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Photonic band gap analysis using finite-difference frequency-domain method

Shangping Guo, Feng Wu, Sacharia Albin, and Robert S. Rogowski

Open Access Open Access

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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Figures (5)
Fig. 1.
Fig. 1. Yee’s 2D mesh in general coordinates. The dotted components are at the boundaries.

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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.

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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.

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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.

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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.

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Tables (1)

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

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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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