Abstract

In-plane light propagation in two-dimensional (2D) photonic crystals (PCs) has been investigated by using the finite element method (FEM) in frequency domain. Conventionally, the band structures of 2D PCs were calculated by either the plane-wave expansion method (PWEM) or the finite difference time domain method. Here, we solve the eigenvalue equations for the band structures of the 2D PCs using the adaptive FEM in real space. We have carefully examined the convergence of this approach for the desired accuracy and efficiency. The calculated results show some discrepancies when compared to the results calculated by the PWEM. This may be due to the accuracy of the PWEM limited by the discontinuous nature of the dielectric functions. After acquiring the whole information of the dispersion relations within the irreducible Brillouin zone of the 2D PCs, the in-plane photon density of states for both the transverse electric (TE) and transverse magnetic (TM) modes can be calculated, accurately. For the case, the width of the complete band gap predicted by the FEM is much smaller, only about 65 % of that calculated by the PWEM. Therefore, the discrepancy in the prediction of complete band gaps between these two methods can be quite large, although the difference in band structure calculations is only a few percent. These results are relevant to the spontaneous emission by an atom, or to dipole radiation in two-dimensional periodic structures.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  34. B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
    [CrossRef]
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    [CrossRef]
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  37. 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]
  38. 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]
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    [CrossRef]
  40. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2006 (1)

M. C. Lin and R. F. Jao, "Quantitative analysis of photon density of states for a realistic superlattice with omnidirectional light propagation," Phys. Rev. E 74, 046613 (2006).
[CrossRef]

2005 (2)

A. S. Sánchez and P. Halevi, "Spontaneous emission in one-dimensional photonic crystals," Phys. Rev. E 72, 056609 (2005).
[CrossRef]

P. Halevi and A. S. Sánchez, "Spontaneous emission in a high-contrast one-dimensional photonic crystal," Opt. Commun. 251, 109-114 (2005).
[CrossRef]

2004 (2)

W. J. Kim and J. D. O’Brien, "Optimization of a two-dimensional photonic crystal waveguide branch by simulated annealing and the finite-element method," J. Opt. Soc. Am. B 21, 289-295 (2004).
[CrossRef]

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]

2002 (3)

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]

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
[CrossRef]

X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, "Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps," Phys. Rev. Lett. 88, 093902 (2002)
[CrossRef] [PubMed]

2001 (4)

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

D. Hermann, M. Frank, K. Busch, and P. w¨olfle, "Photonic band structure computations," Opt. Express 8, 167-172 (2001).
[CrossRef] [PubMed]

W. Zhang, C. T. Chan, and P. Sheng, "Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres," Opt. Express 8, 203-208 (2001).
[CrossRef] [PubMed]

Z. Y. Li and Y. Xia, "Omnidirectional absolute band gaps in two-dimensional photonic crystals," Phys. Rev. B 64, 153108 (2001).
[CrossRef]

2000 (4)

R. Hillebrand, W. Hergert and W. Harms, "Theoretical band gap studies of two-dimensional photonic crystals with varying column roundness," Phys. Stat. Sol.(b) 217, 981-989 (2000).
[CrossRef]

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightwave Tech. 18, 102-110 (2000).
[CrossRef]

G. Pelosi, A. Cocchi, and A. Monorchio, "A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam," IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An efficient method for band structure calculations in 3D photonic crystals," J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

1999 (3)

D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-376, 1999.
[CrossRef]

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

O. J. F. Martin, C. Girard, D. R. Smith, and S. Schultz, "Generalized field propagator for arbitrary finite-size photonic band gap structures," Phys. Rev. Lett. 82, 315-318 (1999).
[CrossRef]

1998 (4)

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

A. J. Ward and J. B. Pendry, "Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method," Phys. Rev. B 58, 7252-7259 (1998).
[CrossRef]

J. K. Hwang, S. B. Hyun, H. Y. Ryu, and Y. H. Lee, "Resonant modes of two-dimensional photonic bandgap cavities determined by the finite-element method and by use of the anisotropic perfectly matched layer boundary condition," J. Opt. Soc. Am. B 15, 2316-2324 (1998).
[CrossRef]

M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998).
[CrossRef]

1997 (4)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature (London) 386, 143-149 (1997).
[CrossRef]

A. Kamli, M. Babiker, A. Al-Hajry, and N. Enfati, "Dipole relaxation in dispersive photonic band-gap structures," Phys. Rev. A 55, 1454-1461 (1997).
[CrossRef]

G. Tayeb and D. Maystre, "Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A 14, 3323-3332 (1997).
[CrossRef]

A. Figotin and Y. A. Godin, "The Computation of Spectra of Some 2D Photonic Crystals," J. Comput. Phys. 136, 585-598, 1997.
[CrossRef]

1996 (3)

J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715, 1996.
[CrossRef]

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

H. Rigneault and S. Monneret, "Modal analysis of spontaneous emission in a planar microcavity," Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

1995 (4)

T. Suzuki and P. K. L. Yu, "Emission power of an electric dipole in the photonic band structure of the fcc lattice," J. Opt. Soc. Am. B 12, 570-582 (1995).
[CrossRef]

K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

K. Sakoda, "Transmittance and Bragg reflectivity of two-dimensional photonic lattices," Phys. Rev. B 52, 8992-9002 (1995).
[CrossRef]

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

1992 (3)

H. S. Sözüer, J. W. Haus and R. Inguva, "Photonic bands: Convergence problems with the plane-wave method," Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

1991 (1)

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

1987 (1)

A. O. Barut and J. P. Dowling, "Quantum electrodynamics based on self-energy: Spontaneous emission in cavities," Phys. Rev. A 36, 649-654 (1987).
[CrossRef] [PubMed]

1981 (1)

D. Kleppner, "Inhibited Spontaneous Emission," Phys. Rev. Lett. 47, 233-236 (1981).
[CrossRef]

1946 (1)

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Al-Hajry, A.

A. Kamli, M. Babiker, A. Al-Hajry, and N. Enfati, "Dipole relaxation in dispersive photonic band-gap structures," Phys. Rev. A 55, 1454-1461 (1997).
[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. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Axmann, W.

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

Babiker, M.

A. Kamli, M. Babiker, A. Al-Hajry, and N. Enfati, "Dipole relaxation in dispersive photonic band-gap structures," Phys. Rev. A 55, 1454-1461 (1997).
[CrossRef]

Barut, A. O.

A. O. Barut and J. P. Dowling, "Quantum electrodynamics based on self-energy: Spontaneous emission in cavities," Phys. Rev. A 36, 649-654 (1987).
[CrossRef] [PubMed]

Beckett, D. H.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
[CrossRef]

Biswas, R.

M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998).
[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. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

Busch, K.

D. Hermann, M. Frank, K. Busch, and P. w¨olfle, "Photonic band structure computations," Opt. Express 8, 167-172 (2001).
[CrossRef] [PubMed]

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

Chan, C. T.

Cocchi, A.

G. Pelosi, A. Cocchi, and A. Monorchio, "A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam," IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

Cox, S. J.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
[CrossRef]

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]

Dobson, D. C.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An efficient method for band structure calculations in 3D photonic crystals," J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-376, 1999.
[CrossRef]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

A. O. Barut and J. P. Dowling, "Quantum electrodynamics based on self-energy: Spontaneous emission in cavities," Phys. Rev. A 36, 649-654 (1987).
[CrossRef] [PubMed]

Elson, J. M.

J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715, 1996.
[CrossRef]

Enfati, N.

A. Kamli, M. Babiker, A. Al-Hajry, and N. Enfati, "Dipole relaxation in dispersive photonic band-gap structures," Phys. Rev. A 55, 1454-1461 (1997).
[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.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature (London) 386, 143-149 (1997).
[CrossRef]

Figotin, A.

A. Figotin and Y. A. Godin, "The Computation of Spectra of Some 2D Photonic Crystals," J. Comput. Phys. 136, 585-598, 1997.
[CrossRef]

Frank, M.

Generowicz, J. M.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
[CrossRef]

Girard, C.

O. J. F. Martin, C. Girard, D. R. Smith, and S. Schultz, "Generalized field propagator for arbitrary finite-size photonic band gap structures," Phys. Rev. Lett. 82, 315-318 (1999).
[CrossRef]

Godin, Y. A.

A. Figotin and Y. A. Godin, "The Computation of Spectra of Some 2D Photonic Crystals," J. Comput. Phys. 136, 585-598, 1997.
[CrossRef]

Gopalakrishnan, J.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An efficient method for band structure calculations in 3D photonic crystals," J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

Gu, B. Y.

X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, "Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps," Phys. Rev. Lett. 88, 093902 (2002)
[CrossRef] [PubMed]

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]

Halevi, P.

A. S. Sánchez and P. Halevi, "Spontaneous emission in one-dimensional photonic crystals," Phys. Rev. E 72, 056609 (2005).
[CrossRef]

P. Halevi and A. S. Sánchez, "Spontaneous emission in a high-contrast one-dimensional photonic crystal," Opt. Commun. 251, 109-114 (2005).
[CrossRef]

Harms, W.

R. Hillebrand, W. Hergert and W. Harms, "Theoretical band gap studies of two-dimensional photonic crystals with varying column roundness," Phys. Stat. Sol.(b) 217, 981-989 (2000).
[CrossRef]

Haus, J. W.

H. S. Sözüer, J. W. Haus and R. Inguva, "Photonic bands: Convergence problems with the plane-wave method," Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

Hergert, W.

R. Hillebrand, W. Hergert and W. Harms, "Theoretical band gap studies of two-dimensional photonic crystals with varying column roundness," Phys. Stat. Sol.(b) 217, 981-989 (2000).
[CrossRef]

Hermann, D.

Hiett, B. P.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
[CrossRef]

Hikari, M.

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightwave Tech. 18, 102-110 (2000).
[CrossRef]

Hillebrand, R.

R. Hillebrand, W. Hergert and W. Harms, "Theoretical band gap studies of two-dimensional photonic crystals with varying column roundness," Phys. Stat. Sol.(b) 217, 981-989 (2000).
[CrossRef]

Ho, K. M.

M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998).
[CrossRef]

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

Hwang, J. K.

Hyun, S. B.

Inguva, R.

H. S. Sözüer, J. W. Haus and R. Inguva, "Photonic bands: Convergence problems with the plane-wave method," Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

Jao, R. F.

M. C. Lin and R. F. Jao, "Quantitative analysis of photon density of states for a realistic superlattice with omnidirectional light propagation," Phys. Rev. E 74, 046613 (2006).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature (London) 386, 143-149 (1997).
[CrossRef]

John, S.

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

Kamli, A.

A. Kamli, M. Babiker, A. Al-Hajry, and N. Enfati, "Dipole relaxation in dispersive photonic band-gap structures," Phys. Rev. A 55, 1454-1461 (1997).
[CrossRef]

Kim, W. J.

Kleppner, D.

D. Kleppner, "Inhibited Spontaneous Emission," Phys. Rev. Lett. 47, 233-236 (1981).
[CrossRef]

Koshiba, M.

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightwave Tech. 18, 102-110 (2000).
[CrossRef]

Kuchment, P.

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

Lee, Y. H.

Li, Z. Y.

Z. Y. Li and Y. Xia, "Omnidirectional absolute band gaps in two-dimensional photonic crystals," Phys. Rev. B 64, 153108 (2001).
[CrossRef]

Lin, M. C.

M. C. Lin and R. F. Jao, "Quantitative analysis of photon density of states for a realistic superlattice with omnidirectional light propagation," Phys. Rev. E 74, 046613 (2006).
[CrossRef]

MacKinnon, A.

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

Martijn de Sterke, C.

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

Martin, O. J. F.

O. J. F. Martin, C. Girard, D. R. Smith, and S. Schultz, "Generalized field propagator for arbitrary finite-size photonic band gap structures," Phys. Rev. Lett. 82, 315-318 (1999).
[CrossRef]

Maystre, D.

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. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Molinari, M.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
[CrossRef]

Monneret, S.

H. Rigneault and S. Monneret, "Modal analysis of spontaneous emission in a planar microcavity," Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

Monorchio, A.

G. Pelosi, A. Cocchi, and A. Monorchio, "A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam," IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

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]

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. Martijn de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

O’Brien, J. D.

Pasciak, J. E.

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An efficient method for band structure calculations in 3D photonic crystals," J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

Pelosi, G.

G. Pelosi, A. Cocchi, and A. Monorchio, "A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam," IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

Pendry, J. B.

A. J. Ward and J. B. Pendry, "Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method," Phys. Rev. B 58, 7252-7259 (1998).
[CrossRef]

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

Plihal, M.

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

Purcell, E. M.

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Rigneault, H.

H. Rigneault and S. Monneret, "Modal analysis of spontaneous emission in a planar microcavity," Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

Ryu, H. Y.

Sakoda, K.

K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

K. Sakoda, "Transmittance and Bragg reflectivity of two-dimensional photonic lattices," Phys. Rev. B 52, 8992-9002 (1995).
[CrossRef]

Sánchez, A. S.

P. Halevi and A. S. Sánchez, "Spontaneous emission in a high-contrast one-dimensional photonic crystal," Opt. Commun. 251, 109-114 (2005).
[CrossRef]

A. S. Sánchez and P. Halevi, "Spontaneous emission in one-dimensional photonic crystals," Phys. Rev. E 72, 056609 (2005).
[CrossRef]

Schultz, S.

O. J. F. Martin, C. Girard, D. R. Smith, and S. Schultz, "Generalized field propagator for arbitrary finite-size photonic band gap structures," Phys. Rev. Lett. 82, 315-318 (1999).
[CrossRef]

Sheng, P.

Sigalas, M. M.

M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998).
[CrossRef]

Smith, D. R.

O. J. F. Martin, C. Girard, D. R. Smith, and S. Schultz, "Generalized field propagator for arbitrary finite-size photonic band gap structures," Phys. Rev. Lett. 82, 315-318 (1999).
[CrossRef]

Soukoulis, C. M.

M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998).
[CrossRef]

Sözüer, H. S.

H. S. Sözüer, J. W. Haus and R. Inguva, "Photonic bands: Convergence problems with the plane-wave method," Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

Suzuki, T.

Tayeb, G.

Thomas, K. S.

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
[CrossRef]

Tran, P.

J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715, 1996.
[CrossRef]

Tsuji, Y.

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightwave Tech. 18, 102-110 (2000).
[CrossRef]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature (London) 386, 143-149 (1997).
[CrossRef]

Wang, R.

X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, "Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps," Phys. Rev. Lett. 88, 093902 (2002)
[CrossRef] [PubMed]

Wang, X. H.

X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, "Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps," Phys. Rev. Lett. 88, 093902 (2002)
[CrossRef] [PubMed]

Ward, A. J.

A. J. Ward and J. B. Pendry, "Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method," Phys. Rev. B 58, 7252-7259 (1998).
[CrossRef]

Xia, Y.

Z. Y. Li and Y. Xia, "Omnidirectional absolute band gaps in two-dimensional photonic crystals," Phys. Rev. B 64, 153108 (2001).
[CrossRef]

Yang, G. Z.

X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, "Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps," Phys. Rev. Lett. 88, 093902 (2002)
[CrossRef] [PubMed]

Yang, H. Y. D.

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

Yu, P. K. L.

Yu, Q. L.

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

Zhang, W.

IEEE Trans. Antennas Propag. (1)

G. Pelosi, A. Cocchi, and A. Monorchio, "A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam," IEEE Trans. Antennas Propag. 48, 973-980 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

J. Comput. Phys. (4)

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

D. C. Dobson, J. Gopalakrishnan, and J. E. Pasciak, "An efficient method for band structure calculations in 3D photonic crystals," J. Comput. Phys. 161, 668-679 (2000).
[CrossRef]

D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-376, 1999.
[CrossRef]

A. Figotin and Y. A. Godin, "The Computation of Spectra of Some 2D Photonic Crystals," J. Comput. Phys. 136, 585-598, 1997.
[CrossRef]

J. Lightwave Tech. (1)

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightwave Tech. 18, 102-110 (2000).
[CrossRef]

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

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

Nature (London) (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature (London) 386, 143-149 (1997).
[CrossRef]

Opt. Commun. (1)

P. Halevi and A. S. Sánchez, "Spontaneous emission in a high-contrast one-dimensional photonic crystal," Opt. Commun. 251, 109-114 (2005).
[CrossRef]

Opt. Express (2)

Phys. Rev. (1)

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).

Phys. Rev. A (4)

A. O. Barut and J. P. Dowling, "Quantum electrodynamics based on self-energy: Spontaneous emission in cavities," Phys. Rev. A 36, 649-654 (1987).
[CrossRef] [PubMed]

H. Rigneault and S. Monneret, "Modal analysis of spontaneous emission in a planar microcavity," Phys. Rev. A 54, 2356-2368 (1996).
[CrossRef] [PubMed]

J. P. Dowling and C. M. Bowden, "Atomic emission rates in inhomogeneous media with applications to photonic band structures," Phys. Rev. A 46, 612-622 (1992).
[CrossRef] [PubMed]

A. Kamli, M. Babiker, A. Al-Hajry, and N. Enfati, "Dipole relaxation in dispersive photonic band-gap structures," Phys. Rev. A 55, 1454-1461 (1997).
[CrossRef]

Phys. Rev. B (10)

M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, "Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals," Phys. Rev. B 58, 6791-6794 (1998).
[CrossRef]

Z. Y. Li and Y. Xia, "Omnidirectional absolute band gaps in two-dimensional photonic crystals," Phys. Rev. B 64, 153108 (2001).
[CrossRef]

K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

K. Sakoda, "Transmittance and Bragg reflectivity of two-dimensional photonic lattices," Phys. Rev. B 52, 8992-9002 (1995).
[CrossRef]

M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991).
[CrossRef]

C. T. Chan, Q. L. Yu, and K. M. Ho, "Order-N spectral method for electromagnetic waves," Phys. Rev. B 51, 16635-16642 (1995).
[CrossRef]

H. S. Sözüer, J. W. Haus and R. Inguva, "Photonic bands: Convergence problems with the plane-wave method," Phys. Rev. B 45, 13962-13972 (1992).
[CrossRef]

A. J. Ward and J. B. Pendry, "Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method," Phys. Rev. B 58, 7252-7259 (1998).
[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]

J. M. Elson and P. Tran, "Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal," Phys. Rev. B 54, 1711-1715, 1996.
[CrossRef]

Phys. Rev. E (5)

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

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]

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

A. S. Sánchez and P. Halevi, "Spontaneous emission in one-dimensional photonic crystals," Phys. Rev. E 72, 056609 (2005).
[CrossRef]

M. C. Lin and R. F. Jao, "Quantitative analysis of photon density of states for a realistic superlattice with omnidirectional light propagation," Phys. Rev. E 74, 046613 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

D. Kleppner, "Inhibited Spontaneous Emission," Phys. Rev. Lett. 47, 233-236 (1981).
[CrossRef]

O. J. F. Martin, C. Girard, D. R. Smith, and S. Schultz, "Generalized field propagator for arbitrary finite-size photonic band gap structures," Phys. Rev. Lett. 82, 315-318 (1999).
[CrossRef]

X. H. Wang, R. Wang, B. Y. Gu, and G. Z. Yang, "Decay distribution of spontaneous emission from an assembly of atoms in photonic crystals with psudogaps," Phys. Rev. Lett. 88, 093902 (2002)
[CrossRef] [PubMed]

J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
[CrossRef] [PubMed]

Phys. Stat. Sol. (1)

R. Hillebrand, W. Hergert and W. Harms, "Theoretical band gap studies of two-dimensional photonic crystals with varying column roundness," Phys. Stat. Sol.(b) 217, 981-989 (2000).
[CrossRef]

Sci. Meas. Technal. (1)

B. P. Hiett, J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett and K. S. Thomas, "Application of finite element methods to photonic crystal modeling," IEE Proc. -Sci. Meas. Technal. 149, 293-296 (2002).
[CrossRef]

Other (6)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, New Jersey 1995).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin 2001).

A. Taflove, Computational Electrodynamics: The Finite- Difference Time-Domain Method, Artech, Boston, Mass. (1995).

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-Interscience, New York 2002).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York 1984).

C. Kittel, Introduction to Solid State Physics (Wiley, New York 1976).

Cited By

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Figures (8)

Fig. 1.
Fig. 1.

(a) A square lattice with unit cell (A, B, C, D) and (b) the corresponding reciprocal lattice with irreducible Brillouin zone (1/8 of the first Brillouin zone, in shaded region). (c) A triangular lattice with unit cell (E, F, G, H) and (d) the corresponding reciprocal lattice with irreducible Brillouin zone (1/12 of the first Brillouin zone, in shaded region).

Fig. 2.
Fig. 2.

Triangular meshes for (a) a unit cell of square lattice and (c) a unit cell of triangular lattice. More meshes adaptively added in (b) and (d).

Fig. 3.
Fig. 3.

Convergence tests of the first 15 bands for (a) TE and (b) TM modes of a triangular array at the M point.

Fig. 4.
Fig. 4.

Photonic band structure for (a) a square array of dielectric columns with a radius r = 0.38 a and dielectric constant εd = 9 and (b) a triangular array of air cylinders with a radius r = 0.4297 a drilled into silicon substrate (εd ≈ 11.9), respectively. The solid lines represent TE modes and the dashed lines represent TM modes.

Fig. 5.
Fig. 5.

Comparisons of band structure of (a) TE and (b) TM modes calculated by the FEM and the PWEM [21].

Fig. 6.
Fig. 6.

Frequency shift of PWEM from FEM as a function of frequency for some symmetric k-points (Γ,M,K) for both the TE and TM modes.

Fig. 7.
Fig. 7.

Dispersion relations within the irreducible Brillouin zones for (a) TE and (b) TM modes of a square array, and those for (c) TE and (d) TM modes of a triangular array, respectively.

Fig. 8.
Fig. 8.

In-plane PDOS for (a) TE and (b) TM modes of a square array. Comparisons of in-plane PDOS for (c) TE and (d) TM modes of a triangular array calculated by the FEM and the PWEM.

Tables (1)

Tables Icon

Table 1. Paths, ranges of k and ranges of phase changes, k R 1 and k R 2, for each of the segments required to traverse the boundary of the irreducible Brillouin zone for a square array and triangular array.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

× [ 1 μ ( r ) × × E ( r ) ] ω 2 ε ( r ) E ( r ) = 0 ,
× [ 1 ε ( r ) × × H ( r ) ] ω 2 μ ( r ) H ( r ) = 0 ,
ε r ( x , y ) = { ε a , x , y air region ε d , x , y dielectric region
2 E ( r ) + ω 2 C 2 ε a ( d ) E ( r ) = 0 ,
2 H ( r ) + ω 2 c 2 ε a ( d ) H ( r ) = 0 .
[ 2 x 2 + 2 y 2 ] E z ( x , y ) + ω 2 c 2 ε a ( d ) E z ( x , y ) = 0 ,
[ 2 x 2 + 2 y 2 ] H z ( x , y ) + ω 2 c 2 ε a ( d ) H z ( x , y ) = 0 .
E z , k ( r + R ) = E z , k ( r ) exp ( i k . R ) ,
H z , k ( r + R ) = H z , k ( r ) exp ( i k . R ) .
Δ S k = 1 + ( Δω Δ k x ) 2 + ( Δω Δ k y ) 2 Δ k x Δ k y .
S k = ω k 1 + ( ω k x ) 2 + ( ω k y ) 2 d k x d k y .
D ( ω ) = S 4 π 2 ω k 1 + ( ω k x ) 2 + ( ω k y ) 2 d k x d k y .

Metrics