Vector Fields in a Focused Laser Beam

We use the diffraction formalism developed by V. S. Ignatovsky in 1920 to analyze the vector components of a laser field focused by a parabolic mirror.  The Maxwell-respecting fields are expressed in terms of integrals.  We compare the results with various closed-form analytic models for the vector fields of a focused Gaussian beam.  The relatively simple model developed by Erikson and Singh in 1994 provides best agreement while other models show poor agreement, especially outside of the focal region.  The Ignatovsky method assumes the form of the laser beam incident on the mirror, while other approaches assume the form of the beam at the focus.  A smooth beam at the mirror, predicts a structured focus, whereas a smooth focus is associated with a structured beam at the mirror, which explains part of the discrepancy between models.