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The energy flow of light represents a natural way of investigating complex light fields with respect to their applicability. With the generation of a three-dimensional Skyrmionic Hopfion structure in light, which is a topological 3D field configuration with particle-like nature, we paved the way to employ optical, topological constructs. In this work, we present an analysis of the transverse energy flow in the optical Skyrmionic Hopfion, showing the transfer of the topological properties to the mechanical attributes such as the optical angular momentum (OAM). Our findings thus prepare topological structures to be applied in optical traps and data storage or communication.

In the past years, enhanced techniques to tailor amplitude, polarization and phase have established the generation of singularities in all degrees of freedom, as phase or polarization vortex structures and caustics [1] . Structured polarization and phase singularities can have an integer, quantized nature which connects structured light to quantum physics and is likely to improve security in data communication [2] . In optical micromanipulation, the spin and orbital angular momentum of light, defined by its polarization and phase structure, enable the transfer of torque to optically trapped particles [3] , providing additional degrees of freedom to advance optical tweezers [4] .

So-called fully-structured light fields are characterized by a spatial variation in their amplitude, phase, and, simultaneously, polarization. Along with the technological progress of light-shaping methods based on q-plates or spatial light modulators, the variety of accessible structures has significantly increased and has proven being advantageous for high resolution imaging [1] , quantum communication [2] and particle manipulation [3] . For the vast majority of implemented light fields, the actually accessible customization is limited to two dimensions, i.e. the transverse plane of the light field to be modulated. Only a few approaches have recently been developed towards a full control of all degrees of freedom of the light field in all three spatial dimensions [4] , [5] . A common approach to structure light along its propagation direction takes advantage of tight focusing [6] . However, the resulting non-paraxial situation as well as the short longitudinal distance of the focal volume make it difficult to obtain an equal partition of fully structured light in all three dimensions. Thus, tailoring the longitudinal direction of a light field in the same effective manner than the transverse direction is an actual unsolved challenge. Counter-propagating light approaches that structure the longitudinal direction by interference at the same scale than transverse dimensions is a promising approach to this question. However, though being theoretically appealing, its experimental realization and detection meets unsolved challenges since the detection of a field in one direction is disturbed by the counterpropagating field.

We customize a transversely structured, tunable light landscape on the basis of orbital angular momentum (OAM)-carrying beams for the purpose of advanced optical manipulation. Combining Laguerre-Gaussian (LG) modes with helical phase fronts of opposite OAM handedness, counter-rotating transfer of OAM is enabled in a concentric intensity structure, creating a dynamic "grinding" scenario on dielectric microparticles. We demonstrate the ability to trap and rotate silica spheres of various sizes and exploit the light fields’ feature to spatially separate trapped objects by their size. We show the adaptability of the light field depending on the chosen LG mode indices, allowing on-demand tuning of the trapping potential and sorting criteria. The versatility of our approach for biomedical application is examined by spatial discriminating yeast cells and silica spheres of distinct diameter.

Three-dimensional (3D) topological states resemble truly localised, particle-like objects in physical space. Among the richest such structures are 3D skyrmions and hopfions, that realise integer topological numbers in their configuration via homotopic mappings from real space to the hypersphere (sphere in 4D space) or the 2D sphere. They have received tremendous attention as exotic textures in particle physics, cosmology, superfluids, and many other systems. Here we experimentally create and measure a topological 3D skyrmionic hopfion in fully structured light. By simultaneously tailoring the polarisation and phase profile, our beam establishes the skyrmionic mapping by realising every possible optical state in the propagation volume. The resulting light field’s Stokes parameters and phase are synthesised into a Hopf fibration texture. We perform volumetric full-field reconstruction of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{\Pi }}}_{{{3}}}$$\end{document}Π3 mapping, measuring a quantised topological charge, or Skyrme number, of 0.945. Such topological state control opens avenues for 3D optical data encoding and metrology. The Hopf characterisation of the optical hypersphere endows a fresh perspective to topological optics, offering experimentally-accessible photonic analogues to the gamut of particle-like 3D topological textures, from condensed matter to high-energy physics., One way to describe a particle is as a localised, 3-dimensional topological state, such as a skyrmion or hopfion. Here, the authors demonstrate and characterise particle-like skyrmionic hopfions in a free-space structured light beam.

We transfer on-demand structuring of three-dimensional scalar amplitude and phase patterns to polarization-structured, vectorial light fields and its singularities. Our approach allows inheriting non-diffracting as well as self-imaging propagation properties to tailored singular ellipse fields, including self-replicating amplitude, polarization, and singularity configurations. It is experimentally realized by amplitude, phase and polarization modulation of the angular spectrum of the light field. We demonstrate the customization of complex singularity formations embedded in three-dimensionally (3d) tailored vectorial field. Our findings show that embedded networks of polarization singularities can be customized to propagate in a robust way along curved trajectories, creating and annihilating during propagation. This 3d structuring of vectorial singular light fields opens new perspectives for in-depth singularity studies and for advancing applications as optical micro-manipulation and material machining.