Research On Numerical Algorithm Of Bistable Problems In Photonic Crystals

Photonic crystal is a kind of artificial structure formed by the medium with different refractive index arranged in a certain period.The essential feature of this structure is the photonic bandgap.Photonic band gap is the range of forbidden frequencies in such structure,which arise from the destructive interference of multiple reflections of light propagating in a photonic crystal at the interfaces of the high-and low-dielectric constant regions.This unique property has led to the widespread use of photonic crystals in optics.Bistable phenomenon is one of the important nonlinear optical properties.It is widely used in optical limiter,logic gate,optical switch,optical memory and other devices.The introduction of kerr media with third-order nonlinear polarizations in a defected photonic crystal may cause bistability.In this thesis,we focus on the mathematical model,numerical algorithm and simulation for the bistable phenomenon in one-dimentional photonic crystals.In an one-dimensional nonlinear photonic crystal,the governing system of the bistable phenomenon is a nonlinear Helmholtz equation with two-point boundary conditions.The main chanlleges of solving this system is the nonlinear term from the kerr media and the existing of multiple solutions.Based on the two-point boundary value problem,two numerical methods are investigated in this thesis.One is an algorithm called continuous finite element iterative algorithm,and another is a finite element steady-state approximation algorithm.The continuous finite element iterative algorithm is proposed based on the finite element methods and the fixed-point iterative algorithm.In this method,a continuous process is introduced to deal with multiple solutions.The results of numerical simulation show that this algorithm is convergent fast and behaves very well to simulate bistable phenomena in one-dimensional photonic crystals.The finite element steady-state approximation algorithm is proposed based on the finite element methods and a steady-state approximation algorithm.The steady-state approximation is a key technique to deal with the bistable phenomenon.The results of numerical simulation results show that the algorithm can effectively simulate the bistable effect and can be extended to solve the high-dimensional problems.Finally,some numerical experiments using the finite element steady-state approximation algorithm is conducted to invertigate the influencing factors of bistable phenomena in photonic crastals.The results show that the threshold value of bistability in one-dimensional photonic crastals is affected not only by the frequency of incident light but also by other factors such as the thickness of the defect layer and the polarization of the material.The threshold value of bistable can be reduced by increasing periodicity or increasing the thickness of the middle layer,which is consistent with the conclusions of physical theory and experimentals.