Parameterization of anisotropic vortices

Six parameterizations of the wave functions and the phase fields of anisotropic optical vortices (phase singularities) are described and compared. These parameterizations include seventeen different parameters that involve four partial derivatives of the real and the imaginary parts of the wave function, two Hilbert space rotational invariants, four gauge invariants, three geometric parameters, and four phase function parameters. Probability-density functions for all 17 parameters are presented for a Gaussian random wave field. It is shown quite generally that, within any given parameterization, four independent parameters are required to describe the wave function, three independent parameters are needed for the phase function, and two independent parameters suffice for the differential phase morphology. Vortex product wave functions with especially interesting transformation properties are also discussed.