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Highly efficient photonic crystal-based multichannel drop filters of three-port system with reflection feedback

Sangin Kim, Ikmo Park, Hanjo Lim, and Chul-Sik Kee

Open Access Open Access

Abstract

We have derived the general condition to achieve 100% drop efficiency in the resonant tunneling-based channel drop filters of a three-port system with reflection feedback. According to our theoretical modeling based on the coupled mode theory in time, the condition is that the Q-factor due to coupling to a bus port should be twice as large as the Q-factor due to coupling to a drop port and the phase retardation occurring in the round trip between a resonator and a reflector should be a multiple of 2π. The theoretical modeling also shows that the reflection feedback in the three-port channel drop filters brings about relaxed sensitivity to the design parameters, such as the ratio between those two Q-factors and the phase retardation in the reflection path. Based on the theoretical modeling, a five-channel drop filter has been designed in a two-dimensional photonic crystal, in which only a single reflector is placed at the end of the bus waveguide. The performance of the designed filter has been numerically calculated using the finite-difference time domain method. In the designed filter, drop efficiencies larger than 96% in all channels have been achieved.

©2004 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. General structure of the channel drop filter of the three-port system with reflection in the bus waveguide

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Fig. 2.
Fig. 2. (a) Dependence of drop efficiency on the ratio between the decay rates. Peak transmissions of the channel drop filter with (blue) and without reflection feedback (red) are calculated as functions of τb/τd using the coupled mode theory in time. For the filter with reflection feedback, ϕ=2π is assumed. (b) Dependence of drop efficiency on the phase retardation. Peak transmission of the filter with reflection feedback is calculated as a function of ϕ for τb/τd =4.

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Fig. 3.
Fig. 3. Structure of photonic crystal based multi-channel drop filter structure with reflection. The point defect resonators of different resonance frequencies are placed on the side of the bus waveguide, and the drop waveguides are directly coupled to the resonators in perpendicular direction to the bus waveguide.

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Fig. 4.
Fig. 4. Dependence of drop efficiency on location. Peak transmissions have been calculated using the 2D FDTD method for the resonators with defect sizes of r=0, 0.05a, 0.065a, and 0.08a at different distances from the closed end of the bus waveguide.

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Fig. 5.
Fig. 5. (a) Schematic diagram of the designed 5-channel drop filter, where channel drop filters are located on both sides of the bus waveguide and the bus waveguide is terminated with another channel drop filter in order to use the space more efficiently. The dielectric constant of all rods is 11.56. (b) Transmission spectrum of the designed filter. The transmission spectrum has been calculated using the 2D FDTD method.

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Fig. 6.
Fig. 6. (a) The wave propagation at the resonance frequency (f=0.3582 C/a) of the drop filters at port C (r=0.065a). (b) The wave propagation at the resonance frequency (f=0.3458 C/a) of the drop filter at port B (r=0.08a).

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Equations (7)

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d a d t = j ω 0 a − ( 2 τ b + 1 τ d ) a + e j θ 1 + 2 τ b s + 1 + e j θ 2 2 τ b s + 2 + e j θ 3 2 τ b s + 3 ,
s − 1 = s + 2 − e − j θ 1 2 τ b a
s − 2 = s + 1 − e − j θ 1 2 τ b a
s − 3 = − s + 3 − e − j θ 3 2 τ d a .
T ( ω ) = ∣ s ˜ − 3 s ˜ + 1 ∣ 2 = ∣ e j ( θ 1 − θ 3 ) 2 τ b 2 τ d ( 1 + e − j ϕ ) j ( ω − ω o ) + 2 τ b ( 1 + e − j ϕ ) + 1 τ d ∣ 2
R ( ω ) = ∣ s ˜ − 1 s ˜ + 1 ∣ 2 = ∣ e − j ϕ j ( ω − ω o ) − 2 τ b ( 1 + e − j ϕ ) + 1 τ d e − j ϕ j ( ω − ω o ) + 2 τ b ( 1 + e − j ϕ ) + 1 τ d ∣ 2 ,
T ( ω ) = ∣ s ˜ − 3 s ˜ + 1 ∣ 2 = ∣ e j ( θ 1 − θ 3 ) 2 τ b 2 τ d j ( ω − ω o ) + 2 τ b + 1 τ d ∣ 2 .
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