Speckle beams with nonzero vorticity and Poincaré–Cartan invariant

Abstract

The vorticity of a monochromatic speckle beam is introduced as an expectation value of the local difference of densities of right and left vortices, or wave-front dislocations. Gaussian statistics allows for the complete description of a speckle beam on the basis of the correlation function 〈E(r1)E*(r2)〉 only, and this function depends on the coordinate R=(r1+r2)/2 explicitly for statistically inhomogeneous beams. An analytic expression is found both for the vorticity and for the sum of the right and the left vortex densities. The vorticity is shown to be nonzero for inhomogeneous beams only. The Poincare–Cartan invariant of Hamilton’s classical mechanics or of geometrical optics is shown to be the topologically invariant integral of vorticity. An example is given of a beam with finite vorticity, which has Gaussian intensity profiles in both angular and spatial distributions. The conditions on the parameters that describe such a beam are found; these conditions follow from the positive character of probability.