Local phase structure of wave dislocation lines: twist and twirl

Abstract

Generic wave dislocations (phase singularities, optical vortices) in three dimensions have anisotropic local structure, which is analysed, with emphasis on the twist of surfaces of equal phase along the singular line, and the rotation of the local anisotropy ellipse (twirl). Various measures of twist and twirl are compared in specific examples, and a theorem is found relating the (quantized) topological twist and twirl for a closed dislocation loop with the anisotropy C line index threading the loop.

Singular values, nematic disclinations, and emergent biaxiality

“…A motivation for this investigation is the study of similar quasi-defect structures in optics, where topological filaments in the derivative of a complex scalar field determine the topology of optical vortices [27,28]. We demonstrate our technique on numerical models of blue phases and discuss the implications of newly extracted information.…”

Topological Configurations of Optical Phase Singularities

“…Another feature of nodal lines that is not necessarily revealed in the intersection of zero contour surfaces is the anisotropy of the vortex core [4,8,13,14,15]; the phase lines in the neighbourhood of a vortex are usually squeezed, in a way related to the anisotropy of the intensity growing away from the node [15] (Fig. 3b).…”

“…In the plane transverse to the vortex line -taken to be the xy-plane -any unit-strength vortex (with Ω ·ẑ > 0) can be written as a sum a(x + iy) + b(x − iy) with |a| > |b| (the complex ratio a/b stereographically projects to the Poincaré-like sphere describing the vortex anisotropy [4,8]). In fact, local analysis shows that in any solution of the Helmholtz equation, or the paraxial equation (provided the vortex line is not perpendicular toẑ), any high-strength vortex of strength must also be of similar form, a(x + iy) + b(x − iy) , rather than a more general polynomial in x and y [5].…”

“…In particular, as systems of curved lines in three dimensions, oriented by their vorticity (topological current), a range of topological configurations are possible in principle, such as infinite lines, closed loops, knots, links, braids etc. [1,5,6,7,8,9,10]. In linear optical fields, the vortex lines arise purely by interference.…”

“…For instance, a two-component link (Hopf link) of vortex loops can only arise in the intersection of wave surfaces of nonzero genus, such as a torus (as considered in Ref. [12]) or a noncompact torus [8].…”