Analysis of butt coupling in photonic Crystals

We present a detailed analysis of butt coupling from conventional dielectric waveguides into photonic crystal waveguides. Closed-form expressions for the reflection and transmission matrices based on an eigenmode expansion technique are derived and validated by means of simulations. We use them to investigate butt-coupling losses in two kinds of photonic crystal structures: one formed by rods with a higher refractive index than the surrounding medium and the other formed by air holes inserted in a high-refractive-index medium. The origin and difference of coupling losses between the two photonic crystal structures is analyzed and discussed. We show that, although the coupling efficiency is much worse in the former structure, it can be significantly improved by choosing the optimum interface position that minimizes the mode impedance mismatch. Furthermore, the dependence of coupling efficiency on frequency is also analyzed. Finally, we also relate some traditionally used approximate formulas to our rigorous expressions.