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All-frequency effective medium theory of a photonic crystal

Geesche Boedecker and Carsten Henkel

Open Access Open Access

Abstract

We consider light propagation in a finite photonic crystal. The transmission and reflection from a one-dimensional system are described in an effective medium theory, which reproduces exactly the results of transfer matrix calculations.We derive simple formulas for the reflection from a semi-infinite crystal, the local density of states in absorbing crystals, and discuss defect modes and negative refraction.

©2003 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. (Left panel) complex eigenvalue λ+=eika vs. frequency for the Kronig-Penney model, Eq. (9), with point scatterers of polarizability α=(0.2+0i)a. (Right panel) band structure ω(k) of the infinite crystal. The thick dispersion curves correspond to the physical Bloch momentum fixed by the requirements of causality and energy conservation. Even bands exhibit negative refraction (k<0).

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Fig. 2.
Fig. 2. (Left) reflectance |rN | for the Kronig-Penney model with α=(0.2+0i)a, N=4 (blue line). The envelope, Eq. (6), is shown as well. Solid green line: half-space approximation |r|, Eq. (11). Dashed dark green line: |r| for absorbing scatterers (α=(0.2+0.02i)a). (Right) reflection coefficient |r i| for Bloch waves reflected from the end face of a semi-infinite crystal. Solid line: our result, Eq. (13); dashed line: proposed by Sakoda [4]. Kronig-Penney model with α=(0.2+0i)a.

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Fig. 3.
Fig. 3. LDOS, Eq. (15), in an infinite crystal with scatterers at x=…,-a/2,a/2,… (Left) no absorption (α=0.2a). (Right) nonzero absorption (α=(0.2+0.05i)a).

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Fig. 4.
Fig. 4. (Left) reflectance from two N=6 layer crystals (α=0.2a) with a defect in between (dR =dL =a/2). Green line with peak: active defect with scattering strength α d/a≈-0.62-0.04i, given by Eq. (16) for ωa/2πc≈0.462. The same structure with a passive defect (α d≈-0.62a) gives the green line with the transmission dip. Blue line: reflectance for α d=α. (Right) defect frequency in the first band gap vs. real part of scattering strength α d.

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Equations (16)

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E n ( x ) = a n e i ω ( x − na ) + b n e − i ω ( x − na ) ,
( a n + 1 b n + 1 ) = T ( a n b n ) .
T N = M ( e ikaN 0 0 e − ikaN ) M − 1 , M = ( N + N − N + c + N − c − ) .
c ± = e ± ika − T 11 T 12 = T 21 e ± ika − T 22 .
r N = c + c − ( e ikaN − e − ikaN ) c + e ikaN − c − e − ikaN .
0 ≤ ∣ r N ∣ ≤ ∣ 2 c + c − c + + c − ∣ ,
d 2 E ( x ) d x 2 + ω 2 ( 1 + ∑ n = 1 N α n δ ( x − ( n − 1 2 ) a ) ) E ( x ) = 0
T = ( ( 1 + i 2 ω α ) e i ω a i 2 ω α − i 2 ω α ( 1 − i 2 ω α ) e − i ω a )
λ ± = cos ⁡ ( ω a ) − ω α 2 sin ⁡ ( ω a ) ± i sin ⁡ ( ω a ) 1 + ω α cot ⁡ ( ω a ) − 1 4 ω 2 α 2
( t 0 ) = M − 1 ( 1 r ) .
r = c + = e iωa − e ika e iωa e ika − 1 ,
( r i 1 ) = M − 1 ( 0 t ′ )
r i = − N − N + = − N + c + N − c − = − c + .
r N , FP = r + t r i t ′ e 2 ikaN 1 − r 1 2 e 2 ikaN ,
ρ ( x , ω ) = Re ( r L ( ω ) e 2 i d L ω + r R ( ω ) e 2 i d R ω + 2 1 − r L ( ω ) r R ( ω ) e 2 i ( d L + d R ) ω − 1 ) , d L = − d R = x .
− i ω α d = r R ( ω ) e 2 i ω d R − 1 r R ( ω ) e 2 i ω d R + 1 + r L ( ω ) e 2 i ω d L − 1 r L ( ω ) e 2 i ω d L + 1 ,
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