Abstract

We present an efficient method for computing the equifrequency surfaces (EFSs) and density of states of a photonic crystal. The method is based on repeatedly solving a small nonlinear eigenvalue problem formulated using the Dirichlet-to-Neumann map of the unit cell. A simple contouring algorithm is presented for sampling EFSs as well as computing group velocity vectors. We compare our method with several published results to demonstrate its efficiency and accuracy.

© 2011 Optical Society of America

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  1. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
    [CrossRef]
  2. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
    [CrossRef]
  3. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003).
    [CrossRef]
  4. D. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12, 1823–1831 (2004).
    [CrossRef] [PubMed]
  5. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
    [CrossRef] [PubMed]
  6. P. Russel and T. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. 17, 1982–1988 (1999).
    [CrossRef]
  7. Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105, 163901 (2010).
    [CrossRef]
  8. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  9. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals (Princeton Univ. Press, 2008).
  10. Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” arXiv:0910.3981v2 (2009).
  11. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. USA 107, 17491–17496 (2010).
    [CrossRef] [PubMed]
  12. M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced photonic band-gap confinement via van Hove saddle point singularities,” Phys. Rev. Lett. 96, 033904 (2006).
    [CrossRef] [PubMed]
  13. G. Gilat and L. J. Raubenheimer, “Accurate numerical method for calculating frequency-distribution functions in solids,” Phys. Rev. 144, 390–395 (1966).
    [CrossRef]
  14. G. Lehmann and M. Taut, “On the numerical calculation of the density of states and related properties,” Phys. Status Solidi B 54, 469–477 (1972).
    [CrossRef]
  15. J. Hama, M. Watanabe, and T. Kato, “Correctly weighted tetrahedron method for k-space integration,” J. Phys. Condens. Matter 2, 7445 (1990).
    [CrossRef]
  16. R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
    [CrossRef]
  17. C. J. Pickard and M. C. Payne, “Extrapolative approaches to brillouin-zone integration,” Phys. Rev. B 59, 4685–4693 (1999).
    [CrossRef]
  18. E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
    [CrossRef]
  19. J. Yuan and Y. Y. Lu, “Photonic bandgap calculations with Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222(2006).
    [CrossRef]
  20. Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24, 2860–2867 (2007).
    [CrossRef]
  21. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399(2008).
    [CrossRef] [PubMed]
  22. V. Liu, Y. Jiao, D. A. B. Miller, and S. Fan, “Design methodology for compact photonic crystal based wavelength division multiplexers,” Opt. Lett. 36, 591–593 (2011).
    [CrossRef] [PubMed]
  23. Z. Hu and Y. Y. Lu, “Improved bends for two-dimensional photonic crystal waveguides,” Opt. Commun. 284, 2812–2816(2011).
    [CrossRef]
  24. J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comp. Phys. 227, 4617–4629 (2008).
    [CrossRef]
  25. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
  26. Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
    [CrossRef]
  27. T. S. Newman and H. Yi, “A survey of the marching cubes algorithm,” Comput. Graph. 30, 854–879 (2006).
    [CrossRef]
  28. J.-D. Boissonnat and S. Oudot, “Provably good sampling and meshing of surfaces,” Graph. Models 67, 405–451 (2005).
    [CrossRef]
  29. A. Quarteroni, Numerical Mathematics (Springer, 2000).
  30. G. Strang, Linear Algebra and Its Applications (Harcourt Brace Jovanovich, 1988).
  31. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
    [CrossRef]
  32. A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, no. 105 in Other Titles in Applied Mathematics, 2nd ed. (SIAM, 2008).
  33. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef] [PubMed]
  34. K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896–3908 (1998).
    [CrossRef]

2011 (2)

2010 (2)

Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105, 163901 (2010).
[CrossRef]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. USA 107, 17491–17496 (2010).
[CrossRef] [PubMed]

2009 (1)

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

2008 (2)

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399(2008).
[CrossRef] [PubMed]

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comp. Phys. 227, 4617–4629 (2008).
[CrossRef]

2007 (1)

2006 (3)

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations with Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222(2006).
[CrossRef]

T. S. Newman and H. Yi, “A survey of the marching cubes algorithm,” Comput. Graph. 30, 854–879 (2006).
[CrossRef]

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced photonic band-gap confinement via van Hove saddle point singularities,” Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

2005 (1)

J.-D. Boissonnat and S. Oudot, “Provably good sampling and meshing of surfaces,” Graph. Models 67, 405–451 (2005).
[CrossRef]

2004 (2)

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

D. Pustai, S. Shi, C. Chen, A. Sharkawy, and D. Prather, “Analysis of splitters for self-collimated beams in planar photonic crystals,” Opt. Express 12, 1823–1831 (2004).
[CrossRef] [PubMed]

2003 (2)

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003).
[CrossRef]

2002 (2)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

2001 (2)

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

2000 (1)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

1999 (2)

C. J. Pickard and M. C. Payne, “Extrapolative approaches to brillouin-zone integration,” Phys. Rev. B 59, 4685–4693 (1999).
[CrossRef]

P. Russel and T. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. 17, 1982–1988 (1999).
[CrossRef]

1998 (1)

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

1990 (1)

J. Hama, M. Watanabe, and T. Kato, “Correctly weighted tetrahedron method for k-space integration,” J. Phys. Condens. Matter 2, 7445 (1990).
[CrossRef]

1987 (1)

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

1972 (1)

G. Lehmann and M. Taut, “On the numerical calculation of the density of states and related properties,” Phys. Status Solidi B 54, 469–477 (1972).
[CrossRef]

1966 (1)

G. Gilat and L. J. Raubenheimer, “Accurate numerical method for calculating frequency-distribution functions in solids,” Phys. Rev. 144, 390–395 (1966).
[CrossRef]

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Antoine, X.

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comp. Phys. 227, 4617–4629 (2008).
[CrossRef]

Asatryan, A. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Birks, T.

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Blackford, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Boissonnat, J.-D.

J.-D. Boissonnat and S. Oudot, “Provably good sampling and meshing of surfaces,” Graph. Models 67, 405–451 (2005).
[CrossRef]

Botten, L. C.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Busch, K.

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

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Chen, C.

de Sterke, C. M.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Du Croz, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Erni, D.

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Fan, S.

V. Liu, Y. Jiao, D. A. B. Miller, and S. Fan, “Design methodology for compact photonic crystal based wavelength division multiplexers,” Opt. Lett. 36, 591–593 (2011).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. USA 107, 17491–17496 (2010).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003).
[CrossRef]

Gilat, G.

G. Gilat and L. J. Raubenheimer, “Accurate numerical method for calculating frequency-distribution functions in solids,” Phys. Rev. 144, 390–395 (1966).
[CrossRef]

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Griewank, A.

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, no. 105 in Other Titles in Applied Mathematics, 2nd ed. (SIAM, 2008).

Hafner, C.

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Hama, J.

J. Hama, M. Watanabe, and T. Kato, “Correctly weighted tetrahedron method for k-space integration,” J. Phys. Condens. Matter 2, 7445 (1990).
[CrossRef]

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Hu, Z.

Z. Hu and Y. Y. Lu, “Improved bends for two-dimensional photonic crystal waveguides,” Opt. Commun. 284, 2812–2816(2011).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399(2008).
[CrossRef] [PubMed]

Huang, Y.

Ibanescu, M.

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced photonic band-gap confinement via van Hove saddle point singularities,” Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

Jacob, Z.

Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” arXiv:0910.3981v2 (2009).

Jiao, Y.

Joannopoulos, J. D.

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced photonic band-gap confinement via van Hove saddle point singularities,” Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals (Princeton Univ. Press, 2008).

John, S.

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

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a plane wave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals (Princeton Univ. Press, 2008).

Kato, T.

J. Hama, M. Watanabe, and T. Kato, “Correctly weighted tetrahedron method for k-space integration,” J. Phys. Condens. Matter 2, 7445 (1990).
[CrossRef]

Lehmann, G.

G. Lehmann and M. Taut, “On the numerical calculation of the density of states and related properties,” Phys. Status Solidi B 54, 469–477 (1972).
[CrossRef]

Li, S.

Li, Z.-Y.

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Lin, L.-L.

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Liu, V.

Lu, Y. Y.

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

McOrist, J.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals (Princeton Univ. Press, 2008).

Miller, D. A. B.

Moreno, E.

E. Moreno, D. Erni, and C. Hafner, “Band structure computations of metallic photonic crystals with the multiple multipole method,” Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Narimanov, E.

Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” arXiv:0910.3981v2 (2009).

Newman, T. S.

T. S. Newman and H. Yi, “A survey of the marching cubes algorithm,” Comput. Graph. 30, 854–879 (2006).
[CrossRef]

Nicorovici, N. A.

R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[CrossRef]

Oudot, S.

J.-D. Boissonnat and S. Oudot, “Provably good sampling and meshing of surfaces,” Graph. Models 67, 405–451 (2005).
[CrossRef]

Payne, M. C.

C. J. Pickard and M. C. Payne, “Extrapolative approaches to brillouin-zone integration,” Phys. Rev. B 59, 4685–4693 (1999).
[CrossRef]

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Pickard, C. J.

C. J. Pickard and M. C. Payne, “Extrapolative approaches to brillouin-zone integration,” Phys. Rev. B 59, 4685–4693 (1999).
[CrossRef]

Prather, D.

Pustai, D.

Quarteroni, A.

A. Quarteroni, Numerical Mathematics (Springer, 2000).

Raman, A.

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. USA 107, 17491–17496 (2010).
[CrossRef] [PubMed]

Raubenheimer, L. J.

G. Gilat and L. J. Raubenheimer, “Accurate numerical method for calculating frequency-distribution functions in solids,” Phys. Rev. 144, 390–395 (1966).
[CrossRef]

Reed, E. J.

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced photonic band-gap confinement via van Hove saddle point singularities,” Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

Russel, P.

Sharkawy, A.

Shi, S.

Smith, D. R.

Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105, 163901 (2010).
[CrossRef]

Smolyaninov, I.

Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” arXiv:0910.3981v2 (2009).

Sorensen, D.

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Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105, 163901 (2010).
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L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
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X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003).
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Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. USA 107, 17491–17496 (2010).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
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Appl. Phys. Lett. (1)

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003).
[CrossRef]

Comput. Graph. (1)

T. S. Newman and H. Yi, “A survey of the marching cubes algorithm,” Comput. Graph. 30, 854–879 (2006).
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J. Comp. Phys. (1)

J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comp. Phys. 227, 4617–4629 (2008).
[CrossRef]

J. Lightwave Technol. (1)

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

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

J. Phys. Condens. Matter (1)

J. Hama, M. Watanabe, and T. Kato, “Correctly weighted tetrahedron method for k-space integration,” J. Phys. Condens. Matter 2, 7445 (1990).
[CrossRef]

Opt. Commun. (1)

Z. Hu and Y. Y. Lu, “Improved bends for two-dimensional photonic crystal waveguides,” Opt. Commun. 284, 2812–2816(2011).
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Opt. Express (3)

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Phys. Rev. E (4)

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

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R. C. McPhedran, L. C. Botten, J. McOrist, A. A. Asatryan, C. M. de Sterke, and N. A. Nicorovici, “Density of states functions for photonic crystals,” Phys. Rev. E 69, 016609 (2004).
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M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced photonic band-gap confinement via van Hove saddle point singularities,” Phys. Rev. Lett. 96, 033904 (2006).
[CrossRef] [PubMed]

Y. A. Urzhumov and D. R. Smith, “Transformation optics with photonic band gap media,” Phys. Rev. Lett. 105, 163901 (2010).
[CrossRef]

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

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Phys. Status Solidi B (1)

G. Lehmann and M. Taut, “On the numerical calculation of the density of states and related properties,” Phys. Status Solidi B 54, 469–477 (1972).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. USA 107, 17491–17496 (2010).
[CrossRef] [PubMed]

Other (6)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals (Princeton Univ. Press, 2008).

Z. Jacob, I. Smolyaninov, and E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” arXiv:0910.3981v2 (2009).

A. Quarteroni, Numerical Mathematics (Springer, 2000).

G. Strang, Linear Algebra and Its Applications (Harcourt Brace Jovanovich, 1988).

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, no. 105 in Other Titles in Applied Mathematics, 2nd ed. (SIAM, 2008).

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Figures (5)

Fig. 1
Fig. 1

Illustration of a unit cell in a square lattice of dielectric rods (gray). The unit cell is discretized with five points per edge. The points on red edges correspond to field values in the u vector, while the blue points correspond to v. The shapes of the points indicate how they should be matched for the Bloch periodic boundary conditions. The lattice vectors corresponding to each pair of matching edges are indicated by the labeled arrows.

Fig. 2
Fig. 2

Illustration of perpendicular search direction for extending fragments. The point of the fragment in the active set is shown in an open circle while the dashed line shows the line segment over which the next point is searched. The arrows indicate the directions of the group velocity direction (gradient vectors) of the EFS (the vector g in the text).

Fig. 3
Fig. 3

Steps of the EFS sampling algorithm in the IBZ of a square lattice. Open circles indicate active endpoints of fragments.

Fig. 4
Fig. 4

Several EFS for a photonic crystal of air holes in silicon. Frequencies of EFS are given in units of 2 π c / a . The boundary of the first BZ is outlined in gray. The curves are essentially identical to those in originally computed in [2].

Fig. 5
Fig. 5

Density of states for a triangular lattice of GaAs rods ( n = 3.6 , r = 0.43 a ) in air for E polarization (top) and H polarization (bottom).

Equations (25)

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N ( ω ) = m 1 A BZ BZ δ ( ω ω m ( k ) ) d 2 k ,
N ( ω ) = m 1 A BZ EFS m d k d ω d s ,
Λ f | Ω = f n ^ | Ω ,
( Λ D ) i , j = ϕ j ( r i ) ,
[ Λ u u Λ u v Λ v u Λ v v ] [ u v ] = [ n u n v ] ,
v = [ e i k · L 1 e i k · L 1 e i k · L 2 e i k · L 2 ] u
= diag ( e i k · L 1 I , e i k · L 2 I ) = Q u
n v = Q n u ,
[ Λ v u + Λ v v Q + Q Λ u u + Q Λ u v Q ] u = 0 ,
A ( k ; ω ) u = 0.
t i + 1 = t i α det A ( t i ) d d t det A ( t i ) ,
d d t det A ( t ) = Tr [ adj ( A ( t ) ) d d t A ( t ) ] = Tr [ adj ( A ) Δ k · k A ( k ) ] ,
t i + 1 = t i α [ Tr ( A 1 Δ k · k A ( k ) ) ] 1 .
k A ( k ) = A k = k ^ x k x A ( k ) + k ^ y k y A ( k ) = [ k x A ( k ) k y A ( k ) ] [ A k x A k y ] .
Δ k · A k = Δ k x A k x + Δ k y A k y ,
Δ k · A k = Λ v v R Q + Q R Λ u u + Q R Λ u v Q + Q Λ u v R Q ,
t i + 1 = t i α [ Tr ( A 1 Δ k · A k ) m 1 t i r m ] 1 .
Tr ( V Σ 1 U H Δ k · A k ) = Tr [ Σ 1 U H ( Δ k · A k ) V ] = diag ( Σ 1 ) · diag ( U H ( Δ k · A k ) V ) ,
t i + 1 = t i α σ min j σ min σ j u j H ( Δ k · A k ) v j m ( t i r m ) 1 ,
g k det A ( k ) = Tr [ adj ( A ) A k ] .
g = [ Tr [ adj ( A ) A k x ] Tr [ adj ( A ) A k y ] ] .
adj ( Σ ) = diag ( 0 , , s ) where     s = j = 1 n 1 σ j ,
d d ω det A = g · d k d ω + Tr [ adj ( A ) A ω ] = 0 ,
d k d ω = Tr [ adj ( A ) A ω ] g g 2 .
d k d ω = s g u n H A ω v n .

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