  Analytical solutions to the wave equation in spheroidal coordinates in the
short wavelength limit are considered. The asymptotic solutions for the radial
function are significantly simplified, allowing scalar spheroidal wave
functions to be defined in a form which is directly reminiscent of the
Laguerre-Gaussian solutions to the paraxial wave equation in optics.
Expressions for the Cartesian derivatives of the scalar spheroidal wave
functions are derived, leading to a new set of vector solutions to Maxwell's
equations. The results are an ideal starting point for calculations of
corrections to the paraxial approximation.
