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We introduce a layer potential representation for the solution of thetransmission problem defined by two dielectric channels, or open wave-guides,meeting along the straight-line interface, $\{x_1=0\}.$ The main observation isthat the outgoing fundamental solution for the operator $\Delta +k_1^2+q(x_2),$acting on functions defined in ${\mathbb R}^2,$ is easily constructed using theFourier transform in the $x_1$-variable and the elementary theory of ordinarydifferential equations. These fundamental solutions can then be used torepresent the solution to the transmission problem in half planes. Thetransmission boundary conditions lead to integral equations along theintersection of the half planes, which, in our normalization, is the$x_2$-axis. We show that, in appropriate Banach spaces, these integralequations are Fredholm equations of second kind, which are thereforegenerically solvable. We analyze the representation of the guided modes in ourformulation.

Solving the Transmission Problem for Open Wave-Guides, I Fundamental Solutions and Integral Equations