Short- and long-range screening of optical phase singularities and C points

Screening of signed (charged) singularities-phase vortices in scalar fields, C points in vector fields, is discussed for paraxial optical fields with short- and long-range correlations. A circular region of radius R is assumed. Short-range screening is exemplified by a Gaussian field correlator, long-range screening by a J₀ Bessel function. The short-range screening length is obtained analytically; this is found to be in substantial agreement with recent experiments. For long-range screening, an accurate asymptotic formula suitable for quantitative comparison with data (numerical or laboratory) is derived for the variance of the net charge. A J₀ correlation function is not attainable in practice, but it is shown how to generate a pseudo-long-range optical field whose correlation function closely approximates this form; screening in such a field is well described by our theoretical results for J₀. The charge variance can be measured by three different methods: by counting positive and negative singularities inside the region of interest, by counting signed zero crossings on the perimeter of this region; or by measuring phase derivatives along the perimeter. For the first method, the charge variance is calculated by integration over the charge correlation function, for the second (third) by integration over the zero crossing (phase derivative) correlation function. It is proven explicitly that, as expected, all three calculations yield the same result. It is also shown analytically that for short-range screening the zero crossings can be counted along a straight line whose length is 2 π R, but that for long-range screening this useful simplification no longer holds; for this case another formula is given that is suitable for data correction. The effects of boundary smoothing are discussed, and a class of generalized exponential smoothing functions is introduced. Analytical (numerical) results are given for the large R limit of the charge variance for the short (long) range case. Finally, it is shown that for realizable optical fields, both for the short and pseudo-long-range cases, for sufficiently small R the charge variance grows as R², whereas for sufficiently large R it grows as R.