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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Stephen A. Burns
  • Vol. 25, Iss. 11 — Nov. 1, 2008
  • pp: 2693–2703
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Discrete-dipole approximation for periodic targets: theory and tests

Bruce T. Draine and Piotr J. Flatau  »View Author Affiliations


JOSA A, Vol. 25, Issue 11, pp. 2693-2703 (2008)
http://dx.doi.org/10.1364/JOSAA.25.002693


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Abstract

The discrete-dipole approximation (DDA) is a powerful method for calculating absorption and scattering by targets that have sizes smaller than or comparable to the wavelength of the incident radiation. The DDA can be extended to targets that are singly or doubly periodic. We generalize the scattering amplitude matrix and the 4 × 4 Mueller matrix to describe scattering by singly and doubly periodic targets and show how these matrices can be calculated using the DDA. The accuracy of DDA calculations using the open-source code DDSCAT is demonstrated by comparison with exact results for infinite cylinders and infinite slabs. A method for using the DDA solution to obtain fields within and near the target is presented, with results shown for infinite slabs.

© 2008 Optical Society of America

1. INTRODUCTION

Electromagnetic scattering is used to study isolated particles, but increasingly to characterize extended targets ranging from nanostructure arrays in laboratories to planetary and asteroidal regoliths. To model the absorption and scattering, Maxwell’s equations must be solved for the target geometry.

For scattering by isolated particles with complex geometry, a number of different theoretical approaches have been used, including the discrete-dipole approximation (DDA) [1

1. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973). [CrossRef]

, 2

2. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

, 3

3. B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

, 4

4. B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), pp. 131–145.

], also known as the coupled-dipole approximation or coupled-dipole method. The DDA can treat inhomogeneous targets and anisotropic materials, and has been extended to treat targets near substrates [5

5. R. Schmehl, B. M. Nebeker, and E. D. Hirleman, “Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique,” J. Opt. Soc. Am. A 14, 3026–3036 (1997). [CrossRef]

, 6

6. M. Paulus and O. J. F. Martin, “Green’s tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001). [CrossRef]

]. Other techniques have also been employed, including the finite-difference time-domain (FDTD) method [7

7. P. Yang and K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), pp. 173–221. [CrossRef]

, 8

8. A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).

].

For illumination by monochromatic plane waves, the DDA can be extended to targets that are spatially periodic and (formally) infinite in extent. This could apply, for example, to a periodic array of nanostructures in a laboratory setting, or it might be used to approximate a regolith by a periodic array of “target unit cells” with complex structure within each unit cell.

Generalization of the DDA (or coupled-dipole method) to periodic structures was first presented by Markel [9

9. V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281–2291 (1993). [CrossRef]

] for a 1-dimensional chain of dipoles, and more generally by Chaumet et al. [10

10. P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003). [CrossRef]

], who calculated the electric field near a 2-dimensional array of parallelepipeds illuminated by a plane wave. Chaumet and Sentenac [11

11. P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205437 (2005). [CrossRef]

] further extended the DDA to treat periodic structures with a finite number of defects.

From a computational standpoint, solving Maxwell’s equations for periodic structures is only slightly more difficult than calculating the scattering properties of a single “unit cell” from the structure. Since it is now feasible to treat targets with N106 dipoles (target volume 200λ3, where λ is the wavelength), it becomes possible to treat extended objects with complex substructure.

The objective of the present paper is to present the theory of the DDA applied to scattering and absorption by structures that are periodic in one or two spatial dimensions. We also generalize the standard formalism for describing the far-field scattering properties of finite targets (the 2×2 scattering amplitude matrix and 4×4 Mueller matrix) to describe scattering by periodic targets. We show how to calculate the Mueller matrix to describe scattering of arbitrarily polarized radiation. The theoretical treatment developed here has been implemented in the open-source code DDSCAT 7 (see Appendix A).

The theory of the DDA for periodic targets is reviewed in Section 2, and the formalism for describing the far-field scattering properties of periodic targets is presented in Sections 3, 4, 5. Transmission and reflection coefficients for targets that are periodic in two dimensions are obtained in Section 6.

The applicability and accuracy of the DDA method are discussed in Sections 7, 8, where we show scattering properties calculated using DDSCAT 7 for two geometries for which exact solutions are available for comparison: (1) an infinite cylinder and (2) an infinite slab of finite thickness. The numerical comparisons demonstrate that, for given λ, the DDA converges to the exact solution as the interdipole spacing d0.

2. DDA FOR PERIODIC TARGETS

The DDA is a general technique for calculating scattering and absorption of electromagnetic radiation by targets with arbitrary geometry. The basic theory of the DDA has been presented elsewhere [3

3. B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

]. Conceptually, the DDA consists of approximating the target of interest by an array of polarizable points with specified polarizabilities. Once the polarizabilities are specified, Maxwell’s equations can be solved accurately for the dipole array. When applied to finite targets, the DDA is limited by the number of dipoles N for which computations are feasible—the limitations may arise from large memory requirements or the large amount of computing that may be required to find a solution when N is large. In practice, modern desktop computers are capable of solving the DDA equations, as implemented in DDSCAT [3

3. B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

, 12

12. B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT6.1,” http://arXiv.org/abs/astro-ph/0409262 (2004).

], for N as large as 106.

Developed originally to study scattering from isolated, finite structures such as dust grains [1

1. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973). [CrossRef]

], the DDA can be extended to treat singly or doubly periodic structures. Consider a collection of N polarizable points defining a target unit cell (TUC). Now consider a target consisting of a 1-dimensional or 2-dimensional periodic array of identical TUCs, as illustrated in Figs. 1, 2 ; we will refer to these as 1-D or 2-D targets, although the constituent TUC may have an arbitrary 3-dimensional shape. For a monochromatic incident plane wave
Einc(r,t)=E0exp(ik0riωt),
(1)
the polarizations of the dipoles in the target will oscillate coherently. Each dipole will be affected by the incident wave plus the electric field generated by all of the other point dipoles.

Let index j=1,,N run over the dipoles in a single TUC, and let indices m, n run over replicas of the TUC. The (m,n) replica of dipole j is located at
rjmn=rj00+mLu+nLv,
(2)
where Lu and Lv are the lattice vectors for the array. For 1-D targets we let m vary, but set n=0. For 2-D targets, the area per TUC is
ATUCLu×Lv=LuLvsinθuv,
(3)
where θuv is the angle between Lu and Lv.

The replica dipole polarization Pjmn(t) is phase-shifted relative to Pj00(t):
Pjmn(t)=Pj00(t)exp[i(mk0Lu+nk0Lv)].
(4)

Define a matrix A such that Aj,kmnPkmn gives the electric field E at rj00 produced by an oscillating dipole Pkmn located at rkmn. Expressions for the 3×3 tensor elements of A have been presented elsewhere (e.g., [3

3. B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

]); A depends on the target geometry and wavelength of the incident radiation, but not on the target composition nor on the direction or polarization state of the incident wave.

Using Eq. (2) we may construct a matrix à such that, for jk, Ãj,kPk00 gives the electric field at rj00 produced by a dipole Pk00 and all of its replica dipoles Pkmn, and for j=k it gives the electric field at rj00 produced only by the replica dipoles:
Ãj,k=m=n=nmaxnmax(1δjkδm0δn0)Aj,kmnexp[i(mk0Lu+nk0Lv)],
(5)
where nmax=0 for 1-D targets and nmax= for 2-D targets, and δij is the Kronecker delta. For m,n, location j00 is in the radiation zone of dipole kmn, and the electric field falls off in magnitude only as 1r. The sums in Eq. (5) would be divergent were it not for the oscillating phases of the terms, which ensure convergence. Evaluation of these sums can be computationally demanding when k0Ly or k0Lz are small. Chaumet et al. [10

10. P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003). [CrossRef]

] have discussed methods for efficient evaluation of these sums.

We evaluate Eq. (5) numerically by introducing a factor exp[(γk0r)4] to smoothly suppress the contributions from large r, and truncating the sums:
Ãj,km,nAj,kmnexp[i(mk0Lu+nk0Lv)(γk0rj,kmn)4],
(6)
where rj,kmnrkmnrj00 and the summation is over (m,n) with rj,kmn2γk0, i.e., out to distances where the suppression factor exp[(γk0r)4]e16. For given k0, Lu, Lv, the Ãj,k depend only on rj00rk00, and therefore only O(8N) distinct Ãj,k require evaluation.

Ideally, one would use a very small value for the interaction cutoff parameter γ, but the number of terms [γ1 for 1-D or γ2 for 2-D] in Eq. (6) diverges as γ0. We show that setting γ0.001 ensures accurate results for the cases studied here.

The polarizations Pj00 of the dipoles in the TUC must satisfy the system of equations
Pj00=αj[Einc(rj)kjÃj,kPk00].
(7)
If there are N dipoles in one TUC, then Eq. (7) is a system of 3N linear equations where the polarizability tensors αj are obtained from lattice-dispersion-relation theory [13

13. B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti—Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993). [CrossRef]

, 14

14. D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://arXiv.org/abs/astro-ph/0403082 (2004).

]. After à has been calculated, Eq. (7) can be solved for Pj00 using iterative techniques when N1.

3. IN THE RADIATION ZONE

In the radiation zone kr1, the electric field due to dipole jmn is
Ejmn=k02exp(ik0rrjmn)rrjmn[1(rrjmn)(rrjmn)rrjmn2]Pjmn,
(8)
rrjmn=[r22rrjmn+rjmn2]12r{1rrjmnr2+12r2[rjmn2(rrjmnr)2]+}.
(9)
Define the unit vector k̂sksk0. We seek to sum the contribution of all the dipoles to the electric field propagating in direction k̂s. At location r=rk̂s, the dominant contribution will be from dipoles located within the Fresnel zone (see, e.g., [15

15. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

]), which will have a transverse radius RF(rk0)12. For dipoles within the Fresnel zone,
1rrjmn[1(rrjmn)(rrjmn)rrjmn2]Pjmn1r[1k̂sk̂s]Pjmn.
(10)
Thus, in the radiation zone,
E(r)=k03k0rexp(ik0r)[1k̂sk̂s]jPj00m,nexp(iψjmn),
(11)
with
ψjmnmk0Lu+nk0Lvksrjmn+k02r[rjmn2(k̂srjmn)2]
(12)
ksrj00+m(k0ks)Lu+n(k0ks)Lv+12k0r[m2(k02ksu0)Lu2+n2(k02ksv2)Lv2+2mn(k02LuLvksuksvLuLv)]+O(mLr),
(13)
where ksuksLuLu, ksvksLvLv, and terms of order (mLr) may be neglected because mLrRFrr12 as r. Thus, for r, the electric field produced by the oscillating dipoles is
Es{k02rexp(ik0r)[1k̂sk̂s]jPj00exp(iksrj00)}G(r,ks),
(14)
G(r,ks)m,nexp(iΦmn),
(15)
Φmnm(k0ks)Lu+n(k0ks)Lv+12k0r[m2(k02ksu2)Lu2+n2(k02ksv2)Lv2+2mn(k02LuLvksuksvLuLv)].
(16)
It is convenient to define
FTUC(k̂s)k03[1k̂sk̂s]j=1NPj00exp(iωtiksrj00),
(17)
so that the electric field produced by the dipoles is
Es=exp(iksriωt)k0rFTUC(k̂s)G(r,ks).
(18)
FTUC depends on the scattering direction k̂s and also on the direction of incidence k̂0 and polarization E0 of the incident wave.

3A. Isolated Finite Target: ν=0

We will refer to finite isolated targets—consisting only of the dipoles in a single TUC—as targets that are periodic in ν=0 dimensions. For this case, we simply set G=1 in Eq. (14); the scattered electric field in the radiation zone is
Es=exp(ik0riωt)k0rFTUC(k̂s).
(19)
The time-averaged scattered intensity is
Is=cEs28π=c8πk02r2FTUC2,
(20)
and the differential scattering cross section is
dCscadΩ=1k02FTUC2E02.
(21)

3B. Target Periodic in One Dimension: ν=1

Without loss of generality, we may assume that targets with 1-D periodicity repeat in the ŷ direction. It is easy to see from Eqs. (15, 16) that G=0 except for scattering directions satisfying
ksy=k0y+M2πLyM=0,±1,±2,,
(22)
where energy conservation (ks2=k02) limits the allowed values of the integer M:
(k0yk0)Ly2πM(k0y+k0)Ly2π.
(23)
If (k0+k0y)Ly<2π, then only M=0 scattering is allowed. Define polar angles α0 and αs for the incident and scattered radiation, so that
k0y=k0cosα0,
(24)
ksy=k0cosαs.
(25)
For each allowed value of ksy, the scattering directions define a cone:
ks=ksyŷ+(k02ksy2)12sinαssinα0[(k̂0ŷcosα0)cosζ+ŷ×k̂0sinζ],
(26)
where ζ is an azimuthal angle measured around the target axis ŷ. The sum for G(r,ks) is [since M must be an integer; see Eq. (22)]
G=m=exp(iΦm0)=m=exp[2πiMm+im22k0r(k02ksy2)Ly2]
(27)
limϵ0+dmexp[i(1+iϵ)2k0rm2(k02ksy2)Ly2]
(28)
=(2πik0r)12(k02ksy2)12Ly=(2πik0r)12k0Lysinαs,
(29)
and the scattered electric field
Es=(2πik0r)12exp(ik0riωt)k0LysinαsFTUC(k̂s)
(30)
shows the expected r12 behavior far from the scatterer (the distance from the cylinder axis is R=rsinαs).

For each allowed value of k̂s, the total time-averaged scattered power P¯sca, per unit length of the target, per unit azimuthal angle ζ, may be written
d2P¯scadLdζ=E028πcd2CscadLdζ,
(31)
where the differential scattering cross section is
d2CscadLdζ=8πE021cd2P¯scadLdζ=8πE02cE2c8πRsinαs
(32)
=2πk03Ly2FTUC2E02.
(33)

3C. Target Periodic in Two Dimensions: ν=2

For targets that are periodic in two dimensions, it is apparent from Eqs. (15, 16) that G=0 unless
(ksk0)Lu=2πM,M=0,±1,±2,,
(34)
(ksk0)Lv=2πN,N=0,±1,±2,.
(35)
The 2-D target constitutes a diffraction grating, with scattering allowed only in directions given by Eqs. (34, 35). It is convenient to define the reciprocal lattice vectors
u2πx̂×Lvx̂(Lu×Lv),v2πx̂×Lux̂(Lv×Lu).
(36)
The wave vector transverse to the surface normal is
ksk0+Mu+Nv.
(37)
Energy conservation requires that
ksx2=k02k0+Mu+Nv2>0.
(38)
For any (M,N) allowed by Eq. (38), there are two allowed values of ksx differing by a sign; one (with ksxk0x>0) corresponds to the (M,N) component of the transmitted wave, and the other (with ksxk0x<0) to the (M,N) component of the reflected wave. Define
sinα0k0xk0,
(39)
sinαsksxk0.
(40)
Note that α0=π2 for normal incidence, and α00 for grazing incidence. For ks satisfying Eqs. (34, 35), we have
G=m,nexp(iΦmn)=m,nexp{i2k0r[(k02ksu2)Lu2m2+(k02ksv2)Lv2n2+2(k02Lu:LvksuksvLuLv)mn]}limϵ0+dmdnexp{i(1+iϵ)2k0r[(k02ksu2)Lu2m2+(k02ksv2)Lv2n2+2(k02LuLvksuksvLuLv)mn]}=limϵ0+1ATUCdydzexp{i(1+iϵ)2k0r[k02(y2+z2)(ksyy+kszz)2]}=2πirk0ATUCsinαs.
(41)

The scattered electric field is
Es=2πiexp(iksriωt)k02ATUCsinαsFTUC(k̂s).
(42)
Note that Es is independent of distance x=rsinαs from the target, as expected for a target that is infinite in two directions. For pure forward scattering, ks=k0, we must sum the incident wave Einc and the radiated wave Es:
E=exp(ik0riωt)[E0+2πiFTUC(k̂s=k̂0)k02ATUCsinαs].
(43)
The cross section per unit target area A for scattering into direction (M,N) is (for ksk0)
dCsca(M,N)dA=E2sinαsE02sinα0
(44)
=4π2k04ATUC2sinα0sinαsFTUC(k̂s)2E02,
(45)
where Csca can be evaluated for either transmitted or reflected waves. For the special case ks=k0, the transmission coefficient T(M,N) is obtained from the total forward-propagating wave (43):
T(0,0)=1E02E0+2πiFTUC(k̂s=k̂0)k02ATUCsinα02.
(46)

4. SCATTERING AMPLITUDE MATRICES Si(νd)

4A. Isolated Finite Targets: ν=0

where
êi=êsk̂s×k̂0k̂s×k̂0=k̂s×k̂01(k̂sk̂0)2=ϕ̂s,
(48)
êik̂0×êi=k̂s(k̂sk̂0)k̂01(k̂sk̂0)2,
(49)
êsk̂s×ês=k̂0+(k̂sk̂0)k̂s1(k̂sk̂0)2=θ̂s
(50)
are the usual conventions for the incident and scattered polarization vectors parallel and perpendicular to the scattering plane (see, e.g., Sec. 3.2 of [16

16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

]).

4B. Target Periodic in One Dimension: ν=1

For targets with 1-D periodicity, it is natural to generalize the scattering amplitude matrix, so that—for directions k̂s for which scattering is allowed—the scattered electric field at a distance R=rsinαs from the target is
[EsêsEsês]=iexp(iksriωt)(k0R)12[S2(1d)S3(1d)S4(1d)S1(1d)][E0êiE0êi]
(51)
for ks satisfying Eqs. (22, 23, 24, 25, 26).

4C. Target Periodic in Two Dimensions: ν=2

For targets with 2-D periodicity, it is natural to generalize the scattering amplitude matrix so that, for directions ksk̂0 for which scattering is allowed, we write
[EsêsEsês]=iexp(iksriωt)[S2(2d)S3(2d)S4(2d)S1(2d)][E0êiE0êi]
(52)
for ks satisfying Eqs. (37, 38).

For the special case of forward scattering (M=N=0 and ks=k0), where the scattering plane is not simply defined by k0 and ks, it is natural to use k0 and the target normal x̂ to define the scattering plane. Thus
êi=êsk0×ksk0×ks,
(53)
with êi and ês defined by Eqs. (49, 50). For r
[EêsEês]=iexp(ik0riωt)[(S2(2d)i)00(S1(2d)i)][E0êiE0êi].
(54)

4D. Far-Field Scattering Amplitude Matrices

5. FAR-FIELD SCATTERING MATRIX FOR STOKES VECTORS

For a given scattering direction k̂s, the 2×2 complex amplitude matrix Si(νd)(ks) fully characterizes the far-field scattering properties of the target. The far-field scattering properties of an isolated finite target are characterized by the 4×4 dimensionless Mueller matrix Sαβ(0d), with the Stokes vector of radiation scattered into direction k̂s at a distance r from the target given by
Isca,α1(k0r)2β=14Sαβ(0d)Iinc,β,
(62)
where Iinc,β=(I,Q,U,V)inc is the Stokes vector for the radiation incident on the target. For 1-D targets, we define the dimensionless scattering matrix Sαβ(1d) by
Isca,α1k0Rβ=14Sαβ(1d)Iinc,β,
(63)
where R is the distance from the 1-D target. For 2-D targets, we define Sαβ(2d) by
Isca,αβ=14Sαβ(2d)Iinc,β.
(64)
The 4×4 scattering intensity matrix Sαβ(νd) is obtained from the scattering amplitude matrix elements Si(νd). Except for the special case of forward scattering (ks=k0) for 2-D targets, the equations are the same as Eq. (3.16) of Bohren and Huffman [16

16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

]. For example,
S11(νd)=12(S1(νd)2+S2(νd)2+S3(νd)2+S4(νd)2),
(65)
S21(νd)=12(S2(νd)2S1(νd)2S4(νd)2+S3(νd)2),
(66)
S14(νd)=Im(S2(νd)S3(νd)*S1(νd)S4(νd)*).
(67)
For the special case of forward scattering (k̂s=k̂0) for 2-D targets, it is necessary to replace S1(2d) and S2(2d) with (S1(2d)i) and (S2(2d)i) [cf. Eq. (54)]. Thus, for example,
S11(2d)(ks=k0)=12(S1(2d)i2+S2(2d)i2+S3(2d)2+S4(2d)2).
(68)

6. TRANSMISSION AND REFLECTION COEFFICIENTS FOR 2-D TARGETS

For unpolarized incident radiation, R11(M,N) is the fraction of the incident power that is reflected in diffraction component (M,N), T11(M,N) is the fraction that is transmitted in component (M,N), and 1M,N[R11(M,N)+T11(M,N)] is the fraction of the incident power that is absorbed.

7. EXAMPLE: INFINITE CYLINDER

DDSCAT 7 has been used to calculate scattering and absorption by an infinite cylinder consisting of a periodic array of disks of thickness d and period Ly=d (where d is the interdipole spacing). Figure 3a shows S11(1d) for refractive index m=1.33+0.01i, πDλ=50 (D is the cylinder diameter and λ the wavelength of the incident radiation), and incidence angle α0=60°. Because k0(1+cosα0)d<2π, Eqs. (22, 23) allow only M=0 scattering, with αs=α0. Also shown is the exact solution, calculated using a code written by Mackowski [17

17. D. Mackowski, private communication (2007).

]. Light scattering by cylinders is generally described by scattering amplitudes Ti; in Appendix B we provide expressions relating these Ti to the Si used here. Figure 3b shows the fractional error in S11(1d) calculated using DDSCAT. As d is decreased, the errors decrease. Excellent accuracy is obtained when the validity criterion [3

3. B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

] mkd0.5 is satisfied: the fractional error in S11 is typically less than a few%, except near deep minima in S11.

Figure 3c shows S21(1d), characterizing scattering of unpolarized light into the Stokes parameter Q (S21<0 corresponds to linear polarization perpendicular to the scattering plane). DDSCAT 7 and the exact solution are in very good agreement when mkd0.5. Note that although the error in S21(1d)(θ=0)2 is large compared to S21(0)=6, this is small compared to S11(1d)(0)1500: the scattered radiation is only slightly polarized.

The results in Fig. 3 were obtained using γ=0.001 to truncate the integrations. To see how the results depend on γ, Fig. 4 shows S11(1d) computed for the problem of Fig. 3 but using different values of γ. For azimuthal angles ζ>20°, the results for γ=0.005 and 0.001 are nearly indistinguishable; the difference between the computed result and the exact solution is evidently due to the finite number of dipoles used, rather than the choice of cutoff parameter γ. However, the results for forward scattering are more sensitive to the choice of γ, as is seen in Figs. 4c, 4d: it is necessary to reduce γ to 0.001 to attain high accuracy in the forward-scattering directions.

Table 1 gives the CPU times to calculate Ã, to then iteratively solve the scattering problem to a fractional error <105 (using double-precision arithmetic), and finally to evaluate the scattering intensities for several of the cases shown in Figs. 3, 4. For most cases the CPU time is dominated by the iterative solution using the conjugate gradient algorithm. While the time required to evaluate à might be reduced using the strategies suggested by [10

10. P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003). [CrossRef]

], this step is generally a subdominant part of the computation for targets with kd0.1.

The above results have been for a weakly absorbing cylinder. To confirm that the DDA can be applied to strongly absorbing material, Fig. 5 shows scattering calculated for a cylinder with m=2+i and x=πDλ=25. Once again, the accuracy is very good, with small fractional errors provided mkd0.5.

8. EXAMPLE: PLANE-PARALLEL SLAB

Consider a homogeneous plane-parallel slab with thickness h and refractive index m. Radiation incident on it at angle of incidence θi will either be specularly reflected or transmitted. The reflection and transmission coefficients R and T can be calculated analytically, taking into account multiple reflections within the slab [15

15. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

]. With an exact solution in hand, we can evaluate the accuracy of the DDA applied to this problem. Figure 6 shows results for two cases: a dielectric slab with m=1.50, and an absorbing slab with m=1.50+0.02i.

DDSCAT 7 was used to calculate reflection, transmission, and absorption by an infinite slab, generated from a TUC consisting of a single line of dipoles extending in the x direction, with Ly=Lz=d. The selection rules (34, 35, 38) allow only M=N=0: transmission or specular reflection. The reflection and transmission coefficients for radiation polarized parallel or perpendicular to the plane containing k̂ and the surface normal are
R=S11(1d)(ksx=k0x)+S12(1d)(ksx=k0x),
(72)
R=S11(1d)(ksx=k0x)S12(1d)(ksx=k0x),
(73)
T=S11(1d)(ksx=k0x)+S12(1d)(ksx=k0x),
(74)
T=S11(1d)(ksx=k0x)S12(1d)(ksx=k0x).
(75)
The DDA results are in excellent agreement with the exact results when the validity condition mk0d<0.5 is satisfied, but results with moderate accuracy are obtained even when mk0d1.

9. NEAR-FIELD EVALUATION

The polarizations Pj00 can be used to calculate the electric and magnetic fields at any point, including within or near the target, using the exact expression for E and B from a point dipole, modified by a function ϕ:
E(r,t)=eiωtjm,nexp(ik0Rjmn)Rjmn3ϕ(Rjmn){k02Rjmn×(Pjmn×Rjmn)+(1ik0Rjmn)Rjmn2[3Rjmn(RjmnPjmn)Rjmn2Pjmn]}+E0exp(ik0riωt),
(76)
B(r,t)=eiωtjm,nk2exp(ik0Rjmn)Rjmn2ϕ(Rjmn)(Rjmn×Pjmn)(11ik0Rjmn)+k̂0×E0exp(ik0riωt),
(77)
Rjmnrrjmn,
(78)
ϕ(R)exp[γ(k0R)4]×{1forRd(Rd)4forR<d}.
(79)

The function ϕ(R) smoothly suppresses the (oscillating) contribution from distant dipoles in order to allow the summations to be truncated, just as in Eq. (6) for evaluation of Ãj,k. If r is within the target or near the target surface, the summations over (m,n) are limited to Rjmn2γk0. The (Rd)4 factor suppresses the R3 divergence of E as r approaches the locations of individual dipoles, and at the dipole locations results in E that is exactly equal to the field that is polarizing the dipoles in the DDA formulation. Evaluation of Eqs. (76, 77) is computationally intensive, because the summations jm,n typically have many terms.

To illustrate the accuracy, we consider the infinite slab of Fig. 6b, with refractive index m=1.5+0.02i and radiation incident at an angle θi=40°. Figure 7 shows the time-averaged E2E02 for slab thickness h=0.2λ—near a minimum in transmission, and a maximum in reflection [see Fig. 6b]. The program DDFIELD (see Appendix A) was used to evaluate E along two lines normal to the slab: track 1 passes directly through dipole sites, and track 2 passes midway between the four nearest dipoles as it crosses each dipole layer. The E fields calculated along tracks 1 and 2 are very similar, although of course not identical. Within the slab, E along track 2 tends to be slightly smaller than along track 1, but for this example the difference is typically less than 1%. Figure 7 shows results for the slab represented by hd=Nx=10 and 20 dipole layers (with mkd=0.19 and 0.094, respectively).

Even for Nx=10, the electric field at points more than a distance d from the edge is obtained to within 2% accuracy at worst, which is perhaps not surprising because, as seen in Fig. 6, the calculated transmission and reflection coefficients are very accurate. The discontinuity in E2 at the boundary is spread out over a distance d. The DDA obviously cannot reproduce field structure near the target surface on scales smaller than the dipole separation d, but fields on scales larger than d appear to be quite accurate. DDSCAT and DDFIELD should be useful tools for studying electromagnetic fields around arrays of nanostructures, such as gold nanodisks [18

18. Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007). [CrossRef]

, 19

19. W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).

].

10. SUMMARY

The principal results of this study are as follows:
  1. The DDA is generalized to treat targets that are periodic in one or two spatial dimensions. Scattering and absorption of monochromatic plane waves can be calculated using algorithms that parallel those used for finite targets.
  2. A general formalism is presented for description of far-field scattering by targets that are periodic in one or two dimensions using scattering amplitude matrices and Mueller matrices that are similar in form to those for finite targets.
  3. The accuracy of the DDA for periodic targets is tested for two examples: infinite cylinders and infinite slabs. The DDA, as implemented in DDSCAT 7, is accurate provided the validity criterion mkd0.5 is satisfied.
  4. We show how the DDA solution can be used to evaluate E and B within and near the target, with calculations for an infinite slab used to illustrate the accuracy of near-field calculations.

APPENDIX A: DDSCAT 7 AND DDFIELD

The theoretical developments reported here have been implemented in a new version of the open-source code DDSCAT [20]. DDSCAT 7 is written in Fortran 90 with dynamic memory allocation and the option to use either single- or double-precision arithmetic. DDSCAT 7 includes options for various target geometries, including a number of periodic structures. A program DDFIELD for near-field calculations is also provided.

DDSCAT 7 offers the option of using an implementation of BICGSTAB with enhancement to maintain convergence in finite precision arithmetic [21

21. M. A. Botchev, SUBROUTINE ZBCG2, http://www.math.uu.nl/people/vorst/zbcg2.f90 (2001).

]. The matrix-vector multiplications ÃP are accomplished efficiently using FFTs [22

22. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier transform techniques to the discrete dipole approximation,” Opt. Lett. 16, 1198–1200 (1990). [CrossRef]

]. Documentation for DDSCAT is available from ArXiv [23

23. B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.0,” http://arXiv.org/abs/0809.0337 (2008).

], with additional information available from [24].

In addition to differential scattering cross sections, DDSCAT reports dimensionless “efficiency factors” QxCx(TUC)πaeff2 for scattering and absorption, where Cx(TUC) is the total cross section for scattering or absorption per TUC, normalized by πaeff2, where aeff(3VTUC4π)13 is the radius of a sphere with volume equal to the solid volume VTUC in one TUC.

In the case of 1-D targets with periodicity Ly in the y direction, the absorption, scattering, and extinction cross sections per unit target length are
dCxdL=1LyQxπaeff2
(A1)
for x=ext, sca, and abs, where Qx are the efficiency factors calculated by DDSCAT.

In the case of 2-D targets with periodicities Lu and Lv, the absorption, scattering, and extinction cross sections per unit target area are
dCxdA=Qxπaeff2LuLvsinθuv.
(A2)

APPENDIX B: RELATION BETWEEN Si AND Ti FOR INFINITE CYLINDERS

The analytic solution for infinite cylinders decomposes the incident and scattered radiation into components polarized parallel and perpendicular to planes containing the cylinder axis and the propagation vector k̂0 or k̂s. These polarization basis states differ from the choice that is usual for scattering by finite particles, where it is customary to decompose the incident and scattered waves into components polarized parallel and perpendicular to the scattering plane—the plane containing k̂0 and k̂s.

ACKNOWLEDGMENTS

Table 1. CPU Time to Calculate Scattering by m=1.33+0.01i Infinite Cylinders on Single-Core 2.4GHz AMD Opteron Model 250

table-icon
View This Table
Fig. 1 (a) Target consisting of a 1-D array of TUCs and (b) showing how an infinite cylinder can be constructed from disklike TUCs (lower). The M=0 scattering cone with αs=α0 is illustrated.
Fig. 2 (a) Target consisting of a 2-D array of TUCs and (b) showing how an infinite slab is created from TUCs consisting of a single “line” of dipoles.
Fig. 3 Scattering by an infinite cylinder with diameter D and m=1.33+0.01i, for radiation with x=πDλ=50 and incidence angle α0=60°. (a) S11(1d): solid curve, exact solution; broken curves, DDA results for Dd=256, 360, and 512 (N=51,676, 102,036, 206,300 dipoles per TUC). (b) Fractional error in S11(1d)(DDA). (c) S21(1d). (d) Error in S21(1d).
Fig. 4 Scattering by an infinite cylinder with diameter D and m=1.33+0.01i, for radiation with πDλ=50 and incidence angle α0=60°. (a) Exact solution (solid curve) and DDA results for Dd=512 and various values of the interaction cutoff parameter γ. (b) Fractional error in S11(1d). (c), (d) Same as (a), (b), but expanding the region 0<ζ<20°. For this case, results computed with γ=0.002 and 0.001 are nearly indistinguishable.
Fig. 5 Light scattered by an infinite cylinder with m=2+i for radiation with x=2πRλ=25 and incidence angle α0=60°. (a) S11(1d): solid curve, exact solution; broken curves, DDA results for Dd=128, 180, and 256 (N=12,972, 25,600, 51,676 dipoles per TUC). (b) Fractional error in S11(1d).
Fig. 6 Transmission and reflection coefficients for radiation of wavelength λ incident at angle θi=(π2α0)=40° relative to the normal on a slab with thickness h, incident E and ⊥ to the scattering plane, as a function of Re(m)hλ. (a) Nonabsorbing slab with m=1.5. (b) Absorbing slab with m=1.5+0.02i. Solid curves exact solution; symbols, results calculated with the DDA using dipole spacing d=h10, h20, and h40.
Fig. 7 E2E02 within and near the dielectric slab of Fig. 6b for slab thickness h=0.2λ, incidence angle αi=40°, and incident polarizations ∥ and ⊥ to the scattering plane. Results were calculated using Eq. (76) with the slab represented by Nx=10 and Nx=20 dipole layers (i.e., dipole spacing d=0.1h and 0.05h). The circles along track 1 are at points where dipoles are located.
1.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973). [CrossRef]

2.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

3.

B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

4.

B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), pp. 131–145.

5.

R. Schmehl, B. M. Nebeker, and E. D. Hirleman, “Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique,” J. Opt. Soc. Am. A 14, 3026–3036 (1997). [CrossRef]

6.

M. Paulus and O. J. F. Martin, “Green’s tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001). [CrossRef]

7.

P. Yang and K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), pp. 173–221. [CrossRef]

8.

A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).

9.

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281–2291 (1993). [CrossRef]

10.

P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003). [CrossRef]

11.

P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205437 (2005). [CrossRef]

12.

B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT6.1,” http://arXiv.org/abs/astro-ph/0409262 (2004).

13.

B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti—Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993). [CrossRef]

14.

D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://arXiv.org/abs/astro-ph/0403082 (2004).

15.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

16.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

17.

D. Mackowski, private communication (2007).

18.

Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007). [CrossRef]

19.

W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).

20.

http://www.astro.princeton.edu/~draine/DDSCAT.html (2008).

21.

M. A. Botchev, SUBROUTINE ZBCG2, http://www.math.uu.nl/people/vorst/zbcg2.f90 (2001).

22.

J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier transform techniques to the discrete dipole approximation,” Opt. Lett. 16, 1198–1200 (1990). [CrossRef]

23.

B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.0,” http://arXiv.org/abs/0809.0337 (2008).

24.

http://ddscat.wikidot.com.

OCIS Codes
(260.0260) Physical optics : Physical optics
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.5298) Diffraction and gratings : Photonic crystals
(290.5825) Scattering : Scattering theory

ToC Category:
Diffraction and Gratings

History
Original Manuscript: June 24, 2008
Manuscript Accepted: August 19, 2008
Published: October 14, 2008

Citation
Bruce T. Draine and Piotr J. Flatau, "Discrete-dipole approximation for periodic targets: theory and tests," J. Opt. Soc. Am. A 25, 2693-2703 (2008)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-25-11-2693


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References

  1. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973). [CrossRef]
  2. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848-872 (1988). [CrossRef]
  3. B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491-1499 (1994). [CrossRef]
  4. B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 131-145.
  5. R. Schmehl, B. M. Nebeker, and E. D. Hirleman, “Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique,” J. Opt. Soc. Am. A 14, 3026-3036 (1997). [CrossRef]
  6. M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001). [CrossRef]
  7. P. Yang and K. N. Liou, “Finite difference time domain method for light scattering by nonspherical and inhomogeneous particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 173-221. [CrossRef]
  8. A. Taflove and S. C. Hagness, Advances in Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).
  9. V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281-2291 (1993). [CrossRef]
  10. P. C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, 165404 (2003). [CrossRef]
  11. P. C. Chaumet and A. Sentenac, “Numerical simulations of the electromagnetic field scattered by defects in a double-periodic structure,” Phys. Rev. B 72, 205437 (2005). [CrossRef]
  12. B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT6.1,” http://arXiv.org/abs/astro-ph/0409262 (2004).
  13. B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti--Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685-697 (1993). [CrossRef]
  14. D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://arXiv.org/abs/astro-ph/0403082 (2004).
  15. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  16. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  17. D. Mackowski, private communication (2007).
  18. Z. N. Utegulov, J. M. Shaw, B. T. Draine, S. A. Kim, and W. L. Johnson, “Surface-plasmon enhancement of Brillouin light scattering from gold-nanodisk arrays on glass,” Proc. SPIE 6641, 66411M (2007). [CrossRef]
  19. W. L. Johnson, S. A. Kim, Z. N. Utegulov, and B. T. Draine, “Surface-plasmon fields in two-dimensional arrays of gold nanodisks,” Proc. SPIE7032(2008) (to be published).
  20. http://www.astro.princeton.edu/~draine/DDSCAT.html (2008).
  21. M. A. Botchev, SUBROUTINE ZBCG2, http://www.math.uu.nl/people/vorst/zbcg2.f90 (2001).
  22. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier transform techniques to the discrete dipole approximation,” Opt. Lett. 16, 1198-1200 (1990). [CrossRef]
  23. B. T. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.0,” http://arXiv.org/abs/0809.0337 (2008).
  24. http://ddscat.wikidot.com.

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