Optical Vortices in Random Wavefields

Vortices (topological phase singularities), which may be either positive or negative in sign, are found in many different types of optical fields. On every zero crossing of the real or imaginary part of the wavefield, adjacent vortices must be of opposite sign. This new "sign principle", which is unaffected by boundaries, leads to the surprising results that for a given set of zero crossings: (i) fixing the sign of any given vortex automatically fixes the signs of all other vortices in the wavefield, (ii) the sign of the first vortex created during the evolution of a wavefield determines the signs of all subsequent vortices, and (iii) the sign of this first vortex places additional strong constraints on the future development of the wavefunction. The sign principle also constrains how contours of equal phase thread through the wavefield from one vortex to another. Amplitude topological (AT) singularities are defined (for the first time) in terms of the gradient of the field amplitude. Such singularities correspond to stationary points of the amplitude and are located at the intersections of the zero crossings of the x- and y-partial derivatives of the amplitude. Amplitude maxima and minima are positive AT singularities and saddle points are negative AT singularities. The sign principle implies that adjacent AT singularities on any given zero crossing must be of opposite sign, that in an unbounded wavefield the total number of maxima and minima must equal the number of saddle points, that in free space stationary points of the amplitude must first appear as positive-negative AT singularity twins, and that there must exist strong correlations between neighboring maxima, minima and saddle points. The numerous, far reaching implications of the sign principle have been verified using a computer simulation that generates a random Gaussian wavefield.