Abstract

The convergence feature of two types of plane-wave expansion methods commonly used for photonic crystals is analyzed. It is shown that the reason for the slow convergence of these plane-wave expansion methods is not the slow convergence of the Fourier series for the permittivity profile of the photonic crystal but the inappropriate formulation of the eigenproblem. A new formulation of the eigenproblem is presented to improve the convergence in the one-dimensional case.

© 2002 Optical Society of America

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References

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  1. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction: E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  3. M. Plihal, A. A. Maradudin, “Photonic band structure of a two-dimensional system: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
    [CrossRef]
  4. K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  5. H. S. Sözüer, J. W. Haus, R. Inguva, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
    [CrossRef]
  6. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  7. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  8. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
    [CrossRef]
  9. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

1996 (2)

1993 (2)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

1992 (1)

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

1991 (1)

M. Plihal, A. A. Maradudin, “Photonic band structure of a two-dimensional system: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

1990 (1)

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

1983 (1)

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Chan, C. T.

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

Gaylord, T. K.

Haggans, C. W.

Haus, J. W.

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

Ho, K. M.

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

Inguva, R.

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

Joannopoulos, J. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Lalanne, P.

Li, L.

Maradudin, A. A.

M. Plihal, A. A. Maradudin, “Photonic band structure of a two-dimensional system: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Moharam, M. G.

Morris, G. M.

Plihal, M.

M. Plihal, A. A. Maradudin, “Photonic band structure of a two-dimensional system: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Soukoulis, C. M.

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

Sözüer, H. S.

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

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Phys. Rev. B (3)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

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

M. Plihal, A. A. Maradudin, “Photonic band structure of a two-dimensional system: the triangular lattice,” Phys. Rev. B 44, 8565–8571 (1991).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (1)

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

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Figures (4)

Fig. 1
Fig. 1

One-dimensional photonic crystal with period Λ.

Fig. 2
Fig. 2

Convergence of the normalized eigenfrequencies at (a) the first band and (b) the second band for H polarization calculated with the different formulations. Here β=0 and kx=π/2Λ.

Fig. 3
Fig. 3

Convergence of the normalized eigenfrequencies at (a) the first band and (b) the second band for H polarization calculated with the three different formulations. Here β=π/Λ and kx=π/2Λ.

Fig. 4
Fig. 4

Convergence (in terms of the relative error) of the eigenfrequencies calculated with the conventional formulation and Ho’s formulation for (a) E polarization and (b) H polarization on the first band.

Equations (15)

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

Exz-Ezx=iωμ0Hy,
-Hyz=-iω0Ex,
Hyx=-iω0Ez,
Ex=mSx,m exp[i(qm+kx)x]exp(iβz),
Ez=mSz,m exp[i(qm+kx)x]exp(iβz),
Hy=mUm exp[i(qm+kx)x]exp(iβz),
β[Sx]-K[Sz]=ωc[U],
β[U]=ωc[Sx],
K[U]=-ωc[Sz],
(K-1K+β2-1)[U]=ω2c2[U].
K1K+β21[U]=ω2c2[U].
β[Sx]-K[Sz]=ωc[U],
β[U]=ωc1-1[Sx],
K[U]=-ωc[Sz].
K-1K+β21[U]=ω2c2[U].

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