Singularities in Speckled Speckle

Speckle patterns produced by random optical fields with two (or more) widely different correlation lengths exhibit speckle spots that are themselves highly speckled. Using computer simulations and analytic theory we present results for the point singularities of speckled speckle fields: optical vortices in scalar (one polarization component) fields; C points in vector (two polarization component) fields. In single correlation length fields both types of singularities tend to be more{}-or{}-less uniformly distributed. In contrast, the singularity structure of speckled speckle is anomalous: for some sets of source parameters vortices and C points tend to form widely separated giant clusters, for other parameter sets these singularities tend to form chains that surround large empty regions. The critical point statistics of speckled speckle is also anomalous. In scalar (vector) single correlation length fields phase (azimuthal) extrema are always outnumbered by vortices (C points). In contrast, in speckled speckle fields, phase extrema can outnumber vortices, and azimuthal extrema can outnumber C points, by factors that can easily exceed $10^{4}$ for experimentally realistic source parameters

The Machete Number

Knot theory is a branch of topology that deals with the structure and properties of links. Employing a variety of tools, including surfaces, graph theory, and polynomials, we develop and explore classical link invariants. From this foundation, we de fine two novel link invariants, braid height and machete number, and investigate their properties and connection to classical invariants

Short and Long Range Screening of Optical Singularities

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

Klein Link Multiplicity and Recursion

The (m,n)-Klein links are formed by altering the rectangular representation of an (m,n)-torus link. Using the braid representation of a (m,n)-Klein link, we generalize a previous braid word result and show that the (m, 2m)-Klein link can be expressed recursively. Applying braid permutations, we determine a formula for the number of components for an (m,n)-Klein link and classify the Klein links that are equivalent to knots

Klein Links and Braids

We introduce the construction of Klein links through an alteration to the orientation on the rectangular representation of a torus knot. We relate the resulting Klein links to their corresponding braid representations, and use these representations to understand the relationship between Klein links and torus knots as well as to prove relationships between several different Klein links

Singularities in Speckled Speckle: Screening

We study screening of optical singularities in random optical fields with two widely different length scales. We call the speckle patterns generated by such fields speckled speckle, because the major speckle spots in the pattern are themselves highly speckled. We study combinations of fields whose components exhibit short- and long-range correlations, and find unusual forms of screening