Abstract

We calculate the effective electric permittivity of photonic crystals of spherical particles with substitutional disorder. The effective permittivity is obtained by using a coherent-potential approximation scheme for photonic crystals in the context of multiple-scattering theory. The method is applied to the case of dielectric spheres in air as well as to inverted structures, i.e., air spheres in a dielectric medium. The results are compared against the standard Maxwell–Garnett effective-medium theory and its variations.

© 2006 Optical Society of America

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    [CrossRef]
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  6. F. Zolla and S. Guenneau, "Duality relation for the Maxwell system," Phys. Rev. E 67, 026610 (2003).
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  7. D. Felbacq and G. Bouchittè, "Negative refraction in periodic and random photonic crystals," New J. Phys. 7, 159 (2005).
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  8. P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  32. F. S. Ham and B. Segall, "Energy bands in periodic lattices—Green's function method," Phys. Rev. 124, 1786-1796 (1961).
    [CrossRef]
  33. K. Kambe, "A multiple-scattering theory of LEED intensities," Surf. Sci. 20, 213-219 (1970).
    [CrossRef]
  34. J. L. Beeby, "The diffraction of low-energy electrons by crystals," J. Phys. C 1, 82-87 (1968).
    [CrossRef]
  35. K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
    [CrossRef]
  36. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
    [CrossRef] [PubMed]
  37. A. Liebsch and B. J. Persson, "Optical properties of small metallic particles in a continuous dielectric medium," J. Phys. C 16, 5375-5391 (1983).
    [CrossRef]
  38. W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
    [CrossRef]
  39. R. Ruppin, "Evaluation of extended Maxwell-Garnett theories," Opt. Commun. 182, 273-279 (2000).
    [CrossRef]
  40. B. U. Felderhof and R. B. Jones, "Effective dielectric constant of dilute suspensions of spheres," Phys. Rev. B 39, 5669-5677 (1989).
    [CrossRef]
  41. L. G. Grechko, V. N. Pustovit, and K. W. Whites, "Dielectric function of aggregates of small metallic particles embedded in host insulating matrix," Appl. Phys. Lett. 76, 1854-1856 (2000).
    [CrossRef]

2005

D. Felbacq and G. Bouchittè, "Negative refraction in periodic and random photonic crystals," New J. Phys. 7, 159 (2005).
[CrossRef]

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

2003

J. P. Hoogenboom, A. K. van Langen-Suurling, H. Romijn, and A. van Blaaderen, "Hard-sphere crystals with hcp and non-close-packed structure grown by colloidal epitaxy," Phys. Rev. Lett. 90, 138301 (2003).
[CrossRef] [PubMed]

V. Yannopapas, N. Stefanou, and A. Modinos, "Anderson localization of light in inverted opals," Phys. Rev. B 68, 193205 (2003).
[CrossRef]

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, "Optical properties of inverted opal photonic band gap crystals with stacking disorder," Phys. Rev. E 67, 016612 (2003).
[CrossRef]

F. Zolla and S. Guenneau, "Duality relation for the Maxwell system," Phys. Rev. E 67, 026610 (2003).
[CrossRef]

2002

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

2001

V. Yannopapas, N. Stefanou, and A. Modinos, "Effect of stacking faults on the optical properties of inverted opals," Phys. Rev. Lett. 86, 4811-4814 (2001).
[CrossRef] [PubMed]

Yu. A. Vlasov, B. Xiang-Zheng, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

2000

Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Z.-Y. Li and Z.-Q. Zhang, "Fragility of photonic band gaps in inverse-opal photonic crystals," Phys. Rev. B 62, 1516-1519 (2000).
[CrossRef]

R. Biswas, M. M. Sigalas, G. Subramania, C. M. Soukoulis, and K.-M. Ho, "Photonic band gaps of porous solids," Phys. Rev. B 61, 4549-4553 (2000).
[CrossRef]

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, "Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E 62, 5711-5720 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000).
[CrossRef]

A. Modinos, V. Yannopapas, and N. Stefanou, "Scattering of electromagnetic waves by nearly periodic structures," Phys. Rev. B 61, 8099-8107 (2000).
[CrossRef]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

R. Ruppin, "Evaluation of extended Maxwell-Garnett theories," Opt. Commun. 182, 273-279 (2000).
[CrossRef]

L. G. Grechko, V. N. Pustovit, and K. W. Whites, "Dielectric function of aggregates of small metallic particles embedded in host insulating matrix," Appl. Phys. Lett. 76, 1854-1856 (2000).
[CrossRef]

1999

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

1998

B. T. Holland, C. F. Blanford, and A. Stein, "Synthesis of macroporous minerals with highly ordered three-dimensional arrays of spheroidal voids," Science 281, 538-540 (1998).
[CrossRef] [PubMed]

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: frequency bands and transmission coefficients," Comput. Phys. Commun. 113, 49-77 (1998).
[CrossRef]

1997

D. Felbacq and G. Bouchittè, "Homogenization of a set of parallel fibres," Waves Random Media 7, 245-256 (1997).
[CrossRef]

1995

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

1994

A. Moroz, "Inward and ouward integral equations and the KKR method for photons," J. Phys. Condens. Matter 6, 171-182 (1994).
[CrossRef]

1993

N. Stefanou and A. Modinos, "Scattering of electromagnetic waves by a disordered two-dimensional array of spheres," J. Phys. Condens. Matter 5, 8859-8868 (1993).
[CrossRef]

1992

N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys. Condens. Matter 4, 7389-7400 (1992).
[CrossRef]

1989

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

B. U. Felderhof and R. B. Jones, "Effective dielectric constant of dilute suspensions of spheres," Phys. Rev. B 39, 5669-5677 (1989).
[CrossRef]

1983

A. Liebsch and B. J. Persson, "Optical properties of small metallic particles in a continuous dielectric medium," J. Phys. C 16, 5375-5391 (1983).
[CrossRef]

1970

K. Kambe, "A multiple-scattering theory of LEED intensities," Surf. Sci. 20, 213-219 (1970).
[CrossRef]

1968

J. L. Beeby, "The diffraction of low-energy electrons by crystals," J. Phys. C 1, 82-87 (1968).
[CrossRef]

1967

P. Soven, "Coherent-potential model of substitutional disordered alloys," Phys. Rev. 156, 809-813 (1967).
[CrossRef]

1961

F. S. Ham and B. Segall, "Energy bands in periodic lattices—Green's function method," Phys. Rev. 124, 1786-1796 (1961).
[CrossRef]

Adawi, A. M.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

Asatryan, A. A.

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, "Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E 62, 5711-5720 (2000).
[CrossRef]

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

Astratov, V. N.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Baryshev, A. V.

Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Beeby, J. L.

J. L. Beeby, "The diffraction of low-energy electrons by crystals," J. Phys. C 1, 82-87 (1968).
[CrossRef]

Biswas, R.

R. Biswas, M. M. Sigalas, G. Subramania, C. M. Soukoulis, and K.-M. Ho, "Photonic band gaps of porous solids," Phys. Rev. B 61, 4549-4553 (2000).
[CrossRef]

Blanco, A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Blanford, C. F.

B. T. Holland, C. F. Blanford, and A. Stein, "Synthesis of macroporous minerals with highly ordered three-dimensional arrays of spheroidal voids," Science 281, 538-540 (1998).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Botten, L. C.

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, "Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E 62, 5711-5720 (2000).
[CrossRef]

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

Bouchittè, G.

D. Felbacq and G. Bouchittè, "Negative refraction in periodic and random photonic crystals," New J. Phys. 7, 159 (2005).
[CrossRef]

D. Felbacq and G. Bouchittè, "Homogenization of a set of parallel fibres," Waves Random Media 7, 245-256 (1997).
[CrossRef]

Busch, K.

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Chan, C. T.

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, "Optical properties of inverted opal photonic band gap crystals with stacking disorder," Phys. Rev. E 67, 016612 (2003).
[CrossRef]

Chen, Z.

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, "Optical properties of inverted opal photonic band gap crystals with stacking disorder," Phys. Rev. E 67, 016612 (2003).
[CrossRef]

Chomski, E.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

de Sterke, C. M.

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, "Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E 62, 5711-5720 (2000).
[CrossRef]

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

Doyle, W. T.

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

Felbacq, D.

D. Felbacq and G. Bouchittè, "Negative refraction in periodic and random photonic crystals," New J. Phys. 7, 159 (2005).
[CrossRef]

D. Felbacq and G. Bouchittè, "Homogenization of a set of parallel fibres," Waves Random Media 7, 245-256 (1997).
[CrossRef]

Felderhof, B. U.

B. U. Felderhof and R. B. Jones, "Effective dielectric constant of dilute suspensions of spheres," Phys. Rev. B 39, 5669-5677 (1989).
[CrossRef]

Fricker, S.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

Gonis, A.

A. Gonis, Green Functions for Ordered and Disordered Systems (North-Holland, 1992).

Grabtchak, S.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Grechko, L. G.

L. G. Grechko, V. N. Pustovit, and K. W. Whites, "Dielectric function of aggregates of small metallic particles embedded in host insulating matrix," Appl. Phys. Lett. 76, 1854-1856 (2000).
[CrossRef]

Guenneau, S.

F. Zolla and S. Guenneau, "Duality relation for the Maxwell system," Phys. Rev. E 67, 026610 (2003).
[CrossRef]

Guérin, C. A.

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

Györffy, B. L.

B. L. Györffy and G. M. Stocks, Electrons in Disordered Metals and at Metallic Surfaces, P.Phariseau, B.L.Györffy, and L.Scheire, eds. (Plenum, 1979), p. 89.

Ham, F. S.

F. S. Ham and B. Segall, "Energy bands in periodic lattices—Green's function method," Phys. Rev. 124, 1786-1796 (1961).
[CrossRef]

Ho, K.-M.

R. Biswas, M. M. Sigalas, G. Subramania, C. M. Soukoulis, and K.-M. Ho, "Photonic band gaps of porous solids," Phys. Rev. B 61, 4549-4553 (2000).
[CrossRef]

Holland, B. T.

B. T. Holland, C. F. Blanford, and A. Stein, "Synthesis of macroporous minerals with highly ordered three-dimensional arrays of spheroidal voids," Science 281, 538-540 (1998).
[CrossRef] [PubMed]

Hoogenboom, J. P.

J. P. Hoogenboom, A. K. van Langen-Suurling, H. Romijn, and A. van Blaaderen, "Hard-sphere crystals with hcp and non-close-packed structure grown by colloidal epitaxy," Phys. Rev. Lett. 90, 138301 (2003).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Ibisate, M.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Jikov, V. V.

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, 1994).
[CrossRef]

John, S.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Jones, R. B.

B. U. Felderhof and R. B. Jones, "Effective dielectric constant of dilute suspensions of spheres," Phys. Rev. B 39, 5669-5677 (1989).
[CrossRef]

Kambe, K.

K. Kambe, "A multiple-scattering theory of LEED intensities," Surf. Sci. 20, 213-219 (1970).
[CrossRef]

Kaplyanskii, A. A.

Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Karathanos, V.

N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys. Condens. Matter 4, 7389-7400 (1992).
[CrossRef]

Karimov, O. Z.

Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Kozlov, S. M.

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, 1994).
[CrossRef]

Leonard, S. W.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Li, Z.-Y.

Z.-Y. Li and Z.-Q. Zhang, "Fragility of photonic band gaps in inverse-opal photonic crystals," Phys. Rev. B 62, 1516-1519 (2000).
[CrossRef]

Liebsch, A.

A. Liebsch and B. J. Persson, "Optical properties of small metallic particles in a continuous dielectric medium," J. Phys. C 16, 5375-5391 (1983).
[CrossRef]

Limonov, M. F.

Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Lopez, C.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Mallet, P.

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

McPhedran, R. C.

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, "Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E 62, 5711-5720 (2000).
[CrossRef]

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

Meseguer, F.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Miguez, H.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Milton, G. W.

G. W. Milton, The Theory of Composites (Cambridge U. Press, 2002).
[CrossRef]

Ming, N. B.

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, "Optical properties of inverted opal photonic band gap crystals with stacking disorder," Phys. Rev. E 67, 016612 (2003).
[CrossRef]

Modinos, A.

V. Yannopapas, N. Stefanou, and A. Modinos, "Anderson localization of light in inverted opals," Phys. Rev. B 68, 193205 (2003).
[CrossRef]

V. Yannopapas, N. Stefanou, and A. Modinos, "Effect of stacking faults on the optical properties of inverted opals," Phys. Rev. Lett. 86, 4811-4814 (2001).
[CrossRef] [PubMed]

A. Modinos, V. Yannopapas, and N. Stefanou, "Scattering of electromagnetic waves by nearly periodic structures," Phys. Rev. B 61, 8099-8107 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: frequency bands and transmission coefficients," Comput. Phys. Commun. 113, 49-77 (1998).
[CrossRef]

N. Stefanou and A. Modinos, "Scattering of electromagnetic waves by a disordered two-dimensional array of spheres," J. Phys. Condens. Matter 5, 8859-8868 (1993).
[CrossRef]

N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys. Condens. Matter 4, 7389-7400 (1992).
[CrossRef]

Mondia, J.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Moroz, A.

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

A. Moroz, "Inward and ouward integral equations and the KKR method for photons," J. Phys. Condens. Matter 6, 171-182 (1994).
[CrossRef]

Nicorovici, N. A.

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, "Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E 62, 5711-5720 (2000).
[CrossRef]

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

Norris, D. J.

Yu. A. Vlasov, B. Xiang-Zheng, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

Oleinik, O. A.

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, 1994).
[CrossRef]

Ozin, G. A.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Persson, B. J.

A. Liebsch and B. J. Persson, "Optical properties of small metallic particles in a continuous dielectric medium," J. Phys. C 16, 5375-5391 (1983).
[CrossRef]

Pusey, P. N.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

Pustovit, V. N.

L. G. Grechko, V. N. Pustovit, and K. W. Whites, "Dielectric function of aggregates of small metallic particles embedded in host insulating matrix," Appl. Phys. Lett. 76, 1854-1856 (2000).
[CrossRef]

Robinson, P. A.

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, "Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E 62, 5711-5720 (2000).
[CrossRef]

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

Romijn, H.

J. P. Hoogenboom, A. K. van Langen-Suurling, H. Romijn, and A. van Blaaderen, "Hard-sphere crystals with hcp and non-close-packed structure grown by colloidal epitaxy," Phys. Rev. Lett. 90, 138301 (2003).
[CrossRef] [PubMed]

Ruppin, R.

R. Ruppin, "Evaluation of extended Maxwell-Garnett theories," Opt. Commun. 182, 273-279 (2000).
[CrossRef]

Segall, B.

F. S. Ham and B. Segall, "Energy bands in periodic lattices—Green's function method," Phys. Rev. 124, 1786-1796 (1961).
[CrossRef]

Sentenac, A.

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

Sheng, P.

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, "Optical properties of inverted opal photonic band gap crystals with stacking disorder," Phys. Rev. E 67, 016612 (2003).
[CrossRef]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Sigalas, M. M.

R. Biswas, M. M. Sigalas, G. Subramania, C. M. Soukoulis, and K.-M. Ho, "Photonic band gaps of porous solids," Phys. Rev. B 61, 4549-4553 (2000).
[CrossRef]

Skolnick, M. S.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

Soukoulis, C. M.

R. Biswas, M. M. Sigalas, G. Subramania, C. M. Soukoulis, and K.-M. Ho, "Photonic band gaps of porous solids," Phys. Rev. B 61, 4549-4553 (2000).
[CrossRef]

Soven, P.

P. Soven, "Coherent-potential model of substitutional disordered alloys," Phys. Rev. 156, 809-813 (1967).
[CrossRef]

Stefanou, N.

V. Yannopapas, N. Stefanou, and A. Modinos, "Anderson localization of light in inverted opals," Phys. Rev. B 68, 193205 (2003).
[CrossRef]

V. Yannopapas, N. Stefanou, and A. Modinos, "Effect of stacking faults on the optical properties of inverted opals," Phys. Rev. Lett. 86, 4811-4814 (2001).
[CrossRef] [PubMed]

A. Modinos, V. Yannopapas, and N. Stefanou, "Scattering of electromagnetic waves by nearly periodic structures," Phys. Rev. B 61, 8099-8107 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: frequency bands and transmission coefficients," Comput. Phys. Commun. 113, 49-77 (1998).
[CrossRef]

N. Stefanou and A. Modinos, "Scattering of electromagnetic waves by a disordered two-dimensional array of spheres," J. Phys. Condens. Matter 5, 8859-8868 (1993).
[CrossRef]

N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys. Condens. Matter 4, 7389-7400 (1992).
[CrossRef]

Stein, A.

B. T. Holland, C. F. Blanford, and A. Stein, "Synthesis of macroporous minerals with highly ordered three-dimensional arrays of spheroidal voids," Science 281, 538-540 (1998).
[CrossRef] [PubMed]

Stocks, G. M.

B. L. Györffy and G. M. Stocks, Electrons in Disordered Metals and at Metallic Surfaces, P.Phariseau, B.L.Györffy, and L.Scheire, eds. (Plenum, 1979), p. 89.

Sturm, J. C.

Yu. A. Vlasov, B. Xiang-Zheng, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

Subramania, G.

R. Biswas, M. M. Sigalas, G. Subramania, C. M. Soukoulis, and K.-M. Ho, "Photonic band gaps of porous solids," Phys. Rev. B 61, 4549-4553 (2000).
[CrossRef]

Toader, O.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

van Blaaderen, A.

J. P. Hoogenboom, A. K. van Langen-Suurling, H. Romijn, and A. van Blaaderen, "Hard-sphere crystals with hcp and non-close-packed structure grown by colloidal epitaxy," Phys. Rev. Lett. 90, 138301 (2003).
[CrossRef] [PubMed]

van Driel, H. M.

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

van Langen-Suurling, A. K.

J. P. Hoogenboom, A. K. van Langen-Suurling, H. Romijn, and A. van Blaaderen, "Hard-sphere crystals with hcp and non-close-packed structure grown by colloidal epitaxy," Phys. Rev. Lett. 90, 138301 (2003).
[CrossRef] [PubMed]

Vlasov, Yu. A.

Yu. A. Vlasov, B. Xiang-Zheng, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

Wang, Z. L.

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, "Optical properties of inverted opal photonic band gap crystals with stacking disorder," Phys. Rev. E 67, 016612 (2003).
[CrossRef]

Whites, K. W.

L. G. Grechko, V. N. Pustovit, and K. W. Whites, "Dielectric function of aggregates of small metallic particles embedded in host insulating matrix," Appl. Phys. Lett. 76, 1854-1856 (2000).
[CrossRef]

Whittaker, D. M.

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

Xiang-Zheng, B.

Yu. A. Vlasov, B. Xiang-Zheng, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

Yannopapas, V.

V. Yannopapas, N. Stefanou, and A. Modinos, "Anderson localization of light in inverted opals," Phys. Rev. B 68, 193205 (2003).
[CrossRef]

V. Yannopapas, N. Stefanou, and A. Modinos, "Effect of stacking faults on the optical properties of inverted opals," Phys. Rev. Lett. 86, 4811-4814 (2001).
[CrossRef] [PubMed]

A. Modinos, V. Yannopapas, and N. Stefanou, "Scattering of electromagnetic waves by nearly periodic structures," Phys. Rev. B 61, 8099-8107 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: frequency bands and transmission coefficients," Comput. Phys. Commun. 113, 49-77 (1998).
[CrossRef]

Zhang, W. Y.

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, "Optical properties of inverted opal photonic band gap crystals with stacking disorder," Phys. Rev. E 67, 016612 (2003).
[CrossRef]

Zhang, Z.-Q.

Z.-Y. Li and Z.-Q. Zhang, "Fragility of photonic band gaps in inverse-opal photonic crystals," Phys. Rev. B 62, 1516-1519 (2000).
[CrossRef]

Zolla, F.

F. Zolla and S. Guenneau, "Duality relation for the Maxwell system," Phys. Rev. E 67, 026610 (2003).
[CrossRef]

Appl. Phys. Lett.

L. G. Grechko, V. N. Pustovit, and K. W. Whites, "Dielectric function of aggregates of small metallic particles embedded in host insulating matrix," Appl. Phys. Lett. 76, 1854-1856 (2000).
[CrossRef]

Comput. Phys. Commun.

N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: frequency bands and transmission coefficients," Comput. Phys. Commun. 113, 49-77 (1998).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000).
[CrossRef]

J. Phys. C

J. L. Beeby, "The diffraction of low-energy electrons by crystals," J. Phys. C 1, 82-87 (1968).
[CrossRef]

A. Liebsch and B. J. Persson, "Optical properties of small metallic particles in a continuous dielectric medium," J. Phys. C 16, 5375-5391 (1983).
[CrossRef]

J. Phys. Condens. Matter

A. Moroz, "Inward and ouward integral equations and the KKR method for photons," J. Phys. Condens. Matter 6, 171-182 (1994).
[CrossRef]

N. Stefanou and A. Modinos, "Scattering of electromagnetic waves by a disordered two-dimensional array of spheres," J. Phys. Condens. Matter 5, 8859-8868 (1993).
[CrossRef]

N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys. Condens. Matter 4, 7389-7400 (1992).
[CrossRef]

Nature

Yu. A. Vlasov, B. Xiang-Zheng, J. C. Sturm, and D. J. Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John, S. W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. Mondia, G. A. Ozin, O. Toader, and H. M. van Driel, "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres," Nature 405, 437-440 (2000).
[CrossRef] [PubMed]

New J. Phys.

D. Felbacq and G. Bouchittè, "Negative refraction in periodic and random photonic crystals," New J. Phys. 7, 159 (2005).
[CrossRef]

Opt. Commun.

R. Ruppin, "Evaluation of extended Maxwell-Garnett theories," Opt. Commun. 182, 273-279 (2000).
[CrossRef]

Phys. Rev.

P. Soven, "Coherent-potential model of substitutional disordered alloys," Phys. Rev. 156, 809-813 (1967).
[CrossRef]

F. S. Ham and B. Segall, "Energy bands in periodic lattices—Green's function method," Phys. Rev. 124, 1786-1796 (1961).
[CrossRef]

Phys. Rev. B

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

W. T. Doyle, "Optical properties of a suspension of metal spheres," Phys. Rev. B 39, 9852-9858 (1989).
[CrossRef]

A. Modinos, V. Yannopapas, and N. Stefanou, "Scattering of electromagnetic waves by nearly periodic structures," Phys. Rev. B 61, 8099-8107 (2000).
[CrossRef]

P. Mallet, C. A. Guérin, and A. Sentenac, "Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy," Phys. Rev. B 72, 014205 (2005).
[CrossRef]

V. Yannopapas, N. Stefanou, and A. Modinos, "Anderson localization of light in inverted opals," Phys. Rev. B 68, 193205 (2003).
[CrossRef]

Z.-Y. Li and Z.-Q. Zhang, "Fragility of photonic band gaps in inverse-opal photonic crystals," Phys. Rev. B 62, 1516-1519 (2000).
[CrossRef]

R. Biswas, M. M. Sigalas, G. Subramania, C. M. Soukoulis, and K.-M. Ho, "Photonic band gaps of porous solids," Phys. Rev. B 61, 4549-4553 (2000).
[CrossRef]

V. N. Astratov, A. M. Adawi, S. Fricker, M. S. Skolnick, D. M. Whittaker, and P. N. Pusey, "Interplay of order and disorder in the optical properties of opal photonic crystals," Phys. Rev. B 66, 165215 (2002).
[CrossRef]

B. U. Felderhof and R. B. Jones, "Effective dielectric constant of dilute suspensions of spheres," Phys. Rev. B 39, 5669-5677 (1989).
[CrossRef]

Phys. Rev. E

Z. L. Wang, C. T. Chan, W. Y. Zhang, Z. Chen, N. B. Ming, and P. Sheng, "Optical properties of inverted opal photonic band gap crystals with stacking disorder," Phys. Rev. E 67, 016612 (2003).
[CrossRef]

Yu. A. Vlasov, V. N. Astratov, A. V. Baryshev, A. A. Kaplyanskii, O. Z. Karimov, and M. F. Limonov, "Manifestation of intrinsic defects in optical properties of self-organized opal photonic crystals," Phys. Rev. E 61, 5784-5793 (2000).
[CrossRef]

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

A. A. Asatryan, P. A. Robinson, L. C. Botten, R. C. McPhedran, N. A. Nicorovici, and C. M. de Sterke, "Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals," Phys. Rev. E 62, 5711-5720 (2000).
[CrossRef]

F. Zolla and S. Guenneau, "Duality relation for the Maxwell system," Phys. Rev. E 67, 026610 (2003).
[CrossRef]

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Phys. Rev. Lett.

J. P. Hoogenboom, A. K. van Langen-Suurling, H. Romijn, and A. van Blaaderen, "Hard-sphere crystals with hcp and non-close-packed structure grown by colloidal epitaxy," Phys. Rev. Lett. 90, 138301 (2003).
[CrossRef] [PubMed]

V. Yannopapas, N. Stefanou, and A. Modinos, "Effect of stacking faults on the optical properties of inverted opals," Phys. Rev. Lett. 86, 4811-4814 (2001).
[CrossRef] [PubMed]

Science

B. T. Holland, C. F. Blanford, and A. Stein, "Synthesis of macroporous minerals with highly ordered three-dimensional arrays of spheroidal voids," Science 281, 538-540 (1998).
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Surf. Sci.

K. Kambe, "A multiple-scattering theory of LEED intensities," Surf. Sci. 20, 213-219 (1970).
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Waves Random Media

D. Felbacq and G. Bouchittè, "Homogenization of a set of parallel fibres," Waves Random Media 7, 245-256 (1997).
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Other

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, 1994).
[CrossRef]

G. W. Milton, The Theory of Composites (Cambridge U. Press, 2002).
[CrossRef]

B. L. Györffy and G. M. Stocks, Electrons in Disordered Metals and at Metallic Surfaces, P.Phariseau, B.L.Györffy, and L.Scheire, eds. (Plenum, 1979), p. 89.

A. Gonis, Green Functions for Ordered and Disordered Systems (North-Holland, 1992).

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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Figures (4)

Fig. 1
Fig. 1

Dispersion lines for an fcc photonic crystal of silicon spheres in air with volume filling fraction f = 0.55 . The dashed line, a periodic fcc crystal; the dotted line, a disordered crystal with 75% occupied lattice sites; and the solid line, the MG result [Eq. (8)]. α 0 is the first-neighbor distance of the fcc crystal. PC, photonic crystal.

Fig. 2
Fig. 2

Effective dielectric constant ϵ eff as a function of the volume filling fraction f for an fcc photonic crystal of silicon spheres in air. Circles, a periodic fcc crystal; filled triangles, a disordered crystal with 75% occupied lattice sites; squares, the MG result [Eq. (8)]; and open triangles, the cluster expansion result [Eq. (9)].

Fig. 3
Fig. 3

Same as in Fig. 2 but for an fcc photonic crystal of air spheres in silicon.

Fig. 4
Fig. 4

Effective permittivity ϵ eff as a function of the permittivity of the spheres ϵ s for an fcc photonic crystal with volume filling fraction f = 0.55 . The spheres are surrounded by air (cermet topology). Circles, a periodic fcc crystal; filled triangles, a disordered crystal with 75% occupied lattice sites; squares, the MG result [Eq. (8)]; and open triangles, the CE result [Eq. (9)].

Equations (9)

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E ( r , t ) = Re [ E ( r ) exp ( i ω t ) ] .
E ( r ) = l = 1 m = l l { a l m H f l ( q r ) X l m ( r ̂ ) + a l m E i q × [ f l ( q r ) X l m ( r ̂ ) ] } ,
a L + = L T L L a L 0 ,
j = 1 N C j P L L 1 ( T L L j T L L c ) = 0 ,
P L L = δ L L L ( T L L j T L L c ) D L L c 00 .
j = 1 N C j = 1 .
T L L c = C A T L L A + C B T L L B L , L Δ T L L A D L L c 00 Δ T L L B ,
ϵ eff = ( 1 + 2 f ) ϵ s + 2 ( 1 f ) ϵ h ( 1 f ) ϵ s + ( 2 + f ) ϵ h ϵ h ,
ϵ eff = ϵ h ( 1 + 3 f B 1 f B 2 3 f B ln 8 + B 8 2 B ) ,

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