Virtual sources for coherent combination beams consisting of off-axis Gaussian beams.

Based on the operator transformation technique, the multiple complex point sources required to generate a coherent superposition of waves are introduced and a closed-form analytical expression is derived for this composite wave. From the expression of the composite wave, the paraxial approximation and the nonparaxial corrections of all orders for the corresponding paraxial beam are determined. The paraxial composite beam uniformly represents off-axis Gaussian beams (GBs), sin(cos)-GBs, sinh(cosh)-GBs, nth-order modified Bessel-GBs, and Bessel-GBs with topological charge.