Abstract

Among the most popular approaches used for simulating plasmonic systems, the discrete dipole approximation suffers from poorly scaling volume discretization and limited near-field accuracy. We demonstrate that transformation to a surface integral formulation improves scalability and convergence and provides a flexible geometric approximation allowing, e.g., to investigate the influence of fabrication accuracy. The occurring integrals can be solved quasi-analytically, permitting even rapidly changing fields to be determined arbitrarily close to a scatterer. This insight into the extreme near-field behavior is useful for modeling closely packed particle ensembles and to study “hot spots” in plasmonic nanostructures used for plasmon-enhanced Raman scattering.

© 2009 Optical Society of America

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    [CrossRef]
  8. T. R. Jensen, G. C. Schatz, and R. P. Van Duyne, “Nanosphere lithography: surface plasmon resonance spectrum of a periodic array of silver nanoparticles by ultraviolet-visible extinction spectroscopy and electrodynamic modeling,” J. Phys. Chem. B 103, 2394-2401 (1999).
    [CrossRef]
  9. E. Hao, S. Li, R. C. Bailey, S. Zou, G. C. Schatz, and J. T. Hupp, “Optical properties of metal nanoshells,” J. Phys. Chem. B 108, 1224-1229 (2004).
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  14. P. Monk, Finite Element Methods for Maxwell's Equations (Oxford U. Press, 2003).
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  16. J. M. Neilson and R. Bunger, “Surface integral equation analysis of quasi-optical launchers,” IEEE Trans. Plasma Sci. 30, 794-799 (2002).
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  17. Y.-H. Chu and W. C. Chew, “Large-scale computation for electrically small structures using surface-integral equation method,” Microwave Opt. Technol. Lett. 47, 525-530 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. G. R. Cowper, “Gaussian quadrature formulas for triangles,” Int. J. Numer. Methods Eng. 7, 405-408 (1973).
    [CrossRef]
  29. I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromagn. Res. PIER 63, 243-278 (2006).
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  30. R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green's function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag. 41, 1448-1455 (1993).
    [CrossRef]
  31. P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n×RWG functions,” IEEE Trans. Antennas Propag. 51, 1837-1846 (2003).
    [CrossRef]
  32. X. Q. Sheng, J. M. Jin, J. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718-1726 (1998).
    [CrossRef]
  33. P. Ylä-Oijala, “Application of a novel CFIE for electromagnetic scattering by dielectric objects,” Microwave Opt. Technol. Lett. 35, 3-5 (2002).
    [CrossRef]
  34. P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168-1173 (2005).
    [CrossRef]
  35. A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics (Permagon, 1973).
  36. Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag. 25, 789-795 (1977).
    [CrossRef]
  37. T.-K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709-718 (1977).
    [CrossRef]
  38. L. N. Medgyesi-Mitschang, J. M. Putnam, and M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 11, 1383-1398 (1994).
    [CrossRef]
  39. C. Bohren and D. Huffmann, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  40. O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909-3915 (1998).
    [CrossRef]
  41. J. Schöberl, “Netgen: an advancing front 2d/3d-mesh generator based on abstract rules,” Comput. Visualization Sci. 1, 41-52 (1997).
    [CrossRef]
  42. L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523-5525 (1976).
    [CrossRef]
  43. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for periodic targets: theory and tests,” J. Opt. Soc. Am. A 25, 2693-2703 (2008).
    [CrossRef]
  44. H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16, 9144-9154 (2008).
    [CrossRef] [PubMed]
  45. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]

2008 (3)

2006 (2)

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromagn. Res. PIER 63, 243-278 (2006).
[CrossRef]

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. García de Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14, 9988-9999 (2006).
[CrossRef] [PubMed]

2005 (2)

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168-1173 (2005).
[CrossRef]

Y.-H. Chu and W. C. Chew, “Large-scale computation for electrically small structures using surface-integral equation method,” Microwave Opt. Technol. Lett. 47, 525-530 (2005).
[CrossRef]

2004 (1)

E. Hao, S. Li, R. C. Bailey, S. Zou, G. C. Schatz, and J. T. Hupp, “Optical properties of metal nanoshells,” J. Phys. Chem. B 108, 1224-1229 (2004).
[CrossRef]

2003 (1)

P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n×RWG functions,” IEEE Trans. Antennas Propag. 51, 1837-1846 (2003).
[CrossRef]

2002 (3)

P. Ylä-Oijala, “Application of a novel CFIE for electromagnetic scattering by dielectric objects,” Microwave Opt. Technol. Lett. 35, 3-5 (2002).
[CrossRef]

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

J. M. Neilson and R. Bunger, “Surface integral equation analysis of quasi-optical launchers,” IEEE Trans. Plasma Sci. 30, 794-799 (2002).
[CrossRef]

2000 (1)

J. P. Kottmann and O. J. F. Martin, “Accurate solution of the volume integral equation for high-permittivity scatterers,” IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

1999 (2)

T. R. Jensen, G. C. Schatz, and R. P. Van Duyne, “Nanosphere lithography: surface plasmon resonance spectrum of a periodic array of silver nanoparticles by ultraviolet-visible extinction spectroscopy and electrodynamic modeling,” J. Phys. Chem. B 103, 2394-2401 (1999).
[CrossRef]

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824-1832 (1999).
[CrossRef]

1998 (4)

P. J. Valle, F. Moreno, and J. M. Saiz, “Comparison of real- and perfect-conductor approaches for scattering by a cylinder on a flat substrate,” J. Opt. Soc. Am. A 15, 158-162 (1998).
[CrossRef]

X. Q. Sheng, J. M. Jin, J. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718-1726 (1998).
[CrossRef]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909-3915 (1998).
[CrossRef]

J. Song, C. Lu, W. Chew, and S. Lee, “Fast illinois solver code (FISC),” IEEE Antennas Propag. Mag. 40, 27-34 (1998).
[CrossRef]

1997 (1)

J. Schöberl, “Netgen: an advancing front 2d/3d-mesh generator based on abstract rules,” Comput. Visualization Sci. 1, 41-52 (1997).
[CrossRef]

1996 (1)

1995 (1)

W.-H. Y. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

1994 (2)

1993 (1)

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green's function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag. 41, 1448-1455 (1993).
[CrossRef]

1991 (1)

C. W. Trueman and S. J. Kubina, “Fields of complex surfaces using wire grid modelling,” IEEE Trans. Magn. 27, 4262-4267 (1991).
[CrossRef]

1988 (1)

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

1982 (1)

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409-418 (1982).
[CrossRef]

1977 (2)

Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag. 25, 789-795 (1977).
[CrossRef]

T.-K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709-718 (1977).
[CrossRef]

1976 (1)

L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523-5525 (1976).
[CrossRef]

1975 (1)

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations,” IEEE Trans. Microwave Theory Tech. 23, 623-630 (1975).
[CrossRef]

1973 (2)

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

G. R. Cowper, “Gaussian quadrature formulas for triangles,” Int. J. Numer. Methods Eng. 7, 405-408 (1973).
[CrossRef]

1972 (1)

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

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 25, 377-445 (1908) (in German).
[CrossRef]

Aizpurua, J.

Averitt, R. D.

Bailey, R. C.

E. Hao, S. Li, R. C. Bailey, S. Zou, G. C. Schatz, and J. T. Hupp, “Optical properties of metal nanoshells,” J. Phys. Chem. B 108, 1224-1229 (2004).
[CrossRef]

Bohren, C.

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

Brodwin, M. E.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations,” IEEE Trans. Microwave Theory Tech. 23, 623-630 (1975).
[CrossRef]

Bryant, G. W.

Bunger, R.

J. M. Neilson and R. Bunger, “Surface integral equation analysis of quasi-optical launchers,” IEEE Trans. Plasma Sci. 30, 794-799 (2002).
[CrossRef]

Chang, Y.

Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag. 25, 789-795 (1977).
[CrossRef]

Chew, W.

J. Song, C. Lu, W. Chew, and S. Lee, “Fast illinois solver code (FISC),” IEEE Antennas Propag. Mag. 40, 27-34 (1998).
[CrossRef]

Chew, W. C.

Y.-H. Chu and W. C. Chew, “Large-scale computation for electrically small structures using surface-integral equation method,” Microwave Opt. Technol. Lett. 47, 525-530 (2005).
[CrossRef]

X. Q. Sheng, J. M. Jin, J. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718-1726 (1998).
[CrossRef]

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

Christy, R. W.

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

Chu, Y.-H.

Y.-H. Chu and W. C. Chew, “Large-scale computation for electrically small structures using surface-integral equation method,” Microwave Opt. Technol. Lett. 47, 525-530 (2005).
[CrossRef]

Cowper, G. R.

G. R. Cowper, “Gaussian quadrature formulas for triangles,” Int. J. Numer. Methods Eng. 7, 405-408 (1973).
[CrossRef]

Davis, L. C.

L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523-5525 (1976).
[CrossRef]

Draine, B. T.

Fischer, H.

Flatau, P. J.

García de Abajo, F. J.

Gedera, M. B.

Glisson, A.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409-418 (1982).
[CrossRef]

González, F.

Graglia, R. D.

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green's function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag. 41, 1448-1455 (1993).
[CrossRef]

Halas, N. J.

Hänninen, I.

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromagn. Res. PIER 63, 243-278 (2006).
[CrossRef]

Hao, E.

E. Hao, S. Li, R. C. Bailey, S. Zou, G. C. Schatz, and J. T. Hupp, “Optical properties of metal nanoshells,” J. Phys. Chem. B 108, 1224-1229 (2004).
[CrossRef]

Harrington, R.

Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag. 25, 789-795 (1977).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Macmillan, 1968).

Howie, A.

F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Huffmann, D.

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

Hupp, J. T.

E. Hao, S. Li, R. C. Bailey, S. Zou, G. C. Schatz, and J. T. Hupp, “Optical properties of metal nanoshells,” J. Phys. Chem. B 108, 1224-1229 (2004).
[CrossRef]

Jackson, J. D.

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

Jensen, T. R.

T. R. Jensen, G. C. Schatz, and R. P. Van Duyne, “Nanosphere lithography: surface plasmon resonance spectrum of a periodic array of silver nanoparticles by ultraviolet-visible extinction spectroscopy and electrodynamic modeling,” J. Phys. Chem. B 103, 2394-2401 (1999).
[CrossRef]

Jin, J. M.

X. Q. Sheng, J. M. Jin, J. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718-1726 (1998).
[CrossRef]

Johnson, P. B.

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

Jung, J.

J. Jung and T. Sondergaard, “Green's function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77, 245310 (2008).
[CrossRef]

Kottmann, J. P.

J. P. Kottmann and O. J. F. Martin, “Accurate solution of the volume integral equation for high-permittivity scatterers,” IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

Kubina, S. J.

C. W. Trueman and S. J. Kubina, “Fields of complex surfaces using wire grid modelling,” IEEE Trans. Magn. 27, 4262-4267 (1991).
[CrossRef]

Lee, S.

J. Song, C. Lu, W. Chew, and S. Lee, “Fast illinois solver code (FISC),” IEEE Antennas Propag. Mag. 40, 27-34 (1998).
[CrossRef]

Li, S.

E. Hao, S. Li, R. C. Bailey, S. Zou, G. C. Schatz, and J. T. Hupp, “Optical properties of metal nanoshells,” J. Phys. Chem. B 108, 1224-1229 (2004).
[CrossRef]

Lu, C.

J. Song, C. Lu, W. Chew, and S. Lee, “Fast illinois solver code (FISC),” IEEE Antennas Propag. Mag. 40, 27-34 (1998).
[CrossRef]

Lu, C. C.

X. Q. Sheng, J. M. Jin, J. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718-1726 (1998).
[CrossRef]

Martin, O. J. F.

H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express 16, 9144-9154 (2008).
[CrossRef] [PubMed]

J. P. Kottmann and O. J. F. Martin, “Accurate solution of the volume integral equation for high-permittivity scatterers,” IEEE Trans. Antennas Propag. 48, 1719-1726 (2000).
[CrossRef]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909-3915 (1998).
[CrossRef]

Medgyesi-Mitschang, L. N.

Mie, G.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 25, 377-445 (1908) (in German).
[CrossRef]

Miller, E. K.

A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics (Permagon, 1973).

Monk, P.

P. Monk, Finite Element Methods for Maxwell's Equations (Oxford U. Press, 2003).
[CrossRef]

Moreno, F.

Neilson, J. M.

J. M. Neilson and R. Bunger, “Surface integral equation analysis of quasi-optical launchers,” IEEE Trans. Plasma Sci. 30, 794-799 (2002).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Scientific, 2006).

Ortiz, E. M.

Pennypacker, C. R.

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

Piller, N. B.

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909-3915 (1998).
[CrossRef]

Poggio, A. J.

A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics (Permagon, 1973).

Purcell, E. M.

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

Putnam, J. M.

Rao, S.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409-418 (1982).
[CrossRef]

Romero, I.

Saiz, J. M.

Sarvas, J.

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromagn. Res. PIER 63, 243-278 (2006).
[CrossRef]

Schatz, G. C.

E. Hao, S. Li, R. C. Bailey, S. Zou, G. C. Schatz, and J. T. Hupp, “Optical properties of metal nanoshells,” J. Phys. Chem. B 108, 1224-1229 (2004).
[CrossRef]

T. R. Jensen, G. C. Schatz, and R. P. Van Duyne, “Nanosphere lithography: surface plasmon resonance spectrum of a periodic array of silver nanoparticles by ultraviolet-visible extinction spectroscopy and electrodynamic modeling,” J. Phys. Chem. B 103, 2394-2401 (1999).
[CrossRef]

W.-H. Y. Yang, G. C. Schatz, and R. P. Van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

Schöberl, J.

J. Schöberl, “Netgen: an advancing front 2d/3d-mesh generator based on abstract rules,” Comput. Visualization Sci. 1, 41-52 (1997).
[CrossRef]

Sheng, X. Q.

X. Q. Sheng, J. M. Jin, J. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718-1726 (1998).
[CrossRef]

Sondergaard, T.

J. Jung and T. Sondergaard, “Green's function surface integral equation method for theoretical analysis of scatterers close to a metal interface,” Phys. Rev. B 77, 245310 (2008).
[CrossRef]

Song, J.

J. Song, C. Lu, W. Chew, and S. Lee, “Fast illinois solver code (FISC),” IEEE Antennas Propag. Mag. 40, 27-34 (1998).
[CrossRef]

X. Q. Sheng, J. M. Jin, J. Song, W. C. Chew, and C. C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag. 46, 1718-1726 (1998).
[CrossRef]

Taflove, A.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations,” IEEE Trans. Microwave Theory Tech. 23, 623-630 (1975).
[CrossRef]

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, IEEE Series on Electromagnetic Waves, 2nd ed. (IEEE, 1994).

Taskinen, M.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the considered two-region problem. Region 1 is bounded on the inside by S and on the outside by S inf .

Fig. 2
Fig. 2

RWG basis function f n is nonzero on the two adjacent triangles T n + and T n and zero everywhere else. The normal component of f n is continuous across all edges, i.e., zero on all outer edges.

Fig. 3
Fig. 3

Integrated error Σ of simulations using SIE and VIE formulations compared to analytic Mie solution. The simulated system was a dielectric sphere with radius a = λ 2 and refractive index n = 2 illuminated by a plane wave with wavelength λ.

Fig. 4
Fig. 4

Bistatic scattering cross section of a high permittivity scatterer (sphere with radius a = λ 2 and refractive index n = 4 ) determined from simulations using SIE and VIE formulations as well as analytic Mie solution.

Fig. 5
Fig. 5

Extinction cross sections of a truncated tetrahedron as described in [8] determined from SIE calculations with increasing numbers of DOF. The arrow indicates the convergence with an increase in DOF.

Fig. 6
Fig. 6

Electric field intensity E 2 around and inside a resonantly excited metal dipole antenna as described in [44], determined using the PMCHW formulation. Illumination intensity is E inc 2 = 1 .

Equations (45)

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× × E i ( r ) k i 2 E i ( r ) = i ω μ i j ( r ) , r V i ,
× × G ¯ i ( r , r ) k i 2 G ¯ i ( r , r ) = 1 ¯ δ ( r r ) ,
a ( b × C ¯ ) = b ( a × C ¯ ) = ( a × b ) C ¯ .
× × E i ( r ) G ¯ i ( r , r ) E i ( r ) × × G ¯ i ( r , r ) = i ω μ i j ( r ) G ¯ i ( r , r ) E i ( r ) δ ( r r ) .
V i d V ( [ × E i ( r ) ] × G ¯ i ( r , r ) + E i ( r ) × [ × G ¯ i ( r , r ) ] ) = i ω μ i V i d V j ( r ) G ¯ i ( r , r ) { E i ( r ) : r V i 0 : otherwise } .
i ω μ i V i d V j ( r ) G ¯ i ( r , r ) = i ω μ i V i d V G ¯ i ( r , r ) j ( r ) = E i inc ( r ) ,
V i d S n ̂ i ( r ) ( [ × E i ( r ) ] × G ¯ i ( r , r ) + E i ( r ) × [ × G ¯ i ( r , r ) ] ) = E i inc ( r ) { E i ( r ) : r V i 0 : otherwise } ,
n ̂ i ( r ) [ × E i ( r ) ] × G ¯ i ( r , r ) = n ̂ i ( r ) × [ × E i ( r ) ] G ¯ i ( r , r ) = i ω μ i G ¯ i ( r , r ) [ n ̂ i ( r ) × H i ( r ) ] ,
n ̂ i ( r ) E i ( r ) × [ × G ¯ i ( r , r ) ] = [ n ̂ i ( r ) × E i ( r ) ] [ × G ¯ i ( r , r ) ] = [ × G ¯ i ( r , r ) ] [ n ̂ i ( r ) × E i ( r ) ] ,
ω μ i i S d S G ¯ i ( r , r ) J ( r ) S d S [ × G ¯ i ( r , r ) ] M ( r ) = { E 1 inc ( r ) : i = 1 and r V 2 \ S 0 : i = 2 and r V 1 \ S } .
n ̂ i ( r ) × ( E 1 ( r ) E 2 ( r ) ) = 0 ,
n ̂ i ( r ) × ( H 1 ( r ) H 2 ( r ) ) = 0 ,
( ω μ i i S d S G ¯ i ( r , r ) J ( r ) S d S [ G ¯ i ( r , r ) ] M ( r ) ) tan = { ( E 1 inc ( r ) ) tan : i = 1 0 : i = 2 } ,
× × H i ( r ) k i 2 H i ( r ) = × j ( r ) ,
H i inc ( r ) = V i d V [ × G ¯ i ( r , r ) ] j ( r ) = V i d V [ × j ( r ) ] G ¯ i ( r , r ) ,
( ω ϵ i i S d S G ¯ i ( r , r ) M ( r ) + S d S [ G ¯ i ( r , r ) ] J ( r ) ) tan = { ( H 1 inc ( r ) ) tan : i = 1 0 : i = 2 } ,
f n ( r ) = { ± L n 2 A n ± ( r p n ± ) : r T n ± 0 : otherwise } ,
J ( r ) = n = 1 N α n f n ( r ) ,
M ( r ) = n = 1 N β n f n ( r ) ,
S m d S f m ( r ) n = 1 N ( α n ω μ i i S n d S G ¯ i ( r , r ) f n ( r ) β n S n d S [ × G ¯ i ( r , r ) ] f n ( r ) ) = { S m d S f m ( r ) E 1 inc ( r ) : i = 1 0 : i = 2 } ,
[ D 1 K 1 D 2 K 2 ] ψ = q E ,
D m n i = ω μ i i S m d S f m ( r ) S n d S G ¯ i ( r , r ) f n ( r ) ,
K m n i = S m d S f m ( r ) S n d S [ × G ¯ i ( r , r ) ] f n ( r ) ,
ψ = ( α 1 , , α N , β 1 , , β N ) T ,
q m E = { S m d S f m ( r ) E 1 inc ( r ) : m = 1 N 0 : m = N + 1 2 N } .
[ K 1 1 Z 1 2 D 1 K 2 1 Z 2 2 D 2 ] ψ = q H ,
q m H = { S m d S f m ( r ) H 1 inc ( r ) : m = 1 N 0 : m = N + 1 2 N } .
G ¯ i ( r , r ) = ( 1 ¯ + k i 2 ) exp ( i k i r r ) 4 π r r = ( 1 ¯ + k i 2 ) G i ( r , r ) ,
S m d S f m ( r ) ( k i 2 + 1 ¯ ) S n d S G i ( r , r ) f n ( r ) = 1 k i 2 S m d S f m ( r ) S n d S G i ( r , r ) f n ( r ) + S m d S f m ( r ) S n d S G i ( r , r ) f n ( r ) = 1 k i 2 S m d S [ f m ( r ) ] S n d S G i ( r , r ) f n ( r ) + S m d S f m ( r ) S n d S G i ( r , r ) f n ( r ) .
× G ¯ i ( r , r ) = [ G i ( r , r ) ] × 1 ¯ = [ G i ( r , r ) ] × 1 ¯ ,
S m d S f m ( r ) S n d S [ × G ¯ i ( r , r ) ] f n ( r ) = S m d S f m ( r ) S n d S [ G i ( r , r ) ] × f n ( r ) .
G i ( r , r ) = 1 4 π ( 1 R + i k i k i 2 R 2 + ) , R = r r .
G i s ( r , r ) = 1 4 π ( e i k i R 1 R + k i 2 R 2 ) ,
G i ( r , r ) G i s ( r , r ) + 1 4 π ( 1 R k i 2 R 2 )
S n d S R q f n ( r ) , S n d S R q f n ( r ) ,
S n d S [ R q ] × f n ( r ) , q = 1 , 1 , 3 , ,
[ D 1 + D 2 K 1 K 2 K 1 + K 2 1 Z 1 2 D 1 + 1 Z 2 2 D 2 ] ψ = q ,
q m = { S m d S f m ( r ) E 1 inc ( r ) : m = 1 N S m N d S f m N ( r ) H 1 inc ( r ) : m = N + 1 2 N } .
E i ( r ) = { + } n [ α n ω μ i i S n d S G ¯ i ( r , r ) f n ( r ) + β n S n d S [ G i ( r , r ) ] × f n ( r ) ] + { E 1 inc ( r ) : i = 1 and r V 1 0 : i = 2 and r V 2 } ,
H i ( r ) = { + } n [ β n ω ϵ i i S n d S G ¯ i ( r , r ) f n ( r ) α n S n d S [ G i ( r , r ) ] × f n ( r ) ] + { H 1 inc ( r ) : i = 1 and r V 1 0 : i = 2 and r V 2 } .
S n d S G ¯ i ( r , r ) f n ( r ) = 1 k i 2 S n d S G i ( r , r ) f n ( r ) + S n d S G i ( r , r ) f n ( r ) .
σ sca φ ( θ ) = 4 π R 2 E sca ( r φ ( θ ) ) 2 E inc 2 , φ = , ,
Σ φ = [ 1 π 0 π d θ ( σ sim φ ( θ ) σ Mie φ ( θ ) ) 2 σ Mie φ ( θ ) 2 ] 1 2 .
σ ext = 1 I inc A d S n ̂ ( r ) S ext ( r ) ,
S ext = 1 2 Re { E inc × H sca * + E sca × H inc * }

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