Propagator model including multipole fields for discrete random media

Vol. 2, No. 12/December 1985/J. Opt. Soc. Am. A 2195

Propagator model including multipole fields for discreterandom media

V. V. Varadan, Y. Ma, and V. K. Varadan

Laboratory for Electromagnetic and Acoustic Research, Department of Engineering Science and Mechanics, ThePennsylvania State University, University Park, Pennsylvania 16802

Received May 29, 1985; accepted August 14, 1985

A propagator model using Feynman diagrams is presented for studying the first and the second moments of theelectromagnetic field in a discrete random medium. The major difference between our work and previous treat-ments of this type is that all diagrams are in a basis of vector spherical functions. Each propagator or infinite-medium Green's function is the translation matrix for spherical functions, and each scatterer is characterized by a Tmatrix that, in turn, is a representation of the Green's function of the scatterer in a basis of spherical functions. Allorders of multipoles are formally retained, in contrast to previous work involving the dipole approximation. Partialresummations of the scattering diagrams are shown to be related to the quasi-crystalline approximation and thefirst-order smoothing approximation. The lowest-order term of the ladder approximation for the incoherentintensity is evaluated. Sample numerical results are presented and compared with available experimental results.

INTRODUCTIONWe consider the propagation of plane coherent electromag-netic waves in an infinite medium containing identical, loss-less, randomly distributed particles. Our aim here is tocharacterize the random medium by an effective complexwave number K (which would be a function of particle con-centration, the electrical size, and the statistical descriptionof the random positions of the scatterers) and to study bothcoherent and incoherent intensities as a function of frequen-cy for various values of the concentration c (the fractionalvolume occupied by the scatterers). Although the formula-tion is generally valid for nonspherical, aligned, or randomlyoriented scatterers, initial calculations are confined tospherical scatterers, which generally give us a better pictureof the order of magnitude of the different contributions tothe intensity without the additional complications of non-spherical geometry and orientation.

Extensive work by Twerskyl-5 has laid the foundation formultiple-scattering theory in discrete random media. Arelated approach using the T matrix of a single scatterer6

together with configurational averaging procedures havebeen used by the authors to develop a computational meth-od for the electromagnetic-wave-propagation problem in in-homogeneous media.7 -9 Lax's1 0 quasi-crystalline approxi-mation (QCA) is used in conjunction with suitable modelsfor the pair correlation function to obtain an effective wavenumber K(= K, + iK2) that is complex and frequency de-pendent. In a classic paper, Frisch" has demonstrated for-mally the relationship between the present problem and itsanalog in quantum mechanics. He used Feynman diagramsto show that the mean Green's function for the randommedium is analogous to the Dyson equation, whereas thesecond moment of the Green's function is analogous to theBethe-Salpeter equation. More interestingly, he showedthat retaining only two-body correlations and summing allterms involving sequential scattering and correlation isequivalent to the first-order smoothing approximation first

introduced by Bourret1 2 and used by Keller13 and his co-workers.

In this paper, we represent the multiple-scattering seriesin a basis of vector spherical functions using the T matrix tocharacterize each scatterer. Formally, the T matrix in-cludes a detailed description of the scatterer to all orders in amultipole expansion, this in contrast to previous treatmentsthat invoked the dipole approximation. The so-called prop-agator or Green's function that propagates the signal fromone scatterer to the next is again represented in a basis ofvector spherical functions. This again is a consequence ofnot invoking the dipole approximation to describe the scat-terer. Thus instead of simply using a full-space Green'sfunction of the form exp(ikr)/kr, as would be appropriate forpoint scatterers, the propagation matrix used here describesthe propagation of a complex, multipole field to the nextscatterer.

The partial resummations that can be performed by re-taining only two-body correlations including only sequentialscattering lead to so-called dressed propagators. Thesepropagators describe a medium with a different propagationconstant. One of the new observations we make is that theQCA first used by Lax10 and the first-order smoothing ap-proximation of Bourret12 and Keller13 appear to be the re-summation of the same class of diagrams. This does notseem to have been commented on before.

In addition, the intensity of the electromagnetic field isalso represented with the help of Feynman diagrams. Theso-called coherent intensity is the summation of all diagramsthat involve two independent field lines, whereas the inco-herent intensity is the summation of all diagrams that in-volve two distinct field lines intercepting different scatterersbut that are, however, coupled by positional correlationsbetween scatterers. The lowest approximation to the inco-herent intensity leads to the ladder diagrams. We also referto the work by Tsang and Kong,14 who have computed thebackscattered intensity in single-scattering approximation.

0740-3232/85/122195-07$02.00 © 1985 Optical Society of America

Varadan et al.

2196 J. Opt. Soc. Am. A/Vol. 2, No. 12/December 1985

Sample numerical results are presented for the first andthe second moments of the field and compared with theexperiments of Killey and Meeten. 15

MULTIPLE-SCATTERING FORMULATION

Consider wave propagation in an infinite medium of volumeV - containing a random distribution of N scatterers, N

a, such that no = N/V, the number density of scatterers,is finite. Plane harmonic waves of frequency X propagate inthe medium and undergo multiple scattering. Let E, EO,Ee, and Eis denote, respectively, the total field, the incidentfield, the field exciting the ith scatterer, and the field scat-tered by the ith scatterer. Then self-consistency requiresthe following relationships among the fields7-9 :

N

E = E0 + > Eisi=l

(1)

andN

Ee = E° + E E sj~i

(2)

Let °44,, generally denote outgoing functions (Hankelfunctions) and functions regular at the origin (Bessel func-tions). We dispense with vector notation, and the abbrevi-ated index may denote n - T, 1, m, a; T = 2, 3; 1 e [0, -)]; m e

[0, 1]; see Refs. 7-9.At a field point r in the host medium, the incident, scat-

tered, and exciting fields are expanded as follows:

E0 (r) = E3 an Re tn(r), (3)

r= fi + r+ jp j

A B t

C

p

iHi E

E

Fig. 1.cesses.

P4. idj

k

D

4° kpJ

P

E

A diagrammatic representation of multiple-scattering pro-

If we substitute Eqs. (7) and (5) in Eq. (1) and iterate, weobtain

E(r) = E0 (r) + >3 Ou V,4,(r -ri)Tnna,

ii

Ee(r) = E an' Re ,n(r - ri), I -ri - <2a,n

ES(r) =Efni OU An(r - r), (r -ri), Jr -ril| > 2a,n

(5)

where ri denotes the center of the ith scatterer and a is theradius of the sphere circumscribing any scatterer. The coef-ficients an are known, whereas the coefficients fni and ani areunknown but are, however, related through the T matrix6 :

fni- Tnnan'. (6)n,

Substituting Eqs. (3)-(6) into Eq. (2) and using the trans-lation-addition theorems for spherical wave functions andthe orthogonality properties of spherical harmonics, 6 we ob-tain

an2 = an +> 33 nn'(ro - rdTn'nn- (7)jni n'

where

Ou 1n(r - ri) = >3 ann,(r, - rj)Rei,, 0 (r - r,) (8)

n'

and an,, is the translation matrix for spherical wave func-tions.

+ (4)'OU ,n(r-ri)Tnn, . . + .k i i k

where rij = (rj - ri).The first term in Eq. (9) is the incident field reaching the

observation point r denoted by P in Fig. 1A. The secondterm of Eq. (9) is a sum of N contributions, each of which canbe represented by a diagram of the type shown in Fig. 1B.The thin line represents the incident field Eo, and the thicksolid line represents the propagator Ou 4n(r - ri)Tnn' thatpropagates the field from scatterer i to observation point r.The sum of all N diagrams of this type is termed singlescattering. The third term of Eq. (9) is a sum of N(N - 1)contributions, each involving a pair of particles, and is repre-sented by the diagram of Fig. 1C. There are also N(N - 1)terms of the form given in Fig. iD that involve only a pair ofparticles. There are N(N - 1)(N - 2) terms of the typeshown in Fig. 1E. As seen from Fig. 1, the three-body pro-cess can include any number of scattering in any orderamong the three objects.

FIRST MOMENT OF THE FIELD AND THEQUASI-CRYSTALLINE APPROXIMATION

Equation (9) can be averaged over the positions of the parti-cles to yield the coherent, average, or first moment of thefield:

Varadan et al.

__Ple +...to


(E(r)) = E0 (r) + E | Ou 4n(r - ri)a ip(ri)dri

+>33 Tnn'Tnn J Ou Vn(r - ri) J _n'n(ri)an-JP(ri)iijX p(rJ Iri)drjdr + . .. , (10)

which involves all orders of joint probability functions, p(ri),'p(rj I ri), etc.

To solve Eq. (10) is a formidable task, and it is not surpris-ing that the QCA was introduced at an early stage by Lax'0

and Twersky. To show the connection between Eq. (10) andthe QCA, we now place some severe restrictions on the al-lowed multiple-scattering processes. First, we require thateach particle contribute only once to any term of the multi-ple-scattering series. Further, we do not permit back andforth scattering between a pair of scatterers. Finally, the N-particle joint probability function is factored as follows:

p(rl, r2, . . , rN) = p(rl)p(r2 I rl)p(r3 I r2) ... p(rN I rN-l).

(lla)

With the above assumptions, Eq. (10) can be representeddiagrammatically as

(EQCA) = E(r) +o+ 4 + -+ +

+ U S E R + .. (lib)where - denotes the incident plane wave, * denotes a scat-terer, *-@ denotes scattering from particle 2 to particle 1,*-0 denotes the correlation between the positions of parti-cles 1 and 2, and, finally, f-* denotes the propagation fromparticle 1 to the observation point r. In Eq. (lib) *- willbe replaced by an, each 0 0 will be replaced by ToT,where a, the transition matrix, accounts for the propagationof waves from one scatterer to another, and *- will bereplaced by p(l, 2). Hence the explicit form of Eq. (11) isthen

(Etot(r)) = EO(r) + N J Ou V(r - r1)Ta'p(r1)dr1

+ N2 J Ou V(r - r1)To(r12)Ta2 p(1, 2)drdr 2

note them by a(k) and g(k), respectively. Using the convo-lution theorem, Eq. (12) can be simplified to

(Et`t(r)) = E0(r) + no f Ou4i(r - rl)T[1 + n0oog(K)T

+ no2og(K)Thg(K)T

+ no3qrg(K)Thg(K)Tqg(K)T

+ ---]exp [iKko- (r, -r2 )]a2dKdrdr2 , (13)

where

rg(K) = S a(x)g(JJ4)eiKxdx. (14)

The terms on the right-hand side of Eq. (13) can be summedformally, and we can rewrite Eq. (13) as

(Etot(r)) = E4(r) + f Oul,'t(r -r)Tnn'

X no S [1 - noag(K)T]nn'

X exp[-iK- (r, - r2)]an2 dKdr~dr2. (15)

This new form of the average field can be interpreted as anincident plane wave propagating through an effective medi-um of propagation constant K and propagator (1 - nogT)'undergoing scattering from a particle at r, and then propa-gating to the observation point r with the wave number ofthe host medium. In Eq. (15) we can write

Hn'n-(r - r2) = X [1 -n f(K)n1n1

X exp[-iK- (r, - r2)]dK;

then

(Etot(r)) = E0 + no f Ou ,n(r - r')-Tnnnn(r -r 2)

X an,,2 drdr 2 .

(16)

(17)

The dispersion equation in the model medium is given bythe zeros of H(K) that yield the effective propagation con-stant of the medium. We recall that the propagator in thehost medium has a Fourier transform of the form 1/(k2 -W

2/c2) that has a pole at k = wic. The poles of the newpropagator are then determined by the roots of the deter-minantal equation:

1- noo-g(K)T = 0. (18)

+ N 3 J Ou #(r - r1)Ta(r12 )p(1, 2)To(r 2 3)p(2, 3)Ta 3

X drldr2 dr3 + N4 Ou 4(r - r1)Tu(r12 )p(1, 2)

X To(r2 3 )p(2, 3)To(r34 )p(3, 4)Ta 4 dr, .. dr4 + .... (12)

In Eq. (12), we have removed the restrictions in sums, suchas hi X2/ Lk', by noting that p(1, 2) is automatically zero if r2= r1. Thus the propagators in Eq. (12) may be interpretedas cut-out propagators that vanish if the argument is lessthan the hard-core diameter of the scatterers. For sphericalstatistics we note that

p(r1) = V X p(1, 2) = p(rl, r2) =VgqI r - r2 D )

We now introduce spatial Fourier transforms of the trans-lation matrix and the radial distribution functions and de-

In our previous papers we derived a dispersion equationfor the random medium by directly invoking the QCA in theequation for the field exciting a particular scatterer. Thus,if Eq. (7) were averaged directly, holding the ith scattererfixed, then

(an') i= ani +>3> 3 (nn'(ri)Tnn (an,2 )ij)i. (19)in nn

This immediately leads to an infinite hierarchy in which theconditional average of an' with p scatterers held fixed in-volves a conditional average on the right-hand side with p +1 scatterers held fixed. Thus a knowledge of all orders of thecorrelation function is required. As is well known, Lax trun-cated this hierarchy by approximating

(an )%j - (an2)p (20)

This is the famous QCA. Twerskyl-5 has already stated that

Varadan et al.


the QCA neglects back and forth scattering and includesonly sequential scattering. By assuming that the averagefield propagates with an effective propagation constant K,i.e.,

(and') ~ X,1 exp(iKh 0 - r,), (21)

and using expressions (20) and (21) in Eq. (19), we formallyobtain

Xn = no 5 rnn'(rij)Tn'ntP(rJri) exp[iK- (rj - ri)]drjXn.(22)

This is a homogeneous set of equations for Xn. For a non-trivial solution, the determinant of the coefficient matrixshould vanish, leading to

I1-nnoog(K)TI =0, (23)

which is identical to Eq. (18).Thus invoking the QCA is identical to a partial resumma-

tion of the multiple-scattering series represented by thediagrams in Eq. (lib). Although this has been generallyknown, no formal proof has been given before, especiallywhen the full multipole description for each scattering isused. The more interesting observation is that Frisch" hasshown that Eqs. (11) are also equivalent to the first-ordersmoothing approximation. Thus it would seem that thefirst-order smoothing and the QCA are equivalent. Thereappears to have been no discussion of this in previous litera-ture on the subject.

SECOND MOMENT OF THE FIELD INTENSITY

The intensity or the second moment of the field with polar-ization a at r is simply defined as

I,(r, w) = a - (E(r, w)E(r, c)) * a. (24)

If Eq. (9) is substituted forE in Eq. (24), we get the multiple-scattering series for the intensity. Diagrammatically eachfield line in Eq. (24) can be represented as in Fig. 1. Whenboth fields are multiplied and then averaged together we getcorrelations between scatterers on both field lines. Usingthe same notation as before, the following diagrams result:

I. (r,) = (-u El2)1 + i +

_+ IL~~~~~~+ +~~~~~~1---1

-� L-�--�

izi7��Iz

II��Iz�II

}3

I 4 , (25)

where we have allowed only sequential two-body correla-tions and grouped the terms into four categories. The firstterm is, of course, the incident intensity. The second groupcontains two field lines that are uncorrelated from one toanother but contain correlations within themselves. Thethird group contains two field lines that, however, interactwith the same particles. The last group contains the ladderdiagrams but also allows sequential correlations within eachfield line. All the other possible terms that contain dia-grams of the type

have been neglected.The first two groups of terms contribute to the coherent

intensity, and the last two groups contribute to the incoher-ent intensity or spectral density of the field fluctuations.The diagrams can be resummed by introducing the so-calleddressed propagators, if we refer to the translation operatoror0'(rij) in the host medium as a bare propagator. Thedressed propagator is the propagator in a medium that al-lows only QCA-type sequential-scattering terms with se-quential correlations and is identical to Eq. (16). Suchpropagators are represented by bold lines in contrast to thebare propagators. Thus

IuJr, cam) = E = X + { = C +

= : }W

��iz p

(26)

The terms in (b) can be summed by using Fourier trans-forms and convolution techniques, but the ladder diagramsin (c) do not lend themselves very easily to resummation byFourier-transform techniques. The relative contribution ofthe terms in (b) and (c) is studied numerically in the nextsection.

NUMERICAL RESULTS

Three types of results are presented in this section. Thefirst type is the calculation of effective propagation constantby solving the roots of the determinantal equation [Eq. (18)].The numerical procedure for doing this has been describedin detail in Refs. 7 and 8 and will not be repeated here. Thecomputation requires the T matrix of the scatterer; here wehave used the multipole field for spherical scatterers. Theconcentration c of scatterers (fractional volume occupied bythe scatterers) and the frequency are the other parametersrequired. In addition, the pair correlation function is re-quired. We have used values generated by Monte Carlosimulation for hard or impenetrable spheres as a function ofdistance between the spheres and the concentration ofspheres. The details may be found in Ref. 16.

.W

-4

uz

Varadan et al.

-5

I-_-_

QI

Varadan et al. Vol. 2, No. 12/December 1985/J. Opt. Soc. Am. A 2199

In Figs. 2 and 3, the real and the imaginary part of the i*C0

effective propagation constant in a distribution of Revacrylspheres is plotted as a function of concentration for twodifferent frequencies denoted by the wavelengths X = 410 0.8nm and X = 546 nm. These wavelengths were chosen be-cause our calculations could be compared with the experi-ments of Killey and Meeten. 1 5 The agreement is very good, < 06 -as can be seen from the graphs. The refractive index for <o /Revacryl spheres used in the computation was taken to be nv kG =0.1= 1.48, and the host medium is distilled water with n = I1- 0.4 - 01.334. l-3

In Fig. 4, the lowest approximation to the coherent inten-sity that is given by the first term in Eq. (26) was used to 02

compute the coherent intensity. In lowest approximation,

o I l l l l 00 20 40 60

1.41 REVACRYL DISPERSIONS IN DISTILLED WATER (a)

Y . EXPERIMENT (KILLEY a MEETEN)

-THEORY (VARADAN .* 01.) 0.6

1.37 - a= 100 nm 0.1

)<~~~~~~X= 546 nm\\

0.51.33 L @ |' 0.4 -

0 10 20 30 40 50c(%)

Fig. 2. Phase velocity versus concentration c for Revacryl disper- <c =0.I0sions in distilled water at X = 546 nm.

0

'1_ 0.2 -2.0-

2D~~~~~~~~~~~~~~~~qY \ 9~~~~~~~~~~~~~~~~~~~~~~~~~~~~1.0o X

S 1.0 / a_{_ . . -- _%__ 410nm 0 2.0

E546nm () 50° 100 15e0 200°

OBSERVATION ANGLE &_ _ (b)

0 1 0 20 30 40 50c %)

Fig. 3. Coherent attenuation versus concentration c for Revacryl 0.6 -dispersions in distilled water at X = 410 and 546 nm.

\ ~~~~0= 180°0

* EXPERIMENT( KILLEY a MEETEN) 0 C=O.I0

- THEORY (VARADAN .e .1.)0

a=I -Onm --l ~~~~~~~~~~~~~~~~~~~~0

X546nmc=5

1-0.2c\ 0.046 2~ -2

N..~ ~ ~ ~ ~~=0

0-3 - 0 0.5 1.0 1.5 2.0

ka\ 0.ca59\ (c)

C=0.279 K'o o (ic-0.279 I \ Fig. 5. (a) The spatial Fourier transform (Klo-k5) ofg(l) - 1

4 200 400 600 as a function of concentration c for ka = 0.1 and 0 = 30°. (b) Thespatial Fourier transform f(Kl&o - kr) of g(Ixi) - 1 as a function of

z(nm) observation angle 0 for c = 0.10 and ka = 0.1, 0.5, 1.0, and 2.0. (c)

Fig. 4. Coherent intensity as a function of propagation depth z for The spatial Fourier transform f(Klho - r) of g(1 xl) - 1 as a functionvarious values of c at X = 546 nm. of ka in the backscattering direction for c = 0.05 and 0.10.


I.0- terms in (b) of Eq. (26) without also retaining higher-order

>_ c ladder diagrams.Ci)~~~~~~~~~~~~~~

08 /In Fig. 5(b), 7(K1ko - k) in Eq. (27) is plotted as a functionS 0.8 ~ / of observation angle for different values of ka. It can be

Z c = 0.10 seen that for 0 = 180°, the contribution from f to the incoher-Z O.6_ 0= 30 / A ent intensity is greatly dependent on the value of ha under

0 / A consideration. In Fig. 5(c), f is plotted for two values of c =M / 0.05 and 0.1 and 0 = 1800 (backscattering) as a function of

Z 0.4 _a.O / / B The conclusion from Figs. 5(a)-5(c) is that the incoherent

intensity has contributions from both (b) and (c) in Eq. (26),< 0.2 - // and more studies are indicated to neglect one in favor of the

other.z In Fig. 6, the normalized incoherent intensity is plotted as

0 a function of ha at c = 0.1 for an observation angle of 300.1.0 1.2 1.4 1.6 1.8 2.0 The curve labeled C is the total contribution that is due to

ka single scattering; the curve A is due to completely correlatedFig. 6. Normalized incoherent intensity that is due to single scat- or single scattering from the same particle I and B is due totering from correlated scatterers as a function of ha for c = 0.10 and 0 I. Iisglea tattBris not nlie .cmpard t A.- 3Q0OI. It is clear that B is not negligible compared to A.

the intensity decays exponentially with penetration into therandom medium, the rate of decay being twice the imaginarypart of the effective propagation constant (K = K1 + iK2) inthe medium. The rate is a function of the frequency, thesize of scatterers, the concentration of scatterers, and thematerial properties. Again the results normalized with re-spect to the incident intensity in Fig. 3 for Revacryl spherescomes quite favorably with the results of Killey and Meeten.

In Figs. 5 and 6, the leading term for the incoherent inten-sity involving only single scattering from one scatterer (A) ortwo correlated scatterers (B) is plotted for various parame-ters. The total single-scattering result is denoted by C inFig. 6. The exact expression used in the computations isgiven below:

I0(r, w) = . (E(r, co)E*(r, )D

no ~~~~~~~~~2V(hr)2 | * 0 An(r)Tnan(ho) Vs

X {1 + no J (g x I-1)exp[-i(Kh0 - kh) * x]dx}

) | E A, 0)Tnan(ho) [1 + A(K -)].O ~~ ~~~A B

(27)

ho is the direction of propagation of the incident wave, an(ho)are the incident-wave field coefficients, 7(Klho - kr) is thespatial Fourier transform of [g(I x - 1], and An = A~lm, arevector spherical harmonics with r = 1, 2, 3.

In order to study the difference in the order of magnitudeof corresponding terms (equal number of T matrices), onecan simply plot expressions A and B in Eq. (27) as a functionof concentration. Although these terms will be a function offrequency for higher-order terms of the multiple-scatteringseries, we can get a feel for the relative contribution of theseterms. They are plotted in Fig. 5(a) and one can see that,even at a concentration of 10%, the value of the correlationintegral [term B of Eq. (27)] is approximately 36% of A.This implies that there is no point in summing higher-order

ACKNOWLEDGMENT

This research was supported by the U.S. Army ResearchOffice under contract no. DAAG 29-84-K-0187 to The Penn-sylvania State University.

REFERENCES

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2. V. Twersky, "Coherent electromagnetic waves in pair-correlat-ed random distributions of aligned scatterers," J. Math. Phys.19, 215-230 (1978).

3. V. Twersky, "Multiple scattering of waves by periodic and byrandom distributions," in Electromagnetic Scattering, P. L. E.Uslenghi, ed. (Academic, New York, 1978), pp. 221-251.

4. V. Twersky, "Acoustic bulk parameters in distribution of pair-correlated scatterers," J. Acoust. Soc. Am. 64, 1710-1719 (1978).

5. V. Twersky, "Propagation and attenuation in composite me-dia," in Macroscopic Properties of Disordered Media, Vol. 154of Lecture Notes in Physics, R. Burridge and S. Childress, eds.(Springer-Verlag, New York, 1982), pp. 258-271.

6. V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromag-netic and Elastic Wave Scattering-Focus on the T-MatrixApproach (Pergamon, New York, 1980).

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8. V. N. Bringi, V. K. Varadan and V. V. Varadan, "Coherent waveattenuation by a random distribution of particles," Radio Sci.17, 946-952 (1982).

9. V. K. Varadan, V. N. Bringi, V. V. Varadan, and A. Ishimaru,"Multiple scattering theory for waves in discrete random mediaand comparison with experiments," Radio Sci. 18, 321-327(1983).

10. M. Lax, "Wave propagation and conductivity in random me-dia," in Proceedings of the SIAM-AMS Conference on Sto-chastic Differential Equations (American Mathematical Soci-ety, Providence, R.I., 1970), Vol. VI, pp. 35-95.

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Po

4)

3

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Propagator model including multipole fields for discrete random media