物理光学模型中特征值问题的分析 计算及优化-analysis, calculation and optimization of eigenvalue problems in physical optical models.docx

ii 物理光学模型中特征值问题的分析、计算及优化 化后，问题转化为多项式特征值问题，最后将其线性化，最终数值结果显示算 法能够高效，精确的一次解出多个特征值。 第四章对全文的工作做了一下总结，并提出了未来的研究方向。 关键词：特征值问题，分片常数水平集方法，反散射问题，有限元方法，敏感 度分析 万方数据  Abstract 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，tile 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 pho