ournal of iomedical o ptical c oherence p ropagation … · ded in an isotropically disordered,...

OPTICAL COHERENCE PROPAGATIONAND IMAGING IN A MULTIPLE SCATTERINGMEDIUM

Sajeev John, Gendi Pang, and Yumin YangUniversity of Toronto, Department of Physics, Toronto, Ontario, Canada M5S 1A7(Paper JBO-012 received May 18, 1995; revised manuscript received Nov. 29, 1995 accepted for publication Nov. 29, 1995)

ABSTRACTWe present a formal, microscopic, solution of the wave propagation problem for an inhomogeneity embed-ded in an isotropically disordered, multiple scattering, homogeneous background. The inhomogeneity isdescribed by a local change in the complex, dielectric autocorrelation function B(r,r8) [v4/c4^e* (r)e(r8)&ensemblefor a wave of frequency w and velocity c . For the homogeneous background, we consider a dielectricautocorrelation function Bh(r−r8) arising from a colloidal suspension of small dielectric spheres. This autocor-relation function can be determined using a newly developed technique called phase space tomography foroptical phase retrieval. This technique measures the optical Wigner distribution function I(R,k) defined as theFourier transform, with respect to r, of the electric field mutual coherence function ^E* (R+r/2)E(R−r/2)&ensemble . The Wigner distribution function is the wave analog of the specific light intensity, Ic(R,k), inradiative transfer theory which describes the number of photons in the vicinity of R propagating in directionk . The Wigner function describes coherence properties of the electromagnetic field which can propagate muchlonger than the transport mean-free-path l* and which are not included in radiative transfer theory. Given thenature of the homogeneous background, repeated light intensity measurements, which determine the opticalphase structure at different points along the tissue surface, may be used to determine the size, shape, andinternal structure of the inhomogeneity. In principle, this method improves the resolution of optical tomog-raphy to the scale of several optical wavelengths in contrast to methods based on diffusion approximationwhich have a resolution on the scale of several transport mean-free-paths. Our theory, which describesmicroscopically the wave characteristics of the light, is more fundamental than conventional radiative transfertheory, which treats photons as classical particles. This distinction remains important on scales longer than l* .Multiple light scattering tomography based on the propagation and measurement of the Wigner distributionfunction may be useful for the characterization of near-surface tumors.

Keywords multiple light scattering; inhomogeneity; optical tomography.

1 INTRODUCTION

Near-infrared optical tomography has recentlyemerged as a potentially powerful tool in diagnos-tic medical imaging.1–9 Unlike well-established xray imaging and magnetic resonance imaging,which utilize very short wavelength or very longwavelength radiation respectively, the opticalmethod probes an intermediate frequency regime.As emphasized by Chance,1 this intermediate fre-quency window facilitates the detection of abnor-mal metabolic processes leading to tumor forma-tion. This distinguishes the optical method as anearly diagnostic tool, from its more establishedcounterparts that respond to structural damage re-sulting from abnormal metabolism. The descriptionof near-infrared electromagnetic wave propagationin biological tissue is, however, much more com-plex than for x rays or radio waves. Light exhibitsmultiple scattering in tissues with a scattering

mean-free-path, l , on the scale of 10−2 to 10−1 mmand a transport mean-free-path, l * , on the scale ofa millimeter. In conventional radiative transfertheory,9,10 it is convenient to define a specific lightintensity, Ic(R,k), for the number of photons at thepoint R travelling in a direction k . This is analogousto a classical phase space distribution function. Itsatisfies a phenomenological Boltzmann transportequation. In this model, photons are treated as clas-sical particles that undergo random multiple scat-tering. It is assumed that the coherent wave natureof the electromagnetic field is lost on the scale ofmany scattering mean-free-paths l , and from thisassumption it follows that on the scale of the trans-port mean-free-path l * , photons exhibit classicaldiffusion.From a microscopic point of view, the electro-

magnetic field satisfies a wave equation. In this pic-ture, the classical phase space distribution function(specific intensity) Ic(R,k) is not well defined.

Address all correspondence to Sajeev John. E-mail:[email protected] 1083-3668/96/$6.00 © 1996 SPIE

JOURNAL OF BIOMEDICAL OPTICS 1(2), 180–191 (APRIL 1996)

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Wave-particle duality suggests that if the wave vec-tor k of the photon is specified, its position is un-certain and likewise knowledge of the particle po-sition R leads to uncertainty in its momentum \k.The nonexistence of a true phase space density mo-tivated Wigner to define the first-order coherencefunction:11,12

I~R,k![E d3r exp@ ik•r#^E0~R1r/2!

3E~R2r/2!&ensemble . (1)

Here, E is the complex electric field amplitude ofthe propagating radiation field and ^ &ensemble de-notes a statistical averaging over all possible real-izations of the dielectric microstructure. In this pa-per we use the terms specific intensity and Wignercoherence function interchangeably. It must be bornein mind, however, that I(R,k) defined in (1) is notpositive definite and that the analogy with conven-tional Boltzmann transport theory is not exact. It isshown in the appendix that a coarse-grained ver-sion of the Wigner function is in fact positive defi-nite and may be identified as a true specific inten-sity.It is our aim in this paper to describe the precise

integro-differential equation satisfied by the propa-gator G(R−R8;k,k8) of first-order coherence, whichis defined as

I~R,k!5E d3R8d3k8G~R2R8;k,k8!I0~R8,k8!,

where I0(R8,k8) is the source coherence function.This G(R−R8;k,k8) is equal to an ensemble averageof a product of two Green’s functions [see Eq. (9)below). It is the Wigner function detected at R aris-ing from a source at R8 with specified coherenceproperties. The transport equation satisfied by Gturns out to be similar to the classical Boltzmannequation except with nonlocal interactions arisingfrom wave coherence. This result suggests that fun-damental distinctions exist between optical tomog-raphy based on conventional radiative transfertheory and coherence propagation theory. We find,very remarkably, that for a simple model ofmultiple-light scattering in a colloidal suspension ofpolystyrene spheres in water, certain features of theWigner function propagate coherently on lengthscales which are on an order of magnitude longerthan the transport mean-free-path l * . This is incontrast to a purely particlelike picture of photonsin which purely ballistic propagation is attenuatedon the scale of the scattering length l !l * . Thissuggests that measurement of the Wigner function(optical coherence imaging) may provide a higherdegree of sensitivity to properties of the dielectricmicrostructure than measurement of the total dif-fuse intensity. Coherence tomography also offersthe possibility of higher resolution imaging.

Classical diffusion theory requires coarse grain-ing of the electric field amplitude on sufficientlylarge length scales that the true wave characteristicsof transport are no longer apparent. The Wignerfunction, however, allows resolution on the scale ofthe optical wavelength. These features of coherencetomography are of value to both time- andfrequency-domain studies5,7 of biological tissue. Inthe case of multiple light scattering from an inho-mogeneity (tumor) in an otherwise homogeneouslydisordered medium, the Wigner function behaveslike a quantum mechanical wave function whichscatters from the effective potential presented bythe inhomogeneity. This facilitates a description oftransport which in many respects resembles that ofscattering of a quantum particle in a uniform back-ground by a fixed impurity.The possibility of measuring the Wigner coher-

ence function using purely intensity measurementsand refractive optics (lenses) has been demon-strated by Raymer, Beck, and McAlister.13 Themethod uses a series of measurements of the lightintensity at a point on the sample surface imagedby a pair of lenses to reconstruct uniquely, by to-mographic reconstruction, the full Wigner coher-ence function at that point. Passage of the lightthrough a pair of lenses effects a Fresnel integraltransformation of the wave amplitude on thesample surface. This is equivalent to a projectionintegral of the Wigner coherence function. By vary-ing the position of the lenses, a variety of differenttomographic projections may be obtained, leadingto reconstruction of the Wigner function by inverseRadon transform. This method can be repeated at avariety of different points along the sample surface,thereby yielding a detailed map of optical absorp-tion and dielectric microstructure within thesample.Light scattering from an inhomogeneity in an

otherwise nonscattering background mediumreadily yields valuable information about the loca-tion and nature of the inhomogeneity. In the case ofa weak inhomogeneity, the single scattering differ-ential cross-section is related to Fourier componentsof the dielectric inhomogeneity evaluated at thephoton wave-vector transfer q=kf−ki . For a dielec-tric fluctuation described by the function efluct(r),the light intensity scattered from the incident wavevector ki to the final wave vector kf is given by (inDirac notation) u^kfuefluct(r)uki&u

2 in the first Born ap-proximation.A similar situation is encountered in the scatter-

ing of an electron from a central potential V(uru).14The scattering of the coherent de Broglie wave ofthe electron can be characterized by a series ofphase shifts, dl , for each of the angular momentumpartial waves of an incident plane wave. As a func-tion of the electron energy, the set of phase shiftsprovides a fingerprint of the scattering potentialV(uru). This is based on the decomposition of the

OPTICAL COHERENCE IMAGING

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incident coherent wave of wave vector ki[kz intoangular momentum components:

exp@ ikz#5 (l 50

`

~2l 11 !i l j l ~kr !P l ~ k i• r !. (2)

This is independent of the strength of the potential,and the scattering is usually dominated by a finitenumber l <l max of partial waves.The simplicity of the scattering problem when the

background medium is uniform (and nonscatter-ing) arises from the fact that Green’s function forthe amplitude of a coherent wave of frequency v=ckis given by14

G06~r,r8!52

14p

exp@6ikur2r8#

ur2r8u(3)

and that this is in turn has a straightforward partialwave expansion. If the background is itself a scat-tering medium, the propagation of energy in thewave field may be described in a statistical sense.This follows from multiple scattering theory andensemble averaging over the possible configura-tions of the dielectric disorder eh(r) with an appro-priate statistical weight. The statistical properties ofthe background are given by the translationally in-variant, ensemble-averaged autocorrelation func-tion:

Bh~r2r8![v4

c4 êh0~r!eh~r8!&ensemble . (4)

Green’s function of wave intensity I(R)[ûE(R)u2&ensemble can then be derived using themethods of multiple scattering theory. Here E(R) isthe electric field amplitude of the electromagneticwave point R. For a source at point R8 and a sourceto detector separation L[uR−R8u@l * , where l * isthe transport mean-free-path, the total light inten-sity detected at R is given by Green’s function ofthe diffusion equation:

I~R!51

4pD0uR2R8u. (5)

Here D0 is the optical diffusion coefficient. In thisdescription, the coherent wave properties of the un-derlying electromagnetic field have been integratedover. The intensity (5) described by the diffusionmodel is the wave-vector integral of the Wigner co-herence function11 I(R,k):

I~R![E d3k~2p!3

I~R,k!. (6)

Here k is a wave vector describing local wavepropagation in the direction k and R describes acoarse-grained region with linear dimensionsgreater than the optical wavelength, l, but smallerthan a few transport mean-free-paths, l * . Asshown in the appendix, a generalized coarse-

graining procedure applied to I(R,k), which imple-ments the fundamental uncertainty relationDRDk>1, leads to the conventional specific intensityIc(R,k).It is our aim to generalize the diffusion Green’s

function (5) for the total light intensity, I(R), to theWigner distribution function, I(R,k) and to eluci-date the relationship of Green’s function for theWigner coherence to the autocorrelation functionBh(r−r8). To the extent that I(R,k) can be measuredexperimentally, the nature of the homogeneousscattering medium can be determined with a reso-lution scale given by the wavelength of light l. Thiscan be several orders of magnitude better than theresolution scale (*l * ) afforded by the diffusionmodel.A tumor in biological tissue can be described as a

localized statistical inhomogeneity:

B~r1 ,r2![v4

c4 ê0~r1!e~r2!&ensemble

5Bh~r!1V~R!Bi~r!, (7)

where

r5r12r2 , R5~r11r2!/2.

Here V(R) is a function that vanishes outside of theregion of the tumor and describes its shape andstrength. Bi(r) describes the change in scatteringand absorption within the tumor region from theirvalues in the background. In this paper, we demon-strate that multiple light scattering from a statisticalinhomogeneity of the form (7) can be reduced to aquantum scattering problem and can be describedin terms of a series of partial waves and scatteringamplitudes. Given the nature of the homoge-neously disordered background Bh(r), it is possibleto relate the scattering amplitudes to the overall tu-mor profile V(R) as well as its internal characteristicfunction Bi(r). This leads to a much more detailedcharacterization cell structure within the tumorthan is possible using either a diffusion model orradiative transfer theory. The functions Bi(r) andBh(r) describe in detail absorption and cell struc-ture, inside and outside the tumor region.

2 WAVE PROPAGATIONIN A HOMOGENEOUSLY DISORDEREDBACKGROUND TISSUEWe first review the multiple scattering theory oflight in an isotropic, homogeneously disordered di-electric material.15,16 This is a microscopic wavepropagation theory which is more fundamentalthan radiative transfer theory or Boltzmann trans-port theory.10 It directly relates Green’s function forWigner coherence, Gh(R−R8,k,k8), to the micro-scopic correlation function Bh(r). Here Gh is the cor-relation function of Wigner coherence for lightemitted with wave vector k8 from a source at R8

JOHN, PANG, AND YANG


and the light detected at point R, with wave vectork. This provides much greater information than thediffusion theory that relates the total light intensitycorrelations to the transport mean-free-path l * . Ac-cordingly, multiple scattering theory requires themeasurement of the two-point electric field auto-correlation function Ê* (r1)E(r2)&ensemble rather thanthe one-point intensity ûE(r)u2&ensemble where thetwo-point separation ur1−r2u is small compared to afew transport mean-free-paths. This autocorrelationfunction Ê* (r1)E(r2)&ensemble can be measured ex-perimentally by phase space tomography and thetechnique of fractional-order Fourier trans-forms.11–13

We consider a model dielectric medium consist-ing of a random liquidlike arrangement of identicaldielectric spheres in water.17 The Fourier transformof Bh(r) is given by

Bh~q!5v4

c4ub~q!u2S~q!. (8)

Here ub(q)u2 is the form factor (sometimes referredto as the phase function) describing scattering fromthe individual dielectric spheres and S(q) is thestructure factor describing the statistical arrange-ment of the collection of spheres in the medium.We employ a Percus-Yevick approximation17 to de-scribe S(q). The position and height of variouspeaks in S(q) as a function of q[uqu are sensitivelydetermined by the volume-filling fraction, f, ofspheres. For a dilute (f&0.25) collection of sphereswith refractive index n.1.5, and diameter compa-rable to the optical wavelength, the function Bh(q)contains detailed microscopic information aboutthe dielectric medium on scales small compared tol * . The diffusion model is insensitive to this infor-mation. We describe how multiple scattering spec-troscopy may resolve the medium characteristics atthe scale of the optical wavelength.Consider an extended source in the vicinity of

point R8 containing the points r18 and r28 . The electricfield generated by the extended source at point r8 isdenoted by E0(r8) and R8[(r18+r28)/2. The resultingelectric field amplitude is measured by an extendeddetector in the vicinity of point R=(r1+r2)/2. Theresulting electric field autocorrelation function isgiven by

Ê~r1!E0~r2!&ensemble

5v4

c4 Er18r28^G1~r1 ,r18!G2~r2 ,r28!&ensemble

3E0~r18!E00~r28! . (9)

Here, and throughout this article, we use the no-tation *d3r[*r for coordinate space integrals andthe notation *[d3k/(2p)3][*k for wave-vector inte-grals. For a source-emitting light with initial wavevector k8, the product of electric fields in the sourceregion may be rewritten as

E0~r18!E00~r28!5I0~R8,k8!exp@ ik8•~r182r28!# , (10)

where I0 is the Wigner coherence function of thesource. Green’s functions in Eq. (9) are solutions ofthe wave equation:

F2¹22v62

c2eh~r!GG6~r,r1!5d~r2r18!, (11)

where v6[v+ih with h→0. It follows from Eqs. (9)and (10) that

I~R,k!5ER8,k8

Gh~R2R8;k,k8!I0~R8,k8!. (12)

The microscopic form of the kernel Gh followsfrom carrying out perturbation theory in the effec-tive ‘‘scattering potential’’ (v2/c2)eh(r) and takingthe ensemble average defined by Eq. (4). The detailsof this derivation may be found elsewhere.15,16 Thederivation of the transport equation involves thesummation of an infinite series of terms. Theseterms describe different multiple scattering pathstaken by the photons. In the case of weak scatter-ing, defined by the criterion l!l * , it is reasonableto neglect the interference between different scatter-ing paths on length scales L@l * . Consider for in-stance, the electric field amplitude arriving at thedetector from two independent paths labelled 1 and2, with amplitudes A1 and A2 respectively. Theseamplitudes can be expressed as

An5fn expF i(jqj

~n !•rj

~n !~t !G , n51,2,

where $rj(n)(t)% are the positions of the scattering

events on path n , which can be a function of thetime variable t due to random thermal motion ofthe scatterers. In biological tissue, in addition tothermal fluctuations, nonequilibrium processessuch as the flow of fluids and blood cells lead totime variations in rj

(n). Here, qj(n)s are the related

scattering wave vectors, and fns are constant factorsthat account for the scattering strength of each scat-tering event. The intensity detected is given by timeaveraging of uA1+A2u

2 over the time scale of mea-surement, which we denote as t. In particular,

ûA11A2u2& time5ûA1u21uA2u2& time

1Â1A201A1

0A2& time. (13)

The interference term A1A201A1

0A2 can be eitherpositive or negative with equal likelihood. For themodel of Brownian motion, the result of averagingis given by16

Â1A201A1

0A2& time5A12~0 !exp@2N^q2&Dst# .

Here A12(0) is the value of the interference term att50, ^q2& is the average value of qj

(n)·qj(n) with re-

spect to the scatterer form factor ub(q)u2, Ds is thediffusion coefficient for the scatterers, and N is the



total number of scattering events on both paths.The time scale t0[(^q2&Ds)

−1 is the time for the scat-tering particles to diffuse a distance on the order ofan optical wavelength. It is clear that for Nt@t0 thisinterference term vanishes (with the exception ofpaths contributing to coherent backscattering).18,19

For a diffusive scattering path, N.(L/l )2, where Lis the thickness of the illuminated sample and l isthe scattering mean-free-path. The criterion,t@t0(l /L)

2, for neglecting interference effects maybe violated in certain exceptional cases such as inthe case of ultrashort optical pulse duration t or inthe case of relatively static tissue such as the humantooth.The neglect of interference effects between differ-

ent scattering paths leads to an elementary partialwave expansion for the transport kernel Gh . This isanalogous to the partial wave expansion (2) of theplane wave amplitude in a nonscattering medium.In the present case, it is an expansion of the coher-ence function ^E(r1)E* (r2)&ensemble rather than directamplitude E(r). The result20 of this expansion is

Gh~R2R8;k,k8!

. (m52l max

l max

(l ,l 85umu

l max exp@2uR2R8u/l l l 8@m# ]

4pD l l 8@m# uR2R8u~11ul 2l 8u!

3c l m0 ~k!c l 8m~k8!. (14)

The l =l 8=m50 term corresponds to the isotropicdiffusion mode. D00

[0] is the optical diffusion coeffi-cient, l00

[m]=` and c00(k) is a function that is highlypeaked in the vicinity of k0=uku=v/c and indepen-dent of the direction k .The terms with l =l 8>1 correspond to higher an-

gular momentum partial waves of the coherenceGreen’s function and, in general, the spectral func-tions take the form

c l m~k!5R l ~k !Y lm~ k !, (15)

where Y lm( k) is a spherical harmonic and R l (k) is a

‘‘radial’’ function that is highly peaked in the vicin-ity of k0 . For l =l 8>1, the length scale parametersl l l 8

@m# are finite. For instance, l11[0] is on the order of

several transport mean-free-paths and, in general,l l l

@m# decreases when l is increased. The l Þl 8terms in (14) correspond to the transitions amongdifferent modes, and the parameters l l l 8

@m# are finite,in general, except for those with l or l 8=0. (Whenone of the l and l 8 is equal to zero, l l l 8

@0#5 `). The

parameters l l l 8@m# for l ,l 8>1 describe the length

scale on which nondiffusive coherence effectspropagate through the medium on average. Onvery long length scales uR−R8u@l11

[0] , only the diffu-sion mode persists, whereas on shorter lengthscales coherence properties are observable. Equa-tion (14) therefore provides a microscopic interpo-lation scheme from the ‘‘diffusive,’’ to ‘‘snakelike,’’

to ‘‘ballistic’’ photons (in the purely particle pic-ture) as the source to detector separation is de-creased, or equivalently the time gating6 of detectedphotons is changed. The length scales l l l 8

@m# , theweight factor D l l 8

@m# , and the spectral functionsc l m(k) can be microscopically related to the char-acteristic function of the medium Bh(r).Equation (14) is derived by using the multiple

scattering theory.15,16 We first define

Gh~Q;k8,k![k024E d3R exp@2iQ•R#Gh~R,k8,k!,

k0[v/c . (16)

Here, the input and output photon wave vectors k8and k may be thought of as continuous indices ofthe matrix Gh , and Q as the ‘‘inverse’’ source-detector vector in reciprocal space. The matrix Ghdescribes the transfer of radiation at wave vector k8to wave vector k through a sequence of intermedi-ate wave vectors. Multiple scattering theory15,16 (ne-glecting the interference of different radiative trans-fer paths) leads to the result that Gh is given by theinverse of the operator (matrix):

H[fQ~k!2Bh~r!, (17)

where

fQ~k!5Fk022S k1Q2 D 22S1S k1

Q2 D G

3Fk022S k2Q2 D 22S2S k2

Q2 D G

and

S6~k!5EqBh~k2q!/@k0

22q22S6~q!# .

In particular,

Gh~Q;k8,k!5^k8uH21uk& . (18)

The operator H is analogous to the Hamiltonian ofa quantum mechanical particle moving in the po-tential well 2Bh(r) with an energy versus wave vec-tor (dispersion) relation given by fQ(k). The trans-port kernel (14) follows from solving (18) by usingthe eigenfunctions of H at Q=0 as basis functions.The ‘‘ground state’’ energy of H at Q50 is rigor-ously equal to zero (in the absence of absorption).This corresponds to the isotropic and undampeddiffusion mode.We find that the eigenvalue spectrum of H at

Q50 is especially simple in the case of (weak) scat-tering pertinent to biological tissue. In general, theeigenvalue spectrum of a three-dimensional poten-tial well consists of bound states and continuumstates. The bound states are labelled by a principalquantum number n and angular momentum quan-



tum numbers l and m. In general there are manypossible values of n in the bound state spectrum.For the model of identical dielectric spheres in wa-ter, with statistical correlations described by thePercus-Yevick approximation, we find that thebound state spectrum contains only the n50 prin-ciple quantum number in a certain parameter re-gion, as shown in Figure 1 (the region under eachcurve) for spheres with a relative refractive index1.09 (the solid line) and 1.193 (the dashed line). Inthis figure, a is the radius of the spheres and f is thevolume-filling fraction of spheres. For a 10% vol-ume fraction of polystyrene spheres in water (witha relative refractive index 1.193), it requires thata&4l (where l=2p/k0 is the optical wavelength) inorder to have only the n50 bound states. This es-sentially requires that the scattering be weak andthat the scattering microstructures be no more thanan order of magnitude larger than the wavelengthof light. In constructing the transport kernel (14) wehave included only the n50 bound states. The con-tribution from continuum states of H to the kernel(14) is negligible, since these modes are exponen-tially damped on the scale of the wavelength.The parameters l l l 8

@m# and D l l 8@m# in the kernel (14)

are determined by the correlation function Bh(r)and they can be calculated numerically once thefunction Bh(r) is given. Table 1 gives an example forthis calculation. In this example, we consider themodel of identical polystyrene spheres* in water,with statistical correlations described by the Percus-Yevick approximation. We choose the radius of the

spheres a520k021 and the volume-filling fraction

f=0.1 in the calculation. The l and l * in Table 1are the related scattering mean-free-path and trans-port mean-free-path, respectively. We see that l11

[0] isthe largest among all l l l 8

@m# and it is about twentytimes larger than the transport mean-free-path! Thismeans that the l =l 8=1 and m50 term in (14) canpropagate for a distance much longer than thetransport mean-free-path in this example. For thisparticular choice of a and f, we find thatD00

[0]/D11[0]. .054. In other words, the l =1 mode am-

plitude is only 1.5 orders of magnitude smaller thanthe l =0 mode. From Table 1, we also see that l22

[m]

and l33[m] are much smaller than l11

[0] , while someoff-diagonal elements, such as l13

[0] , are close to l11[0]

in value. If we change the function Bh(r) by chang-ing, for instance, the radius or the refractive indexof the spheres, the l11

[0] and the other coherencelengths will change their values accordingly. Table2 shows how the quantities l11

[0]/l * and l33[0]/l *

change when we change the parameters a, f and De(where De is the ratio of the dielectric constant ofthe spheres to that of water). We see from Table 2that, for fixed f and De, l11

[0]/l * first increases andthen decreases when we increase parameter a, anda520k0

21 (with f=0.1 and De=1.423) corresponds tothe maximum value of l11

[0]/l * in this table. We alsosee that, in Table 2, l11

[0]/l * increases when we in-crease f or De. Owing to the dependence of l l l 8

@m# onthe function Bh(r), it is possible to use the observ-able parameters l l l 8

@m# as well as D l l 8@m# to characterize

the medium properties contained in Bh(r).We mention that the neglect of interference be-

tween different radiative transfer paths in (13) is agood approximation with the exception of the back-scattering direction. Here it is well known that thecoherent interference of ‘‘time-reversed’’ opticalpaths can give rise to an intensity peak in retrore-flection from a disordered, dielectric half-space.18,19

This peak intensity is up to twice the ‘‘diffuse’’background and has an angular width that is of theorder (l/l * ) radians. Accordingly, the interferencecorrections to the transport theory we have out-lined are of the order (l/l * )2 . For biological tissue,with a transport mean-free-path l *;1 mm and atwavelengths l;1 mm, this correction is very small.Nevertheless, a generalization of (17) to include theeffects of coherent backscattering is possible. This,however, leads to the complication that the local‘‘potential’’ Bh(r) must be replaced by a nonlocal‘‘potential.’’20

In the above paragraphs we described wavepropagation in an isotropically disordered homoge-neous background without absorption. In the pres-ence of absorption, the dielectric constant eh(r) is nolonger real and has an imaginary part. It can beshown20 that the corresponding Green’s functionfor Wigner coherence in this case is of the sameform as (14) but the coherence lengths l l l 8

@m# , espe-*The ratio of the dielectric constant of polystyrene spheres tothat of water is 1.423.

Fig. 1 Parameter region (the one under each curve) in which thebound state spectrum contains only the n50 principal quantumnumber. The calculation was carried out for the model of identicaldielectric spheres in water, with relative refractive indexes 1.09(the solid line) and 1.193 (the dashed line). f is the volume-fillingfraction and a is the radius of the spheres.



cially, l00[0] , l0l

@0# and l l 0@0#, now are all finite. This is

because the ground state energy of H0 is no longerequal to zero in dissipative media.We note, finally, that in this paper we have con-

sidered only the case of time-independent, steady-state imaging. For frequency-domain or time-domain imaging, we need to consider the time-dependent generalization of Eqs. (16) to (18). Forexample, to describe modulation of the source in-tensity at the frequency V, it is necessary to evalu-ate the advanced and retarded self-energies S6 atfrequencies ck0+V/2 and ck0−V/2 respectively,rather than at the same frequency ck0 . A detaileddiscussion of frequency-domain and time-domainmeasurements is given in Ref. 20.

3 COMPARISON WITH CONVENTIONALRADIATIVE TRANSFER THEORYIn this section we compare our wave theory de-scribed in the previous section with conventionalradiative transfer theory. As mentioned earlier, ourtheory is more fundamental than conventional ra-diative transfer theory.10 Here, we elucidate the im-portant differences. For this purpose, we first re-write (18) as the following integral equation20

2k·QGh~Q;k,k8!

5DGk~Q!dkk81Ek1

DGk~Q!Bh~k2k1!Gh~Q;k1,k8!

2S Ek1

DGk1~Q!Bh~k2k1! D Gh~Q;k,k8!, (19)

where

DGk~Q![G1~k1Q/2!2G2~k2Q/2!,(20)

G6~k![@k022k22S6~k!#21.

Equation (19) is equivalent to (18) and can be de-rived by using (17) and (20).On the other hand, in conventional radiative

transfer theory, the (time-independent) specificlight intensity Ic(R,k) (where the index c denotesconventional radiative transfer theory) of a homo-geneous medium without absorption satisfies thefollowing phenomenological Boltzmann transportequation9,10

Table 1 Numerical results of the coherent lengths ll l 8@m# , the scattering mean-free-path l , and the trans-

port mean-free-path l * for the model of identical polystyrene spheres in water, with statistical correlationsdescribed by the Percus-Yevick approximation.

l l * m l11[m] l22

[m] l33[m] l12

[m] l13[m] l23

[m]

7.4 6.03102 0 .1.13104 2.03102 2.03102 .7.63103 .8.53103 2.03102

1 3.63102 4.73102 1.93102 4.63102 .1.23103 4.63102

Note: The radius of the spheres and the volume-filling fraction are taken to be 20k021 and 0.1, respectively. We setl max=3 in the calculation. All lengths are in units of k021 .

Table 2 Ratios of coherence lengths l11[0] and l33

[0] to transport mean-free-path l * versus parametersf—the volume-filling fraction, a—the radius of the spheres, and De—the ratio of dielectric constant of thespheres to that of water.

Parameters

l k0 l *k0 l11[0]/l * l33

[0]/l *De f ak0

1.423 0.1 10 2.03101 4.93102 12.8 0.21

1.423 0.1 20 7.4 6.03102 18.0 0.33

1.423 0.1 27 8.9 1.23103 11.4 0.26

1.193 0.1 20 4.53101 3.73103 3.0 0.21

1.1 0.1 20 1.83102 1.53104 0.7 0.15

1.193 0.3 20 2.73101 1.43103 5.0 0.24

1.193 0.05 20 7.93101 7.23103 1.5 0.20



~k•“R!Ic~R,k!1ks~k !Ic~R,k!

5I0c ~R,k!1E

k8k8s~k8→k!Ic~R,k8!, (21)

where s(k) and s(k8→k) are the total and angularscattering coefficients, respectively, and satisfy therelation

Ek8

s~k→k8!5s~k !. (22)

In (21), I0c (R,k) is the source-specific intensity. Now,

if we define

Ic~R,k!5ER8k8

Ghc ~R2R8;k,k8!I0

c ~R8,k8!, (23)

then we can show from (21) that the Fourier trans-form of Gh

c (R−R8;k,k8) satisfies

ik·QGhc ~Q;k,k8!5dkk81E

k1k1s~k1→k!

3Ghc ~Q;k1 ,k8!2ks~k !

3Ghc ~Q;k,k8!. (24)

It can be easily seen that (19) and (24) are equiva-lent if we ignore the Q dependence of DGk(Q) in(19) and identify

Gh~Q;k,k8!↔Ghc ~Q;k,k8!@ iDGk8~0 !#/2,

s~k !↔12kEk1Bh~k2k1!@ iDGk1

~0 !#52ImS1~k !/k ,

s~k8→k!↔12k8

@ iDGk~0 !#Bh~k2k8!. (25)

Therefore, the conventional radiative transfertheory ignores† the Q dependence of DGk(Q),which corresponds to a nonlocal interaction in po-sition space, as shown in Fig. 2 where we plot theFourier transform of iDGk(Q) for some fixed k’s inthe forward direction [in the backward direction,with k·R<0, the Fourier transform of iDGk(Q) os-cillates rapidly on the scale of the wavelength]. Ow-ing to the neglect of Q dependence, the propagatorGh(R−R8;k,k8) given by conventional radiativetransfer theory differs from that obtained from ourwave theory in the following two respects20: (1) Therelated terms with m50,l =1 and l 8>1 as well asthe terms with m=0,l 8=1 and l >1 in (14) are com-pletely missing in the conventional radiative trans-fer theory. In particular, the term with m50 andl =l 8=1, which can usually propagate for a much

longer distance than the transport mean-free-path,is absent in radiative transfer theory. This is themost significant difference between the results ofthese two theories. (2) The values of parametersl l l 8

@m# and D l l 8@m# given by the radiative transfer

theory for other remaining terms are, in general,not the same as those given by our wave theoryexcept for the terms with l or l 8=0. When one of l

†One consequence of ignoring the Q dependence of DGk(Q) inthe conventional radiative transfer theory is that the diffusioncoefficient D0 , in the limit of length scales much larger than l* ,comes only from the coupling between c00 and c01 , the groundstate and the first excited state of H at Q50.

Fig. 2 Fourier transform of iDGk(Q), namely, DGk(uRu)[ *Qexp@iQ • R#@ iDGk(Q)# for the case when: (a) k·R/uRu=1.0k0 ;(b) k·R/uRu=0.99k0 ; (c) k·R/uRu=0.80k0 . It was calculated for themodel of identical polystyrene spheres in water with f=0.1 anda520k021 . Note that DGk(uRu) is real because from Eq. (20) wehave@DGk(Q)#1 5 2DGk(2Q).



and l 8 is equal to zero, the radiative transfer theoryalso gives l l l 8

@0#5 ` (in the absence of absorption).

The above discussion shows that the conven-tional radiative transfer theory is an approximationto our wave theory, and the difference betweenthese two theories remains important on scaleslonger than the transport mean-free-path. Nonlocalcorrections to particlelike transport of photons ap-pear even without consideration of the interferenceof different diffusion paths. These nonlocal correc-tions arise from the fact that wavelike coherencepersists on long length scales in a medium with anontrivial underlying structure. This would be ab-sent in a ‘‘white noise’’ model consisting of spa-tially uncorrelated point scatterers. The diffusionmodel is insensitive to the important distinctionsbetween white noise disorder and a scattering me-dium with specific cellular structures arranged in aspecific order. An extreme illustration of this effectis that of a periodic dielectric microstructure. Here,wavelike coherence persists on an infinite lengthscale due to perfect coherent interference betweendifferent single scattering events, giving rise to apropagating Bloch wave. Biological tissue is an in-termediate case in which this wave coherence mayextend on scales that are large compared with thetransport mean-free-path.We mention finally that coherent backscattering

effects18,19 may be systematically incorporated intowave theory by replacing Bh(k−k1) in Eq. (19) by an‘‘irreducible vertex function,’’ Uk,k1

(Q).16 This ver-tex function has a perturbation expansion in thesmall parameter (l/l * )2. The leading term in thisexpansion is in fact Bh(k−k1). The Q dependence,obtained at a higher order in perturbation theory,gives rise to further nonlocality in the resulting ra-diative transfer equation.

4 MULTIPLE LIGHT SCATTERING NEARAN INHOMOGENEITY

In Sec. 2 we described the relationship between thecoherence propagator Gh and the underlying dielec-tric microstructure characterized by the translation-ally invariant function Bh(r−r8)[v4/c4^eh

0(r)3eh(r8)&ensemble . In the presence of a localized inho-mogeneity, the dielectric autocorrelation is nolonger translationally invariant. This loss of transla-tional symmetry is related to the function V(R)which appears in Eq. (7) and which describes thefixed location and shape of the inhomogeneity. Weconsider, for simplicity, an inhomogeneous regionthat is located near some central point R0 . The in-homogeneity may absorb light and may have dif-ferent scattering characteristics than the homoge-neous background. We assume that the internalcorrelations of the inhomogeneity, like the back-ground, are isotropic after ensemble averaging.That is to say, Bi(r)=Bi(r) where r[uru. It is useful todefine the matrix

b l m ,l 8m8[^c l muBiuc l 8m8&. (26)

From the isotropy assumption for Bi(r), it followsthat this matrix is diagonal. We define

b l m ,l 8m8[b l d l l 8dmm8 . (27)

This together with the fact that only the n50 prin-cipal quantum number states contribute to Gh(weak scattering), leads to the result that the inho-mogeneity does not mix the partial waves with dif-ferent n or m indexes of the coherence propagatorGh . It is useful to introduce a (l max−umu)3(l max−umu)matrix Gh

[m](R−R8) whose (l ,l 8)th element is de-fined by

~ Gh@m#~R2R8!! l l 8[

1

4pD l l 8@m#

exp@2uR2R8u/l l l 8@m#

#

uR2R8u~11ul 2l 8u!.

(28)

It follows from the above assumptions that the fullcoherence propagator (including the inhomogene-ity) can be expressed in the form

G~R,R8;k,k8!5 (m52lmax

l max

(l ,l 85umu

l max

G l l 8@m#

~R,R8!

3c l m0 ~k!c l 8m~k8!, (29)

where G l l 8@m# (R,R8) is the (l ,l 8)th element of a

(l max−umu)3(l max−umu) matrix G [m](R,R8), whichwill be determined.In the absence of the inhomogeneity [V(R)=0],

G [m](R,R8) reduces to Gh[m](R−R8). The presence of

the inhomogeneity, however, removes the transla-tional symmetry of the problem. Using the same setof approximations which were used in the deriva-tion of Gh , namely the noninterference of differentradiative transfer paths, it is straightforward toshow20 that G [m](R,R8) satisfies the following linearintegral equation:

G@m#~R,R8!5Gh@m#~R2R8!1E

R9Gh

@m#~R2R9!

3V~R9!b @m#G@m#~R9,R8!, (30)

where b [m] is a constant diagonal matrix whose l thdiagonal element is b l , which describes the inter-nal, ensemble-averaged microstructure of the inho-mogeneity. For a spherical tumor of radius b, cen-tered at R0 , the ‘‘potential’’ V can be described bythe function

V~R!5Q~b2uR2R0u!, (31)

where the step function is defined by Q(x)=1 forx>0 and Q(x)=0 for x,0.The integral equation (3) for the inhomogeneous

propagator is remarkably similar to the Lippmann-Schwinger equation14 describing the quantum me-chanical scattering of a particle from a static poten-tial well. For the quantum particle, Gh

[m](R−R8) in



Eq. (30) would be replaced by the free-particlepropagator (3) and V(R9) would be replaced by thequantum mechanical scattering potential. Theensemble-averaged, multiple scattering problemdiffers from the conventional quantum scatteringproblem in two fundamental respects. First of all,Eq. (30) constitutes a set of (l max−um u)3(l max−um u)coupled equations. Second, the ‘‘background’’propagators Gh

[m] are damped rather than purely os-cillatory, since the higher order partial waves of thecoherence function fail to propagate in the disor-dered medium on scales much, much greater thanthe transport mean-free-path l * . It is significant tonote, however, that the damping length scales areas large as twenty times the transport mean-free-path in some of the models we have studied. Fi-nally, we mention that in the case of strong scatter-ing by the homogeneous background (violating thefact that only the n50 principle quantum numberstates contribute to Gh) or in the case of anisotropicinternal correlations within the inhomogeneity[Bi(r)ÞBi(r)], the matrix bnl m ,n8l 8m8 [generaliza-tion of Eq. (26)] has off-diagonal elements. Thisleads to a coupling between the partial wave propa-gators in Eq. (30) for different values of m and n .The solution of the coupled system of integral

equations is similar in structure to the solution ofthe Lippmann-Schwinger equation in quantum me-chanics. We consider the asymptotic behavior ofEq. (30) when the source and detectors are far fromthe inhomogeneity (uR−R0u@b and uR8−R0@b). Forthis purpose, we rewrite Eq. (30) as follows:

G@m#~R,R8!

5Gh@m#~R2R8!1E

R9Gh

@m#~R2R9!

3V~R9!b @m#Gh@m#~R92R8!

1ER9R-

Gh@m#~R2R9!V~R9!b @m#G @m#~R9,R-!

3V~R-!b @m#Gh@m#~R-2R8!. (32)

In the limit uR−R0u@b , we may use the fact thatuR9−R0u!uR−R0u to rewrite uR−R9u=uR−R0−(R9−R0)uin (32), and then Taylor expand this expression inthe small parameter uR9−R0u. Similar expansion alsoapplies to the limit uR8−R0u@b .These expansions yield the asymptotic behavior

of the partial-wave coherence propagator whenboth source and detector are far from the inhomo-geneity:

G l l 8@m#

~R,R8!

.1

4pD l l 8@m#

exp@2uR2R8u/l l l 8@m#

#

uR2R8u11ul 2l 8u

1 (i ,j5umu

l max

f l l 8,ij@m# (u ,i/l l i

@m# ,i/l jl 8@m#)

3exp@2uR2R0u/l l i

@m#2uR82R0u/l jl 8@m#

#

4pD l i@m#Djl 8

@m#uR2R0u11ul 2iuuR82R0u11ul 82ju.

(33)

Here, cos u[(R0−R8)·(R−R0)/(uR0−R8uuR−R0u) de-scribes the angle between the line connecting thesource to inhomogeneity and the line connectingthe inhomogeneity to the detector. The functionf l l 8,ij@m# (u ,i/l l i

@m# ,i/l jl 8@m#) is the analog of a scattering

amplitude in quantum mechanics. A straightfor-ward calculation leads to the following formal ex-pression for this scattering amplitude:

f l l 8,ij@m#

~u ,i/l l i@m# ,i/l jl 8

@m#!514pEr9

exp@r9•R/l l i@m##

3V~R01r9!@ b @m#

3C~m ,l 8!~R01r9!# ij ,

(34)

where R[(R−R0)/uR−R0u and the ‘‘scattering wavefunction’’ satisfies the matrix integral equation:

C~m ,l 8!~R!5E ~m ,l 8!~z !1ER9

Gh@m#~R2R9!

3V~R9!b @m#C~m ,l 8!~R9!. (35)

Here, z[(R8−R0)·(R−R0)/uR8−R0u, and @ E (m ,l 8)

3(z)# ij 5 exp@z/ljl 8@m#d ij# . Equation (35) is a matrix

type of Lippmann-Schwinger equation analyticallycontinued to imaginary wave vectors i/l l l 8

@m# .The above analysis demonstrates the formal rela-

tionship between scattering from a statistical inho-mogeneity in a homogeneous multiple-scatteringbackground and a quantum mechanical type ofscattering governed by Eq. (35). The measurementof the scattering amplitude f l l 8,ij

@m# (u ,i/l l i@m# ,i/l jl 8

@m#)coupled with knowledge of the parameters l l l 8

@m#

and D l l 8@m# (from measurement of transport in the

homogeneous background) leads to a completecharacterization of the inhomogeneity (tumor). Thescattering amplitude in this illustration may bemeasured by using a coherent light source at onespot on the sample surface and then performingphase-space tomography at a large number of otherspots along the sample surface. Reconstruction ofthe Wigner coherence function at any given spotrequires a large number of intensity measurementsat the spot. A second level of reconstruction con-sists of evaluating V(R) and Bi(r) from comparisonof the reconstructed Wigner function between alarge number of different spots along the samplesurface. Reconstruction of the tumor profile V(R)



and the tumor’s internal characteristic functionBi(r) reduces to a quantum inverse scatteringproblem.21

5 DISCUSSION

We have outlined a formal analogy between wavepropagation in a multiple scattering medium with astatistical inhomogeneity and the quantum me-chanical scattering of a particle by a localized po-tential. Our multiple scattering theory is more fun-damental than conventional radiative transfertheory in which photons are treated as classical par-ticles. In this analogy, the disorder-averaged coher-ence function Ê* (r1)E(r2)&ensemble plays the role of ascattering wave function. The coordinateR[(r1+r2)/2 corresponds to the center of mass ofthe quantum particle and the relative coordinater[r1−r2 corresponds to some internal degree offreedom of the quantum particle. By taking theFourier transform of the coherence function withrespect to r, we obtain the Wigner function I(R,k).This function is the wave analog of the specific lightintensity that is used in a Boltzmann-type transporttheory. Alternatively, the spectral content of the ra-diation field at R may be described in terms of a setof ‘‘eigenfunctions’’ C l m(k). The precise form ofthese eigenfunctions is determined by the homoge-neous dielectric autocorrelation functionBh(r)[v4/c4êh

0(r)eh(0)&ensemble . In particularc l m(k) are the Fourier transforms of the bound-state wave functions in the ‘‘potential’’ 2Bh(r). Ifthe initial mode at point R8 is c l m(k), then, as theradiation propagates from point R8 to R, the inten-sity remaining in the initial mode c l m(k) decreasesby the factor exp@ 2 uR 2 R8u/l l l

@m##/(4pD l l@m#uR

2 R8u). In many cases of physical importance, thecoherence length l11

0 may exceed the transportmean-free-path by an order of magnitude. The pa-rameters l l l

@m# and D l l@m# are related to the ‘‘energy

eigenvalues’’ of the potential 2Bh(r). Here, l00[0]=`

and D00[0] is the optical diffusion coefficient. In the

quantum mechanical analogy, Bh(r) plays the roleof an internal binding potential for the (composite)particle and the c l m are the ground and excitedstates of the particle. Depending on the nature ofthe coherent light source, some superposition of theground (diffusion mode) and excited states may becreated. As the particle propagates from point R8 toR, the higher excited states are damped and theinternal state of the particle relaxes to the groundstate.In the presence of a statistical inhomogeneity,

V(R)Bi(r), the center of mass of the composite par-ticle experiences an external potential proportionalto V(R). When the internal state of the particle isgiven by c l m, the external potential from which theparticle scatters is given by ^c l muBiuc l m&V(R).The above analogy and the mathematical formal-

ism associated with it opens the possibility of high-

resolution multiple light scattering tomography.The propagation of the total intensity fieldûE(r)u2&ensemble probes the nature of the dielectricmedium on scales that are large compared with thetransport mean-free-path l * . The internal structureof the coherence function Ê* (r1)E(r2)&ensemble probesthe structure of the dielectric medium on scalesranging from l * down to the optical wavelength.These considerations clearly underscore the en-hanced sensitivity and resolving power of coherentmultiple light scattering tomography beyond thatallowed by the diffusion model.

APPENDIX: RELATIONSHIP OF THEWIGNER DISTRIBUTION FUNCTION TOTHE SPECIFIC LIGHT INTENSITY

The Wigner distribution function, I(R,k), defined inEq. (1) as the electric field mutual coherence func-tion is not positive definite. This differs from thespecific light intensity, Ic(R,k), of conventional ra-diative transfer theory, defined by Eq. (21). The spe-cific intensity is a measure of the number of pho-tons in a vicinity of the point R travelling in thedirection of the wave vector k and is by definitionpositive definite. As a consequence of the wave na-ture of the photon, the specific intensity Ic(R,k), re-fers to coarse-grained regions in coordinate space Rand momentum space k, such that the fundamentaluncertainty relation DRiDki>1, i5x ,y ,z . We dem-onstrate briefly in the following paragraphs that asuitable coarse graining of the Wigner distributionleads to a positive definite distribution which weidentify with the specific light intensity. More de-tails on the subject of coarse graining of the Wignerfunction can be found in Refs. 22 and 23.Consider the coarse-graining function

r~R,k;R8,k8![1

p3 exp@2~R82R!2/~DR !2!]

3exp@2~k82k!2/~Dk !2# (36a)

with the constraint that

DRDk51. (36b)

It is straightforward to verify that

E d3Rd3kr~R,k;R8,k8!5~DRDk !351. (37)

Applying this coarse-graining operation to theWigner distribution, we obtain



Ic~R,k!5E d3R8d3k8r~R,k;R8,k8!I~R8,k8!

51

p3E d3Rd3rd3k8r~R,k;R8,k8!

3exp@2ik8•r#Ê0~R81r/2!

3E~R82r/2!&ensemble . (38)

Using the definition of the coarse-graining function(36) and making the change of variables u=R8+r/2and v=R8−r/2, we may integrate (38) with respectto k8 to obtain:

Ic~R,k!5F ~Dk !2

p G3/2E d3ud3vÊ0~u!E~v!&ensemble

3exp@2ik•~u2v!#

3expF2~Dk !2

4~u2v!22

~u1v22R!2

4~DR !2 G .(39)

Using the fact that (Dk)2=1/(DR)2, it follows that

Ic~R,k!51

p3/2~DR !3E d3ud3vÊ0~u!E~v!&ensemble

3exp@2ik•~u2v!#

3expF2~u2R!21~v2R!2

2~DR !2G

51

p3/2~DR !3K U E d3uE~u!exp@2ik•u

2~u2R!2/@2~DR !2##U2Lensemble

,

which is indeed positive definite. We identify thiscoarse-grained version of the Wigner distributionfunction with the specific light intensity of conven-tional radiative transfer theory.

AcknowledgmentsWe are grateful to Prof. Michael Raymer for a num-ber of stimulating discussions. This work was sup-ported in part by the Ontario Laser and LightwaveResearch Centre and the Natural Sciences and En-gineering Research Council of Canada.

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