Abstract

We present a numerical formalism for solving the Lippmann–Schwinger equation for the electric field in three dimensions. The formalism may be applied to scatterers of different shapes and embedded in different background media, and we develop it in detail for the specific case of spherical scatterers in a homogeneous background medium. In addition, we show how several physically important quantities may readily be calculated with the formalism. These quantities include the extinction cross section, the total Green’s tensor, the projected local density of states, and the Purcell factor as well as the quasi-normal modes of leaky resonators with the associated resonance frequencies and quality factors. We demonstrate the calculations for the well-known plasmonic dimer consisting of two silver nanoparticles and thus illustrate the versatility of the formalism for use in modeling of advanced nanophotonic devices.

© 2013 Optical Society of America

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2012 (2)

J. R. de Lasson, P. T. Kristensen, and J. Mørk, “Multiple-scattering formalism beyond the quasistatic approximation: analyzing resonances in plasmonic chains,” AIP Conf. Proc. 1475, 158–160 (2012).
[CrossRef]

P. T. Kristensen, C. V. Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012).
[CrossRef]

2011 (3)

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011).
[CrossRef]

K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photon. Rev. 5, 773–809 (2011).
[CrossRef]

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011).
[CrossRef]

2010 (5)

2009 (3)

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113, 2784–2791 (2009).
[CrossRef]

M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009).
[CrossRef]

J. Kellendonk and S. Richard, “Weber–Schafheitlin-type integrals with exponent 1,” Integral Transforms Spec. Funct. 20, 147–153 (2009).
[CrossRef]

2008 (4)

M. Chen, Y.-F. Chau, and D. Tsai, “Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres,” Plasmonics 3, 157–164 (2008).
[CrossRef]

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

J. Z. Zhang and C. Noguez, “Plasmonic optical properties and applications of metal nanostructures,” Plasmonics 3, 127–150 (2008).
[CrossRef]

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–21800 (2008).
[CrossRef]

2007 (2)

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7, 2080–2088 (2007).
[CrossRef]

2004 (1)

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
[CrossRef]

2003 (1)

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

2000 (1)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

1999 (3)

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
[CrossRef]

F. J. García de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B 59, 3095–3107 (1999).
[CrossRef]

K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. I. Completeness relation,” J. Opt. Soc. Am. B 16, 1409–1417 (1999).
[CrossRef]

1998 (3)

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

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

P. A. Martin, “Multiple scattering and the Rehr–Albers–Fritzsche formula for the propagator matrix,” J. Phys. A 31, 8923 (1998).
[CrossRef]

1996 (2)

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

1995 (2)

1994 (2)

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

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys 114, 185–200 (1994).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

1980 (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

1975 (1)

V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861–871 (1975).
[CrossRef]

1948 (1)

H. Levine and J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948).
[CrossRef]

1946 (1)

E. M. Purcell, “Proceedings of the American Physical Society, b10. Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 674 (1946).
[CrossRef]

1908 (1)

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Alivisatos, A. P.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011).
[CrossRef]

Arnold, M. D.

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113, 2784–2791 (2009).
[CrossRef]

Asatryan, A. A.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys 114, 185–200 (1994).
[CrossRef]

Bischoff, S.

Blaber, M. G.

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113, 2784–2791 (2009).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Pergamon, 1981).

Botten, L. C.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Busch, K.

K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photon. Rev. 5, 773–809 (2011).
[CrossRef]

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Bykov, V. P.

V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861–871 (1975).
[CrossRef]

Capolino, F.

F. Capolino, Theory and Phenomena of Metamaterials, 1st ed. (CRC Press, 2009).

Catchpole, K. R.

Chau, Y.-F.

M. Chen, Y.-F. Chau, and D. Tsai, “Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres,” Plasmonics 3, 157–164 (2008).
[CrossRef]

Chen, M.

M. Chen, Y.-F. Chau, and D. Tsai, “Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres,” Plasmonics 3, 157–164 (2008).
[CrossRef]

Chen, Y.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[CrossRef]

Ching, E. S. C.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Chung, I.-S.

de Lasson, J. R.

J. R. de Lasson, P. T. Kristensen, and J. Mørk, “Multiple-scattering formalism beyond the quasistatic approximation: analyzing resonances in plasmonic chains,” AIP Conf. Proc. 1475, 158–160 (2012).
[CrossRef]

de Sterke, C. M.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Dems, M.

Draine, B. T.

El-Sayed, M. A.

P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7, 2080–2088 (2007).
[CrossRef]

Flatau, P. J.

Ford, M. J.

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113, 2784–2791 (2009).
[CrossRef]

Funston, A. M.

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

García de Abajo, F. J.

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

F. J. García de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B 59, 3095–3107 (1999).
[CrossRef]

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
[CrossRef]

Gay-Balmaz, P.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

Gaylord, T. K.

Giessen, H.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011).
[CrossRef]

Grann, E. B.

Gray, S. K.

M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009).
[CrossRef]

Gregersen, N.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[CrossRef]

Guyot-Sionnest, P.

M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009).
[CrossRef]

Hagness, S.

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

Hao, E.

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
[CrossRef]

Harris, N.

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113, 2784–2791 (2009).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006).

Hentschel, M.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011).
[CrossRef]

Huang, W.

P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7, 2080–2088 (2007).
[CrossRef]

Hughes, S.

Jackson, J. D.

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

Jain, P. K.

P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7, 2080–2088 (2007).
[CrossRef]

Jauho, A.-P.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Kellendonk, J.

J. Kellendonk and S. Richard, “Weber–Schafheitlin-type integrals with exponent 1,” Integral Transforms Spec. Funct. 20, 147–153 (2009).
[CrossRef]

Koenderink, A. F.

König, M.

K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photon. Rev. 5, 773–809 (2011).
[CrossRef]

Kristensen, P. T.

Lagendijk, A.

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

Langtry, T. N.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Lee, K. M.

Lee, T.-W.

M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009).
[CrossRef]

Leung, P. T.

K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. I. Completeness relation,” J. Opt. Soc. Am. B 16, 1409–1417 (1999).
[CrossRef]

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Levine, H.

H. Levine and J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948).
[CrossRef]

Lipschutz, S.

M. R. Spiegel, S. Lipschutz, and J. Liu, Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 3rd ed. (McGraw-Hill, 2008).

Liu, J.

M. R. Spiegel, S. Lipschutz, and J. Liu, Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 3rd ed. (McGraw-Hill, 2008).

Liu, M.

M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009).
[CrossRef]

Liu, N.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011).
[CrossRef]

Liz-Marzan, L. M.

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

Lodahl, P.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[CrossRef]

P. T. Kristensen, P. Lodahl, and J. Mørk, “Light propagation in finite-sized photonic crystals: multiple scattering using an electric field integral equation,” J. Opt. Soc. Am. B 27, 228–237 (2010).
[CrossRef]

Mackowski, D. W.

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications, 1st ed. (Springer, 2007).

Martin, O. J. F.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

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

Martin, P. A.

P. A. Martin, “Multiple scattering and the Rehr–Albers–Fritzsche formula for the propagator matrix,” J. Phys. A 31, 8923 (1998).
[CrossRef]

P. A. Martin, Multiple Scattering. Interaction of Time-Harmonic Waves with N Obstacles, 1st ed. (Cambridge University, 2006).

McPhedran, R. C.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Mie, G.

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Mishchenko, M. I.

Mittra, R.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, 1st ed. (IEEE, 1998).

Moharam, M. G.

Mørk, J.

J. R. de Lasson, P. T. Kristensen, and J. Mørk, “Multiple-scattering formalism beyond the quasistatic approximation: analyzing resonances in plasmonic chains,” AIP Conf. Proc. 1475, 158–160 (2012).
[CrossRef]

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[CrossRef]

P. T. Kristensen, P. Lodahl, and J. Mørk, “Light propagation in finite-sized photonic crystals: multiple scattering using an electric field integral equation,” J. Opt. Soc. Am. B 27, 228–237 (2010).
[CrossRef]

Mortensen, N. A.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011).
[CrossRef]

Mulvaney, P.

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

Myroshnychenko, V.

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

Nicorovici, N. A.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Niegemann, J.

K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photon. Rev. 5, 773–809 (2011).
[CrossRef]

Nielsen, T. R.

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[CrossRef]

Noguez, C.

J. Z. Zhang and C. Noguez, “Plasmonic optical properties and applications of metal nanostructures,” Plasmonics 3, 127–150 (2008).
[CrossRef]

Novo, C.

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006).

Nyakas, P.

Panajotov, K.

Pang, K. M.

Pastoriza-Santos, I.

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

Paulus, M.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

Pelton, M.

M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009).
[CrossRef]

Peterson, A. F.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, 1st ed. (IEEE, 1998).

Piller, N. B.

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

Polman, A.

Pommet, D. A.

Purcell, E. M.

E. M. Purcell, “Proceedings of the American Physical Society, b10. Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 674 (1946).
[CrossRef]

Pustovit, V. N.

V. N. Pustovit and T. V. Shahbazyan, “Plasmon-mediated superradiance near metal nanostructures,” Phys. Rev. B 82, 075429 (2010).
[CrossRef]

Ray, S. L.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, 1st ed. (IEEE, 1998).

Raza, S.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011).
[CrossRef]

Reddy, J.

J. Reddy, An Introduction to the Finite Element Method, 3rd ed. (McGraw-Hill Science/Engineering/Math, 2005).

Richard, S.

J. Kellendonk and S. Richard, “Weber–Schafheitlin-type integrals with exponent 1,” Integral Transforms Spec. Funct. 20, 147–153 (2009).
[CrossRef]

Robinson, P. A.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Rodriguez-Fernandez, J.

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

Schatz, G. C.

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
[CrossRef]

Schwinger, J.

H. Levine and J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948).
[CrossRef]

Shahbazyan, T. V.

V. N. Pustovit and T. V. Shahbazyan, “Plasmon-mediated superradiance near metal nanostructures,” Phys. Rev. B 82, 075429 (2010).
[CrossRef]

Shore, R. A.

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

Smith, G. H.

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Spiegel, M. R.

M. R. Spiegel, S. Lipschutz, and J. Liu, Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 3rd ed. (McGraw-Hill, 2008).

Sprik, R.

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

Suen, W. M.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Taflove, A.

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

Tai, C.-T.

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

Tong, S. S.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Toscano, G.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011).
[CrossRef]

Tsai, D.

M. Chen, Y.-F. Chau, and D. Tsai, “Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres,” Plasmonics 3, 157–164 (2008).
[CrossRef]

van den Brink, A. Maassen

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

van Tiggelen, B. A.

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

Vlack, C. V.

Weiss, T.

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Pergamon, 1981).

Wubs, M.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011).
[CrossRef]

Xu, Y. L.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Yaghjian, A. D.

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Young, K.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Zhang, J. Z.

J. Z. Zhang and C. Noguez, “Plasmonic optical properties and applications of metal nanostructures,” Plasmonics 3, 127–150 (2008).
[CrossRef]

AIP Conf. Proc. (1)

J. R. de Lasson, P. T. Kristensen, and J. Mørk, “Multiple-scattering formalism beyond the quasistatic approximation: analyzing resonances in plasmonic chains,” AIP Conf. Proc. 1475, 158–160 (2012).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Appl. Opt. (1)

Chem. Soc. Rev. (1)

V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008).
[CrossRef]

Europhys. Lett. (1)

R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996).
[CrossRef]

Integral Transforms Spec. Funct. (1)

J. Kellendonk and S. Richard, “Weber–Schafheitlin-type integrals with exponent 1,” Integral Transforms Spec. Funct. 20, 147–153 (2009).
[CrossRef]

J. Chem. Phys. (1)

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
[CrossRef]

J. Comput. Phys (1)

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys 114, 185–200 (1994).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

J. Phys. A (1)

P. A. Martin, “Multiple scattering and the Rehr–Albers–Fritzsche formula for the propagator matrix,” J. Phys. A 31, 8923 (1998).
[CrossRef]

J. Phys. Chem. C (1)

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113, 2784–2791 (2009).
[CrossRef]

Laser Photon. Rev. (1)

K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photon. Rev. 5, 773–809 (2011).
[CrossRef]

Nano Lett. (1)

P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7, 2080–2088 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (2)

H. Levine and J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948).
[CrossRef]

E. M. Purcell, “Proceedings of the American Physical Society, b10. Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 674 (1946).
[CrossRef]

Phys. Rev. B (5)

V. N. Pustovit and T. V. Shahbazyan, “Plasmon-mediated superradiance near metal nanostructures,” Phys. Rev. B 82, 075429 (2010).
[CrossRef]

Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010).
[CrossRef]

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011).
[CrossRef]

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999).
[CrossRef]

F. J. García de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B 59, 3095–3107 (1999).
[CrossRef]

Phys. Rev. E (2)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[CrossRef]

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

Phys. Rev. Lett. (3)

M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009).
[CrossRef]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Plasmonics (2)

M. Chen, Y.-F. Chau, and D. Tsai, “Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres,” Plasmonics 3, 157–164 (2008).
[CrossRef]

J. Z. Zhang and C. Noguez, “Plasmonic optical properties and applications of metal nanostructures,” Plasmonics 3, 127–150 (2008).
[CrossRef]

Proc. IEEE (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Prog. Electromagn. Res. (1)

L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003).
[CrossRef]

Radio Sci. (1)

R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
[CrossRef]

Rev. Mod. Phys. (1)

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[CrossRef]

Science (1)

N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861–871 (1975).
[CrossRef]

Other (11)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Pergamon, 1981).

P. A. Martin, Multiple Scattering. Interaction of Time-Harmonic Waves with N Obstacles, 1st ed. (Cambridge University, 2006).

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

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

L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006).

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, 1st ed. (IEEE, 1998).

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

J. Reddy, An Introduction to the Finite Element Method, 3rd ed. (McGraw-Hill Science/Engineering/Math, 2005).

M. R. Spiegel, S. Lipschutz, and J. Liu, Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 3rd ed. (McGraw-Hill, 2008).

S. A. Maier, Plasmonics: Fundamentals and Applications, 1st ed. (Springer, 2007).

F. Capolino, Theory and Phenomena of Metamaterials, 1st ed. (CRC Press, 2009).

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Figures (10)

Fig. 1.
Fig. 1.

Example of scattering geometry where an incoming field, E B , impinges on N = 3 spherical scatterers embedded in a homogeneous material of permittivity ϵ B . The scatterers have permittivities ϵ j , and the local coordinate systems are indicated.

Fig. 2.
Fig. 2.

Two scatterers with indices j and j , sketched in 2D. The centers of the scatterers, r j 0 and r j 0 , and two arbitrary local points inside the scatterers, r j and r j , are indicated. The displacement vectors between the two points, R , and between the centers of the scatterers, b , are shown.

Fig. 3.
Fig. 3.

Integration domains for evaluation of self terms. Gray shading indicates a volume that is excluded from the r j integration. Left panel: integration domain for A j , j α α , extending over all space minus the principal volume. Right panel: integration domain for B j , j α α , extending over all space minus the scatterer volume.

Fig. 4.
Fig. 4.

Schematic of plasmonic dimer consisting of two Ag spheres aligned along the y axis. Each particle has radius R = 25 nm , and the surface-to-surface distance between the spheres is d .

Fig. 5.
Fig. 5.

Relative enhancement of the electric field intensity, log 10 ( | E | 2 / | E B | 2 ) , for scattering of a plane wave on a dimer, shown in the x y plane at z = 0 . The spacing between the particles is d = 10 nm . The incoming plane wave propagates along the z direction and is polarized parallel (top panel, λ 0 = 505 nm ) or perpendicular (bottom panel, λ 0 = 412 nm ) to the dimer axis. Note that different color scales are used in the two plots.

Fig. 6.
Fig. 6.

Extinction efficiency Q ext versus excitation wavelength for scattering of a plane wave on two Ag spheres spaced by d = 50 nm . The incoming field is polarized parallel (top panel) or perpendicular (bottom panel) to the dimer axis. The spectra have been obtained using the full formalism with l max = 8 and the DA.

Fig. 7.
Fig. 7.

Global relative error of the electric field as a function of distance between two Ag spheres, d / R . Results for three different truncations of the set of basis functions are shown. Illumination by plane waves at λ 0 = 800 nm with oblique incidence and polarization.

Fig. 8.
Fig. 8.

Imaginary part of Green’s tensor Im ( G y y ( r , r ) ) relative to Im ( G B y y ( r , r ) ) for the dimer with two Ag spheres ( d / R = 0.4 ). The two panels show results with different positions of the dipole emitter, r , (black dots) and different wavelengths λ 0 = 505 nm (top) and λ 0 = 447 nm (bottom). Note that different color scales are used in the two plots.

Fig. 9.
Fig. 9.

Spectra of Purcell factor for dipole emitter at two positions, r 1 (top panel) and r 2 (bottom panel), in the z = 0 plane in the vicinity of the Ag dimer with d / R = 0.4 . Two orientations of the dipole moment of the emitter, α { x , y } , have been employed. Note that different scalings are used in the two plots.

Fig. 10.
Fig. 10.

Real part of y components of quasi-normal modes of plasmonic dimer, as specified in Fig. 9, in the x y plane. The modes, found at Re ( λ 0 bright ) = 505 nm (top panel) and Re ( λ 0 dark ) = 447 nm (bottom panel), respectively, are bright and dark modes, respectively. Note that different scalings are used in the two plots.

Equations (63)

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× × E ( r ; ω ) k 0 2 ϵ B ( ω ) E ( r ; ω ) = k 0 2 Δ ϵ ( r ; ω ) E ( r ; ω ) ,
E ( r ) = E B ( r ) + k 0 2 V scat G B ( r , r ) Δ ϵ ( r ) E ( r ) d r ,
G B α α ( r , r ) = ( δ α α + 1 k B 2 α α ) g B ( r , r ) ,
g B ( r , r ) = exp ( i k B | r r | ) 4 π | r r | ,
E ( r ) = E B ( r ) + k 0 2 V scat δ V G B ( r , r ) Δ ϵ ( r ) E ( r ) d r L Δ ϵ ( r ) ϵ B E ( r ) .
E ( r j ) = α , l , m a j α l m ψ l , m j ( r j ) e α ,
E B ( r j ) = α , l , m a j α l m B ψ l , m j , B ( r j ) e α ,
f | g { f ( r ) } Y g ( r ) d r ,
ψ l , m j | ψ l , m j = δ j j δ l l δ m m ,
ψ l , m j , B | ψ l , m j , B = δ j j δ l l δ m m ,
ψ l , m j | ψ l , m j , B = M l j δ j j δ l l δ m m ,
a = M B a B + ( k 0 2 G Δ ϵ L ϵ B Δ ϵ ) a ,
[ G j , j α α ] l , l m , m V j V j δ V { ψ l , m j ( r j ) } Y G B α α ( r j , r j ) ψ l , m j ( r j ) d r j d r j .
g B ( r , r ) = i k B p , t ν , μ S p , ν t , μ ( b ) { ψ ˜ p , t j , B ( r j ) } Y ψ ˜ ν , μ j , B ( r j ) ,
G B α α ( r , r ) = i k B p , t ν , μ S p , ν t , μ ( b ) { ψ ˜ p , t j , B ( r j ) } Y × ( δ α α + 1 k B 2 α α ) ψ ˜ ν , μ j , B ( r j ) .
α α ψ ˜ ν , μ j , B ( r j ) = γ α , α g γ α , α ψ ˜ ν ( γ α , α ) , μ ( γ α , α ) j , B ( r j ) ,
[ G j , j α α ] l , l m , m = i k B M l j M l j / ( N l j , B N l j , B ) × ( δ α α S l , l m , m ( b ) + 1 k B 2 γ α , α g γ α , α S l , l γ α , α m , m γ α , α ( b ) ) .
[ G j , j α α ] l , l m , m = [ A j , j α α ] l , l m , m [ B j , j α α ] l , l m , m ,
[ A j , j α α ] l , l m , m V j R 3 δ V j { ψ l , m j ( r j ) } Y G B α α ( r j , r j ) ψ l , m j ( r j ) d r j d r j ,
[ B j , j α α ] l , l m , m V j R 3 V j { ψ l , m j ( r j ) } Y G B α α ( r j , r j ) ψ l , m j ( r j ) d r j d r j .
ψ l , m j ( r j ) = N l j ν , μ ( 1 ) μ S ^ l , ν m , μ ( r j ) { ψ ˜ ν , μ j ( R ) } Y
g B ( r , r ) = g B ( R ) = i k B 4 π φ 0 , 0 B ( R ) .
[ A j , j α α ] l , l m , m = i k B 4 π N l j ( δ α α ψ l , m j | S ^ l , 0 m , 0 I 0 R 3 δ V j + 1 k B 2 γ α , α g γ α , α ( 1 ) γ α , α × ψ l , m j | S ^ l , γ α , α m , γ α , α I γ α , α R 3 δ V j ) ,
g B ( R ) = i k B ν , μ ψ ˜ ν , μ j , B ( r j ) { φ ν , μ B ( r j ) } Y .
[ B j , j α α ] l , l m , m = i k B M l j I l R 3 V j N l j / ( N l j , B ) × ( δ α α δ l l δ m m + 1 k B 2 γ α , α g γ α , α δ l γ α , α l δ m γ α , α m ) ,
E α ( r ) = E B α ( r ) + k 0 2 j = 1 N Δ ϵ j α H j α α ( r ) ,
H j α α ( r ) V j G B α α ( r , r j ) E α ( r j ) d r j .
E B ( r ) = E 0 exp ( i k B · r ) e B ,
a j α l m B = [ E 0 e B α 4 π i l { Y l m ( θ k , ϕ k ) } * / ( N l j , B ) ] exp ( i k B · r j 0 ) ,
a j α l m B = i k B ( 1 ) m / ( N l j , B ) ( δ α α φ l , m B ( r j ) + 1 k B 2 γ α , α g γ α , α φ l ( γ α , α ) , m * ( γ α , α ) B ( r j ) ) .
ρ α ( r ; λ 0 ) = 4 c λ 0 Im ( G α α ( r , r ) ) .
f i ( r ; ω ˜ i ) = k 0 2 ( ω ˜ i ) V G B ( r , r ; ω ˜ i ) Δ ϵ ( r ; ω ˜ i ) f i ( r ; ω ˜ i ) d r .
E scat ( r ) f ( θ , ϕ ) E 0 exp ( i k B r ) r , 1 k B r ,
h l ( 1 ) ( k B r ) ( i ) l + 1 k B exp ( i k B r ) r , 1 k B r ,
C ext = 4 π k B Im ( f ( θ k , ϕ k ) · e B * ) .
E L ( r ) | E B ( r ) E ( r ) L Δ ϵ ( r ) ϵ B E ( r ) + k 0 2 V scat δ V G B ( r , r ) Δ ϵ ( r ) E ( r ) d r | .
E G V scat E L ( r ) d r V scat | E ( r ) | d r ,
ψ l , m j ( r j ) S j ( r ) N l j j l ( k j r j ) Y l m ( θ j , ϕ j ) ,
ψ l , m j , B ( r j ) S j ( r ) N l j , B j l ( k B r j ) Y l m ( θ j , ϕ j ) ,
S j ( r ) { 1 r V j 0 otherwise ,
φ l , m B ( r ) h l ( 1 ) ( k B r ) Y l m ( θ , ϕ ) ,
ψ ˜ l , m j ( r j ) ψ l , m j ( r j ) N l j ,
ψ ˜ l , m j , B ( r j ) ψ l , m j , B ( r j ) N l j , B ,
Ω l m ( r ) { ψ ˜ l , m j ( r j ) , ψ ˜ l , m j , B ( r j ) , φ l , m B ( r ) } .
D 1 ± 1 k ( x ± i y ) ,
D 1 0 1 k z ,
D 1 ± Ω l m ( r ) = ( ( l ± m + 2 ) ( l ± m + 1 ) ( 2 l + 1 ) ( 2 l + 3 ) Ω l + 1 m ± 1 ( r ) + ( l m ) ( l m 1 ) 4 l 2 1 Ω l 1 m ± 1 ( r ) ) ,
D 1 0 Ω l m ( r ) = ( l + 1 ) 2 m 2 ( 2 l + 1 ) ( 2 l + 3 ) Ω l + 1 m ( r ) l 2 m 2 4 l 2 1 Ω l 1 m ( r ) .
S p , ν t , μ ( b ) = 4 π ( 1 ) ν + μ + Q q = 0 Q ( 1 ) q φ q 0 + 2 q , t μ B ( b ) × G ( p , t ; ν , μ ; q 0 + 2 q ) ,
Q = p + ν q 0 2 ,
q 0 = q 0 ( p , t ; ν , μ ) ,
q 0 ( p , t ; ν , μ ) = { | p ν | if | p ν | | t + μ | | p + μ | if | p ν | < | t + μ | and p + ν + | t + μ | is even | p + μ | + 1 if | p ν | < | t + μ | and p + ν + | t + μ | is odd ,
G ( p , t ; ν , μ ; q ) ( 1 ) t + μ Ω Y p t ( θ , ϕ ) Y ν μ ( θ , ϕ ) Y q t μ ( θ , ϕ ) d Ω
G ( p , t ; ν , μ ; q ) = ( 1 ) t + μ ( 2 p + 1 ) ( 2 ν + 1 ) ( 2 q + 1 ) 4 π × ( p ν q 0 0 0 ) ( p ν q t μ t μ ) ,
S ^ p , ν t , μ ( b ) = 4 π ( 1 ) ν + μ + Q q = 0 Q ( 1 ) q ψ ˜ q 0 + 2 q , t μ j ( b ) × G ( p , t ; ν , μ ; q 0 + 2 q ) .
I l R 3 δ V j lim δ R 0 ( δ R h l ( 1 ) ( k B r ) j l ( k j r ) r 2 d r ) .
lim r 0 ( h l ( 1 ) ( k B r ) j l ( k j r ) r 2 ) = 0 .
I l R 3 δ V j = 0 h l ( 1 ) ( k B r ) j l ( k j r ) r 2 d r = π 2 k B k j 0 H l + 1 / 2 ( k B r ) J l + 1 / 2 ( k j r ) r d r .
I l R 3 δ V j = i k B 1 k j 2 k B 2 ( k j k B ) l .
I l V j 0 R j h l ( 1 ) ( k B r ) j l ( k j r ) r 2 d r = M l j N j l N j l , B + i π 2 1 k j k B 1 k j 2 k B 2 × { [ k j R j Y l + 1 / 2 ( k B R j ) J l + 3 / 2 ( k j R j ) k B R j J l + 1 / 2 ( k j R j ) Y l + 3 / 2 ( k B R j ) ] k j l + 1 / 2 k B l 1 / 2 2 π } .
I l R 3 V j R j h l ( 1 ) ( k B r ) j l ( k j r ) r 2 d r = I l R 3 δ V j I l V j .
g B ( r , r ) = i k B ν , μ ( 1 ) μ ψ ˜ ν , μ j , B ( r j ) φ ν , μ B ( r j ) .
G B α α ( r , r d ) = i k B ν , μ ( 1 ) μ ψ ν , μ j , B ( r j ) / ( N ν j , B ) × ( δ α α φ ν , μ B ( r j ) + 1 k B 2 γ α , α g γ α , α φ ν ( γ α , α ) , μ * ( γ α , α ) B ( r j ) ) ,

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