A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides

TL;DR

This paper introduces a new radiation condition for ensuring the uniqueness of solutions to the Helmholtz equation in 2-D open waveguides, accounting for inhomogeneous refractive indices and guided wave components.

Contribution

It proposes an explicit uniqueness condition tailored for waveguides with inhomogeneous indices, extending classical radiation conditions to more complex propagation scenarios.

Findings

The new condition reduces to the classical Sommerfeld-Rellich condition in standard cases.

The condition is satisfied by known solutions in the literature.

It effectively distinguishes guided and non-guided wave components.

Abstract

We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. Our main result provides an explicit condition for uniqueness which takes into account the physically significant components, corresponding to guided and non-guided waves; this condition reduces to the classical Sommerfeld-Rellich condition in the relevant cases. Finally, we also show that our condition is satisfied by a solution, already present in literature, of the problem under consideration.