Abstract

The Cell Method, a new efficient numerical method suitable for working with periodic structures having anisotropic inhomogeneous media with curved shapes, is proposed in order to calculate the band gap of 2D photonic crystals for in-plane propagation of TM and TE waves. Moreover some numerical comparisons with other numerical methods will be provided.

© 2002 Optical Society of America

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References

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  1. J. D. Joannopulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton University Press, 1995).
  2. H. Y. D. Yang, "Finite Di.erence Analysis of 2-D Photonic Crystals," IEEE Trans. Microwave Theory Tech. 44 2688-2695 (1996).
    [CrossRef]
  3. E. Tonti, "Finite Formulation of the Electromagnetic Field," in Geometric Methods for Computational Electromagnetics, PIER 32, F. L.Teixeira, J. A.Kong, ed.(EMW Publishing 2001) 1-44.
  4. M. Marrone, �??Computational Aspects of Cell Method in Electrodynamics,�?? in Geometric Methods for Computational Electromagnetics, PIER 32, F. L.Teixeira, J. A. Kong, ed. (EMW Publishing 2001), 317-356.
  5. S. G. Johnson, S. Fan, P.R. Villeneuve, J. D. Joannopulos, L. A. Kolodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 3, 5751-5758 (1999)
    [CrossRef]
  6. Z. Y. Li, B. Y. Gu, G. Z. Yang, �??Improvement of absolute band gaps in 2D photonic crystals by anisotropy in dielectricity,�?? Eur. Phys. J. B 11, 65-73 (1999).
  7. M. Clemens, T. Weiland, �??Discrete Electromagnetism with the Finite Integration Technique,�?? in Geometric Methods for Computational Electromagnetics, PIER 32, F. L. Teixeira, J. A. Kong, ed. (EMW Publishing 2001), 65-87.

Eur. Phys. J. B (1)

Z. Y. Li, B. Y. Gu, G. Z. Yang, �??Improvement of absolute band gaps in 2D photonic crystals by anisotropy in dielectricity,�?? Eur. Phys. J. B 11, 65-73 (1999).

IEEE Trans. Microwave Theory Tech. (1)

H. Y. D. Yang, "Finite Di.erence Analysis of 2-D Photonic Crystals," IEEE Trans. Microwave Theory Tech. 44 2688-2695 (1996).
[CrossRef]

Phys. Rev. B (1)

S. G. Johnson, S. Fan, P.R. Villeneuve, J. D. Joannopulos, L. A. Kolodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 3, 5751-5758 (1999)
[CrossRef]

Other (4)

J. D. Joannopulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton University Press, 1995).

E. Tonti, "Finite Formulation of the Electromagnetic Field," in Geometric Methods for Computational Electromagnetics, PIER 32, F. L.Teixeira, J. A.Kong, ed.(EMW Publishing 2001) 1-44.

M. Marrone, �??Computational Aspects of Cell Method in Electrodynamics,�?? in Geometric Methods for Computational Electromagnetics, PIER 32, F. L.Teixeira, J. A. Kong, ed. (EMW Publishing 2001), 317-356.

M. Clemens, T. Weiland, �??Discrete Electromagnetism with the Finite Integration Technique,�?? in Geometric Methods for Computational Electromagnetics, PIER 32, F. L. Teixeira, J. A. Kong, ed. (EMW Publishing 2001), 65-87.

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Figures (6)

Fig. 1.
Fig. 1.

(a)Inner and outer orientations of the space cells in the 3D space (b)Primal and dual complexes and global variables associated with their primal and dual space cells respectively.

Fig. 2.
Fig. 2.

Examples of 2D photonic crystals and periodicity of the unit cells in the xy-plane (a) for rectangular unit cells (b) for quadrangular unit cells. (c) Dependence of some voltages on the boundary of a rectangular unit cell by the periodic boundary condition.

Fig. 3.
Fig. 3.

TMz case

Fig. 4.
Fig. 4.

Examples of 2Dprimal and dual grids (a) Triangular primal cell complex, barycentric dual cell complex for the TMz cases (b) Barycentric primal cell complex, triangular dual cell complex for the TEz cases

Fig. 5.
Fig. 5.

(a) Results of the test 1 (b) A comparison of some CM and FEM results in the test 1.

Fig. 6.
Fig. 6.

(a) Results of the test 2 (b) Results of the test 3.

Equations (8)

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G T M ν GV = ω 2 M ε V
V ( x + a , y + b ) = e j β x a j β y b V x y
V = T ( β x , β y ) V
AV = ω 2 BV
M TM V = ω 2 V
G ˜ T M ε 1 G ˜ V = ω 2 M μ F
F ( x + a , y + b ) = e j β x a j β y b F x y
M TM F = ω 2 F

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