We consider a system governed by the wave equation with index of refraction @math , taken to be variable within a bounded region @math and constant in @math . The solution of the time-dependent wave equation with initial data, which is localized in @math , spreads and decays with advancing time. This rate of decay can be measured (for @math , and more generally, @math odd) in terms of the eigenvalues of the scattering resonance problem, a non--self-adjoint eigenvalue problem governing the time-harmonic solutions of the wave (Helmholtz) equation which are outgoing at @math . Specifically, the rate of energy escape from @math is governed by the complex scattering eigenfrequency, which is closest to the real axis. We study the structural design problem: Find a refractive index profile @math within an admissible class which has a scattering frequency with minimal imaginary part. The admissible class is defined in terms of the compact suppo...
An important, related class of problems is to find photonic structures with large spectral band gaps. For the one-dimensional case, see the further discussion in Appendix . Structures with optimally large band gaps have been proven to exist @cite_28 @cite_41 and numerical methods have been applied to finding them @cite_46 @cite_7 @cite_17 . In @cite_34 , topology optimization was used to find photonic crystals with optimally large bandgaps and also which optimally damp or guide waves. In @cite_35 , properties of photonic crystals with optimally large bandgaps are investigated.
@cite_35 @cite_7 @cite_28 @cite_41 @cite_46 @cite_34 @cite_17
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Photonic crystals are periodic structures composed of dielectric materials and designed to exhibit band gaps, i.e., ranges of frequencies in which electromagnetic waves cannot propagate, or other interesting spectral behavior. Structures with large band gaps are of great interest for many important applications. In this paper, the problem of designing structures that exhibit maximal band gaps is considered. Admissible structures are constrained to be composed of "mixtures" of two given dielectric materials. The optimal design problem is formulated, existence of a solution is proved, a simple optimization algorithm is described, and several numerical examples are presented. This paper provides a review on the optimal design of photonic bandgap structures by inverse problem techniques. An overview of inverse problems techniques is given, with a special focus on topology design methods. A review of first applications of inverse problems techniques to photonic bandgap structures and waveguides is given, as well as some model problems, which provide a deeper insight into the structure of the optimal design problems. Photonic crystals can be designed to control and confine light. Since the introduction of the concept by Yablonovitch and John two decades ago, there has been a quest for the optimal structure, i.e., the periodic arrangement of dielectric and air that maximizes the photonic band gap. Based on numerical optimization studies, we have discovered some surprisingly simple geometric properties of optimal planar band gap structures. We conjecture that optimal structures for gaps between bands n and n + 1 correspond to n elliptic rods with centers defined by the generators of an optimal centroidal Voronoi tessellation (transverse magnetic polarization) and to the walls of this tessellation (transverse electric polarization). Optimization of a truncated, dielectric photonic crystal cavity leads to configurations that are far from truncated crystal cavities, and which have significantly better radiation confinement. Starting from a two-dimensional truncated photonic crystal cavity with optimal Q-factor, moving the rods from the lattice positions can increase the Q-factor by orders of magnitude, e.g., from 130 to 11 000 for a cavity constructed from 18 rods. In the process, parity symmetry breaking occurs. Achieving the same Q-factor with a regular lattice requires 60 rods. Therefore, using optimized irregular structures for photonic cavities can greatly reduce material requirements and device size.