Abstract

We describe methods of investigating the behavior of photonic crystals. Our approach establishes a link between the dispersion relation of the Bloch modes for an infinite crystal (which describes the intrinsic properties of the photonic crystal in the absence of an incident field) and the diffraction problem of a grating (finite photonic crystal) illuminated by an incident field. We point out the relationship between the translation operator of the first problem and the transfer matrix of the second. The eigenvalues of the transfer matrix contain information about the dispersion relation. This approach enables us to answer questions such as When does ultrarefraction occur? Can the photonic crystal simulate a homogeneous and isotropic material with low effective index? This approach also enables us to determine suitable parameters to obtain ultrarefractive or negative refraction properties and to design optical devices such as highly dispersive microprisms and ultrarefractive microlenses. Rigorous computations add a quantitative aspect and demonstrate the relevance of our approach.

© 2000 Optical Society of America

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  1. S. Y. Lin, V. M. Hietala, L. Wang, E. D. Jones, “Highly dispersive photonic band-gap prism,” Opt. Lett. 21, 1771–1773 (1996).
    [CrossRef] [PubMed]
  2. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
    [CrossRef]
  3. R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
    [CrossRef]
  4. K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  5. M. Plihal, A. A. Maradudin, “Photonic band structure of two-dimensional systems: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [CrossRef]
  6. H. S. Sözüer, J. W. Haus, R. Inguva, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
    [CrossRef]
  7. J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  8. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993). See also the erratum in Phys. Rev. B 55, 15942 (1997).
    [CrossRef]
  9. A. Moroz, C. Sommers, “Photonic band gaps of three-dimensional face-centered cubic lattices,” J. Phys. Condens. Matter 11, 997–1008 (1999).
    [CrossRef]
  10. D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
    [CrossRef]
  11. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994).
    [CrossRef]
  15. J. M. Elson, 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]
  16. R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
    [CrossRef]
  17. M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
    [CrossRef]
  18. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  19. L. Li, “Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
    [CrossRef]
  20. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  21. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
  22. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  23. D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet; applications à la spectroscopie et au filtrage des ondes électromagnétiques,” Ph.D. thesis (Université Aix-Marseille 3, Marseille, France, 1974).
  24. L. Li, “Justification of matrix truncation in the modal methods of diffraction gratings,” J. Opt. A Pure Appl. Opt. 1, 531–536 (1999).
    [CrossRef]
  25. P. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals:  mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [CrossRef]
  26. A. Sentenac, J. J. Greffet, F. Pincemin, “Structure of the electromagnetic field in a slab of photonic crystal,” J. Opt. Soc. Am. B 14, 339–347 (1997).
    [CrossRef]
  27. G. Tayeb, 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]
  28. Z. Yuan, J. W. Haus, K. Sakoda, “Eigenmode symmetry for simple cubic lattices and the transmission spectra,” Opt. Express 3, 19–27 (1998).
    [CrossRef] [PubMed]
  29. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  30. J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
    [CrossRef]
  31. R. C. McPhedran, N. A. Nicorovici, L. C. Botten, “The TEM mode and homogenization of doubly periodic structures,” J. Electromagn. Waves Appl. 11, 981–1012 (1997).
    [CrossRef]
  32. D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
    [CrossRef]
  33. G. Guida, D. Maystre, G. Tayeb, P. Vincent, “Mean-field theory of two-dimensional metallic photonic crystals,” J. Opt. Soc. Am. B 15, 2308–2315 (1998).
    [CrossRef]

1999 (3)

A. Moroz, C. Sommers, “Photonic band gaps of three-dimensional face-centered cubic lattices,” J. Phys. Condens. Matter 11, 997–1008 (1999).
[CrossRef]

S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

L. Li, “Justification of matrix truncation in the modal methods of diffraction gratings,” J. Opt. A Pure Appl. Opt. 1, 531–536 (1999).
[CrossRef]

1998 (3)

1997 (4)

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, “The TEM mode and homogenization of doubly periodic structures,” J. Electromagn. Waves Appl. 11, 981–1012 (1997).
[CrossRef]

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

A. Sentenac, J. J. Greffet, F. Pincemin, “Structure of the electromagnetic field in a slab of photonic crystal,” J. Opt. Soc. Am. B 14, 339–347 (1997).
[CrossRef]

G. Tayeb, 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]

1996 (4)

P. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals:  mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

S. Y. Lin, V. M. Hietala, L. Wang, E. D. Jones, “Highly dispersive photonic band-gap prism,” Opt. Lett. 21, 1771–1773 (1996).
[CrossRef] [PubMed]

J. M. Elson, 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]

1994 (8)

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

L. Li, “Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
[CrossRef]

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

1993 (2)

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993). See also the erratum in Phys. Rev. B 55, 15942 (1997).
[CrossRef]

1992 (1)

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

1991 (1)

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

1990 (1)

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

1988 (1)

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

1987 (1)

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

1979 (1)

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993). See also the erratum in Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Botten, L. C.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, “The TEM mode and homogenization of doubly periodic structures,” J. Electromagn. Waves Appl. 11, 981–1012 (1997).
[CrossRef]

Bouchitté, G.

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

Bowden, C. M.

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993). See also the erratum in Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Chan, C. T.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

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

Dowling, J. P.

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

Economou, E. N.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Elson, J. M.

J. M. Elson, 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]

Enoch, S.

S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

Fan, S.

P. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals:  mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Felbacq, D.

D. Felbacq, G. Bouchitté, “Homogenization of a set of parallel fibers,” Waves Random Media 7, 245–256 (1997).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

Greffet, J. J.

Guida, G.

Hall, R. C.

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Haus, J. W.

Z. Yuan, J. W. Haus, K. Sakoda, “Eigenmode symmetry for simple cubic lattices and the transmission spectra,” Opt. Express 3, 19–27 (1998).
[CrossRef] [PubMed]

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

Hietala, V. M.

Ho, K. M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

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

Inguva, R.

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

Joannopoulos, J.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Joannopoulos, J. D.

P. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals:  mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993). See also the erratum in Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Jones, E. D.

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Li, L.

Lin, S. Y.

Maradudin, A. A.

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

Maystre, D.

S. Enoch, G. Tayeb, D. Maystre, “Numerical evidence of ultrarefractive optics in photonic crystals,” Opt. Commun. 161, 171–176 (1999).
[CrossRef]

G. Guida, D. Maystre, G. Tayeb, P. Vincent, “Mean-field theory of two-dimensional metallic photonic crystals,” J. Opt. Soc. Am. B 15, 2308–2315 (1998).
[CrossRef]

G. Tayeb, 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]

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet; applications à la spectroscopie et au filtrage des ondes électromagnétiques,” Ph.D. thesis (Université Aix-Marseille 3, Marseille, France, 1974).

McCall, S. L.

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, “The TEM mode and homogenization of doubly periodic structures,” J. Electromagn. Waves Appl. 11, 981–1012 (1997).
[CrossRef]

Meade, R.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993). See also the erratum in Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Mittra, R.

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Mitzner, K. M.

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

Montiel, F.

Moroz, A.

A. Moroz, C. Sommers, “Photonic band gaps of three-dimensional face-centered cubic lattices,” J. Phys. Condens. Matter 11, 997–1008 (1999).
[CrossRef]

Nevière, M.

Nicorovici, N. A.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, “The TEM mode and homogenization of doubly periodic structures,” J. Electromagn. Waves Appl. 11, 981–1012 (1997).
[CrossRef]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

Pincemin, F.

Platzmann, P. M.

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

Plihal, M.

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

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993). See also the erratum in Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Sakoda, K.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

Sentenac, A.

Sigalas, M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

Smith, D. R.

D. R. Smith, S. Schultz, S. L. McCall, P. M. Platzmann, “Defect studies in a two-dimensional periodic photonic lattice,” J. Mod. Opt. 41, 395–404 (1994).
[CrossRef]

Sommers, C.

A. Moroz, C. Sommers, “Photonic band gaps of three-dimensional face-centered cubic lattices,” J. Phys. Condens. Matter 11, 997–1008 (1999).
[CrossRef]

Soukoulis, C. M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, K. M. Ho, “Photonic band gaps and defects in two dimensions: studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[CrossRef]

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

Sözüer, H. S.

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

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Tayeb, G.

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[CrossRef]

Tran, P.

J. M. Elson, 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]

Villeneuve, P.

P. Villeneuve, S. Fan, J. D. Joannopoulos, “Microcavities in photonic crystals:  mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[CrossRef]

Vincent, P.

Wang, L.

Winn, J.

J. Joannopoulos, R. Meade, J. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Yeh, P.

Yuan, Z.

Zengerle, R.

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34, 1589–1617 (1987).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. C. Hall, R. Mittra, K. M. Mitzner, “Analysis of multilayered periodic structures using generalized scattering matrix theory,” IEEE Trans. Antennas Propag. 36, 511–517 (1988).
[CrossRef]

J. Electromagn. Waves Appl. (1)

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, “The TEM mode and homogenization of doubly periodic structures,” J. Electromagn. Waves Appl. 11, 981–1012 (1997).
[CrossRef]

J. Mod. Opt. (4)

J. P. Dowling, C. M. Bowden, “Anomalous index of refraction in photonic bandgap materials,” J. Mod. Opt. 41, 345–351 (1994).
[CrossRef]

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Dispersion diagram for E polarization in a 2D crystal with square lattice. The abscissa represents the Bloch wave vector on the edge of the first reduced Brillouin zone shown in the small insert: Γ, X, and M stand for the points with coordinates (0, 0), (π/d, 0), and (π/d, π/d), respectively, in the (kx, ky) plane.

Fig. 2
Fig. 2

Crystal with finite extent with respect to the y direction, made of N gratings (N=3 in this example). The structure is z independent and infinite along the x direction. The periodicity along x is d. The distance between two grids is Δy. The x shift between two grids is Δx. Thus the two elementary translations are d=dex and Δ=Δxex-Δyey. Each grating is characterized by x-periodic electromagnetic parameters, which are not necessarily piecewise constant.

Fig. 3
Fig. 3

Single grating extracted from Fig. 2.

Fig. 4
Fig. 4

Three-dimensional dispersion diagram. The horizontal plane gives the Bloch wave vector k. The vertical axis gives the normalized frequency ωd/(2πc)=d/λ. The bottom sides of the sheets are represented in darker gray. The triangle corresponding to the first reduced Brillouin zone is drawn in the (kx, ky) plane. The parameters are the same as in Fig. 1. The diagram was computed with the plane-wave expansion method.

Fig. 5
Fig. 5

Enlarged view of the 3D dispersion diagram near the upper edge of the second gap. The intersection between the sheets and the horizontal plane corresponding to the wavelength λ=2.545 is the curve that limits the two differently shaded regions of the sheets. Owing to the slow convergence of the plane-wave expansion method, the levels of the sheets are not perfectly accurate, and this diagram gives only qualitative information.

Fig. 6
Fig. 6

Constant-frequency dispersion diagram and energy-flow direction for some particular values of kx. The dotted curves are the intersection of the sheets (Fig. 5) and the horizontal plane corresponding to λ=2.545. There is no propagating solution for 0.6<kx<1.2. This quantitative diagram is obtained by using the T matrix.

Fig. 7
Fig. 7

Map of the total field modulus for a crystal made of 697 rods lying in vacuum and illuminated by an E polarized Gaussian beam with θ0=6.4°. The crystal parameters are the same as in Fig. 1. Above the crystal the beam reflected by the crystal interferes with the incident beam and generates a system of stationary waves. Straight lines show the locus of the maximum incident (black), transmitted, and reflected (white) fields.

Fig. 8
Fig. 8

Negative refraction. Same parameters as Fig. 7, except that θ0=40°.

Fig. 9
Fig. 9

Same as Fig. 6 for two different wavelengths. Only the central region is presented.

Fig. 10
Fig. 10

Microlens illuminated in normal incidence from the top by a Gaussian beam in E polarization.

Fig. 11
Fig. 11

Microprism illuminated from the left by a Gaussian beam with λ1=2.56.

Fig. 12
Fig. 12

Scattered intensity at infinity for λ1=2.56 and λ2=2.55.

Fig. 13
Fig. 13

Exponential decay of the transmission for a photonic crystal made of N layers of perfectly conducting rods.

Equations (35)

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(r+pd+qΔ)=(r)forallintegerspandq.
uk(r)=exp(ikr)v(r),
v(r+pd+qΔ)=v(r)forallintegerspandq.
Ve=cell (Poyntingvector) drcell (energydensity) dr,
Vg=gradk(ω)=ωkx ex+ωky ey.
Ve=Vg.
u(x, y)=-+uˆ(α, y)exp(iαx)dα.
u(x, y)=02π/duα(x, y)dα,
uα(x, y)=m=--uˆα+m 2πd , yexpiα+m 2πdx
uα(x+d, y)=exp(iαd)uα(x, y).
uα(x, y)=m=-+[Am- exp(-iβmy)+Am+ exp(+iβmy)]exp(iαmx);
uα(x, y)=m=-+{Bm- exp[-iβm(y+Δy)]+Bm+ exp[+iβm(y+Δy)]}×exp[iαm(x-Δx)],
αm=α+m 2πd
αm2+βm2=ω2c2,with arg(βm){0, π/2}.
B-B+=TA-A+=T11T12T21T22 A-A+.
A+B-=S11S12S21S22 A-B+.
T11=S21-S22S12-1S11,
T12=S22S12-1,
T21=-S12-1S11,
T22=S12-1.
Aμ=Aμ-Aμ+.
Bμ=Bμ-Bμ+=TMAμ-Aμ+=TMAμ=μAμ=μAμ-Aμ+.
uα,μ(r+d)=exp(iαd)uα,μ(r),
uk(r+d)=exp(ikxd)uk(r),
kx=α.
uα,μ(r+Δ)=exp[i arg(μ)]uα,μ(r),
uk(r+Δ)=exp(ikxΔx-ikyΔy)uk(r),
ky=kxΔx-arg(μ)Δy.
α=ki sin θ.
ui(x, y)=-+A(α)exp[iαx-iβ(α)y]dα,
A(α)=W2π exp-(α-α0)2W24.
α0=ki sin θ0.
kx2+ky2=k02neff2,
f=11-n1 R=54.7,
n1 sin 45°=sin(45° - θ1)  θ1=41.5°.

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