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Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics

L. C. Botten, T. P. White, C. Martijn de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry

Open Access Open Access

Abstract

A rigorous semi-analytic approach to the modelling of coupling, guiding and propagation in complex microstructures embedded in two-dimensional photonic crystals is presented. The method, which is based on Bloch mode expansions and generalized Fresnel coefficients, is shown to be able to treat photonic crystal devices in ways which are analogous to those used in thin film optics with uniform media. Asymptotic methods are developed and exemplified through the study of a serpentine waveguide, a potential slow wave device.

©2004 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. A typical three segment photonic crystal device showing the component regions M1, M2 and M3, three lateral supercells of the model and a constituent grating bounded by dashed lines.

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Fig. 2.
Fig. 2. Two equivalent serpentine waveguide geometries (assuming no tunneling through the guide ends). Both are characterized by the period Dy and the double and single guide lengths, L 1 and L 2.

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Fig. 3.
Fig. 3. (a) Band diagram for a serpentine tine waveguide with L 1=L 2=5d, where q is the Bloch coefficient along the waveguide, and Dy =L 1+L 2. (b) Transmission through an FDC with same parameters (dashed), one period of the serpentine waveguide (dotted) and two periods (solid). Note that for frequencies below ωd/(2πc)=0.3064, the double guide cavity only supports a single, odd mode, and thus the analytic result of (19) does not apply.

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Fig. 4.
Fig. 4. (a) Band diagram for a serpentine waveguide with L 1=L 2=7d. The solid curve is calculated with the full numerical simulation while the dashed curve is calculated using the approximation (19) L 1=7.5d, L 2=6.7d (b) Transmission through a FDC with L=7d (dashed), 2 periods of the serpentine guide (dotted) and 3 periods of the serpentine guide.

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Equations (22)

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𝓣 𝓯 = μ 𝓯 where 𝓣 = ( T − R T − 1 R R T − 1 − T − 1 R T − 1 ) , 𝓯 = ( f − f + ) ,
𝓣 T 𝓠 𝓣 = 𝓠 , where 𝓠 = [ 0 Q − Q 0 ] .
𝓣 = 𝓕 ℒ 𝓕 − 1 with 𝓕 = [ F − F + F + F − ] , ℒ = [ Λ 0 0 Λ − 1 ] , Λ = diag ( μ i ) .
𝓕 T 𝓠 𝓕 = ℐ where ℐ = ( 0 I − I 0 )
𝓕 H 𝓣 p 𝓕 = 𝓣 m where 𝓣 p = ( I r − i I e i I e − I r ) and 𝓣 m = ( I m − i I m ¯ − i I m ¯ − I m ) ,
R ij = ( F i − ) − 1 ( I − R j R i ) − 1 ( R j − R i ) F i − , ⋱
T ij = ( F j − ) − 1 ( I − R i R j ) − 1 ( I − R i 2 ) F i − ,
r = R 12 δ + T 21 Λ L c + , c − = T 12 δ + R 21 Λ L c + ,
c + = R 23 Λ L c − , t = T 23 Λ L c − ,
R = R 13 = R 12 + T 21 Λ L R 23 Λ L ( I − R 21 Λ L R 23 Λ L ) − 1 T 12 ,
T = T 13 = T 23 Λ L ( I − R 21 Λ L R 23 Λ L ) − 1 T 12 .
S 12 2 = I , where S 12 = ( R 12 T 21 T 12 R 21 )
R 13 = T 12 − 1 ( − R 21 + Λ L R 23 Λ L ) ( I − R 21 Λ L R 23 Λ L ) − 1 T 12 ,
= T 21 − 1 ( I − Λ L R 23 Λ L R 21 ) − 1 ( − R 21 + Λ L R 23 Λ L ) T 21 .
S 13 H S 13 = I 13 , where S 13 = ( R 13 T 31 T 13 R 31 ) , I 13 = ( I 1 0 0 I 3 ) ,
w 1 T = w 3 T = [ 1 0 … 0 0 ] , w 2 T = [ 1 0 … 0 0 0 1 … 0 0 ] .
T ˜ 13 = T ˜ 23 Λ ˜ L ( I − R ˜ 21 Λ ˜ L R ˜ 23 Λ ˜ L ) − 1 T ˜ 12 ,
R 13 = ρ f = cos 2 ( Δ β L ) exp ( 2 i β ¯ L ) 1 + sin 2 ( Δ β L ) exp ( 2 i β ¯ L ) , T 13 = τ f = i exp ( i β ¯ L ) sin ( Δ β L ) ( 1 + exp ( 2 i β ¯ L ) ) 1 + sin 2 ( Δ β L ) exp ( 2 i β ¯ L ) ,
𝓣 s = ( τ s − ρ s 2 ⁄ τ s ρ s ⁄ τ s − ρ s ⁄ τ s 1 ⁄ τ s ) ,
𝓣 s = 𝓣 f 2 , where 𝓣 f = ( τ f ′ − ρ f ′ 2 ⁄ τ f ′ ρ f ′ ⁄ τ f ′ − ρ f ′ ⁄ τ f ′ 1 ⁄ τ f ′ )
μ + 1 ∓ μ 1 ⁄ 2 = τ f ′ 2 − ρ f ′ 2 + 1 τ f , i. e . ∓ cos ( q D y ⁄ 2 ) = ℜ ( 1 τ f ′ ) ,
cos ( q D ⁄ 2 ) = sin ( Δ β L 1 ) sin ( β ¯ L 1 + β L 2 ) + cos 2 ( Δ β L 1 ) sin ( 2 β ¯ L 1 + β L 2 ) 2 sin ( Δ β L 1 ) cos ( β ¯ L 1 ) .

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