;t2), (7)

where zF~ is the usual hypergeometric function and 2= IA~2I / (A11A22)1/2 is the modulus of the temporal

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degree of coherence (2~<P) [6]. When a and fl are integers (let or= 2p, fl=2q for instance), then we obtain after tedious calculations

( IEI 12pIE2 I 2 q ) = F ( p + 1 )F(q+ 1 )

×( ( IEI[Z) )P( ( IE21)2) q ~ (P)(q~22r. (8) r=O \ r / \ r /

This equation is useful to characterize intensity correlations (i.e. fourth-order field correlations).

From (8) and the usual properties of the complete elliptic integrals o f the first and second kind (re- spectively K and E) [ 1 1 ], one may derive the covariance of al and a2:

c o v ( a l , a2) =½( ( l E a 12 ) ( [ E 2 12) )

× [ 2 E ( 2 ) - ( 1 - 2 2 ) K ( 2 ) - ½ r t ] . (9)

A similar analysis holds for the statistical characterization of the phases. The joint density function of phases is

p ( O , , O 2 ) = i i a l a 2 e x p [ - ( A , l a 2 + A 2 2 a 2 ) ] o o

X exp[ - 2 IA12 lal a2 cos(01 --02 --012) ] dal da2.

(10)

It can be also written using simple algebra (see footnote 1 ) as the closed form

p(Ol ,02)- det (A) (1 __,H2) _3/2 8K 2All A22

× [/~ s i n - ' (/z) + ½ ~ / t+ (1 - /z2)1 /2] ,

- ~ <<. 01, 02 ~ ~,

= 0, otherwise, (11)

with/~ being defined as/1 = 2 cos ( 01 - - 0 2 - - 012 )" Since 01 and 02 appear only as 01-02, we may write p(01, 02 ) = p ( 0 1 - 02) as a function of the phase difference only.

As was pointed out in ref. [12 ] ,2, 01 and 02 are

,2 Note that these results depend critically on the utilization of a particular Gaussian statistics (due to N.R. Goodman). A recent work (see ref. [ 131 ) has shown that a more general treatment gives different results from those presented by Barakat.

uniformly distributed on the interval [ - 7t, ~ ] as any other possible density function would induce non- stationarity.

From (1 1 ) and making use of the Jacobi-Anger expression [ 1 1 ], one can check that the covariance of the phases is expressed as

1 , m F 2 ( 1 + ½ m ) coy (01 ,0~ ) = 2 2. ~ A

,n=l m- F ( I + r n )

Xcos(mOI2)2Fl(½m, ½m, m + l ; 2 2 ) • (12)

At this point, two comments are in order. First, it is instructive to specialize these results to the case of unpolarized light (i.e. A is a multiple of the 2 × 2 unit matrix [ 14 ] ). Then it follows from eqs. (5) and (10) that the aj and O? are uncorrelated. Statistical properties of the unpolarized light do not change on in- troducing any phase difference between the field components Ej and any rotation of these two orthogonal components by an angle about the direction of propagation [ 6 ]: this can be appreciated from the rotational invariants of the coherency matrix. This is in conformity with the results o f ref. [ 13 ]. Then, we add a note on the fact that few published experimental data on the subject have come to our attention except for the complementary problem of the spatial statistics [ 15 ].

The next problem we will now consider is the characterization of the statistical behavior of a set of waves as they pass through a random medium. This question is critical for understanding systems with spatio-temporal disorder since field fluctuations carry information on the dynamics of these systems.

A number of restrictions are placed at the outset: in addit ion to constraint (ii), we only consider the influence of temporal fluctuations in the medium and assume that the spatial fluctuations vary slowly enough to be neglected in order to employ the usual 2 × 2 input /output Jones matrix representation [ 16 ] ; moreover we simplify our task by considering only forward scattering. Kim et al. [ 17 ] have recently ex- amined other aspects of this problem (i.e. the rela- t ionship between Jones and Miiller matrices for random media) .

Now the effect o f a non-image optical med ium on light is to relate the input q~i (subscript i ) and output qbo (subscript o) coherency matrices by a congruent t ransformat ion [14]. As anticipated above, the

34


problem of characterizing the statistics of scattered waves reduces to the determination of @o. Let us suppose a Gaussian wave field is incident onto a slab- shaped temporally random medium (non-mag- netic) characterized by a Jones polarization transfer matrix T that we decompose as a sum of a deter- ministic part T D and a stationary zero mean Gaus- sian random process TR. To proceed further, we shall make the additional assumption that the fluctuations in the medium are much slower than the fluctuations in the incident wave, so that the fluctuations are uncorrelated. From the recent theoretical analysis of Brosseau and Barakat [ 16 ], we write the output coherency matrix as

Oo =a(09) + A(09), (13)

where B(09) =To(09)Oi(09)Ti~ (09), and

A(09) = l im ~T (TR (09)Ei(09) XE~- (09)Tff (09) ) ,

X being the notation for the direct product. It follows from the Wiener-Khintchine theorem

that @o (o9) represents the power spectral density of • o(r) (eq. (2) ) [18]. Once Oo is known, then the corresponding matrix & can be computed (eq. (2) ) and the above results apply (thanks to constraint ( i i)) .

Two quantities of experimental interest are the transmittance of the medium (at frequency 09):

1 3

g= 2 (So) i k=o ~ O ° ( S k ) i ' (14)

where (Sk) ~ denote the input spectral Stokes parameters [16] (the coefficients D O are explicitly given in ref. [ 16 ] ), and the output degree of polarization which takes the form

(So)o 2( 1 - P o z) = (So)2( 1 - e 2 ) I det (TD)l 2

+4 [det (A) + (B~ ~Az2 +B22Al~ )

- (BI2AT2-BT2A12) ] • (15)

In order to fix our ideas, we shall compare these findings with the recent results of Cohen et al. [ 2 ]. These authors produced interesting experimental data concerning intensity scattering from a diffuse coat- 'ing albeit concerning spatial correlations of coherent waves. The distribution function they choose to specify the polarization ellipse was the ellipticity

(i.e. the ratio between the minor and major axes of the eUiptically polarized light). Their analysis is equivalent to the special unpolarized case of the present temporal statistics: the two field components are spatially independent (see eq. (5) of ref. [ 2 ] ). The ellipse is inscribed in a rectangle whose lengths are 2a~ and 2a2. The notation for the minor (major) axis is a (b). Then ~=a/b with 0~<~< 1. Detailed calculations are given elsewhere [19]. The deriva- tion of p(~) proceeds in two steps. First, recast eq. (3) in the usual ellipsometric notations [ 6 ] in terms of the intensity (So) , the angle ~u which the major axis makes with the 1-axis and the ellipticity ~ by Ja- cobian transformations. Then, specialize these results to the case P = 0: it follows that the joint distribution function reads

p(go, N ,~ )=p(go )p (~ )p (~ ) ,

2(1 - e 2) p (E)= (1+~2) 2. (16)

It is interesting to remark that the explicit calcula- tion ofp(E) was first done by Hurwitz [20] using a different approach than ours (having also a different motivation since his work was strictly limited to the characterization of the statistical properties of unpolarized light); however, with a~, a2 being statistically independent Gaussian processes and 01, 02 uniformly distributed between 0 and 2~, which is the case of interest here. Eq. (16) is the distribution function found experimentally by Cohen et al. [2]. Note that this is a surprising result since it gives more weight to linear than to circular polarization states (i.e. (~) =0.307).

In summary, we have obtained analytic expres- sions concerning the statistics of the field fluctuations for studying optical media that change the polarization state of the incident light during the scattering process. The predictions made by the present work can be tested against experiments (e.g. to derive the form of the Jones matrix TD and its fluctuation T~). So far, we have been concerned exclu- sively with random scattering media whose statistics is Gaussian. It has to be said that in many situations of interest scatterers are random but not Gaussian (e.g. K distributions), we hope to present work in this area at a later data. Our treatment should also eventually be extended to include (a) multiple scat-

35


t e r i n g effects a n d ( b ) n o n - l i n e a r i t y effects.

I t is a p l e a s u r e to t h a n k D i c k B a r a k a t a t H a r v a r d

U n i v e r s i t y for i n t r o d u c i n g m e to th i s p r o b l e m a n d

for g iv ing t he f i rs t i m p u l s e for t h e ca l cu la t ions . Spe-

cial t h a n k s to P r o f e s s o r Ed. R o c k o w e r a t t he N a v a l

P o s t g r a d u a t e School , M o n t e r e y for s t i m u l a t i n g a n d

he lp fu l d i s c u s s i o n s o n t he subjec t .

References

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Rev. A 43 (1991) 5748. [ 3 ] R. Berkovits and M. Kaveh, Europhys. Lett. 13 (1990) 97. [ 4 ] E. Akkermans, P.E. Wolf, R. Maynard and G. Maret, J. Phys.

(Paris) 49 (1988) 77; M.P. van Albada and A. Lagendijk, Phys. Rev. B 36 ( 1987 ) 2353.

[5]A. lshimaru, Wave propagation and scattering in random media (Academic Press, New York, 1978).

[6] M. Born and E. Wolf, Principles of optics, 6th Ed. (Pergamon, Oxford, 1980); L. Mandel and E. Wolf, Rev. Mod. Phys. 32 ( 1965 ) 231.

[ 7 ] W.B. Davenport and W.L. Root, Random signals and noise ( McGraw-Hill, New York, 1958 ).

[ 8 ] E. Wolf, J. Opt. Soc. Am. 72 (1982) 343; A 3 ( 1986 ) 76. [ 9 ] L. Mandel, J. Opt. Soc. Am. 51 ( 1961 ) 1342.

[10] B.E.A. Saleh, Photoelectron statistics (Springer, Berlin, 1978).

[11] M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Dover, New York, 1983 ); G.A. Korn and T.M. Korn, eds., Mathematical handbook for scientists and engineers (McGraw-Hill, New York, 1961) ch. 21.8.

[12] R. Barakat, J. Opt. Soc. Am. A 4 (1987) 1256; Opt. Acta 32 (1985) 295.

[ 13] L. Mandel, Proc. Phys. Soc. 81 (1963) 1104; C. Brosseau, R. Barakat and E. Rockower, Opt. Commun. 82 (1991) 204.

[ 14] C. Brosseau, Optik 88 ( 1991 ) 109. [ 15 ] P.F. Steeger, T. Asakura, K. Zocha and A.F. Fercher, J. Opt.

Soc. Am. A 1 (1984) 677: P.F. Steeger and A.F. Fercher, Opt. Acta 29 (1982) 1395: A.F. Fercher and P.F. Steeger, Opt. Acta 28 ( 1981 ) 443.

[ 16] C. Brosseau and R. Barakat, Opt. Commun. 84 ( 1991 ) 127. [ 17] K. Kim, L. Mandel and E. Wolf, J. Opt. Soc. Am. A 4 (1987)

433. [18] B. Crosignani, P. di Porto and M. Bertolotti, Statistical

properties of scattered light (Academic Press, New York, 1975).

[ 19 ] C. Brosseau, unpublished. [20] H. Hurwitz, J. Opt. Soc. Am. 35 (1945) 525.

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effect of polarization on the statistics of scattered waves from linear random media

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