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

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˙Q 2¸i n a circular region A with radius R grows linearly with R, instead of with R 2 as expected in the absence of screening; for long range screening˙Q 2¸g rows faster than R: for a field whose autocorrelation function is the zero order Bessel function J0,˙Q 2¸∼ R ln R. A J0 correlation function is not attainable in practice, but we show how to generate an optical field whose correlation function closely approximates this form; screening in such a field is well described by our theoretical results for J0.˙Q 2¸c an be measured by counting positive and negative singularities inside A, or more easily by counting signed zero crossings on the perimeter P of A. For the first method˙Q 2¸i s calculated by integration over the charge correlation function C (r), for the second by integration over the zero crossing correlation function Γ (r). Using the explicit forms of C (r) and of Γ (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,˙Q 2¸∼ R 2 , whereas for sufficiently large R,˙Q 2¸∼ R. These universal laws are applicable to both short and pseudo-long range correlation functions.

Geometry and scaling of tangled vortex lines in three-dimensional random wave fields

“…The fact that the space-filling regime begins at this scale seems to be an indicator of the previously discussed rigidity and regularity of random fields satisfying the Helmholtz equation [16,35]. This property has been related in 2D to the infinite screening length of the vortex points as topological charges [8,36]. Given that the screening length of distributions of wavenumbers (such as a Gaussian), we anticipate that vortices in these fields would display n b = 3 only at larger values of δ.…”

Singularities in speckled speckle: Statistics

“…whereas the autocorrelation function of the finite width annulus, Eq. (7), is [3] W (R) (pr) = 1 pεr [(p + ε/2) J 1 (pr + εr/2)…”

Singularities in speckled speckle: screening

“…Nonsingular random sources produce random fields that exhibit short range screening [4 − 19]. In such systems positive (negative) topological charges are surrounded by a local net excess of negative (positive) charge, leading to charge neutrality within a characteristic distance, the screening length, that can be less than the average separation between charges [19].…”

“…Singular sources, such as a ring of finite radius but zero width [8], produce random fields that exhibits longrange screening [8,16,19]. The singularities in the field produced by a ring form a quasi-lattice in which positive/negative singularities occupy alternate corners of a square cell, Fig.…”

“…Q 2 ∼ R ∼ √ N , in contrast to the case of no screening, where Q 2 grows with the area, 8,19]. For the long-range screening produced by a singular ring source of radius p and zero width, the large R limit is [19 (Eq.…”