Abstract

The dispersion relation of Bloch waves is derived from the properties of a single grating layer. A straightforward way to impose the Bloch condition leads to the calculation of the eigenvalues of the transfer matrix through the single grating layer. Unfortunately, the transfer-matrix algorithm is known to be unstable as a result of the growing evanescent waves. This problem appears again in the calculation of the eigenvalues, making unusable the transfer matrix in numerous practical problems. We propose two different algorithms to circumvent this problem. The first one takes advantage of scattering matrices, while the second one takes advantage of impedance matrices. Numerical evidence of the efficiency of the algorithms is given. Dispersion diagrams of simple cubic and woodpile photonic crystals are obtained by using, respectively, the scattering and impedance matrices.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  4. A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
    [CrossRef]
  5. A. Moroz, C. Sommers, “Photonic band gaps of three-dimensional face-centered cubic lattices,” J. Phys. Condens. Matter 11, 997–1008 (1999).
    [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. B. Gralak, S. Enoch, G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).
    [CrossRef]
  8. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  28. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
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2001 (4)

2000 (3)

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]

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]

J. G. Fleming, S.-Y. Lin, “Three-dimensional photonic crystal with a stop band from 1.35 to 1.95 µm,” Opt. Lett. 24, 49–51 (1999).
[CrossRef]

1998 (1)

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

1997 (1)

1996 (2)

1995 (1)

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
[CrossRef]

1994 (3)

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

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

1993 (2)

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

1992 (1)

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

1990 (1)

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

1985 (1)

M. Nevière, R. Reinisch, D. Maystre, “Surface enhanced second harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[CrossRef]

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

1979 (1)

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

1963 (1)

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. Theor. Instrum. 6, 263–281 (1963).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Asatryan, A. A.

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

Biswas, R.

S.-Y. Lin, J. G. Fleming, R. Lin, M. M. Sigalas, R. Biswas, K. M. Ho, “Complete three-dimensional photonic bandgap in a simple cubic structure,” J. Opt. Soc. Am. B 18, 32–35 (2001).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Botten, L. C.

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Bur, J.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Chan, C.

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

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

de Sterke, C. M.

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

Enoch, S.

Fleming, J. G.

S.-Y. Lin, J. G. Fleming, R. Lin, M. M. Sigalas, R. Biswas, K. M. Ho, “Complete three-dimensional photonic bandgap in a simple cubic structure,” J. Opt. Soc. Am. B 18, 32–35 (2001).
[CrossRef]

J. G. Fleming, S.-Y. Lin, “Three-dimensional photonic crystal with a stop band from 1.35 to 1.95 µm,” Opt. Lett. 24, 49–51 (1999).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Gralak, B.

Haus, J.

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

Hetherington, D. L.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Ho, K.

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

Ho, K. M.

S.-Y. Lin, J. G. Fleming, R. Lin, M. M. Sigalas, R. Biswas, K. M. Ho, “Complete three-dimensional photonic bandgap in a simple cubic structure,” J. Opt. Soc. Am. B 18, 32–35 (2001).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Inguva, R.

H. Sözüer, J. 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).

Kurtz, S. R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Li, L.

Lin, R.

Lin, S. Y.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Lin, S.-Y.

Maystre, D.

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

M. Nevière, R. Reinisch, D. Maystre, “Surface enhanced second harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[CrossRef]

D. Maystre, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

McPhedran, R. C.

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

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Meade, R.

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

Modinos, A.

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]

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
[CrossRef]

Nevière, M.

Nicorovici, N. A.

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

Pendry, J. B.

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

Petit, R.

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. Theor. Instrum. 6, 263–281 (1963).

Popov, E.

Reinisch, R.

M. Nevière, R. Reinisch, D. Maystre, “Surface enhanced second harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[CrossRef]

Robinson, P. A.

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]

Sandström, S.-E.

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

Sigalas, M. M.

S.-Y. Lin, J. G. Fleming, R. Lin, M. M. Sigalas, R. Biswas, K. M. Ho, “Complete three-dimensional photonic bandgap in a simple cubic structure,” J. Opt. Soc. Am. B 18, 32–35 (2001).
[CrossRef]

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Smith, B. K.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[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.

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

Sözüer, H.

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

Stefanou, N.

Tayeb, G.

B. Gralak, S. Enoch, G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).
[CrossRef]

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

Vincent, P.

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

Whittaker, D. M.

Winn, J.

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

Yannopapas, V.

Zubrzycki, W.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Appl. Phys. (1)

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

J. Electromagn. Waves Appl. (1)

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

J. Mod. Opt. (2)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

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

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

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

J. Phys. Condens. Matter (1)

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

Nature (London) (1)

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature (London) 394, 351–353 (1998).
[CrossRef]

Opt. Acta (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (3)

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

M. Nevière, R. Reinisch, D. Maystre, “Surface enhanced second harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[CrossRef]

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
[CrossRef]

Phys. Rev. E (2)

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]

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

Phys. Rev. Lett. (1)

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

Pure Appl. Opt. (1)

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

Rev. Opt. Theor. Instrum. (1)

R. Petit, “Contribution à l’étude de la diffraction d’une onde plane par un réseau métallique,” Rev. Opt. Theor. Instrum. 6, 263–281 (1963).

Other (3)

D. Maystre, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

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Figures (10)

Fig. 1
Fig. 1

(a) 3D photonic crystal with spatial periods d 1 , d 2 , and d 3 , (b) grating layer extracted from this 3D crystal with spatial periods d 1 and d 2 .

Fig. 2
Fig. 2

Value of the electromagnetic field at the planes x 3 = 0 and x 3 = 1 delimiting the grating layer.

Fig. 3
Fig. 3

Definition of the S matrix.

Fig. 4
Fig. 4

Unit cell V * of the dual lattice and the reduced unit cell V r * (hatched volume).

Fig. 5
Fig. 5

Projected reduced unit cell of the dual lattice (hatched triangle) and its boundary Γ XM Γ .

Fig. 6
Fig. 6

Schematic representation of the face-centered-cubic woodpile photonic crystal. The horizontal spatial periods d 1 , 1 and d 2 , 2 are d 1 , 1 = d 2 , 2 = a / 2 . The shifts δ 1 and δ 2 are equal to a half-period: δ 1 = δ 2 = a / ( 2 2 ) . The silicón rods are of width w 1 = w 2 = 0.28 d 1 , 1 = 0.28 d 2 , 2 and of height d 3 , 3 / 2 = a / 4 .

Fig. 7
Fig. 7

Representation of the dispersion relation of the face-centered-cubic woodpile photonic crystal. Abscissa, projection of the Bloch vector k onto Γ XM Γ ; ordinate, normalized frequency.

Fig. 8
Fig. 8

Unit cell of the simple cubic photonic crystal.

Fig. 9
Fig. 9

Representation of the dispersion relation of the simple cubic photonic crystal. Abscissa, projection of the Bloch vector k onto Γ XM Γ ; ordinate, normalized frequency.

Fig. 10
Fig. 10

Transmission through six identical grating layers for normal incidence (Γ point). The dashed lines [ ω a / ( 2 π c ) = 0.273 and 0.426] are the limits of the gap corresponding to the infinite crystal in normal incidence, deduced from Fig. 9.

Tables (2)

Tables Icon

Table 1 Upper and Lower Band Edges for Different Values of the Number of Functions Retained for the Computation

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Table 2 Upper Band Edge at the Γ Point (Normal Incidence) for Different Values of the Number of Functions Retained for the Computation

Equations (33)

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V = { x = x 1 d 1 + x 2 d 2 + x 3 d 3 3 | x 1 ,   x 2 ,   x 3 [ 0 , 1 ] } ,
L = { γ = γ 1 d 1 + γ 2 d 2 + γ 3 d 3 3 | γ 1 ,   γ 2 ,   γ 3 } .
d 1 * = ( 2 π / | V | ) d 2 × d 3 ,
d 2 * = ( 2 π / | V | ) d 3 × d 1 ,
d 3 * = ( 2 π / | V | ) d 1 × d 2 .
x 3 , γ L : ϵ ( x + γ ) = ϵ ( x ) .
F ( x + n 1 d 1 + n 2 d 2 ) = exp [ i k · ( n 1 d 1 + n 2 d 2 ) ] F ( x ) ,
k V * = { k 1 d 1 * + k 2 d 2 * + k 3 d 3 * 3 | k 1 ,   k 2 ,   k 3 [ - 1 / 2 ,   1 / 2 ] }
F u ( x 1 ,   x 2 ) = F ( x 1 ,   x 2 ,   1 ) ,
F d ( x 1 ,   x 2 ) = F ( x 1 ,   x 2 ,   0 ) ,
F σ ( x 1 ,   x 2 ) = n 1 , n 2 F n 1 , n 2 σ   exp [ i ( k 1 + n 1 ) x 1 ] × exp [ i ( k 2 + n 2 ) x 2 ] ,
F n 1 , n 2 σ = F 1 , n 1 , n 2 σ d 1 + F 2 , n 1 , n 2 σ d 2 + F 3 , n 1 , n 2 σ d 3 3 .
F σ = F 1 , n 1 , n 2 σ F 2 , n 1 , n 2 σ n 1 , n 2
E u H u = T ( ω ,   k 1 ,   k 2 ) E d H d ,
E u H u = exp ( i k · d 3 ) E d H d .
det [ T ( ω ,   k 1 ,   k 2 ) - I   exp ( i k · d 3 ) ] = 0 ,
det { [ T ( ω ,   k 1 ,   k 2 ) ± I ] - 1 - I [ exp ( i k · d 3 ) ± 1 ] - 1 } = 0 .
E u E d = R ( ω ,   k 1 ,   k 2 ) H u H d = R 11 R 12 R 21 R 22 H u H d ,
( T ± I ) - 1 = ( R 22 R 21 ) X ( R 22 R 21 ) XY ( R 22 R 21 ) - 1 R 22 X - X ( R 11 ± R 21 ) ,
X = [ ( R 12 - R 21 ) ± ( R 22 - R 11 ) ] - 1 ,
Y = R 12 R 22 - 1 R 21 - R 11 .
E ( x ) = n 1 , n 2 [ E n 1 , n 2 u , +   exp ( i k n 1 , n 2 + · x ) + E n 1 , n 2 u , -   exp ( i k n 1 , n 2 - · x ) ]
E ( x ) = n 1 , n 2 [ E n 1 , n 2 d , +   exp ( i k n 1 , n 2 + · x ) + E n 1 , n 2 d , -   exp ( i k n 1 , n 2 - · x ) ]
E n 1 , n 2 σ , ± = E 1 , n 1 , n 2 σ , ± d 1 + E 2 , n 1 , n 2 σ , ± d 2 + E 3 , n 1 , n 2 σ , ± d 3 3 ,
k n 1 , n 2 ± = ( k 1 + n 1 ) d 1 * + ( k 2 + n 2 ) d 2 * ± k 3 , n 1 , n 2 d 3 * ,
| k n 1 , n 2 ± | = ω / c , arg ( k 3 , n 1 , n 2 ) { 0 ,   π / 2 } .
E σ , ± = E 1 , n 1 , n 2 σ , ± E 2 , n 1 , n 2 σ , ± n 1 , n 2 .
E u , + E d , - = S ( ω ,   k 1 ,   k 2 ) E u , - E d , + = S 11 S 12 S 21 S 22 E u , - E d , + .
E u , + E u , - = T ˜ ( ω ,   k 1 ,   k 2 ) E d , + E d , - .
( T ˜ - 1 ± I ) - 1 = S 11 - 1 ( I 11 ± S 12 ) X ˜ - S 11 - 1 ( I 11 ± S 12 ) X ˜ S 22 ( I 11 ± S 12 ) - 1 X ˜ X ˜ ( S 21 S 11 - 1 S 12 - S 22 ± S 11 - 1 S 12 ) ,
X ˜ = ( S 21 S 11 - 1 ± S 11 - 1 ± S 21 S 11 - 1 S 12 S 22 + S 11 - 1 S 12 ) - 1 .
V r * = { k = k 1 d 1 * + k 2 d 2 * + k 3 d 3 * 3 | k 1 [ 0 ,   1 / 2 ] , k 2 [ 0 ,   k 1 ] , k 3 [ - 1 / 2 ,   1 / 2 ] } .
ω a / ( 2 π c ) [ 0.468 ± 0.001 , 0.569 ± 0.001 ] .

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