Abstract

We introduce a novel algorithm for band structure computations based on multigrid methods. In addition, we demonstrate how the results of these band structure calculations may be used to compute group velocities and effective photon masses. The results are of direct relevance to studies of pulse propagation in such materials.

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References

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  1. C.M. Soukoulis (Ed.), Photonic Band Gap Materials, NATO ASI Series E 315, (Kluwer Academic Publishers, 1996).
    [CrossRef]
  2. A. Birner et al., "Macroporous Silicon: A Two-Dimensional Photonic Bandgap Material Suitable for the Near-Infrared Spectral Range," phys. stat. sol (a) 165, 111-117 (1998)
    [CrossRef]
  3. K.M. Ho, C.T. Chan, and C.M. Soukoulis, "Existence of a photonic band gap in periodic structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
    [CrossRef] [PubMed]
  4. K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
    [CrossRef]
  5. K. Busch and S. John. "Liquid crystal photonic band gap materials: The tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
    [CrossRef]
  6. P. Wesseling, An Introduction to Multigrid Methods, (John Wiley & Sons, 1992).
  7. A. Brandt, S. McCormick, and J. Ruge, "Multigrid methods for differential eigenproblems," SIAM J. Sci. Stat. Comput. 4, 244-260 (1983).
    [CrossRef]
  8. C. Martijn de Sterke and J.E. Sipe, "Envelope-function approach for the electrodynamics of non-linear periodic media," Phys. Rev. A 38, 5149-5165 (1988)
    [CrossRef]

Other

C.M. Soukoulis (Ed.), Photonic Band Gap Materials, NATO ASI Series E 315, (Kluwer Academic Publishers, 1996).
[CrossRef]

A. Birner et al., "Macroporous Silicon: A Two-Dimensional Photonic Bandgap Material Suitable for the Near-Infrared Spectral Range," phys. stat. sol (a) 165, 111-117 (1998)
[CrossRef]

K.M. Ho, C.T. Chan, and C.M. Soukoulis, "Existence of a photonic band gap in periodic structures," Phys. Rev. Lett. 65, 3152-3155 (1990).
[CrossRef] [PubMed]

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

K. Busch and S. John. "Liquid crystal photonic band gap materials: The tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

P. Wesseling, An Introduction to Multigrid Methods, (John Wiley & Sons, 1992).

A. Brandt, S. McCormick, and J. Ruge, "Multigrid methods for differential eigenproblems," SIAM J. Sci. Stat. Comput. 4, 244-260 (1983).
[CrossRef]

C. Martijn de Sterke and J.E. Sipe, "Envelope-function approach for the electrodynamics of non-linear periodic media," Phys. Rev. A 38, 5149-5165 (1988)
[CrossRef]

Supplementary Material (1)

» Media 1: MOV (813 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

Graphical illustration of the V-cycle used in the multigrid iteration of Eq. (5) depicting four levels of grids. A movie (881.mov(0.8M)) shows the multigrid iteration using two levels for the third band at the x-point in the TM-polarized case (see Fig. 2).

Fig. 2.
Fig. 2.

Photonic band structure for a square lattice of dielectric rods (a =13, r/a=0.45) in air (b =1) for TM-polarization (blue) and TE-polarization (green).

Fig. 3.
Fig. 3.

Group velocity of the first three bands (first band: black; second band: red; third band: green) for a square lattice of dielectric rods (a =13, r/a=0.45) in air (b =1) for TM-polarized light. The corresponding band structure is displayed in Fig. 2.

Tables (1)

Tables Icon

Table 1. Comparison of numerical data obtained from PWM (upper part) and the multigrid method (lower part) using different numbers N of plane waves and different mesh-sizes (128×128 and 256×256), respectively. The values for the defect give an estimate of the absolute error of the Bloch-functions [7]. The computations have been performed at the X-point for TM-polarization using a 466 MHz Celeron processor.

Equations (13)

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1 ( r ) ( x 2 + y 2 ) E ( r ) + ω 2 c 2 E ( r ) = 0 ,
x + ( 1 ( r ) x H ( r ) ) + y ( 1 ( r ) y H ( r ) ) + ω 2 c 2 H ( r ) = 0 .
E k ( r + a i ) = e i k a i E k ( r )
H k ( r + a i ) = e i k a i H k ( r ) ,
k U i = Λ i U i ,
1 V w s c d 2 r E n k * ( r ) ( r ) E m k ( r ) = δ n m
1 V w s c d 2 r H n k * ( r ) H m k ( r ) = δ n m ,
Λ i = k 1 U i 1 , U i 1 U i 1 , U i 1 ,
E k ( r ) = e i k r u k ( r ) ,
( Δ + 2 i · k k 2 ) u k ( r ) + ω k 2 c 2 ( r ) u k ( r ) ) ) = 0 .
H ̂ ( k ) u k + q ( r ) + ( 2 q · Ω q 2 ) u k + q ( r ) + ω k + q 2 c 2 ( r ) u k + q ( r ) = 0 .
ω k + q = ω k + ( 1 st order kp PT ) + ( 2 nd order kp PT ) +
= ω k + q · υ k + q · M k · q + .

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