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A general family of structured Gaussian beams naturally emerges from a consideration of families of rays. These ray families, with the property that their transverse profile is invariant upon propagation (except for cycling of the rays and a global rescaling), have two parameters, the first giving a position on an ellipse naturally represented by a point on the Poincar\'e sphere (familiar from polarization optics), and the other determining the position of a curve traced out on this Poincar\'e sphere. This construction naturally accounts for the familiar families of Gaussian beams, including Hermite-Gauss, Laguerre-Gauss and Generalized Hermite-Laguerre-Gauss beams, but is far more general. The conformal mapping between a projection of the Poincar\'e sphere and the physical space of the transverse plane of a Gaussian beam naturally involves caustics. In addition to providing new insight into the physics of propagating Gaussian beams, the ray-based approach allows effective approximation of the propagating amplitude without explicit diffraction calculations.

Structured light refers to the generation and application of custom light fields. As the tools and technology to create and detect structured light have evolved, steadily the applications have begun to emerge. This roadmap touches on the key fields within structured light from the perspective of experts in those areas, providing insight into the current state and the challenges their respective fields face. Collectively the roadmap outlines the venerable nature of structured light research and the exciting prospects for the future that are yet to be realized.

Reconfigurable, ordered matter offers great potential for future low-power computer memory by storing information in energetically stable configurations. Among these, skyrmions—which are topologically protected, robust excitations that have been demonstrated in chiral magnets1–4 and in liquid crystals5–7—are driving much excitement about potential spintronic applications⁸. These information-encoding structures topologically resemble field configurations in many other branches of physics and have a rich history⁹, although chiral condensed-matter systems so far have yielded realizations only of elementary full and fractional skyrmions. Here we describe stable, high-degree multi-skyrmion configurations where an arbitrary number of antiskyrmions are contained within a larger skyrmion. We call these structures skyrmion bags. We demonstrate them experimentally and numerically in liquid crystals and numerically in micromagnetic simulations either without or with magnetostatic effects. We find that skyrmion bags act like single skyrmions in pairwise interaction and under the influence of current in magnetic materials, and are thus an exciting proposition for topological magnetic storage and logic devices.

As the size of an optical vortex knot, imprinted in a coherent light beam, is decreased, nonparaxial effects alter the structure of the knotted optical singularity. For knot structures approaching the scale of wavelength, longitudinal polarization effects become non-negligible, and the electric and magnetic fields differ, leading to intertwined
knotted nodal structures in the transverse and longitudinal polarization components, which we call a knot bundle of polarization singularities. We analyze their structure using polynomial beam approximations and numerical diffraction theory. The analysis reveals features of spin–orbit effects and polarization topology in tightly focused geometry, and we propose an experiment to measure this phenomenon.

A simple non-interferometric approach for probing the geometric phase of a structured Gaussian beam is proposed. Both the Gouy and Pancharatnam-Berry phases can be determined from the intensity distribution following a mode transformation if a part of the beam is covered at the initial plane. Moreover, the trajectories described by the centroid of the resulting intensity distributions following these transformations resemble those of ray optics, revealing an optical analogue of Ehrenfest's theorem associated with changes in geometric phase.

As the size of an optical vortex knot, imprinted in a coherent light beam, is decreased, nonparaxial effects alter the structure of the knotted optical singularity. For knot structures approaching the scale of wavelength, longitudinal polarization effects become non-negligible and the electric and magnetic fields differ, leading to intertwined knotted nodal structures in the transverse and longitudinal polarization components which we call a knot bundle of polarization singularities. We analyze their structure using polynomial beam approximations, and numerical diffraction theory. The analysis reveals features of spin-orbit effects and polarization topology in tightly-focused geometry, and we propose an experiment to measure this phenomenon.

Knots are topological structures describing how a looped thread can be arranged in space. Although most familiar as knotted material filaments, it is also possible to create knots in singular structures within three-dimensional physical fields such as fluid vortices¹ and the nulls of optical fields2–4. Here we produce, in the transverse polarization profile of optical beams, knotted lines of circular transverse polarization. We generate and observe both simple torus knots and links as well as the topologically more complicated figure-eight knot. The presence of these knotted polarization singularities endows a nontrivial topological structure on the entire three-dimensional propagating wavefield. In particular, the contours of constant polarization azimuth form Seifert surfaces of high genus⁵, which we are able to resolve experimentally in a process we call seifertometry. This analysis reveals a level of topological complexity, present in all experimentally generated polarization fields, that goes beyond the conventional reconstruction of polarization singularity lines.

Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein backbone chains, for which these mechanical constraints are of fundamental importance to their molecular functionality, despite their being open curves in which the knots are not mathematically well defined; knotting can only be identified by closing the termini of the chain somehow. We introduce a new method for resolving knotting in open curves using virtual knots, a wider class of topological objects that do not require a classical closure and so naturally capture the topological ambiguity inherent in open curves. We describe the results of analysing proteins in the Protein Data Bank by this new scheme, recovering and extending previous knotting results, and identifying topological interest in some new cases. The statistics of virtual knots in protein chains are compared with those of open random walks and Hamiltonian subchains on cubic lattices, identifying a regime of open curves in which the virtual knotting description is likely to be important.

The connection between Poincaré spheres for polarization and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic two-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite–Gaussian, Laguerre–Gaussian and generalized Hermite–Laguerre–Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the two-dimensional oscillator, which acquire an extra significance as families of classical ellipses upon semiclassical quantization.
This article is part of the themed issue ‘Optical orbital angular momentum’.

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.