  Picard-Lefschetz theory is applied to solutions of the Helmholtz equation,
formulated in terms of sums of integrals of a proper-time, or `einbein', wave
function $\Psi(\Lambda) = \exp(i\mathbb S(\Lambda))$ along complex contours
bounded by essential singularities of $\Psi$. There is a one to one map between
steepest descent paths connecting essential singularities and real or complex
eigenrays. Residues of finite $\Lambda$ poles of $\mathbb S(\Lambda)$ are shown
to vanish at spatial points corresponding to sources, provided that the pole
bounds only one steepest descent path. If the sum includes two such paths, with
one beginning and the other ending at the same pole, points of vanishing
residue are not sources, but are argued to be the locus on which caustic curves
may have singularities such as cusp points. The map between $\mathbb S$ and the
generating function in the Thom--Arnold classification of catastrophes is
discussed. Monodromies of the solution set with respect to complexified
parameters defining the index of refraction, or spatial endpoints of Green's
functions, are trivially determined from the singularities of $\mathbb
S(\Lambda)$. We construct a variant of a Laurent series expansion of $\mathbb
S$ about a pole at finite $\Lambda$. Expressions for the coefficients of each
order in this expansion can often be given exactly. Based on the Laurent series
expansion, we propose a variation of a Pad\'e approximant for $\mathbb S$, with
the intent of capturing additional poles and the associated cusp caustics which
are not visible in the Laurent series expansion.
