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Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals

Chin-ping Yu and Hung-chun Chang

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Abstract

A finite-difference frequency-domain method based on the Yee’s cell is utilized to analyze the band diagrams of two-dimensional photonic crystals with square or triangular lattice. The differential operator is replaced by the compact scheme and the index average scheme is introduced to deal with the curved dielectric interfaces in the unit cell. For the triangular lattice, the hexagonal unit cell is converted into a rectangular one for easier mesh generation. The band diagrams for both square and triangular lattices are obtained and the numerical convergence of computed eigen frequencies is examined and compared with other methods.

©2004 Optical Society of America

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Figures (8)
Fig. 1.
Fig. 1. The cross-sectional view of a 2-D PC and its unit cell with a being the lattice distance and r being the radius of the circles for (a) the square lattice and (b) the triangular lattice.

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Fig. 2.
Fig. 2. Yee’s mesh for the (a) TE and (b) TM modes.

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Fig. 3.
Fig. 3. (a) The unit cell of the PC with triangular lattice and its corresponding PBCs. (b) The modified unit cell.

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Fig. 4.
Fig. 4. Band diagrams for the 2-D PC formed by square-arranged alumina rods with r/a=0.2 and ε=8.9 in the air. (a) TE mode and (b) TM mode.

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Fig. 5.
Fig. 5. The first eigen frequency versus the number of grid points for (a) the TE and (b) the TM modes as k is at the M point. The lines with circles and rectangles are the results obtained by our FDFD method without and with the index average scheme, respectively.

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Fig. 6.
Fig. 6. The convergence properties of our method for the 2-D square-lattice PC compared with other methods. (a) TE first band; (b) TE second band; (c) TM first band; (d) TM second band.

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Fig. 7.
Fig. 7. Band diagrams of (a) the TE mode and (b) the TM mode for the 2-D PC formed by triangular-arranged dielectric cylinders with r/a=0.2 and ε=11.4 in the air. Our results (triangles) are compared with the results from the FDTD method (circles) and the PWE method (solid lines).

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Fig. 8.
Fig. 8. The convergence properties of our method for the 2-D triangular-lattice PC compared with other methods. (a) TE first band; (b) TE second band; (c) TM first band; (d) TM second band.

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Equations (42)

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∇ × E = − j ω μ 0 μ H
∇ × H = j ω ε 0 ε E
− j ω μ 0 μ H z = ∂ E y ∂ x − ∂ E x ∂ y
j ω ε 0 ε E x = ∂ H z ∂ y
j ω ε 0 ε E y = − ∂ H z ∂ x .
− j ω μ 0 ( μ z H z ) ∣ i + 1 2 , j + 1 2 = ( ∂ E y ∂ x − ∂ E x ∂ y ) ∣ i + 1 2 , j + 1 2
j ω ε 0 ( ε x E x ) ∣ i + 1 2 , j = ( ∂ H z ∂ y ) ∣ i + 1 2 , j
j ω ε 0 ( ε y E y ) ∣ i , j + 1 2 = − ( ∂ H z ∂ x ) ∣ i , j + 1 2
∂ H z ∂ x ∣ i + 1 , j + 1 2 = ∂ H z ∂ x ∣ i + 1 2 , j + 1 2 + ( Δ x ⁄ 2 ) 1 ! ∂ 2 H z ∂ x 2 ∣ i + 1 2 , j + 1 2
+ ( Δ x ⁄ 2 ) 2 2 ! ∂ 3 H z ∂ x 3 ∣ i + 1 2 , j + 1 2 + ( Δ x ⁄ 2 ) 3 3 ! ∂ 4 H z ∂ x 4 ∣ i + 1 2 , j + 1 2 + H. O . T .
∂ H z ∂ x ∣ i , j + 1 2 = ∂ H z ∂ x ∣ i + 1 2 , j + 1 2 − ( Δ x ⁄ 2 ) 1 ! ∂ 2 H z ∂ x 2 ∣ i + 1 2 , j + 1 2
+ ( Δ x ⁄ 2 ) 2 2 ! ∂ 3 H z ∂ x 3 ∣ i + 1 2 , j + 1 2 − ( Δ x ⁄ 2 ) 3 3 ! ∂ 4 H z ∂ x 4 ∣ i + 1 2 , j + 1 2 + H. O . T .
∂ H z ∂ x ∣ i − 1 , j + 1 2 = ∂ H z ∂ x ∣ i + 1 2 , j + 1 2 − ( 3 Δ x ⁄ 2 ) 1 ! ∂ 2 H z ∂ x 2 ∣ i + 1 2 , j + 1 2
+ ( 3 Δ x ⁄ 2 ) 2 2 ! ∂ 3 H z ∂ x 3 ∣ i + 1 2 , j + 1 2 − ( 3 Δ x ⁄ 2 ) 3 3 ! ∂ 4 H z ∂ x 4 ∣ i + 1 2 , j + 1 2 + H. O . T .
H z ∣ i − 1 2 , j + 1 2 = H z ∣ i + 1 2 , j + 1 2 − ( Δ x ) 1 ! ∂ H z ∂ x ∣ i + 1 2 , j + 1 2 + ( Δ x ) 2 2 ! ∂ 2 H z ∂ x 2 ∣ i + 1 2 , j + 1 2
− ( Δ x ) 3 3 ! ∂ 3 H z ∂ x 3 ∣ i + 1 2 , j + 1 2 + ( Δ x ) 4 4 ! ∂ 4 H z ∂ x 4 ∣ i + 1 2 , j + 1 2 + H. O . T .
H z ∣ i + 1 2 , j + 1 2 − H z ∣ i − 1 2 , j + 1 2 =
Δ x 24 ( ∂ H z ∂ x ∣ i + 1 , j + 1 2 + 22 ∂ H z ∂ x ∣ i , j + 1 2 + ∂ H z ∂ x ∣ i − 1 , j + 1 2 ) .
H z ∣ i + 1 2 , j + 1 2 − H z ∣ i + 1 2 , j − 1 2 =
Δ y 24 ( ∂ H z ∂ y ∣ i + 1 2 , j + 1 + 22 ∂ H z ∂ y ∣ i + 1 2 , j + ∂ H z ∂ y ∣ i + 1 2 , j − 1 )
E x ∣ i + 1 2 , j + 1 − E x ∣ i + 1 2 , j =
Δ y 24 ( ∂ E x ∂ y ∣ i + 1 2 , j + 3 2 + 22 ∂ E x ∂ y ∣ i + 1 2 , j + 1 2 + ∂ E x ∂ y ∣ i + 1 2 , j − 1 2 )
E y ∣ i + 1 , j + 1 2 − E y ∣ i − 1 , j + 1 2 =
Δ x 24 ( ∂ E y ∂ x ∣ i + 3 2 , j + 1 2 + 22 ∂ E y ∂ x ∣ i + 1 2 , j + 1 2 + ∂ E y ∂ x ∣ i − 1 2 , j + 1 2 ) .
U · [ ∂ H z ∂ x ] = V x H z
V · [ ∂ H z ∂ y ] = V y H z
V · [ ∂ E x ∂ y ] = U y E x
U · [ ∂ E y ∂ x ] = U x E y
− j ω [ μ 0 μ z 0 0 0 − ε 0 ε x 0 0 0 − ε 0 ε y ] [ H z E x E y ] =
[ 0 − V − 1 U y U − 1 U x V − 1 V y 0 0 − U − 1 V x 0 0 ] [ H z E x E y ]
k 0 2 H z = − μ z − 1 { V − 1 U y ε x − 1 V − 1 V y + U − 1 U x ε y − 1 U − 1 V x } H z
j ω ε 0 ε E z = ∂ H y ⁄ ∂ x − ∂ H x ⁄ ∂ y
− j ω μ 0 μ H x = ∂ E z ⁄ ∂ y
− j ω μ 0 μ H y = − ∂ E z ⁄ ∂ x
j ω [ ε 0 ε z 0 0 0 − μ 0 μ x 0 0 0 − μ 0 μ y ] [ E z H x H y ] =
[ 0 − V − 1 V y U − 1 V x V − 1 U y 0 0 − U − 1 U x 0 0 ] [ E z H x H y ]
k 0 2 E z = − ε z − 1 { V − 1 V y μ x − 1 V − 1 U y + U − 1 V x μ y − 1 U − 1 U x } E z .
Ψ ( x + a , y ) = e − jk x a Ψ ( x , y )
Ψ ( x , y + a ) = e − jk y a Ψ ( x , y )
PBC 1 : Ψ ( x + 3 a 2 , y − a 2 ) = e − j ( k x 3 a ⁄ 2 − k y a ⁄ 2 ) Ψ ( x , y )
PBC 2 : Ψ ( x + 3 a 2 , y + a 2 ) = e − j ( k x 3 a ⁄ 2 + k y a ⁄ 2 ) Ψ ( x , y )
PBC 3 : Ψ ( x , y + a ) = e − jk y a Ψ ( x , y ) .
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