Coupling, Scattering, and Perturbation Theory: Semi-analytical Analyses of Photonic-Crystal Waveguides Steven G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy,* M. Soljaˇ ci´ c, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos Dept. of Physics, Massachusetts Institute of Technology, stevenj@alum.mit.edu * OmniGuide Communications, Inc., Cambridge MA 02139 ABSTRACT Although brute-force simulations of Maxwell’s equations, such as FDTD methods, have enjoyed wide success in modeling 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. 1 INTRODUCTION Photonic crystals, periodic dielectric structures with a band gap that prohibits the propagation of light in a range of wavelengths, offer tantalizing new possibilities for controlling and designing optical phenomena [1]. In order to employ them in practical devices, especially in potential high-density integrated optical systems, however, one must gain a greater understanding of scattering loss mechanisms in such crystal structures. This task that is made more challenging by the use of high index contrasts, tight confinement, and strong periodic modulation, which invalidate many semi-analytical tools that were previously employed in more conventional optical waveguides. One approach that has been successful in many cases to employ brute-force simulation, such as FDTD methods, which model the full Maxwell’s equations with approximations only in the finite resolution and in the boundary conditions. Brute force simulation becomes more challenging and less illuminating, however, when applied to problems such as disorder-induced scattering and losses from slow transitions, due to the high spatial resolution and weak effects that those phenomena embody, as well as the large variety of scatterers that one might like to consider. It is these situations that we study in this paper, and we exploit their key property of involving small perturbations to an ideal system in order to develop efficient semi-analytical tools and even general analytical predictions. We are able to show, for example, that an adiabatic theorem applies to slow transitions in photonic crystals and periodic waveguides (and what scaling laws the losses follow as the length and group velocity of the waveguide are changed). For roughness/disorder losses, we are able to prove that, compared to an equivalent conventional waveguide (equal mode size, group velocity, and disorder strength), radiative scattering is suppressed and reflections are no worse in a photonic-crystal waveguide (rather than being increased as radiation is suppressed as one might fear). For a perturbation Δε(x) in the dielectric function ε(x), a key quantity is the volume current J ∼ Δε · E, where E is an electric field of the unperturbed system (whose solution is already known). This current is central to the computation of low-order corrections to electromagnetic eigenstates and eigenvalues in perturbation theory, is used for the coupling matrix element in coupled-wave theory, and acts as an oscillating source term for a Green’s function formalism. As we shall discuss below, all three of these are promising approaches in the analytical and semi-analytical study of small perturbations and slow waveguide transitions. First, however, we address an important problem in high index-contrast systems, that of the proper evaluation of Δε · E when Δε comes from a perturbed boundary between two dielectrics—in this case, the Δε · E current must be modified to avoid problems (incorrect, or even ill-defined expressions) due to the discontinuity in the electric field at dielectric interfaces.