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Intensity correlation in absorbing random media
Volume 157, number 4,5
PHYSICS LETTERS A
29 July 1991
Intensity correlation in absorbing random media R. P n i n i a n d B. S h a p i r o
Department of Physics, Technion - Israel Institute of Technology, 32000 Haifa, Israel Received 1 July 1991; accepted for publication 4 July 1991 Communicated by A. Lagendijk
We consider the effect of absorption on intensity correlation in a speckle pattern. We derive a general expression for the correlation function, and discuss in some detail two experimentally relevant geometries: a wide slab and a narrow tube. Our calculations show that strong absorption modifies but not destroys the long-range correlations in intensity: these correlations exist even at distances much larger than the absorption length.
1. Waves undergoing multiple scattering in a rand o m medium, produce a complicated intensity speckle pattern, I(r). This pattern is highly irregular but not entirely random. The intensity-intensity correlation function ( I ( r ) I ( r ' ) ) , for large distances I r- r' I, decays according to a power law rather than exponentially (the angular brackets denote averaging over the ensemble o f samples). The more complicated correlation function (I0,(r)I0,, (r') ), where ca and o9' correspond to two different frequencies o f the source, also exhibits slow decay, for either large distances I r - r' I or large frequency shifts I c a - ca' I. These and other related correlation functions have been studied in recent years, both theoretically [ 1 10 ] and experimentally [ 11-16 ]. In this Letter, we study the effect o f absorption on long-range correlation in the speckle pattern. The effect of absorption was considered previously by Stephen [2], who calculated the fluctuations in the total transmission (~T 2), and recently, by de Boer et al. [16] who derived an expression for the frequency correlation function <~T0,~T,o,). Correlations in the presence of absorption have also been studied by Kogan and Kaveh [8 ], for a tube geometry. Their approach deals only with intensity integrated over a cross-section o f the tube, rather than with intensity at a given point in space. Below we present an approach which allows one to study the effect of absorption on the correlation function
ometry. The approach is based on the Langevin method, whose use for this kind of problems was first proposed by Zyuzin and Spivak [ 3]. 2. We start with a wave equation which describes a scalar monochromatic field 9'o,(r), of frequency ca, propagating in a weakly absorbing medium: {VE+k2[ 1 + a ( r ) + 2i~] }9"0,(r) = 0 .
( 1)
Here 7<< 1 is the attenuation index, ko=ca/Co where Co is the speed o f propagation in the average medium, and a ( r ) is the fluctuating part of the refraction index. The r a n d o m function/2(r) is assumed to obey white-noise Gaussian statistics: <#(r)) =0,
(2)
where ~ describes the strength of the scattering potential. We consider weak disorder, such that the elastic mean free path l =-4n/~kg is much larger than the wavelength o f the radiation. The basic equations, describing fluctuations and correlations in the presence of absorption, are derived in close analogy with those o f ref. [ 7 ]: (i) First, from the Bethe-Salpeter equation, in the ladder approximation, one derives equations for the average intensity,
(3a)
V. ( j 0 , ( r ) ) = - 2ca~,(I0,(r) ) ,
(3b) 265
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PHYSICS LETTERS A
where ~=col/3 is the diffusion coefficient ~. (ii) Next, one considers the intensity-intensity correlation function, (Io~(r)lo~, (r') ) for short distances, i.e. for I r - r ' l - A r < l . For this one can use the factorization approximation [ 1,9 ] to obtain
29 July 1991
1
(V2-a2)8I~o(r)= -ffjV.g~,,(r) .
The formal solution of this equation can be written as
if
( 8Lo(r)8I, o, (r') )
8Ion(r) = - ~
= I(l, oo,,(r) )12 sin2(k° Ar)
(ko Ar) 2 e x p ( - A r / l ) ,
(7)
d3r ' v ' g ( r , r')'go~(r' )
(4) + ~ dS'-[8Io~(r' )V'K(r, r ' )
where 5 I = I - ( I ) ,
and the field autocorrelation ( l,ooj,(r) ) = ( q/,o(r) ~/*, ( r ) ) obeys the equation
V2(I, oo~(r))+(2iflZ-otZ)(Io,,o,(r))=O.
(5)
Here a - (6ko7/l) lie is the inverse absorption length and f12_=3 (~o- ~o' )/2col. This equation is similar but not identical to the one given in ref. [2]. (iii) Finally, according to the Langevin approach [3], the large local fluctuations (discussed in the previous paragraph) act as external random sources in the diffusion equation for the fluctuating quantities, 5 1 - 1 - ( I ) and ~ j - j - ( j ) . This leads to the following set of equations,
8j,o(r) = - ~VSI, o(r) +go~(r) ,
(6a)
V. 8j,o(r) = - 2~o78Lo(r),
(6b)
where the correlator for the external random fluxes is
(g~)(r)g~)(r')) =~is
c 2 × 2~l 3k 2 I ( I , o , o ' ( r ) ) 1 2 ~ ( r - r ' ) .
+ ( 1/~)Sj~(r' )K(r, r' ) ] ,
(8)
where f~ is the volume occupied by the sample and E is its confining boundary. The kernel K(r, r') satisfies (V2-a2)K(r,
r ' ) =t~(r--r' ) .
(9)
Eq. ( 9 ) does not define K(r, r' ) completely since the boundary conditions for K have not yet been specified. We shall require that on absorbing boundaries K(r, r' ) = 0, while on reflecting boundaries the normal derivative ti. V'K(r, r' ) =0. The surface term in eq. (8) then drops out and one ends up with the following expression for the intensity-intensity correlation function, Co~o~(r~, r2) -- (8Io~(rl)8I~, (r2)):
6hi C~oos(r,, r z ) = (kol)2 X [ d3rVK(rl, r)'VK(r2, r) l(I~o~,(r))12 .
f~
(lO)
(6c)
These equations, supplemented by the appropriate boundary conditions, enable one to calculate the longrange behaviour for various correlation functions. Usually, one requires that either 5I or the normal component of 8j disappear at the boundary (absorbing or reflecting wall, respectively). 3. After eliminating 8j~o(r) from eqs. (6a) and (6b) one obtains an equation for the intensity fluctuation 8Ion(r): ~ We do not distinguish here between the phase velocity Co and the energy transport velocity v. The difference between the two velocities is unimportant for our model of weak, point-like scatterers (the white-noise limit ), but may become important for more realistic scatterers (see ref. [ 16 ], as well as ref. [ 17 ] ).
266
z
This is a general expression, valid for any sample geometry. The problem is thus reduced to the calculation of ( I ~ , ( r ) ) (eq. ( 5 ) ) and K(r, r') (eq. ( 9 ) ) for specific geometry and given sources (i.e. given incident fluxes). 4. Consider a slab of infinite extent in the x, y directions and of finite thickness L (L>>I) in the z direction. A beam of radiation with an arbitrary intensity profile F(x, y) impinges on the slab along the z axis. K(r, r ' ) for this problem, satisfies zero boundary conditions on the planes z = 0, z - - L and is invariant under translations in the x - y plane:
K(r, r' ) = (2rt) - - 2
f dZp Kp(z, z' ) exp(ip'R)
,
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PHYSICS LETTERS A
where R denotes the projection of r - r' on the x - y plane, and/('p(z, z' ) is equal to /~'p(Z, z ' ) = - sinh(kz< ) s i n h [ k ( L - z > ) ] k sinh (kL)
( 11 )
Here z>_=max[z, z ' ] ; z < - m i n [ z , z' ], k 2 = p 2 + a 2 and p 2= p x2+ p y 2. ( I o ~ , ( r ) ) is obtained by solving eq. (5) with boundary conditions such that (Io~o~,( r ) ) = F(x, y) on the incident plane z = 0, and (Io~, ( r ) ) = 0 on the outgoing plane z = L:
of radius W (i.e. 3 ~ ) = 2 J , ( p W ) / p W where J~ is the Bessel function of first order) and consider the two extreme cases: (i) Broad beam, i.e. W>> L (this corresponds to the plane wave limit). For this case 3~l
~(Ato) = 2 (kol) 2 ×
(Io~o,,( r ) ) f
-
d2p
(2n) 2
sinh[x(L-z)] sinh(xL)
if(p) e x p ( i p . R ) , (12a)
where F ( p ) is the Fourier transform o f F ( x , y) and x = ( p 2 + a 2 - 2 i f l 2) ~/2=-bi-a, with a, b positive and a 2=- ½[x/(P: + a z) 2+4B4 + (p2 + ot 2) ] =_ ½[r+
(p~+a ~) ],
b2= 1 [x/(p2+ot2)2+4f14 _ ( p 2 + a 2 ) ] I [/"
(p2+a2)
(12b)
] .
Let us, first, consider the fluctuations and correlations in total transmission q~(A~) - ( 8T,,,STo~, ) / ( T ) 2, where ( T ) is the average transmission and 8To~ is the deviation from the average. In order to relate q~(A09) to the correlation function Co~, (r~, rE) (eq. (10) ), one has to integrate the latter over transverse coordinates x,, YL, x2, Y2 and set z] = z z = L - l . The resulting expression is ~2 3nl
tib(A¢o)_
(2kolfl)2 f ~d2p
[Zl(p) 12
X [cosh(2aL) - cos(2bL) ]-1 ( a 2 p 2 + f14) -] X {b sinh (2aL) [c~4+ 4fl4 + ot 2'(3p2 + / ' ) ] - 2fl2c~F sinh ( 2 a L )
-asin(2bL)[o~4+4fl4+otZ(3p2-1")]},
(13)
where A(p) _=if(p)/P(O ). We assume a circular beam ,2 An expression for ~(Aco) was also derived in ref. [ 16], for a Gaussian beam. Our eq. ( 13 ) differs from that expression by a term 2(f14+o~2p2) ] in the curly brackets. There are also other, minor, differences between the two expressions.
[bsinh(2aL)-asin(2bL)
29 July 1991
N/a4+4fl 4
f14A
ao sinh (2aoL) - b o sin (2boL) - a sinh ( 2 a L ) cosh(2aoL)-cos(2boL) (14)
where ao and bo are the values of a and b, respectively, at p = 0 (eq. (12b)), and A = n W 2. This expression coincides with the one obtained in ref. [ 8 ] for a tube geometry. It is worthwhile to emphasize that, as long as the sample is illuminated by a plane wave, eq. (14) is valid irrespective of the ratio between the transverse dimension, v/A, to the longitudinal one, L. In the limit c ~ 0 one recovers the result of ref. [ 7 ] which, up to a numerical factor, coincides with the earlier result of ref. [ 2 ]. The limit fl~ 0 describes the relative variance fluctuation in total transmission qb(0) _= ( ~ T 2 ) / ( T ) 2: 3n l ~ ( 0 ) - 8(ko/) 2 aELA
a L sinh ( 2 a L ) - 2a ZL 2 [ 2 - cosh ( 2 a L ) ] X
sinh 2 ( a L )
' (15)
which differs from the corresponding expression of ref. [2]. The quantity ~ ( A c o ) / ~ ( 0 ) was measured (as a function of Aco) in ref. [ 14], for a plane wave incident on a slab. The result was compared to the theoretical curve calculate in the absence of absorption. The theoretical curve came out somewhat below the experimental one. One can, however, verify (from eq. (14)) that absorption will lift the theoretical curve up, possibly bringing it closer to the experiment (it was not possible to make a detailed comparison without knowledge of the absorbing properties of the sample). Additional measurements are reported in ref. [ 16 ]. There, averaging over disorder was replaced by averaging over frequency range, so that a direct comparison with eq. (14) is not possible (the required corrections are discussed in ref. [ 16 ] ). 267
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(ii) Narrow beam, i.e. ( W<< L, a - l, fl-- l ). In this case eq. (13) reduces to [ 7 ] 3
29 July 1991
/~(p)=(2n)26(p) leads to [ 13 ]
(plane wave geometry), which
I(lo~o~,(r))[ 2
l
Ob(Aog) _ 2(kol) 2 W '
(16)
where small corrections of order (olW)% (/~W) 4, have been neglected. Thus, absorption has only a minor effect on the correlations in total transmission, for a narrow beam. We shall not go into a detailed discussion of the intensity correlation function Coo,(r~, r2) (for the slab geometry), but only present the final expression for one particular case: the incident beam is a plane wave, the frequency difference is low (ill << 1 ), the absorption is strong (ctL >> 1 ), and the two points are taken at the output plane, separated by a distance I rl - r21 = R ~
Go,o, (rl, r2) -= C "~ (R) ( a l ) 3 3L Jl ( a R ) exp( - 2 a L ) . (kol) 2 R
(17)
For large R, this gives a power law decay, R --3/2, with oscillations (in the absence of absorption [ 1 ], it gives R - l decay).
= cosh[2ao(L-z) ] - c o s [ 2 b o ( L - z ) ] cosh ( 2aoL ) - c o s ( 2 b o L )
In order to calculate the correlation function Co,o, (r~, r2), one has to substitute eqs. ( 18 ) and (19) into eq. (10). Some simplification for Co,o, (rl, r2) occurs if the two points of observation rl, r2, are chosen on the axis of the cylinder: only the m = 0 term in eq. ( 18 ) survives for this case. Let us set co= o9', and introduce the normalized correlation function C ( z l , z2) =
(6I, o(p=0, zl )6Io~(p=O, z2) )
( I(zl) ) ( I(z2) )
In addition, let us choose z~ = L - l , z z = L - I - R , and consider C ( L - I , L - I - R ) = - C ( R ) for two special cases: (i) A long narrow tube (L>> W). In this case (and for R >> W), the main contribution to ~ ( R ) comes from the s = 0 term of eq. (18). The expression for ( ' ( R ) is rather cumbersome and we only give the resuits for two limiting cases, weak absorption ( ~ L << 1 ) and strong absorption (c~L >> 1, a R > 1 ): 1
L
t ~ t n ) ~ (koW)2 l X 2 [ 1 - ( R / L ) 2] 5. We consider now a cylinder of radius W and length L. The walls of the cylinder are reflecting and its ends are assumed to be open. A plane wave impinges on the left end of the cylinder and propagates in the z direction. The kernel K(r, r' ) for this problem, written in cylindrical coordinates (p, O, z), is
~ 6,,o - 2 K(r, r ' ) = ,,s n W 2 J,n(PmsP)Jm(PmsP' ) X j2m(pm s W ) [ 1 - (m/Pms W) 2] c o s [ m ( ¢ - O '
×
sinh (km~z<) sinh[k,.s(L-z> kms sinh (k,,sL)
)]
)] '
(18)
where Jm(X) is the Bessel function of order m, Pros is the sth root ofdJm(xW)/dx--O (m, s = 0 , 1, ...), and k~s =p2s + a 2. The field autocorrelation function (Io,,o, ( r ) ) is obtained from eq. (12) by setting 268
(19)
×](1-R/L)
( a L < < 1), (aL>> 1), (20)
which coincide with the corresponding expressions for the intensity correlation functions integrated over the cross-section of the tube [ 8 ]. The linear decay of ~ ( R ) in the case of strong absorption was observed by Genack et al. [15]. (ii) A wide cylinder (L<< W). In this case, higher terms of the kernel K(rl, r2) (eq. (18) ) become important. One can take the limit W--. oo and recover the case of an infinite slab (eq. ( 11 ) ). In the absence of absorption one obtains 3/8 l f ( X ) C ( R ) ~ (ko/)ZL X ' where X= ( R + I ) / L , and
(21)
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PHYSICS LETTERSA
oo
dq f ( X ) =- f q s~-nh2q [2q2X 2 c o s h [ q ( 2 - X ) ] 0
+ 2 q X sinh [ q (2 - X) ] - 2q (2 - X) sinh (qX) - 2q 2X2 cosh (qX) } . I f R is not too close to zero or L, the function f ( X ) is of order I. Hence, for a given L, the strength of the intensity correlation in a tube of radius W is larger than in a slab by a factor of order L 2 / W 2 > > 1. 6. In conclusion, we derived a general expression for the i n t e n s i t y - i n t e n s i t y correlation function Co,o, (r~, r2) - ( 8I, o(rl )81,o, ( r 2 ) ) , which is valid for any geometry in the presence of absorption. We considered two specific cases: correlation in an infinitely wide slab and in a narrow tube. We find that absorption can significantly affect fluctuations and correlations for various quantities. However, long-range correlations exist even for distances R larger than the absorption length a -~. We are grateful to A.Z. Genack for m a n y illuminating discussions, which initiated the present work. We are also grateful to the authors of refs. [ 8,16 ] for sending us preprints of their work prior to publication. The research was supported by the U n i t e d States-Israel Binational Science F o u n d a t i o n (BSF) a n d by the F u n d for P r o m o t i o n of Research at the Technion.
29 July 1991
References [1] M.J. Stephen and G. Cwilich, Phys. Rev. Len. 59 (1987) 285. [2] M.J. Stephen, Phys. Lett. A 127 (1988) 371. [3] B.Z. Spivak and A.Yu. Zyuzin, Solid State Commun. 65 (1988) 311; A.Yu. Zyuzin and B.Z. Spivak, Zh. Eksp. Teor. Fiz. 93 (1987) 994 l Sov. Phys. JETP 66 ( 1987 ) 560]. [4 ] S. Feng, C. Kane, P.A. Lee and A.D. Stone, Phys. Rev. Lett. 61 (1988) 834. [5] P.A. Mello, E. Akkermansand B. Shapiro, Phys. Rev. Lett. 61 (1988) 459. [6] I. Edrei and M. Kaveh, Phys. Rev. B 38 (1988) 950. [7] R. Pnini and B. Shapiro, Phys. Rev. B 39 (1989) 6986. [8] E. Kogan and M. Kaveh, unpublished. [9] B. Shapiro, Phys. Rev. Lett. 57 (1986) 2168. [ 10] R. Berkovitz, M. Kavehand S. Feng,Phys. Rev. B 40 (1989) 737. [ 11 ] A.Z. Genack, Phys. Rev. Lett. 58 (1987) 2043. [ 12] I. Freund, M. Rosenbluth and S. Feng, Phys. Rev. Lett. 61 (1988) 2328. [13]N. Garcia and A.Z. Genack, Phys. Rev. Lett. 63 (1989) 1678. [ 14] M.P. van Albada, J.F. de Boer and A. Lagendijk,Phys. Rev. Lett. 64 (1990) 2787. [ 15] A.Z. Genack, N. Garcia and W. Polkosnik, Phys. Rev. Lett. 65 (1990) 2129. [16]J.F. de Boer, M.P. van Albada and A. Lagendijk, unpublished. [ 17 ] M.P. van Albada et al., Phys. Rev. Lett. 66 ( 1991 ) 3132.
269
PHYSICS LETTERS A
29 July 1991
Intensity correlation in absorbing random media R. P n i n i a n d B. S h a p i r o
Department of Physics, Technion - Israel Institute of Technology, 32000 Haifa, Israel Received 1 July 1991; accepted for publication 4 July 1991 Communicated by A. Lagendijk
We consider the effect of absorption on intensity correlation in a speckle pattern. We derive a general expression for the correlation function, and discuss in some detail two experimentally relevant geometries: a wide slab and a narrow tube. Our calculations show that strong absorption modifies but not destroys the long-range correlations in intensity: these correlations exist even at distances much larger than the absorption length.
1. Waves undergoing multiple scattering in a rand o m medium, produce a complicated intensity speckle pattern, I(r). This pattern is highly irregular but not entirely random. The intensity-intensity correlation function ( I ( r ) I ( r ' ) ) , for large distances I r- r' I, decays according to a power law rather than exponentially (the angular brackets denote averaging over the ensemble o f samples). The more complicated correlation function (I0,(r)I0,, (r') ), where ca and o9' correspond to two different frequencies o f the source, also exhibits slow decay, for either large distances I r - r' I or large frequency shifts I c a - ca' I. These and other related correlation functions have been studied in recent years, both theoretically [ 1 10 ] and experimentally [ 11-16 ]. In this Letter, we study the effect o f absorption on long-range correlation in the speckle pattern. The effect of absorption was considered previously by Stephen [2], who calculated the fluctuations in the total transmission (~T 2), and recently, by de Boer et al. [16] who derived an expression for the frequency correlation function <~T0,~T,o,). Correlations in the presence of absorption have also been studied by Kogan and Kaveh [8 ], for a tube geometry. Their approach deals only with intensity integrated over a cross-section o f the tube, rather than with intensity at a given point in space. Below we present an approach which allows one to study the effect of absorption on the correlation function
ometry. The approach is based on the Langevin method, whose use for this kind of problems was first proposed by Zyuzin and Spivak [ 3]. 2. We start with a wave equation which describes a scalar monochromatic field 9'o,(r), of frequency ca, propagating in a weakly absorbing medium: {VE+k2[ 1 + a ( r ) + 2i~] }9"0,(r) = 0 .
( 1)
Here 7<< 1 is the attenuation index, ko=ca/Co where Co is the speed o f propagation in the average medium, and a ( r ) is the fluctuating part of the refraction index. The r a n d o m function/2(r) is assumed to obey white-noise Gaussian statistics: <#(r)) =0,
(2)
where ~ describes the strength of the scattering potential. We consider weak disorder, such that the elastic mean free path l =-4n/~kg is much larger than the wavelength o f the radiation. The basic equations, describing fluctuations and correlations in the presence of absorption, are derived in close analogy with those o f ref. [ 7 ]: (i) First, from the Bethe-Salpeter equation, in the ladder approximation, one derives equations for the average intensity,
(3a)
V. ( j 0 , ( r ) ) = - 2ca~,(I0,(r) ) ,
(3b) 265
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PHYSICS LETTERS A
where ~=col/3 is the diffusion coefficient ~. (ii) Next, one considers the intensity-intensity correlation function, (Io~(r)lo~, (r') ) for short distances, i.e. for I r - r ' l - A r < l . For this one can use the factorization approximation [ 1,9 ] to obtain
29 July 1991
1
(V2-a2)8I~o(r)= -ffjV.g~,,(r) .
The formal solution of this equation can be written as
if
( 8Lo(r)8I, o, (r') )
8Ion(r) = - ~
= I(l, oo,,(r) )12 sin2(k° Ar)
(ko Ar) 2 e x p ( - A r / l ) ,
(7)
d3r ' v ' g ( r , r')'go~(r' )
(4) + ~ dS'-[8Io~(r' )V'K(r, r ' )
where 5 I = I - ( I ) ,
and the field autocorrelation ( l,ooj,(r) ) = ( q/,o(r) ~/*, ( r ) ) obeys the equation
V2(I, oo~(r))+(2iflZ-otZ)(Io,,o,(r))=O.
(5)
Here a - (6ko7/l) lie is the inverse absorption length and f12_=3 (~o- ~o' )/2col. This equation is similar but not identical to the one given in ref. [2]. (iii) Finally, according to the Langevin approach [3], the large local fluctuations (discussed in the previous paragraph) act as external random sources in the diffusion equation for the fluctuating quantities, 5 1 - 1 - ( I ) and ~ j - j - ( j ) . This leads to the following set of equations,
8j,o(r) = - ~VSI, o(r) +go~(r) ,
(6a)
V. 8j,o(r) = - 2~o78Lo(r),
(6b)
where the correlator for the external random fluxes is
(g~)(r)g~)(r')) =~is
c 2 × 2~l 3k 2 I ( I , o , o ' ( r ) ) 1 2 ~ ( r - r ' ) .
+ ( 1/~)Sj~(r' )K(r, r' ) ] ,
(8)
where f~ is the volume occupied by the sample and E is its confining boundary. The kernel K(r, r') satisfies (V2-a2)K(r,
r ' ) =t~(r--r' ) .
(9)
Eq. ( 9 ) does not define K(r, r' ) completely since the boundary conditions for K have not yet been specified. We shall require that on absorbing boundaries K(r, r' ) = 0, while on reflecting boundaries the normal derivative ti. V'K(r, r' ) =0. The surface term in eq. (8) then drops out and one ends up with the following expression for the intensity-intensity correlation function, Co~o~(r~, r2) -- (8Io~(rl)8I~, (r2)):
6hi C~oos(r,, r z ) = (kol)2 X [ d3rVK(rl, r)'VK(r2, r) l(I~o~,(r))12 .
f~
(lO)
(6c)
These equations, supplemented by the appropriate boundary conditions, enable one to calculate the longrange behaviour for various correlation functions. Usually, one requires that either 5I or the normal component of 8j disappear at the boundary (absorbing or reflecting wall, respectively). 3. After eliminating 8j~o(r) from eqs. (6a) and (6b) one obtains an equation for the intensity fluctuation 8Ion(r): ~ We do not distinguish here between the phase velocity Co and the energy transport velocity v. The difference between the two velocities is unimportant for our model of weak, point-like scatterers (the white-noise limit ), but may become important for more realistic scatterers (see ref. [ 16 ], as well as ref. [ 17 ] ).
266
z
This is a general expression, valid for any sample geometry. The problem is thus reduced to the calculation of ( I ~ , ( r ) ) (eq. ( 5 ) ) and K(r, r') (eq. ( 9 ) ) for specific geometry and given sources (i.e. given incident fluxes). 4. Consider a slab of infinite extent in the x, y directions and of finite thickness L (L>>I) in the z direction. A beam of radiation with an arbitrary intensity profile F(x, y) impinges on the slab along the z axis. K(r, r ' ) for this problem, satisfies zero boundary conditions on the planes z = 0, z - - L and is invariant under translations in the x - y plane:
K(r, r' ) = (2rt) - - 2
f dZp Kp(z, z' ) exp(ip'R)
,
Volume 157, number 4,5
PHYSICS LETTERS A
where R denotes the projection of r - r' on the x - y plane, and/('p(z, z' ) is equal to /~'p(Z, z ' ) = - sinh(kz< ) s i n h [ k ( L - z > ) ] k sinh (kL)
( 11 )
Here z>_=max[z, z ' ] ; z < - m i n [ z , z' ], k 2 = p 2 + a 2 and p 2= p x2+ p y 2. ( I o ~ , ( r ) ) is obtained by solving eq. (5) with boundary conditions such that (Io~o~,( r ) ) = F(x, y) on the incident plane z = 0, and (Io~, ( r ) ) = 0 on the outgoing plane z = L:
of radius W (i.e. 3 ~ ) = 2 J , ( p W ) / p W where J~ is the Bessel function of first order) and consider the two extreme cases: (i) Broad beam, i.e. W>> L (this corresponds to the plane wave limit). For this case 3~l
~(Ato) = 2 (kol) 2 ×
(Io~o,,( r ) ) f
-
d2p
(2n) 2
sinh[x(L-z)] sinh(xL)
if(p) e x p ( i p . R ) , (12a)
where F ( p ) is the Fourier transform o f F ( x , y) and x = ( p 2 + a 2 - 2 i f l 2) ~/2=-bi-a, with a, b positive and a 2=- ½[x/(P: + a z) 2+4B4 + (p2 + ot 2) ] =_ ½[r+
(p~+a ~) ],
b2= 1 [x/(p2+ot2)2+4f14 _ ( p 2 + a 2 ) ] I [/"
(p2+a2)
(12b)
] .
Let us, first, consider the fluctuations and correlations in total transmission q~(A~) - ( 8T,,,STo~, ) / ( T ) 2, where ( T ) is the average transmission and 8To~ is the deviation from the average. In order to relate q~(A09) to the correlation function Co~, (r~, rE) (eq. (10) ), one has to integrate the latter over transverse coordinates x,, YL, x2, Y2 and set z] = z z = L - l . The resulting expression is ~2 3nl
tib(A¢o)_
(2kolfl)2 f ~d2p
[Zl(p) 12
X [cosh(2aL) - cos(2bL) ]-1 ( a 2 p 2 + f14) -] X {b sinh (2aL) [c~4+ 4fl4 + ot 2'(3p2 + / ' ) ] - 2fl2c~F sinh ( 2 a L )
-asin(2bL)[o~4+4fl4+otZ(3p2-1")]},
(13)
where A(p) _=if(p)/P(O ). We assume a circular beam ,2 An expression for ~(Aco) was also derived in ref. [ 16], for a Gaussian beam. Our eq. ( 13 ) differs from that expression by a term 2(f14+o~2p2) ] in the curly brackets. There are also other, minor, differences between the two expressions.
[bsinh(2aL)-asin(2bL)
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N/a4+4fl 4
f14A
ao sinh (2aoL) - b o sin (2boL) - a sinh ( 2 a L ) cosh(2aoL)-cos(2boL) (14)
where ao and bo are the values of a and b, respectively, at p = 0 (eq. (12b)), and A = n W 2. This expression coincides with the one obtained in ref. [ 8 ] for a tube geometry. It is worthwhile to emphasize that, as long as the sample is illuminated by a plane wave, eq. (14) is valid irrespective of the ratio between the transverse dimension, v/A, to the longitudinal one, L. In the limit c ~ 0 one recovers the result of ref. [ 7 ] which, up to a numerical factor, coincides with the earlier result of ref. [ 2 ]. The limit fl~ 0 describes the relative variance fluctuation in total transmission qb(0) _= ( ~ T 2 ) / ( T ) 2: 3n l ~ ( 0 ) - 8(ko/) 2 aELA
a L sinh ( 2 a L ) - 2a ZL 2 [ 2 - cosh ( 2 a L ) ] X
sinh 2 ( a L )
' (15)
which differs from the corresponding expression of ref. [2]. The quantity ~ ( A c o ) / ~ ( 0 ) was measured (as a function of Aco) in ref. [ 14], for a plane wave incident on a slab. The result was compared to the theoretical curve calculate in the absence of absorption. The theoretical curve came out somewhat below the experimental one. One can, however, verify (from eq. (14)) that absorption will lift the theoretical curve up, possibly bringing it closer to the experiment (it was not possible to make a detailed comparison without knowledge of the absorbing properties of the sample). Additional measurements are reported in ref. [ 16 ]. There, averaging over disorder was replaced by averaging over frequency range, so that a direct comparison with eq. (14) is not possible (the required corrections are discussed in ref. [ 16 ] ). 267
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(ii) Narrow beam, i.e. ( W<< L, a - l, fl-- l ). In this case eq. (13) reduces to [ 7 ] 3
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/~(p)=(2n)26(p) leads to [ 13 ]
(plane wave geometry), which
I(lo~o~,(r))[ 2
l
Ob(Aog) _ 2(kol) 2 W '
(16)
where small corrections of order (olW)% (/~W) 4, have been neglected. Thus, absorption has only a minor effect on the correlations in total transmission, for a narrow beam. We shall not go into a detailed discussion of the intensity correlation function Coo,(r~, r2) (for the slab geometry), but only present the final expression for one particular case: the incident beam is a plane wave, the frequency difference is low (ill << 1 ), the absorption is strong (ctL >> 1 ), and the two points are taken at the output plane, separated by a distance I rl - r21 = R ~
Go,o, (rl, r2) -= C "~ (R) ( a l ) 3 3L Jl ( a R ) exp( - 2 a L ) . (kol) 2 R
(17)
For large R, this gives a power law decay, R --3/2, with oscillations (in the absence of absorption [ 1 ], it gives R - l decay).
= cosh[2ao(L-z) ] - c o s [ 2 b o ( L - z ) ] cosh ( 2aoL ) - c o s ( 2 b o L )
In order to calculate the correlation function Co,o, (r~, r2), one has to substitute eqs. ( 18 ) and (19) into eq. (10). Some simplification for Co,o, (rl, r2) occurs if the two points of observation rl, r2, are chosen on the axis of the cylinder: only the m = 0 term in eq. ( 18 ) survives for this case. Let us set co= o9', and introduce the normalized correlation function C ( z l , z2) =
(6I, o(p=0, zl )6Io~(p=O, z2) )
( I(zl) ) ( I(z2) )
In addition, let us choose z~ = L - l , z z = L - I - R , and consider C ( L - I , L - I - R ) = - C ( R ) for two special cases: (i) A long narrow tube (L>> W). In this case (and for R >> W), the main contribution to ~ ( R ) comes from the s = 0 term of eq. (18). The expression for ( ' ( R ) is rather cumbersome and we only give the resuits for two limiting cases, weak absorption ( ~ L << 1 ) and strong absorption (c~L >> 1, a R > 1 ): 1
L
t ~ t n ) ~ (koW)2 l X 2 [ 1 - ( R / L ) 2] 5. We consider now a cylinder of radius W and length L. The walls of the cylinder are reflecting and its ends are assumed to be open. A plane wave impinges on the left end of the cylinder and propagates in the z direction. The kernel K(r, r' ) for this problem, written in cylindrical coordinates (p, O, z), is
~ 6,,o - 2 K(r, r ' ) = ,,s n W 2 J,n(PmsP)Jm(PmsP' ) X j2m(pm s W ) [ 1 - (m/Pms W) 2] c o s [ m ( ¢ - O '
×
sinh (km~z<) sinh[k,.s(L-z> kms sinh (k,,sL)
)]
)] '
(18)
where Jm(X) is the Bessel function of order m, Pros is the sth root ofdJm(xW)/dx--O (m, s = 0 , 1, ...), and k~s =p2s + a 2. The field autocorrelation function (Io,,o, ( r ) ) is obtained from eq. (12) by setting 268
(19)
×](1-R/L)
( a L < < 1), (aL>> 1), (20)
which coincide with the corresponding expressions for the intensity correlation functions integrated over the cross-section of the tube [ 8 ]. The linear decay of ~ ( R ) in the case of strong absorption was observed by Genack et al. [15]. (ii) A wide cylinder (L<< W). In this case, higher terms of the kernel K(rl, r2) (eq. (18) ) become important. One can take the limit W--. oo and recover the case of an infinite slab (eq. ( 11 ) ). In the absence of absorption one obtains 3/8 l f ( X ) C ( R ) ~ (ko/)ZL X ' where X= ( R + I ) / L , and
(21)
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oo
dq f ( X ) =- f q s~-nh2q [2q2X 2 c o s h [ q ( 2 - X ) ] 0
+ 2 q X sinh [ q (2 - X) ] - 2q (2 - X) sinh (qX) - 2q 2X2 cosh (qX) } . I f R is not too close to zero or L, the function f ( X ) is of order I. Hence, for a given L, the strength of the intensity correlation in a tube of radius W is larger than in a slab by a factor of order L 2 / W 2 > > 1. 6. In conclusion, we derived a general expression for the i n t e n s i t y - i n t e n s i t y correlation function Co,o, (r~, r2) - ( 8I, o(rl )81,o, ( r 2 ) ) , which is valid for any geometry in the presence of absorption. We considered two specific cases: correlation in an infinitely wide slab and in a narrow tube. We find that absorption can significantly affect fluctuations and correlations for various quantities. However, long-range correlations exist even for distances R larger than the absorption length a -~. We are grateful to A.Z. Genack for m a n y illuminating discussions, which initiated the present work. We are also grateful to the authors of refs. [ 8,16 ] for sending us preprints of their work prior to publication. The research was supported by the U n i t e d States-Israel Binational Science F o u n d a t i o n (BSF) a n d by the F u n d for P r o m o t i o n of Research at the Technion.
29 July 1991
References [1] M.J. Stephen and G. Cwilich, Phys. Rev. Len. 59 (1987) 285. [2] M.J. Stephen, Phys. Lett. A 127 (1988) 371. [3] B.Z. Spivak and A.Yu. Zyuzin, Solid State Commun. 65 (1988) 311; A.Yu. Zyuzin and B.Z. Spivak, Zh. Eksp. Teor. Fiz. 93 (1987) 994 l Sov. Phys. JETP 66 ( 1987 ) 560]. [4 ] S. Feng, C. Kane, P.A. Lee and A.D. Stone, Phys. Rev. Lett. 61 (1988) 834. [5] P.A. Mello, E. Akkermansand B. Shapiro, Phys. Rev. Lett. 61 (1988) 459. [6] I. Edrei and M. Kaveh, Phys. Rev. B 38 (1988) 950. [7] R. Pnini and B. Shapiro, Phys. Rev. B 39 (1989) 6986. [8] E. Kogan and M. Kaveh, unpublished. [9] B. Shapiro, Phys. Rev. Lett. 57 (1986) 2168. [ 10] R. Berkovitz, M. Kavehand S. Feng,Phys. Rev. B 40 (1989) 737. [ 11 ] A.Z. Genack, Phys. Rev. Lett. 58 (1987) 2043. [ 12] I. Freund, M. Rosenbluth and S. Feng, Phys. Rev. Lett. 61 (1988) 2328. [13]N. Garcia and A.Z. Genack, Phys. Rev. Lett. 63 (1989) 1678. [ 14] M.P. van Albada, J.F. de Boer and A. Lagendijk,Phys. Rev. Lett. 64 (1990) 2787. [ 15] A.Z. Genack, N. Garcia and W. Polkosnik, Phys. Rev. Lett. 65 (1990) 2129. [16]J.F. de Boer, M.P. van Albada and A. Lagendijk, unpublished. [ 17 ] M.P. van Albada et al., Phys. Rev. Lett. 66 ( 1991 ) 3132.
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