Abstract

We present an accurate, stable, and efficient solution to the Lippmann–Schwinger equation for electromagnetic scattering in two dimensions. The method is well suited for multiple scattering problems and may be applied to problems with scatterers of arbitrary shape or non-homogenous background materials. We illustrate the method by calculating light emission from a line source in a finite-sized photonic crystal waveguide.

© 2010 Optical Society of America

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  1. V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861-871 (1975).
    [CrossRef]
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  3. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  4. K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
    [CrossRef] [PubMed]
  5. C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, 1990).
  6. A. Tavlove, Computational Electromagnetics: The Finite-Difference Time-Domain Method (Artech House, 1995).
  7. P. Šolín, Partial Differential Equations and the Finite Element Method (Wiley Interscience, 2006).
  8. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322-2330 (2002).
    [CrossRef]
  9. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation: errata,” J. Opt. Soc. Am. B 20, 1581-1581 (2003).
    [CrossRef]
  10. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331-2340 (2002).
    [CrossRef]
  11. S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919-1928 (2004).
    [CrossRef]
  12. A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 13, 9-25 (2003).
    [CrossRef]
  13. D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, “Three-dimensional Green's tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders,” Phys. Rev. E 70, 066608 (2004).
    [CrossRef]
  14. X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161-4167 (1993).
    [CrossRef]
  15. V. Yannopapas and N. V. Vitanov, “Electromagnetic Green's tensor and local density of states calculations for collections of spherical scatterers,” Phys. Rev. B 75, 115124 (2007).
    [CrossRef]
  16. A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (IEEE, 1998).
  17. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973).
    [CrossRef]
  18. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491-1499 (1994).
    [CrossRef]
  19. R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” EPL 35, 265-270 (1996).
    [CrossRef]
  20. K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
    [CrossRef]
  21. R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
    [CrossRef]
  22. I. S. Nikolaev, W. L. Vos, and A. F. Koenderink, “Accurate calculation of the local density of optical states in inverse-opal photonic crystals,” J. Opt. Soc. Am. B 26, 987-997 (2009).
    [CrossRef]
  23. A. F. Koenderink, M. Kafesaki, C. M. Soukoulis, and V. Sandoghdar, “Spontaneous emission rates of dipoles in photonic crystal membranes,” J. Opt. Soc. Am. B 23, 1196-1206 (2006).
    [CrossRef]
  24. V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
    [CrossRef]
  25. A. R. Cowan and J. F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606 (2003).
    [CrossRef]
  26. S. Hughes, “Quantum emission dynamics from a single quantum dot in a planar photonic crystal nanocavity,” Opt. Lett. 30, 1393-1395 (2005).
    [CrossRef] [PubMed]
  27. L.-M. Zhao, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Green's function for photonic crystal slabs,” Phys. Rev. E 72, 026614 (2005).
    [CrossRef]
  28. 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]
  29. 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]
  30. M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
    [CrossRef]
  31. O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909-3915 (1998).
    [CrossRef]
  32. A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68, 248-263 (1980).
    [CrossRef]
  33. P. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves withNObstacles (Cambridge U. Press, 2006).
    [CrossRef]

2009 (1)

2007 (2)

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[CrossRef]

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green's tensor and local density of states calculations for collections of spherical scatterers,” Phys. Rev. B 75, 115124 (2007).
[CrossRef]

2006 (1)

2005 (2)

S. Hughes, “Quantum emission dynamics from a single quantum dot in a planar photonic crystal nanocavity,” Opt. Lett. 30, 1393-1395 (2005).
[CrossRef] [PubMed]

L.-M. Zhao, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Green's function for photonic crystal slabs,” Phys. Rev. E 72, 026614 (2005).
[CrossRef]

2004 (2)

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, “Three-dimensional Green's tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders,” Phys. Rev. E 70, 066608 (2004).
[CrossRef]

S. Campbell, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919-1928 (2004).
[CrossRef]

2003 (4)

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 13, 9-25 (2003).
[CrossRef]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation: errata,” J. Opt. Soc. Am. B 20, 1581-1581 (2003).
[CrossRef]

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

A. R. Cowan and J. F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606 (2003).
[CrossRef]

2002 (2)

2001 (1)

M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
[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]

1998 (2)

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

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

1996 (1)

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

1994 (1)

1993 (1)

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161-4167 (1993).
[CrossRef]

1990 (1)

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

1987 (2)

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

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

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]

1973 (1)

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

Asatryan, A. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 13, 9-25 (2003).
[CrossRef]

Botten, L. C.

Busch, K.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 13, 9-25 (2003).
[CrossRef]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[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]

Campbell, S.

Cowan, A. R.

A. R. Cowan and J. F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606 (2003).
[CrossRef]

de Sterke, C. M.

Draine, B. T.

Flatau, P. J.

Fussell, D. P.

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, “Three-dimensional Green's tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders,” Phys. Rev. E 70, 066608 (2004).
[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]

Gu, B. -Y.

L.-M. Zhao, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Green's function for photonic crystal slabs,” Phys. Rev. E 72, 026614 (2005).
[CrossRef]

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

Hafner, C.

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, 1990).

Harmon, B. N.

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161-4167 (1993).
[CrossRef]

Hughes, S.

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[CrossRef]

S. Hughes, “Quantum emission dynamics from a single quantum dot in a planar photonic crystal nanocavity,” Opt. Lett. 30, 1393-1395 (2005).
[CrossRef] [PubMed]

John, S.

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

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

Kafesaki, M.

Koenderink, A. F.

Kuhlmey, B. T.

Lagendijk, A.

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

Leung, K. M.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

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]

Liu, Y. F.

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

Manga Rao, V. S. C.

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[CrossRef]

Martin, O. J. F.

M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
[CrossRef]

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.

P. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves withNObstacles (Cambridge U. Press, 2006).
[CrossRef]

Maystre, D.

McPhedran, R. C.

Mittra, R.

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

Nicorovici, N. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 13, 9-25 (2003).
[CrossRef]

Nikolaev, I. S.

Paulus, M.

M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
[CrossRef]

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]

Pennypacker, C. R.

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

Peterson, A. F.

A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics (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]

Purcell, E. M.

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

Ray, S. L.

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

Renversez, G.

Sandoghdar, V.

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]

Šolín, P.

P. Šolín, Partial Differential Equations and the Finite Element Method (Wiley Interscience, 2006).

Soukoulis, C. M.

Sprik, R.

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

Tavlove, A.

A. Tavlove, Computational Electromagnetics: The Finite-Difference Time-Domain Method (Artech House, 1995).

van Tiggelen, B. A.

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

Vitanov, N. V.

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green's tensor and local density of states calculations for collections of spherical scatterers,” Phys. Rev. B 75, 115124 (2007).
[CrossRef]

Vos, W. L.

Wang, R.

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

Wang, X.

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161-4167 (1993).
[CrossRef]

Wang, X. -H.

L.-M. Zhao, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Green's function for photonic crystal slabs,” Phys. Rev. E 72, 026614 (2005).
[CrossRef]

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

White, T. P.

Yablonovitch, E.

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

Yaghjian, A. D.

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

Yang, G. -Z.

L.-M. Zhao, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Green's function for photonic crystal slabs,” Phys. Rev. E 72, 026614 (2005).
[CrossRef]

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

Yannopapas, V.

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green's tensor and local density of states calculations for collections of spherical scatterers,” Phys. Rev. B 75, 115124 (2007).
[CrossRef]

Young, J. F.

A. R. Cowan and J. F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606 (2003).
[CrossRef]

Yu, Q.

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161-4167 (1993).
[CrossRef]

Zhang, X. -G.

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161-4167 (1993).
[CrossRef]

Zhao, L. -M.

L.-M. Zhao, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Green's function for photonic crystal slabs,” Phys. Rev. E 72, 026614 (2005).
[CrossRef]

Astrophys. J. (1)

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

EPL (1)

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

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (6)

Opt. Lett. (1)

Phys. Rev. (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]

Phys. Rev. B (4)

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[CrossRef]

R. Wang, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Local density of states in three-dimensional photonic crystals: calculation and enhancement effects,” Phys. Rev. B 67, 155114 (2003).
[CrossRef]

X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161-4167 (1993).
[CrossRef]

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green's tensor and local density of states calculations for collections of spherical scatterers,” Phys. Rev. B 75, 115124 (2007).
[CrossRef]

Phys. Rev. E (7)

D. P. Fussell, R. C. McPhedran, and C. M. de Sterke, “Three-dimensional Green's tensor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders,” Phys. Rev. E 70, 066608 (2004).
[CrossRef]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

L.-M. Zhao, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, “Green's function for photonic crystal slabs,” Phys. Rev. E 72, 026614 (2005).
[CrossRef]

A. R. Cowan and J. F. Young, “Optical bistability involving photonic crystal microcavities and Fano line shapes,” Phys. Rev. E 68, 046606 (2003).
[CrossRef]

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]

M. Paulus and O. J. F. Martin, “Green's tensor technique for scattering in two-dimensional stratified media,” Phys. Rev. E 63, 066615 (2001).
[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)

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

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

K. M. Leung and Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646-2649 (1990).
[CrossRef] [PubMed]

Proc. IEEE (1)

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

Waves Random Media (1)

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Media 13, 9-25 (2003).
[CrossRef]

Other (5)

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, 1990).

A. Tavlove, Computational Electromagnetics: The Finite-Difference Time-Domain Method (Artech House, 1995).

P. Šolín, Partial Differential Equations and the Finite Element Method (Wiley Interscience, 2006).

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

P. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves withNObstacles (Cambridge U. Press, 2006).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Sketch of the local coordinates used for the calculation of the self-term in scattering domain D.

Fig. 2
Fig. 2

Sketch of local coordinates for r and r in two independent scatterers.

Fig. 3
Fig. 3

Example calculation: A TE plane wave of unit amplitude, E B ( r ) = e   exp ( i n B k 0 r ) , is incident from the top left on a crystallite consisting of seven air holes ( n d = 1 ) in a high-index dielectric background ( n B = 3.5 ) . Parameters are k 0 = ( 3 / 2 , 1 / 2 ) and R PC = 0.3 a , where R PC is the radius of the cylindrical holes and a = 0.3 λ 0 is the distance in between. Top: Absolute square, | E ( r ) | 2 , of the resulting field as a function of position in the x y -plane. Bottom: Absolute value of the components E x ( x ) (red solid line) and E y ( x ) (blue dashed line) along the line y = 0 .

Fig. 4
Fig. 4

Real (top) and imaginary (bottom) parts of the total TM Green’s tensor G z z ( r , r ) as functions of r with k 0 r = ( 1 , 1 / 4 ) (as indicated by the red dot) in a structure consisting of four dielectric rods ( n d = 3.5 ) of square cross section in air. Parameters are a = 2 L , where a is the distance between the rods and L = λ 0 / 4 is the side length.

Fig. 5
Fig. 5

Global error as a function of the number of basis functions used in the expansion of the electric fields (controlled by Q m a x ). Circular markers correspond to the problem in Fig. 3 with different curves corresponding to different fixed errors on the relevant matrix elements as indicated. Square markers correspond to the problem in Fig. 4 calculated for the Green’s tensor ( G z z ) and plane waves (PW) as the background field.

Fig. 6
Fig. 6

Top: Absolute value | G z z ( r , r ) | of the TM Green’s tensor for a finite-sized photonic crystal waveguide consisting of 80 rods of refractive index n d = 3.4 in a background with an interface between a low-index dielectric ( n B = 1.5 ) and air ( n A = 1 ) . The results are calculated as functions of r with k 0 r = ( 0 , 7.58 ) (indicated by the red dot and vertical dashed line). Bottom: Real (red solid line) and imaginary (blue dashed curve) parts of G z z ( y , r ) along the line x = 0 . Parameters are R PC = 0.25 a where R PC is the radius of the cylindrical holes and a = 0.28 λ 0 the distance in between.

Fig. 7
Fig. 7

Contour plot of emission pattern, | G z z ( r , r ) | , of the system in Fig. 6, but for positions outside the photonic crystal.

Fig. 8
Fig. 8

Sketch of relative coordinates as used in the expression for Graf’s addition theorem.

Equations (37)

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× × E ( r ) k 0 2 ϵ ( r ) E ( r ) = 0 ,
E ( r ) = E B ( r ) + D G B ( r , r ) k 0 2 Δ ε ( r ) E ( r ) d r ,
G 2 D B ( r , r ) = ( I + k B 2 ) i 4 H 0 ( k B R ) ,
E ( r ) = E B ( r ) + lim δ A 0 D δ A G B ( r , r ) k 0 2 Δ ε ( r ) E ( r ) d r L Δ ϵ ( r ) ϵ B E ( r ) ,
ψ n = K n J q ( k d r d ) e i q φ d S d ( r ) ,
ψ n B = K n B J q ( k B r d ) e i q φ d S d ( r ) ,
ψ m | ψ n = ψ m ( r ) ψ n ( r ) d r ,
E ( r ) = n e n ψ n ( r ) e n ,
E B ( r ) = n e n B ψ n B ( r ) e n ,
( 1 + L m n Δ ϵ ϵ B ) n ψ m | ψ n e n = n ψ m | ψ n B e n B + k 2 Δ ε n G m n α β e n ,
G m n α β = D ψ m ( r ) lim δ A 0 D δ A G α β B ( r , r ) ψ n ( r ) d r d r ,
A m n α β = D ψ m ( r ) lim δ A 0 P δ A G α β B ( r , r ) ψ n ( r ) d r d r ,
B m n α β = D ψ m ( r ) P D G α β B ( r , r ) ψ n ( r ) d r d r .
A α β = K m K n μ D J m ( k R r ) J n + μ ( k R r ) e i ( n + μ m ) φ P δ A G α β B ( 0 , R ) J μ ( k R R ) ( 1 ) μ e i μ θ R d R d r = K m K n μ D J m ( k R r ) J n + μ ( k R r ) e i ( n + μ m ) φ d r I μ α β ,
b m n α β ( r , r ) = ψ m ( r ) G α β B ( r , r ) ψ n ( r ) = K m K n μ J m ( k R r ) e i m φ L α β { i 4 J μ ( k B r ) e i μ φ } H μ ( k B r ) J n ( k R r ) e i ( n + μ ) φ ,
b m n α β ( r , r ) = K m K n μ J m ( k R r ) e i m φ L α β { i 4 H μ ( k B r ) e i μ φ } J μ ( k B r ) J n ( k R r ) e i ( n μ ) φ ,
G m n α β = i 4 μ , λ H μ + λ ( k B L ) ( 1 ) μ e i ( μ + λ ) θ D m K m J m ( k m r ) e i m φ L α β { J λ ( k B r ) e i λ φ } d r D n K n J μ ( k B r ) J n ( k n r ) e i ( n μ ) φ d r ,
G ( r , r ) = G B ( r , r ) + D G B ( r , r ) k 0 2 Δ ε ( r ) G ( r , r ) d r ,
E ( r ) = E B ( r ) + D G B ( r , r ) k 0 2 Δ ε ( r ) n e n ψ n ( r ) e n d r .
E ( r ) = E B ( r ) + i 4 k 0 2 Δ ε μ , n L { H μ ( k B L ) e i μ θ } ( 1 ) μ e n e n D K n J n ( k R r ) J μ ( k B r ) e i ( n μ ) φ d r ,
E L ( r ) = | E B ( r ) E n u m ( r ) + G B ( r , r ) k 0 2 Δ ε ( r ) E n u m ( r ) d r | ,
E G = E L ( r ) d r | E B ( r ) | d r ,
G z z B ( r , r ) = y ̂ y ̂ k 2 2 δ ( R ) + i 4 π 1 k B , y e i k x ( x x ) e i k B , y | y y | d k x + i 4 π F B A S k B , y e i k x ( x x ) e i k B , y ( y + y ) d k x ,
F B A S = k B , y k A , y k B , y + k A , y = k B 2 k x 2 k A 2 k x 2 k B 2 k x 2 + k A 2 k x 2
G m n S = i 4 π F B A S ( k x ) k B , y ( k x ) D m D n ψ m ( r ) e i k x ( x x ) e i k B , y ( k x ) ( y + y ) ψ n ( r ) d r d r d k x .
G m n S = i 4 π F B A S ( k x ) k B , y ( k x ) e i ( k x ( X X ) k B , y ( k x ) ( Y + Y ) ) D m K m J m ( k m r ) e i m φ e i k B r   cos ( φ θ ( k x ) ) d r D n K n J n ( k n r ) e i n φ e i k B r   cos ( φ θ ( k x ) ) d r d k x ,
G m n S = i 4 π λ , γ i λ + γ F B A S ( k x ) k B , y ( k x ) e i ( k x ( X X ) k B , y ( k x ) ( Y + Y ) ) e i ( λ θ ( k x ) + γ θ ( k x ) ) d k x D m K m J m ( k m r ) J λ ( k B r ) e i ( λ m ) φ d r D n K n J n ( k n r ) J λ ( k B r ) e i ( γ + n ) φ d r .
e i k 0 r   cos ( φ θ ) = n = i n e i n θ J n ( k 0 r ) e i n φ ,
Z n ( k r ) e i n ( φ θ ) = μ = Z n + μ ( k L ) J μ ( k r ) ( 1 ) μ e i μ ( θ φ ) ,
I μ α β = P δ A G α β B ( 0 , R ) J μ ( k R R ) ( 1 ) μ e i μ θ R d R ,
I μ x x = i 4 0 0 2 π { sin 2 θ R H 0 ( k B R ) + cos ( 2 θ R ) k B R H 1 ( k B R ) + 2 i π cos ( 2 θ R ) k B 2 R 2 } J μ ( k R R ) ( 1 ) μ e i μ θ R R d θ R d R i 4 lim δ R 0 δ R 0 2 π 2 i π cos ( 2 θ R ) k B 2 R 2 J μ ( k R R ) ( 1 ) μ e i μ θ R R d θ R d R = i π 4 0 { ( 1 2 δ μ , 2 + δ μ , 0 1 2 δ μ , 2 ) H 0 ( k B R ) + ( δ μ , 2 + δ μ , 2 ) ( H 1 ( k B R ) k B R + 2 i π k B 2 R 2 ) } J μ ( k R R ) R d R + 1 2 lim δ R 0 δ R ( δ μ , 2 + δ μ , 2 ) J μ ( k R R ) k B 2 R 2 R d R ,
lim δ R 0 δ R J 2 ( K R R ) k B 2 R 2 R d R = 1 2 k B 2 .
I μ y y = i π 4 0 { ( 1 2 δ μ , 2 + δ μ , 0 + 1 2 δ μ , 2 ) H 0 ( k B R ) ( δ μ , 2 + δ μ , 2 ) ( H 1 ( k B R ) k B R + 2 i π k B 2 R 2 ) } J μ ( k R R ) R d R 1 4 k B 2 ( δ μ , 2 + δ μ , 2 ) ,
I μ x y = π 4 0 ( δ μ , 2 δ μ , 2 ) ( 1 2 H 2 ( k B R ) + 2 i π k B 2 R 2 ) J μ ( k R R ) R d R + i 4 k B 2 ( δ μ , 2 δ μ , 2 ) .
2 x 2 { Z λ ( k r ) e i λ φ } = k 2 4 { Z λ + 2 ( k r ) e i ( λ + 2 ) φ + Z λ 2 ( k r ) e i ( λ 2 ) φ 2 Z λ ( k r ) e i λ φ } ,
2 y 2 { Z λ ( k r ) e i λ φ } = k 2 4 { Z λ + 2 ( k r ) e i ( λ + 2 ) φ + Z λ 2 ( k r ) e i ( λ 2 ) φ + 2 Z λ ( k r ) e i λ φ } ,
2 x y { Z λ ( k r ) e i λ φ } = i k 2 4 { Z λ + 2 ( k r ) e i ( λ + 2 ) φ Z λ 2 ( k r ) e i ( λ 2 ) φ } .

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