A detailed statistical representation of the local structure of optical vortices in random wavefields

Abstract

The statistical properties near phase singularities in a complex wavefield are here studied by means of the conditional distributions of the real and imaginary Gaussian components, given a common zero crossing point. The exact distribution is expressed as a Slepian model, where a regression term provides the main structure, with parameters given by the gradients of the Gaussian components at the singularity, and Gaussian non-stationary residuals that provide local variability. This technique differs from the linearization (Taylor expansion) technique commonly used. The empirically and theoretically verified elliptic eccentricity of the intensity contours in the vortex core is a property of the regression term, but with different normalization compared to the classical theory. The residual term models the statistical variability around these ellipses. The radii of the circular contours of the current magnitude are similarly modified by the new regression expansion and also here the random deviations are modeled by the residual field.

The relation between wave asymmetry and particle orbits analysed by Slepian models

“…Advanced applications of the Slepian model are found in signal processing (Szajnowski 1996) for bandwidth estimation, in seismic design to describe plastic deformation (Lazarov & Ditlevsen 2005; Feau 2008), optics to describe an optical vortex, also called a ‘black hole’, (Lindgren 2012), cosmology to describe high minima and low maxima in energy fields (Blanco-Pillado et al. 2020) and frequently over many years in safety studies in mechanical and structural engineering by Ditlevsen (1985) and subsequently Sobczyk (1993, §§ 4.4–4.5), Randrup-Thomsen & Ditlevsen (1997), Ditlevsen & Tarp-Johansen (1999), van de Lindt & Niedzwecki (2005) and Grigoriu (2020).…”