Short and Long Range Screening of Optical Singularities

TL;DR

This paper investigates how optical singularities are screened in fields with different correlation ranges, deriving universal laws for charge variance growth and proposing practical measurement methods.

Contribution

It introduces a comprehensive analysis of charge screening in optical fields with short and long range correlations, including new formulas and measurement techniques.

Findings

Charge variance grows linearly with radius R in short range screening.

For long range screening with J0 correlation, variance grows as R ln R.

Measurement methods using zero crossings and charge counting are validated.

Abstract

Screening of topological charges (singularities) is discussed for paraxial optical fields with short and with long range correlations. For short range screening the charge variance in a circular region with radius $R$ grows linearly with $R$, instead of with $R^{2}$ as expected in the absence of screening; for long range screening it grows faster than $R$: for a field whose autocorrelation function is the zero order Bessel function J_{0}, the charge variance grows as R ln R$. A J_{0} correlation function is not attainable in practice, but we show how to generate an optical field whose correlation function closely approximates this form. The charge variance can be measured by counting positive and negative singularities inside the region A, or more easily by counting signed zero crossings on the perimeter of A. \For the first method the charge variance is calculated by integration over the charge correlation function C(r), for the second by integration over the zero crossing correlation function Gamma(r). Using the explicit forms of C(r) and of Gamma(r) we show that both methods of calculation yield the same result. We show that for short range screening the zero crossings can be counted along a straight line whose length equals P, but that for long range screening this simplification no longer holds. We also show that for realizable optical fields, for sufficiently small R, the charge variance goes as R^2, whereas for sufficiently large R, it grows as R. These universal laws are applicable to both short and pseudo-long range correlation functions.