Abstract

We demonstrate an enhancement of the plane wave expansion method treating two-dimensional photonic crystals by applying Fourier factorization with generally elliptic polarization bases. By studying three examples of periodically arranged cylindrical elements, we compare our approach to the classical Ho method in which the permittivity function is simply expanded without changing coordinates, and to the normal vector method using a normal–tangential polarization transform. The compared calculations clearly show that our approach yields the best convergence properties owing to the complete continuity of our distribution of polarization bases. The presented methodology enables us to study more general systems such as periodic elements with an arbitrary cross-section or devices such as photonic crystal waveguides.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  40. K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. M. de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
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2009 (5)

2008 (2)

2007 (3)

2006 (2)

S. Noda, "Recent progresses and future prospects of two- and three-dimensional photonic crystals," J. Lightwave Technol. 24, 4554-4567 (2006).
[CrossRef]

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

2005 (4)

N. Bonod, E. Popov, and M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, and M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

S. Kinoshita and S. Yoshioka, "Structural colors in nature: The role of regularity and irregularity in the structure," ChemPhysChem 6, 1442-1459 (2005).
[CrossRef] [PubMed]

E. Reyes, A. A. Krokhin, and J. Roberts, "Effective dielectric constants of photonic crystal of aligned anisotropic cylinders and the optical response of a periodic array of carbon nanotubes," Phys. Rev. B 72, 155118 (2005).
[CrossRef]

2004 (2)

2003 (2)

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).

P. Vukusic and J. R. Sambles, "Photonic structures in biology," Nature 424, 852-855 (2003).
[CrossRef] [PubMed]

2002 (3)

2001 (3)

S. Visnovsky, and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
[CrossRef]

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gosele, "Large birefringence in two-dimensional silicon photonic crystals," Phys. Rev. B 63, 161101 (2001).
[CrossRef]

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

2000 (1)

1998 (3)

L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274-276 (1998).
[CrossRef] [PubMed]

1997 (2)

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

L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
[CrossRef]

1996 (1)

1993 (1)

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

1987 (1)

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

1984 (1)

D. Maystre, "Rigorous vector theories of diffraction gratings," Prog. Opt. 21, 1-67 (1984).
[CrossRef]

Antos, R.

Arriaga, J.

A. A. Krokhin, P. Halevi, and J. Arriaga, "Long-wavelength limit (homogenization) for two-dimensional photonic crystals," Phys. Rev. B 65, 115208 (2002).
[CrossRef]

Benisty, H.

H. Benisty, "Dark modes, slow modes, and coupling in multimode systems," J. Opt. Soc. Am. B 26, 718-724 (2009).
[CrossRef]

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Birner, A.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gosele, "Large birefringence in two-dimensional silicon photonic crystals," Phys. Rev. B 63, 161101 (2001).
[CrossRef]

Block, I. D.

I. D. Block, N. Ganesh, M. Lu, and B. T. Cunningham, "Sensitivity model for predicting photonic crystal biosensor performance," IEEE Sens. J. 8, 274-280 (2008).
[CrossRef]

Bonod, N.

N. Bonod, E. Popov, and M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, and M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

Borel, P. I.

Botten, L. C.

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. M. de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009), http: //www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19629.
[CrossRef] [PubMed]

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

Boyer, P.

Chan, C. T.

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Chernov, B.

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

Chow, E.

S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274-276 (1998).
[CrossRef] [PubMed]

Cunningham, B. T.

I. D. Block, N. Ganesh, M. Lu, and B. T. Cunningham, "Sensitivity model for predicting photonic crystal biosensor performance," IEEE Sens. J. 8, 274-280 (2008).
[CrossRef]

Datta, S.

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

David, A.

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

de Sterke, C. M.

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. M. de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009), http: //www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19629.
[CrossRef] [PubMed]

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

Dossou, K. B.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009), http: //www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19629.
[CrossRef] [PubMed]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. M. de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

Fan, S.

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

Frandsen, L. H.

Frenner, K.

Ganesh, N.

I. D. Block, N. Ganesh, M. Lu, and B. T. Cunningham, "Sensitivity model for predicting photonic crystal biosensor performance," IEEE Sens. J. 8, 274-280 (2008).
[CrossRef]

Genereux, F.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gosele, "Large birefringence in two-dimensional silicon photonic crystals," Phys. Rev. B 63, 161101 (2001).
[CrossRef]

Gosele, U.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gosele, "Large birefringence in two-dimensional silicon photonic crystals," Phys. Rev. B 63, 161101 (2001).
[CrossRef]

Gotz, P.

Halevi, P.

A. A. Krokhin, P. Halevi, and J. Arriaga, "Long-wavelength limit (homogenization) for two-dimensional photonic crystals," Phys. Rev. B 65, 115208 (2002).
[CrossRef]

Hietala, V.

S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274-276 (1998).
[CrossRef] [PubMed]

Ho, K. M.

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Hugonin, J. P.

Joannopoulos, J. D.

S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274-276 (1998).
[CrossRef] [PubMed]

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

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Kerwien, N.

Kinoshita, S.

S. Kinoshita and S. Yoshioka, "Structural colors in nature: The role of regularity and irregularity in the structure," ChemPhysChem 6, 1442-1459 (2005).
[CrossRef] [PubMed]

Kjems, J.

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Krauss, T. F.

Kristensen, M.

Krokhin, A. A.

E. Reyes, A. A. Krokhin, and J. Roberts, "Effective dielectric constants of photonic crystal of aligned anisotropic cylinders and the optical response of a periodic array of carbon nanotubes," Phys. Rev. B 72, 155118 (2005).
[CrossRef]

A. A. Krokhin and E. Reyes, "Homogenization of magnetodielectric photonic crystals," Phys. Rev. Lett. 93, 023904 (2004).
[CrossRef] [PubMed]

A. A. Krokhin, P. Halevi, and J. Arriaga, "Long-wavelength limit (homogenization) for two-dimensional photonic crystals," Phys. Rev. B 65, 115208 (2002).
[CrossRef]

Lalanne, P.

Leonard, S. W.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gosele, "Large birefringence in two-dimensional silicon photonic crystals," Phys. Rev. B 63, 161101 (2001).
[CrossRef]

Li, L.

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).

L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
[CrossRef]

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

Lin, S.-Y.

S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274-276 (1998).
[CrossRef] [PubMed]

Lu, M.

I. D. Block, N. Ganesh, M. Lu, and B. T. Cunningham, "Sensitivity model for predicting photonic crystal biosensor performance," IEEE Sens. J. 8, 274-280 (2008).
[CrossRef]

Mahmoodian, S.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009), http: //www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19629.
[CrossRef] [PubMed]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. M. de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

Maystre, D.

D. Maystre, "Rigorous vector theories of diffraction gratings," Prog. Opt. 21, 1-67 (1984).
[CrossRef]

McPhedran, R. C.

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009), http: //www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19629.
[CrossRef] [PubMed]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. M. de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

Neviere, M.

N. Bonod, E. Popov, and M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, and M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

P. Boyer, E. Popov, M. Neviere, and G. Tayeb, "Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section," J. Opt. Soc. Am. A 21, 2146-2153 (2004).
[CrossRef]

K. Watanabe, R. Petit, and M. Neviere, "Differential theory of gratings made of anisotropic materials," J. Opt. Soc. Am. A 19, 325-334 (2002).
[CrossRef]

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

E. Popov, and M. Neviere, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

Noda, S.

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Osten, W.

Petit, R.

Popov, E.

N. Bonod, E. Popov, and M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, and M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

P. Boyer, E. Popov, M. Neviere, and G. Tayeb, "Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section," J. Opt. Soc. Am. A 21, 2146-2153 (2004).
[CrossRef]

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

E. Popov, and M. Neviere, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

Poulton, C. G.

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. M. de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of shallow photonic crystal waveguides: semi-analytic treatment," Opt. Express 17, 19629-19643 (2009), http: //www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-19629.
[CrossRef] [PubMed]

Rafler, S.

Reyes, E.

E. Reyes, A. A. Krokhin, and J. Roberts, "Effective dielectric constants of photonic crystal of aligned anisotropic cylinders and the optical response of a periodic array of carbon nanotubes," Phys. Rev. B 72, 155118 (2005).
[CrossRef]

A. A. Krokhin and E. Reyes, "Homogenization of magnetodielectric photonic crystals," Phys. Rev. Lett. 93, 023904 (2004).
[CrossRef] [PubMed]

Roberts, J.

E. Reyes, A. A. Krokhin, and J. Roberts, "Effective dielectric constants of photonic crystal of aligned anisotropic cylinders and the optical response of a periodic array of carbon nanotubes," Phys. Rev. B 72, 155118 (2005).
[CrossRef]

Ruoff, J.

Sambles, J. R.

P. Vukusic and J. R. Sambles, "Photonic structures in biology," Nature 424, 852-855 (2003).
[CrossRef] [PubMed]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Schuster, T.

Skivesen, N.

Soukoulis, C. M.

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Tayeb, G.

Tetu, A.

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

van Driel, H. M.

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gosele, "Large birefringence in two-dimensional silicon photonic crystals," Phys. Rev. B 63, 161101 (2001).
[CrossRef]

Villeneuve, P. R.

S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274-276 (1998).
[CrossRef] [PubMed]

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

Visnovsky, S.

S. Visnovsky, and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
[CrossRef]

Vukusic, P.

P. Vukusic and J. R. Sambles, "Photonic structures in biology," Nature 424, 852-855 (2003).
[CrossRef] [PubMed]

Watanabe, K.

Weisbuch, C.

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

White, T. P.

Yablonovitch, E.

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

Yasumoto, K.

S. Visnovsky, and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
[CrossRef]

Yoshioka, S.

S. Kinoshita and S. Yoshioka, "Structural colors in nature: The role of regularity and irregularity in the structure," ChemPhysChem 6, 1442-1459 (2005).
[CrossRef] [PubMed]

ChemPhysChem (1)

S. Kinoshita and S. Yoshioka, "Structural colors in nature: The role of regularity and irregularity in the structure," ChemPhysChem 6, 1442-1459 (2005).
[CrossRef] [PubMed]

Czech. J. Phys. (1)

S. Visnovsky, and K. Yasumoto, "Multilayer anisotropic bi-periodic diffraction gratings," Czech. J. Phys. 51, 229-247 (2001).
[CrossRef]

IEEE Sens. J. (1)

I. D. Block, N. Ganesh, M. Lu, and B. T. Cunningham, "Sensitivity model for predicting photonic crystal biosensor performance," IEEE Sens. J. 8, 274-280 (2008).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

L. Li, "Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

J. Opt. A (1)

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A 5, 345-355 (2003).

J. Opt. Soc. Am. A (7)

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

Nature (2)

P. Vukusic and J. R. Sambles, "Photonic structures in biology," Nature 424, 852-855 (2003).
[CrossRef] [PubMed]

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

Opt. Commun. (3)

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

N. Bonod, E. Popov, and M. Neviere, "Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications," Opt. Commun. 245, 355-361 (2005).
[CrossRef]

N. Bonod, E. Popov, and M. Neviere, "Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect," Opt. Commun. 244, 389-398 (2005).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (2)

S. Mahmoodian, R. C. McPhedran, C. M. de Sterke, K. B. Dossou, C. G. Poulton, and L. C. Botten, "Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps," Phys. Rev. A 79, 013814 (2009).
[CrossRef]

K. B. Dossou, C. G. Poulton, L. C. Botten, S. Mahmoodian, R. C. McPhedran, and C. M. de Sterke, "Modes of symmetric composite defects in two-dimensional photonic crystals," Phys. Rev. A 80, 013826 (2009).
[CrossRef]

Phys. Rev. B (6)

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, "Effective dielectric constant of periodic composite structures," Phys. Rev. B 48, 14936-14943 (1993).
[CrossRef]

F. Genereux, S. W. Leonard, H. M. van Driel, A. Birner, and U. Gosele, "Large birefringence in two-dimensional silicon photonic crystals," Phys. Rev. B 63, 161101 (2001).
[CrossRef]

A. A. Krokhin, P. Halevi, and J. Arriaga, "Long-wavelength limit (homogenization) for two-dimensional photonic crystals," Phys. Rev. B 65, 115208 (2002).
[CrossRef]

E. Reyes, A. A. Krokhin, and J. Roberts, "Effective dielectric constants of photonic crystal of aligned anisotropic cylinders and the optical response of a periodic array of carbon nanotubes," Phys. Rev. B 72, 155118 (2005).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

A. A. Krokhin and E. Reyes, "Homogenization of magnetodielectric photonic crystals," Phys. Rev. Lett. 93, 023904 (2004).
[CrossRef] [PubMed]

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

K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

Prog. Opt. (1)

D. Maystre, "Rigorous vector theories of diffraction gratings," Prog. Opt. 21, 1-67 (1984).
[CrossRef]

Science (1)

S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274-276 (1998).
[CrossRef] [PubMed]

Other (4)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ., 1995).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980).

M. Neviere and E. Popov, Light Propagation in Periodic Media: Diffraction Theory and Design (Marcel Dekker, New York, 2003).

R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1997).

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Figures (7)

Fig. 1
Fig. 1

Two studied configurations of 2D PhCs, with square (a) and hexagonal (b) symmetry, with denoted square and rectangular unit cells, respectively. The corresponding first Brillouin zones of the reciprocal space are depicted in (c) and (d), respectively, where the emphasized triangles denote irreducible fractions. Note that for calculations we use a rectangular unit supercell (solid rectangle) rather than a hexagonal primitive cell (dashed hexagon) in the case of the hexagonal lattice (b) so that the corresponding first Brillouin zone is the solid rectangle instead of the dashed hexagon in (d), which creates a folded band structure, from where we choose values corresponding to the usual convention.

Fig. 2
Fig. 2

Schematic description of the polarization distributions in Models A (a), B (b), C (c), and C’ (d) within the first quadrant of the periodic cell, as functions of the polar coordinates re = x + iy. In (a) the xy Cartesian basis is uniform; in (b) the polarization vectors are normal (u) and tangential (v) to the cylindrical element and constant along lines coming from the center; in (c) and (d) the u, v polarizations are in general elliptic which enables their continuity.

Fig. 3
Fig. 3

Distribution of the rotation and ellipticity of the basis polarization vector u for the presented models. The rotation for Models B and C is in (a) [(e) for the hexagonal periodicity]; the ellipticity for Model C is in (b) [(f) for the hexagonal periodicity]; and the rotation and ellipticity for Model C’ are in (c) and (d), respectively. The color scales for both the rotation and the ellipticity are on the right (in degrees). The structure discontinuities (the circular boundaries of the periodic elements) are plotted as black or white circles. Notice that u is always linear along and normal to the circle.

Fig. 4
Fig. 4

Amplitude distribution of the scaled magnetic field |Hz(x, y)| within one cell of the real space for Samples S1 (a), S2 (b) and H (c). The color scale, same for each sample, is in the top left corner of (a). The subfigures represent eigenmodes distinguished by the symmetry point letters (Γ, X, M) and band numbers (1–4). The structure discontinuities (boundaries of the rods or holes) are plotted as white or black circles.

Fig. 5
Fig. 5

Convergence properties of normalized eigenfrequencies (ωa/2πc) calculated for selected bands of Sample S1. The modes Γ2, Γ3 are on the left part of the figure; X1, X4 in the middle; and M2, M4 on the right. The Models A, B, and C are compared to each other, plotted as crosses, squares, and circles, respectively.

Fig. 6
Fig. 6

Convergence properties of normalized eigenfrequencies (ωa/2πc) calculated for the modes M1 (left), M2 (middle), and M3 (right) of Sample H. The Models A, B, and C are plotted as crosses, squares, and circles, respectively.

Fig. 7
Fig. 7

Convergence properties of normalized eigenfrequencies (ωa/2πc) calculated for the modes X1 (left), X2 (middle), and X3 (right) of Sample S2. The Models A, B, C, and C’ are plotted as crosses, squares, circles, and triangles, respectively.

Equations (52)

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

ɛ ( x , y ) = m , n = + ɛ mn e i ( m G x x + n G y y ) ,
y H z = i ω ɛ 0 ɛ E x ,
x H z = i ω ɛ 0 ɛ E y ,
x E y y E x = i ω μ 0 H z ,
[ y , x ] [ E x E y ] = i ω μ 0 H z ,
ɛ [ E x E y ] = 1 i ω ɛ 0 [ y x ] H z .
[ D ˜ x D ˜ y ] = ɛ [ E x E y ] = [ ɛ xx ɛ xy ɛ yx ɛ yy ] [ E x E y ] ,
( y η xx y + x η yx y + y η xy x x η yy x ) H z = ω 2 c 2 H z ,
H z ( x , y ) = m , n = + h z , mn e i ( p m x + q n y ) ,
[ D ˜ x ] = [ ɛ E x ] = [ [ ɛ ] ] [ E x ] ,
[ D ˜ y ] = [ ɛ E y ] = [ [ ɛ ] ] [ E y ] .
[ [ η ] ] A = [ [ [ ɛ ] ] 1 [ [ 0 ] ] [ [ 0 ] ] [ [ ɛ ] ] 1 ] .
[ E x E y ] = F [ E u E v ] ,
[ D ˜ u ] = [ [ 1 / ɛ ] ] 1 [ E u ] ,
[ D ˜ v ] = [ [ ɛ ] ] [ E v ] .
F = [ cos ϕ sin ϕ sin ϕ cos ϕ ] ,
[ [ E x ] [ E y ] ] = [ [ F ] ] [ [ E u ] [ E v ] ] ,
[ [ F ] ] = [ [ [ c ] ] [ [ s ] ] [ [ s ] ] [ [ c ] ] ] .
[ [ E x ] [ E y ] ] = [ [ F ] ] [ [ [ 1 ɛ ] ] [ [ 0 ] ] [ [ 0 ] ] [ [ ɛ ] ] 1 ] [ [ F 1 ] ] [ [ D ˜ x ] [ D ˜ y ] ] ,
[ [ η ] ] B = [ [ F ] ] [ [ [ 1 ɛ ] ] [ [ 0 ] ] [ [ 0 ] ] [ [ ɛ ] ] 1 ] [ [ F 1 ] ] = [ [ [ 1 ɛ ] ] [ [ c 2 ] ] + [ [ ɛ ] ] 1 [ [ s 2 ] ] , [ [ 1 ɛ ] ] [ [ c s ] ] [ [ ɛ ] ] 1 [ [ c s ] ] [ [ 1 ɛ ] ] [ [ cs ] ] [ [ ɛ ] ] 1 [ [ cs ] ] , [ [ 1 ɛ ] ] [ [ s 2 ] ] + [ [ ɛ ] ] 1 [ [ c 2 ] ] ] ,
u = [ ξ ζ ] , v = [ ζ * ξ * ]
u = e i θ [ cos θ sin θ sin θ cos θ ] [ cos i sin ] ,
θ ( r , ϕ ) = ϕ ,
( r , ϕ ) = { π 8 ( 1 + cos π r R ) ( r R ) π 8 { 1 + cos π [ r + D ( ϕ ) 2 R ] D ( ϕ ) R } ( r > R ) .
D ( ϕ ) = a / 2 max ( | cos ϕ | , | sin ϕ | )
lim r 0 u = lim r D ( ϕ ) u = 1 2 [ 1 i ] ,
[ [ η ] ] C = [ [ [ 1 ɛ ] ] [ [ ξ ξ * ] ] + [ [ ɛ ] ] 1 [ [ ζ ζ * ] ] , [ [ 1 ɛ ] ] [ [ ξ ζ * ] ] [ [ ɛ ] ] 1 [ [ ξ ζ * ] ] [ [ 1 ɛ ] ] [ [ ξ * ζ ] ] [ [ ɛ ] ] 1 [ [ ξ * ζ ] ] , [ [ 1 ɛ ] ] [ [ ζ ζ * ] ] + [ [ ɛ ] ] 1 [ [ ξ ξ * ] ] ] .
D ( ϕ ) = a / 2 max n = 0 , , 5 [ cos ( ϕ n π 3 ) ] ,
θ b ( ϕ ) = θ ( D ( ϕ ) , ϕ ) = π 2 round ( ϕ / π 2 ) ,
b ( ϕ ) = ( D ( ϕ ) , ϕ ) = π 8 ( 1 cos 4 ϕ )
θ ( r , ϕ ) = 1 2 { θ b ( ϕ ) + ϕ + [ θ b ( ϕ ) ϕ ] cos π [ r + D ( ϕ ) 2 R ] D ( ϕ ) R } ,
( r , ϕ ) = b ( ϕ ) 2 { 1 + cos π [ r + D ( ϕ ) 2 R ] D ( ϕ ) R } .
ɛ ( x , y ) = m , n = + ɛ mn e i ( m G x x + n G y y ) ,
f ( x , y ) = m , n = + f mn e i ( p m x + q n y ) .
h ( x , y ) = ɛ ( x , y ) f ( x , y ) ,
g x ( x , y ) = x f ( x , y ) ,
g y ( x , y ) = y f ( x , y ) ,
h ( x , y ) = m , n = + h mn e i ( p m x + q n y ) ,
g x ( x , y ) = m , n = + g x , mn e i ( p m x + q n y ) ,
g y ( x , y ) = m , n = + g y , mn e i ( p m x + q n y ) ,
h mn = k , l = + ɛ m k , n l f kl ,
g x , mn = i p m f mn ,
g y , mn = i q n f mn .
α ( m , n ) = m + M + 1 + ( n + N ) ( 2 M + 1 ) ,
n ( α ) = ( α 1 ) div ( 2 M + 1 ) N ,
m ( α ) = ( α 1 ) mod ( 2 M + 1 ) M ,
[ h ] = [ [ ɛ ] ] [ f ] ,
[ g x ] = i p [ f ] ,
[ g y ] = i q [ f ] ,
[ [ ɛ ] ] α β = ɛ m ( α ) m ( β ) , n ( α ) n ( β ) ,
p α β = p m ( α ) δ α β ,
q α β = q n ( α ) δ α β ,

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