Mode Field Analysis Method Of The Planar Optical Waveguides Based On An Integral Eguation

With the rapid development of optical fiber sensing and communication technology,the study of the integrated optics and optoelectronic devices which are based on the theory of guidewave optics has got more and more attention. In the integrated optics, planar dielectric optical waveguide is the basic construction unit, and the analysis of the mode field requires to solve the Maxwell’s equations. Except of a few of simple structures of the planar optical waveguide, the planar optical waveguide of arbitrary refractive index profile has no analytical solution, so it needs to be analyzed by using the numerical algorithm.This thesis puted forward a numerical algorithm of the Fredholm integral equation to study the mode field of planar optical waveguide. The algorithm can be used to solve the mode fields and dispersion relation of planar dielectric waveguide with any refractive index profiles. The algorithm can get high accuracy results by using a small calculation quantity and a low order matrix. At the same time, this method can also be generalized to the two-dimensional dielectric optical waveguide.Firstly,this thesis introduced the concept, development and research status of the integrated optics in this paper. The one dimensional wave equation of the planar optical waveguide mode field was expounded, and the numerical algorithm of the planar optical waveguide mode field was summarized.Secondly, this paper introduced the concept, the types of integral equation, and the several commonly used numerical algorithms to solved the second kind Fredholm integral equations. The Nystr?m method was discussed in detail, and the second kind Fredholm integral equations were transformed into matrix eigenvalue by using the gauss-legendre integral rule; At last, the way of using the LAPACK package to solve the linear algebra problem was explained.Thirdly, the one dimensional wave equation of the time domain was transformed into the second kind Fredholm integral equation of space-frequency domain by using Fourier transform. The second type of Legendre function was introduced the singular integral of the kernel function that might occur in the integral equation. Nystr?m method was used to solve the integral equation and it turned into a matrix eigenvalue problem. The eigenvalueand eigenvector of the matrix was solved by using the LAPACK packages. The mode fields and dispersion relation of a planar dielectric waveguide could be obtained simultaneously according to the given eigenvalue and eigenvector.Finally, the accuracy of the method in this thesis was verified by comparing with its analytical solutions. And the calculation accuracy was futher improved by the method of partitioned matrix; The feasibility and accuracy of this method was proved by calculating the mode fields and dispersion relation of planar optical waveguides with different refractive index distribution.