  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.
