Abstract

We introduce a layer potential representation for the solution of the scattering problem defined by two dielectric channels, or open wave-guides, meeting along a straight line, orthogonal to both channels, which is well adapted to numerical implementation. This is a simple example of a wave-guide network. The main observation is that the outgoing fundamental solution for the operator ${\partial }_{x_1}^2+{\partial }_{x_2}^2+k_1^2+q(x_2)$, acting on functions defined in ${ℝ}^2$, is easily constructed using the Fourier transform in the $x_1$-variable and the classical theory of ordinary differential equations. These fundamental solutions can then be used to represent the solution to the open wave-guide network scattering problem in half planes. The $H_{\mathrm{loc}<!--FUNCTION\; APPLICATION-->}^2$-regularity of the solution to the scattering problem imposes transmission boundary conditions along the common boundary, which then leads to integral equations along the intersection of the half planes. We show that, in appropriate Banach spaces, these integral equations are Fredholm equations of index zero, which are therefore generically solvable. We also analyze the representation of the guided modes in our formulation.

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