Towards Realistic Simulation of Photonic Crystal Waveguide Losses

Transcription

1 Towards Realistic Simulation of Photonic Crystal Waveguide Losses R.Kappeler 1, P.Kaspar 1, G.Stark 1, F.Robin 1, Ch.Hafner 2 and H.Jäckel 1 1 Communication Photonics Group, Electronics Laboratory, ETH Zurich 2 Computational Optics Group, Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zurich ETH Zürich Electronics Laboratory

2 Outline Goal, motivation Photonic band gap calculations using finite elements Computation of radiation leakage in 2D using the open super cell approach Conclusion Outlook 2

3 Overall Goal: Active Photonic Crystal Devices Losses are a crucial parameters for active photonic crystal devices, since the effective gain is g = g α eff MAT WG Requires 3D simulations design/layer structure, such that waveguide losses are minimal 3

4 Motivation Indication, but no quantitative numbers 4

5 Goal: Method to Compute the Losses Input: structure/design material properties ε ( ω) r Method: Finite Elements - MPB: no dispersion, no radiation leakage/out of plane losses - maybe better scaling for complex structures 5

6 3D PhC Waveguide H ( x, t) = H (0) e ω β α z z i( t x i x) λ=1395nm, d=100nm 6

7 3D PhC Waveguide λ=1395nm, d=450nm Problem: - transient effects - multiple k-vectors - fitting: needs more data for better results Longer waveguides even more resource 7

8 Using the Periodicity of PhC Instead of a long waveguide, compute just a slice Periodic boundary condition quite efficient with FEM, good convergence* Open super cell approach combination of periodic boundary conditions and absorbing boundary conditions complex eigenvalue problem *A. von Rhein et al., Proceedings of the COMSOL Users Conference 2006 Frankfurt 8

9 Photonic Band Gap Calculations Using a Unit Cell for a Triangular Lattice A. von Rhein et al., Proceedings of the COMSOL Users Conference 2006 Frankfurt COMSOL MPB (Res 32) MPB (Res 256) 9

10 W1 Waveguides COMSOL MPB (Res 32) 10

11 Open Super Cell Approach J.Smaijc, et al., Progress in Electromagnetic Research Symposium 2004, Pisa Absorbing Boundary Condition complex k-vector Cx Periodic Boundary Condition H ( a) = H (0) e z z i(re{ Cx} + iim{ Cx}) a Eigenvalue Equation 2 1 ω H () r = H() r 2 ε r () r c Absorbing Boundary Condition Absorbing Boundary Condition Propagation direction Radiation leakage Periodic Boundary Conditions a 11

12 Open vs. Closed Super Cell 12

13 Eigen-Frequency Solver He z ωi ν = Im{ ω}/ 2π i( ω + iω ) t α = i Re{ ω} i Conversion to db/cm 1cm = v t gr 1cm α = ω α = 1 cm db / cm vgr 20log e 1 ω v gr 13

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17 Radiation Leakage 17

18 Improvements Drawbacks: Inaccurate results for low group velocities Tracking of the propagating modes Improvements: Solve the eigenvalue equation for fixed ω and solve for k instead 18

19 Conclusion Successful implementation of photonic band gap computation using FEM Computation of radiation leakage for 2D models with FEM Inaccurate results for some regions in the band diagram Many? left 19

20 Outlook Include material dispersion Include absorption of metallic layers Extension to 3D determine out of plane losses Comparison to measurements Structure Optimization 20

21 Thank you for your attention! Questions? 21