Modal Analysis Of Nonhoniogeneous Open Waveguide And The Application Of PML

In this thesis, the modal analysis of a kind of nonhomogeneous waveguide which is unbounded and with continuous refractive index profile is considered. And the applications of the Perfectly Matched Layers (PML) in the simulation of open waveguide are also studied. The influence of PML on modal analysis, and the problems of PML in computation of nonlinear Schrodinger equation are included in the study.An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. The equation of the optical waveguide is the Maxwell’s equation, which can also be reduced into Helmholtz equation under some certain assumptions. The eigemnode expansion technique (EET) is a fre-quently used tool to solve the Helmholtz equation. The field can be expressed as the expansion of the modes, and a mode corresponds to an unique propagation constant, which is solved by the eigenvalue problem of a Sturm-Liouville oper-ator. The eigenvalues of the unbounded Sturm-Liouville operator are divided into two sets:the propagation modes, and the radiation modes. The latter is uncountable. As a result, the sum of a finite number of leaky modes are used to approximate the continuum of radiation modes.If the refractive index profile of the waveguide is a piecewise-constant func-tion, the propagation constants of propagation modes and leaky modes satisfy a nonlinear equation, which is also named as the dispersion relation of the waveg-uide. However, a more general situation is considered in this thesis, when the refractive index profile is a continuous function in the core. The differential trans-fer matrix method is used, and the dispersion relation for the exact solutions as leaky modes are given. Though it is not easy to solve, based on the assump-tion that the waveguide is slowly varied, the approximated dispersion relation is solvable by Newton’s method.Since the problem is unbounded, in practical computations, the perfectly matched layer (PML) technique is applied to truncate the infinite domain. A new kind of mode which only depends on the parameter of the PML can be observed besides the leaky modes, which is named as the Berenger modes or the PML modes. For the problems with PML, the differential transfer matrix method is also useful to derive the dispersion relation for the propagation constants of Berenger modes. Under the assumption of slowly varying, the approximated dis-persion relation can also be solved by Newton’s method. Furthermore, based on the different behaviors of leaky modes and Berenger modes, different formula-tions of asymptotic solutions are deduced for leaky modes and Berenger modes separately, which are used as initial guesses of the iteration.In the computation of nonlinear Schrodinger equation, PML is also very useful. In practical computation, the domain inside the PML is discrete, and errors appear due to the discretization. The method for reducing the errors is proposed in this thesis, which is based on the redistribution of finite difference grids, and is very efficient for time-dependent equations.The most important work of this thesis is that the dispersion relations of the exact solutions as leaky modes and Berenger modes are deduced by the differen-tial transfer matrix method. The methods here are efficient for the waveguides in which the refractive index profiles are slowly varying. Other methods are al-so applicable for the computation of the propagation constants of leaky modes or Berenger modes, since the dispersion relations have been built. For waveg-uides in which the refractive index profiles are not slowly varied, the dispersion relations for exact solutions still hold, but more efficient methods need further investigations.