Evolution properties of four-petal Gaussian vortex beam propagating in uniaxial crystals orthogonal to the optical axis

The analytical propagation formulae of the four-petal Gaussian vortex beam propagating in uniaxial crystals orthogonal to the optical axis are derived, and some analyses are illustrated by numerical examples related to the evolution properties (intensity and phase) of four-petal Gaussian vortex beams in detail. The influences of beam order N, topological charge M and the ratio of refractive index ne/no on the normalized intensity distribution and phase distribution are discussed