# Geometry and Perturbative Sensitivity of non-Smooth Caustics of the Helmholtz Equation

## TL;DR

This paper investigates the geometry and stability of non-smooth caustics in Helmholtz equation solutions, revealing how their structure relates to complex singularities and remains invariant under certain perturbations.

## Contribution

It introduces a novel complex proper-time framework linking caustic geometry to singularities of the einbein action, demonstrating their invariance under index of refraction perturbations.

## Key findings

- Cusps are linked to poles of the einbein action in the complex plane.
- Ghost sources are spatial curves where pole residues vanish.
- Singularities are invariant under broad classes of perturbations.

## Abstract

The geometry of non-smooth $A_{n>2}$ caustics in solutions of the Helmholtz equation is analyzed using a Fock-Schwinger proper-time formulation. In this description, $A_3$ or cusp caustics are intimately related to poles of a quantity called the einbein action in the complex proper-time, or einbein, plane. The residues of the poles vanish on spatial curves known as ghost sources, to which cusps are bound. The positions of cusps along the ghost sources is related to the value of the poles. A similar map is proposed to relate essential singularities of the einbein action to higher order caustics. The singularities are shown to originate from degenerations of a certain Dirichlet problem as the einbein is varied. It follows that the singularities of the einbein action, along with the associated aspects of caustic geometry, are invariant with respect to large classes of perturbations of the index of refraction.