Abstract
This paper describes the algorithm for the numerical solution of the diffraction problem of waveguide modes at the joint of two open planar waveguides. For planar structures under consideration, we can formulate a scalar diffraction problem, which is a boundary value problem for the Helmholtz equation with a variable coefficient in two-dimensional space. The eigenmode problem for an open three-layer waveguide is the Sturm-Liouville problem for a second-order operator with piecewise constant potential on the axis, where the potential is proportional to the refractive index. The described problem is singular and has a mixed spectrum and therefore the Galerkin method can not be used in this definition. One way to adapt the Galerkin method for the problem solution is to artificially limit the area, which is equivalent to placing the open waveguide in question in a hollow closed waveguide whose boundaries are remote from the real boundaries of the waveguide layer of the open waveguide. Thus, we obtain a diffraction problem on a finite interval and with a discrete spectrum, which can be solved by the projection method, as done in this paper.
Highlights
The paper presents the definition of the diffraction problem for waveguide modes at the joint of two open planar three-layer waveguides
The waveguide joint is the simplest model of waveguide transition, and the description of the diffraction problem at the interface is similar to the definition of the problem for a waveguide transition [1,2,3]
In this paper we described the algorithm for solving the diffraction problem for waveguide modes at the joint of two planar three-layer waveguides
Summary
The paper presents the definition of the diffraction problem for waveguide modes at the joint of two open planar three-layer waveguides. The classical method for solving diffraction problems in closed waveguides is the so-called incomplete Galerkin method [4,5,6,7]. The propagation of radiation in open waveguides is described by Maxwell's equations, boundary conditions, material equations, and asymptotic conditions. The boundary conditions on the boundaries of dielectric waveguide layers, imply the continuity of the tangential c ompo nEe n t sx o0f ,h th e e lHe c t rox m0,ah gn e0tic field [1,2,3]: (3). The asymptotic conditions at an infinite distance from the upper boundary of the waveguide layer and, respectively, the lower boundary, are described assuming the field are boun ded [1,2,3]: E x CE. Where CE and CH are some non-negative constants, and for guided waveguide modes C E C H 0
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