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Comprehensive FDTD modelling of photonic crystal waveguide components

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Abstract

Planar photonic crystal waveguide structures have been modelled using the finite-difference-time-domain method and perfectly matched layers have been employed as boundary conditions. Comprehensive numerical calculations have been performed and compared to experimentally obtained transmission spectra for various photonic crystal waveguides. It is found that within the experimental fabrication tolerances the calculations correctly predict the measured transmission levels and other major transmission features.

©2004 Optical Society of America

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Figures (6)

Fig. 1.
Fig. 1. Transmission spectra for a PhCW calculated for (a) three different spatial resolutions in 2D (b) two different spatial resolutions in 3D.
Fig. 2.
Fig. 2. 2D Band diagram (left) shown for modes of different parities in a W1 PhCW. Even guided modes are shown in red, odd modes in blue, and slab modes in black. Transmission spectrum (right) shown for excitation with even modes.
Fig. 3.
Fig. 3. Scanning electron micrographs of (a) a straight PhCW of length 10 µm, and (b) a PhCW containing two modified 60° bends (details shown in zoom), which are separated by a 20Λ long straight PhCW.
Fig. 4.
Fig. 4. (a) 3D FDTD transmission spectra for different lengths of PhCW. (b) The measured (gray) and calculated (dashed black) transmission spectra for a 10 µm PhCW.
Fig. 5.
Fig. 5. The measured (gray) and calculated (dashed black) propagation losses for the TE polarization.
Fig. 6.
Fig. 6. Measured (gray) and calculated (dashed black) bend loss in two consecutive 60° bends for the TE polarization.

Equations (50)

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Δ t + E ( r , t ) = ε ( r ) 1 q × H ( r , t ) , Δ t H ( r , t ) = Q H μ ( r ) 1 q + × E ( r , t ) ,
ε ij = ε q x q y q z q i q j q 0 , μ ij = μ q x q y q z q i q j q 0 ,
ε ˜ = ε Λ ˜ , μ ˜ = μ Λ ˜ ,
Λ ˜ = ( s y s z s x 0 0 0 s x s z s y 0 0 0 s x s y s z ) ,
Λ ˜ ( r , ω ) = Λ ˜ x ( x , ω ) Λ ˜ y ( y , ω ) Λ ˜ z ( z , ω ) ,
Λ ˜ x = ( 1 s x 0 0 0 s x 0 0 0 s x )
E i ( t + t ) = 1 1 + σ j t { E i ( t ) + 1 ε ii [ q × H ( t ) ] i + σ i t ε ii n = 0 [ q × H ( t n t ) ] i }
E j ( t + t ) = 1 1 + σ i t { E j ( t ) + 1 ε jj [ q × H ( t ) ] j + σ j t ε jj n = 0 [ q × H ( t n t ) ] j }
E k ( t + t ) = E k ( t ) σ i σ j t 2 n = 0 E k ( t n t ) + ( × H ( t ) ) k ε kk 1 + ( σ i + σ j ) t + σ i σ j t 2
H i ( t ) = 1 1 + σ j t ×
{ H i ( t t ) Q H μ ii [ q × E ( t ) ] i σ i t μ ii Q H n = 0 [ q × E ( t n t ) ] i }
H j ( t ) = 1 1 + σ i t ×
{ H j ( t t ) Q H μ jj [ q × E ( t ) ] j σ j t μ jj Q H n = 0 [ q × E ( t n t ) ] j }
H k ( t ) = H k ( t t ) σ i σ j t 2 n = 0 H k ( t t n t ) Q H ( × E ( t ) ) k μ kk 1 + ( σ i + σ j ) t + σ i σ j t 2 .
ε ˜ xx = ε xx ( 1 + i σ z ω ) , ε ˜ yy = ε yy ( 1 + i σ z ω ) , ε ˜ zz = ε zz ( 1 + i σ z ω ) 1 ,
E x ( t + t ) E x ( t ) = ( ε ˜ 1 ) xx [ q × H ( t ) ] x = 1 ε xx ( 1 + i σ z ω ) 1 [ q × H ( t ) ] x .
[ q × H ] x = i t ε xx ω + ( 1 + i σ z ω ) E x
E x ( t + t ) = E x ( t ) + 1 ε xx ( 1 + i σ z ω ) 1 [ q × H ( t ) ] x
= 1 + σ z t 1 + σ z t E x ( t ) + 1 ε xx ( 1 + i σ z ω ) 1 [ q × H ( t ) ] x
= 1 1 + σ z t E x ( t ) + σ z t 1 + σ z t E x ( t ) + 1 ε xx ω ω + i σ z [ q × H ( t ) ] x
1 1 + σ z t E x ( t ) + [ q × H ( t ) ] x ε xx ( 1 + σ z t ) × [ i ω σ z ω + ( ω + i σ z ) + ω ( 1 + σ z t ) ω + i σ z ]
= 1 1 + σ z t ( E x ( t ) + [ q × H ( t ) ] x ε xx ) .
E z ( t + t ) = E z ( t ) + 1 ε zz ( 1 + i σ z ω ) [ q × H ( t ) ] z
= E z ( t ) + 1 ε zz [ q × H ( t ) ] z + i σ z ε zz · i t 1 e i ω t [ q × H ( t ) ] z
= E z ( t ) + 1 ε zz [ q × H ( t ) ] z + σ z t ε zz · n = 0 e in ω t [ q × H ( t ) ] z
= E z ( t ) + 1 ε zz [ q × H ( t ) ] z + σ z t ε zz · n = 0 [ q × H ( t n t ) ] z .
ε ˜ = ( ε 11 1 + i σ y ω 1 + i σ x ω 0 0 0 ε 22 1 + i σ x ω 1 + i σ y ω 0 0 0 ε 33 ( 1 + i σ x ω ) ( 1 + i σ y ω ) ) .
E x ( t + t ) E x ( t ) = ( ε ˜ 1 ) xx [ q × H ( t ) ] x
= 1 ε xx ( 1 + i σ x ω ) ( 1 + i σ y ω ) 1 [ q × H ( t ) ] x .
E x ( t + t ) = 1 1 + σ y t ×
{ E x ( t ) + 1 ε xx [ q × H ( t ) ] x + σ x t ε xx · n = 0 [ q × H ( t n t ) ] x }
E z ( t + t ) E z ( t ) = ( ε ˜ 1 ) zz [ q × H ( t ) ] z
= 1 ε zz ( 1 + i σ x ω ) 1 ( 1 + i σ y ω ) 1 [ q × H ( t ) ] z .
( 1 + i σ x ω ) ( 1 + i σ y ω ) ( E z ( t + t ) E 2 ( t ) ) =
( 1 + i ( σ x + σ y ) ω σ x σ y ( ω ) 2 ) ( E z ( t + t ) E z ( t ) ) .
i ( σ x + σ y ) ω ( E z ( t + t ) E z ( t ) ) = i ( σ x + σ y ) ( i t ) 1 e i ω σ t ( E z ( t + t ) E z ( t ) )
= ( σ x + σ y ) t k = 0 e ki ω t ( E z ( t + t ) E z ( t ) )
= ( σ x + σ y ) t ×
( k = 0 E z ( t + t k t ) k = 0 E z ( t k t ) )
= ( σ x + σ y ) t E z ( t + t ) ;
σ x σ y ( ω ) 2 ( E z ( t + t ) E z ( t ) ) = σ x σ y ω i t 1 e i ω t ( E z ( t + t ) E z ( t ) )
= i t σ x σ y ω E z ( t + t )
= ( i t ) 2 σ x σ y 1 e i ω t E z ( t + t )
= t 2 σ x σ y k = 0 E z ( t + t k t )
= t 2 σ x σ y ( E z ( t + t ) + k = 0 E z ( t k t ) ) .
1 ε zz [ q × H ( t ) ] z = E z ( t + t ) E z ( t ) + t ( σ x + σ y ) E z ( t + t ) +
t 2 σ x σ y ( E z ( t + t ) + k = 0 E z ( t k t ) ) .
E z ( t + t ) = E z ( t ) σ x σ y t 2 k = 0 E z ( t k t ) + ( × H ( t ) ) z ε zz 1 + ( σ x + σ y ) t + σ x σ y t 2 .
ε ˜ = ( ε xx ( 1 + i σ y ω ) ( 1 + i σ z ω ) 1 + i σ x ω 0 0 0 ε yy ( 1 + i σ x ω ) ( 1 + i σ z ω ) 1 + i σ y ω 0 0 0 ε zz ( 1 + i σ x ω ) ( 1 + i σ y ω ) 1 + i σ z ω ) .
ε zz E z t + ε zz ( σ x + σ y ) E z + ε zz σ x σ y 0 t E ˜ z ( t ) dt = ( × H ) z .
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