1994 Draine Discrete Dipole

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    Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1491

    Discrete-dipole approximation for scattering calculationsBruce T. Draine

    Princeton University Observatory, Princeton, New Jersey 08544-1001Piotr J. Flatau

    Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California92093-0221Received July 20, 1993; revised manuscript received October 12, 1993; accepted October 18, 1993

    The discrete-dipole approximation (DDA)for scattering calculations, including the relationship between theDDA and other methods, is reviewed. Computational considerations, i.e., the use of complex-conjugategradient algorithms and fast-Fourier-transform methods, are discussed. We test the accuracy of the DDAby using the DDA to compute scattering and absorption by isolated, homogeneous spheres as well as bytargets consisting of two contiguous spheres. It is shown that, for dielectric materials (Iml c 2), the DDApermits calculations of scattering and absorption that are accurate to within a few percent.

    1. INTRODUCTIONThe discrete-dipole approximation (DDA) s a flexible andpowerful technique for computing scattering and absorp-tion by targets of arbitrary geometry. The developmentof efficient algorithms and the availability of inexpen-sive computing power together have made the DDA themethod of choice for many scattering problems. Bohrenand Singham' reviewed recent advances in calculations ofbackscattering by nonspherical particles, including DDAresults. In this paper we review the DDA, with particu-lar attention to recent developments.In Section 2 we briefly summarize the conceptual basisfor the DDA. The relation of the DDA to other finite-element methods is discussed.DDA calculations require choices for the locations andthe polarizabilities of the point dipoles that are to repre-sent the target. In Section 3 we discuss these choices,with attention to the important question of dipole polar-izabilities. Criteria for the validity of the DDA are alsoconsidered.Recent advances in numerical methods now permit thesolution of problems involving large values ofN, the num-ber of point dipoles. These developments, particularlythe use of complex-conjugate gradient (CCG)methods andfast-Fourier-transform (FFT) techniques, are reviewed inSection 4.We illustrate the accuracy of the method in Section 5by using the DDA to compute scattering by a singlesphere and by targets consisting of two spheres in contactand by comparing our results with the exact results forthese geometries. Although these are only examples,they make clear that the DDA can be used to computehighly accurate results for targets with dielectric con-stants that are not too large (Iml c 2).

    A portable FORTRANmplementation of the DDA, theprogram DDSCAT,as been developed by the authors andis freely available.2

    2. WHAT IS THE DISCRETE-DIPOLEAPPROXIMATION?A. ApproximationGiven a target of arbitrary geometry, we seek to calculateits scattering and absorption properties. Exact solutionsto Maxwell's equations are known only for special geome-tries such as spheres, spheroids, or infinite cylinders, soapproximate methods are in general required. The DDAis one such method.The basic idea of the DDA was introduced in 1964by DeVoe,3' 4 who applied it to study the optical prop-erties of molecular aggregates; retardation effects werenot included, so DeVoe's treatment was limited to ag-gregates that were small compared with the wavelength.The DDA, including retardation effects, was proposed in1973 by Purcell and Pennypacker,5 who used it to studyinterstellar dust grains. Simply stated, the DDA is anapproximation of the continuum target by a finite array ofpolarizable points. The points acquire dipole moments inresponse to the local electric field. The dipoles of courseinteract with one another via their electric fields,5' 6 so theDDA is also sometimes referred to as the coupled dipoleapproximation.7'8 The theoretical basis for the DDA, in-cluding radiative reaction corrections, is summarized byDraine.6Nature provides the physical inspiration for the DDA:in 1909 Lorentz showed9 that the dielectric properties ofa substance could be directly related to the polarizabil-ities of the individual atoms of which it was composed,with a particularly simple and exact relationship, theClausius-Mossotti (or Lorentz-Lorenz) relation, whenthe atoms are located on a cubic lattice. We may expectthat, just as a continuum representation of a solid isappropriate on length scales that are large compared withthe interatomic spacing, an array of polarizable pointscan accurately approximate the response of a continuum

    0740-3232/94/041491-09$06.00 1994Optical Society of America

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    1492 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994target on length scales that are large compared with theinterdipole separation.For a finite array of point dipoles the scattering problemmay be solved exactly, so the only approximation that ispresent in the DDA is the replacement of the continuumtarget by an array of N-point dipoles. The replacementrequires specification of both the geometry (location rjof the dipoles j = 1, ... , N) and the dipole polarizabili-ties aj.For monochromatic incident waves the self-consistentsolution for the oscillating dipole moments Pj maybe found, as is discussed in Section 4 below; fromthese P the absorption and scattering cross sectionsare computed.6 If DDA solutions are obtained fortwo independent polarizations of the incident wave,then the complete amplitude-scattering matrix can bedetermined.6With the recognition that the polarizabilities ajmay be tensors, the DDA can readily be applied toanisotropic materials.6" 0 The extension of the DDA totreat materials with nonzero magnetic susceptibility isalso straightforward, "" 2 although for most applicationsmagnetic effects are negligible.B. Relation to Other MethodsThe DDA is one particular discretization method for solv-ing Maxwell's equations in the presence of a target. TheDDA was anticipated in the engineering literature by themethod of moments,'3 -'5 which is a general prescriptionfor discretization of the integral form of Maxwell's equa-tions. There is a close correspondence between the DDAand discretizations that are based on the digitized Green'sfunction (DGF) method' 6 or the volume-integral equationformulation (VIEF).1-22 In the low-frequency limit kd -0 (where d is the interdipole spacing or the side of a cubi-cal subvolume and k = /c), it has been shown"" 2 2' thatthe DDA is indeed equivalent to the DGF/VIEF. How-ever, researchers developing the DGF/VIEF approachencountered difficulties in trying to calculate O[(kd)2 ]corrections, whereas the DDA, because the fundamentalapproximation is simple and physically clear, permits rig-orous derivation of the O[(kd)2] correction (see Subsection3.B below). On the other hand, the integral form ofMaxwell's equations often serves as a convenient pointof departure, and there is now a strong interplay betweenboth approaches.The principal limitation of the DDA involves the han-dling of target boundaries, since the geometry of theDDA array has a minimum length scale that is equalto the interdipole spacing d. For targets with largevalues of the refractive index m, the accuracy of the DDAsuffers. In principle we can make the DDA as accurateas we desire by increasing N, the number of dipolesrepresenting the target (thus decreasing d), but very largevalues of N become computationally prohibitive.3. SPECIFICATION OF THE DIPOLE ARRAYA. GeometryEach dipole may be thought of as representing the polar-izability of a particular subvolume of target material. Ifwe wished to approximate a certain target geometry (e.g.,a sphere) with a finite number of dipoles, then we might

    consider using some number of closely spaced, weakerdipoles in regions near the target boundaries to do a bet-ter job of approximating the boundary geometry. Someresearchers have followed this approach2324 ; although itclearly offers benefits, there is also a substantial cost: aswe show below, FFT methods can greatly accelerate thecomputations that are required for solving the scatteringproblem, but only if the dipoles are located on a periodiclattice. With FFT methods we can perform calculations(on a workstation) for numbers N of dipoles (e.g., 105)that are much greater than the largest values (- 104)that could readily be handled without FFT methods.We therefore elect to use FFT methods, accepting therequirement that the dipoles be located on a periodiclattice. We further restrict ourselves to cubic lattices.This option still permits considerable latitude in rep-resentation of the continuum target. For a given targetgeometry we use the following algorithm to generate thedipole array. We assume that the target orientation isfixed relative to a coordinate system x, 9, i.1. Generate a trial lattice that is defined by the latticespacing d and coordinates (xo, yo, zo) of the lattice pointnearest the origin.2. Let the array now be defined as all lattice siteslocated within the volume V of the continuum target.3. Try different values of d and (x0, yo,zo) and maxi-mize some goodness-of-fit criterion, subject to the con-straint that the number of points in the dipole array notbe so large as to be computationally prohibitive. Thegoodness-of-fit criteria are basically geometric in nature,as well as somewhat arbitrary, 6 and are not discussedhere.4. We now have a list of occupied sites j = 1, ... , N.

    We imagine that each of these occupied sites represents acubic subvolume d3 of material centered on the site. Inthe current implementation of DDSCAT we do not distin-guish between lattice sites near the surface and those inthe interior. We therefore rescale the array by requiringthat d = (V/N)" 3 , so that the volume Nd3 of the occu-pied lattice sites is equal to the volume V of the origi-nal target.5. For each occupied site j assign a dipole polarizabil-ity aj.B. Dipole PolarizabilitiesThere has been some controversy concerning the bestmethod of assigning the dipole polarizabilities. Purcelland Pennypacker 5 used the Clausius-Mossotti polar-izabilities:

    CM=3d3 ej-1aj 47r e + 2 (1)where e is the dielectric function of the target materialat location rj. As is well known,2 5 for an infinite cubiclattice the Clausius-Mossotti prescription is exact in thedc limit kd - 0. Draine6 and Goedecke and O'Brien 6showed that the polarizabilities should also include aradiative reaction correction of O[(kd)3]. Goedecke andO'Brien16 proposed a correction term of 0[(kd) 2]; the samecorrection term was independently proposed by Iskanderet al.2 0 and by Hage and Greenberg.2 ' Unfortunately thederivation of this correction assumed that the electric

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    Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1493field could be taken to be uniform over cubical regionsof volume d3, and this assumption itself introduces errorsof O[(kd)].26 Another approach was taken by Dungeyand Bohren,2 7 who arrived at a different O[(kd)2 ] correc-tion by requiring that each point dipole have the samepolarizability as a finite sphere of diameter d, but with amodified dielectric constant.To clarify this issue Draine and Goodman 6 consideredthe following problem: for what polarizability a(Co)will an infinite lattice of polarizable points have thesame dispersion relation as a continuum of refractiveindex m(co)? The lattice dispersion relation (LDR) maybe found analytically; in the long-wavelength limitkd > 1 the DDAcan seriously overestimate absorption cross sections, evenin the dc limit Imlkd >1 itappears that other techniques (e.g., the method of Rouleauand Martin29 ) may be superior to the DDA.4. COMPUTATIONALCONSIDERATIONSA. Scattering ProblemThe electromagnetic scattering problem must be solvedfor the target array of point dipoles (j = 1, ... , N) withpolarizabilities aj, located at positions r. Each dipolehas a polarization Pj = aEj, where Ej is the electricfield at r that is due to the incident wave En =Eo exp(ik r - it) plus the contribution of each of the

    aCM1 + (aCM/d3)[(bi+ m2b2 + m2b3S)(kd)2 - (2/3)i(kd)3]' (2)bi = -1.891531,b3 = -1.7700004,

    b2 = 0.1648469,3_ 1 (e)2j=1

    other N - 1 dipoles:(3) Ej = Eincj AjkPk,k~jwhere and are unit vectors defining the incidentdirection and the polarization state. The O[(kd)2] termin the LDR expansion differs from previously proposedcorrections.6' 2 0' 21'27 The LDR prescription for a(w) is,by construction, optimal for wave propagation on an infi-nite lattice, and it is reasonable to assume that it willalso be a good choice for finite dipole arrays. Exten-sive DDA calculations for spheres, comparing differentprescriptions for the dipole polarizability, confirm that

    the LDR prescription appears to be best for mlkd < 1.26The different prescriptions have also been compared incalculations of scattering by two touching spheres2 8 (seeFig. 9 below).C. Validity CriteriaThere are two obvious criteria for validity of the DDA:(1) lmlkd 1 (so that the lattice spacing d is small com-pared with the wavelength of a plane wave in the targetmaterial), and (2) d must be small enough (N must belarge enough) to describe the target shape satisfactorily.Define the effective radius aeff of a target of volume Vby aeff (3V/4v-)"3 . The first criterion is then equiva-lent to

    N > (47r/3)lml 3 (kaeff)3 . (4)Thus targets with large values of ml or scattering prob-lems with large values of kaeff will require that largenumbers of dipoles be used to represent the targets.Unfortunately the second criterion has not yet beenformulated precisely. Draine6 shows that even in thekd - 0 limit the polarizations are too large in the surfacemonolayer of dipoles in a pseudosphere, and similar errorsmust also occur for other target shapes. As a result therate of energy absorption by the dipoles in the surfacemonolayer is too large, which leads to an error in theoverall absorption cross section in proportion to the frac-tion -N- 3 of the total volume that is contributed by the

    where -AjkPk is the electric field at r that is due todipole Pk at location rk, including retardation effects.Each element Ajk is a 3 X 3 matrix:= exp(ikrjk)A jk, = ~ 3 1rjk

    X Fk2jk r jk r13+ - (31Pk jk -13)j k, (6)

    where k coc, rjk rj - rk, rjk (rj - rk)/rjk, and13 is the 3 X 3 identity matrix. Defining Ajj = aj- re-duces the scattering problem to finding the polarizationsPj that satisfy a system of 3N complex linear equations:NY AjkPk = Emcj.k=1 (7)

    Once Eq. (7) has been solved for the unknown polariza-tions Pj, the extinction and absorption cross sections Cextand Cabsmay be evaluated6 :4irk NCext Y m(Ein * j)IEoJ2j (8)

    Cabs= 4Jl- { [P * aj-7)*Pj*1- 2 k3IPi2}. (9)The scattering cross section Csca = Cext - Cabs. Dif-ferential scattering cross sections may also be directlyevaluated once the Pj are known.6 In the far field the

    (5)

    B. T. Draine and P. J. Flatau

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    1494 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994scattered electric field is given by

    Eca = exp(ikr) exp(-iki rj)(PP - 13 )Pj- (10)The problem is that matrix A is large and full, i.e., amatrix that in general has few zero elements. We would

    like (1) to solve the linear system of Eq. (7) efficiently and(2) to solve efficiently for multiple cases of the incidentplane wave Enc. LU decomposition0 is the method ofchoice for small problems. It solves problem (2) becauseonce the LU decomposition has been obtained the so-lution for a new right-hand side Eincrequires just onematrix-vector multiplication. However, it is not feasibleor efficient for large problems because of the need to storeA and the LU decomposition and because of computertime that is proportional to N3 .For homogeneous, rectangular targets the matrix Ahas block-Toeplitz symmetry,3' and algorithms exist forfinding A-' in O(N2 ) operations rather than in the O(N 3)operations that are required for general matrix inversion.It seems possible that the fundamental symmetries of Amay permit efficient algorithms for finding A-' even fortargets that are inhomogeneous or nonrectangular.B. Complex-Conjugate Gradient MethodRather than direct methods for solving Eq. (7), CCGmethods for finding P iteratively have proven effectiveand efficient. The particular CCG algorithm that isused in DDSCAT is described by Draine.6 As P has 3Nunknown elements, CCG methods in general are onlyguaranteed to converge in 3N iterations. In fact, how-ever, _ 10_102 iterations are often found to be sufficientto obtain a solution to high accuracy.63 2 The choiceof conjugate-gradient variant may influence the conver-gence rate. 33 4 5When N is large, CCG methods are much faster thanare direct methods for finding P. The fact that CCGmethods in practice converge relatively rapidly is presum-ably a consequence of the basic symmetries of A.As an iterative technique the CCG requires an ini-tial guess for P. The simplest choice is P = 0. At-tempts to improve on this by use of the scattering-orderapproximation3 6 as a first estimate for P were found tooffer little, if any, advantage over starting with P = 0.6The iterative method that is described here mustbe repeated for each incident plane wave [cf. Eq. (7)].Notice that if we had A-' then for each new incident waveEinc he solution P = A-'Emc would be obtained from asingle matrix-vector multiplication. In principle we canrecover eigenvalues and eigenvectors from conjugate-gradient iteration.3 7 3 8 Some attempts to improve theefficiency of the conjugate-gradient method for multipleincident electromagnetic fields have been reported.34 39 'FORTRAN implementations of approximately 15 CCGmethods are available from us.42C. Fast Fourier TransformsThe computational burden in the CCG method primar-ily consists of matrix-vector multiplications of the formA v. Goodman et al.3 2 show that the structure of thematrix A implies that such multiplications are essen-tially convolutions, so that FFT methods can be employed

    to evaluate A v in O(N ln N) operations rather thanin the 0(N 2) operations that are required for generalmatrix-vector multiplication. Since N is large this isan important calculational breakthrough. As mentionedabove, FFT methods require that the dipoles be situatedon a periodic lattice, which is most simply taken to becubic.FFT methods in effect require the computation of three-dimensional FFT's over 8NXNYNZ points, where NXNyNzis the number of sites in a rectangular region of the latticecontaining all the N occupied lattice sites. Therefore, inthe case of a fractal structure with a large volume-fillingfactor for vacuum, FFT methods may lose some of theiradvantage over conventional techniques for evaluatingA *v, since NX.NNZ may be much larger than the actualnumber of dipoles N.We note that CCG and FFT techniques are routinelyused for numerical solutions of electromagnetic problemsin engineering.3 4 4 3 ,4 4

    D. Memory and CPU RequirementsIf the N dipoles were located at arbitrary positions rj,then the 9N2 elements of A would be nondegenerate.Storage of these elements (with 8 bytes/complex number)would require 72(N/103 )2 Mbytes. By locating thedipoles on a lattice, the elements of A become highlydegenerate, since they depend only on the displacementrj - r,. As a result the memory requirements dependapproximately linearly on N rather than on N 2 . Theprogram DDSCAT has a total memory requirement of-0.58(N.NN./1000)'Mbytes, where N. X N, X N isthe rectangular volume containing all the N dipoles.Thus for a target fitting into a 32 32 x 32 portion ofthe lattice (e.g., an N = 17,904 pseudosphere), DDSCATrequires -19 Mbytes of memory. We therefore see thatmemory requirements begin to be a consideration fortargets that are much larger than -104 dipoles.The CPU requirements also are significant for largetargets. On a Sun 4/50 (Sparcstation IPX), a singleCCG iteration requires 3.0(N.NyN,/10 3 ) CPU s; thusone iteration for a N = 17,904 pseudosphere requires-100 CPU s. Between 10 and 102 iterations are typi-cally necessary to solve for a single incident directionand polarization; thus orientational averaging with theDDA can be time consuming if many target orientationsare required. It is for this reason that T-matrixmethods,

    5 46which exploit efficient procedures for ori-entational averaging,4 7 are competitive for some targetgeometries as well as recursive T-matrix algorithmscurrently being developed by Chew and Lu4 8 and Chewet al. 4 9

    5. ACCURACY OF THE DISCRETE-DIPOLEAPPROXIMATIONAt this time there is no known way to predict preciselythe accuracy of DDA calculations. Instead we rely on ex-amples to guide us. Spheres are most convenient sinceexact solutions are readily available for comparison.Note that the DDA does not give preferential treatment toany geometry (with the possible exception of rectangulartargets), so spheres constitute a representative test of

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    Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1495of the DDA. Clusters of spheres, for whichcan now also be computed, are also useful.Several earlier studies have examined the accuracy ofDDA,5,6,3 ,2 7 5 0,5' but only one26 employed the LDRThe DDA is most accurate for targetsindices m near unity, with the accuracyfixed kaeff and N) as Im - 11ncreases.

    the LDR polarizabilities and concentratetwo cases: m = 1.33 + 0.01i, representative of moder-material, and m = 2 + i, representativestrongly absorbing material inthe ultraviolet. As is well known, veryindices (Imi >> 1) occur for conductingin the infrared; the DDA s not particularly wellregime, and we do not consider any suchhere. We will examine the dependence of thekaeff and N.

    Total Cross Sectionsa given incident wave the scattering problem is solvediteration until a specified level of accuracy isAll the results shown here are converged untilP - EincI< lO-3lEmc. Figures 1 and 2 show theg efficiency fac-Qabs 3 Cabs/7ra 2 and Qsca 3 Csca/ra2 , as functions2ra/A, for pseudospheres containing N dipoles;labeled by N. Each curve terminates atImikd - 1. The pseudospheres arenot precisely spherically symmetric; the DDAFigs. 1-8 are for radiation incident along1, 1) direction. Figure 1 shows results for m =+ 0.01ie = 1.769 + 0.0266i). It is clear that for

    Qscamay be obtained to 2% accuracy provided thatis large enough that the criterion Imlkd< 1 is satisfied.Figure 2 shows results for m = 2 + i (e = 3 + 4i). Ine larger refractive index leads to larger errors,errors in Qscaare still quite small. Somewhatthe largest errors tend to occur in theka ' 0.5; here Qsca s in error by - 3%,Qabs is in error by 5% for N = 1064. For fixed kaas N is increased. It is apparent thata refractive index m = 2 + i, which is moderatelyand absorptionof - 1%- 4%, respectively, for pseudospheres containing104 dipoles.

    Scattering1 and 2 show that the total scattering crosscan be calculated fairly accurately. DifferentialdQ/dfl = S, 1/r(ka) 2 for unpolar-light incident upon a sphere are shown in Figs. 3-8indices (m = 1.33 + 0.01ind m =ka = 3, 5, and 7. The DDA results are shownof dipoles N. The lower panel in3-8 displays the fractional error:

    eo dQ/dfl(DDA) - dQ/dfl(Mie) (11)error dQ/dfl (Mie) (1

    The fractional error becomes quite large for scatteringangle 0 > 1500because dQ/dfl itself becomes quite smallfor backscattering; hence small absolute errors in dQ/dflcorrespond to large fractional errors.From Figs. 1-8we may draw the following conclusions:1. For pseudospheres with Iml ' 3, total scattering

    and absorption can be computed to accuracies of a few

    Q)a(50.C.o0.01

    01

    -3CY~2

    13

    040

    0 1 2 3 4 5 6 7 8 9 10 11 12 13x=kaFig. 1. Scattering and absorption for a sphere with refractiveindex m = 1.33 + 0.01i. The upper panel shows the exact valuesof the scattering and absorption efficiency factors Qsca and Qabs,obtained with Mie theory, as functions of ka, where k = 2X/A.The middle and lower panels show fractional errors in Qsca andQabs, obtained from DDA pseudospheres, labeled by the numberN of dipoles in the pseudosphere.

    a5 0.1a0.01

    IR 4a 00_O

    V -2

    1 6

    a 4.502a)

    0 L0 1 2 3 4 5 6 7x=kaame as Fig. 1, but for the refractivea 9 10

    T. Draine and P. J. Flatau

    Fig. 2. index m= 2 + i.

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    1496 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994 B. T. Draine and P. J. Flatau

    aI-0.l

    0.01

    1510= 5

    .5 -5-1 0-15-20

    0 20 40 60 80 0(.)100 120 140 160 180Fig. 3. Differential scattering cross section for pseudosphereswith m = 1.33 + 0.01i and ka = 3. The curves are labeled byN, the number of dipoles in the pseudosphere.

    10

    0

    C.

    0.01

    FN

    ._'00)

    20151050

    -5

    Mie theoryN= 17,904=7664N=2320~~~~_ \ ~~~~~~~~~~Qsc./47r

    m=1.33+0.Oli27ra/X=5

    rms, N=17,904: 1.8%7664: 3.1%2320: 3.4%

    ,jj,'~, I' I.0 20 40 60 80 5(I)100 120 140Fig. 4. Same as Fig. 3 but for ka = 5. 160 180percent if the validity criterion [relation (4)] is satisfied,and N ' 103, so that the surface geometry is reasonablywell approximated.2. For pseudospheres, if the validity criterion [relation(4)] is satisfied, differential scattering cross sections canbe calculated to reasonable accuracy.

    For m = 1.33 + 0.01i, fractional errors in dQsca/dl arewithin 20% for all scattering directions, with rms errorsof G6%. For m = 2 + i, fractional errors in dQ 4Ca/dQare within - 30% for all scattering directions, with rmsvalues of S 11%.

    B. Two Contiguous SpheresRecent investigations have studied the general solutionto Maxwell's equations for clusters of spheres.5 2 5 3 Theavailability of this exact solution offers a unique opportu-nity to test the accuracy of the DDA when it is applied tononspherical targets. 28 54Figure 9 presents a comparison of a two-sphere, multi-pole solution with the DDA as a function of x = kaeff-Results are shown for three different prescriptions for thedipole polarizabilities: CMRR (Clausius-Mossotti plusradiative reaction), DGF-VIEF, and LDR. The refrac-tive index is m = 1.33 + 0.01i, and the dipoles are placedon a 64 X 32 X 32 cubic lattice. The calculations are for

    10

    c: 1'0IC.'X 0.1

    0.0115

    ig 10C'0 5

    ._0

    - 50 20 40 60

    Fig. 5. Same

    C-0' 0.1'0

    0.01

    cr\ 5

    .531

    0'--l

    -200 20 40

    80 (0)l0as Fig. 3 but

    60 80 0(.100 120Fig. 6. Same as Fig. 3 but for m = 2 + i.

    Mie heoryN=59,728N=17,904N=7664-- N=2320Q.../4Tr \m=2+i

    - 2nra/X=3

    rmNsN=59,728: 4.5%17,904: 6.6%7664: 8.1% i2320: 10.7%

    ~~ \ /

    120 140 160 180for ka = 7.

    140 160 180

    ..........................

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    Draine and P. J. Flatau Vol. 11, No. 4/April 1994/J. Opt. Soc. Am. A 1497

    1

    VI 0.1

    0.015

    3U

    o-10

    -15_s

    Mie theoryN=59728N=17,904N=7664

    -m=2+i\2Ta/ =5

    /\/17,904: 7.9%7664: 8.3%0 0 0 0 0 100 |120 140,160,180,,,1,

    0 20 40 60 80 or()100 120 140 160 180Fig. 7. Same as Fig. 3, but for m = 2 + i and a = 5.

    10

    1ca.

    0.1

    0.015

    '0c -5C.- 100-1

    h-20q)-25-30

    0 20 40 60 80 0(.)100 120 140 160 180Fig. 8. Same as Fig. 3, but for m = 2 + i and ka = 7.

    6. RECENT RESEARCHThe DDA has recently been applied to a number of diverseproblems, including (since 1990)calculations ofscattering

    0.10.01

    0.0012c 1

    Ic03- -104) -2

    6" 4

    o 20 -2

    2 6 a 10

    2 4 6 8 10x=ka. 1 4Fig.9. (a) Qsca CCa/raeff and Qabs a Cabs/7aeff (calculatedby modal analysis) for two contiguous spheres of refractive indexm = 1.33 + 0.01i, for radiation incident at an angle of 30relative to the axis passing through the center of the two spheres.(b) Fractional error in computed value of Qsca and (c) Qabsfor a DDA representation containing N = 2 X 17256 dipoles.Results are shown for three different prescriptions for the dipolepolarizabilities: LDR, DGF/VIEF, and CMRR as functions ofx = kaeff (source, Ref. 28).

    0 20 40 60 80 100 120 14010102

    = 10')2

    0.1

    10-2103102

    light incident at an angle a = 30 relative topassing through the centers of the two spheres.in Section 3, the LDR method for specifyingthe polarizabilities results in very accurate values forand Qabs and is clearly preferable to the CMRR andIn Fig. 10 we compare the exact solution for scatteringtwo touching spheres (solid curve) with the solutionderived from the DDA (X's) for m = 1.33 + 0.001i and

    kaeff = 10. The exact results for S11 and S22 are com-pared with results that were computed with the DDA.The overall agreement is very good, although errors canbe seen for scattering angle 0 ' 500.

    S32)2 10l

    0.110-2

    160 180

    0 20 40 60 80 100 120 140 160 180O()Fig. 10. Scattering by two contiguous spheres with refractiveindex m = 1.33 + 0.001i for x = kaeff = 10. Radiation is in-cident along the axis passing through the centers of the twospheres. Results are shown for two elements (S1u and S 22 ) ofthe 4 x 4 scattering matrix.55 S and S22 computed with theDDA are compared with exact results (source, Ref. 28).

    -X,

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    1498 J. Opt. Soc. Am. A/Vol. 11, No. 4/April 1994and absorption by rough and porous particles,5 6 inter-stellar graphite particles,5 7 5 8 aggregate particles,0'5 9 -6 5inhomogeneous particles, 6 6 structures on surfaces, 6 7 mi-crowave scattering by ice crystals,68 and single scatteringproperties of ice crystals. 6 97. SUMMARYThe discrete-dipole approximation, or the coupled-dipolemethod, is a flexible and powerful tool for computing scat-tering and absorption by arbitrary targets. For targetswith Iml S2 (with m being the complex refractive index),scattering and absorption cross sections can be evaluatedto accuracies of a few percent provided that the number Nof dipoles satisfies relation (4). The use of CCG and FFTalgorithms permits calculations for N as large as - 105,so that scattering problems with kaff ' 10 can be studiedwith scientific workstations.ACKNOWLEDGMENTSB. T. Draine's research has been supported in partby National Science Foundation grant AST-9017082.P. J. Flatau has been supported in part by Atmo-spheric Radiation Measurement grant DOE DE-FG03-91ER61198 and Western Regional Center for GlobalEnvironmental Change grant UCSD93-1220.REFERENCES AND NOTES

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    B. T. Draine and P. J. Flatau