Abstract

We propose a method for the determination of the band structure, reflectance, and transmittance of one- and two-dimensional photonic crystals that is based on the solution of integral equations. The results of this method are compared with those obtained by other well-known algorithms, and good agreement between them is found. The method is also tested by considering systems that possess a complex structure in their unit cell such as fractal geometries.

© 2006 Optical Society of America

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  1. R. C. Schroden, M. Al-Daous, C. F. Blanford, and A. Stein,"Optical properties of inverse opal photonic crystals," Chem. Mater. 14, 3305-3315 (2002).
    [CrossRef]
  2. A. F. Koenderink and W. L. Vos, "Optical properties of real photonic crystals," J. Opt. Soc. Am. B 22, 1075-1084 (2005).
    [CrossRef]
  3. L. Florescu, K. Busch, and S. John, "Semiclassical theory of lasing in photonic crystals," J. Opt. Soc. Am. B 19, 2215-2223 (2002).
    [CrossRef]
  4. P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
    [CrossRef]
  5. M. Mengens, J. E. G. J. Wijnhoven, A. Lagendijk, and W. L. Vos, "Light sources inside photonic crystals," J. Opt. Soc. Am. B 16, 1403-1408 (1999).
    [CrossRef]
  6. M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, "Nonlinear photonic crystal microdevices for optical integration," Opt. Lett. 28, 637-639 (2005).
    [CrossRef]
  7. S. F. Mingaleev and K. Busch, "Scattering matrix approach to large-scale photonic crystal circuits," Opt. Lett. 28, 619-621 (2003).
    [CrossRef] [PubMed]
  8. K. Busch, S. F. Mingaleev, A. García-Martin, M. Schillinger, and D. Hermann, "The Wannier functions approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-R1256 (2003).
    [CrossRef]
  9. D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
    [CrossRef]
  10. F. Villa-Villa and J. A. Gaspar-Armenta, "Brewster angle and optical tunneling in one-dimensional photonic crystals composed of left- and right-handed materials," J. Opt. Soc. Am. B 23, 375-380 (2006).
    [CrossRef]
  11. F. Ramos-Mendieta and P. Halevi, "Surface modes in a 2D array of square dielectric cylinders," Solid State Commun. 100, 311-314 (1996).
    [CrossRef]
  12. J. B. Pendry, "Calculating the photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
    [CrossRef]
  13. A. J. Ward and J. B. Pendry, "A program for calculating the photonic band structures, Green's functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 128, 590-621 (2000).
    [CrossRef]
  14. R. M. Josephand and A. Taflove, "FDTD Maxwell's equations models for nonlinear electrodynamics and optics," IEEE Trans. Antennas Propag. 45, 364-374 (1997).
    [CrossRef]
  15. A. Lavrinenko, P. I. Borel, L. H. Fransen, M. Thorhauge, A. Harpoth, M. Kristensen, and T. Niemi, "Comprehensive FDTD modeling of photonic crystal waveguide components," Opt. Express 12, 234-248 (2004).
    [CrossRef] [PubMed]
  16. 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]
  17. M. Qui and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000).
    [CrossRef]
  18. S. Fan, P. R. Villeeuve, and J. D. Joannopoulos, "Large omnidirectional band gaps in metallodielectric photonic crystals," Phys. Rev. B 54, 11245-11251 (1996).
    [CrossRef]
  19. O. J. F. Martin and N. B. Piller, "Electromagnetic scattering in polarizable backgrounds," Phys. Rev. E 58, 3909-3915 (1998).
    [CrossRef]
  20. 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]
  21. A. I. Rahachou and I. V. Zozoulenko, "Light propagation in finite photonic crystals: the recursive Green's function technique," Phys. Rev. B 72, 155117 (2005).
    [CrossRef]
  22. L.-M. Zhao, X.-H. Wang, B.-Y. Gu, and G.-Z. Yang, "Green's functions for photonic crystal slabs," Phys. Rev. E 72, 026614 (2005).
    [CrossRef]
  23. F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
    [CrossRef]
  24. J. A. Gaspar-Armenta and F. Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am. B 21, 405-412 (2004).
    [CrossRef]
  25. A. A. Maradudin, E. R. Mendez, and T. Michel, "Enhanced backscattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
    [CrossRef]
  26. A. Mendoza-Suárez and E. R. Mendez, "Light scattering by a reentrant fractal surface," Appl. Opt. 36, 3521-3531 (1997).
    [CrossRef] [PubMed]
  27. A. Mendoza-Suárez, R. Espinosa-Luna, J. Cruz-Mandujano, and J. Espinosa-Luna, "Numerical technique to calculate modes in waveguides of arbitrarily cross-sectional shape," J. Opt. Soc. Am. A 18, 961-965 (2001).
    [CrossRef]
  28. M. Plihal, A. Shamrock, A. A. Maradudin, and P. Sheng, "Two-dimensional photonic band structures," Opt. Commun. 80, 199-204 (1991).
    [CrossRef]

2006 (1)

2005 (4)

A. F. Koenderink and W. L. Vos, "Optical properties of real photonic crystals," J. Opt. Soc. Am. B 22, 1075-1084 (2005).
[CrossRef]

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, "Nonlinear photonic crystal microdevices for optical integration," Opt. Lett. 28, 637-639 (2005).
[CrossRef]

A. I. Rahachou and I. V. Zozoulenko, "Light propagation in finite photonic crystals: the recursive Green's function technique," Phys. Rev. B 72, 155117 (2005).
[CrossRef]

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

2004 (3)

2003 (3)

S. F. Mingaleev and K. Busch, "Scattering matrix approach to large-scale photonic crystal circuits," Opt. Lett. 28, 619-621 (2003).
[CrossRef] [PubMed]

K. Busch, S. F. Mingaleev, A. García-Martin, M. Schillinger, and D. Hermann, "The Wannier functions approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-R1256 (2003).
[CrossRef]

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

2002 (2)

R. C. Schroden, M. Al-Daous, C. F. Blanford, and A. Stein,"Optical properties of inverse opal photonic crystals," Chem. Mater. 14, 3305-3315 (2002).
[CrossRef]

L. Florescu, K. Busch, and S. John, "Semiclassical theory of lasing in photonic crystals," J. Opt. Soc. Am. B 19, 2215-2223 (2002).
[CrossRef]

2001 (2)

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

A. Mendoza-Suárez, R. Espinosa-Luna, J. Cruz-Mandujano, and J. Espinosa-Luna, "Numerical technique to calculate modes in waveguides of arbitrarily cross-sectional shape," J. Opt. Soc. Am. A 18, 961-965 (2001).
[CrossRef]

2000 (2)

M. Qui and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

A. J. Ward and J. B. Pendry, "A program for calculating the photonic band structures, Green's functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 128, 590-621 (2000).
[CrossRef]

1999 (2)

M. Mengens, J. E. G. J. Wijnhoven, A. Lagendijk, and W. L. Vos, "Light sources inside photonic crystals," J. Opt. Soc. Am. B 16, 1403-1408 (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 (1)

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

1997 (2)

A. Mendoza-Suárez and E. R. Mendez, "Light scattering by a reentrant fractal surface," Appl. Opt. 36, 3521-3531 (1997).
[CrossRef] [PubMed]

R. M. Josephand and A. Taflove, "FDTD Maxwell's equations models for nonlinear electrodynamics and optics," IEEE Trans. Antennas Propag. 45, 364-374 (1997).
[CrossRef]

1996 (3)

F. Ramos-Mendieta and P. Halevi, "Surface modes in a 2D array of square dielectric cylinders," Solid State Commun. 100, 311-314 (1996).
[CrossRef]

J. B. Pendry, "Calculating the photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

S. Fan, P. R. Villeeuve, and J. D. Joannopoulos, "Large omnidirectional band gaps in metallodielectric photonic crystals," Phys. Rev. B 54, 11245-11251 (1996).
[CrossRef]

1995 (1)

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]

1991 (1)

M. Plihal, A. Shamrock, A. A. Maradudin, and P. Sheng, "Two-dimensional photonic band structures," Opt. Commun. 80, 199-204 (1991).
[CrossRef]

1990 (1)

A. A. Maradudin, E. R. Mendez, and T. Michel, "Enhanced backscattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Agio, M.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

Akjouj, A.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Al-Daous, M.

R. C. Schroden, M. Al-Daous, C. F. Blanford, and A. Stein,"Optical properties of inverse opal photonic crystals," Chem. Mater. 14, 3305-3315 (2002).
[CrossRef]

Birner, A.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

Blanford, C. F.

R. C. Schroden, M. Al-Daous, C. F. Blanford, and A. Stein,"Optical properties of inverse opal photonic crystals," Chem. Mater. 14, 3305-3315 (2002).
[CrossRef]

Borel, P. I.

Bria, D.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Busch, K.

Chan, C. T.

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]

Cruz-Mandujano, J.

Djafari-Rouhani, B.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Dobrzynski, L.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

El Boudoti, E. H.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Espinosa-Luna, J.

Espinosa-Luna, R.

Fan, S.

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, "Nonlinear photonic crystal microdevices for optical integration," Opt. Lett. 28, 637-639 (2005).
[CrossRef]

S. Fan, P. R. Villeeuve, and J. D. Joannopoulos, "Large omnidirectional band gaps in metallodielectric photonic crystals," Phys. Rev. B 54, 11245-11251 (1996).
[CrossRef]

Florescu, L.

Fransen, L. H.

García-Martin, A.

K. Busch, S. F. Mingaleev, A. García-Martin, M. Schillinger, and D. Hermann, "The Wannier functions approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-R1256 (2003).
[CrossRef]

Gaspar-Armenta, J. A.

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]

Gösele, U.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

Gu, B.-Y.

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

Halevi, P.

F. Ramos-Mendieta and P. Halevi, "Surface modes in a 2D array of square dielectric cylinders," Solid State Commun. 100, 311-314 (1996).
[CrossRef]

Harpoth, A.

He, S.

M. Qui and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

Hermann, D.

K. Busch, S. F. Mingaleev, A. García-Martin, M. Schillinger, and D. Hermann, "The Wannier functions approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-R1256 (2003).
[CrossRef]

Ho, K. M.

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]

Joannopoulos, J. D.

M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, "Nonlinear photonic crystal microdevices for optical integration," Opt. Lett. 28, 637-639 (2005).
[CrossRef]

S. Fan, P. R. Villeeuve, and J. D. Joannopoulos, "Large omnidirectional band gaps in metallodielectric photonic crystals," Phys. Rev. B 54, 11245-11251 (1996).
[CrossRef]

John, S.

Josephand, R. M.

R. M. Josephand and A. Taflove, "FDTD Maxwell's equations models for nonlinear electrodynamics and optics," IEEE Trans. Antennas Propag. 45, 364-374 (1997).
[CrossRef]

Koenderink, A. F.

Kramper, P.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

Kristensen, M.

Lagendijk, A.

Lavrinenko, A.

Luo, C.

Maradudin, A. A.

M. Plihal, A. Shamrock, A. A. Maradudin, and P. Sheng, "Two-dimensional photonic band structures," Opt. Commun. 80, 199-204 (1991).
[CrossRef]

A. A. Maradudin, E. R. Mendez, and T. Michel, "Enhanced backscattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[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]

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

Mendez, E. R.

A. Mendoza-Suárez and E. R. Mendez, "Light scattering by a reentrant fractal surface," Appl. Opt. 36, 3521-3531 (1997).
[CrossRef] [PubMed]

A. A. Maradudin, E. R. Mendez, and T. Michel, "Enhanced backscattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Mendoza-Suárez, A.

Mengens, M.

Michel, T.

A. A. Maradudin, E. R. Mendez, and T. Michel, "Enhanced backscattering of light from a random grating," Ann. Phys. (N.Y.) 203, 255-307 (1990).
[CrossRef]

Mingaleev, S. F.

K. Busch, S. F. Mingaleev, A. García-Martin, M. Schillinger, and D. Hermann, "The Wannier functions approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-R1256 (2003).
[CrossRef]

S. F. Mingaleev and K. Busch, "Scattering matrix approach to large-scale photonic crystal circuits," Opt. Lett. 28, 619-621 (2003).
[CrossRef] [PubMed]

Mlynek, J.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

Müller, F.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

Niemi, T.

Nougaoui, A.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Pendry, J. B.

A. J. Ward and J. B. Pendry, "A program for calculating the photonic band structures, Green's functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 128, 590-621 (2000).
[CrossRef]

J. B. Pendry, "Calculating the photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

Piller, N. B.

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

Plihal, M.

M. Plihal, A. Shamrock, A. A. Maradudin, and P. Sheng, "Two-dimensional photonic band structures," Opt. Commun. 80, 199-204 (1991).
[CrossRef]

Qui, M.

M. Qui and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000).
[CrossRef]

Rahachou, A. I.

A. I. Rahachou and I. V. Zozoulenko, "Light propagation in finite photonic crystals: the recursive Green's function technique," Phys. Rev. B 72, 155117 (2005).
[CrossRef]

Ramos-Mendieta, F.

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, "Surface modes in a 2D array of square dielectric cylinders," Solid State Commun. 100, 311-314 (1996).
[CrossRef]

Sandoghdar, V.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

Schillinger, M.

K. Busch, S. F. Mingaleev, A. García-Martin, M. Schillinger, and D. Hermann, "The Wannier functions approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-R1256 (2003).
[CrossRef]

Schroden, R. C.

R. C. Schroden, M. Al-Daous, C. F. Blanford, and A. Stein,"Optical properties of inverse opal photonic crystals," Chem. Mater. 14, 3305-3315 (2002).
[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]

Shamrock, A.

M. Plihal, A. Shamrock, A. A. Maradudin, and P. Sheng, "Two-dimensional photonic band structures," Opt. Commun. 80, 199-204 (1991).
[CrossRef]

Sheng, P.

M. Plihal, A. Shamrock, A. A. Maradudin, and P. Sheng, "Two-dimensional photonic band structures," Opt. Commun. 80, 199-204 (1991).
[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]

Soljacic, M.

Soukoulis, C. M.

P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[CrossRef]

Stein, A.

R. C. Schroden, M. Al-Daous, C. F. Blanford, and A. Stein,"Optical properties of inverse opal photonic crystals," Chem. Mater. 14, 3305-3315 (2002).
[CrossRef]

Taflove, A.

R. M. Josephand and A. Taflove, "FDTD Maxwell's equations models for nonlinear electrodynamics and optics," IEEE Trans. Antennas Propag. 45, 364-374 (1997).
[CrossRef]

Thorhauge, M.

Vigneron, J. P.

D. Bria, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, J. P. Vigneron, E. H. El Boudoti, and A. Nougaoui, "Band structure and omnidirectional photonic band gap in lamellar structures with left-handed materials," Phys. Rev. E 69, 066613 (2004).
[CrossRef]

Villa, F.

J. A. Gaspar-Armenta and F. Villa, "Band-structure properties of one-dimensional photonic crystals under the formalism of equivalent systems," J. Opt. Soc. Am. B 21, 405-412 (2004).
[CrossRef]

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, "One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function," Opt. Commun. 216, 361-367 (2003).
[CrossRef]

Villa-Villa, F.

Villeeuve, P. R.

S. Fan, P. R. Villeeuve, and J. D. Joannopoulos, "Large omnidirectional band gaps in metallodielectric photonic crystals," Phys. Rev. B 54, 11245-11251 (1996).
[CrossRef]

Vos, W. L.

Wang, X.-H.

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

Ward, A. J.

A. J. Ward and J. B. Pendry, "A program for calculating the photonic band structures, Green's functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 128, 590-621 (2000).
[CrossRef]

Wijnhoven, J. E. G. J.

Yang, G.-Z.

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Zozoulenko, I. V.

A. I. Rahachou and I. V. Zozoulenko, "Light propagation in finite photonic crystals: the recursive Green's function technique," Phys. Rev. B 72, 155117 (2005).
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A. J. Ward and J. B. Pendry, "A program for calculating the photonic band structures, Green's functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 128, 590-621 (2000).
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P. Kramper, A. Birner, M. Agio, C. M. Soukoulis, F. Müller, U. Gösele, J. Mlynek, and V. Sandoghdar, "Direct spectroscopy of a deep two-dimensional photonic crystal microresonator," Phys. Rev. B 64, 233102 (2001).
[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]

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[CrossRef]

A. I. Rahachou and I. V. Zozoulenko, "Light propagation in finite photonic crystals: the recursive Green's function technique," Phys. Rev. B 72, 155117 (2005).
[CrossRef]

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

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[CrossRef]

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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).
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Figures (10)

Fig. 1
Fig. 1

Unit cell of a one-dimensional photonic crystal (1DPC) composed of two different materials with electrical permittivities ε 1 and ε 2 and thicknesses d 1 and d 2 , respectively. The materials are limited by the contours Γ a , Γ b , and Γ c whose normal vectors were chosen as shown. Dashed lines represent closed contours inside each material.

Fig. 2
Fig. 2

1DPC including periodic defects on the surfaces of the unit cell.

Fig. 3
Fig. 3

Band structure under E polarization and normal incidence ( β ¯ = 0 ) for a 1DPC with d 1 = 0.4545 D , d 2 = 0.5454 D , ε 1 = 2 , and ε 2 = 1.1 . Solid curves indicate the results obtained with the proposed method, and dashed curves correspond to results obtained by the characteristic matrix method.

Fig. 4
Fig. 4

Schematic representation of a finite 1DPC where light is incident from vacuum on a surface whose profile is represented by Γ a . The angle θ s is the scattering angle, and θ 0 is the incidence angle.

Fig. 5
Fig. 5

Reflectance of a single-period truncated 1DPC (solid curve) determined by the integral method compared with those obtained by using the characteristic matrix (dashed curve).

Fig. 6
Fig. 6

Unit cell of a 2DPC with a squared lattice and inclusions of arbitrary shape. As in the 1DPC, we have two different materials. The materials are limited by the contours Γ a , Γ b , Γ c , Γ d , and Γ e whose normal vectors are indicated in the figure.

Fig. 7
Fig. 7

Band structure of a 2DPC with a square Bravais lattice and cylindrical inclusions. The left inset shows the unit cell in real space; the right inset shows the first Brillouin zone in k space.

Fig. 8
Fig. 8

Band structure of a 2DPC with bars of square sections. The left inset shows the unit cell in real space, and the right inset shows the first Brillouin zone in k space.

Fig. 9
Fig. 9

Band structure of a 2DPC with bars whose transversal sections are constituted by Koch prefractals of the first order.

Fig. 10
Fig. 10

Band structure of a 2DPC with bars whose transversal sections are constituted by Koch prefractals of the second order (solid curves). In this case, the band structure corresponding to the prefractal of first order (previous figure) is overlapped (dashed curves).

Equations (39)

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2 Ψ j ( r ) + ε j ω 2 c 2 Ψ j ( r ) = 0 .
2 G j ( r , r ) + ε j ω 2 c 2 G j ( r , r ) = 4 π δ ( r r ) .
G j ( r , r ) = i π H 0 ( 1 ) ( ε j ω c r r ) ,
4 π S j δ ( r r ) Ψ j ( r ) d A = C j [ G j ( r , r ) Ψ j ( r ) n j G j ( r , r ) n j Ψ j ( r ) ] d s .
1 4 π lim ξ 0 + { Γ a [ G 1 ( R a + ξ n ̂ a , r ) Ψ a n a G 1 ( R a + ξ n ̂ a , r ) n a Ψ a ] d s Γ b [ G 1 ( R a + ξ n ̂ a , r ) Ψ b n b G 1 ( R a + ξ n ̂ a , r ) n b Ψ b ] d s } = 0 ,
1 4 π lim ξ 0 + { Γ a [ G 1 ( R b + ξ n ̂ b , r ) Ψ a n a G 1 ( R b + ξ n ̂ b , r ) n a Ψ a ] d s Γ b [ G 1 ( R b + ξ n ̂ b , r ) Ψ b n b G 1 ( R b + ξ n ̂ b , r ) n b Ψ b ] d s } = 0 ,
1 4 π lim ξ 0 + { Γ b [ G 2 ( R b + ξ n ̂ b , r ) Ψ ̃ b n b G 2 ( R b + ξ n ̂ b , r ) n b Ψ ̃ b ] d s Γ c [ G 2 ( R b + ξ n ̂ b , r ) Ψ ̃ c n c G 2 ( R b + ξ n ̂ b , r ) n c Ψ ̃ c ] d s } = Ψ ̃ b ,
1 4 π lim ξ 0 + { Γ b [ G 2 ( R c + ξ n ̂ c , r ) Ψ ̃ b n b G 2 ( R c + ξ n ̂ c , r ) n b Ψ ̃ b ] d s Γ c [ G 2 ( R c + ξ n ̂ c , r ) Ψ ̃ c n c G 2 ( R c + ξ n ̂ c , r ) n c Ψ ̃ c ] d s } = Ψ ̃ c ,
R a n = ( X n , Y n ) = [ X ( s n ) , Y ( s n ) ] ,
1 4 π lim ξ 0 + Γ a G 1 ( s m , s ; ξ ) Φ a d s = 1 4 π n = Φ a n [ lim ξ 0 + s n ( Δ s 2 ) s n + ( Δ s 2 ) G 1 ( s m , s ; ξ ) d s ] ,
G 1 ( s m , s ; ξ ) = i π H 0 ( 1 ) ( ε 1 ω c { [ X m + ξ n ̂ a x X ( s ) ] 2 + [ Y m + ξ n ̂ a y Y ( s ) ] 2 } 1 2 ) ,
1 4 π lim ξ 0 + Γ a G 1 ( s m , s ; ξ ) Φ a d s n = L m n 1 Φ a n .
1 4 π lim ξ 0 + Γ a G 1 ( s m , s ; ξ ) n a Ψ a d s n = N m n 1 Ψ a n ,
L m n j = i Δ s 4 H 0 ( 1 ) ( ε j ω c d m n ) ( 1 δ m n ) + i Δ s 4 H 0 ( 1 ) ( ε j ω c Δ s 2 e ) δ m n ,
N m n j = i Δ s 4 ε j ω c H 1 ( 1 ) ( ε j ω c d m n ) D m n d m n ( 1 δ m n ) + ( 1 2 + Δ s 4 π D n ) δ m n ,
d m n = ( X m X n ) 2 + ( Y m Y n ) 2 ,
D m n = Y n ( X m X n ) + X n ( Y m Y n ) ,
D n = X n Y n X n Y n ,
Ψ ̃ b = Ψ b , 1 f 2 Ψ ̃ b n b = 1 f 1 Ψ b n b ,
f j = { 1 for E polarization ε j for H polarization } .
Ψ ( r + D ι ̂ ) = Ψ ( r ) exp ( i k D ) ,
Ψ ̃ c = Ψ a exp ( i k D ) , Ψ ̃ c n c = f 2 f 1 Ψ a n a exp ( i k D ) .
Ψ a n = Ψ a ( r ) r = R a n , Φ a n = Ψ a ( r ) n a r = R a n , Ψ b n = Ψ b ( r ) r = R b n , Φ b n = Ψ b ( r ) n b r = R a n .
n = 1 N a N m n 1 Ψ a n n = 1 N a L m n 1 Φ a n + n = N a + 1 N a + N b N m n 1 Ψ b n n = N a + 1 N a + N b L m n 1 Φ b n = 0 .
exp ( i k D ) n = N a + N b + 1 N a b c N m n 2 Ψ a n exp ( i k D ) n = N a + N b + 1 N a b c f 2 f 1 L m n 2 Φ a n + n = N a + 1 N a + N b ( δ m n + N m n 2 ) Ψ b n n = N a + 1 N a + N b f 2 f 1 L m n 2 Φ b n = 0 .
exp ( i k D ) n = N a + N b + 1 N a b c ( δ m n + N m n 2 ) Ψ a n exp ( i k D ) n = N a + N b + 1 N a b c f 2 f 1 L m n 2 Φ a n + n = N a + 1 N a + N b N m n 2 Ψ b n n = N a + 1 N a + N b f 2 f 1 L m n 2 Φ b n = 0 .
D ( k , ω ) = ln [ det ( M ) ] ,
Ψ j ( x , y ) = exp ( i β y ) ψ j ( x ) ,
n = L m n 1 Φ a n = [ n = L 0 n 1 exp ( i β Y n ) ] Φ a 0 = M 12 ( β , ω ) Φ a 0 ,
A ( β , ω ) [ exp ( i k D ) ] 2 + B ( β , ω ) exp ( i k D ) + C ( β , ω ) = 0 ,
Ψ ( r ) = Ψ inc ( r ) + 1 4 π Γ a [ G 0 ( r , r ) n a Ψ a ( r ) G 0 ( r , r ) Ψ a ( r ) n a ] d s ,
A ( θ s , ω ) = Γ a [ i ω c ( n ̂ a r ̂ ) Ψ a ( r ) Ψ a ( r ) n a ] exp ( i ω c r r ̂ ) d s ,
R θ s = h A ( θ s , ω ) 2 ,
R ( ω ) = π 2 π 2 R θ s d θ s .
Ψ c = Ψ a exp ( i k x D x ) , Φ c = Φ a exp ( i k x D x ) ,
Ψ d = Ψ b exp ( i k y D y ) , Φ d = Φ b exp ( i k y D y ) ,
Ψ ̃ e = Ψ e , Φ ̃ e = f 2 f 1 Φ e .
n = 1 N a N m n 1 Ψ a n + exp ( i k x D x ) n = N a + N b + 1 N a b c d N d N m n 1 Ψ a n n = 1 N a L m n 1 Φ a n + exp ( i k x D x ) n = N a + N b + 1 N a b c d N d L m n 1 Φ a n + n = N a + 1 N a + N b N m n 1 Ψ b n + exp ( i k y D y ) n = N a b c d N d + 1 N a b c d N m n 1 Ψ b n n = N a + 1 N a + N b L m n 1 Φ b n + exp ( i k y D y ) n = N a b c d N d + 1 N a b c d L m n 1 Φ b n + n = N a b c d + 1 N a b c d + N e N m n 1 Ψ e n n = N a b c d + 1 N a b c d + N e L m n 1 Φ e n = 0 .
n = N a b c d + 1 N a b c d + N e ( δ m n + N m n 2 ) Ψ e n f 2 f 1 n = N a b c d + 1 N a b c d + N e L m n 2 Φ e n = 0 .

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