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 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). 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 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. Abstract Periodic media are routinely used in optical devices and, in particular, photonic crystals to create spectral gaps, prohibiting the propagation of waves with certain temporal frequencies. In one dimension, Bragg structures, also called quarter-wave stacks, are frequently used because they are relatively easy to manufacture and the spectrum exhibits large spectral gaps at explicitly computable frequencies. In this short work, we use variational methods to demonstrate that within an admissible class of pointwise-bounded, periodic media, the Bragg structure uniquely maximizes the first spectral gap-to-midgap ratio. In the study of photonic crystals, the question arises naturally: Which crystals produce the largest band gaps? This question is investigated by means of an optimization-based evolution algorithm which, given two dielectric materials, seeks to produce a material distribution within the fundamental cell which produces a maximal band gap at a given point in the spectrum. The case of H-polarization in two dimensions is considered. Several numerical examples are presented. Phononic band–gap materials prevent elastic waves in certain frequency ranges from propagating, and they may therefore be used to generate frequency filters, as beam splitters, as sound or vibration protection devices, or as waveguides. In this work we show how topology optimization can be used to design and optimize periodic materials and structures exhibiting phononic band gaps. Firstly, we optimize infinitely periodic band–gap materials by maximizing the relative size of the band gaps. Then, finite structures subjected to periodic loading are optimized in order to either minimize the structural response along boundaries (wave damping) or maximize the response at certain boundary locations (waveguiding). The optimal design of photonic band gaps for two-dimensional square lattices is considered. We use the level set method to represent the interface between two materials with two different dielectric constants. The interface is moved by a generalized gradient ascent method. The biggest gap of GaAs in air that we found is 0.4418 for TM (transverse magnetic field) and 0.2104 for TE (transverse electric field).