Abstract

The band structure for an absorptive two-dimensional photonic crystal made from cylinders consisting of a Drude material is calculated. Absorption causes the spectrum to become complex and form islands in the negative complex half-plane. The boundaries of these islands are not always formed by the eigenvalues calculated for Bloch vectors on the characteristic path, and we find a hole in the spectrum. For realistic parameter values, the real part of the spectrum is hardly influenced by absorption, typically less than 0.25%. The employed method uses a Korringa–Kohn–Rostoker procedure together with analytical continuation. This results in an efficient approach that allows these band-structure calculations to be done on a Pentium III personal computer.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. P. Bykov, “Spontaneous emission in a periodic structure,” Sov. Phys. JETP 35, 269–273 (1972).
  2. V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861–871 (1975).
    [CrossRef]
  3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  4. C. M. Soukoulis, ed., Photonic Crystals and Localization in the 21st Century, NATO ASI Ser. Ser. C 563 (2001).
  5. A. Moroz, “Three-dimensional complete photonic-band-gap structures in the visible,” Phys. Rev. Lett. 83, 5274–5277 (1999).
    [CrossRef]
  6. A. Moroz, “Photonic crystals of coated metallic spheres,” Europhys. Lett. 50, 466–472 (2000).
    [CrossRef]
  7. A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66, 115109 (2002).
    [CrossRef]
  8. H. van der Lem and A. Moroz, “Towards two-dimensional complete photonic bandgap structures below infrared wavelengths,” J. Opt. A: Pure Appl. Opt. 2, 395–399 (2000).
    [CrossRef]
  9. A. Tip, “Linear absorptive dielectrics,” Phys. Rev. A 57, 4818–4841 (1998).
    [CrossRef]
  10. A. Tip, A. Moroz, and J. M. Combes, “Band structure for absorptive photonic crystals,” J. Phys. A 33, 6223–6252 (2000).
    [CrossRef]
  11. A. M. Ozorio de Almeida, “Real-space methods for interpreting electron micrographs in cross-grating orientations. I. exact wave formulation,” Acta Crystallogr. Sect. A 31, 435–442 (1975).
    [CrossRef]
  12. J. S. Faulkner, “Multiple-scattering calculations in two dimensions,” Phys. Rev. B 38, 1686–1694 (1988).
    [CrossRef]
  13. A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B 51, 2068–2081 (1995).
    [CrossRef]
  14. V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
    [CrossRef]
  15. B. Gralak, FOM-Instituut voor Atoom en Molecuulfysica, Amsterdam, The Netherlands (personal communication, 2002).
  16. J. Ziman, “The T matrix, the K matrix, d bands and l-dependent pseudo-potentials in the theory of metals,” Proc. Phys. Soc. London 86, 337–353 (1965).
    [CrossRef]
  17. A. Moroz and A. Tip, “Resonance-induced effects in photonic crystals,” J. Phys. Condens. Matter 11, 2503–2512 (1999).
    [CrossRef]
  18. W. H. Zachariasen, Theory of X-ray Diffraction in Crystals (Dover, New York, 1945).
  19. P. S. J. Russel, “Photonic band gaps,” Phys. World 37, 37–41 (1992).
  20. K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
    [CrossRef]
  21. J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica (Utrecht) 13, 392–400 (1947).
    [CrossRef]
  22. W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
    [CrossRef]

2002 (1)

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66, 115109 (2002).
[CrossRef]

2001 (1)

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

2000 (3)

A. Moroz, “Photonic crystals of coated metallic spheres,” Europhys. Lett. 50, 466–472 (2000).
[CrossRef]

H. van der Lem and A. Moroz, “Towards two-dimensional complete photonic bandgap structures below infrared wavelengths,” J. Opt. A: Pure Appl. Opt. 2, 395–399 (2000).
[CrossRef]

A. Tip, A. Moroz, and J. M. Combes, “Band structure for absorptive photonic crystals,” J. Phys. A 33, 6223–6252 (2000).
[CrossRef]

1999 (2)

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

A. Moroz and A. Tip, “Resonance-induced effects in photonic crystals,” J. Phys. Condens. Matter 11, 2503–2512 (1999).
[CrossRef]

1998 (1)

A. Tip, “Linear absorptive dielectrics,” Phys. Rev. A 57, 4818–4841 (1998).
[CrossRef]

1997 (1)

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[CrossRef]

1995 (1)

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

1992 (1)

P. S. J. Russel, “Photonic band gaps,” Phys. World 37, 37–41 (1992).

1988 (1)

J. S. Faulkner, “Multiple-scattering calculations in two dimensions,” Phys. Rev. B 38, 1686–1694 (1988).
[CrossRef]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1975 (2)

V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861–871 (1975).
[CrossRef]

A. M. Ozorio de Almeida, “Real-space methods for interpreting electron micrographs in cross-grating orientations. I. exact wave formulation,” Acta Crystallogr. Sect. A 31, 435–442 (1975).
[CrossRef]

1972 (1)

V. P. Bykov, “Spontaneous emission in a periodic structure,” Sov. Phys. JETP 35, 269–273 (1972).

1965 (1)

J. Ziman, “The T matrix, the K matrix, d bands and l-dependent pseudo-potentials in the theory of metals,” Proc. Phys. Soc. London 86, 337–353 (1965).
[CrossRef]

1954 (1)

W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
[CrossRef]

1947 (1)

J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica (Utrecht) 13, 392–400 (1947).
[CrossRef]

Bykov, V. P.

V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861–871 (1975).
[CrossRef]

V. P. Bykov, “Spontaneous emission in a periodic structure,” Sov. Phys. JETP 35, 269–273 (1972).

Chutinan, A.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Combes, J. M.

A. Tip, A. Moroz, and J. M. Combes, “Band structure for absorptive photonic crystals,” J. Phys. A 33, 6223–6252 (2000).
[CrossRef]

de Almeida, A. M. Ozorio

A. M. Ozorio de Almeida, “Real-space methods for interpreting electron micrographs in cross-grating orientations. I. exact wave formulation,” Acta Crystallogr. Sect. A 31, 435–442 (1975).
[CrossRef]

Faulkner, J. S.

J. S. Faulkner, “Multiple-scattering calculations in two dimensions,” Phys. Rev. B 38, 1686–1694 (1988).
[CrossRef]

Hirao, K.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Ito, T.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Kawai, N.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Kohn, W.

W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
[CrossRef]

Korringa, J.

J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica (Utrecht) 13, 392–400 (1947).
[CrossRef]

Kuzmiak, V.

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[CrossRef]

Maradudin, A. A.

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[CrossRef]

Mitsuyu, T.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Moroz, A.

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66, 115109 (2002).
[CrossRef]

H. van der Lem and A. Moroz, “Towards two-dimensional complete photonic bandgap structures below infrared wavelengths,” J. Opt. A: Pure Appl. Opt. 2, 395–399 (2000).
[CrossRef]

A. Moroz, “Photonic crystals of coated metallic spheres,” Europhys. Lett. 50, 466–472 (2000).
[CrossRef]

A. Tip, A. Moroz, and J. M. Combes, “Band structure for absorptive photonic crystals,” J. Phys. A 33, 6223–6252 (2000).
[CrossRef]

A. Moroz and A. Tip, “Resonance-induced effects in photonic crystals,” J. Phys. Condens. Matter 11, 2503–2512 (1999).
[CrossRef]

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

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

Noda, S.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Rostoker, N.

W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
[CrossRef]

Russel, P. S. J.

P. S. J. Russel, “Photonic band gaps,” Phys. World 37, 37–41 (1992).

Sakoda, K.

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Tip, A.

A. Tip, A. Moroz, and J. M. Combes, “Band structure for absorptive photonic crystals,” J. Phys. A 33, 6223–6252 (2000).
[CrossRef]

A. Moroz and A. Tip, “Resonance-induced effects in photonic crystals,” J. Phys. Condens. Matter 11, 2503–2512 (1999).
[CrossRef]

A. Tip, “Linear absorptive dielectrics,” Phys. Rev. A 57, 4818–4841 (1998).
[CrossRef]

van der Lem, H.

H. van der Lem and A. Moroz, “Towards two-dimensional complete photonic bandgap structures below infrared wavelengths,” J. Opt. A: Pure Appl. Opt. 2, 395–399 (2000).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Ziman, J.

J. Ziman, “The T matrix, the K matrix, d bands and l-dependent pseudo-potentials in the theory of metals,” Proc. Phys. Soc. London 86, 337–353 (1965).
[CrossRef]

Acta Crystallogr. Sect. A (1)

A. M. Ozorio de Almeida, “Real-space methods for interpreting electron micrographs in cross-grating orientations. I. exact wave formulation,” Acta Crystallogr. Sect. A 31, 435–442 (1975).
[CrossRef]

Europhys. Lett. (1)

A. Moroz, “Photonic crystals of coated metallic spheres,” Europhys. Lett. 50, 466–472 (2000).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

H. van der Lem and A. Moroz, “Towards two-dimensional complete photonic bandgap structures below infrared wavelengths,” J. Opt. A: Pure Appl. Opt. 2, 395–399 (2000).
[CrossRef]

J. Phys. A (1)

A. Tip, A. Moroz, and J. M. Combes, “Band structure for absorptive photonic crystals,” J. Phys. A 33, 6223–6252 (2000).
[CrossRef]

J. Phys. Condens. Matter (1)

A. Moroz and A. Tip, “Resonance-induced effects in photonic crystals,” J. Phys. Condens. Matter 11, 2503–2512 (1999).
[CrossRef]

Phys. Rev. (1)

W. Kohn and N. Rostoker, “Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium,” Phys. Rev. 94, 1111–1120 (1954).
[CrossRef]

Phys. Rev. A (1)

A. Tip, “Linear absorptive dielectrics,” Phys. Rev. A 57, 4818–4841 (1998).
[CrossRef]

Phys. Rev. B (5)

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66, 115109 (2002).
[CrossRef]

J. S. Faulkner, “Multiple-scattering calculations in two dimensions,” Phys. Rev. B 38, 1686–1694 (1988).
[CrossRef]

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

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[CrossRef]

K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

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

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Phys. World (1)

P. S. J. Russel, “Photonic band gaps,” Phys. World 37, 37–41 (1992).

Physica (Utrecht) (1)

J. Korringa, “On the calculation of the energy of a Bloch wave in a metal,” Physica (Utrecht) 13, 392–400 (1947).
[CrossRef]

Proc. Phys. Soc. London (1)

J. Ziman, “The T matrix, the K matrix, d bands and l-dependent pseudo-potentials in the theory of metals,” Proc. Phys. Soc. London 86, 337–353 (1965).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861–871 (1975).
[CrossRef]

Sov. Phys. JETP (1)

V. P. Bykov, “Spontaneous emission in a periodic structure,” Sov. Phys. JETP 35, 269–273 (1972).

Other (3)

C. M. Soukoulis, ed., Photonic Crystals and Localization in the 21st Century, NATO ASI Ser. Ser. C 563 (2001).

W. H. Zachariasen, Theory of X-ray Diffraction in Crystals (Dover, New York, 1945).

B. Gralak, FOM-Instituut voor Atoom en Molecuulfysica, Amsterdam, The Netherlands (personal communication, 2002).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Complex eigenvalues and the real part of the band structure calculated on the characteristic path for the E polarization. The lower graph shows the eigenvalues in the complex plane. The upper graph shows the real part of the band structure.

Fig. 2
Fig. 2

Complex eigenvalues and the real part of the band structure calculated on the characteristic path for the H polarization. The lower graph shows the eigenvalues in the complex plane. The upper graph shows the real part of the band structure.

Fig. 3
Fig. 3

Detail of Fig. 1 showing the eigenvalues in the complex plane. The black dots correspond to eigenvalues calculated for a Bloch vector on the characteristic path. The gray area represents the eigenvalues calculated for Bloch vectors that are not on the characteristic path. Note the small white area inside the gray, where no spectrum is present.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

tD(x, t)=x×B(x, t)-J(x, t),
tB(x, t)=-x×E(x, t),
xD(x, t)=ρ(x, t),xB(x, t0)=0,
D(x, t)=ε0E(x, t)+P(x, t),
P(x, t)=-tdsχ(x, t-s)E(x, s),
u(x, ω, t)=exp[-iωt]u(x, ω)
[ω2ε(x, ω)-H0]u(x, ω)=0,H0=-x2U+xx,
ε(x, ω)=1+χˆ(x, ω),
χˆ(x, ω)=0dt exp[iωt]χ(x, t).
ε(ω)=1+0dt exp[iωt]χ(t),
ε(ω)=1-ωp2ω(ω+iγ),
D(ω, k)=det[1-t(ω)g(ω, k)],
u(x, z, t)=exp[-iz(k)t]u(x, z)
C={z(k)|kB},
ΓD-1(z, k)dz=2πi Res(D-1; z0),
ΓzD-1(z, k)dz=2πi Res((zD-1); z0)=2πiz0Res(D-1; z0).
E(x, t)=(2π)-1dω exp[-izt]x|[z2ε(x, z)-H0]-1|y{izD(y, 0)-y×B(y, 0)}=(2π)-1dω exp[-izt]G(x, y, z){izD(y, 0)-y×B(y, 0)}.
ψ(r+rs)=ψ(r)exp(ikrs).
[Δ+σ2]ψ=0,
G0Λ(σ, k, R)=rsΛG0(σ, R-rs)exp(ik  rs)=rsΛG0(σ, R+rs)exp(-ik  rs),
G0(σ, R)=G0(σ, r, r)=-i4 H0+(σR),
det[1-t(σ)g(σ, k)]=0,
G0Λ(σ, k, R)-G0(σ, R)
=m,mgm,m(σ, k)Jm(σr)exp(imφr)Jm(σr)
×exp(-imφr),

Metrics