Physics Letters A 162 (1992) 493-496 North-Holland

PHYSICS LETTERS A

Spectral and angular correlations in disordered wave guides

Eugene Kogan and Moshe Kaveh Department of Physics, Bar-Ilan University, Ramat-Gan. Israel and Department of Physics, Cavendish Laboratory, University of Cambridge, Cambridge, UK

Received 14 November 199 1; accepted for publication 9 January 1992 Communicated by A. Lagendijk

We study the intensity-intensity auto-cormlation function of a disordered medium as a simultaneous function of the wave frequency change Aw, the change in incident wave vector Aq. and the change in scattered wave vector Aq,,. For a restricted geom- etry such as a long narrow disordered wave guide, we find that the auto-correlation function C(Aw, Aq,, Aqs) decouples in the following way: C(Aw, Aqa, ~)=C1(~)F(~,)F(~)+C2(A~)[F(~,)+F(~)], where C,(Aw) and C,(Aw) are the short- range and the long-range spectral auto-correlation functions, respectively, and Fis a geometrical form factor. When the width of the wave guide becomes larger than its length the decoupling breaks down.

Multiple scattering of waves in random media has recently attracted much attention (for recent reviews see ref. [ I] ). One important aspect in this activity is the study of the long-range character of the intensity fluc- tuations [ 2-5 1. The spectral intensity-intensity auto-correlation function C(Aw) has been calculated by many authors [ 6- 111. It was predicted theoretically [ 9, lo] and observed experimentally [ 12,13 ] that for a restricted geometry, in which the width W of a disordered wave guide is much smaller than its length L, the long-range character of C(Ao) is enhanced and indeed its slow decay to zero as (Am) - ‘I2 was recently observed. In par- allel to these studies, it was predicted [ 141 that the (far-field) angular intensity-intensity auto-correlation function C( Aq) contains an oscillatory dependence on Aq,, or Aq,, (both for the short-range and the long-range contribution). This oscillatory dependence has not yet been studied experimentally but was confirmed by nu- merical simulations [ 14 1.

In this paper we study the intensity-intensity auto-correlation function C(Ao, L\q,, Aqb) in the far field as a function of all three arguments. Our main result is that for the geometry of a long-tube wave guide, both the short-range and the long-range contributions to C(Ao, Aqa, Aqb) decouple in the following way,

C(Ao,Aq,,Aqb)=Cr(A~)C,(Aqa>Aqb)+G(A~)G(Aq=,Aqt,) > (1)

where C1 (AU) and C2 ( AU) are the short-range [ 4-8 ] and long-range [ 2,9,10,12,13 ] spectral auto-correlation functions and Cl (Aqa, Aq,,) and C2(Aqa, Aqb) are the corresponding angular correlation functions with the os- cillatory [ 141 character. When W> L, this decoupling breaks down and the auto-correlation function is a mixed function of Ao, Aqa and Aq,,.

We consider the case where the scattered intensity is measured in the far field (Fraunhofer diffraction) and we are interested in the intensity-intensity correlation function

C(Aw, ADZ,, hb) = <1o(&, !?b)zw’(dv 6)) - (z”(!k, qb)) (I”‘(!?:, 6) > , (2)

where ( > denotes an ensemble average, Zw(qa, qb) is the intensity of a wave with frequency w and incident wave vector q. and scattered wave vector qb and Aq=q-q’ is the change in the incident Aq. or scattered Aqb wave vector and Ao = w - w’ .

According to the Fresnel-Huygens principle,

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C( Aw, Aq, Aq’ ) = E,(r) exp(iq,*r) dr EL. (r) exp( iqi*r) dr I)

2

EJ I) exp( iqi,*r) dr I m 2

- EL. (r) exp(iqi,-r) dr I> ,

where E is the scalar electric field on the output face (the electric field on the input face being EE (r) = exp (i&-r) and E$ (r) =exp(iq= *r), respectively). Let us make the simplest decoupling assumption for the cot-relators:

(L(r)-%(r’ ) > = (L(r) )6(+-r ) , (4a)

(E,(r)E~(r’)E~.(r”)E~.(r”‘))=(Z,(r)Z,.(r”))6(r-r’)S(r”-r”’)

+(E,(r)Ez.(r)E:(r’)E;,(r’))d(r-r”’)d(r’-r”) (4b)

(for simplicity we have omitted the constants of proportionality in eqs. (3a) and (3b) which are later absorbed into C, and C,). From (4a) and (4b) we see explicitly the additivity of the long-range and short-range con- tributions to the auto-correlation function:

C(Aw,Aqa,Aqb)=Ci(Aw,Aqa,AqiJ+C2(Ac%Aqa,Aqb) 7 (5)

where

2

C,(Ao,&a,&d= (&JrbG(r)) exp(-ibgbsr) dr (6)

and

C,(AW,&~,&~=~ (WAr)VG~(r”))) drdr”

+ (G(E,(r)E:,(r))d(E:(r’)Ek.(r’))) exp[-&*(r-r’)] drdr' . (7)

To calculate the correlator (E,(r)E;s (r) ), we may use the real-space approach. That is, we may write

(E,(rE~~~))=~ G( r, r’)Ez(r’)E$‘?(r’) dr’ , (8)

where

G(r,r’)= CP,exp[i@,(r,r’)-i@&(r,r’)] ( > 01

(9)

and the index (Y enumerates all the trajectories that lead from the point r’ to the point r, @a is the phase shift which the wave packet acquires when travelling along the trajectory (Y, and Pa is the amplitude of the electric field in a given trajectory. We first consider the case of a long-tube geometry (L z+ IV). The many intersections of any two trajectories strongly correlate the intensity at any two points on a given surface. Thus we assume that G( r, r’ ) is independent of r and r’. It also follows that the correlator ( Eo( r)E;* (r) ) is independent of r and we obtain

C,(Ao,Aq=,~b)=1(E,(r)E~(r))12F(Aqb),

where F is the traditional geometric form factor in diffraction theory and is given by

(10)

F(~) = sin2( WAq”/2) sin2( WAqY/2)

( WAqX/2)2 (WAqYI2)’ ’ (11)

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Carrying out the integration in (8 ), we obtain

I (Ko(~b%=.~)) I*=G(AwF’(4,) 3 (12)

where C, is the short-range auto-correlation function which is calculated for qa =q; and hence, in our ap- proximation, is independent of L\q,. When both Aq,=O and Aq,=O, we get

C,(Ao,Aqa,Aqb)=C,(AW)ZYAqa)F(AqJ. (13)

For Ao=O eq. ( 13) coincides with the short-range correlation C1 (Aqa, Aqb) obtained by Eliyahu et al. [ 141. We now prove the decoupling of the spectral and angular long-range auto-correlation function C,. In order

to obtain the connection between this correlator and the long-range correlator in the near field, C,(Ao), we may use either the real-space approach, or the Langevin approach. In the real-space approach, the first cor- relator in eq. (6) can be written in the following form [ 141,

(6(I,(r))S(~.(r”)))=lI j

2

G(R, r’)E~(r’)E~(r’) dr’ dR, (14)

where only trajectories which intersect with each other make a contribution to G and R is the intersection point. We first consider the case of a long tube and suppose again that complete mixing of trajectories makes G

independent of r’. Carrying out the integration in ( 14) we obtain

(J(Z,(r) )d(I:.(rn)) > =Cz(Am)F(&,) 9

where C,(Aw) is exactly the long-range spectral correlator.

(15)

Treating in the same spirit the second correlator in eq. (6), we obtain

(G(&,(r)E$(r))6(E:,(r’)E~,(r’)))=C2(Ao).

Eq. (6) leads to our final result:

(16)

C~(A~,A~~,A~~)=CZ(AW)[F(~~)+F(A~~)I. (17)

We see that the long-range decoupling arises because the angular dependence of the intensity-intensity auto- correlation function is purely geometrical. Thus, the oscillatory angular dependence of the auto-correlation function is purely geometrical and does not affect the monotonic dependence on Aw. Moreover, if we include absorption in the calculation, we find that it will only affect the spectral correlation functions C, (Ao) and C,(Am). The geometrical form factors will not be affected. The explicit form of C,(Aw) with absorption was recently obtained in ref. [ lo].

Eqs. ( 15 ) and ( 16) can also be derived using the Langevin approach [ 15 1. This can be seen from the fact that in this approach, the long-range correlators (for example (6( ZJ r) )d( IL, (r” ) ) ) ) are connected with the short-range correlator

I <Urb% (f-1 > I2 .

Hence eq. ( 15) immediately follows from eq. ( 12). We now consider the case of a wide sample for which WB L. In this case, it is evident that G( r, r’ ) = G (r- r’ ) .

Carrying out the integration in eqs. (6) and (8) we obtain

C,(Ao,4a,Aqb)=C1(AO,Aqa)6(Aqa_Aqb). (18)

We see that the “memory effect” [ 4,16,17] is preserved even when the frequencies of the two waves differ. However, the amplitude of this effect is reduced. In this case, the Llq, dependence of C, is not a geometrical dependence but rather follows from the properties of the random medium through G( r- r’ ). Using the Langevin approach, we obtain for the long-range contribution:

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(19)

Thus, we no longer have the factorization of the Ao and the Aq dependences but we still retain the separation of Aq.. and Aqt, dependences.

In summary, we have shown that for a narrow disordered wave guide, the spectral and angular parts of the auto-correlation function decouple both for the short-range and the long-range contributions. In this case we show that the angular oscillatory dependence of the correlation function is purely geometrical. When the wave guide becomes wide, the decoupling property disappears because the angular correlations become non-geo- metrical. In this case, the “memory-effect” is preserved but with a reduced amplitude.

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