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Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing

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Abstract

We show that the self-imaging principle still holds true in multi-mode photonic crystal (PhC) line-defect waveguides just as it does in conventional multi-mode waveguides. To observe the images reproduced by this self-imaging phenomenon, the finite-difference time-domain computation is performed on a multi-mode PhC line-defect waveguide that supports five guided modes. From the computed result, the reproduced images are identified and their positions along the propagation axis are theoretically described by self-imaging conditions which are derived from guided mode propagation analysis. We report a good agreement between the computational simulation and the theoretical description. As a possible application of our work, a photonic crystal 1-to-2 wavelength de-multiplexer is designed and its performance is numerically verified. This approach can be extended to novel designs of PhC devices.

©2004 Optical Society of America

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Figures (7)

Fig. 1.
Fig. 1. Schematic illustration of a multi-mode waveguide. Input image is reproduced at x=Lm and at x=Ld .
Fig. 2.
Fig. 2. Computational setup for observation of self-imaging phenomena. The black dots represent dielectric rods (n=3.4) in air and their radius is 0.18a, where n is the refractive index of the rods and a is the lattice constant of the PhC.
Fig. 3.
Fig. 3. (a) The dispersion curve for the access PCW and the computational super-cell (inset). The access PCW ensures single-mode operation from 0.312(a/λ) to the top of band gap. (b) The dispersion curve for the multi-mode PCW and the computational super-cell (inset). The multi-mode PCW supports 4 guided modes at 0.37(a/λ) and 5 guided modes at 0.43(a/λ).
Fig. 4.
Fig. 4. Modal patterns of electric field z-component for each mode at the operation frequency 0.37(a/λ) presented in Fig. 3. (a) Input image for the access PCW, (b) the 0th mode, (c) the 1st mode, (d) the 2nd mode, and (e) the 3rd mode at 0.37(a/λ).
Fig. 5.
Fig. 5. (a) Steady-state electric field distribution at 0.37(a/λ). (b) Time-averaged Poynting vector distribution at 0.37(a/λ).
Fig. 6.
Fig. 6. The designed 1-to-2 PhC de-multiplexer.
Fig. 7.
Fig. 7. Steady-state electric field distributions in the designed PhC wavelength de-multiplexer (a) at 0.37(a/λ) and (b) at 0.43(a/λ).

Tables (3)

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Table 1. Parameters Used to Calculate Lm at 0.37(a/λ)

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Table 2. Parameters Used to Calculate Ld at 0.37(a/λ)

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Table 3. Output Power Normalized to Total Input Power

Equations (11)

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Ψ ( x , y ) = n = 0 p 1 c n φ n ( y ) e j β n x ,
Ψ ( L m , y ) = n = 0 p 1 c n φ n ( y ) e j β n L m
= c 0 φ 0 ( y ) e j β 0 L m + c 2 φ 2 ( y ) e j β 2 L m + c 4 φ 4 ( y ) e j β 4 L m + ,
+ c 1 φ 1 ( y ) e j β 1 L m + c 3 φ 3 ( y ) e j β 3 L m + c 5 φ 5 ( y ) e j β 5 L m + · ·
= Ψ ( 0 , y ) e j Δ m
Ψ ( 0 , y ) e j Δ m = n = 0 p 1 c n φ n ( y ) e j Δ m
= c 0 φ 0 ( y ) e j Δ m + c 2 φ 2 ( y ) e j Δ m + c 4 φ 4 ( y ) e j Δ m + .
c 1 φ 1 ( y ) e j Δ m c 3 φ 3 ( y ) e j Δ m c 5 φ 5 ( y ) e j Δ m
φ n ( y ) = { φ n ( y ) for n even φ n ( y ) for n odd
β n L m = { 2 k n π + Δ m for n even ( 2 k n 1 ) π + Δ m for n odd with k n = 1 , 2 , 3 .
β n L d = 2 k n π + Δ d with k n = 1 , 2 , 3 ,
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