Complex source point theory of paraxial and nonparaxial elliptical Gaussian beams

Using the operator transformation technology, we extend the circular Gaussian beam based virtual (complex) sources method to investigate the paraxial and nonparaxial propagation properties of the elliptical Gaussian beams (EGBs) with planar or cylindrical wavefronts travelling in free space. The paraxial approximation analysis reveals the self-reappearance and self-focusing propagation features for the EGBs with cylindrical wavefront under proper parameter conditions. We further introduce the nonparaxial theory to derive the analytical expressions for the field distribution of an EGB in free space, and confirm that these intriguing propagation features can still be observed with added nonparaxial correction. Comparing with the paraxial approximation results, it is worth noting that there is a clear deviation of the on-axial intensity and phase distributions near the self-focusing position on the basis of nonparaxial correction solution. Our results reveal that the anisotropic diffraction of light propagating through homogeneous medium or free space is possible. The approach in this work can easily be generalized to other beam models with elliptical geometry, which allows us to correctly predict some important information about their near field propagation characteristics for various applications.