Vortex derivatives

It has long been known that optical vortices are located at intersections of the zero crossings of the real and imaginary parts of the wave function. By examining singular (divergent) phase and other derivatives of all orders, we now show that there are infinitely many additional zero crossings passing through every vortex. Each zero crossing makes its own topological demands on the wave function, leading to infinitely many topological constraints on the wave field structure.