Photonic crystal waveguide analysis using interface boundary conditions

Devices based on combinations of photonic bandgap materials are understood intuitively in terms of the dispersion relations of the constituent periodic and locally homogeneous media. Quantitatively, though, photonic crystal-based devices are analyzed using numerical simulations which take no advantage of the a priori understanding of underlying periodic building-block materials. Here we unite the quantitative and qualitative pictures of photonic crystal devices and their design. We describe photonic crystals as effective media and impose boundary conditions between photonic crystals and homogeneous materials. We express optical field profiles as superpositions of plane waves in the homogeneous parts and propagating or decaying Bloch modes in the crystals, connected by transmission, reflection, and diffraction coefficients at the interfaces. We calculate waveguide modes, coupling lengths in directional couplers, and coupling between waveguides and point defects, achieving agreements of approximately 1% in frequencies and around 2% in quality factors. We use the new approach to optimize waveguide properties in a forward-going method, instead of the usual iterative optimizations.