Topological Configurations of Optical Phase Singularities

Abstract

Geometric and topological properties of phase singularity lines in threedimensional complex scalar wavefields are discussed. In particular, their role as the intersections of the zero contour surfaces of the real and imaginary parts of the field gives numerous insights into 3D vortex topology. In addition, complex scalar wavefields (i.e. solutions of the three-dimensional Helmholtz and paraxial equations) are compared to more general complex scalar fields, including those arising naturally from algebraic geometry.

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

“…Statistical quantities such as the density of vortex lines per unit volume [6,8,9] and the probability distribution of the vortex lines' curvature [8] can be computed analytically, treating the wave field as an isotropic Gaussian random function. Calculation of the statistical distribution of other geometric quantities, such as the lines' torsion, seems analytically intractable by these methods [10].…”

Phase Singularities to Polarization Singularities

“…ere are also sign rules that describe the distribution of these critical points [12,17,[174][175][176][177]. Vortex trajectories with the help of topological manifolds, leading to formation of knots, links, and loops, were studied [170,171,[178][179][180][181][182][183][184]. Other types of phase defects such as edge and mixed-type phase defects [32,34], anisotropic vortices [185], and perfect [76,[186][187][188][189][190][191][192][193][194] and fractional vortices came under study [195][196][197][198][199][200][201][202][203][204].…”

A random wave model for the Aharonov–Bohm effect

“…For large R, g(R) ∼ 1 + 4 sin(2R)/πR [31], contrasting with ∆(R, β)/∆ irw ∼ 1 + 2 sin(βπ) cos(2R)/πR; in addition to the β-dependence, the peaks of maximum and minimum density are shifted by π/4 in R. Similarly, the asymptotic form g s (R) ∼ 8 cos(2R)/πR 2 , has a similar relationship with ρ(R, β)/∆ irw ∼ 2(1 − 2β) sin(βπ) sin(2R)/πR 2 ; in both cases, the charge oscillations decay more rapidly than the number oscillations, and are out of phase with them. Both g(R) and g s (R) are finite and nonzero at the origin, such that g(R) + g s (R) ≈ O(R 2 ) [3,26]. This is because, statistically, vortices of like charge repel, as can be seen from the like-charge correlation func-…”

“…Therefore g s (R) is analogous to ρ(R)/∆ irw , giving the average topological charge density at distance R from the fixed +1 vortex, and likewise g(R) to ∆/∆ irw . g(R) can be expressed as an integral representation [3] or in closed form as a complicated expression involving elliptic integrals [26]. The charge correlation function g s (R) is more straightforward [3, 23] and is…”

“…When N 1, by the central limit theorem Ψ irw (r) is a complex gaussian random variable at each r, so averages can often be calculated analytically. The statistical distribution of the vortices in this system has been studied in depth [3,19,23,24,[26][27][28][29]; the isotropic vortex density ∆ irw = 1/4π (in dimensionless units) and charge density ρ irw = 0 are obviously independent of position; 2-point vortex correlation functions have a richer structure, and will be discussed below.…”