Abstract

A numerical implementation and generalized conservation properties of a formulation for calculating wave propagation through stacked gratings comprising metallic and dielectric cylinders are presented. The basic formulation of the method was given in a companion paper [J. Opt. Soc. Am. A. 17, 2165 (2000)]. Here, details of the numerical implementation of the method are discussed and are illustrated for the ensemble average of a strongly scattering structure with refractive index and radius disorder. Also presented are a comprehensive treatment of energy conservation and generalized phase relations, as well as reciprocity.

© 2000 Optical Society of America

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  1. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
    [CrossRef]
  2. M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par une réseau infiniment conducteur,” Opt. Commun. 3, 44–52 (1971).
  3. R. C. McPhedran, D. Maystre, “On the theory and solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
    [CrossRef]
  4. L. C. Botten, J. L. Adams, R. C. McPhedran, G. H. Derrick, “Symmetry properties of lossless diffraction gratings,” J. Opt. (Paris) 11, 43–52 (1980).
    [CrossRef]
  5. L. C. Botten, R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985).
    [CrossRef]
  6. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
    [CrossRef]
  7. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  8. F. Gadot, E. Akmansoy, T. Brillat, A. de Lustrac, J.-M. Lourtioz, “Band gap engineering in metallic PBG materials at microwave frequencies using composite material and defect lattice,” presented at the International Union of Theoretical and Applied Mechanics Symposium 99/4, Mechanical and Electromagnetic Waves in Structured Media, Sydney, Australia, January 18–22, 1999.
  9. A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
    [CrossRef]
  10. M. M. Sigalas, C.-T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
    [CrossRef]
  11. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
    [CrossRef]
  12. W. T. Perrins, D. R. McKenzie, R. C. McPhedran, “Transport properties of regular arrays of cylinders,” Proc. R. Soc. London, Ser. A 369, 207–225 (1979).
    [CrossRef]
  13. V. P. Shestopalov, Smith-Purcell Effect (Nova Science, New York, 1998).
  14. R. C. McPhedran, L. C. Botten, “Phase constraints for lossy symmetric structures,” Opt. Acta 32, 595–605 (1985).
    [CrossRef]
  15. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, 22, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 1–52.
    [CrossRef]
  16. V. Twersky, “Elementary function representations of Schlömilch series,” Arch. Ration. Mech. Anal. 8, 323–332 (1961).
    [CrossRef]
  17. R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48, 928–936 (1935).
    [CrossRef]
  18. J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
    [CrossRef]
  19. J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
    [CrossRef] [PubMed]
  20. D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
    [CrossRef]
  21. U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
    [CrossRef]
  22. D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
    [CrossRef]

2000 (1)

1999 (3)

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

1997 (1)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

1996 (1)

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

1995 (1)

M. M. Sigalas, C.-T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

1994 (2)

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

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

1992 (1)

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

1985 (2)

L. C. Botten, R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985).
[CrossRef]

R. C. McPhedran, L. C. Botten, “Phase constraints for lossy symmetric structures,” Opt. Acta 32, 595–605 (1985).
[CrossRef]

1980 (1)

L. C. Botten, J. L. Adams, R. C. McPhedran, G. H. Derrick, “Symmetry properties of lossless diffraction gratings,” J. Opt. (Paris) 11, 43–52 (1980).
[CrossRef]

1979 (1)

W. T. Perrins, D. R. McKenzie, R. C. McPhedran, “Transport properties of regular arrays of cylinders,” Proc. R. Soc. London, Ser. A 369, 207–225 (1979).
[CrossRef]

1977 (1)

R. C. McPhedran, D. Maystre, “On the theory and solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

1971 (1)

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par une réseau infiniment conducteur,” Opt. Commun. 3, 44–52 (1971).

1970 (1)

J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

1961 (1)

V. Twersky, “Elementary function representations of Schlömilch series,” Arch. Ration. Mech. Anal. 8, 323–332 (1961).
[CrossRef]

1935 (1)

R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48, 928–936 (1935).
[CrossRef]

Adams, J. L.

L. C. Botten, J. L. Adams, R. C. McPhedran, G. H. Derrick, “Symmetry properties of lossless diffraction gratings,” J. Opt. (Paris) 11, 43–52 (1980).
[CrossRef]

Akmansoy, E.

F. Gadot, E. Akmansoy, T. Brillat, A. de Lustrac, J.-M. Lourtioz, “Band gap engineering in metallic PBG materials at microwave frequencies using composite material and defect lattice,” presented at the International Union of Theoretical and Applied Mechanics Symposium 99/4, Mechanical and Electromagnetic Waves in Structured Media, Sydney, Australia, January 18–22, 1999.

Asatryan, A. A.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

Benisty, H.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Botten, L. C.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

L. C. Botten, R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985).
[CrossRef]

R. C. McPhedran, L. C. Botten, “Phase constraints for lossy symmetric structures,” Opt. Acta 32, 595–605 (1985).
[CrossRef]

L. C. Botten, J. L. Adams, R. C. McPhedran, G. H. Derrick, “Symmetry properties of lossless diffraction gratings,” J. Opt. (Paris) 11, 43–52 (1980).
[CrossRef]

Bousquet, J.

J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

Brillat, T.

F. Gadot, E. Akmansoy, T. Brillat, A. de Lustrac, J.-M. Lourtioz, “Band gap engineering in metallic PBG materials at microwave frequencies using composite material and defect lattice,” presented at the International Union of Theoretical and Applied Mechanics Symposium 99/4, Mechanical and Electromagnetic Waves in Structured Media, Sydney, Australia, January 18–22, 1999.

Busch, K.

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Cadilhac, M.

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par une réseau infiniment conducteur,” Opt. Commun. 3, 44–52 (1971).

Cerutti-Maori, G.

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par une réseau infiniment conducteur,” Opt. Commun. 3, 44–52 (1971).

Chan, C.-T.

M. M. Sigalas, C.-T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

de Lustrac, A.

F. Gadot, E. Akmansoy, T. Brillat, A. de Lustrac, J.-M. Lourtioz, “Band gap engineering in metallic PBG materials at microwave frequencies using composite material and defect lattice,” presented at the International Union of Theoretical and Applied Mechanics Symposium 99/4, Mechanical and Electromagnetic Waves in Structured Media, Sydney, Australia, January 18–22, 1999.

de Sterke, C. M.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

Derrick, G. H.

L. C. Botten, J. L. Adams, R. C. McPhedran, G. H. Derrick, “Symmetry properties of lossless diffraction gratings,” J. Opt. (Paris) 11, 43–52 (1980).
[CrossRef]

Gadot, F.

F. Gadot, E. Akmansoy, T. Brillat, A. de Lustrac, J.-M. Lourtioz, “Band gap engineering in metallic PBG materials at microwave frequencies using composite material and defect lattice,” presented at the International Union of Theoretical and Applied Mechanics Symposium 99/4, Mechanical and Electromagnetic Waves in Structured Media, Sydney, Australia, January 18–22, 1999.

Grüning, U.

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Ho, K. M.

M. M. Sigalas, C.-T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Houdre, R.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Krauss, T. F.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Kroll, N.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Labilloy, D.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Lehmann, V.

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Lourtioz, J.-M.

F. Gadot, E. Akmansoy, T. Brillat, A. de Lustrac, J.-M. Lourtioz, “Band gap engineering in metallic PBG materials at microwave frequencies using composite material and defect lattice,” presented at the International Union of Theoretical and Applied Mechanics Symposium 99/4, Mechanical and Electromagnetic Waves in Structured Media, Sydney, Australia, January 18–22, 1999.

MacKinnon, A.

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

Maystre, D.

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

R. C. McPhedran, D. Maystre, “On the theory and solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

McKenzie, D. R.

W. T. Perrins, D. R. McKenzie, R. C. McPhedran, “Transport properties of regular arrays of cylinders,” Proc. R. Soc. London, Ser. A 369, 207–225 (1979).
[CrossRef]

McPhedran, R. C.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

L. C. Botten, R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985).
[CrossRef]

R. C. McPhedran, L. C. Botten, “Phase constraints for lossy symmetric structures,” Opt. Acta 32, 595–605 (1985).
[CrossRef]

L. C. Botten, J. L. Adams, R. C. McPhedran, G. H. Derrick, “Symmetry properties of lossless diffraction gratings,” J. Opt. (Paris) 11, 43–52 (1980).
[CrossRef]

W. T. Perrins, D. R. McKenzie, R. C. McPhedran, “Transport properties of regular arrays of cylinders,” Proc. R. Soc. London, Ser. A 369, 207–225 (1979).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the theory and solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

Nevière, M.

M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par une réseau infiniment conducteur,” Opt. Commun. 3, 44–52 (1971).

Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

Oesterle, U.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Ottow, S.

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Pavageau, J.

J. Pavageau, J. Bousquet, “Diffraction par un réseau conducteur nouvelle méthode de résolution,” Opt. Acta 17, 469–478 (1970).
[CrossRef]

Pendry, J. B.

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

Perrins, W. T.

W. T. Perrins, D. R. McKenzie, R. C. McPhedran, “Transport properties of regular arrays of cylinders,” Proc. R. Soc. London, Ser. A 369, 207–225 (1979).
[CrossRef]

Petit, R.

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, 22, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 1–52.
[CrossRef]

Robinson, P. A.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method,” J. Opt. Soc. Am. A 17, 2165–2176 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Shestopalov, V. P.

V. P. Shestopalov, Smith-Purcell Effect (Nova Science, New York, 1998).

Sigalas, M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Sigalas, M. M.

M. M. Sigalas, C.-T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

Smith, D. R.

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[CrossRef]

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[CrossRef]

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

Appl. Phys. (1)

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[CrossRef]

Appl. Phys. Lett. (3)

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

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[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Arch. Ration. Mech. Anal. (1)

V. Twersky, “Elementary function representations of Schlömilch series,” Arch. Ration. Mech. Anal. 8, 323–332 (1961).
[CrossRef]

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R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

J. Opt. (Paris) (1)

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[CrossRef]

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R. C. McPhedran, L. C. Botten, “Phase constraints for lossy symmetric structures,” Opt. Acta 32, 595–605 (1985).
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M. Nevière, G. Cerutti-Maori, M. Cadilhac, “Sur une nouvelle méthode de résolution du problème de la diffraction d’une onde plane par une réseau infiniment conducteur,” Opt. Commun. 3, 44–52 (1971).

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R. W. Wood, “Anomalous diffraction gratings,” Phys. Rev. 48, 928–936 (1935).
[CrossRef]

Phys. Rev. B (1)

M. M. Sigalas, C.-T. Chan, K. M. Ho, C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B 52, 11744–11751 (1995).
[CrossRef]

Phys. Rev. E (2)

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, C. M. de Sterke, “Effects of disorder on wave propagation in two-dimensional photonic crystals,” Phys. Rev. E 60, 6118–6127 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614–7617 (1999).
[CrossRef]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Experimental data (dotted curve) and our numerical results (solid curve) for a square array of 7 gratings, each composed of 14 copper rods, for E polarization. The constant of the array is 6.0 mm, and the radius of the rods is a=0.75 mm.

Fig. 2
Fig. 2

Natural logarithm of the average transmittance versus normalized wavelength λ/d for E polarization for a ten-layer stack with D=20, with refractive index and radius disorder as discussed in the text. Solid curve, total transmittance; dashed curve, transmittance in the specular order (p=0).

Fig. 3
Fig. 3

As Fig. 2, but for H polarization.

Fig. 4
Fig. 4

Normalized localization length l/d versus normalized wavelength λ/d for E polarization for the same structure as in Fig. 2. Solid curve, D=20; short-dashed curve, D=10; long-dashed curve, D=5.

Fig. 5
Fig. 5

As Fig. 4, but for H polarization.

Tables (2)

Tables Icon

Table 1 Convergence Study for Hz Polarization, with Ten Layers (NL=10) and Nc=5 Cylinders per Unit Cella

Tables Icon

Table 2 As Table 1, but for Nc=20

Equations (119)

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νl=ν+δν,ν=3.0,δν[-1.5, 1.5],
al=a+δa,a=0.3,δa[-0.1, 0.1],
lh=-2NLlog T,
U\C(V2V¯-V¯2V)dA=0=(U\C)VV¯n-V¯Vnds,
U+-U-VV¯y-V ¯Vydx
=CVV¯r-V ¯Vrds,
2iCext ImVV¯rds=2iCint ImVV¯rds
2iCext ImVV¯rds=-2ik2C Im[ε]|V|2dA,
2iCext ImVV¯rds=-2iC Im1ε|V|2dA,
·VV¯ε¯=|V|2ε¯+V·V¯ε¯
V(r)=n=-[AnJn(kr)+BnYn(kr)]exp(inθ),
2iCext ImVV¯rds=4n=-(AnBn¯-An¯Bn)=4(BHA-AHB)=4BH(M-MH)B,
2iU+-U-ImVV¯ydx
=2iDpΩr(|δp|2-|rp|2-|tp|2)+ipΩe(rp¯δp-rpδp¯),
pΩr(|rp|2+|tp|2)-4Dn=- Im(Mn)|Bn|2
=pΩr|δp|2+ipΩe(rp¯δp-rpδp¯),
pΩr(|rp|2+|tp|2)
=pΩr|δp|2+ipΩe(rp¯δp-rpδp¯),
δHZδ=0,
Z=RHIrR+THIrT-Ir-i(RHIe-IeR).
(δ1, Zδ1)+(δ2, Zδ2)-i(δ2, Zδ1)+i(δ1, Zδ2)=0.
RHIrR+THIrT=Ir+i(RHIe-IeR).
U\C(V2V¯-V¯2V)dA
=UCVV¯n-V¯Vnds.
0DVV¯n-V¯Vndx,
pΩr(rp¯tp+tp¯rp)=-ipΩe(tpδp¯-tp¯δp),
RHIrT+THIrR=i(THIe-IeT).
RIrTH+TIrRH=i(THIe-IeT).
RHIrR+THIrT=Ir+i(RHIe-IeR),
R=R,T=T,
SHIrS=Ir+i(SHIe-IeS),
S˜H I˜rS˜=I˜r+i(S˜H I˜e-I˜eS˜),
S˜=RTTR,I˜r=Ir00Ir,I˜e=Ie00Ie.
S˜rHS˜r=I˜r.
S/=±(I-2X/),
X=1Dχ-1/2E˜HbfK˜(σ˜+iM˜)-1J˜E˜χ-1/2,
XH(IriIe)+(Ir±iIe)X=2XHIr X,
Kpm=exp(-imθp)+(-1)m exp(imθp)=Jmp¯.
Kpm=imkm[(|χp|-αp)m+(-1)m(|χp|+αp)m]=Jmp¯.
K=JH,K=(Ir-Ie)JH.
σ/˜+σ/˜H=2DJ/˜E˜χr-1E˜HJ/˜H,
2XHIrX=2D2χ-1/2¯E˜HJ˜H(σ˜H-iM˜)-1J˜E˜χr-1×E˜HJ˜H(σ˜+iM˜)-1J˜E˜χ-1/2
=1Dχ-1/2¯E˜HJ˜H(σ˜H-iM˜)-1J˜E˜χ-1/2+1Dχ-1/2¯ E˜HJ˜H(σ˜+iM˜)-1J˜E˜χ-1/2,
SHIrS=(I-2X)HIr(I-2X)
=Ir±2iIeX2iXHIe
=Ir-iIeS+iSHIe,
 R=R1+T1R2(I-R1R2)-1T1,
T=T2(I-R1R2)-1T1,
R=R2+T2R1(I-R2R1)-1T2,
T=T1(I-R2R1)-1T2,
RHIrT+THIrR
=[R2H+T2H(I-R1HR2H)-1R1HT2H]IrT2(I
-R1R2)-1T1+T2H(I-R1HR2H)-1T1HIr[R1
+T1R2(I-R1R2)-1T1],
R2HIrT2(I-R1R2)-1T1
=[-T2HIrR2+iT2HIe-iIeT2]
×(I-R1R2)-1T1.
T2H(I-R1HR2H)-1T1HIrR1
 =T2H(I-R1HR2H)-1
×[-R1HIrT1+iT1HIe-iIeT1].
RHIrT+THIrR=iTHIe-iIeT,
s=δ+2iDχ-1/2E˜HJ˜HD˜,
sHIrs=δHIrδ+isHIeδ-δHIes+2iDD˜H(MH-M)B˜,
sHIrs=δHIrδ+2iDD˜HJ˜E˜χ¯-1/2Irδ
-2iDδHIrχ-1/2E˜HJ˜HD˜H
+4D2D˜HJ˜E˜χr-1E˜HJ˜HD˜
4D2D˜HJ˜E˜χr-1E˜HJ˜HD˜
=4D2D˜H(σ˜+σ˜H)D˜
=2D{D˜H(σ˜+iM˜)D˜+[(σ˜+iM˜)D˜]HD˜
+iD˜H(M˜H-M˜)D˜}.
2DD˜H(σ˜+iM˜)D˜=(δ-s)H(Ir-iIe)δ,
2D[(σ˜+iM˜)D˜]HD˜=δH(Ir+iIe)(δ-s).
sHIrs+A=δHIrδ+isHIeδ-iδHIes.
A=-4DD˜H Im(M˜)D˜.
sHIrs+A=δHIrδ+isHIeδ-iδHIes,
A=-4DD˜H Im(M˜)D˜.
rHIrr+tHIrt+A=δHIrδ+irHIeδ-δHIer,
A=12(A+A)=-4DBH Im(M)B.
B=[(B1)T,, (BNc)T]T,M=diag(Ml).
rp=rp,tp=tp.
αs=-αp-s,χs=-χp-s,
σ˜=σ˜T
sp=δp0-2Dχp-1/2χ0-1/2wpH(σ˜+iM)-1w0,
wqT=[vqT exp(iαqc1),vqT exp(iαqc2),,vqT exp(iαqcN)],
sp=δp0-2D(χp)-1/2(χ0)-1/2wpH(σ˜+iM)-1w0,
wpH(σ˜+iM)-1w0=w0T[(σ˜+iM)-1]Twp¯
=(w0T[(σ˜+iM)-1]Twp¯)T
=wpH(σ˜+iM)-1w0,
ST=I-2Dχ-1/2(J˜E˜)T(σ˜T+IM)-1(J˜E˜)¯χ-1/2.
J˜E˜=[(Je1)T,(Je2)T,, (JeN)T]T,
(J˜E˜)T=[e1JT,, eNJT]=[eN¯JH,, e1¯JH]U˜=[e1¯JH,, eN¯JH]P˜U˜=(J˜E)HP˜U˜,
U˜=diag{U|l=1,, N},
P˜=00I0I0I00.
ST=I-2Dχ-1/2E˜HJ˜H[P˜U˜(σ˜+iM)TU˜P˜]-1J˜E˜χ-1/2.
(P˜U˜σ˜TU˜P˜)ij=(U˜σ˜TU˜)N+1-i,N+1-j=U(σ˜N+1-j,N+1-i)TU=σ˜N+1-j,N+1-i=σ˜ij.
TT=[T2(I-R1R2)-1T1]T
=T1T(I-R2TR1T)-1T2T
=T1(I-R2R1)-1T2=T.
σ, ll+σ, llH=2DJχr-1JH,
(σ,ll+σ,llH)nm=[Sn-mll+(-1)n-mSn-mll¯]+(-1)m[Sn+mll+(-1)n+mSn+mll¯].
Srll+(-1)rSrll¯=2δr,0+2SrJ,reven2iSrJ,rodd.
S2sJ=-δs0+2DpΩrcos(2sθp)χp,
S2s+1J=2DpΩrsin[(2s+1)θp]χp,
Srll+(-1)rSrll¯=2DpΩr1χp[exp(irθp)+(-1)r×exp(-irθp)]=2DpΩr1χpJrp,
(σ,ll+σ,llH)nm=2DpΩr1χp[Jn-m,p+(-1)mJn+m,p]=2DpΩr1χpJnpJmp¯,
σ,ll+σ,llH=2DJχr-1JH.
σ,ij+σ,jiH=2DJEiχr-1EjHJH,
(σ,ij+σ, jiH)nm=[Sn-mij+(-1)n-mSn-mji¯]+(-1)m[Sn+mij+(-1)n+mSn+mji¯].
 Srij=Hr(1)(kc)+p=-Sr+pJp(kc)(-1)p,
Srji=(-1)rHr(1)(kc)+p=-Sr+pJp(kc),
Srij+(-1)rSrji¯
=p=-(-1)p[Sr+p+(-1)r+pSr+p¯] Jp(kc),
exp(-iαsc)=p=-(-1)pJp(kc)exp(ipθs),
Srij+(-1)rSrji¯=2DpΩr1χpexp(-iαpc)Jrp.
(σ, ij+σ, jiH)
=2DpΩr1χpexp(-iαpc)[Jn-m,p+(-1)mJn+m,p]
=2DpΩr1χpexp(-iαpc)JnpJmp¯,
σ, ij+σ, jiH=2DJEiχr-1EjHJH.
σ˜+σ˜H=2DJ˜E˜χr-1E˜HJ˜H.

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