Effects of localization of light in one-dimensional random media with weak nonlinearity

Volume 155, number 2,3

PHYSICS LETTERS A

6 May 1991

Effects of localization of light in one-dimensional random media with weak nonlinearity K.I. Grigonshin and E.I. Ogievetsky Institute of Spectroscopv, USSR Academy ofSciences, Troitsk, Moscow Region 142092, USSR Received 1 February 1991; accepted for publication 4 March 1991 Communicated by V.M. Agranovich

The problem of light propagation in a 1D random medium with the Kerr nonlinearity is solved with the Berezinskii technique modified to take into account nonlinear interactions. The mean free path length associated with the scattering by the impurities is considered to be much smaller than the characteristic length ofphoton—photon scattering. It is shown that the presence of weak nonlinearity results in a change of the preexponential factorofthe light intensity correlation function and does not affect localization.

At the present time the optical properties of random media are the object of special interest as a very convenient object for both theoretical research and the experimental investigation of localization effects. The concept of localization was first introduced by Anderson to describe the propagation of electrons in dirty metals [1]. Localization is due to the interference of electron waves scattered by impurities. The same effect takes place when light propagates in an opaque medium. The most well-known consequence of light localization is the enhancement of backscattering of light from random media [2,3]. Nevertheless up to now the question of the coexistence of disorder and nonlinearity is open. A set of interesting predictions of the influence of nonlinearity upon the light propagation in weakly disordered media have been made in refs. [4,51.Some results concerning localization of light in a ID random medium with the Kerr nonlinearity were obtained in refs. [6,7]. In ref. [7] the transmission coefficient of light through a 1 D disordered nonlinear layer is derived by means of an imbedding method: the nonlinear differential equation for the transmission coefficient of the layer is solved exactly in the absence of disorder while the disorder is taken into account as a perturbation. In the present paper the disorder is considered to be stronger than nonlinearity and thus making use of the Berezinskii technique [8,9] the averaging over disorder with summation of all contributions in all orders of nonlinearity is carried out. The master equation for the propagation of monochromatic light in the infinite 1 D random medium with the Kerr nonlinearity a I El2 is d2E [e(z)E+aIEI2E]=J(z), (1) ~—~+

where z is the coordinate, J(z) is a pointlike source of monochromatic light with the intensity I~at the point z= 0 switched on at 0, the electric field E(z) is considered to be a scalar due to the one-dimensionality of the problem. The disordered medium is described by the dielectric function t=

(2) Here ~ is the homogeneous part of the dielectric function while its random part associated with the impurities is assumed to be a Gaussian random function with zero average and correlator 0375-9601/91/$ 03.50 © 1991

—

Elsevier Science Publishers B.V. (North-Holland)

189

Volume 155, number 2,3

PHYSICS LETTERS A

6 May 1991

(3)

<&(z’),&(z)>= ~ö(z’—z),

where the angular brackets mean the averaging over impurity configurations, and I is the photon mean free path in a disordered medium. (Here and afterwards it will be assumed that the velocity of light c and the light wavelength), are equal to unity.) In what follows the case 1>> 1 is considered. In this paper the spatial distribution of the light intensity in the medium will be investigated. In solving the problem under discussion it turns out that the time and spatial dependence of light intensity at large times (t—~oo)and large distances (~zI>>l) is factorized, i.e. (4)

<1(z, t)>=fi(z)J~(t) .

Thus, here the function f1 (z) will be derived. The function describing the time evolution of the intensity distribution will diverge in the limit t—+ cx. This divergence is due to the presence of a time-independent source of light and is cut off1NL by the observation time. Thelight mean free path be assumed 1/al determined by the intensity at will the cutoff: 1NL>> to 1. be much greater than theThe nonlinear length calculation technique for the electron correlation function in the limit 1>> 1 in a 1 D disordered medium without nonlinearity was developed by Berezinskii [8] and consists in singling out and subsequent summation of diagrams of a special type. One of these diagrams yielding the main contribution to the electron density— density correlator in the 1/i approximation is shown in fig. 1. An analogous retarded correlator in the considered light propagation problem is <(G(z,0)J(0)), (G*(z,0)J*(0))>w; J( 0) is a pointlike source and the subscript w denotes the respective component of the temporal Fourier transform. The physical meaning of the previous correlator is the Fourier component of light intensity at the point z: <1(z)> To take into account the Kerr nonlinearity the light source at z =0 and additional vertices associated with nonlinear interaction have to be introduced. This leads to a modification ofthe diagrams yielding the main contribution to the correlator. An example of the modified graph is presented in fig. 2. In this figure the circles denote the light source at the point z =0, the dashed and wavy lines denote the averaged scattering by imperfections and nonlinear interaction, respectively, the single and the double lines designate respectively the Green function G and its conjugate G ~.

‘~,

G(z 1,z2)=—

2exp[iIz1—z21(l+w/2)}’

(5)

Im w> 0, I w I ~z 1, w is the characteristic frequency corresponding to the observation time. Following Berezinskii [8] after space ordering the diagrams containing only slowly oscillating vertices have been chosen. The selection of the impurity vertices is the same as in refs. [8,91 (fig. 3). The introduced nonlinear slowly oscillating vertices are presented in fig. 4. The diagrams (fig. 2) are divided into three parts: the left one (to the left of point 0), the right one (to the right of point z) and the central one (between points 0 and z), the same as in ref. [81. The cross section of the diagram in fig. 2 containing N nonlinear interactions (k1 interactions to the left of 0, k2 interactions to the right of z, q interactions between points 0 and z, k1+k2+q=N) is shown in fig. 5. Let .~)(z1, X2N+ ~ z) be the sum of all contributions of all left-hand parts containing k1 nonlinear interactions to the left of 0. These contributions have at the boundary with the central part x1, x3 X2N_ 1 pairs of single lines originating from N sources denoted by the bright circles and x2, x4, X2N pairs of double lines originating from N conjugated sources denoted by the shaded circles. X2N+l denotes the total number of single line pairs at the boundary. Since the changes of the total number of single and double line pairs are the same for all slowly oscillating vertices, X2N± is the total number of double line pairs at the boundary as well [8]. Derivation of the equation for k is analogous to that presented in refs. [8,91. But in this case one has to ...,

...,

190

Volume 155, number 2,3

PHYSICS LETTERS A

6 May 1991

1,:

~ ~

X

3~ X ~ X4 ~ X5 ~

X5~X ~ X~X8

Fig. 1. An example ofa spatially ordered Berezinskii diagram for the ~,, correlator in the absence of nonlinear interaction. z. are the coordinates ofthe impurities,

Fig. 2. An example of the modified graph with N=2 nonlinear interactions. The bright and dashed circles denote the pointlike source of light at the point z=0 and the conjugated one, respectively; the dashed and wavy lines denote the averaged scattering by the imperfections and nonlinear interactions, respectively. The single and the double lines designate respectively the Green function G and its conjugate G.

operate with 2N+ 1 variables and, besides, there is a set of new vertices (fig. 4). As the correlator will be derived in the asymptotic limit (a—p0, l— v = 4aiI << 1) and, furthermore, at INL>> 1, every photon is multiply scattered due to disorder in the dielectric function before the nonlinear interaction and thus all xi>> 1. Consequently, only the highest powers ofx1— s have to be retained in the equation for R. (Account of the neglected terms results in O(l/INL) corrections.) In these assumptions the equation for ~,

R~)(xl,...,x2N+I)=.~i(xj

x2N+l;z)exp(iwx2N÷lz)

in the continuous limit is 2N+I ÔR~) +2k ~ xj—~-—+k(k—l)R~)+Lvx2N+IR~

2N+I 2N+I

~ j~’1 ,n=I

(JX1UXm

j~1

(JXj

=y[(xf—x2)R~’(x3,x4,...,x2N,xI,x2,x2N+I)+(xs—x4)R~,~(xl,x2,x5,...,x2N,xs,x4,x2N+l)+...

+(x2k_I—x2k)R~H(xI,x2,...,x2k_l,x2k,x2N+I)], where y = a101, and a is the nonlinear interaction coefficient. The function by the relation

(6) Q~Vc2(x1

X2N+

~) is determined 191

Volume 155, number 2,3

PHYSICS LETTERS A

6 May 1991

>~ Fig. 3. Types of slowly oscillating vertices of the impurity. The primed vertices not shown in the figure differ from the a, b, c vertices by interchange of the single and double lines.

l~

Q~k2(xl.x

2N+l~w)~j 0

XC~~2(xi

jr

dx2N+I

Fig. 4. Types of additional nonlinear slowly oscillating vertices. The primed vertices not shown in the figure differ from the a, b, c vertices by interchange ofthe single and double lines.

jr

2N

0

~7IX2j_I~
~~lX2j~X2N+I

X2N÷1,X~,..., X~f~ z)R~(x~, ..., ~

1’

fldx~j dze~”~exp(iwx2N+lz)

(7)

,

where C~2(x1, X2N+ ~, X’1, X’2N+ ~ z) is the sum of all central parts of the diagrams (fig. 2) containing k2 nonlinear interactions to the right of z and q interactions in the central part. These contributions have x1, X2N_ and x’1, x’3 X’2N_ 1 pairs of single lines originating from N sources denoted by the bright circles in the 0 and z cross sections, respectively, and x2, ..., X2N and x’2 X’2N pairs of double lines originating from N shaded circles in the same cross sections. 2 in the same assumptions as those for the function R in the continuous limit are The equations 2N+ 1 2N+ 1 a 2forq,k2Q~ 2N+ i a q,k2 ...,

~

j=1 — —

~

—

...,

x

3x,,, Dk2) N q,0

N

+

ax1ax~,

ri YLkX2kI+I

f

~1~~X2kl÷3—X2k1+4;~N

2(k2 +q+l)

~ ~=i

\r)q—1,k2( X2ki÷2J~N k~i

~X1

N

+(k2+q)(k2

~

X2k1,X2k1+1, X2k1+2, ..., X2N+1

X2k1,X2k1±3,X2k1±4,X2k1÷1 X2N÷l X2kl,X2ki+2q_1,x2kI+2q,x2kI+l,..,~v2N÷1)]

where ~i=4Kl<<1. In terms of 192

Q

and R the correlator

<1(z) >~,,is

,

(8)

Volume 155, number 2,3

PHYSICS LETTERS A

6 May 1991

X2~1{”:

x

2Nl{(

—

--

______

x2N÷,_~x2.{ X2N{~

-

_______

K +2

X2

•

Fig. 5. Cross section at the point z=0—ö, ö—.0 of the diagram containing N nonlinear interactions. This diagram has k nonlinear interactions to the left of the cross section and N—k interactions to the right. x2~_1and x2, are respectively the number of pairs of the single and double lines originating from the jth source. (For 1=1 k the number of lines in the cross section is odd, nevertheless, one can neglect unity compared with X2j_I, x21.) X2N+ is the total number of single line pairs (the number of double line pairs is the same). x2N+l—X~,! X2~_I and X2N+ I — are respectively the number of single and double lines originating from the source and propagating towards the point z.

~


________

-. ______

I

-.

1.

<1(z)>~=

where I(w,

J

(9)

~1(w,K)e~,

i~) is

determined by

I(w,~)~Jdx2N+l 0

S ~~LX2j~X2N+I

S ~=lX2j—I

fl~JR~(xI,...,x2N+l;w) ~X2N+l

(10) To solve eqs. (6) and (8) the variables are changed as follows: 2N+l

u=(fl

~)

l/(2N+l)

1/(2N+1)

;

U_(2N÷1)

U_—l,...,2N).

(11)

This substitution of variables reduces the system (6), (8) to a pair ofreccurence differential equation systems,

193

Volume 155, number 2,3

1

1

=

YUVN[( (u)2N÷1(u)2N)RN~3,u4

+( /

(u

1

2Q~2

(u1, u2, u5, u6

)2N+l)RN

1

2N~

U2N, u3, u4 u)+...

(12)

~

(u2k)

—

+2(q+k 2+l

2d

dQ~2 )u

5qo =

u2N,uI,u2;u)

1 (u

3)2N±1 —

~

6 May 1991

dJ?~ 2ku— + k(k—l)R~+iuvVNR~ du

+

du

PHYSICS LETTERS A

—R~/(u1,

du

+

~

U2N, U)

...,

+YVNU[(() 2N+J

—

(U2k,

1

U2k1+1, U2k1+2, U2k1+3

2N+I)Q~_1k2ul

u2N; u)+...

+2)

1 \(U2k1+2q1)

(u2k1+2Q)2N+1)

\2N+l

xQ~_lk2(ut

where

VN=

R~(u1

—

U2k1, U2k1+2q1, U2k1+2q, U2ki+1

j=1

q!k2! q+k2 fl

q+ k2

\iVJ

Substituting r~(u)and

U

1 ‘2N+

u2N;u)=(—~k!fl(

Q~k2(UU;U)(i~

du

u2N; U)],

...~

(13)

1J]~u1. The solution of eqs. (12) and (13) are sought in the form \1VJ

u2~~

U2k1+2a_2, U2k1+2q+l,

q~k2(U)

j=1

1

(u

)2N+l)r (u),

(/

1 \ (u2N_21+

2N+1 i)

1

—

2~

(U2N_2j)

)

(14) ~ q,k~ ( u).

(15)

in (12), (13) one obtains (16)

+2ku~ +k(k—l)r~+iuVNPr~=iuvVNr~,

du

2 +2(q+k 2dq~1 dq~ +(k2 +q)(k2 ~ d 2 +l)u— du

~

(17)

~

And finally, introducing a new variable (18)

P—1UVNP,

the following system of differential over p and recurrent over k(q) r~(p)(q~k2(p)) is derived:

194

equations

for the functions

Volume 155, number 2,3

6 May 1991

+2kp~ +k(k—l)r~—pr~=—pr~,

~2i

~

PHYSICS LETTERS A

d2 ~ dp2

d

q,k2

+2(q±k 2+1)p__+(q+k2)(q+k2+l)q%~2_i,7q~2_pq~~2

q—1,k211 11qO r _rN~p,u~1~J—pqN ‘~‘ The systems of equations (16), — k2(

\~

(17) should be complemented with boundary conditions. Now these conditions for r~(p)are derived. The boundary conditions for the function r~(p)at infinity are obvious, r~(p)
(20)

p—~’cx.

The function r~(p)coincides with the function R introduced by Berezinskii [81. Thus, the boundary condition for it at p=0 is easily imposed, r~(p=0)=l

(21)

.

The imposition of the boundary conditions at zero for the functions r~(p) (k>0) turns out to be more difficult. The behaviour of these functions in the region u>> 1, VNV << 1 so that I P1 = u VNV ~z 1, has to be investigated. For this in the left-hand side of (16) the terms of the order of v will be omitted. The solutions of these equations without the neglected terms are applicable if u VNV << 1 and their asymptotics merge together with the r~,(p)asymptotics at p—’0. Consequently, p~0.

(22)

The addition of the solutions regular at infinity to the respective homogeneous equation violates the asymptotic condition at p—~0and, thus, there exists a unique solution for eq. (16). Taking into account r~(p)= 1 at p—*0 the missing boundary condition is r~(p)xp~,

p—~0.

(23)

The boundary conditions for q~k2(p_,ç~j)<~

Q

are imposed after analogous speculations:

q~k2(p, 0)~p...q...k2...1/2_P/2,

~2

1 +4ii~, Rep> 0.

(24)

Substitution of expressions (14), (15) in (10) yields the correlator in terms of the functions r~.,(p)and

~

1(w,K)=~ ~

It N=O k~+q+k2 =N

(2N+l)2NJdp

5 5

2J~IU2j ~JIU2j_I

N

N

1 ~ (u fldl~~fl(

j=

1 2N+1 2~..~)

—

1

(U21)

2N)

0

~

(25)

—~)J,

where the function q~2 (p, ,~) corresponds to a definite sign of ~in eq. (17). In expression (10) the integration over [xJ is replaced by integration over [u, p1. The integration over [u] can be carried out immediately: 1(w,K)=

~

~

ItNOki+q+k2

S(N)(iv)_2~1(~) 1V

kl!k 2!q!Jdpp2Nr~(p)[q~2(p,?,)+q~2(p,

—ti)],

(26)

where N (_l)m S(N)= m~om!(Nm)!(N+m)!(2Nm)!

(27)

195

Volume 155, number 2,3

PHYSICS LETTERS A

It is clear that S(N) is zero at odd N whereas for even N (= 2n) S(N) = lation into eq. (26) one finds I(w, K) =

IS(2n) I ~

It ,~=0ki+k ~ 2+q=2,,

1.11

(~) $ k, !k2!q!

6 May 1991 (—

4~r~(p) [q~2(p,

1)” I S(2n) ~)

+q~2(p,

I.

Substituting this re_~)

}.

(28)

dpp

V

All terms of (28) with odd powers of the nonlinear interaction coefficient are eliminated. Thus, the results obtained are in no way dependent upon the sign of the nonlinearity coefficient. The function to be integrated in (28) tends to zero in both cases p—’0 and p—+oo. A detailed investigation has clarified that for given n the main contribution to this integral is made by the region p—. n. The solution of the systems (19) together with the boundary conditions (20), (23), (24) results, for the vicinity of p in the following formulae, 2 Slfl(ltyk/2) F 13/22/3 IP \ 213k’ rN P ~2e k 1k (Yk ~2p)\yk/ F(lYk/2)exPt~ —

ak2( qN

P

—

‘~

~ 2~ / ~ ~3q(5—p) 2It

2 F(ji) COS[~It(Yq+~)] (k 2 2_(f~Y~)/2()yq/2+2 P —k T((~+l_yq)/2) 2— I —yq/ 2)

e —q—k2

xexp[—(3/2213)p213q’’3]

(29)

.

Here Yk= (p12k)’ ~ Y~= (p/2q) l~’3, Substituting (29) into (28), integrating over p and performing the inverse spatial Fourier transformation, one obtains <1(z)>,,~—_.f(w)(4I/IzI)”2exp(—IzI/4I) .

(30)

Formula (30) shows the factorization df the spatial and time (frequency w) dependence of the correlator. The complexity of the time dependence is not surprising: it is due to constant light pumping. As is evident from the coordinate dependence of the correlator (30), the presence of weak nonlinearity does not destroy localization and, moreover, weak nonlinear interaction conserves its exponential form and the characteristic localization length. The only difference ofthis result from that obtained in the absence of nonlinearity is a change in the preexponential factor power [9]. All calculations in the present Letter were carried out in the main order of the perturbation theory over l//NL. The account of the next orders seems to result in renormalization of the localization length 4/. In the most interesting case when the localization and nonlinear lengths are of the same order the localization law can drastically change. The authors are thankful for fruitful discussions and valuable remarks to Professors V.M. Agranovich, A.G. and V.1. Yudson.

Mal’shukov

References [11 P.W. Anderson, Phys. Rev. 109 (1958)1492. [2] VI. Tatarski, The effects of a turbulent atmosphere on wave propagation (National Technical Information Service, Washington, DC, 1971). [3] E. Akkermans, P.E. Wolf, R. Maynard and G. Marit, J. Phys. (Paris) 49 (1988) 77. [4] V.M. Agranovich and V.E. Kravtsov, Phys. Lett. A 131(1988)378, 386; Soy. Phys. JETP 68 (1989) 272. [5] V.E. Kravtsov, V.1. Yudson and V.M. Agranovich, Phys. Rev. B 41(1990) 2794. [6] Boulivare, J. Stat. Phys. 43(1986) 423. [7] B. Doucot and R. Rammal, Europhys. Lett. 3 (1987) 969; J. Phys. (Paris) 48 (1987) 509, 525. [8] V.L. Berezinskii, Zh. Eksp. Teor. Fiz. 65 (1973) 1251. [91 A.A. Gogolin, VI. Mel’nikov and E.J. Rashba, Zh. Eksp. Teor. Fiz. 69 (1975) 328; A.A. Gogolin, Phys. Rep. 86 (1982) 2.

196