A Numerical Approach for Defect Modes Localization in an Inhomogeneous Medium

Abstract

Some optical design problems arise from the study of photonic bandgap structure, including defect modes localization, that is, computing the optimal dielectric property to highly localize particular eigenfunctions of a Dirichlet model problem. The steepest descent method has been studied for this problem. In this paper, we present a new approach for the defect modes localization. Rather than focusing on the original objective and optimizing the structure along the gradient, a variant of the original problem is put forward with its corresponding method. Although the original problem and the variant presented in this work are not equivalent, our method is shown to solve both of them in numerical experiments. Furthermore, the algorithm in this paper can restart the optimization if the original gradient descent method gets stuck during the iteration. Introduction.In recent years, there has been some research involving photonic crystals (first introduced in [8, 15]) and the bandgap phenomenon, in which the wave whose frequency is in some band cannot propagate in this medium. The phenomenon is related to loss mechanisms in optical and mechanical systems, thus leading to a series of optimal problems about the medium. For example, we can derive the eigenvalue problem from wave equations, then set up the objective whose variables are the eigenpairs and search for the optimal solution in an admissible set. Cox and Dobson [13,14] have considered the optimization of bandgap in two-dimensional periodic structures composed of two given dielectric materials, including the E-polarization and H-polarization cases. Another level set method can be used in this optimization [5]; Lipton, Shipman, and Venakides [12] have optimized the electromagnetic resonant properties in periodic photonic crystal slabs, from the relation between resonance and transmission; Kao and Santosa [6] have derived the maximization of quality factor from wave equations in an inhomogeneous medium, whose index of refraction is the design variable. This optimization is related to a nonlinear eigenvalue problem, and the quality factor is defined as the ratio between the real part of the complex eigenfrequency and the imaginary part. Numerical results demonstrate that the optimal solution is a piecewise constant function, which is analytically proved by Karabash [9]; Heider et al. [10] have optimized scattering resonances in micro-and nano-scale components, thus decreasing the radiative loss, which is the magnitude of the imaginary part of scattering resonances; Osting [3] considered the optimization of two spectral functions from the Dirichlet-Laplacian eigenvalue problem. The variable domains are

Pareto optimization of resonances and minimum-time control

“…To address these goals several other approaches were proposed. One of directions [18,28] suggests to study resonant properties with the help of certain associated selfadjoint spectral problems avoiding in this way the difficulties of nonselfadjoint spectral optimization. One more direction employs 'solvable' models of Schrödinger operators with point interactions [32,4].…”

“…Such problems are much less studied in comparison with the selfadjoint spectral optimization, which go back to the Faber-Krahn solution of Lord Rayleigh's problem on the lowest tone of a drum. We refer to [16,18,27,37] for reviews and more recent studies of variational problems for eigenvalues of selfadjoint operators and would like to note that some of these studies [16,18,27] are directly or indirectly connected with resonance optimization, in particular, because square roots κ P iR `:" tic : c P R `u of nonpositive eigenvalues κ 2 are often considered to be resonances [39] for associated selfadjoint operators.…”

Nonlinear Bang–Bang Eigenproblems and Optimization of Resonances in Layered Cavities

“…[45,34,35,40]). The recent progress in fabrication of small size optical resonators [1,26,29] attracted considerable interest to numerical [19,15,3,11,29,31] and analytical [21,20,23] aspects of resonance optimization.…”

Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions

“…The present growth of interest in numerical [21,24,25,37,41] and analytical [26,27,28] aspects of resonance optimization is stimulated by a number of optical engineering studies of resonators with high quality factor (high-Q cavities), see [12,34,37,39] and references therein.…”