Abstract

Numerical formulations based on surface integral equations (SIEs) provide an accurate and efficient framework for the solution of the electromagnetic scattering problem by three-dimensional plasmonic nanostructures in the frequency domain. In this paper, we present a unified description of SIE formulations with both singular and nonsingular kernel and we study their accuracy in solving the scattering problem by metallic nanoparticles with spherical and nonspherical shape. In fact, the accuracy of the numerical solution, especially in the near zone, is of great importance in the analysis and design of plasmonic nanostructures, whose operation critically depends on the manipulation of electromagnetic hot spots. Four formulation types are considered: the N-combined region integral equations, the T-combined region integral equations, the combined field integral equations and the null field integral equations. A detailed comparison between their numerical solutions obtained for several nanoparticle shapes is performed by examining convergence rate and accuracy in both the far and near zone of the scatterer as a function of the number of degrees of freedom. A rigorous analysis of SIE formulations and their limitations can have a high impact on the engineering of numerous nano-scale optical devices such as plasmon-enhanced light emitters, biosensors, photodetectors, and nanoantennas.

© 2012 Optical Society of America

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    [CrossRef]
  51. D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
    [CrossRef]
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    [CrossRef]
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2012 (3)

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

G. Iadarola, C. Forestiere, L. Dal Negro, F. Villone, and G. Miano, “GPU-accelerated T-matrix algorithm for light-scattering simulations,” J. Comput. Phys. 231, 5640–5652 (2012).
[CrossRef]

M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express 20, 9161–9171 (2012).
[CrossRef]

2011 (6)

M. Karamehmedović, R. Schuh, V. Schmidt, T. Wriedt, C. Matyssek, W. Hergert, A. Stalmashonak, G. Seifert, and O. Stranik, “Comparison of numerical methods in near-field computation for metallic nanoparticles,” Opt. Express 19, 8939–8953 (2011).
[CrossRef]

J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28, 1341–1348 (2011).
[CrossRef]

J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express 19, 23386–23399 (2011).
[CrossRef]

C. Forestiere, G. Iadarola, L. Dal Negro, and G. Miano, “Near-field calculation based on the T-matrix method with discrete sources,” J. Quant. Spectrosc. Radiat. Transfer. 112, 2384–2394 (2011).
[CrossRef]

P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nano-particles,” Nano Lett. 11, 2329–2333 (2011).
[CrossRef]

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for plasmonics nanoparticles,” Int. J. Appl. Electromag. Mech. 35, 79–91 (2011).

2010 (5)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for the analysis of plasmon oscillations in metallic nanoparticles,” IEEE Trans. Antennas Propag. 58, 2920–2933 (2010).
[CrossRef]

A. Doicu and T. Wriedt, “Near-field computation using the null-field method,” J. Quant. Spectrosc. Radiat. Transfer 111, 466–473 (2010).
[CrossRef]

C. Forestiere, M. Donelli, G. Walsh, E. Zeni, G. Miano, and L. Dal Negro, “Particle-swarm optimization of broadband nanoplasmonic arrays,” Opt. Lett. 35, 133–135 (2010).
[CrossRef]

B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010).
[CrossRef]

2009 (3)

A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
[CrossRef]

C. Geuzaine and J. Remacle, “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,” Int. J. Numer. Methods Eng. 79, 1309–1331 (2009).

L. Dal Negro, G. Miano, G. Rubinacci, A. Tamburrino, and S. Ventre, “A fast computation method for the analysis of an array of metallic nanoparticles,” IEEE Trans. Magn. 45, 1618–1621 (2009).
[CrossRef]

2008 (1)

G. W. Bryant, F. J. Garcia de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef]

2005 (3)

P. Yla-Oijala and M. Taskinen, “Well-conditioned Muller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antennas Propag. 53, 3316–3323 (2005).
[CrossRef]

P. Yla-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]

P. M. Taskinen, P. Ylä-Oijala, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, 1–19 (2005).

2004 (1)

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

2003 (1)

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef]

2002 (1)

B. H. Jung, T. K. Sarkar, and Y.-S. Chung, “A survey of various frequency domain integral equations for the analysis of scattering from three-dimensional dielectric objects,” Prog. Electromagn. Res. 33, 193–245 (2002).
[CrossRef]

1999 (1)

1998 (2)

X. Sheng, J.-M. Jin, J. Song, W. 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]

C. Leat, N. Shuley, and G. Stickley, “Triangular-patch model of bowtie antennas: validation against Brown and Woodward,” IEEE Proc. Microw. Antennas Propag. 145, 465–470 (1998).
[CrossRef]

1997 (1)

K. Yee and J. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving maxwell’s equations,” IEEE Trans. Antennas Propag. 45, 354–363 (1997).
[CrossRef]

1994 (1)

M. Mishchenko and L. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

1993 (1)

R. 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]

1988 (1)

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

1983 (1)

D. Lindholm, “Automatic triangular mesh generation on surfaces of polyhedra,” IEEE Trans. Magn. 19, 2539–2542 (1983).
[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]

1979 (1)

J. Mautz and R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Uebertraeg. 33, 71–80 (1979).

1975 (1)

1973 (1)

E. Purcell and C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (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]

1969 (2)

R. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromag. Waves Appl. 3, 1–15, (1969).
[CrossRef]

P. C. Waterman, “Scattering by dielectric obstacles,” Alta Freq. 38, 348–352 (1969).

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

1901 (1)

A. E. H. Love, “The integration of equation of propagation of electric waves,” Proc. R. Soc. London 197, 1–45 (1901).
[CrossRef]

Aizpurua, J.

G. W. Bryant, F. J. Garcia de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef]

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef]

Araújo, M. G.

Asano, S.

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Berkovitch, N.

P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nano-particles,” Nano Lett. 11, 2329–2333 (2011).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Bryant, G. W.

G. W. Bryant, F. J. Garcia de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef]

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef]

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Capretti, A.

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

Chen, J.

K. Yee and J. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving maxwell’s equations,” IEEE Trans. Antennas Propag. 45, 354–363 (1997).
[CrossRef]

Chew, W.

X. Sheng, J.-M. Jin, J. Song, W. 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]

Chew, W. C.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics.(Artech House, 2001).

Christy, R. W.

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

Chung, Y.-S.

B. H. Jung, T. K. Sarkar, and Y.-S. Chung, “A survey of various frequency domain integral equations for the analysis of scattering from three-dimensional dielectric objects,” Prog. Electromagn. Res. 33, 193–245 (2002).
[CrossRef]

Dal Negro, L.

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

G. Iadarola, C. Forestiere, L. Dal Negro, F. Villone, and G. Miano, “GPU-accelerated T-matrix algorithm for light-scattering simulations,” J. Comput. Phys. 231, 5640–5652 (2012).
[CrossRef]

C. Forestiere, G. Iadarola, L. Dal Negro, and G. Miano, “Near-field calculation based on the T-matrix method with discrete sources,” J. Quant. Spectrosc. Radiat. Transfer. 112, 2384–2394 (2011).
[CrossRef]

C. Forestiere, M. Donelli, G. Walsh, E. Zeni, G. Miano, and L. Dal Negro, “Particle-swarm optimization of broadband nanoplasmonic arrays,” Opt. Lett. 35, 133–135 (2010).
[CrossRef]

L. Dal Negro, G. Miano, G. Rubinacci, A. Tamburrino, and S. Ventre, “A fast computation method for the analysis of an array of metallic nanoparticles,” IEEE Trans. Magn. 45, 1618–1621 (2009).
[CrossRef]

Ding, K.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).

Doicu, A.

A. Doicu and T. Wriedt, “Near-field computation using the null-field method,” J. Quant. Spectrosc. Radiat. Transfer 111, 466–473 (2010).
[CrossRef]

A. Doicu and T. Wriedt, “Calculation of the T matrix in the null-field method with discrete sources,” J. Opt. Soc. Am. A 16, 2539–2544 (1999).
[CrossRef]

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).

Donelli, M.

Draine, B.

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

B. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.1” (2010).

Eremin, Y.

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).

Flatau, P.

B. Draine and P. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT 7.1” (2010).

Forestiere, C.

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

G. Iadarola, C. Forestiere, L. Dal Negro, F. Villone, and G. Miano, “GPU-accelerated T-matrix algorithm for light-scattering simulations,” J. Comput. Phys. 231, 5640–5652 (2012).
[CrossRef]

C. Forestiere, G. Iadarola, L. Dal Negro, and G. Miano, “Near-field calculation based on the T-matrix method with discrete sources,” J. Quant. Spectrosc. Radiat. Transfer. 112, 2384–2394 (2011).
[CrossRef]

C. Forestiere, M. Donelli, G. Walsh, E. Zeni, G. Miano, and L. Dal Negro, “Particle-swarm optimization of broadband nanoplasmonic arrays,” Opt. Lett. 35, 133–135 (2010).
[CrossRef]

Fromm, D. P.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

Gallinet, B.

Garcia de Abajo, F. J.

G. W. Bryant, F. J. Garcia de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef]

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef]

Gedney, S.

A. Zhu, and S. Gedney, “Comparison of the Muller and PMWCHT surface integral formulations for the locally corrected Nystrom method,” in Proceedings of 2004 Antennas and Propagation Society International Symposium (IEEE, 2004), pp. 3871–3874.

Geuzaine, C.

C. Geuzaine and J. Remacle, “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,” Int. J. Numer. Methods Eng. 79, 1309–1331 (2009).

Ginzburg, P.

P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nano-particles,” Nano Lett. 11, 2329–2333 (2011).
[CrossRef]

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]

Graglia, R.

R. 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]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hanarp, P.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef]

Harrington, R.

J. Mautz and R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Uebertraeg. 33, 71–80 (1979).

R. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromag. Waves Appl. 3, 1–15, (1969).
[CrossRef]

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

Hergert, W.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Iadarola, G.

G. Iadarola, C. Forestiere, L. Dal Negro, F. Villone, and G. Miano, “GPU-accelerated T-matrix algorithm for light-scattering simulations,” J. Comput. Phys. 231, 5640–5652 (2012).
[CrossRef]

C. Forestiere, G. Iadarola, L. Dal Negro, and G. Miano, “Near-field calculation based on the T-matrix method with discrete sources,” J. Quant. Spectrosc. Radiat. Transfer. 112, 2384–2394 (2011).
[CrossRef]

Järvenpää, S.

P. M. Taskinen, P. Ylä-Oijala, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, 1–19 (2005).

Jin, J. M.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics.(Artech House, 2001).

Jin, J.-M.

X. Sheng, J.-M. Jin, J. Song, W. 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]

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Jung, B. H.

B. H. Jung, T. K. Sarkar, and Y.-S. Chung, “A survey of various frequency domain integral equations for the analysis of scattering from three-dimensional dielectric objects,” Prog. Electromagn. Res. 33, 193–245 (2002).
[CrossRef]

Käll, M.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef]

Karamehmedovic, M.

Kauranen, M.

Kern, A. M.

Kino, G.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Landesa, L.

Leat, C.

C. Leat, N. Shuley, and G. Stickley, “Triangular-patch model of bowtie antennas: validation against Brown and Woodward,” IEEE Proc. Microw. Antennas Propag. 145, 465–470 (1998).
[CrossRef]

Lee, S. Y.

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

Lindholm, D.

D. Lindholm, “Automatic triangular mesh generation on surfaces of polyhedra,” IEEE Trans. Magn. 19, 2539–2542 (1983).
[CrossRef]

Love, A. E. H.

A. E. H. Love, “The integration of equation of propagation of electric waves,” Proc. R. Soc. London 197, 1–45 (1901).
[CrossRef]

Lu, C.-C.

X. Sheng, J.-M. Jin, J. Song, W. 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]

Maier, S.

S. Maier, Plasmonics: Fundamental and Applications (Springer, 2007).

Mäkitalo, J.

Martin, O. J. F.

Matyssek, C.

Mautz, J.

J. Mautz and R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Uebertraeg. 33, 71–80 (1979).

Miano, G.

G. Iadarola, C. Forestiere, L. Dal Negro, F. Villone, and G. Miano, “GPU-accelerated T-matrix algorithm for light-scattering simulations,” J. Comput. Phys. 231, 5640–5652 (2012).
[CrossRef]

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

C. Forestiere, G. Iadarola, L. Dal Negro, and G. Miano, “Near-field calculation based on the T-matrix method with discrete sources,” J. Quant. Spectrosc. Radiat. Transfer. 112, 2384–2394 (2011).
[CrossRef]

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for plasmonics nanoparticles,” Int. J. Appl. Electromag. Mech. 35, 79–91 (2011).

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for the analysis of plasmon oscillations in metallic nanoparticles,” IEEE Trans. Antennas Propag. 58, 2920–2933 (2010).
[CrossRef]

C. Forestiere, M. Donelli, G. Walsh, E. Zeni, G. Miano, and L. Dal Negro, “Particle-swarm optimization of broadband nanoplasmonic arrays,” Opt. Lett. 35, 133–135 (2010).
[CrossRef]

L. Dal Negro, G. Miano, G. Rubinacci, A. Tamburrino, and S. Ventre, “A fast computation method for the analysis of an array of metallic nanoparticles,” IEEE Trans. Magn. 45, 1618–1621 (2009).
[CrossRef]

Michielssen, E.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics.(Artech House, 2001).

Mishchenko, M.

M. Mishchenko and L. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Moerner, W. E.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

Muller, C.

C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, 1969).

Nevet, A.

P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nano-particles,” Nano Lett. 11, 2329–2333 (2011).
[CrossRef]

Obelleiro, F.

Orenstein, M.

P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nano-particles,” Nano Lett. 11, 2329–2333 (2011).
[CrossRef]

Pasquale, A. J.

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

Pennypacker, C. R.

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

Purcell, E.

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

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]

Reinhard, B. M.

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

Remacle, J.

C. Geuzaine and J. Remacle, “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,” Int. J. Numer. Methods Eng. 79, 1309–1331 (2009).

Rivero, J.

Rubinacci, G.

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for plasmonics nanoparticles,” Int. J. Appl. Electromag. Mech. 35, 79–91 (2011).

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for the analysis of plasmon oscillations in metallic nanoparticles,” IEEE Trans. Antennas Propag. 58, 2920–2933 (2010).
[CrossRef]

L. Dal Negro, G. Miano, G. Rubinacci, A. Tamburrino, and S. Ventre, “A fast computation method for the analysis of an array of metallic nanoparticles,” IEEE Trans. Magn. 45, 1618–1621 (2009).
[CrossRef]

Sarkar, T. K.

B. H. Jung, T. K. Sarkar, and Y.-S. Chung, “A survey of various frequency domain integral equations for the analysis of scattering from three-dimensional dielectric objects,” Prog. Electromagn. Res. 33, 193–245 (2002).
[CrossRef]

Schmidt, V.

Schuck, P. J.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

Schuh, R.

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Seifert, G.

Sheng, X.

X. Sheng, J.-M. Jin, J. Song, W. 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]

Shor, I.

P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nano-particles,” Nano Lett. 11, 2329–2333 (2011).
[CrossRef]

Shuley, N.

C. Leat, N. Shuley, and G. Stickley, “Triangular-patch model of bowtie antennas: validation against Brown and Woodward,” IEEE Proc. Microw. Antennas Propag. 145, 465–470 (1998).
[CrossRef]

Solís, D. M.

Song, J.

X. Sheng, J.-M. Jin, J. Song, W. 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, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics.(Artech House, 2001).

Stalmashonak, A.

Stickley, G.

C. Leat, N. Shuley, and G. Stickley, “Triangular-patch model of bowtie antennas: validation against Brown and Woodward,” IEEE Proc. Microw. Antennas Propag. 145, 465–470 (1998).
[CrossRef]

Stranik, O.

Sundaramurthy, A.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

Sutherland, D. S.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef]

Suuriniemi, S.

Taboada, J. M.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Tamburrino, A.

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for plasmonics nanoparticles,” Int. J. Appl. Electromag. Mech. 35, 79–91 (2011).

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for the analysis of plasmon oscillations in metallic nanoparticles,” IEEE Trans. Antennas Propag. 58, 2920–2933 (2010).
[CrossRef]

L. Dal Negro, G. Miano, G. Rubinacci, A. Tamburrino, and S. Ventre, “A fast computation method for the analysis of an array of metallic nanoparticles,” IEEE Trans. Magn. 45, 1618–1621 (2009).
[CrossRef]

Taskinen, M.

P. Yla-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]

P. Yla-Oijala and M. Taskinen, “Well-conditioned Muller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antennas Propag. 53, 3316–3323 (2005).
[CrossRef]

P. Yla-Oijala and M. Taskinen, “Electromagnetic scattering analysis with combined field integral equations,” in Proceedings of Antennas and Propagation Society International Symposium (IEEE, 2007), pp. 4869–4872.

Taskinen, P. M.

P. M. Taskinen, P. Ylä-Oijala, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, 1–19 (2005).

Travis, L.

M. Mishchenko and L. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Tsang, L.

L. Tsang, J. A. Kong, and K. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, 2000).

Ventre, S.

L. Dal Negro, G. Miano, G. Rubinacci, A. Tamburrino, and S. Ventre, “A fast computation method for the analysis of an array of metallic nanoparticles,” IEEE Trans. Magn. 45, 1618–1621 (2009).
[CrossRef]

Villone, F.

G. Iadarola, C. Forestiere, L. Dal Negro, F. Villone, and G. Miano, “GPU-accelerated T-matrix algorithm for light-scattering simulations,” J. Comput. Phys. 231, 5640–5652 (2012).
[CrossRef]

Walsh, G.

Waterman, P. C.

P. C. Waterman, “Scattering by dielectric obstacles,” Alta Freq. 38, 348–352 (1969).

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Wilton, D.

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

Wriedt, T.

Yamamoto, G.

Yee, K.

K. Yee and J. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving maxwell’s equations,” IEEE Trans. Antennas Propag. 45, 354–363 (1997).
[CrossRef]

Yla-Oijala, P.

P. Yla-Oijala and M. Taskinen, “Well-conditioned Muller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antennas Propag. 53, 3316–3323 (2005).
[CrossRef]

P. Yla-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]

P. Yla-Oijala and M. Taskinen, “Electromagnetic scattering analysis with combined field integral equations,” in Proceedings of Antennas and Propagation Society International Symposium (IEEE, 2007), pp. 4869–4872.

Ylä-Oijala, P.

P. M. Taskinen, P. Ylä-Oijala, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, 1–19 (2005).

Zeni, E.

Zhu, A.

A. Zhu, and S. Gedney, “Comparison of the Muller and PMWCHT surface integral formulations for the locally corrected Nystrom method,” in Proceedings of 2004 Antennas and Propagation Society International Symposium (IEEE, 2004), pp. 3871–3874.

Alta Freq. (1)

P. C. Waterman, “Scattering by dielectric obstacles,” Alta Freq. 38, 348–352 (1969).

Appl. Opt. (1)

Arch. Elektron. Uebertraeg. (1)

J. Mautz and R. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektron. Uebertraeg. 33, 71–80 (1979).

Astrophys. J. (2)

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

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

IEEE Proc. Microw. Antennas Propag. (1)

C. Leat, N. Shuley, and G. Stickley, “Triangular-patch model of bowtie antennas: validation against Brown and Woodward,” IEEE Proc. Microw. Antennas Propag. 145, 465–470 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (7)

X. Sheng, J.-M. Jin, J. Song, W. 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]

R. 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]

K. Yee and J. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving maxwell’s equations,” IEEE Trans. Antennas Propag. 45, 354–363 (1997).
[CrossRef]

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for the analysis of plasmon oscillations in metallic nanoparticles,” IEEE Trans. Antennas Propag. 58, 2920–2933 (2010).
[CrossRef]

P. Yla-Oijala and M. Taskinen, “Well-conditioned Muller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antennas Propag. 53, 3316–3323 (2005).
[CrossRef]

P. Yla-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]

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

IEEE Trans. Magn. (2)

L. Dal Negro, G. Miano, G. Rubinacci, A. Tamburrino, and S. Ventre, “A fast computation method for the analysis of an array of metallic nanoparticles,” IEEE Trans. Magn. 45, 1618–1621 (2009).
[CrossRef]

D. Lindholm, “Automatic triangular mesh generation on surfaces of polyhedra,” IEEE Trans. Magn. 19, 2539–2542 (1983).
[CrossRef]

Int. J. Appl. Electromag. Mech. (1)

G. Miano, G. Rubinacci, and A. Tamburrino, “Numerical modeling for plasmonics nanoparticles,” Int. J. Appl. Electromag. Mech. 35, 79–91 (2011).

Int. J. Numer. Methods Eng. (1)

C. Geuzaine and J. Remacle, “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,” Int. J. Numer. Methods Eng. 79, 1309–1331 (2009).

J. Comput. Phys. (1)

G. Iadarola, C. Forestiere, L. Dal Negro, F. Villone, and G. Miano, “GPU-accelerated T-matrix algorithm for light-scattering simulations,” J. Comput. Phys. 231, 5640–5652 (2012).
[CrossRef]

J. Electromag. Waves Appl. (1)

R. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromag. Waves Appl. 3, 1–15, (1969).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Quant. Spectrosc. Radiat. Transfer (1)

A. Doicu and T. Wriedt, “Near-field computation using the null-field method,” J. Quant. Spectrosc. Radiat. Transfer 111, 466–473 (2010).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer. (1)

C. Forestiere, G. Iadarola, L. Dal Negro, and G. Miano, “Near-field calculation based on the T-matrix method with discrete sources,” J. Quant. Spectrosc. Radiat. Transfer. 112, 2384–2394 (2011).
[CrossRef]

Nano Lett. (4)

G. W. Bryant, F. J. Garcia de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008).
[CrossRef]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

P. Ginzburg, N. Berkovitch, A. Nevet, I. Shor, and M. Orenstein, “Resonances on-demand for plasmonic nano-particles,” Nano Lett. 11, 2329–2333 (2011).
[CrossRef]

C. Forestiere, A. J. Pasquale, A. Capretti, G. Miano, A. Tamburrino, S. Y. Lee, B. M. Reinhard, and L. Dal Negro, “Genetically engineered plasmonic nanoarrays,” Nano Lett. 12, 2037–2044 (2012).
[CrossRef]

Nat. Mater. (1)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef]

Opt. Commun. (1)

M. Mishchenko and L. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. B (1)

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

Phys. Rev. Lett. (1)

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 057401 (2003).
[CrossRef]

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Proc. R. Soc. London (1)

A. E. H. Love, “The integration of equation of propagation of electric waves,” Proc. R. Soc. London 197, 1–45 (1901).
[CrossRef]

Prog. Electromagn. Res. (1)

B. H. Jung, T. K. Sarkar, and Y.-S. Chung, “A survey of various frequency domain integral equations for the analysis of scattering from three-dimensional dielectric objects,” Prog. Electromagn. Res. 33, 193–245 (2002).
[CrossRef]

Radio Sci. (1)

P. M. Taskinen, P. Ylä-Oijala, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, 1–19 (2005).

Other (12)

A. Zhu, and S. Gedney, “Comparison of the Muller and PMWCHT surface integral formulations for the locally corrected Nystrom method,” in Proceedings of 2004 Antennas and Propagation Society International Symposium (IEEE, 2004), pp. 3871–3874.

C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer, 1969).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).

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Figures (13)

Fig. 1.
Fig. 1.

Illustration of the scattering problem.

Fig. 2.
Fig. 2.

Illustration of the RWG basis function associated to the p th edge and defined on the triangle pair T p + , T p . In T p + and T p f p is proportional to the vector ( r v p + ) and ( v p r ), respectively.

Fig. 3.
Fig. 3.

Bistatic scattering cross section σ θ / λ 2 (dB) of a gold sphere with D = 200 nm for three mesh densities with N e = (a) 228 ( R ¯ = 28 nm , Q ¯ = 0.85 ), (b) 888 ( R ¯ = 14 nm , Q ¯ = 0.86 ), and (c) 2700 ( R ¯ = 8 nm , Q ¯ = 0.87 ). (d)  C ext spectrum of the considered scatterer obtained by the T-PMCHWT for the three meshes compared to the Mie solution (continuous line).

Fig. 4.
Fig. 4.

Scattered electric field magnitude E e ( s ) of a gold sphere ( D = 200 nm ) along the segment shown in the inset for three mesh densities with N e = (a) 228 ( R ¯ = 28 nm , Q ¯ = 0.85 ), (b) 888 ( R ¯ = 14 nm , Q ¯ = 0.86 ), and (c) 2700 ( R ¯ = 8 nm , Q ¯ = 0.87 ). (d) Surface distribution of the total field magnitude E e on the surface of the sphere for N e = 2700 obtained by the T-PMCHWT.

Fig. 5.
Fig. 5.

Error in the far zone ξ f (a) and in the near zone ξ n (b) of a gold sphere ( D = 200 nm ) obtained with the mN-Muller, T-PMCHWT, and JMCFIE formulations as a function of the number of mesh edges N e .

Fig. 6.
Fig. 6.

(a) Number of GMRES iterations required to achieve an error on the solution less than ε = 10 6 as a function of the number of edges. (b) Condition number of the investigated SIE formulations as a function of the number of edges.

Fig. 7.
Fig. 7.

Scattered electric field magnitude E e ( s ) of a gold oblate spheroid ( a = 100 nm , b = 50 nm ) along the segment shown in the inset for three meshes densities with N e = (a) 276 ( R ¯ = 21 nm , Q ¯ = 0.85 ), (b) 516 ( R ¯ = 15 nm , Q ¯ = 0.86 ), and (c) 1884 ( R ¯ = 7.8 nm , Q ¯ = 0.89 ). (d) Distribution of the total field magnitude E e on the surface of the oblate spheroid obtained by the T-PMCHWT for N e = 1884 .

Fig. 8.
Fig. 8.

Scattered electric field magnitude E e ( s ) of a gold prolate spheroid ( a = 100 nm , b = 200 nm ) along the segment shown in the inset for three mesh densities with N e = (a) 624 ( R ¯ = 34 nm , Q ¯ = 0.86 ), (b) 1284 ( R ¯ = 21 nm , Q ¯ = 0.91 ), and (c) 2544 ( R ¯ = 21 nm , Q ¯ = 0.91 ). (d) Distribution of the total field magnitude E e on the surface of the gold prolate spheroid obtained with the T-PMCHWT for N e = 2544 .

Fig. 9.
Fig. 9.

Error in the far zone ξ f and in the near zone ξ n of an oblate ( a = 100 nm , b = 50 nm ) (a)–(b) and a prolate ellipsoid ( a = 100 nm , b = 200 nm ) (c)–(d) obtained with the mN-Muller, T-PMCHWT, and JMCFIE formulations as a function of the number of mesh edges N e . The NFM solution has been assumed to be the reference.

Fig. 10.
Fig. 10.

Bistatic scattering cross section σ θ / λ 2 (dB) for a gold cube with rounded edges ( L = 200 nm , R c = 20 nm ) for three mesh densities with N e = (a) 606 ( R ¯ = 22 nm , Q ¯ = 0.87 ), (b) 2424 ( R ¯ = 11 nm , Q ¯ = 0.90 ), and (c) 3777 ( R ¯ = 8.6 nm , Q ¯ = 0.92 ). (d)  C ext spectrum of the rounded cube obtained by the T-PMCHWT for the three meshes compared to the NFM solution (continuous line).

Fig. 11.
Fig. 11.

Scattered electric field magnitude E e ( s ) of a gold cube with rounded edges ( L = 200 nm , R c = 20 nm ) along the segment shown in the inset for three mesh densities with N e = (a) 606 ( R ¯ = 22 nm , Q ¯ = 0.87 ), (b) 2424 ( R ¯ = 11 nm , Q ¯ = 0.90 ), and (c) 3777 ( R ¯ = 8.6 nm , Q ¯ = 0.92 ). (d) Distribution of the total field magnitude E e on the surface of the rounded cube obtained by the T-PMCHWT with N e = 3777 .

Fig. 12.
Fig. 12.

Bistatic scattering cross section σ θ / λ 2 (dB) for a gold triangle with rounded edges ( L = 200 nm , R c = 10 nm , h = 40 nm ) for three mesh densities with number of edges (a)  N e = 477 ( R ¯ = 12 nm , Q ¯ = 0.75 ), (b)  N e = 1908 ( R ¯ = 6.2 nm , Q ¯ = 0.77 ), and (c)  N e = 2979 ( R ¯ = 4.7 nm , Q ¯ = 0.92 ). (d)  C ext spectrum of the rounded triangular prism scatterer obtained by the T-PMCHWT for the three meshes comparted to the DDA solution (continuous line).

Fig. 13.
Fig. 13.

Scattered electric field magnitude E e ( s ) of a triangular prism with rounded edges ( L = 200 nm , R c = 10 nm , h = 40 nm ) for three mesh densities with number of edges (a)  N e = 477 ( R ¯ = 12 nm , Q ¯ = 0.75 ) (b)  N e = 1908 ( R ¯ = 6.2 nm , Q ¯ = 0.77 ), and (c)  N e = 2979 ( R ¯ = 4.7 nm , Q ¯ = 0.92 ). (d) Distribution of the total field magnitude E e on the rounded triangular prism obtained by the T-PMCHWT with N e = 2979 .

Equations (79)

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{ × E l = j ω μ l H l × H l = j ω ε l E l + J l ( 0 ) in Ω l , with l = i , e ,
{ n × ( H e H i ) = 0 n × ( E e E i ) = 0 on S ,
{ E l ( s ) = E l E l ( 0 ) H l ( s ) = H l H l ( 0 ) in Ω l , with l = i , e ,
{ × E l ( s ) = j ω μ l H l ( s ) × H l ( s ) = j ω ε l E l ( s ) in Ω l , with l = i , e ,
{ n × ( H e ( s ) H i ( s ) ) = n × ( H e ( 0 ) H i ( 0 ) ) n × ( E e ( s ) E i ( s ) ) = n × ( E e ( 0 ) E i ( 0 ) ) on S ,
E l ( r ) = E l { j ( e ) , j ( m ) } ( r ) ,
H l ( r ) = H l { j ( e ) , j ( m ) } ( r ) ,
E l { j ( e ) , j ( m ) } ( r ) = ζ l T l { j ( e ) } ( r ) + K l { j ( m ) } ( r ) + { 0 if r S + ζ l 2 j k l [ S · j ( e ) ( r ) ] n 1 2 n × j ( m ) ( r ) if r S i ζ l 2 j k l [ S · j ( e ) ( r ) ] n + 1 2 n × j ( m ) ( r ) if r S e ,
H l { j ( e ) , j ( m ) } ( r ) = K l { j ( e ) } ( r ) + 1 ζ l T l { j ( m ) } ( r ) + { 0 if r S + 1 2 n × j ( e ) ( r ) + 1 2 j ζ l k l [ S · j ( m ) ( r ) ] n if r S i 1 2 n × j ( e ) ( r ) 1 2 j ζ l k l [ S · j ( m ) ( r ) ] n if r S e .
K l { w } ( r ) = S w ( r ) × g l ( r r ) d S ,
T l { w } ( r ) = j k l S g l ( r r ) w ( r ) d S 1 j k l S g l ( r r ) S · w ( r ) d S ,
g l ( r r ) = e j k l | r r | 4 π | r r | ,
E e { j e ( e ) , j e ( m ) } ( r ) + E e ( 0 ) ( r ) = { 0 if r Ω i E e ( r ) if r Ω e , H e { j e ( e ) , j e ( m ) } ( r ) + H e ( 0 ) ( r ) = { 0 if r Ω i H e ( r ) if r Ω e ,
{ j e ( e ) = + n × H e | S e , j e ( m ) = n × E e | S e ,
E i { j i ( e ) , j i ( m ) } ( r ) + E i ( 0 ) ( r ) = { E i ( r ) if r Ω i 0 if r Ω e , H i { j i ( e ) , j i ( m ) } ( r ) + H i ( 0 ) ( r ) = { H i ( r ) if r Ω i 0 if r Ω e ,
{ j i ( e ) = n × H i | S i , j i ( m ) = + n × E i | S i ,
{ j i ( e ) = j e ( e ) , j i ( m ) = j e ( m ) .
( N-EFIE ) i : n × ( E i { j e ( e ) , j e ( m ) } ( r ) + E i ( 0 ) ( r ) ) | S i + j e ( m ) ( r ) = 0 ,
( N-EFIE ) e : n × ( + E e { j e ( e ) , j e ( m ) } ( r ) + E e ( 0 ) ( r ) ) | S e + j e ( m ) ( r ) = 0 ,
( N-MFIE ) i : n × ( H i { j e ( e ) , j e ( m ) } ( r ) + H i ( 0 ) ( r ) ) | S i + j e ( e ) ( r ) = 0 ,
( N-MFIE ) e : n × ( + H e { j e ( e ) , j e ( m ) } ( r ) + H e ( 0 ) ( r ) ) | S e + j e ( e ) ( r ) = 0 ,
( T-EFIE ) i : n × n × ( E i { j e ( e ) , j e ( m ) } ( r ) + E i ( 0 ) ( r ) ) | S i n × j e ( m ) ( r ) = 0 ,
( T-EFIE ) e : n × n × ( + E e { j e ( e ) , j e ( m ) } ( r ) + E e ( 0 ) ( r ) ) | S e n × j e ( m ) ( r ) = 0 ,
( T-MFIE ) i : n × n × ( H i { j e ( e ) , j e ( m ) } ( r ) + H i ( 0 ) ( r ) ) | S i + n × j e ( e ) ( r ) = 0 ,
( T-MFIE ) e : n × n × ( + H e { j e ( e ) , j e ( m ) } ( r ) + H e ( 0 ) ( r ) ) | S e + n × j e ( e ) ( r ) = 0 ,
a e ( N-MFIE ) e + a i ( N-MFIE ) i ,
b e ( N-EFIE ) e + b i ( N-EFIE ) i .
a e ( H e ( n ) { j e ( e ) , j e ( m ) } + h e ( 0 , n ) j e ( e ) ) a i ( H i ( n ) { j e ( e ) , j e ( m ) } + h i ( 0 , n ) j e ( e ) ) = 0 ,
b e ( E e ( n ) { j e ( e ) , j e ( m ) } + e e ( 0 , n ) + j e ( m ) ) + b i ( E i ( n ) { j e ( e ) , j e ( m ) } + e i ( 0 , n ) + j e ( m ) ) = 0 ,
e l ( 0 , n ) = n × E l ( 0 ) | S l , h l ( 0 , n ) = n × H l ( 0 ) | S l ,
H l ( n ) { j ( e ) , j ( m ) } = n × H l { j ( e ) , j ( m ) } | S l , E l ( n ) { j ( e ) , j ( m ) } = n × E l { j ( e ) , j ( m ) } | S l .
C ( n ) x = y ( n ) ,
C ( n ) = | a e ( 1 2 I + K e ( n ) ) a e ζ e 1 T e ( n ) b e ζ e T e ( n ) b e ( 1 2 I + K e ( n ) ) | + | a i ( 1 2 I K i ( n ) ) a i ζ i 1 T i ( n ) b i ζ i T i ( n ) b i ( 1 2 I K i ( n ) ) | ,
K l ( n ) { · } = n × K l { · } | S l ,
T l ( n ) { · } = n × T l { · } | S l .
x = | j e ( e ) j e ( m ) | ,
y ( n ) = | a e h e ( 0 , n ) + a i h i ( 0 , n ) b e e e ( 0 , n ) b i e i ( 0 , n ) | .
N-PMCHWT ( a e = 1 , a i = 1 , b e = 1 , b i = 1 ) ; CNF ( a e = 1 , a i = 1 , b e = 1 , b i = 1 ) ; N-Muller ( a e = μ e , a i = μ i , b e = ε e , b i = ε i ) ; mN-Muller ( a e = μ e ( μ e + μ i ) 1 , a i = μ i ( μ e + μ i ) 1 , b e = ε e ( ε e + ε i ) 1 , b i = ε i ( ε e + ε i ) 1 ) .
c e ( T-EFIE ) e + c i ( T-EFIE ) i ,
d e ( T-MFIE ) e + d i ( T-MFIE ) i .
c e ( E e ( t ) { j e ( e ) , j e ( m ) } + e e ( 0 , t ) N { j e ( m ) } ) + c i ( E i ( t ) { j e ( e ) , j e ( m ) } + e i ( 0 , t ) N { j e ( m ) } ) = 0 ,
d e ( H e ( t ) { j e ( e ) , j e ( m ) } + h e ( 0 , t ) + N { j e ( e ) } ) + d i ( H i ( t ) { j e ( e ) , j e ( m ) } + h i ( 0 , t ) + N { j e ( e ) } ) = 0 ,
e l ( 0 , t ) = n × n × E l ( 0 ) | S l , h l ( 0 , t ) = n × n × H l ( 0 ) | S l ,
H l ( t ) { j ( e ) , j ( m ) } = n × n × H l { j ( e ) , j ( m ) } | S l , E l ( t ) { j ( e ) , j ( m ) } = n × n × E l { j ( e ) , j ( m ) } | S l , N { · } = n × { · } | S l .
C ( t ) x = y ( t ) ,
C ( t ) = | c e ζ e T e ( t ) c e ( 1 2 N K e ( t ) ) d e ( 1 2 N K e ( t ) ) d e ζ e 1 T e ( t ) | + | c i ζ i T i ( t ) c i ( 1 2 N + K i ( t ) ) d i ( 1 2 N + K i ( t ) ) d i ζ i 1 T i ( t ) | ,
K l ( t ) { · } = n × n × K l { · } | S l ,
T l ( t ) { · } = n × n × T l { · } | S l .
y ( t ) = | c e e e ( 0 , t ) + c i e i ( 0 , t ) d e h e ( 0 , t ) + d i h i ( 0 , t ) | .
T-PMCHWT ( c e = 1 , c i = 1 , d e = 1 , d i = 1 ) ; CTF ( c e = ζ e 1 , c i = ζ i 1 , d e = ζ e , d i = ζ i ) .
α e ζ e 1 ( T-EFIE ) e + β e ζ e 1 ( N-EFIE ) e γ e ( T-MFIE ) e + δ e ( N-MFIE ) e ,
α i ( T-EFIE ) i + β i ( N-EFIE ) i γ i ζ i ( T-MFIE ) i + δ i ζ i ( N-MFIE ) i .
( N-MFIE ) e + ( N-MFIE ) i ζ e 1 ( T-EFIE ) e + ζ i 1 ( T-EFIE ) i ,
( N-EFIE ) e + ( N-EFIE ) i ζ e ( T-MFIE ) e + ζ i ( T-MFIE ) i .
C JMCFIE x = ( C JMCFIE ( t ) + C JMCFIE ( n ) ) x = ( y JMCFIE ( n ) y JMCFIE ( t ) ) = y JMCFIE ,
C JMCFIE ( t ) = C ( t ) { c e = ζ e 1 , c i = ζ i 1 , d e = ζ e , d i = ζ i } , C JMCFIE ( n ) = C ( n ) { a e = a i = b e = 1 = b i = 1 } ,
y JMCFIE ( t ) = y ( t ) { c e = ζ e 1 , c i = ζ i 1 , d e = ζ e , d i = ζ i } , y JMCFIE ( n ) = y ( n ) { a e = a i = b e = b i = 1 } .
ζ e T e { j e ( e ) } ( r ) + K e { j e ( m ) } ( r ) + E e ( 0 ) ( r ) = 0 r Ω i .
{ K e { · } = + 1 j k e × T e { · } , T e { · } = 1 j k e × K e { · } .
E e ( 0 ) ( r ) = × S j e ( m ) ( r ) g e ( k e , r , r ) d S + j ζ e k e × S j e ( e ) ( r ) g e ( k e , r , r ) d S r Ω i .
{ j e ( e ) ( r ) p = 1 N e α p f p ( r ) j e ( m ) ( r ) p = 1 N e β p f p ( r ) ,
f p ( r ) = { + l p 2 A p + ( r v p + ) r T p + l p 2 A p ( r v p ) r T p 0 otherwise ,
f , g = S f · g d S .
K l , p q ( n ) = f p , K l ( n ) f q , T l , p q ( n ) = f p , T l ( n ) f q , K l , p q ( t ) = f p , K l ( t ) f q , T l , p q ( t ) = f p , T l ( t ) f q , I l , p q = f p , I l f q , N l , p q = f p , N l f q .
h l , p ( 0 , n ) = f p , h l ( 0 , n ) , e l ( 0 , n ) = f p , e l ( 0 , n ) , h l , p ( 0 , t ) = f p , h l ( 0 , t ) , e l ( 0 , t ) = f p , e l ( 0 , t ) .
{ p p ( e ) = α p l p j ω ( c p c p + ) , p p ( m ) = β p l p j ω ( c p c p + ) .
{ B e , ( s ) ( k ^ s ) = p = 1 N e k 0 2 4 π [ ζ 0 k ^ s × p p ( e ) k ^ s × k ^ s × p p ( m ) ] e ( + j k 0 k ^ s · e p ) E e , ( s ) ( k ^ s ) = p = 1 N e k 0 2 4 π [ 1 ε 0 k ^ s × k ^ s × p p ( e ) c k ^ s × p p ( m ) ] e ( + j k 0 k ^ s · e p ) ,
C ext = 4 π k 0 { E e , ( s ) ( k ^ i ) · b ^ } ,
E e ( 0 ) ( r ) = ν = 1 N tot a ν M ν 1 ( k e r ) + b ν N ν 1 ( k e r ) ,
ν = 1 N tot n = 1 N m = n n .
N tot = N 2 + 2 N .
j k e 2 π S [ j e ( m ) ( r ) · ( N μ ¯ 3 ( k e r ) M μ ¯ 3 ( k e r ) ) + j ζ e j e ( e ) ( r ) · ( M μ ¯ 3 ( k e r ) N μ ¯ 3 ( k e r ) ) ] d S = ( a μ b μ ) μ N tot ,
{ j e ( m ) ( r ) = ν = 1 N tot c ν n × M ν 1 ( k i r ) + d ν n × N ν 1 ( k i r ) j e ( e ) ( r ) = j ζ i ν = 1 N tot c ν n × N ν 1 ( k i r ) + d ν n × M ν 1 ( k i r ) on S .
Q 31 ( k e , k i ) J = B ,
Q t = 2 r t R t .
R ¯ = 1 N t t = 1 N t R t , Q ¯ = 1 N t t = 1 N t Q t .
E s ( 0 ) ( r ) = ν = 1 N tot a ν M ν 3 ( k e r ) + b ν N ν 3 ( k e r ) ,
ξ f = | C ext C ¯ ext | C ¯ ext ,
ξ n = S n × ( E e ( s ) E ¯ e ( s ) ) | S e 2 n × E ¯ e ( s ) | S e 2 d S ,

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