Physica A 200 (1993) 469-475

North-Holland

Distribution function of the intensity of optical waves in random systems

Eugene Rene Baumgartner, Richard Berkovits Moshe Kaveh

and Pearl Technology, University, Ramat-Gan 52900, Israel

Statistics of coherent radiation in medium is analyzed in the

diagram The distribution function for

lated and is shown, only for values of the the distribution function is

exponential, as statistics. For larger of the

distribution function differs from the simple exponential, and the asymptotical

behavior is stretched exponential. The obtained are numerical

simulations.

When a coherent wave propagates in a random medium the density of energy in a given point (intensity) Z is a strongly fluctuating quantity. These fluctuations are traditionally described by Rayleigh statistics [ 11. This statistics means the following distribution function PR(Z):

PRV) = -& exp ( -- i, ’ )

which corresponds to the equation for the moments

(I”) = n!(zy . (2)

In eqs. (1) and (2) the averaging is with respect to all macroscopically equivalent random samples. This statistics is in fact the manifestation of the central limit theorem. If we suppose the field of the wave in a given point to be the sum of a large number of independent random complex terms (“contribu- tions from different trajectories”) then for the intensity, which is proportional to the square of the modulus of the field, negative exponential statistics can be very easily obtained [l]. However in measurements of the distribution of the intensity of polarized radiation the departures from this simple statistics are found [2]. The reason for these departures is obvious. There is interference

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470 E. Kogan et al. I Intensity distribution of optical waves

between different trajectories. It is this interference which leads to strong non-classical “mesoscopical” fluctuations, first discovered for electrons [3,4] and then for classical waves [5].

To study the issue of statistics from scratch we will present a systematical approximation theory for the distribution function Z’(Z) and show that Rayleigh statistics corresponds to the first order approximation in this theory. The second order approximation would be explicitly calculated. This approximation gives a negative exponential distribution function for small values of argument and a stretched exponent for large values of argument. The consideration would be in the framework of the traditional diagram technique.

First let us remember how Rayleigh statistics is obtained in this framework [6]. For simplicity we’ll consider the case of a point source situated at the origin; the intensity is measured in the point r. In the diagrammatic representa- tion (I) is given by the diagrams with a pair of wave propagators G,R, and GE, summed with respect to all possible interconnections by interaction lines. The nth moment (I”) is given by the set of diagrams with n propagators G,R, and it propagators G $. Important for us is the following property of diagrams: if the diagram consists of several disconnected parts, then the contribution of that diagram is equal to the product of contributions of disconnected parts. It means, that if we consider for (I”) only the diagrams consisting of IZ disconnected parts, each part being the set of diagrams with a pair of propagators (advanced and retarded), we immediately obtain eq. (2). The multiplier n! which appears in the nth moment is of combinatorial origin; it is simply the number of possible pairings between propagators.

The generalization of this result is straightforward. An arbitrary diagram for (I” ) is generally speaking disconnected and consists of several connected parts. That is, it has m, parts with 1 pair of propagators (an advanced and a retarded one), m2 parts with 2 pairs and so on up to II pairs (the connected diagram where the number of advanced propagators does not coincide with the number of retarded propagators would give a contribution of higher order with respect to the parameter exp(-L/8), where 8 is a mean free path and L is a sample thickness, and we neglect such diagrams). So we can classify all the diagrams according to their topology, which is given by the numbers {m,, . . . , m,}, and the sum of all diagrams can be written down in the following way:

(I”) = n!M, ) (3)

where

M, = c fYm,,m,,..., m,) (Zc)m1(Z’)~2-~ * (I”):” ; (4) m,+2m*+~..+nm,=n

E. Kogan et al. I Intensity distribution of optical waves 471

the connected diagram contribution (Ii), is the sum of all connected diagrams with i pairs of propagators and

n! fym,,m,,...,m,)= (1!)“‘(2!)“2. . . (n!)““m,!m,!~ . . m,! (5)

is the number of partitions of IZ different objects into m, groups of 1 object, m2 groups of 2 objects and so on. The summation in eq. (4) is with respect to all possible non-negative integers satisfying the equation given. A single term in the sum in eq. (4) gives the contribution from all the diagrams with the topology given by the numbers {m,, . . . , m,}, and the multipliers n! and

Z+,, m2,. . . ,m,) are of purely combinatorical origin. The former is the number of ways we can “couple” propagators and the latter is the number of partitions of n pairs between connected diagrams.

The distribution function Z’(Z) is connected with its moments in the usual way:

P(Z) = i exp(itZ) 2 9 n=O

(I”) 2. -m

(6)

Using integral representation for n!,

m

n! = I

du un exp(-u) , (7) 0

for the distribution function Z’(Z) we have, changing integration variable

5=&,

Z’(Z)= 1 exp(ilZ/u) ~du~~o~M.,~exp(-u)$. -CC 0

From eq. (5) follows

5 (-il)” m (-il)y (I) 9 m (-i[),” (1’) “2

n=O yp%= c m,!(y) c m,!(y) -** ml=0 m*=o

( m (-i[)”

= exp C n! n=l

(Z”),) *

(8)

(10)

So from eq. (8) after changing the integration variable u = Z/u we finally obtain

472 E. Kogan et al. I Intensity distribution of optical waves

P(z) = 1 du F’(U) $ exp (- ;) ,

0

(11)

where we have introduced an auxiliary function p(u) given by the equation

F(u) = 7 exp(i[u) exp (nY$l 9 (I”),) $. -cc

The distribution function (11) can be described as the Rayleigh distribution function but with some effective averaged value which in turn fluctuates around the real averaged value.

So we obtained an exact equation which expresses the distribution function through the connected diagrams contribution only. Of course by itself, this does not solve the problem of finding a distribution function like finding the expression of a one particle Green function through the self-energy does not solve the problem of finding the Green function, but like in the former case it gives an opportunity to get reasonable approximations. First we notice that if we retain in the sum Cz=, [(-il)“/n!](Z”), only the first term, we immediately get Rayleigh statistics from eq. (11). This prompts that in an approximation of the order m one should take into account only m types of connected diagrams, that is retain in the sum Cr=, [(-i[)“ln!](Z”), only the first m terms. Hence we obtain the distribution function in terms of {(I) c, (Z’ ) c, . . . , (I” ) ,} , which should be treated in our theory as m input parameters. Because we classify the diagrams only according to their topological properties it is natural to call such approximations “topological” ones.

We shall present explicit results for the second order approximation. In this approximation the distribution function is determined by (I), and (Z*) =, which can be easily calculated; the first is determined mainly by ladder diagrams, the second by Hikami box diagrams [5]. Eq. (ll), however, can be looked at from another point of view. From this point of view the system of equations (3) should be understood as expressing the connected diagram contributions through the moments. For example the first two equations give

(Z), = (0 3

(z2), =i(Z2) - (z)2. (12)

So freely speaking we make an expansion of the distribution function near Rayleigh statistics and restore the distribution function on the basis of only two moments known:

E. Kogan et al. I Intensity distribution of optical waves

P(Z) - i dv exp (- (“2~Z!~~))2) i exp (- f> ,

0 c

and eq. (3) in this approximation can be written in the form

n/2

(I”) = (I)” msO ,,(~~)~~), (2A)-*” = n! ($$-)nZZn(iA),

473

(13)

(14)

where A* = (Z)*/2(Z*), and H,, is the Hermite polynomial. (In general we express the distribution function through the m moments known).

The functional dependence of the distribution function given by eq. (13) is determined in fact by a single parameter A). For tube geometry A: -g,, for slab geometry A: - Ltk*, where k is the radiation wave vector and g, is the classical conductance (the intensity is supposed to be measured on the output face). We are considering the “metallic phase”, i.e. go 9 1. That means that in the exponent in eq. (13) there is a large parameter, and the integral can be calculated by the steepest descend method. We consider explicitly only the limiting cases:

1) Zl (Z) G A, where we get

fv) = P,(Z) ; (15)

it is interesting that our distribution function has weak singularity: when Z goes to zero, P(Z) - -1nZ; we cannot say whether this is a real effect or an artifact due to approximations made. Anyhow this singularity manifests itself only for extremely low values of argument Zl (Z) < exp(-exp A*).

2) Z/(Z) %A’, where we get a stretched exponential,

P(Z) -exp - (16)

This tail is close to what was obtained experimentally in ref. [2]. To check the validity of the theory proposed, we have performed numerical

simulations based on a widely used model by Edrei et al. [7] in order to confirm the validity of our previous assumptions. A wide W % L two-dimen- sional sample was used. The samples length was L = 6b, and width W= lOOb, where b is the averaged distance between the point scatterers. Since in our particular realization the scatterers were chosen to be relatively strong, the mean free path e-b. The wavelength of the incoming wave is A = 0.012348, therefore, kl- 500. The results of the simulations are presented in fig. 1.

In conclusion a general “topological” approximation theory for the angular

474 E. Kogan et al. I Intensity distribution of optical waves

10’

10”

10-l

lo.*

10-l

lo4

1o’5

1o-6

10”

i “““““““1

E

‘. ‘\

f ‘\

‘\ ‘,

‘\ ‘\

‘\ : ‘\ \-

I , I , ,

0 5 10 15

I(r)/<I(r)>

Fig. 1. Intensity distribution function in a semilogarithmic scale. The full line corresponds to eq. (13), the dotted line to Rayleigh statistics and the x-symbols to the results of the numerical simulation.

transmission coefficient and intensity distribution functions is formulated. In the framework of this perturbation theory the approximation of the order m corresponds to taking into account all the diagrams which consist of the connected parts with no more than 2m propagators. We show that Rayleigh statistics corresponds to the first order approximation and calculate explicitly the statistics in the second order approximation. It is shown that only for small values of the argument the distribution functions for intensity in a random media Z’(Z) is a simple exponential, as predicted by Rayleigh statistics. For larger values the distribution functions differ drastically from the simple exponentials, and the asymptotical behavior is a stretched exponential. The results obtained are confirmed by numerical simulations.

References

[l] J.W. Goodman, Statistical Optics (Wiley, New York, 1985). [2] N. Garcia and A.Z. Genack, Phys. Rev. Lett. 63 (1989) 1678;

A.Z. Genack and N. Garcia, Europhys. Lett., to be published.

E. Kogan et al. I Intensity distribution of optical waves 475

[3] P.A. Lee and A.D. Stone, Phys. Rev. Lett. 55 (1985) 1623. [4] B.L. Altshuler, Pis’ma Zh. Eksp. Teor. Fiz. 51 (1981) 530 [JETP Lett. 41 (1981) 6481 [5] S. Feng, C.L. Kane, P.A. Lee and A.D. Stone, Phys. Rev. Lett. 61 (1988) 834. [6] B. Shapiro, Phys. Rev. Lett. 57 (1986) 2168. [7] I. Edrei, M. Kaveh and B. Shapiro, Phys. Rev. Lett. 62 (1989) 2120.