The Analysis of Ridge Waveguides Using Green's Functions

[摘要] In the theory of integrated optics, a central role is played by the rib waveguide geometry. The electromagnetics problems consists of the determination of the transverse mode profile and propagation constant of the dominant mode in the guide. Since this problem is not soluble in closed form by any known analytical methods, several numerical techniques have been used, amongst which the simple finite-difference (FD) method is very attractive because of the rectilinear geometry of the rib waveguide. However, the problem requires the imposition of a boundary condition at infinity in the transverse cross-section, which is not possible on a FD mesh of finite extent. The Green's function (GF) is an alternative formulation for this open-boundary problem. A suitable GF for a planar waveguide (with no rib) can be found analytically by Fourier transform methods, which can then be used to transform the Helmholtz equation for the region outside the rib from an elliptical partial differential equation into an integral equation on a contour surrounding the rib only, via Kirchhoff's theorem. This integral equation is matched to a FD solution for the field inside the rib by an iterative method. In this way the correct boundary conditions at infinity are automatically incorporated. In actual implementation of the method, a fully discretised analogue of Green's theorem on the FD mesh is used, in which the GF is the true inverse of the discrete FD operator, rather than discretising the exact boundary integral equation, which permits a systematic treatment of the singularity of the GF to be carried out. The method is explored fully by initially considering slab waveguides, i.e. with no rib. Application to the ridge waveguide is however not investigated to due time contraints, but a possible method is discussed.

[发布机构] University:University of Glasgow

[关键词] Electrical engineering