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Sharp bends in photonic crystal waveguides as nonlinear Fano resonators

Andrey E. Miroshnichenko and Yuri S. Kivshar

Open Access Open Access

Abstract

We demonstrate that high transmission through sharp bends in photonic crystal waveguides can be described by a simple model of the Fano resonance where the waveguide bend plays a role of a specific localized defect. We derive effective discrete equations for two types of the waveguide bends in two-dimensional photonic crystals and obtain exact analytical solutions for the resonant transmission and reflection. This approach allows us to get a deeper insight into the physics of resonant transmission, and it is also useful for the study and design of power-dependent transmission through the waveguide bends with embedded nonlinear defects.

©2005 Optical Society of America

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Figures (5)
Fig. 1.
Fig. 1. (a) Spatial structure of the coupling coefficients Jnm (ω) of the effective discrete model (3) at ω=0.4×2πc/a. (b) Dependence of the specific coupling coefficients (marked) on the frequency ω.

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Fig. 2.
Fig. 2. Schematic view of two designs of the waveguide bends studied in the paper. Empty circles correspond to removed rods, dashed lines denote the effective coupling. Yellow circles mark the defects with different dielectric constant εd (|E|), which can be nonlinear.

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Fig. 3.
Fig. 3. Transmission coefficient of the waveguide bend for different values of the dielectric constant εd . The Fano resonance is observed when the value of the dielectric constant of the defect rod εd approaches the value of the dielectric constant of the lattice rod ε rod. The plot εd =εbg corresponds to the case when a rod is removed from the bend corner, whereas the plot εd =ε rod corresponds to the case when the lattice rod remains at the corner. For comparison, the crosses (×) show the results of the direct FDTD numerical calculations.

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Fig. 4.
Fig. 4. Transmission coefficient through the waveguide bend with a (yellow) defect rod placed outside the corner. In this case, there exist two Fano resonances, one of them is characterized by an asymmetric profile and corresponds to the perfect transmission.

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Fig. 5.
Fig. 5. Nonlinear transmission calculated for two types of the waveguide bends shown in Fig. 2. In both the cases, the Fano resonance is observed as the perfect reflection. The waveguide bend of the type B allows the perfect transmission that can be also tuned.

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Equations (20)

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[ ∇ 2 + ( ω c ) 2 ε ( x ) ] E ( x ∣ ω ) = 0 .
E ( x ∣ ω ) = ( ω c ) 2 ∫ d 2 y G ( x , y ∣ ω ) δ ε ( y ) E ( y ∣ ω ) ,
E n , m = ∑ k , l J n − k , m − l ( ω ) δ ε k , l E k , l
J n , m ( ω ) = ( ω c ) 2 ∫ r d d 2 y G ( x n , x m + y ∣ ω )
δ ε n , m = ε n , m − ε rod ,
ε n , m = ε bg , for n ≤ − 1 and m = 0 or n = 0 and m ≥ 1 .
( 1 − J 0 , 0 δ ε 0 ) E n , 0 = δ ε 0 J 1 , 0 ( E n + 1 , 0 + E n − 1 , 0 ) , n < − 1 ,
( 1 − J 0 , 0 δ ε 0 ) E 0 , m = δ ε 0 J 0 , 1 ( E 0 , m + 1 + E m , n − 1 ) , m > 1 ,
( 1 − J 0 , 0 δ ε 0 ) E − 1 , 0 = J 1 , 0 ( δ ε 1 E 0 , 0 + δ ε 0 E − 2 , 0 ) + J 1 , 1 δ ε 0 E 0 , 1 ,
( 1 − J 0 , 0 δ ε 1 ) E 0 , 0 = δ ε 0 ( J 0 , 1 E 0 , 1 + J 1 , 0 E − 1 , 0 ) ,
( 1 − J 0 , 0 δ ε 0 ) E 0 , 1 = J 0 , 1 ( δ ε 1 E 0 , 0 + δ ε 0 E 0 , 2 ) + J 1 , 1 δ ε 0 E − 1 , 0 ,
cos ⁡ k = 1 − δ ε 0 J 0 , 0 δ ε 0 J 0 , 1 ,
T = 4 a 2 sin ⁡ 2 k ∣ b ( c − b ) ∣ 2 ,
a = ( J 11 + J 0 , 1 2 δ ε 1 − J 0 , 0 J 1 , 1 δ ε 1 ) J 0 , 1 δ ε 0 2 , b = ( J 0 , 0 + exp ( i k ) J 0 , 1 − J 1 , 1 ) δ ε 0 − 1 ,
c = ( J 0 , 0 2 − 2 J 0 , 1 2 + J 0 , 0 J 1 , 1 + exp ( i k ) J 0 , 0 J 0 , 1 ) δ ε 0 δ ε 1 − J 0 , 0 δ ε 1 .
J 0 , 0 J 1 , 1 δ ε 1 = J 11 + J 0 , 1 2 δ ε 1 .
J 0 , 0 = ω − ω d , J 0 , 1 = C , J 1 , 1 = V J 0 , 0 = V ( ω − ω d ) ,
δ ε 1 V ω 2 − ( 2 δ ε 1 ω d + 1 ) V ω + ( V δ ε 1 ω d 2 + V ω d − C 2 δ ε 1 ) = 0 ,
ω F = ω d + 1 2 δ ε 1 ± [ C 2 V + 1 4 δ ε 1 2 ] 1 2 .
ε d ( ∣ E ∣ ) = ε d + λ ∣ E ∣ 2 .
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