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Vector light beams, characterised by a spatially varying polarisation, can exhibit localised structures reminiscent of the Skyrmions familiar from the study of magnetic media. We present a theory of such Skyrmions within paraxial optics, exploiting mathematical analogies with the study of superfluids, especially the A phase of superfluid $\textrm{He}^3$. The key feature is the Skyrmion field which, together with the underlying Skyrmion vector potential, determines the properties of the Skyrmions and, more generally, the polarisation structure of every paraxial vector beam. In addition to structures with integer Skyrmion number we find polarisation patterns with non-integer Skyrmion number; these seem to have no analogue in other fields of physics.

The Skyrmion number of paraxial optical Skyrmions can be defined solely via their polarization singularities and associated winding numbers, using a mathematical derivation that exploits Stokes's theorem. It is demonstrated that this definition provides a robust way to extract the Skyrmion number from experimental data, as illustrated for a variety of optical (N\'eel-type) Skyrmions and bimerons, and their corresponding lattices. This method generates not only an increase in accuracy, but also provides an intuitive geometrical approach to understanding the topology of such quasi-particles of light, and their robustness against smooth transformations.

We introduce a simple mathematical expression based on rational maps to construct ideal paraxial optical skyrmions fields including Neel-type and Bloch-type skyrmions, anti-skyrmions, bimerons and multi-skyrmions, including skyrmion lattices. We review the rules that fully polarized paraxial light fields must obey to be considered as optical skyrmions. This work provides guidelines for the experimental generation of general skyrmion fields, beyond conventional single skyrmion beams. This lays the foundation for the exploration of nucleation and annihilation mechanisms in multiskyrmions fields

Geometric phases are ubiquitous in physics; they act as memories of the transformation of a physical system. In optics, the most prominent examples are the Pancharatnam-Berry phase and the spin-redirection phase. Recent technological advances in phase and polarization structuring have led to the discovery of additional geometric phases of light. The underlying mechanism for all of these is provided by fibre bundle theory. In this colloquium, we review how fibre bundle theory does not only shed light on the origin of geometric phases of light, but also lays the foundations for the exploration of high dimensional state spaces, with implications for topological photonics and quantum communications., Comment: Submitted to Reviews of Modern Physics

When focusing a light beam at high numerical aperture, the resulting electric field profile in the focal plane depends on the transverse polarisation profile, as interference between different parts of the beam needs to be taken into account. It is well known that radial polarised light produces a longitudinal polarisation component and can be focused below the conventional diffraction limit for homogeneously polarised light, and azimuthally polarised light that carries one unit of angular momentum can achieve even tighter focal spots. This is of interest for example for enhancing resolution in scanning microscopy. There are numerous ways to generate such polarisation structures, however, setups can be expensive and usually rely on birefringent components, hence prohibiting broadband operation. We have recently demonstrated a passive, low-cost technique using a simple glass cone (Fresnel cone) to generate beams with structured polarisation. We show here that the polarisation structure generated by Fresnel cones focuses better than radial polarised light at all numerical apertures. Furthermore, we investigate in detail the application of polarised light structures for two-photon microscopy. Specifically we demonstrate a method that allows us to generate the desired polarisation structure at the back aperture of the microscope by pre-compensating any detrimental phase shifts using a combination of waveplates.

We report the transfer of phase structure, and in particular of orbital angular momentum, from near-infrared pump light to blue light generated in a four-wave-mixing process in 85Rb vapour. The intensity and phase profile of the two pump lasers at 780nm and 776nm, shaped by a spatial light modulator, influences the phase and intensity profile of light at 420nm which is generated in a subsequent coherent cascade. In particular we oberve that the phase profile associated with orbital angular momentum is transferred entirely from the pump light to the blue. Pumping with more complicated light profiles results in the excitation of spatial modes in the blue that depend strongly on phase-matching, thus demonstrating the parametric nature of the mode transfer. These results have implications on the inscription and storage of phase-information in atomic gases., Comment: 4 pages, 4 figures. Referees' improvements incorporated

The measurement of high-dimensional entangled states of orbital angular momentum prepared by spontaneous parametric down-conversion can be considered in two separate stages: a generation stage and a detection stage. Given a certain number of generated modes, the number of measured modes is determined by the measurement apparatus. We derive a simple relationship between the generation and detection parameters and the number of measured entangled modes., Comment: 6 pages, 4 figures

We report the entanglement of topological features, namely, isolated, linked optical vortex loops in the light from spontaneous parametric down-conversion (SPDC). In three dimensions, optical vortices are lines of phase singularity and vortices of energy flow which percolate through all optical fields. This example of entanglement is between features that extend over macroscopic and finite volumes, furthermore, topological features are robust to perturbation . The entanglement of photons in complex three-dimensional(3D) topological states suggests the possibility of entanglement of similar structures in other quantum systems describable by complex scalar functions, such as superconductors, superfluids and Bose-Einstein condensates.

We demonstrate highly efficient generation of coherent 420nm light via up-conversion of near-infrared lasers in a hot rubidium vapor cell. By optimizing pump polarizations and frequencies we achieve a single-pass conversion efficiency of 260% per Watt, significantly higher than in previous experiments. A full exploration of the coherent light generation and fluorescence as a function of both pump frequencies reveals that coherent blue light is generated close to 85Rb two-photon resonances, as predicted by theory, but at high vapor pressure is suppressed in spectral regions that do not support phase matching or exhibit single-photon Kerr refraction. Favorable scaling of our current 1mW blue beam power with additional pump power is predicted., Comment: 6 pages, 4 figures. Modified to include referees' improvements

The quantum theory of rotation angles (S. M. Barnett and D. T. Pegg, Phys. Rev. A, 41, 3427-3425 (1990)) is generalised to non-integer values of the orbital angular momentum. This requires the introduction of an additional parameter, the orientation of a phase discontinuity associated with fractional values of the orbital angular momentum. We apply our formalism to the propagation of light modes with fractional orbital angular momentum in the paraxial and non-paraxial regime., Comment: 14 pages, 6 figure (figures are of low quality due to jpeg2ps conversion), submitted to J Mod Opt special issue for the Photon 06 conference

We propose a versatile optical ring lattice suitable for trapping cold and quantum degenerate atomic samples at discrete angular positions. We demonstrate the realisation of intensity patterns generated from Laguerre-Gauss ($\exp(i \ell\theta)$) modes with different $\ell$ indices. The ring lattice can have either intensity maxima or minima, suitable for trapping in red or blue detuned light, and it can be rotated by introducing a frequency shift between the Laguerre Gauss modes. The potential wells can be joined to form a uniform ring trap, making it ideal for studying persistent currents and the Mott insulator transition in a ring geometry., Comment: 4 pages, 4 figures

Light propagation in an atomic medium whose coupled electronic levels form a diamond-configuration exhibits a critical dependence on the input conditions. In particular, the relative phase of the input fields gives rise to interference phenomena in the electronic excitation whose interplay with relaxation processes determines the stationary state. We integrate numerically the Maxwell-Bloch equations and observe two metastable behaviors for the relative phase of the propagating fields corresponding to two possible interference phenomena. These phenomena are associated to separate types of response along propagation, minimize dissipation, and are due to atomic coherence. These behaviors could be studied in gases of isotopes of alkali-earth atoms with zero nuclear spin, and offer new perspectives in control techniques in quantum electronics., Comment: 16 pages, 11 figures, v2: typos corrected, v3: final version, to appear in Phys. Rev. A