Coupling, scattering, and perturbation theory: Semi-analytical analyses of photonic-crystal waveguides

Although brute-force simulations of Maxwell´s equations, such as FDTD methods, have enjoyed wide success in modelling photonic-crystal systems, they are not ideally suited for the study of weak perturbations, such as surface roughness or gradual waveguide transitions, where a high resolution and/or large computational cells are required. Instead, we suggest that these important problems are ideally suited for semi-analytical methods, which employ perturbative corrections (typically only needing the lowest order) to the exactly understood perfect waveguide. However, semi-analytical methods developed for the study of conventional waveguides require modification for high index-contrast, strongly periodic photonic crystals, and we have developed corrected forms of coupled-wave theory, perturbation theory, and the volume-current method for this situation. In this paper, we survey these new developments and describe the most significant results for adiabatic waveguide transitions and disorder losses. We present design rules and scaling laws for adiabatic transitions. In the case of disorder, we show both analytically and numerically that photonic crystals can suppress radiation loss without any corresponding increase in reflection, compared to a conventional strip waveguide with the same modal area, group velocity, and disorder strength.