  Many nonlinear optical technologies require the two-mode spectral amplitude
function that describes them---the \emph{joint spectral amplitude} (JSA)---to
be separable. We prove that the JSA factorizes \emph{only} when the incident
pump field and phase-matching function are Gaussian functions. We show this by
mapping our problem to a known result, in continuous variable quantum
information, that only squeezed states remain unentangled when combined on a
beam splitter. We then conjecture that only a squeezed state minimizes
entanglement when sent through a beam splitter with another pre-specified ket.
This implies that to maximize JSA separability when one of the (pump or
nonlinear medium) functions is non-Gaussian, the other function \emph{must} be
Gaussian. This answers an outstanding question about optimal design of certain
nonlinear processes, and is of practical interest to researchers using
waveguide nonlinear optics to generate and manipulate quantum light.
