An Efficient Solver Of Eigenvalue Problem For The Second Order Linear Differential Operator With Variable Coefficiens

In optical waveguide,light is essentially an electromagnetic wave,which satisfies Maxwell's equation,it can be obtained the Helmholtz equation by Fourier transform.Therefore,the eigenvalue problem of second-order linear differential operators with variable coefficients is to solve the eigenmodes of a complex optical waveguide.This thesis focuses on the efficient solver of eigenvalue problem for the second order linear differential operator with variable coefficiens.First,the first derivative of characteristic equation for second order linear differential operator complicates the problem,in order to eliminate the first derivative,it is necessary to transform the characteristic equation for second order linear differential operator into a simpler Sturm-Liouville eigenvalue problem.Next,since the open waveguide has a unbounded region,so the boundary condition is introduced to obtain a bounded problem.In this thesis,the perfectly matched layer is used to truncate the unbounded region.Mathematically,the PML technique is to introduce a complex coordinate transform.Finally,the refractive-index profile is approximated by a piecewise polynomial of degree two for the Sturm-Liouville eigenvalue problem with variable coefficients,therefore,the original characteristic equation with variable coefficients can be approximated by a simple Sturm-Liouville characteristic equation in each piecewise subinterval.At the same time,the Sturm-Liouville characteristic equation can obtain a solver in each piecewise subinterval by Whittaker function.Adding the interface conditions,we can obtain a nonlinear equation,which is converted into a root-finding problem of the nonlinear algebraic equation(the dispersion relation).The approximate dispersion equations converge fast to the exact ones for the continuous refractive-index function as the maximum value of the subinterval sizes tends to zero.Müller's method is applied to the dispersion relations to compute the eigenmodes in the thesis,numerical simulations show that high-precision modes may be obtained with suitable initial values.