  The intensity peak of a "flying focus" travels at a programmable velocity
over many Rayleigh ranges while maintaining a near-constant profile. Assessing
the extent to which these features can enhance laser-based applications
requires an accurate description of the electromagnetic fields. Here we present
exact analytical solutions to Maxwell's equations for the electromagnetic
fields of a constant-velocity flying focus, generalized for arbitrary
polarization and orbital angular momentum. The approach combines the complex
source-point method, which transforms multipole solutions into beam-like
solutions, with the Lorentz invariance of Maxwell's equations. Propagating the
fields backward in space reveals the space-time profile that an optical
assembly must produce to realize these fields in the laboratory. Comparisons
with simpler paraxial solutions provide conditions for their reliable use when
modeling a flying focus.
