low-coherence interferometry in random media. i. theory
TRANSCRIPT
2024 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 Brodsky et al.
Low-coherence interferometry in random media.I. Theory
A. Brodsky, S. R. Thurber, and L. W. Burgess
Department of Chemistry, University of Washington, Seattle, Washington 98195-1700
Received October 12, 1999; revised manuscript received July 6, 2000; accepted July 6, 2000
We present a new nonperturbative theoretical method for the analytical description of light propagation inrandom multiscattering media. The method is illustrated through the calculation of an expression that de-scribes optical backscattering from a semi-infinite disordered medium. A companion paper [J. Opt. Soc. Am.A 17, 2034 (2000)] compares the theoretical expression with experimental data. © 2000 Optical Society ofAmerica [S0740-3232(00)01612-4]
OCIS codes: 030.1670, 290.4210.
1. INTRODUCTIONThe propagation of classical waves in random media hasbeen the subject of experimental and theoretical investi-gations for several decades in various areas of physics(see Refs. 1–10 and the literature therein). In spite of in-tensive efforts, the subject still presents a challenge, withexperiments providing new and often unexpected results.Interest in this problem is stimulated by the fact that indisordered media, both constructive and destructive in-terference of multiple-scattered fields can occur in somespatial regions. This interference gives rise to such in-teresting effects as strong (quasi) localization, fluctua-tional waveguiding, weak localization (enhanced back-scattering), universal fluctuations in wave transmissions,memory effects, laser action without an external cavity,and local enhancement of nonlinear optical processes.4–9
Recently, substantial coherence effects were observedin low-coherence interferometry (LCI) of nonuniform di-electric media with characteristic nonuniformity dimen-sions less than, or on the order of, the wavelength.10
LCI, which was originally developed to resolve sharp dis-continuities along the wave path,11 is based on a classicaloptical measurement technique, first proposed by SirIsaac Newton. In the LCI device, based on a Michelsoninterferometer, light from the source is split evenly bymeans of an optical fiber splitter (see Fig. 1). One of thefibers directs light to the test sample and the other to amoving reference mirror. The wave packets backscat-tered from the sample and the wave packets reflectedfrom the reference mirror are recombined at the coupler.An interference between two light signals is observedwhen the optical path lengths match to within the corre-sponding coherence length of the packets. LCI is an ex-tremely sensitive technique that allows one to measuresignals with an intensity less than 10210 of an intensity ofincoming light. Our work10,12 showed that the analysisof a LCI interferogram provides substantial informationnot only about the concentration and the shape of nonuni-formities (particles) in a medium but also about the de-gree of uniformity of their spatial distribution and theBrownian motion.
0740-3232/2000/112024-10$15.00 ©
For a complete evaluation of the information containedin LCI and other precise interferometric measurements,it is necessary to rely on the theory of electromagneticwave scattering in inhomogeneous media. The most so-phisticated of the existing approaches to the problem ofwave scattering in random media are based on the replicamethod or the introduction of supersymmetrical func-tional integrals. These approaches are analogous totechniques used in the analysis of electronic properties ofdisordered metals.13 However, to obtain analytical re-sults using these and analogous approaches, it is neces-sary to use approximations based on some versions of per-turbative techniques with partial summation ofperturbative terms. In particular, such partial summa-tion is used to construct kernels of Bethe–Salpeter-typeequations.6,9 Such approximations effectively reduce theproblem to solving generalized equations of photon diffu-sion in which important coherence effects are at least par-tially neglected. Additional difficulties arise when it isnecessary to take into account effects of light absorptionand emission in a medium.9 The theory of systems withone-dimensional randomness is an exception because theformal expression for the corresponding Green functionshas a relatively simple exponential structure and onemight directly use the Fokker–Planck equation.
In this paper we describe the nonperturbative theoret-ical method of solution of Maxwell equations for nonuni-form media. The method is based on an analogy with theexact solution of the Bloch–Nordsiek model in nonrelativ-istic quantum field theory. In this solution Fock’s tech-nique of an introduction of an additional timelike coordi-nate is used.14 This technique allows us to find, in athree-dimensional space, the formal expression of Greenfunctions with an exponential structure analogous to thecorresponding structure of one-dimensional Green func-tions. Such formal expressions can be averaged overboth the wave functions (in quantum theory) and the ran-dom dielectric properties (in our case) without solving theproblem for specific probable realizations of the consid-ered systems. In contrast to the description of wavepropagation based on Bethe–Salpeter equations, which
2000 Optical Society of America
Brodsky et al. Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 2025
were also borrowed from the quantum field theory (albeita much more complicated relativistic one with a possibil-ity of virtual pair production), our method allows us tofind analytical expressions for averaged characteristics ofwave propagation without references to the perturbationtheory. Our approach to the solution of the wave-scattering problem in random media is analogous to themethod recently applied to the analysis of the solutions ofthe scalar Helmholtz equation by Samelsohn andMazar.15 In our calculations we have taken into accountthe vector structure of Maxwell’s equations, because thescalar approximation is not adequate for a quantitativedescription of the coherence effects in multiscattering me-dia with characteristic dimensions of nonuniformitiescomparable with, or smaller than, the wavelength.5
The outline of this work is as follows. In Section 2 weobtain the formal solution of Maxwell equations for a LCIgeometry. In Section 3 this solution is averaged overfluctuating dielectric properties for the case of a statisti-cally homogeneous Gaussian distribution of nonuniformi-ties. Functional integral techniques are used in this av-eraging. Finally, Section 4 is devoted to a summary ofresults and a discussion of possible applications of theproposed theoretical technique.
2. BASIC EXPRESSIONS FOR A LOW-COHERENCE INTERFEROMETRYGEOMETRYIn LCI experiments the light is scattered from a semi-infinite sample under test (SUT) into the half-space x1, 0, where the incident and reference channels and thecoupler of the Michelson interferometer are situated (seeFig. 1). A photocurrent is measured in the coupler as afunction of the overlap between (a replica of) the incidentand scattered wave packets, with the variation in overlapcontrolled by changes in length l of the reference arm ofthe interferometer.
We use the following notation. The magnetic perme-ability is taken to be equal to unity. The optical dielec-tric reflective index is taken in the simplest scalar and lo-cal form:
n~v, x! 5 u~2x1! 1 n~v, x!u~x1!, (1)
where vector x is the coordinate, x1 5 0 is the mean po-sition of the sample boundary, and u(x1) is a step func-tion, given by
Fig. 1. Block diagram of the LCI experiments.
u~x1! 5 H 1 for x1 . 012 for x1 5 0
0 for x1 , 0
. (2)
The fluctuations of dielectric properties and dispersion ef-fects outside the sample, as well as the difference betweenthe structures of the sample bulk and its surface layer,are neglected.
The effect of fluctuating dielectric properties is takeninto account by introducing the following expression forthe sample bulk optical refractive index n(v; x):
n~v; x! 5 n~v! 1 k~v!dr~x!
r, n~v! 5 ^n~v; x!&.
(3)
The angle brackets introduced in Eq. (3) denote the aver-aging over sample dielectric properties. In Eq. (3) dr(x)is a real parameter (or, in the more general case, a set ofparameters) fluctuating about zero. The coefficient k(v)introduced in Eq. (3) is related to the amplitudes of lightscattered by nonuniformities. This coefficient is, in thegeneral case, complex, even if it is possible to disregardinelastic light scattering, with an imaginary part of k(v)resulting from the coherence loss in individual scatteringevents. Correspondingly, an adequate description of co-herence effects in random media in the general case can-not be accomplished by considering only real optical re-fractive indices. The meaning of k(v) can be easilyunderstood in the framework of the coherent phaseapproximation.16 This approximation is valid in cases ofmoderate concentrations of nonuniformities and/or lowoptical contrasts. We will use the coherent phase ap-proximation in Ref. 12 in our comparison of the theorywith the experiment.
We present the electric field in the reference channel,Eref, entering the coupler of the LCI after reflecting fromthe reference mirror (see Fig. 1), in the following form of aGaussian wave packet, with the center frequency v5 v0 and the bandwidth s, moving along the axis x1 inthe negative direction:
E1ref ~2l 1 x; t ! 5
E0ref
2psE dv expH 2iFvt 1
v
c~2l 1 x1!G
2~ uvu 2 v0!2
2s 2 J , (4)
where l is the difference between the distances of the mir-ror and the sample surface to the coupler. In LCI thevalue of s is relatively large, which allows us to measurepositions of small-scale nonuniformities with precision;c/s. The LCI signal JSUT(l) is proportional to themaximum value of an envelope of the interferogram be-tween Eref and the wave packet entering the coupler afterreflecting from the sample. This envelope corresponds tothe signal averaging over a time interval on the order ofthe inverse Doppler frequency shift, which is due to themirror movement in the reference channel of the Michel-son interferometer. We take the expression for JSUT(l)in the following form:
2026 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 Brodsky et al.
JSUT~l ! [ const. 3 @J~l ! 1 Jdev#,
J~l ! 5 Re~E0
ref!*
2ps K E dvEux2x0u,L
d3x E~v; x!
3 expH 2F2iv
c~2l 1 x1! 1
~ uvu 2 v0!2
2s 2 G J Lav
,
(5)
where E(v; x) is the complex component, with the fre-quency v, of the field reflected by the sample in the nor-mal direction with the same polarization as that ofEref, J(l) is the interference signal depending on the scat-tering properties of the SUT, and Jdev is the signal result-ing from the fundamental photon noise and the fluctua-tions in the measuring device. In Eqs. (5) L is a lineardimension of the coupling unit (with the position of itscenter at x0 5 $x1
0, 0, 0% with x10 , 0). For the following
calculations, it is important only that L is much largerthan the wavelength.
To obtain an expression for E(v; x) convenient for theapplication of Fock’s technique, we introduce the follow-ing two vector functions F(1,2)(v, x):
F~1,2!~v, x! 5 E~v, x! 6i
h~v; x !H~v, x!, (6)
where E(v; x) and H(v; x) are electric and magnetic vec-tors. These functions are quoted in the literature as De-bye’s potentials.17 The linear Maxwell equation for F(1,2)
has the following form of two coupled equations:
curl F~1,2! 7v
cn~v; x!F~1,2!
6 H @grad ln n~v; x!#F~1 ! 2 F~2 !
2 J 5 0. (7)
Note that the modulus of the coefficient grad ln n(v; x) inthe last term of Eq. (7) is equal to the inverse principalray curvature at the point x.18 By definition, the inversecurvature is equal to zero at all points for rays with a con-stant direction. Without the last term, the two equationsin expression (7) are decoupled and reduced to two inde-pendent first-order partial differential equations. Insuch a case, the backscattering from bulk nonuniformitiesdisappears.
To carry out the calculations in a more compact form,we introduce the algebra of operators in six-dimensionalspace. This space is a direct product (i 3 a) of a three-dimensional space i 5 1, 2, 3 and a two-dimensional pho-ton spin space a 5 1, 2, with the following matrix ele-ments of relevant operators I, Ikl , Sk , and s1,3 :
^i, auIu j, b& 5 d ijda,b ,
^i, auIklu j, b& 5 d ikd jlda, b ,
^i, auSku j, b& 5 2da,bieijk ,
^i, au s1u j, b& 5 d ij~da1db2 1 da2db1!,
^i, au s3u j, b& 5 d ij~da1db1 2 da2db2!, (8)
where eijk is the completely asymmetric tensor in three-dimensional space with e123 5 1 and d ij is the Kroneckersymbol. Equation (7) can now be rewritten in matrixform as follows:
i(k51
3
Sk
]
]xkF 1 s3
v
cn~v; x!F
1I 2 t1
2i(
k51
3
Sk
] ln n~v; x!
]xkF 5 0, (9)
where F [ F(v, x) is the column
F [ S F~1 !~v, x!
F~2 !~v, x! D [ S F1~1 !
F2~1 !
F3~1 !
F1~2 !
F2~2 !
F3~2 !
D ,
F* ~v, x! 5 s1F~2v, x!. (10)
For the construction of the expression for the scatteredfield E(x), we introduce the retarded matrix Green func-tion G(v; x, x8), which is the solution of the followingmatrix equation:
F(k51
3
iSk
]
]xk 1 s3
v
cn~v; x!
1I 2 t1
2 (k51
3
iSk
] ln n~v; x!
]xkG
3 G~v; x 2 x8! 5 Id 3~x, x8!. (11)
It follows from Eq. (11) and the definition of a retardedGreen function that
S12E
ux2x0u,LG~v; x; x8!d2x i
5 S12
expS 2iv
cx1D
2p F Euxi2xi0u,L
G~v; x, x8!d2x i8Ux150
Gfor x1 < 0, x18 > 0, x i 5 $x2 , x3%. (12)
This result corresponds to the continuity of the tangentialcomponents of the electric and magnetic fields across thesample boundary. With the help of the Green functionG(v8, x, x8), we can now rewrite the second of Eqs. (5) asfollows:
Brodsky et al. Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 2027
J~l ! 5 U K LI0
~2ps!2 Euxi2xi
0u,Ld ~x1!d3xd3x8
3 @F ~0 !#trE dvXexpF2~ uvu 2 v0!2
s 2 G3 expH 2i
v
c@2l 2 n~v!x18#J
3 G~v; x, x8!F u~x18 !iv
ck~v!
dr~x8!
rs3
1I 2 s1
2n~v; x!(k51
3
Sk
]
]xk8u~x18 !k~v!
dr~x8!
rGCL U ,
(13)
where I0 is the electromagnetic energy flow entering thestudied medium, d (x1) is a delta function, and the con-stant column F (0) represents the vector and polarizationstructure of the solution of wave equations in uniform me-dia. In the following calculations, we consider the case oflinear polarization with an electric vector directed alongthe x2 axis when
F ~0 ! 5 s1~F0!* 51
4 S 01
2i01i
D . (14)
Now we introduce, following Fock’s idea,14 the fifth, time-like coordinate n and write the expression for the Greenfunction G in the following form:
G~v; x, x8! 5 F I 1 s1
21
I 2 s1
2
n0~v; x1!
n~v; x!G
3 Gg~v; x, x8!
3 F I 1 s1
21
I 2 s1
2
n~v; x8!
n0~v; x18 !G ,
Gg~v; x, x8! 5 is3E0
`
U~n!dn, (15)
where
U~n! [ U~n, v; x, x8! 5 exp~is3Kn!d 3~x 2 x8!,
K 5 i (k51
3
Sk
]
]xk1
v
cs3n~v; x1!
1 S I 2 d1
2 Dv
cd3
n2~v; x! 2 n2~v; x1!
n~v; x!
n0~v; x1! 5 u~x1!n~v! 1 u~2x1!. (16)
It follows from Eqs. (16) that the matrix U(n) has to sat-isfy the differential equation
2i]U
]n5 s3KU (17)
with the initial condition
U~n! 5 Id 3~x 2 x8!un50 . (18)
It is important to note that the term proportional tod ln n(v; x)/dx present in Eq. (11) is excluded from Eq.(17), as a result of the local gauge transformation in thefirst equality in Eqs. (15), which connects G with Gg.
We write an expression for U(n) in the following form:
U~n! 51
~2p!3/2 E d3p expXivc
@1 2 n~v!#
3 H s3S1
~x1 1 n!u~2x1 2 n! 1 ~x1 2 n!u~2x1 1 n! 2 2x1u~2x1!
2
2 S12Fnu~2x1! 2
~x1 1 n!u~2x1 2 n! 1 ~n 2 x1!u~2x1 1 n!
2 G 1 Inu~2x1!J C3 expH iFp~x 2 x8! 1 ns3 (
k51
3
Skpk 1 nn~v!v
c Gexp@iL~v; n; x!#J , (19)
where L(v; n; x) is a matrix function of v, n, and x suchthat
L~v; 0; x! 5 0. (20)
The form of expression (19) is convenient, since whenL(v; n; x) [ 0, the introduction of Eq. (19) into Eqs. (15)provides, after the integration over dn, the standard ex-pression for a Green function in the case of wave scatter-ing from a uniform semi-infinite medium with dr(x)[ 0. According to Eqs. (17) and (19), the operatorexp(iL) has to obey the equation
F iI]
]n1 is3(
k51
3
Sk
]
]xk, exp~iL!G
1 k~v!v
cu~x1!
dr~x!
rexp~iL! 5 0, (21)
where a commutator of two operators A and B is denotedby a square bracket:
@A, B# 5 AB 2 BA. (22)
2028 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 Brodsky et al.
Equation (21) can be reduced to the following equation forL(v, n, x):
S I]
]n1 s3 (
k51
3
Sk
]
]xkD L 1
v
ck~v!
dr~x!u~x1!
r5 0.
(23)
Note that, according to Eq. (23),
F S I]L
]n1 s3(
k51
3
Sk
]L
]xkD , LG 5 0. (24)
It is possible to check by direct substitution that the fol-lowing expression for L satisfies Eqs. (20) and (23):
L 5vk~v!
c~2p!3/2 E d3q exp~iqx!
3dr~q!
r H is3(k51
3
Skqk
1 2 cos nq
q2
1 @I 2 I~q !#S n 2sin nq
q D 1 Isin nq
q J ,
L* ~v; n; x! 5 2s1L~2v; n; x!s1 , Ls3 2 s3L 5 0,(25)
where the operator I(q) is defined according to
^aiu I~q !ubj& 5 dabS d ij 2qiqj
q2 D , q 5 A~q!2 (26)
and
dr~q! 51
~2p!3/2 E d3x exp~2iqx!dr~x!u~x1!. (27)
In the derivation of Eq. (25), we have taken into accountthe operator identity
(k51
3
Sk
]
]xk S I¹2 2 (k8,k951
3
Sk8Sk9
]2
]xk8]xk9D F~x! [ 0,
(28)
which is a matrix analog of the vector calculus equalitycurl grad div F(x) 5 0. It follows from Eqs. (13), (15),and (19) that
J~l ! 52LI0
~2p!s 2 U K iEuxiu,L
dvd3pd3xd3x8 d ~x1!u~x18 !
3 E0
`
dn expF22iv
cn1~v!lG
3 @F ~0 !#trXexpF2~ uvu 2 v0!2
s 2 1 iL~v; n; x!G3 expH 2
iv
cn@1 2 n~v!#
s3S1
2
1iv
cn
n~1 ! 1 n~v!
2IJ
3 expH iFp~x 2 x8! 1 n(k51
3
s3Skpk
1v
cn~v!x18G J H iv
ck~v!
dr~x8!
r
1I 1 s1
2n~v!
k~v!
rdr~x8!Fk~v!
dr~x8!
r
1 (k51
3
is3Skpk 1 s3n~v!v
cS1
v
c G J CF ~0 !L U .(29)
In Eq. (29) we performed the integration by parts over dx18and have taken into account Eqs. (25) and the followingrelations:
it1t3F I 1 s1
21
I 2 s1
2
n~v; x8!
n0~v; x18 !G v
cs3k~v!u~x18 !
dr~x8!
r
5 s1F1 1I 1 s1
2
dr~v; x8!
n~v!rG i
v
ck~v!u~x18 !
dr~x8!
r
→I 1 s1
2
dr~v; x8!
n~v!r
v
c~v!u~x18 !
dr~x8!
r. (30)
The term dropped in the last line gives a result equal tozero after its introduction into Eq. (29) and averagingover dielectric fluctuations.
The expression under the modulus sign in Eq. (29) isreal, as a result of Eqs. (1), (4), and (25) and the equalities
H expF iL~v, n, x!expS 2ins3(k51
3
SkpkD J *
5 exp@is1L~2v, n, x!s1#expS 2ins1s3(k51
3
Skpks1D5 s1 exp@iL~2v, n, x!#expS ins3(
k51
3
SkpnD s1 ,
I 1 s1
2s1 5
I 1 s1
2,
I 2 s1
2s1 5 2
I 2 s1
2,
Brodsky et al. Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 2029
Sk* 5 2Sk , (31)
because it is possible to substitute v → 2v under the in-tegral over dv. The fact that this expression is realmeans that the famous backscattering enhancement ef-fect related to the time-inversion invariance1,3 has beentaken into account.
3. AVERAGINGIn performing the averaging of Eq. (29) over density fluc-tuations dr(x) in the sample bulk, we assume that thefluctuations can be considered a stationary isotropicGaussian random process. The averaging is defined bythe following equalities:
^dr~x!&
5 ^dr~q!& 5 0,
K dr~x!
r
dr~x8!
rL
5 D expF2~x 2 x8!2
a2 G for x1 , x18 > 0,
K dr~q!
r
dr~q8!
rL
5 ~aAp!3D expS 2q2a2
4 D d 3~q 1 q8!
3 K dr~q1!
r¯
dr~qn!
rexpF iE f~q!
dr~q!
rd3qG L
5 ~2i !nd n
df~q1! ¯ df~qn!
3 expF21
2E f~q!D~q, q8!f~q8!d3qd3q8G , (32)
where dr(q) is the Fourier transform of dr(x), d/df(q)designates the functional derivative,14 the constant D isthe measure of the fluctuation intensity, and the param-eter a is the correlation length. It is assumed in Eqs. (32)that dr(x) is averaged over a small volume less than(c/s)3. In the calculations, we are deriving an analyticalexpression for J(l) in the case when D , 1 and
g ~z ! 5 expS 2z2
2DD ! 1 for z > 1. (33)
The terms on the order of g (1) neglected in the followingcalculations correspond to the far tails of the density dis-tribution, where in real systems deviations from an as-sumed Gaussian distribution can play an importantrole.19 The inequality (33) relieves us also from explicitlyintroducing the condition that the fluctuating density hasto be positive. With the help of Eqs. (32), after integra-tion over d3x, d3x8, and d2p i [ dp2dp3 in Eq. (29), we findthat
J~l ! 5Da3L3A2
s 22piE dvE
0
`
dnFv
ck~v!G2
3 FexpH iv
c Fn1 1 n2~v!
22 2lG 2
~ uvu 2 v0!2
s 2 J
3 E dp1
p1 2v
cn2~v!
@F ~0 !#tr expX2D
3p2 Fva
ck~v!G2
3 H 1 1n2
2a2 2 expF2S n
a D 2G J C3 E expS 2
q2a2
4 D d3qH is3(k51
3
Skqk
1 2 cos nq
q2
1 @I 2 I~q !#S n 2sin nq
q D 1 Isin nq
q J3 expF iv
cn
1 2 n~v!
2s3S1G
3 expS ins3S1p1 2 in(k51
3
s3SkqkD3 S is3S1p1 2 i(
k51
3
s3SkqkD I 2 s1
2F ~0 !G
1 O~D2!. (34)
In Eq. (34) we have taken into account that
E dx18 u~x8!expH iFq1 2 p1 1v
cn~v!Gx8J
5 iFq1 2 p1 1v
cn~v!G21
, (35)
since Im@(v/c)n(v)# . 0. We have also made the shift inthe integrand p1 → p1 1 q1 and used the identity
s3S1F ~0 ! 5 2F ~0 !. (36)
For simplicity, we did not write out in Eq. (34) the explicitexpression for the term proportional to D2. This termmakes a negligible contribution in the case of nonunifor-mities with dimensions less than the wavelength.
The expression (34) looks rather complicated. How-ever, it can be substantially simplified, since the symme-tries destroyed by density fluctuations in the sample me-dium are restored after the averaging over fluctuatingdielectric properties. The expressions under the integralover d3q in Eq. (34) are cylindrically symmetrical, andnonvanishing terms can be proportional only to the mu-tually commuting matrices s3S1 , S1
2, and I. Corre-spondingly, we can drop from the integrand all additionalterms that arise during the shift of matrices proportional
2030 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 Brodsky et al.
to s3S1 to the extreme right-hand position. This allowsus to take the integral over dp1 in Eq. (34) with the helpof the following equalities:
1
2piE dp1 p1
m
p1 2 n~v!exp~iS1t3 p1n!F ~0 !
51
2piE dp1 p1
m
p1 2 n~v!exp~2inp1!F ~0 ! 5 0,
t1
2p1E dp1 p1
m
p1 2 n~v!exp~iS1t3 p1n!t1F ~0 !
51
2piE dp1 p1
m
p1 2 n~v!exp~inp1!F ~0 !
5 @n~v!#mF ~0 !, n > 0, m 5 0, 1, 2 ,... . (37)
In the two equalities (37), we have taken into accountEqs. (31) and (36) and have closed the integration con-tours over dp1 by the half-circles at infinity in the lowerand the upper complex half-space, respectively. After in-tegrating over d3q and performing some matrix algebra,we find from Eq. (34), taking into account Eqs. (36) and(37), that
J~l ! 5L3I0D
3s 2 E0
`
dnE0
`
dvXnFk~v!v
c G2
expF2~v 2 v0!2
s 2 G3 H 1
21 expF2S n
a D 2G J3 expH 2iv
c@n~v!n 2 l# 2 S v
c D 2
k2~v!gS n
a D J C.(38)
In Eq. (38) we have introduced the function g(n/a):
gS n
a D 5D
32a2H 1 1
1
2S n
a D 2
2 expF2S n
a D 2G J
5 5 Dn2 forn
a→ 0
D
3n2 for Re
n
a→ `
. (39)
The integration over dv in Eq. (38) can be done explicitlyif we use the following expressions for n(v) and k(v):
v
cn~v! 5
v0
cn~v0! 1 ~v 2 v0!
1
v,
v
ck~v! 5
v0
ck~v0! 1 ~v 2 v0!
1
v ~0 !, (40)
where v and v (0) represent corresponding group veloci-ties. The approximation (40) is a standard one for thewave-packet description when an interval @v0 2 s, v01 s# does not include frequencies corresponding toanomalous dispersion. It follows from Eq. (40), after in-tegration over dv, that J(l) can be presented, up to expo-nentially small terms, in the following form:
J~l ! 5 2Up2L3I ~0 !s
2E
0
`
dn ln@n#]
]n
3 F 1
F1 1 S s
v D 2
gS nl
a D G1/2
3 5 1
22
v2Fk~v0!iv0
c2
s 2l
v2v S n 2v
c D G2
s 2F1 1 S s
v D 2
gS nl
a D G 63 expX S s
v D 2
gS nl
a D1 1 S s
v D 2
gS nl
a D H S sl
v2D 2S n 2
v
c D2
22iv0l
ck~v0!
v
v2S n 2
v
c D 2 Fv0v
csk~v0!G2J C
3 exp5 2F S sl
v2D 2S n 2
v
c D2
2 2iln2~v0!v0
c
3 S n 2v
c D 12iv0l
c
d ln n~v!
d ln v
1 1d ln n~v!
d ln v
Uv5v0
G 6 G,
(41)
where we have introduced the notation
@n# [ F1 1 S s
nD 2
gS nl
a D G . (42)
In the integrand in Eq. (41), we have used the scalingtransformation n → nl to make the integration variabledimensionless. Also, we have performed the integrationby parts over dn using the identity
4nS l
a D 2 D
3 S s
v D 2H 1
21 expF2S vl
a D 2G J1 1 S s
v D 2
gS nl
a D5
]
]nlnF1 1 S s
v D 2
gS nl
a D G . (43)
The formula (41) provides the expression for J(l) in theform of a single integral over dn. This integration is the
Brodsky et al. Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 2031
price that we have to pay for the presentation of theGreen function in the form in Eqs. (15) convenient for av-eraging. The integral in Eq. (41) can be taken in anasymptotic approximation, since the exponential factorsin the integrand rapidly oscillate and decline when the
value of the integration variable n deviates from
ns 5v
c, (44)
if, as was assumed above, there are no anomalous disper-sion effects and
v0
s
d ln n~v!
d ln v
1 1d ln n~v!
d ln v
Uv5v0
, 1,
sc
vv0k~v0!5 s
d lnv
ck~v!
dvU
v5v0
, 1. (45)
In the asymptotic approximation, we substitute n → nseverywhere in the integrand in Eq. (41), with the excep-tion of the exponents and the main pre-exponential factor.Extending the limits of integration over dn in Eq. (41) to[2`, `], we can perform in the considered approximationsthe integration over dn taking into account Eq. (43). Af-ter substituting the result into Eq. (5), we find the follow-ing final expression for JSUT(l) in the first nonzeroasymptotic approximation:
JSUT~l ! 5 const. 3 Uk2~v0!DF l 1 in~v0!v0v
s 2 GF1 1 S s
nD 2
gS ns
l
a D G5/2 H 1
21 expF2S nsl
a D 2G J expH 2
Fv0
ck~v0!G2
gS nS
l
a D@nS#
J3 expX2H 2l
v0
cIm
@d/~d ln v!#@ln n~v!#
1 1d ln n~v!
d ln vJ CU
v5v0
1 Jdev , (46)
where the definition of @ns# follows from Eqs. (42) and(44). In Eq. (46) we have absorbed all universal and de-vice characterizing factors in one coefficient. The struc-ture of expression (46) corresponds to a smooth matchingof two exponential dependencies on l:
ISUT~l ! ; 5expF2S l
ct2D 2G , t2 5
1
ADnsv0k~v0!for l ,
a
nsAD
expF2l
ct3G , t3 5 S 2v0 Im
d/~d ln v!@ln n~v!#
1 1d ln n~v!
d ln vU
v5v0
D 21
for l @a
nsAD
. (47)
The dependences ISUT(l) on l [Eq. (46) and relation (47)]reflected the lows of coherence loss in the medium and isqualitatively different from that in the case in which theSUT is replaced by a simple mirror. In this case ISUT(l)has the form of an approximate Gaussian peak around l5 0 with a narrow width on the order of c/s. Suchpeaks are observed in LCI of stratified nonscatteringsamples.11 The characteristic (two intervals, nonsym-metrical) dependence (46) also cannot be found in aneffective-medium model as well as in models based onstandard photon diffusion theory. The lows of the coher-ence loss and the dynamics of phase and ray randomiza-tions are beyond the scope of such theories (see also Ref.8). These dynamics define the structure of LCI signalsfrom nonuniform media.
The main parameters entering Eq. (46) are the inten-sity of fluctuation D, the ratio of correlation length a tothe optical coherence length c/s, and the scattering char-acteristic k(v). The signal ISUT(l) is proportional tok2(v) and is correspondingly small for small particles,since k(v) contains the factor R(v/c). However, as isshown in the companion paper,12 it is nevertheless pos-sible to use LCI as a detector even of nanoparticle char-acteristics because of the extreme sensitivity of opticalphase measurements and the accumulation of the effectsof multiple scatterings. Corresponding numerical ex-amples can be found there.12
Expression (47) reproduces the general LCI signalstructure experimentally observed in our work,10 wherethe characteristic times t2 and t3 were introduced to de-
2032 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 Brodsky et al.
scribe the signal shape on the different time intervals.The only difference with the description in Ref. 10 is theappearance of l2 in the exponentials in Eq. (47). The lim-ited experimental data in Ref. 10 did not allow us to dis-tinguish between linear and quadratic dependencies on lin the corresponding exponential. The results of moreprecise experiments in the companion paper12 are inagreement with Eqs. (46) and (47).
4. CONCLUSIONWe have presented in the framework of classical linearelectrodynamics a derivation of the expression describingthe linear (in field) effects in the coherent backscatteringof electromagnetic waves from semi-infinite media withrandomly distributed structureless nonuniformities. Inthe derivation we followed the general approach used inthe solution of the Bloch–Nordsiek quantum fieldmodel.14 An asymptotic analytical expression was foundfor the effect measured in the LCI experiments. The pro-posed theoretical approach can be applied to the calcula-tion of the different bilinear and higher-degree combina-tions of coherently scattered fields in random media. Itcan also be applied to the analysis of the influence of lo-calized bulk and surface modes and resonances, whichcan be formed in the presence of nonuniformities, on lightscattering.
The substantial element of the described calculations isthe application of Gaussian functional integration in av-eraging over fluctuating dielectric properties. Such aver-aging is adequate in the case of the equilibrium distribu-tion of weakly interacting particles as well as in the caseof freely diffusing particles in liquids. It may be possibleto generalize our calculations to more complex cases ofnon-Gaussian distributions. In particular, it is possibleto relatively simply generalize them to the cases of a Pois-sonian distribution or of a finite sum of different Gauss-ian distributions with different correlation lengths bychanging the expression for kernel D in Eqs. (32). All theapproximations and the models used, including that of anabsence of anomalous dispersion effects, have exactly de-fined limits of applicability.
The advantages of the proposed theoretical approachcan also be exploited in the description of a number of dif-ferent classical and quantum wave effects in nonuniformmedia. These include the description of nonlinear opticalprocesses, such as second-harmonic generation, as well asnontrivial effects that accompany light propagation inrandom gain media. In particular, it may be possible touse this approach for the description of mirrorless lasereffects in random gain media in both optical and x-ray fre-quency intervals.
ACKNOWLEDGMENTSWe thank the Center for Process Analytical Chemistrysponsors and The Boeing Company for their support.
Address correspondence to A. Brodsky at the locationon the title page or by e-mail, [email protected].
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14. The solution of the Bloch–Nordsiek model is reviewed in W.Bogolubov and D. Shirkov, Introduction to the Theory ofQuantized Fields (Wiley, New York, 1980).
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16. In the coherent phase approximation,
k~v! 5 2pF c
n~v!G2(
a
^Na&Aa~0! ,
where ^Na& is the mean density of nonuniformities (par-ticles) of type a and Aa(0) are the complex amplitudes offorward scattering caused by these nonuniformities (T ma-trices). The imaginary component of Aa(0) includes an ef-fect of coherence loss during scattering events. See Ref. 17for its applications of coherent phase approximation in op-tics.
17. R. Newton, Scattering Theory of Waves and Particles
Brodsky et al. Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 2033
(McGraw-Hill, New York, 1966), Chap. I; H. van De Hulst,Light Scattering by Small Particles (Wiley, New York,1957).
18. L. Landau and E. Lifshitz, Electrodynamics of CondensedMedia (Pergamon, London, 1980), Chap. 10.
19. The long tail of a distribution of waves scattered by
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