Constructing a polynomial whose nodal set is the three-twist knot 525_2

We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot $5_2$. The construction generalizes a similar approach for lemniscate knots: a braid representation is engineered from finite Fourier series and then considered as the nodal set of a certain complex polynomial which depends on an additional parameter. For sufficiently small values of this parameter, the nodal lines form the three-twist knot. Further mathematical properties of this map are explored, including the relationship of the phase critical points with the Morse-Novikov number, which is nonzero as this knot is not fibred. We also find analogous functions for other knots with six crossings. The particular function we find, and the general procedure, should be useful for designing knotted fields of particular knot types in various physical systems.Comment: 19 pages, 6 figure

Limits to superweak amplification of beam shifts

The magnitudes of beam shifts (Goos-H\"anchen and Imbert-Fedorov, spatial and angular) are greatly enhanced when a reflected light beam is postselected by an analyzer, by analogy with superweak measurements in quantum theory. Particularly strong enhancements can be expected close to angles at which no light is transmitted for a fixed initial and final polarizations. We derive a formula for the angular and spatial shifts at such angles (which includes the Brewster angle), and we show that their maximum size is limited by higher-order terms from the reflection coefficients occurring in the Artmann shift formula.Comment: 3 pages, 2 figures, Optics Letters styl

Topological aberration of optical vortex beams and singularimetry of dielectric interfaces

The splitting of a high-order optical vortex into a constellation of unit vortices, upon total reflection, is described and analyzed. The vortex constellation generalizes, in a local sense, the familiar longitudinal Goos-H\"anchen and transverse Imbert-Federov shifts of the centroid of a reflected optical beam. The centroid shift is related to the centre of the constellation, whose geometry otherwise depends on higher-order terms in an expansion of the reflection matrix. We present an approximation of the field around the constellation of increasing order as an Appell sequence of complex polynomials whose roots are the vortices, and explain the results by an analogy with the theory of optical aberration.Comment: 5 pages, 3 figures, REVTeX 4.

Propagation-invariant beams with quantum pendulum spectra: from Bessel beams to Gaussian beam-beams

We describe a new class of propagation-invariant light beams with Fourier transform given by an eigenfunction of the quantum mechanical pendulum. These beams, whose spectra (restricted to a circle) are doubly-periodic Mathieu functions in azimuth, depend on a field strength parameter. When the parameter is zero, pendulum beams are Bessel beams, and as the parameter approaches infinity, they resemble transversely propagating one-dimensional Gaussian wavepackets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of an operator which interpolates between the squared angular momentum operator and the linear momentum operator. The analysis reveals connections with Mathieu beams, and insight into the paraxial approximation.Comment: 4 pages, 3 figures, Optics Letters styl

Position, spin and orbital angular momentum of a relativistic electron

Motivated by recent interest in relativistic electron vortex states, we revisit the spin and orbital angular momentum properties of Dirac electrons. These are uniquely determined by the choice of the position operator for a relativistic electron. We overview two main approaches discussed in the literature: (i) the projection of operators onto the positive-energy subspace, which removes the zitterbewegung effects and correctly describes spin-orbit interaction effects, and (ii) the use of Newton-Wigner-Foldy-Wouthuysen operators based on the inverse Foldy-Wouthuysen transformation. We argue that the first approach [previously described in application to Dirac vortex beams in K.Y. Bliokh et al., Phys. Rev. Lett. 107, 174802 (2011)] has a more natural physical interpretation, including spin-orbit interactions and a nonsingular zero-mass limit, than the second one [S.M. Barnett, Phys. Rev. Lett. 118, 114802 (2017)].Comment: 10 pages, 1 table, to appear in Phys. Rev.

A random wave model for the Aharonov-Bohm effect

We study an ensemble of random waves subject to the Aharonov-Bohm effect. The introduction of a point with a magnetic flux of arbitrary strength into a random wave ensemble gives a family of wavefunctions whose distribution of vortices (complex zeros) are responsible for the topological phase associated with the Aharonov-Bohm effect. Analytical expressions are found for the vortex number and topological charge densities as functions of distance from the flux point. Comparison is made with the distribution of vortices in the isotropic random wave model. The results indicate that as the flux approaches half-integer values, a vortex with the same sign as the fractional part of the flux is attracted to the flux point, merging with it at half-integer flux. Other features of the Aharonov-Bohm vortex distribution are also explored.Comment: 16 pages, 5 figure

On the Burgers vector of a wave dislocation

Following Nye and Berry's analogy with crystal dislocations, an approach to the Burgers vector of a wave dislocation (phase singularity, optical vortex) is proposed. It is defined to be a regularized phase gradient evaluated at the phase singularity, and is computed explicitly. The screw component of this vector is naturally related to the helicoidal twisting of wavefronts along a vortex line, and is related to the helicity of the phase gradient. The edge component is related to the nearby current flow (defined by the phase gradient) perpendicular to the vortex, and the distribution of this component is found numerically for random two-dimensional monochromatic waves.Comment: 15 pages, 4 figures, IoP styl

Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects

Geometric phases are a universal concept that underpins numerous phenomena involving multi-component wave fields. These polarization-dependent phases are inherent in interference effects, spin-orbit interaction phenomena, and topological properties of vector wave fields. Geometric phases have been thoroughly studied in two-component fields, such as two-level quantum systems or paraxial optical waves. However, their description for fields with three or more components, such as generic nonparaxial optical fields routinely used in modern nano-optics, constitutes a nontrivial problem. Here we describe geometric, dynamical, and total phases calculated along a closed spatial contour in a multi-component complex field, with particular emphasis on 2D (paraxial) and 3D (nonparaxial) optical fields. We present several equivalent approaches: (i) an algebraic formalism, universal for any multi-component field; (ii) a dynamical approach using the Coriolis coupling between the spin angular momentum and reference-frame rotations; and (iii) a geometric representation, which unifies the Pancharatnam-Berry phase for the 2D polarization on the Poincar\'e sphere and the Majorana-sphere representation for the 3D polarized fields. Most importantly, we reveal close connections between geometric phases, angular-momentum properties of the field, and topological properties of polarization singularities in 2D and 3D fields, such as C-points and polarization M\"obius strips.Comment: 21 pages, 11 figures, to appear in Rep. Prog. Phy

Singular Values, Nematic Disclinations, and Emergent Biaxiality

Both uniaxial and biaxial nematic liquid crystals are defined by orientational ordering of their building blocks. While uniaxial nematics only orient the long molecular axis, biaxial order implies local order along three axes. As the natural degree of biaxiality and the associated frame, that can be extracted from the tensorial description of the nematic order, vanishes in the uniaxial phase, we extend the nematic director to a full biaxial frame by making use of a singular value decomposition of the gradient of the director field instead. New defects and degrees of freedom are unveiled and the similarities and differences between the uniaxial and biaxial phase are analyzed by applying the algebraic rules of the quaternion group to the uniaxial phase.Comment: 5 pages, 1 figure, submitted to PR

A three-dimensional degree of polarization based on Rayleigh scattering

A measure of the degree of polarization for the three-dimensional polarization matrix (coherence matrix) of an electromagnetic field is proposed, based on Rayleigh scattering. The degree of polarization, due to dipole scattering of the three-dimensional state of polarization, is averaged over all scattering directions. This gives a well-defined purity measure, which, unlike other proposed measures of the three-dimensional degree of polarization, is not a unitary invariant of the matrix. This is demonstrated and discussed for several examples, including a partially polarized transverse beam.Comment: 17 pages, 3 figures. OSA styl