Numerical Methods For Dispersive Dielectric Photonic Crystals With Ring Shape

Photonic crystals are a class of widely used materials,and their"photonic band gap" can realize the effective regulation of light.Dispersion photonic crystals are a special kind of photonic crystals.The refractive index of a certain medium that constitutes them changes with frequency,which not only has a significant impact on the distribution of the internal light field,thus providing a new way to adjust the photonic bandgap structure At the same time,it brings new challenges for numerical simulation to study the energy band structure of photonic crystals.Designing an efficient numerical calculation method to explore the energy band structure and light field distribution in photonic crystals with annular dispersion media is a research problem that has attracted extensive attention.This paper mainly studies the numerical simulation,mathematical model,calculation method and simulation implementation of the energy band structure of two two-dimensional dispersion medium photonic crystals based on the finite element method.And some properties of the photonic crystal with annular dispersion medium and the microcavity of the photonic crystal with annular dispersion medium are studied by using the numerical method proposed.In this thesis,we deduce and give two numerical calculation methods for solving the band structure of dispersive media in detail.If there is a dispersion material in the photonic crystal we are studying,then a nonlinear eigenvalue problem will be obtained by discretizing it through the finite element method.At this point we have given two methods:the first is to give the frequency ω and find the wave vector k.At this time,ω will be used as a known quantity,the wave vector k will be used as an eigenvalue,and the problem will be transformed into a quadratic eigenvalue problem to be solved.The second way to give the wave vector k and find the frequency ω is to combine the Newton method and the finite element method,starting from the governing equations of photonic crystal electromagnetic wave propagation,deduce the variational form,and then for the discrete nonlinear problem,Select the appropriate solution as the initial value,and use Newton’s inverse iterative method to solve the nonlinear equation,so as to obtain the energy band structure distribution of the photonic crystal.In this thesis,based on Matlab combined with COMSOL software,two numerical methods are implemented,and simulations are carried out to study the energy band structure of the annular dispersion medium photonic crystal and the quality factor of the annular dispersion medium photonic crystal microcavity.Numerical experimental results show that the numerical algorithm proposed in this paper can effectively solve the energy band structure of the annular dispersion medium photonic crystal.For some dispersion medium photonic crystals,it is possible to obtain a wider photonic band gap by using the ring structure.Crystalline microcavities also have a larger quality factor.