Abstract

A finite-difference frequency-domain (FDFD) method is applied for photonic band gap calculations. The Maxwell’s equations under generalized coordinates are solved for both orthogonal and non-orthogonal lattice geometries. Complete and accurate band gap information is obtained by using this FDFD approach. Numerical results for 2D TE/TM modes in square and triangular lattices are in excellent agreements with results from plane wave method (PWM). The accuracy, convergence and computation time of this method are also discussed.

© 2004 Optical Society of America

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References

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IEEE Trans. Antennas Propagat.

K.S Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propagat. 14, 302-307 (1966)
[CrossRef]

IEEE Trans. Microwave Theory Tech.

K. Bierwith, N. Schulz, F. Arndt, �??Finite-difference analysis of rectangular dielectric waveguide structures,�?? IEEE Trans. Microwave Theory Tech. 34, 1104-1113 (1986)
[CrossRef]

H. Y. D. Yang, �??Finite-difference analysis of 2D photonic crystals,�?? IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

IEEE Trans. MTT

P R McIssac, �??Symmetry induced modal characteristics of uniform waveguides-I: Summary of results,�?? IEEE Trans. MTT 23, 421-429 (1975)
[CrossRef]

J. Appl. Phys

M. Qiu, S. He, �??A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions ,�?? J. Appl. Phys. 87, 8268-8275 (2000)
[CrossRef]

J. Lightwave Technol.

P. Lusse, P. Stuwe, J Schule, H. G. Unger, �??Analysis of vectorial mode fields in optical waveguides by a new finite-difference method,�?? J. Lightwave Technol. 12, 487-494 (1994)
[CrossRef]

J. Mod. Opt.

A. J. Ward and J. B. Pendry, �??Refraction and geometry in Maxwell�??s equations,�?? J. Mod. Opt. 43, 773-793 (1996)
[CrossRef]

Opt. Express

Phys. Rev. B

C. T Chan, Y. L. Yu, K. M. Ho, �??Order-N spectral method for electromagnetic waves,�?? Phys. Rev. B 51, 16635-16642 (1995)
[CrossRef]

J. Arriaga, A. J. Ward and J. B. Pendry, �??Order-N photonic band structures for metals and other dispersive materials,�?? Phys. Rev. B 59, 1874-1877 (1999)
[CrossRef]

R. D. Meade, A. M. Rappe et al., �??Accurate theoretical analysis of photonic band gap materials,�?? Phys. Rev. B 48, 8434-8437 (1993)
[CrossRef]

Phys. Rev. Lett

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

Phys. Rev. Lett.

K. M. Leung and Y. F. Liu, �??Full vector wave calculation of photonic band structures in FCC dielectric media,�?? Phys. Rev. Lett. 65, 2646-2649 (1990)
[CrossRef] [PubMed]

E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987)
[CrossRef] [PubMed]

S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987)
[CrossRef] [PubMed]

Other

A. J. Ward, �??Order-N program documentation,�?? <a href="http://www.sst.ph.ic.ac.uk/photonics/ONYX/orderN.html">http://www.sst.ph.ic.ac.uk/photonics/ONYX/orderN.html</a>

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Figures (5)

Fig. 1.
Fig. 1.

Yee’s 2D mesh in general coordinates. The dotted components are at the boundaries.

Fig. 2.
Fig. 2.

The band structure for a 2D square lattice by FDFD (o) and PWM (-). 441 plane waves are used for PWM and mesh resolution is a/80 for FDFD. Left: TM mode, Right: TE mode.

Fig. 3.
Fig. 3.

The calculated band structure of a triangular lattice by FDFD (o) and PWM (-). 441 plane waves are used for PWM and mesh resolution is a/80 for FDFD. Left: TM, Right: TE.

Fig. 4.
Fig. 4.

The convergence of eigen-frequency (the 5th band at k=0) and the computation time vs. the number of grids along each direction.

Fig. 5.
Fig. 5.

The Ez field of a defect mode in a 2D square lattice with alumina rods in air using a 5×5 supercell with the center rod removed. The rods are displayed as black circles.

Tables (1)

Tables Icon

Table 1. Eigen-frequencies for the first five bands of TE wave (k=0) for a triangular lattice with air holes in dielectric materials.

Equations (14)

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q × H ̂ = jk 0 ε ̂ ( r ) E ̂ q × E ̂ = jk 0 μ ̂ ( r ) H ̂ ,
E ̂ i = Q i ε 0 μ 0 E i H ̂ i = Q i H i ,
ε ̂ ij ( r ) = ε ri ( r ) g ij u 1 · u 2 × u 3 Q 1 Q 2 Q 3 Q i Q j Q 0 μ ̂ ij ( r ) = μ ij ( r ) g ij u 1 · u 2 × u 3 Q 1 Q 2 Q 3 Q i Q j Q 0 .
g 1 = [ u 1 · u 1 u 1 · u 2 u 1 · u 3 u 2 · u 1 u 2 · u 2 u 2 · u 3 u 3 · u 1 u 3 · u 2 u 3 · u 3 ]
r 2 = r T [ g 1 ] r .
jk 0 [ E ̂ 1 E ̂ 2 E ̂ 3 ] = [ ε 11 1 ε 12 1 ε 13 1 ε 21 1 ε 22 1 ε 23 1 ε 31 1 ε 32 1 ε 33 1 ] [ 0 U 3 U 2 U 3 0 U 1 U 2 U 1 0 ] [ H ̂ 1 H ̂ 2 H ̂ 3 ] ,
jk 0 [ H ̂ 1 H ̂ 2 H ̂ 3 ] = [ μ 11 1 μ 12 1 μ 13 1 μ 21 1 μ 22 1 μ 23 1 μ 31 1 μ 32 1 μ 33 1 ] [ 0 V 3 V 2 V 3 0 V 1 V 2 V 1 0 ] [ E ̂ 1 E ̂ 2 E ̂ 3 ] ,
H ̂ ( r + R l ) = exp ( ik · R l ) H ̂ ( r ) E ̂ ( r + R l ) = exp ( ik · R l ) E ̂ ( r ) ,
k 0 2 E ̂ z = ε 33 1 { U 1 ( μ 21 1 V 2 μ 22 1 V 1 ) U 2 ( μ 11 1 V 2 μ 12 1 V 1 ) } E ̂ z .
U 1 = 1 Q 1 [ 1 1 u x 1 0 u x 1 0 1 u x 1 1 ] , V 1 = 1 Q 1 [ 1 v x 1 1 0 v x 1 0 1 v x 1 1 ]
u x = exp ( ik · a 1 u 1 ) , v x = exp ( ik · a 1 u 1 )
U 2 = 1 Q 2 [ 1 1 1 1 1 1 u y 1 u y 1 ] , V 2 = 1 Q 2 [ 1 1 1 1 1 −1 v y 1 v y 1 ]
u y = exp ( ik · a 2 u 2 ) , v y = exp ( ik · a 2 u 2 ) .
k 0 2 H ̂ z = { ε 12 1 U 1 μ 33 1 V 2 ε 22 1 U 1 μ 33 1 V 1 } H ̂ z .

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