Exact Boundary Conditions for Periodic Waveguides Containing a Local Perturbation

Abstract

We consider the solution of the Helmholtz equation −∆u(x)−n(x)²ω²u(x) = f(x), x = (x,y), in a domain Ω which is infinite in x and bounded in y. We assume that f(x) is supported in Ω⁰ := {x ∈ Ω |a⁻ < x < a⁺} and that n(x) is x-periodic in Ω\Ω⁰. We show how to obtain exact boundary conditions on the vertical segments, Γ⁻ := {x ∈ Ω |x = a⁻} and Γ⁺ := {x ∈ Ω |x = a⁺}, that will enable us to find the solution on Ω⁰ ∪Γ⁺ ∪Γ⁻. Then the solution can be extended in Ω in a straightforward manner from the values on Γ⁻ and Γ⁺. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell.