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Bloch mode modelling of propagation in photonic crystal waveguide and other devices

Lindsay Botten1 and Martijn de Sterke2

1CUDOS and Department of Mathematical Sciences, University of Technology, Sydney
2CUDOS and School of Physics, University of Sydney

11.00am, Tuesday 25 March 2003, AR103 Seminar Room, Graduate Research Centre

Photonic crystals (PC) will be key building blocks for future micro-optical technology, with PC waveguides being integral components in all-optical photonic processors. The realization of this new technology requires that coupling and guiding mechanisms be well understood. The coupling problem, in particular, is both challenging and general, involving the interfacing of one guide to another (e.g., at a bend or junction) or to some external medium. The majority of investigations have relied almost exclusively on numerical techniques, and while producing accurate results, they are computationally intensive and exploit little of the underlying physics to derive analytic models that can yield real insight into the actual propagation and scattering mechanisms.

In this talk, we describe a semi-analytic approach based on a Bloch mode method which separates the solution of the propagation problem in uniform waveguide sections from the scattering/diffraction of modes at interfaces. The modal basis is formed from the solution of an algebraic eigenvalue problem involving the reflection and transmission scattering matrices of a single crystal layer and we show how propagation in multi-layer structures may be elegantly handled in a manner that is a generalization of the analysis of multi-layer Fabry-Perot interferometers. Mode scattering at boundaries is handled by reflection and transmission matrices that are the analogues of Fresnel interface coefficients. In long waveguides, propagation is governed by only the real (propagating) states, allowing an asymptotic analysis and yielding a major computational simplification which, in the case of a single mode guide, leads to a simple two-parameter model.

We outline the formulation and demonstrate its accuracy with energy conservation and reciprocity results (which hold analytically within the theoretical framework). We demonstrate its computational efficiency and apply it to a range of simulations. In particular, we will cover the transmission properties of a displaced 90 degree waveguide bend and a modified directional coupler, demonstrate a number of surprising properties that arise due to a combination of Fabry-Perot resonances and tunneling effects, and infer simple and accurate models of the phenomena from both coupled mode theory and an asymptotic analysis from the Bloch-mode theory.

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