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

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  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]

2010 (2)

2009 (5)

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

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]

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]

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).

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]

2008 (5)

2007 (2)

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]

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]

2006 (2)

2004 (1)

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]

2002 (1)

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]

1998 (1)

1994 (2)

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]

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

1993 (1)

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]

1988 (1)

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

1983 (1)

1972 (1)

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

Agranovich, V.

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]

Akyurtlu, A.

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]

Albaladejo, S.

Albella, P.

Alu, A.

Alú, A.

Armelles, G.

Bohren, C. F.

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]

Carminati, R.

Chaumet, P. C.

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).

Chong, Y.

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]

Christy, R.

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

Ctistis, G.

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]

Draine, B. T.

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

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]

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

Engheta, N.

Flatau, P. J.

Froufe-Pérez, L. S.

Fuller, K. A.

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]

Fumagalli, P.

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]

García-Cámara, B.

García-Martín, A.

Gartstein, Y.

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]

Giersig, M.

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]

Giles, C. L.

Gómez-Medina, R.

González, F.

González-Díaz, J. B.

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]

Goodman, J.

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]

Gutek, J.

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]

Hoekstra, A.

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]

Joannopoulos, J. D.

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]

Johnson, P.

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

Kattawar, G. W.

Kerker, M.

Lechuga, L. M.

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]

Maltsev, V.

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]

Marinchio, H.

Marqués, R.

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]

Martin, O. J. F.

Medina, F.

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]

Moreno, F.

Mulholland, G. W.

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]

Papaioannou, E.

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]

Patoka, P.

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]

Piller, N. B.

Rafii-El-Idrissi, R.

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]

Rahmani, A.

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).

Sáenz, J. J.

Saiz, J. M.

Salandrino, A.

Sanz, J. M.

Sepúlveda, B.

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]

Smith, D. A.

Soljacic, M.

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]

Stokes, K. L.

Torrado, J. F.

Videen, G.

Wang, D.-S.

Wang, Z.

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]

Werner, D.

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]

Yang, P.

You, Y.

Yurkin, M.

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]

Zhai, P.-W.

Astrophys. J. (2)

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

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]

IEEE Trans. Antenn. Propag. (1)

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]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Quant. Spectrosc. Radiat. Transf. (2)

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).

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]

Langmuir (1)

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]

Metamaterials (Amst.) (1)

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]

Nano Lett. (1)

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]

Nature (1)

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]

Opt. Express (7)

Opt. Lett. (2)

Phys. Rev. B (2)

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]

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

Phys. Rev. Lett. (1)

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]

Other (4)

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.

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

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

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

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Figures (2)

Fig. 1
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.

Fig. 2
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.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

α ¯ ¯ CM = 3 V ( ɛ ¯ ¯ r ɛ m I ¯ ¯ ) ( ɛ ¯ ¯ r + 2 ɛ m I ¯ ¯ ) 1
χ ¯ ¯ CM = 3 V ( μ ¯ ¯ r μ m I ¯ ¯ ) ( μ ¯ ¯ r + 2 μ m I ¯ ¯ ) 1
α ¯ ¯ = α ¯ ¯ CM ( I ¯ ¯ i k 3 α ¯ ¯ CM 6 π ) 1
χ ¯ ¯ = χ ¯ ¯ CM ( I ¯ ¯ i k 3 χ ¯ ¯ CM 6 π ) 1
E j = E inc , j + k j N α ¯ ¯ j C ¯ ¯ j k E k μ 0 μ m ɛ 0 ɛ m k j N χ ¯ ¯ j f ¯ ¯ j k H k
H j = H inc , j + Σ k j N χ ¯ ¯ j C ¯ ¯ j k H k + ɛ 0 ɛ m μ 0 μ m Σ k j N α ¯ ¯ j f ¯ ¯ j k E k
A ¯ ¯ x = b
C ext = k ɛ 0 ɛ m | E 0 | 2 j = 1 N [ E inc , j * p j + μ 0 μ m H inc , j * m j ]
C abs = k ɛ 0 ɛ m | E 0 | 2 j = 1 N { [ E j * p j ] k 3 6 π ɛ 0 ɛ m | p j | 2 + μ 0 μ m [ [ H j * m j ] k 3 6 π | m j | 2 ] }
C sca = k 2 16 π 2 ɛ 0 ɛ m | E 0 | 2 | j = 1 N exp ( i k n r j ) { 1 ɛ 0 ɛ m [ p j ( n p j ) n ] μ 0 μ m n × m j } | 2 d Ω
C pha = k 2 ɛ 0 ɛ m | E 0 | 2 j = 1 N [ E inc , j * p j + μ 0 μ m H inc , j * m j ]
μ ¯ ¯ r = [ 4 I ¯ ¯ ɛ ¯ ¯ r i ( k d ) 3 π ( ɛ ¯ ¯ r I ¯ ¯ ) ] [ 2 I ¯ ¯ + ɛ ¯ ¯ r i ( k d ) 3 π ( ɛ ¯ ¯ r I ¯ ¯ ) ] 1
ɛ ¯ ¯ r 1 = ( 2.0 + 0.01 i 0 0 0 2.0 + 0.01 i 0 0 0 2.0 + 0.01 i ) = ( 2.0 + 0.01 i ) I ¯ ¯
ɛ ¯ ¯ r 2 = ( 2.0 + 0.01 i 0.3 + 0.2 i 0 0.3 0.2 i 2.0 + 0.01 i 0 0 0 2.0 + 0.01 i )
ɛ ¯ ¯ r 3 = ( 2.0 + 0.01 i 0.3 + 0.2 i 0 0.3 0.2 i 2.0 + 0.01 i 0 0 0 2.0 + 0.01 i )
μ ¯ ¯ r 1 = ( 0.4 3.6005 × 10 3 i ) I ¯ ¯
μ ¯ ¯ r 2 = ( 3.9232 × 10 1 2.0508 × 10 2 i 1.0899 × 10 1 6.8489 × 10 2 i 0 1.0899 × 10 1 + 6.8489 × 10 2 i 3.9232 × 10 1 2.0508 × 10 2 i 0 0 0 0.4 3.6005 × 10 3 i )
μ ¯ ¯ r 3 = ( 4.0712 × 10 1 + 1.3870 × 10 2 i 1.0804 × 10 1 7.3802 × 10 2 i 0 1.0804 × 10 1 7.3802 × 10 2 i 4.0712 × 10 1 + 1.3870 × 10 2 i 0 0 0 0.4 3.6005 × 10 3 i )

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