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Coupled mode theory for photonic crystal cavity-waveguide interaction

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Abstract

We derive a coupled mode theory for the interaction of an optical cavity with a waveguide that includes waveguide dispersion. The theory can be applied to photonic crystal cavity waveguide structures. We derive an analytical solution to the add and drop spectra arising from such interactions in the limit of linear dispersion. In this limit, the spectra can accurately predict the cold cavity quality factor (Q) when the interaction is weak. We numerically solve the coupled mode equations for the case of a cavity interacting with the band edge of a periodic waveguide, where linear dispersion is no longer a good approximation. In this regime, the density of states can distort the add and drop spectra. This distortion can lead to more than an order of magnitude overestimation of the cavity Q.

©2005 Optical Society of America

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Figures (8)

Fig. 1.
Fig. 1. Dispersion relation for a periodic waveguide.
Fig. 2.
Fig. 2. Inverted waveguide transmission spectrum for different cavity resonant frequencies. Each spectrum has been normalized to its peak value.
Fig. 3.
Fig. 3. Drop spectrum for cavity with ω 0=0.46 and different quality factors (Q).
Fig. 4.
Fig. 4. SEM image of coupled cavity-waveguide system.
Fig. 5.
Fig. 5. FDTD simulation of cavity mode. Figure shows z-component of the magnetic field at the center of the slab.
Fig. 6.
Fig. 6. Dispersion relation for photonic crystal waveguide.
Fig. 7.
Fig. 7. Calculated coupling strength for cavity-waveguide system.
Fig. 8.
Fig. 8. Transmission spectrum of realistic cavity-waveguide system.

Equations (16)

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× × E + ε ( r ) c 2 2 E t 2 = 0
E w = B k ( r ) e i ( ω ( k ) t k z )
E = a ( t ) A ( r ) e i ω c t + d k B k ( r ) e i ω ( k ) t [ b ( k , t ) e i k z + c ( k , t ) e i k z ]
d a d t = i d k ω 2 ( k ) ω c e i Δ ω ( k ) t [ b ( k , t ) κ b a ( k ) + c ( k , t ) κ b a ( k ) ] v a + P c e i ( ω p ω ( k ) ) t
d b ( k ) d t = i ω c 2 κ a b ( k ) ω ( k ) a e i Δ ω ( k ) t + P w ( k ) e i ( ω p ω ( k ) ) t η b ( k )
d c ( k ) d t = i ω c 2 κ a b ( k ) ω ( k ) a e i Δ ω ( k ) t + P w ( k ) e i ( ω p ω ( k ) ) t η c ( k )
κ b a ( k ) = d r Δ ε w ( r ) c 2 e i k z B k · A * d r 2 ε t ( r ) c 2 A 2
κ a b ( k ) = d r Δ ε c ( r ) c 2 e i k z A · B k *
ω ( k ) = ω 0 + V g ( k k 0 )
1 ( ω c , p ω ( k ) ) + i s P [ 1 ω c , p ω ( k ) ] + i π δ ( ω c , p ω ( k ) )
a ( s ) = 1 s + λ + Γ + i δ ω ( a 0 + P c + J ( s i ( ω p ω c ) ) )
a ( t ) = a ( 0 ) e ( v + Γ ) t
b k ( t ) 2 = π κ a b P c 2 ω ( k ) 2 1 ( ω p ω c + δ ω ) 2 + ( ν + Γ ) 2
b k ( t ) 2 = P w ( k p ) 2 1 J i ( ω p ω c ) + ν + Γ 2
ω k = c n eff ( π a D 2 + ( k π a ) 2 )
P T = k d k b ( k , t f ) 2
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