Boundary Value Problem for an Oblique Paraxial Model of Light Propagation

TL;DR

This paper investigates a Schrödinger equation derived from a paraxial approximation of the Helmholtz equation for tilted propagation directions, focusing on boundary conditions in half-plane and quadrant domains, with implications for numerical methods.

Contribution

It analyzes boundary value problems for an oblique paraxial model, extending previous work to new domain configurations and boundary conditions.

Findings

Established boundary conditions for half-plane and quadrant domains.

Connected the model to numerical methods implemented in the HERA platform.

Provided theoretical insights into the tilted propagation Schrödinger equation.

Abstract

We study the Schr\"odinger equation which comes from the paraxial approximation of the Helmholtz equation in the case where the direction of propagation is tilted with respect to the boundary of the domain. This model has been proposed in (Doumic, Golse, Sentis, CRAS, 2003). Our primary interest here is in the boundary conditions successively in a half-plane, then in a quadrant of R2. The half-plane problem has been used in (Doumic, Duboc, Golse, Sentis, JCP, to appear) to build a numerical method, which has been introduced in the HERA plateform of CEA.