Although perfectly matched layers (PMLs) have been widely used to truncate numerical simulations of electromagnetism and other wave equations, we point out important cases in which a PML fails to be reflectionless even in the limit of infinite resolution. In particular, the underlying coordinate-stretching idea behind PML breaks down in photonic crystals and in other structures where the material is not an analytic function in the direction perpendicular to the boundary, leading to substantial reflections. The alternative is an adiabatic absorber, in which reflections are made negligible by gradually increasing the material absorption at the boundaries, similar to a common strategy to combat discretization reflections in PMLs. We demonstrate the fundamental connection between such reflections and the smoothness of the absorption profile via coupled-mode theory, and show how to obtain higher-order and even exponential vanishing of the reflection with absorber thickness.
We compare several methods for the efficient generation of correlated random sequences (colored noise) by filtering white noise to achieve a desired correlation spectrum. We argue that a class of IIR filter-design techniques developed in the 1970s, which obtain the global Chebyshev-optimum minimum-phase filter with a desired magnitude and arbitrary phase, are uniquely suited for this problem but have seldom been used. The short filters that result from such techniques are crucial for applications of colored noise in physical simulations involving random processes, for which many long random sequences must be generated and computational time and memory are at a premium.
We consider the design of tapers for coupling power between uniform and slow-light waveguides in photonic crystals. We describe new optimization methods for designing robust tapers, which not only perform well under nominal conditions, but also over a given set of parameter variations. When the set of parameter variations models the inevitable variations typical in the manufacture or operation of the coupler, a robust design is one that will have a high yield, despite these parameter variations. We introduce the ideas of successive refinement, and robust optimization based on multi-scenario optimization with iterative sampling of uncertain parameters, using a fast method for approximately evaluating the reflection coefficient. We compare our robust design results to a linear taper, and to optimized tapers that do not take parameter variation into account. We verify the robust performance of our designs using an accurate, but much more expensive, method for evaluating the reflection coefficient.
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