Analysis, Computation And Optimization For Eigenvalue Problems In Optical Models

Optics is very important in both daily life and industry. It is widely used in communication, computing, manufacturing, and storage. Scientists have been trying to establish effective models to explain revelent physical phenomena. For example, in diffractive optics, the focus is on micro-optics, which are structures of scales comparable to the wavelength of the visible light. This tiny structure makes it no longer possible to predict the wave propagation by the classical ge-ometrical optics approximation.As a kind of magnetic wave function, the propagation of light can be char-acterized by the Maxwell equations. We can the change the coefficients in the equation, or the various boundary conditions to model different kinds of materi-als. We wish to give some conclusions on the propagation of light without actually doing physical experiments, such like is the light absorbed? Which frequency is the light of? What is the nature of the material in which the light propagates? These questions can all be answered after a systematic study of the relevant mathematical models. We are concerned about these eigenvalue problems with actually physical application. This thesis is focus on the computation, analysis, the optimization of the eigenvalue problems in two kinds of optical models. The two problems are photonic crystal band structure optimization problem and the transmission eigenvalue problem in scattering theory.In the first chapter, we give a brief introduction for both kinds of eigenvalue problems.For the photonic band gap maximization problems of photonic crystals, we establish two different numerical methods to solve the problem. The first kind is piecewise constant level set method, which is a variant of the traditional level set method. Using the piecewise constant level set function to represent the struc-ture of the photonic crystal is the first step. Next we apply finite element method to solve the discrete eigenvalue problem, then derive the gradient derivatives by sensitivity analysis, finally use a generalization gradient ascent method to update the piecewise constant level set function. By this approach, we get the optimized structure for TM mode, TE mode, and absolute band gaps. The biggest gap is about 0.2922 for TE mode, which is the largest band gap ever reported to our best knowledge. For the absolute band gap maximization problem, we choose several Fourier functions as base function to represent the density function, changing the original structure optimization problem into a unknown coefficient determination problem. We also get some good numerical result.In the third chapter, we consider an important kind of interior/exterior transmission eigenvalue in scattering theory. We give a new numerical method to solve this eigenvalue problem with special structure. Firstly, we reformulate the problem into a fourth-order problem, then reduce it into a second-order or-dinary differential equation. We apply the Hermite finite element to the weak formulation of the equation. With proper rewriting of the matrix-vector form, the original nonlinear eigenvalue problem becomes a quadratic problem, which can be written as a linear system. Numerical results show this numerical method is fast, effective, and can calculate as many transmission eigenvalues as needed at a time.Summary and an outlook on future work are given in the last chapter.