Abstract

A brief introduction of the layer-Korringa-Kohn-Rostoker method for calculations of the frequency band structure of photonic crystals and of the transmission and reflection coefficients of light incident on slabs of such crystals is followed by two applications of the method. The first relates to the frequency band structure of metallodi-electric composites and demonstrates the essential difference between cermet and network topology of such composites at low frequencies. The second application is an analysis of recent measurements of the reflection of light from a slab of a colloidal system consisting of latex spheres in air.

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References

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  1. X. D. Wang, X-G. Zhang, Q. L. Yu, and B. N. Harmon, "Multiple-scattering theory for electro-magnetic waves," Phys. Rev. B 47, 4161-4167 (1993).
    [CrossRef]
  2. A. Moroz, "Inward and outward integral equations and the KKR method for photons," J. Phys.: Condens. Matter 6, 171-182 (1994).
    [CrossRef]
  3. A. Moroz, "Three-dimensional complete photonic-band-gap structures in the visible," Phys. Rev. Lett. 83, 5274-5277 (1999).
    [CrossRef]
  4. K. Ohtaka, "Energy band of photons and low-energy photon diffraction," Phys. Rev. B 19, 5057-5067 (1979).
    [CrossRef]
  5. K. Ohtaka and Y. Tanabe, "Photonic bands using vector spherical waves. I. Various properties of Bloch electric fields and heavy photons," J. Phys. Soc. Jpn. 65, 2265-2275 (1996).
    [CrossRef]
  6. N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys.: Condens. Matter 4, 7389-7400 (1992).
    [CrossRef]
  7. N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: Frequency bands and transmission coefficients," Comput. Phys. Commun. 113, 49-77 (1998).
    [CrossRef]
  8. N. Stefanou, V. Yannopapas, and A. Modinos, "MULTEM2: A new version of a program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000).
    [CrossRef]
  9. J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992).
    [CrossRef] [PubMed]
  10. P. M. Bell, J. B. Pendry, L. Martín-Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
    [CrossRef]
  11. Y. Qiu, K. M. Leung, L. Cavin, and D. Kralj, "Dispersion curves and transmission spectra of a two-dimensional photonic band-gap crystal-Theory and experiment," J. Appl. Phys. 77, 3631-3636 (1995).
    [CrossRef]
  12. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Ordered and disordered photonic band gap materials," Aust. J. Phys. 52, 791-809 (1999).
  13. V. Yannopapas, A. Modinos and N. Stefanou, "Optical properties of metallodielectric photonic crystals," Phys. Rev. B 60, 5359-5365 (1999).
    [CrossRef]
  14. R. C. McPhedran, N. A. Nicorovici, L. C. Botten, C. M. de Sterke, P. A. Robinson, and A. A. Asatryan, "Anomalous absorptance by stacked metallic cylinders," Opt. Commun. 168, 47-53 (1999).
    [CrossRef]
  15. A. Modinos, Field, Thermionic, and Secondary Electron Emission Spectroscopy (Plenum, New York, 1984).
  16. M. P. Pileni, "Nanosized particles made in colloidal assemblies," Langmuir 13, 3266-3276 (1997)
    [CrossRef]
  17. W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, "Robust photonic band gap from tunable scatterers," Phys. Rev. Lett. 84, 2853-2856 (2000).
    [CrossRef] [PubMed]
  18. O. D. Velev and E. W. Kaler, "Structured porous materials via colloidal crystal templating: from inorganic oxides to metals," Adv. Mater. 12, 531-534 (2000).
    [CrossRef]
  19. M. Allard, E. Sargent, E. Kumacheva, and O. Kalinina, "Characterization of internal order of colloidal crystals by optical diffraction," Opt. Quant. Elec. (to be published).

Other (19)

X. D. Wang, X-G. Zhang, Q. L. Yu, and B. N. Harmon, "Multiple-scattering theory for electro-magnetic waves," Phys. Rev. B 47, 4161-4167 (1993).
[CrossRef]

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

A. Moroz, "Three-dimensional complete photonic-band-gap structures in the visible," Phys. Rev. Lett. 83, 5274-5277 (1999).
[CrossRef]

K. Ohtaka, "Energy band of photons and low-energy photon diffraction," Phys. Rev. B 19, 5057-5067 (1979).
[CrossRef]

K. Ohtaka and Y. Tanabe, "Photonic bands using vector spherical waves. I. Various properties of Bloch electric fields and heavy photons," J. Phys. Soc. Jpn. 65, 2265-2275 (1996).
[CrossRef]

N. Stefanou, V. Karathanos, and A. Modinos, "Scattering of electromagnetic waves by periodic structures," J. Phys.: Condens. Matter 4, 7389-7400 (1992).
[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, V. Yannopapas, and A. Modinos, "MULTEM2: A new version of a program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000).
[CrossRef]

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

P. M. Bell, J. B. Pendry, L. Martín-Moreno, and A. J. Ward, "A program for calculating photonic band structures and transmission coefficients of complex structures," Comput. Phys. Commun. 85, 306-322 (1995).
[CrossRef]

Y. Qiu, K. M. Leung, L. Cavin, and D. Kralj, "Dispersion curves and transmission spectra of a two-dimensional photonic band-gap crystal-Theory and experiment," J. Appl. Phys. 77, 3631-3636 (1995).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, "Ordered and disordered photonic band gap materials," Aust. J. Phys. 52, 791-809 (1999).

V. Yannopapas, A. Modinos and N. Stefanou, "Optical properties of metallodielectric photonic crystals," Phys. Rev. B 60, 5359-5365 (1999).
[CrossRef]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, C. M. de Sterke, P. A. Robinson, and A. A. Asatryan, "Anomalous absorptance by stacked metallic cylinders," Opt. Commun. 168, 47-53 (1999).
[CrossRef]

A. Modinos, Field, Thermionic, and Secondary Electron Emission Spectroscopy (Plenum, New York, 1984).

M. P. Pileni, "Nanosized particles made in colloidal assemblies," Langmuir 13, 3266-3276 (1997)
[CrossRef]

W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam, C. T. Chan, and P. Sheng, "Robust photonic band gap from tunable scatterers," Phys. Rev. Lett. 84, 2853-2856 (2000).
[CrossRef] [PubMed]

O. D. Velev and E. W. Kaler, "Structured porous materials via colloidal crystal templating: from inorganic oxides to metals," Adv. Mater. 12, 531-534 (2000).
[CrossRef]

M. Allard, E. Sargent, E. Kumacheva, and O. Kalinina, "Characterization of internal order of colloidal crystals by optical diffraction," Opt. Quant. Elec. (to be published).

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Figures (2)

Fig. 1.
Fig. 1.

Projection of the frequency band structure of a fcc crystal of (a): nonabsorbing Drude spheres (f=0.1, ωp a0/c=1, f is the fractional volume occupied by the spheres and a 0 is the first-neighbor distance) in gelatine (ε=2.37) and (b): silicon (ε=11.9) spheres in a nonabsorbing Drude metal (p/c=0.2, ωpa 0/c=1, S is the radius of the spheres), on the SBZ of the fcc (001) surface along the symmetry lines shown in the inset.

Fig. 2.
Fig. 2.

(a) The photonic band structure normal to the (111) surface of a fcc crystal of close-packed PMMA (ε=2.2231) spheres of radius S=243.415 nm in air. The black/red lines refer to doubly degenerate/nondegenerate bands. (b) The calculated reflectance curve for light incident normally on a slab of the above crystal consisting of 32 planes of spheres parallel to the (111) surface. (c) The solid line was calculated for the same crystal as in (b) but now the dielectric constant of the PMMA spheres contains an imaginary part Imε=0.05 and the crystal consists of 4096 planes of spheres. The blue dotted curve represents the experimental data [19].

Equations (14)

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ψ k α ( r + R n ( 3 ) ) exp [ i ω a ( k ) t ] = exp ( i k · R n ( 3 ) ) ψ k α ( r ) exp [ i ω α ( k ) t ] ,
R n = n 1 a 1 + n 2 a 2 ,
g = m 1 b 1 + m 2 b 2 , m 1 , m 2 = 0 , ± 1 , ± 2 ± ,
k ( k x , k y ) within the SBZ
b 3 2 < k z b 3 2 ,
E ( r ) = g { E g + ( N ) exp [ i K g + · ( r A N ) ] + E g ( N ) exp [ i K g · ( r A N ) ] }
K g ± = ( k + g , ± [ q 2 ( k + g ) 2 ] 1 2 ) ,
E g i ( N ) = g ' i ' Q g i ; g ' i ' IV E g ' i ' ( N + 1 ) + g ' i ' Q g i ; g ' i ' III E g ' i ' + ( N )
E g i + ( N + 1 ) = g ' i ' Q g i ; g ' i ' I E g ' i ' + ( N ) + g ' i ' Q g i ; g ' i ' II E g ' i ' ( N + 1 ) ,
E g ± ( N + 1 ) = exp ( i k · a 3 ) E g ± ( N )
k = ( k , k z ( ω , k ) )
( Q I Q II [ Q IV ] 1 Q III Q I [ Q IV ] 1 [ I Q III Q II ] ) ( E + ( N ) E ( N + 1 ) ) =
exp ( i k · a 3 ) ( E + ( N ) E ( N + 1 ) )
( ω ) = 1 ω p 2 ω 2 ,

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