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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 23865–23871
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Extended discrete dipole approximation and its application to bianisotropic media

R. Alcaraz de la Osa, P. Albella, J. M. Saiz, F. González, and F. Moreno  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 23865-23871 (2010)
http://dx.doi.org/10.1364/OE.18.023865


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Abstract

In this research we introduce the formalism of the extension of the discrete dipole approximation to a more general range of tensorial relative permittivity and permeability. Its performance is tested in the domain of applicability of other methods for the case of composite materials (nanoshells). Then, some early results on bianisotropic nanoparticles are presented, to show the potential of the Extended Discrete Dipole Approximation (E-DDA) as a new tool for calculating the interaction of light with bianisotropic scatterers.

© 2010 Optical Society of America

1. Introduction

2. E-DDA code

2.1. Method description

A radiative correction is also included to take into account the phase lag between the incident light and the scattered light radiated by the dipole [21

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

]:
α¯¯=α¯¯CM(I¯¯ik3α¯¯CM6π)1
(3)
χ¯¯=χ¯¯CM(I¯¯ik3χ¯¯CM6π)1
(4)

Although this radiative correction is precise enough in most of the real situations, further corrections on the dipole polarizabilities are being used by several authors, developing prescriptions aimed to improve the accuracy [22

22. 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]

, 23

23. D. Gutkowicz-Krusin and B. T. Draine,“Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” (2004). arXiv:astro-ph/0403082v1.

].

As in the conventional DDA [17

17. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]

, 24

24. M. Yurkin, V. Maltsev, and A. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 546–557 (2007). [CrossRef]

], each dipole feels an electromagnetic field sum of the incident field at that site, and the contributions due to all the other dipoles. Working with the same formalism used by Mulholland et. al [25

25. G. W. Mulholland, C. F. Bohren, and K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10(8), 2533–2546 (1994). [CrossRef]

], which utilizes the M.K.S. unit’s system, we can express the total electric and magnetic fields at lattice site j as:
Ej=Einc,j+kjNα¯¯jC¯¯jkEkμ0μmɛ0ɛmkjNχ¯¯jf¯¯jkHk
(5)
Hj=Hinc,j+ΣkjNχ¯¯jC¯¯jkHk+ɛ0ɛmμ0μmΣkjNα¯¯jf¯¯jkEk
(6)
, where Einc,j = E0 exp(ik · rjk) and Hinc,j = H0 exp(ik · rjk), with H0=[ɛ0ɛm/(μ0μm)]1/2(k×E0*) and where label[*] denotes the complex conjugate. The matrixes C̿jk and f̿jk involved in Eqs. (5) and (6) can be found in Ref. [25

25. G. W. Mulholland, C. F. Bohren, and K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10(8), 2533–2546 (1994). [CrossRef]

].

In order to solve the set of equations in Eqs. (5) and (6), it is convenient to rewrite them using its associated matrix form. Then, one can easily obtain the system of equations in the form:
A¯¯x=b
(7)
, where A̿ is a 6N × 6N matrix, x is the 6N-dimensional vector of unknowns given by x = (E1, H1, E2, H2, . . . , EN, HN) and b is the 6N-dimensional vector of independent terms containing the incident field at each lattice site: b = (Einc,1, Hinc,1, Einc,2, Hinc,2, . . . , Einc,N, Hinc,N). This system can be solved using some of the iterative methods already available. Initially we used the successive approximations method, proving to be very slow and not convergent on most cases. At this time, E-DDA implements the Complex-Conjugate Gradient (CCG) method through the Botchev’s subroutine [26

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

].

2.2. Code Performance

The main achievement of our code in its current version is versatility, in the sense that it is able to deal with situations involving arbitrary electric and magnetic susceptibilities in a broad sense. In the following section we shall survey the most representative situations at reach for this code, from the most basic one of a dielectric, to the most generals, like bianisotropic materials, metals, metamaterials (in particular left-handed materials, with both real parts of ɛr and μr negative) or magneto-optical materials (with ɛ̿r an antisymmetric tensor). Some of these situations admit direct comparison with past DDA versions, while other can only be treated with other calculations methods, and not in a feasible way. The former will contribute to validate our code, while the latter constitute genuine novel results. Both together, prove the potential of E-DDA as a new and reliable computing tool. It is interesting to remark that the incident wave polarization can be arbitrary (elliptical in the most general case), allowing polarimetric calculations to be performed. As for extinction and absorption cross-sections (and efficiencies), they have been implemented following [9

9. P. C. Chaumet and A. Rahmani, “Coupled-dipole method for magnetic and negative-refraction materials,” J. Quant. Spectrosc. Radiat. Transf. 110(1–2), 22–29 (2009).

]:
Cext=kɛ0ɛm|E0|2j=1N[Einc,j*pj+μ0μmHinc,j*mj]
(8)
Cabs=kɛ0ɛm|E0|2j=1N{[Ej*pj]k36πɛ0ɛm|pj|2+μ0μm[[Hj*mj]k36π|mj|2]}
(9)

The scattering cross-section can be easily obtained by the difference of the extinction and absorption cross-sections: Csca = CextCabs, but a more convenient way is to compute the far-field scattered by the object [27

27. J. D. Jackson, Classical Electrodynamics Third Edition, 3rd ed. (Wiley, 1998).

]:
Csca=k216π2ɛ0ɛm|E0|2|j=1Nexp(iknrj){1ɛ0ɛm[pj(npj)n]μ0μmn×mj}|2dΩ
(10)
, where n is an unit vector in the direction of scattering. Additionally, we can also define the phase-lag cross-section in terms of the imaginary part of the forward-scattering amplitude [21

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

]:
Cpha=k2ɛ0ɛm|E0|2j=1N[Einc,j*pj+μ0μmHinc,j*mj]
(11)

Conventional angular scattering patterns (scattered intensities) can be obtained, as well as the full Mueller Matrix of the system, completing the far-field results list. As for local magnitudes, these include dipole moments, fields or Poynting vector distributions, local phase functions, etc.

3. Testing the code

To check the reliability of E-DDA, we have performed some calculations on systems that conventional methods can solve. Fig. 1 shows the extinction, absorption and scattering efficiencies for both a gold sphere of radius R = 20 nm, and a sphere with a dielectric (ɛc = 2) core (inclusion) of radius R = 12 nm and a metallic (gold, optical constants taken from Johnson and Christy [28

28. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B. 6, 4370–4379 (1972). [CrossRef]

]) shell, for an external radius R = 20 nm. Comparison between our code and the well-proved DDSCAT code from B. T. Draine [17

17. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]

, 18

18. B. T. Draine and P. J. Flatau, User Guide for the Discrete Dipole Approximation Code DDSCAT 7.1 (2010). http://arxiv.org/abs/1002.1505.

] is also presented, finding a very good agreement in both, the spectral shape and the absolute differences for all the efficiencies within the optical range.

Fig. 1 Extinction, absorption and scattering efficiencies for both a gold sphere of radius R = 20 nm, and a sphere with a dielectric (ɛc = 2) core (inclusion) of radius R = 12 nm and a metallic (gold) shell, for an external radius R = 20 nm. Comparison between our code and the well-established DDSCAT code is also provided. The dipole spacing was d = 4 nm, with N = 515.

4. Bianisotropic media

When the E-DDA is used in the most general case of a bianisotropic scatterer, it is possible to find a situation in which we already have a knowledge of the system’s behavior. Such is the case of particles with directional scattering imposed by their optical constants as, for instance, the null-scattering conditions proposed by Kerker et. al. [29

29. M. Kerker, D.-S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73(6), 765–767 (1983). [CrossRef]

]. It will be shown here that these well-known conditions remain valid for bianisotropic media. As a reminder, Kerker’s null scattering conditions are obtained when the electric and magnetic polarizabilities meet certain stipulations. In particular, when α̿ = χ̿ the backscatter gain equals zero, and when α̿ = –χ̿ the forward scatter is zero. These conditions lead to ɛ̿r = μ̿r for the first case, and ɛ̿r = (4I̿μ̿r)(2μ̿r + I̿)−1 for the latter. However, these relations fail to include the radiative correction [21

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

]. It can be shown that the zero-backward scattering condition remains valid, that is, ɛ̿r = μ̿r. But, for the zero-forward scattering condition, we obtain the new relation:
μ¯¯r=[4I¯¯ɛ¯¯ri(kd)3π(ɛ¯¯rI¯¯)][2I¯¯+ɛ¯¯ri(kd)3π(ɛ¯¯rI¯¯)]1
(12)

Let us now consider a set of three different permittivity tensors. The first case (isotropic material) is the scalar case:
ɛ¯¯r1=(2.0+0.01i0002.0+0.01i0002.0+0.01i)=(2.0+0.01i)I¯¯
(13)

The second case is chosen in the form of a typical magneto-optical material, with an antisymmetric relative electric permittivity tensor ɛ̿r2 given by (magnetization along z-axis):
ɛ¯¯r2=(2.0+0.01i0.3+0.2i00.30.2i2.0+0.01i0002.0+0.01i)
(14)

For the third case we propose a symmetric permittivity tensor ɛ̿r3:
ɛ¯¯r3=(2.0+0.01i0.3+0.2i00.30.2i2.0+0.01i0002.0+0.01i)
(15)

For each of these three cases we shall define two values of the relative magnetic permeability tensor in order to fulfill each of the zero scattering conditions (this makes six different materials as a whole). We now consider an sphere of diameter D = 20 nm made of those materials, illuminated with a wavelength of λ = 500 nm, obtaining that the zero-forward scattering condition is, for each case:
μ¯¯r1=(0.43.6005×103i)I¯¯
(16)
μ¯¯r2=(3.9232×1012.0508×102i1.0899×1016.8489×102i01.0899×101+6.8489×102i3.9232×1012.0508×102i0000.43.6005×103i)
(17)
μ¯¯r3=(4.0712×101+1.3870×102i1.0804×1017.3802×102i01.0804×1017.3802×102i4.0712×101+1.3870×102i0000.43.6005×103i)
(18)
and μ̿ri = ɛ̿ri (i = 1, 2, 3) for the zero backward. The scattering patterns show that Kerker’s conditions are being satisfied (Fig. 2). It is worth noticing that, while the isotropic case can be computed by means of conventional numerical methods (producing a perfect match), the bianisotropic ones do not admit such comparison in a feasible way. By using E-DDA, we obtain results that show that the systems behave exactly as expected from the theoretically imposed condition, that is, exhibiting zero-back and zero-forward scattering.

Fig. 2 a) Zero-backward scattering (μ̿r = ɛ̿r). b) Zero-forward scattering. In every case, the dipole spacing was d = 2 nm, with N = 515.

5. Conclusions

We have extended the applicability range of the discrete dipole approximation to the case of anisotropic and magnetic materials, including the bianisotropic case. We have tested the validity of the method for a case where its applicability range overlaps with the one of a well-proved code, which includes inhomogeneities and presence of metallic and dielectric media, finding a very good agreement. We have applied the proposed method to a situation out of reach for current implementations of the DDA, like the null-scattering (backward and forward) conditions for bianisotropic media. We have verified numerically these conditions in its tensorial form.

Acknowledgments

This research has been supported by MICINN (Ministerio de Ciencia e Innovacion) under project #FIS2007-60158. The authors thankfully acknowledge the computer resources provided by the RES (Red Espaola de Supercomputacion) node (Altamira) at Universidad de Cantabria. R. Alcaraz de la Osa also thanks the Ministry of Education of Spain for his FPU grant.

References and links

1.

S. Albaladejo, R. Gómez-Medina, L. S. Froufe-Pérez, H. Marinchio, R. Carminati, J. F. Torrado, G. Armelles, A. García-Martín, and J. J. Sáenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle,” Opt. Express 18(4), 3556–3567 (2010). [CrossRef] [PubMed]

2.

B. Sepúlveda, J. B. González-Díaz, A. García-Martín, L. M. Lechuga, and G. Armelles, “Plasmon-induced magneto-optical activity in nanosized gold disks,” Phys. Rev. Lett. 104(14), 147401 (2010). [CrossRef] [PubMed]

3.

G. Ctistis, E. Papaioannou, P. Patoka, J. Gutek, P. Fumagalli, and M. Giersig, “Optical and magnetic properties of hexagonal arrays of subwavelength holes in optically thin cobalt films,” Nano Lett. 9(1), 1–6 (2008). [CrossRef] [PubMed]

4.

D. A. Smith and K. L. Stokes, “Discrete dipole approximation for magneto-optical scattering calculations,” Opt. Express 14(12), 5746–5754 (2006). [CrossRef] [PubMed]

5.

N. B. Piller and O. J. F. Martin, “Extension of the generalized multipole technique to three-dimensional anisotropic scatterers,” Opt. Lett. 23(8), 579–581 (1998). [CrossRef]

6.

V. Agranovich and Y. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3(1), 1–9 (2009). [CrossRef]

7.

A. Alu and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17(7), 5723–5730 (2009). [CrossRef] [PubMed]

8.

Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express 16(3), 2068–2079 (2008). [CrossRef] [PubMed]

9.

P. C. Chaumet and A. Rahmani, “Coupled-dipole method for magnetic and negative-refraction materials,” J. Quant. Spectrosc. Radiat. Transf. 110(1–2), 22–29 (2009).

10.

A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006). [CrossRef] [PubMed]

11.

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002). [CrossRef]

12.

P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express 15(11), 6857–6867 (2007). [CrossRef] [PubMed]

13.

B. García-Cámara, F. Moreno, F. González, J. M. Saiz, and G. Videen, “Light scattering resonances in small particles with electric and magnetic properties,” J. Opt. Soc. Am. A 25(2), 327–334 (2008). [CrossRef]

14.

B. García-Cámara, F. González, F. Moreno, and J. M. Saiz, “Exception for the zero-forward-scattering theory,” J. Opt. Soc. Am. A 25(11), 2875–2878 (2008). [CrossRef]

15.

P. Albella, F. Moreno, J. M. Saiz, and F. González, “Surface inspection by monitoring spectral shifts of localized plasmon resonances,” Opt. Express 16(17), 12,872–12,879 (2008). [CrossRef]

16.

P. Albella, J. M. Saiz, J. M. Sanz, F. González, and F. Moreno, “Nanoscopic surface inspection by analyzing the linear polarization degree of the scattered light,” Opt. Lett. 34(12), 1906–1908 (2009). [CrossRef] [PubMed]

17.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]

18.

B. T. Draine and P. J. Flatau, User Guide for the Discrete Dipole Approximation Code DDSCAT 7.1 (2010). http://arxiv.org/abs/1002.1505.

19.

A. Akyurtlu and D. Werner, “Modeling of transverse propagation through a uniaxial bianisotropic medium using the finite-difference time-domain technique,” IEEE Trans. Antennas Propag. 52, 3273–3279 (2004). [CrossRef]

20.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009). [CrossRef] [PubMed]

21.

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

22.

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]

23.

D. Gutkowicz-Krusin and B. T. Draine,“Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” (2004). arXiv:astro-ph/0403082v1.

24.

M. Yurkin, V. Maltsev, and A. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 546–557 (2007). [CrossRef]

25.

G. W. Mulholland, C. F. Bohren, and K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10(8), 2533–2546 (1994). [CrossRef]

26.

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

27.

J. D. Jackson, Classical Electrodynamics Third Edition, 3rd ed. (Wiley, 1998).

28.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B. 6, 4370–4379 (1972). [CrossRef]

29.

M. Kerker, D.-S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73(6), 765–767 (1983). [CrossRef]

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(260.2110) Physical optics : Electromagnetic optics
(290.5850) Scattering : Scattering, particles
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(160.3918) Materials : Metamaterials
(160.2710) Materials : Inhomogeneous optical media

ToC Category:
Physical Optics

History
Original Manuscript: August 25, 2010
Revised Manuscript: September 22, 2010
Manuscript Accepted: September 28, 2010
Published: October 28, 2010

Citation
R. Alcaraz de la Osa, P. Albella, J. M. Saiz, F. González, and F. Moreno, "Extended discrete dipole approximation and its application to bianisotropic media," Opt. Express 18, 23865-23871 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-23865


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References

  1. S. Albaladejo, R. Gómez-Medina, L. S. Froufe-Pérez, H. Marinchio, R. Carminati, J. F. Torrado, G. Armelles, A. García-Martín, and J. J. Sáenz, “Radiative corrections to the polarizability tensor of an electrically small anisotropic dielectric particle,” Opt. Express 18(4), 3556–3567 (2010). [CrossRef] [PubMed]
  2. B. Sepúlveda, J. B. González-Díaz, A. García-Martín, L. M. Lechuga, and G. Armelles, “Plasmon-induced magneto-optical activity in nanosized gold disks,” Phys. Rev. Lett. 104(14), 147401 (2010). [CrossRef] [PubMed]
  3. G. Ctistis, E. Papaioannou, P. Patoka, J. Gutek, P. Fumagalli, and M. Giersig, “Optical and magnetic properties of hexagonal arrays of subwavelength holes in optically thin cobalt films,” Nano Lett. 9(1), 1–6 (2008). [CrossRef] [PubMed]
  4. D. A. Smith and K. L. Stokes, “Discrete dipole approximation for magneto-optical scattering calculations,” Opt. Express 14(12), 5746–5754 (2006). [CrossRef] [PubMed]
  5. N. B. Piller and O. J. F. Martin, “Extension of the generalized multipole technique to three-dimensional anisotropic scatterers,” Opt. Lett. 23(8), 579–581 (1998). [CrossRef]
  6. V. Agranovich and Y. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials (Amst.) 3(1), 1–9 (2009). [CrossRef]
  7. A. Alu and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17(7), 5723–5730 (2009). [CrossRef] [PubMed]
  8. Y. You, G. W. Kattawar, P.-W. Zhai, and P. Yang, “Zero-backscatter cloak for aspherical particles using a generalized DDA formalism,” Opt. Express 16(3), 2068–2079 (2008). [CrossRef] [PubMed]
  9. P. C. Chaumet and A. Rahmani, “Coupled-dipole method for magnetic and negative-refraction materials,” J. Quant. Spectrosc. Radiat. Transf. 110(1–2), 22–29 (2009).
  10. A. Alú, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006). [CrossRef] [PubMed]
  11. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002). [CrossRef]
  12. P. Albella, F. Moreno, J. M. Saiz, and F. González, “Backscattering of metallic microstructures with small defects located on flat substrates,” Opt. Express 15(11), 6857–6867 (2007). [CrossRef] [PubMed]
  13. B. García-Cámara, F. Moreno, F. González, J. M. Saiz, and G. Videen, “Light scattering resonances in small particles with electric and magnetic properties,” J. Opt. Soc. Am. A 25(2), 327–334 (2008). [CrossRef]
  14. B. García-Cámara, F. González, F. Moreno, and J. M. Saiz, “Exception for the zero-forward-scattering theory,” J. Opt. Soc. Am. A 25(11), 2875–2878 (2008). [CrossRef]
  15. P. Albella, F. Moreno, J. M. Saiz, and F. González, “Surface inspection by monitoring spectral shifts of localized plasmon resonances,” Opt. Express 16(17), 12,872–12,879 (2008). [CrossRef]
  16. P. Albella, J. M. Saiz, J. M. Sanz, F. González, and F. Moreno, “Nanoscopic surface inspection by analyzing the linear polarization degree of the scattered light,” Opt. Lett. 34(12), 1906–1908 (2009). [CrossRef] [PubMed]
  17. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11(4), 1491–1499 (1994). [CrossRef]
  18. B. T. Draine and P. J. Flatau, User Guide for the Discrete Dipole Approximation Code DDSCAT 7.1 (2010). http://arxiv.org/abs/1002.1505.
  19. A. Akyurtlu and D. Werner, “Modeling of transverse propagation through a uniaxial bianisotropic medium using the finite-difference time-domain technique,” IEEE Trans. Antenn. Propag. 52, 3273–3279 (2004). [CrossRef]
  20. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009). [CrossRef] [PubMed]
  21. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]
  22. 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]
  23. D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” (2004). arXiv:astro-ph/0403082v1.
  24. M. Yurkin, V. Maltsev, and A. Hoekstra, “The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength,” J. Quant. Spectrosc. Radiat. Transf. 106(1–3), 546–557 (2007). [CrossRef]
  25. G. W. Mulholland, C. F. Bohren, and K. A. Fuller, “Light scattering by agglomerates: coupled electric and magnetic dipole method,” Langmuir 10(8), 2533–2546 (1994). [CrossRef]
  26. M. A. Botchev, SUBROUTINE ZBCG2, http://www.math.uu.nl/people/vorst/zbcg2.f90 (2001).
  27. J. D. Jackson, Classical Electrodynamics Third Edition, 3rd ed. (Wiley, 1998).
  28. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
  29. M. Kerker, D.-S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73(6), 765–767 (1983). [CrossRef]

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Figures

Fig. 1 Fig. 2
 

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