Propagation analysis of the Laguerre–Gaussian beam with astigmatism

The entire range of transformations that a Laguerre-Gaussian (LG) beam with astigmatism can go through in free space is clarified. The transformations are governed by the relative phase between the astigmatic Hermite-Gaussian components. Formulas describing the behavior of this relative phase are obtained and used to classify and map the transformation patterns to initial beam parameters. The difference between an LG beam and a phase singular beam generated by a hologram under astigmatic conditions is also investigated.