Tailoring focal plane component intensities of polarization singular fields in a tight focusing system

The scientific community studies tight focusing of radially and azimuthally-polarized vector beams as it is a versatile solution for many applications. We offer a new method to produce tight focusing that ensures a more uniform intensity profile in multiple dimensions, providing a more versatile and stable solution. We manipulate the polarization of the radially and azimuthally polarized vector beams to find an optimal operating point. We examine in detail optical fields whose polarization states lie on the equator of the relevant Poincaré spheres namely, the fundamental Poincaré sphere, the hybrid order Poincaré sphere (HyOPS), and the higher order Poincaré sphere. We find via simulation that the fields falling on these equators have focal plane intensity distributions characterized by a single rotation parameter α determining the individual state of polarization. The strengths of the component field distributions vary with α and can be tuned to achieve equal strengths of longitudinal (z) and transverse (x and y) components at the focal plane. Without control of this parameter (e.g., using α = 0 in radially and α = π in azimuthally-polarized vector beams) intensity in x and y components are at 20% of the z component. In our solution with α = π / 2 , all components are at 80% of the maximum possible intensity of z. In examining the impact of α on a tightly focused beam, we also found that a helicity inversion of HyOPS beams causes a rotation of 180 degree in the axial intensity distribution.

Near-Field Spin Chern Number Quantized by Real-Space Topology of Optical Structures

The Chern number has been widely used to describe the topological properties of periodic structures in momentum space. Here, we introduce a real-space spin Chern number for the optical near fields of finite-sized structures. This new spin Chern number is intrinsically quantized and equal to the structure's Euler characteristic. The relationship is robust against continuous deformation of the structure's geometry and is irrelevant to the specific material constituents or external excitation. Our Letter enriches topological physics by extending the Chern number to real space, opening exciting possibilities for exploring the real-space topological properties of light.

Optical skyrmions in the Bessel profile

Optical skyrmions formed in terms of polarization are topological quasi-particles, and they have garnered much interest in the optical community owing to their unique inhomogeneous polarization structure and simplicity in their experimental realization. These structures belong to the Poincaré beams satisfying the stable topology. We theoretically investigated the non-diffracting and self-healing Poincaré beams based on the superposition of two orthogonal Bessel modes by the longitudinal mode matching technique. These Poincaré beams are topologically protected, and we suggest them as optical skyrmions in the corresponding Stokes vector fields. These optical skyrmions are quasi-skyrmions, and their range of propagation depends on the range of superposed Bessel modes. We have shown longitudinal mode matching of superposed Bessel beams is a necessary condition for the generation of propagation-invariant and non-diffracting skyrmions. The proposed longitudinal mode matching technique facilitates the generation of skyrmions with tunable position and range without any on-axis intensity modulations along the propagation axis. A suitable experimental configuration is suggested to realize variable order skyrmions in Bessel modes. The suggested experimental configuration can produce optical skyrmions even in ultra-short laser pulses with high mode conversion efficacy. This work can provide a new direction for the generation of skyrmions with completely new textures and features with reference to existing skyrmions originating from Laguerre-Gaussian modes.

Multichannel focused higher-order Poincaré sphere beam generation based on a dielectric geometric metasurface

Focused vector beams (VBs) are important topic in the areas of light field manipulation. Geometric metasurfaces provide a convenient platform to facilitate the generation of focused VBs. In this study, we propose a dielectric geometric metasurface to generate multichannel focused higher-order Poincaré sphere (HOP) beams. With identical meta-atoms of half-wave plate, the metasurface comprises two sub-metasurfaces, and each of them includes two sets of rings related to Fresnel zones. For meta-atoms on each set of rings, the hyperbolic geometric phase profile is configured so that the mirror-symmetrical position-flip of the off-axis focal point is enabled under the chirality switch of the illuminating circular polarization. With the design of helical geometric phase profiles for the two sets of rings, a sub-metasurface generate two HOP beams at the symmetrical two focal points. The performance of the two sub-metasurfaces enables the metasurface with four sets of rings to generate the array of four HOP beams. The proposed method was validated by theoretical analyses, numerical simulation and experimental conduction. This research would be significant in miniaturizing and integrating optical systems involving applications of VB generations and applications.

Determining triple half-twist of optical Möbius strips in electromagnetic fields with a non-planar structure

Polarization ellipses in a nonparaxial field form twisted strips when traced along closed contours around a circular polarization singularity line. We found an analytical expression for the twist number of the strip when the contour is coplanar with the polarization ellipse in its center. Necessary and sufficient conditions for strips having one or three half-twists are found. A set of five parameters of electromagnetic field at the polarization singularity point is found which definitely determines the value of the twist number of the strip.

Nanostructure-based orbital angular momentum encryption and multiplexing

The orthogonality among the OAM modes provides a new degree of freedom for optical multiplexing communications. So far, traditional Dammann gratings and spatial light modulators (SLMs) have been widely used to generate OAM beams by modulating electromagnetic waves at each pixel. However, such architectures suffer from limitations in terms of having a resolution of only a few microns and the bulkiness of the entire optical system. With the rapid development of the electromagnetic theory and advanced nanofabrication methods, artificial nanostructures, especially optical metasurfaces, have been introduced which greatly shrink the size of OAM multiplexing devices while increasing the level of integration. This review focuses on the study of encryption, multiplexing and demultiplexing of OAM beams based on nanostructure platforms. After introducing the focusing characteristics of OAM beams, the interaction mechanism between OAM beams and nanostructures is discussed. The physical phenomena of helical dichroism response and spatial separation of OAM beams achieved through nanostructures, setting the stage for OAM encryption and multiplexing, are reviewed. Afterward, the further advancements and potential applications of nanophotonics-based OAM multiplexing are deliberated. Finally, the challenges of conventional design methods and dynamic tunable techniques for nanostructure-based OAM multiplexing technology are addressed.

Generation for high-dimensional caustics and artificially tailored structured caustic beams

We theoretically propose and demonstrate topological parabolic umbilic beams (PUBs) with high-dimensional caustic by mapping catastrophe theory into optics. The PUBs are first experimentally observed via dimensionality reduction. Due to the high-dimensionality, such light beams exhibit rich caustic structures characterized by optical singularities where the high-intensity gradient appears. Further, we propose an improved caustic approach to artificially tailored structured beams which exhibit significant intensity gradient and phase gradient. The properties can trap and drive particles to move along the predesigned trajectory, respectively. The advantages for structured caustic beams likely enable new applications in flexible particle manipulation, light-sheet microscopy, and micromachining.

Dissipative three-dimensional topological optical solitons with crossed localization of polarization components

We present a new, to the best of our knowledge, type of vector three-dimensional dissipative optical solitons with more extended degrees of freedom in a laser or laser medium with saturable absorption. These solitons are reconfigurable, include polarization singularities, and have various mutual orientations of nearly toroidal localization domains of polarization components. Numerical modeling confirms the stability of these solitons and breathers and reveals their symmetry and even "supersymmetry," as well as transformations when parameters leave the stability region. These solitons, which have no scalar analogs, are capable of carrying more than one bit of information. Our results expand the "alphabet" of solitons and can provide a route to breakthroughs in larger-capacity communication and information applications.

Magneto-optical-like effect in tight focusing of azimuthally polarized sine-Gaussian beams

Magneto-optical effects, which have been known for over a century, are among the most fundamental phenomena in physics and describe changes in the polarization state of light when it interacts with magnetic materials. When a polarized plane wave propagates in or through a homogeneous and isotropic transparent medium, it is generally accepted that its transverse polarization structure remains unchanged. However, we show that a strong radial polarization component can be generated when an azimuthally polarized sine-Gaussian plane wave is tightly focused by a high numerical aperture lens, resulting in a magneto-optical-like effect that does not require external magnetic field or magnetic medium. Calculations show that the intensity structure and polarization distribution of the highly confined electric field strongly depend on the parameters m and φ0 in the sinusoidal term, where m can be used to control the number of the multifocal spots and φ0 can be used to control the position of each focal spot. Finally, we show that this peculiar electric field distribution can be used to realize multiple particles trapping with controllable numbers and locations.

Mixed-state ptychography for quantitative optical properties measurement of vector beam

Recent advances in ptychography have extended to anisotropic specimens, but vectorial reconstruction of probes owing to polarization aliasing remains a challenge. A polarization-sensitive ptychography that enables full optical property measurement of vector light is proposed. An optimized reconstruction strategy, first calibrating the propagation direction and then performing faithful retrieval, is established. This method avoids multiple image acquisitions with various polarizer configurations and significantly improves the measurement accuracy by correlating the intensity and position of different polarization components. The capability of the proposed method to quantify anisotropic parameters of optical materials and polarization properties of vector probe is demonstrated by experiment.

Water-Wave Vortices and Skyrmions

Topological wave structures-phase vortices, skyrmions, merons, etc.-are attracting enormous attention in a variety of quantum and classical wave fields. Surprisingly, these structures have never been properly explored in the most obvious example of classical waves: water-surface (gravity-capillary) waves. Here, we fill this gap and describe (i) water-wave vortices of different orders carrying quantized angular momentum with orbital and spin contributions, (ii) skyrmion lattices formed by the instantaneous displacements of the water-surface particles in wave interference, and (iii) meron (half-skyrmion) lattices formed by the spin-density vectors, as well as (iv) spatiotemporal water-wave vortices and skyrmions. We show that all these topological entities can be readily generated in linear water-wave interference experiments. Our findings can find applications in microfluidics and show that water waves can be employed as an attainable playground for emulating universal topological wave phenomena.

Polarization characteristics and transverse spin of Mie scattering

Complicated polarization states in the near field of Mie scattering have aroused wide interest due to their broad potential applications. In this work, we investigated polarization properties, including polarization dimension, degree of nonregularity, and transverse electric-field spin, of scattering of a partially polarized plane wave by a dielectric nanosphere based on the rigorous Mie scattering theory. It is shown that with the decrease of the correlation coefficient, the polarization dimension and degree of nonregularity generally increase. In the limit of unpolarized incident light, a nearly-perfect nonregular polarization state (PN = 0.928) appears in the near field and the spin is transverse to the radial direction everywhere. The rich structure contained by the partially polarized scattered light offers an approach to manipulating the interaction between light and nanoparticles, which may lead to novel designs of nanoantenna, optical trap and sensing.

Longitudinally variable 3D optical polarization structures

Generation and manipulation of three-dimensional (3D) optical polarization structures have received considerable interest because of their distinctive optical features and potential applications. However, the realization of multiple 3D polarization structures in a queue along the light propagation direction has not yet been reported. We propose and experimentally demonstrate a metalens to create longitudinally variable 3D polarization knots. A single metalens can simultaneously generate three distinct 3D polarization knots, which are indirectly validated with a rotating polarizer. The 3D polarization profiles are dynamically modulated by manipulating the linear polarization direction of the incident light. We further showcase the 3D image steganography with the generated 3D polarization structures. The ultrathin nature of metasurfaces and unique properties of the developed metalenses hold promise for lightweight polarization systems applicable to areas such as 3D image steganography and virtual reality.

Excitons Enabled Topological Phase Singularity in a Single Atomic Layer

The nontrivial and rigorous Heaviside phase jump behavior of phase singularities (PSs) empowers exotic topological modes and widely divergent nature compared to neighboring points, which has attracted great attention in condensed matter physics as well as applications in photonics and ultrasensitive sensors. Here we demonstrate the universal existence of a family of topologically protected PSs generated from exciton resonances of single-atom layers. We obtain the PSs by coating the transition metal dichalcogenide (TMDC) monolayers on a nonabsorptive semi-infinite substrate without surface plasmon effect or other assisted resonators, which exploits the benefits of both exciton-dominated enhancement and peculiarities of the singular phase. We show that a refractive indices matched transparent substrate enables TMDC monolayers to exhibit topologically protected zero reflection accompanied by a perfect Heaviside π-phase jump at strong light adsorptions, which can be utilized to radically reduce the thickness of PS-based devices to a single atomic layer. By using the TMDC monolayer-based PSs for refractive index biosensors, we demonstrate its superior phase sensitivity at a level of 104 degrees per refractive index unit and detection of bioactive bacteria, respectively, which is comparable to the cutting-edge surface plasmon and Fabry-Perot resonance sensors. Our proof-of-concept results offer experimental and theoretical insights into a single atomic playground for flat singular optics and label-free biosensing technologies.

Airy-Gaussian vector beam and its application in generating flexible optical chains

In recent years, the manipulation of structured optical beam has become an attractive and promising area. The Gaussian beam is the most common beam as the output beam of the laser, and the Airy beam is recently proposed with fascinating properties and applications. In this paper, for the first time to our knowledge, the polarization is used as a tool to design a new kind of Airy-Gaussian vector beam by connecting the Gaussian and Airy functions, which opens a new avenue in designing new beams based on the existed beams. We realize the Airy-Gaussian vector beam with space-variant polarization distribution in theory and experiment, and find that the vector beam can autofocus twice during propagation. The optical chains with flexible intensity peaks are achieved with the Airy-Gaussian vector beam, which can be applied in trapping and delivering particles including biological cells and Rydberg atoms. Such optical chains can significantly improve the trapping efficiency, reduce the heat accumulation, and sweep away the impurity particles.

Indices of polarization Möbius strips surrounding lines of circular polarization in nonparaxial optical fields

Detailed analysis of polarization ellipse strips constructed on small circular contours surrounding a single polarization singularity line of an electromagnetic field is carried out. It is shown that the strips formed by major and minor axes of polarization ellipses are nonorientable Möbius strips with opposite directions of twist. Explicit analytical expressions relating the direction of the strip twist to the orientation of its construction contour and the parameters of the electromagnetic field at the polarization singularity point are found.

Full characterization of vector eigenstates in symmetrically confined systems with photonic spin-orbit coupling

The photonic spin-orbit (SO) coupling is a widely investigated effect in optical microcavities leading to various interesting physical phenomena and potential applications. We report the full sets of eigenenergies and eigenstates in a symmetrically confined potential under the effect of SO coupling induced by the transverse-electric transverse-magnetic (TE-TM) splitting, which are derived analytically via the degenerate perturbation theory. We obtained the eigenenergies and the eigenstates from the 1st to the 6th orders of excited manifold, and demonstrate unambiguously that universal rules governing the mode formation exist in such complicated photonic systems, making the modes exhibiting the features of solid and hollow skyrmions as well as spin vortices. We show that these eigenstates can be described by the SO coupled hyperspheres that can be decomposed into a series of higher-order Poincare spheres. Our results significantly extend the area of microcavity spin-optronics to the general theory of eigenvalues in confined systems, and provide an efficient theoretical frame for the information processing using microcavity-based high-dimensional vector states.

Interferometric measurement of arbitrary propagating vector beams that are tightly focused

In this work, we demonstrate a simple setup to generate and measure arbitrary vector beams that are tightly focused. The vector beams are created with a spatial light modulator and focused with a microscope objective with an effective numerical aperture of 1.2. The transverse polarization components (E_x, E_y) of the tightly focused vector beams are measured with three-step interferometry. The axial component E_z is reconstructed using the transverse fields with Gauss's law. We measure beams with the following polarization states: circular, radial, azimuthal, spiral, flower, and spider web.

Generation of optical 3D unpolarized lattices in a tightly focused random beam

We study the three-dimensional (3D) polarization properties of a tightly focused partially coherent vector beam whose initial spatial coherence structure exhibits a lattice distribution. By examining the 3D degree of polarization and the polarimetric dimension of the tightly focused field, we demonstrate that this initial spatial coherence structure induces a 3D isotropically unpolarized beam lattice in the focal plane. Along the longitudinal direction, we observe the formation of nearly 3D unpolarized channels spanning 16 wavelengths in length near the focal region. We demonstrate that the spatial distribution of the 3D unpolarized lattice can be conveniently controlled through engineering the spatial coherence structure of the incident beam.

Exceptional points and non-Hermitian photonics at the nanoscale

Exceptional points (EPs) arising in non-Hermitian systems have led to a variety of intriguing wave phenomena, and have been attracting increased interest in various physical platforms. In this Review, we highlight the latest fundamental advances in the context of EPs in various nanoscale systems, and overview the theoretical progress related to EPs, including higher-order EPs, bulk Fermi arcs and Weyl exceptional rings. We peek into EP-associated emerging technologies, in particular focusing on the influence of noise for sensing near EPs, improving the efficiency in asymmetric transmission based on EPs, optical isolators in nonlinear EP systems and novel concepts to implement EPs in topological photonics. We also discuss the constraints and limitations of the applications relying on EPs, and offer parting thoughts about promising ways to tackle them for advanced nanophotonic applications.

Non-orientable order and non-commutative response in frustrated metamaterials

From atomic crystals to animal flocks, the emergence of order in nature is captured by the concept of spontaneous symmetry breaking^1-4. However, this cornerstone of physics is challenged when broken symmetry phases are frustrated by geometrical constraints. Such frustration dictates the behaviour of systems as diverse as spin ices^5-8, confined colloidal suspensions⁹ and crumpled paper sheets¹⁰. These systems typically exhibit strongly degenerated and heterogeneous ground states and hence escape the Ginzburg-Landau paradigm of phase ordering. Here, combining experiments, simulations and theory we uncover an unanticipated form of topological order in globally frustrated matter: non-orientable order. We demonstrate this concept by designing globally frustrated metamaterials that spontaneously break a discrete [Formula: see text] symmetry. We observe that their equilibria are necessarily heteregeneous and extensively degenerated. We explain our observations by generalizing the theory of elasticity to non-orientable order-parameter bundles. We show that non-orientable equilibria are extensively degenerated due to the arbitrary location of topologically protected nodes and lines where the order parameter must vanish. We further show that non-orientable order applies more broadly to objects that are non-orientable themselves, such as buckled Möbius strips and Klein bottles. Finally, by applying time-dependent local perturbations to metamaterials with non-orientable order, we engineer topologically protected mechanical memories^11-19, achieve non-commutative responses and show that they carry an imprint of the braiding of the loads' trajectories. Beyond mechanics, we envision non-orientability as a robust design principle for metamaterials that can effectively store information across scales, in fields as diverse as colloidal science⁸, photonics²⁰, magnetism⁷ and atomic physics²¹.

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Divide and focus: generating novel focal polarization modalities by symmetry-based phase tailoring in one dimension

Symmetry-based tailoring of photonic systems recently heralded the advent of novel concepts, such as photonic topological insulators and bound states in the continuum. In optical microscopy systems, similar tailoring was shown to result in tighter focusing, spawning the field of phase- and polarization-tailored light. Here, we show that even in the fundamental case of 1D focusing using a cylindrical lens, symmetry-based phase tailoring of the input field can result in novel features. Dividing the beam or utilizing a π phase shift for half the input light along the non-invariant focusing direction, these features include a transverse dark focal line and a longitudinally polarized on-axis sheet. While the former can be used in dark-field light-sheet microscopy, the latter, similar to the case of a radially polarized beam focused by a spherical lens, results in a z polarized sheet with reduced lateral size when compared with the thickness of a transversely polarized sheet produced by focusing a non-tailored beam. Moreover, the switching between these two modalities is achieved by a direct 90° rotation of the incoming linear polarization. We interpret these findings in terms of the requirement to adapt the symmetry of the incoming polarization state to match the symmetry of the focusing element. The proposed scheme may find application in microscopy, probing anisotropic media, laser machining, particle manipulation, and novel sensor concepts.

Accelerating Airy beams with particle-like polarization topologies and free-space bimeronic lattices

Phase and polarization singularities in electromagnetic waves are usually attributed to one-dimensional topologies-lines, knots, and braids. Recently, particle-like structures have been predicted and observed: optical Skyrmions, vortices with spherical polarization, etc. In this article, we devise vector Airy beams with point-like singularity in the focal plane, thus leading to the presence of a particle-like topology. We present an extensive analytical analysis of the spatial spectra and focal structure of such beams. We report on the presence of a free-space lattice of bimerons in such vector Airy beams.

Highly-twisted states of light from a high quality factor photonic crystal ring

Twisted light with orbital angular momentum (OAM) has been extensively studied for applications in quantum and classical communications, microscopy, and optical micromanipulation. Ejecting high angular momentum states of a whispering gallery mode (WGM) microresonator through a grating-assisted mechanism provides a scalable, chip-integrated solution for OAM generation. However, demonstrated OAM microresonators have exhibited a much lower quality factor (Q) than conventional WGM resonators (by >100×), and an understanding of the limits on Q has been lacking. This is crucial given the importance of Q in enhancing light-matter interactions. Moreover, though high-OAM states are often desirable, the limits on what is achievable in a microresonator are not well understood. Here, we provide insight on these two questions, through understanding OAM from the perspective of mode coupling in a photonic crystal ring and linking it to coherent backscattering between counter-propagating WGMs. In addition to demonstrating high-Q (10⁵ to 10⁶), a high estimated upper bound on OAM ejection efficiency (up to 90%), and high-OAM number (up to l = 60), our empirical model is supported by experiments and provides a quantitative explanation for the behavior of Q and the upper bound of OAM ejection efficiency with l. The state-of-the-art performance and understanding of microresonator OAM generation opens opportunities for OAM applications using chip-integrated technologies.

Topological Unidirectional Guided Resonances Emerged from Interband Coupling

Unidirectional guided resonances (UGRs) are optical modes in photonic crystal slabs that radiate toward one side without the need for mirrors on the other. In this Letter, we report a mechanism to realize UGRs by tuning the interband coupling effect originating from up-down symmetry breaking. We theoretically find that UGRs that reside along high-symmetric lines correspond to phase singularities of far-field radiation, depicted by phase winding numbers as a type of topological indices. We investigate the phase dislocation lines in three-dimensional parameter space and elaborate on the interplay between UGRs and non-Hermitian degeneracies accordingly. Our findings reveal the topological nature of UGRs about their generation, evolution, and annihilation in general parameter spaces, thus paving the way to new possibilities of light manipulation.

Observation of Polarization Singularities and Topological Textures in Sound Waves

Polarization singularities and topological polarization structures are generic features of inhomogeneous vector wave fields of any nature. However, their experimental studies mostly remain restricted to optical waves. Here, we report the observation of polarization singularities, topological Möbius-strip structures, and skyrmionic textures in 3D polarization fields of inhomogeneous sound waves. Our experiments are made in the ultrasonic domain using nonparaxial propagating fields generated by space-variant 2D acoustic sources. We also retrieve distributions of the 3D spin density in these fields. Our results open the avenue to investigations and applications of topological features and nontrivial 3D vector properties of structured sound waves.

Color-selective three-dimensional polarization structures

Polarization as an important degree of freedom for light plays a key role in optics. Structured beams with controlled polarization profiles have diverse applications, such as information encoding, display, medical and biological imaging, and manipulation of microparticles. However, conventional polarization optics can only realize two-dimensional polarization structures in a transverse plane. The emergent ultrathin optical devices consisting of planar nanostructures, so-called metasurfaces, have shown much promise for polarization manipulation. Here we propose and experimentally demonstrate color-selective three-dimensional (3D) polarization structures with a single metasurface. The geometric metasurfaces are designed based on color and phase multiplexing and polarization rotation, creating various 3D polarization knots. Remarkably, different 3D polarization knots in the same observation region can be achieved by controlling the incident wavelengths, providing unprecedented polarization control with color information in 3D space. Our research findings may be of interest to many practical applications such as vector beam generation, virtual reality, volumetric displays, security, and anti-counterfeiting.

Topological near fields generated by topological structures

The central idea of metamaterials and metaoptics is that, besides their base materials, the geometry of structures offers a broad extra dimension to explore for exotic functionalities. Here, we discover that the topology of structures fundamentally dictates the topological properties of optical fields and offers a new dimension to exploit for optical functionalities that are irrelevant to specific material constituents or structural geometries. We find that the nontrivial topology of metal structures ensures the birth of polarization singularities (PSs) in the near field with rich morphologies and intriguing spatial evolutions including merging, bifurcation, and topological transition. By mapping the PSs to non-Hermitian exceptional points and using homotopy theory, we extract the core invariant that governs the topological classification of the PSs and the conservation law that regulates their spatial evolutions. The results bridge singular optics, topological photonics, and non-Hermitian physics, with potential applications in chiral sensing, chiral quantum optics, and beyond photonics in other wave systems.

Multiwavelength achromatic super-resolution focusing via a metasurface-empowered controlled generation of focused cylindrically polarized vortex beams

Non-invasive imaging beyond the diffraction limit and free from fluorescent labels in the visible is highly desired for microscopy. It remains a challenge to obtain such super-resolution focusing along with multiwavelength achromatic performance in the far field using an integratable and easily designed system. In this work, we demonstrate a straightforward metasurface-based method to realize multiwavelength achromatic generation and focusing of cylindrically polarized vortex beams (CPVBs). Attributed to the extra degrees of freedom of CPVBs and multi-section design, we have realized multiwavelength achromatic super-resolution focusing in the air with focal size tighter than that of normally used schemes like immersion metalenses or focused radially polarized beams. It is expected that this metasurface-empowered ultra-compact design will benefit potential applications which call for high resolution, like optical microscopy, laser processing, etc.

Polarization singularities hidden in a deep subwavelength confined electromagnetic field with angular momentum

Topologies associated with polarization point and line singularities can provide tools for controlling light propagation. By using the Stokes parameter, we demonstrate the emergence of polarization singularities hidden in deep subwavelength confined electromagnetic fields with angular momentum. We show that when the incoming orbital angular momentum is appropriately chosen, highly confined electromagnetic fields with super-diffraction-limited spatial dimensions can be obtained. At the same time, a conversion of orbital to spin angular momentum occurs, leading to a non-trivial topology. Our method provides a platform for developing topological photonics and studying the behavior of polarization singularities under strong focusing.

Topological flowers and spider webs in 3D vector fields

Topological structures currently are of special interest in the integration of singular optics and topological photonics. In this article, the topological flowers and spider webs, which got less attention comparing to the lemon-, star- structures of the same category, are investigated in 3D vector fields. We show that by strongly focusing higher-order singular beams, both the spin density (SD) vectors and the polarization states of the transverse fields on the focal plane exhibit flowers and spider webs structures in topology with 2|m - 1| folds/sectors (m is the beam order), and the topological structures of the SD vectors are demonstrated to have a 90°/|m - 1| rotation. On the other hand, the topological theory also needs to be developed according to the rapid growth of topological photonics. Here, by defining a 'relative topological charge', we have observed and analyzed the topological reactions of the loops (composed of SD singularities) rather than the 'point-type' singularities in conventional reactions. More specially, the 'radial index' and 'azimuthal index' are proposed to characterize the topological features of the flowers and spider webs, and have been verified that the 'radial index' is peculiar to 3D vector fields and proportional to 1/|m - 1| in general. Our work provides a way to describe the topological behaviors of groups of singularities and supplies new parameters for measuring the topological patterns in 3D vector fields, which will rich the topological theory and may have applications in topological photonics.

3D-Printed Möbius Microring Lasers: Topology Engineering in Photonic Microstructures

Manipulating photons in artificially structured materials is highly desired in modern photonic technology. Nontrivial topological structures are rapidly emerging as a state-of-art platform for achieving unprecedented fascinating phenomena of photon manipulation. However, the current studies mainly focus on planar structures, and the fabrication of photonic microstructures with specific topological geometric features still remains a great challenge. Extending the topological photonics to 3D microarchitectures is expected to enrich the photon manipulation capabilities and further advance the topological photonic devices. Here, a femtosecond laser direct writing technique is employed to fabricate 3D topological Möbius microring resonators from dye-doped polymer. The high-quality-factor Möbius microring resonator supports a unique spin-orbit coupled lasing at very low threshold. Due to the spin-orbit coupling induced geometric/Berry phase, the Möbius microrings, in striking contrast with ordinary microrings, output laser signals with all polarization states. The manipulation of miniaturized coherent light sources in the fabricated Möbius microrings represents a significant step forward toward 3D topological photonics that offers a novel design philosophy for functional photonic and optoelectronic devices.

Generation of Stokes singularities using polarization lateral shear interferometer

Lateral shear interferometer, being a self-referenced interferometer, has proven to be an important tool in scalar optics. Here we employ a vectorial counterpart - polarization lateral shear interferometer, in which the two interfering beams apart from being derived from the test wavefront, are in orthogonal states of polarization. Therefore when the test wavefront has spatially varying phase gradient across the beam cross-section, the resulting shearogram produces polarization fringes instead of intensity fringes. Further, the shearogram becomes inhomogeneously polarized. This polarization lateral shear interferometer may have potential uses in metrology, but in this article we demonstrate the ability of the interferometer in the generation of all Stokes singularities in the single beam by launching a phase singular beam into it. It is found that a vortex dipole is formed along with other generic Stokes singularities. Experimental observations support the results and they are discussed in the article.

Extreme Concentration and Nanoscale Interaction of Light

Concentrating light strongly calls for appropriate polarization patterns of the focused light beam and for up to a full 4π solid angle geometry. Focusing on the extreme requires efficient coupling to nanostructures of one kind or another via cylindrical vector beams having such patterns, the details of which depend on the geometry and property of the respective nanostructure. Cylindrical vector beams can not only be used to study a nanostructure, but also vice versa. Closely related is the discussion of topics such as the ultimate diffraction limit, a resonant field enhancement near nanoscopic absorbers, as well as speculations about nonresonant field enhancement, which, if it exists, might be relevant to pair production in vacuum. These cases do require further rigorous simulations and more decisive experiments. While there is a wide diversity of scenarios, there are also conceptually very different models offering helpful intuitive pictures despite this diversity.

Probing coherence Stokes parameters of three-component light with nanoscatterers

We establish a method to determine the spectral coherence Stokes parameters of a random three-component optical field via scattering by two dipolar nanoparticles. We show that measuring the intensity and polarization-state fringes of the scattered far field in three directions allows us to construct all nine coherence Stokes parameters at the dipoles. The method extends current nanoprobe techniques to detection of the spatial coherence of random light with arbitrary three-dimensional polarization structure.

Optical force and torque on small particles induced by polarization singularities

Optical forces in the near fields have important applications in on-chip optical manipulations of small particles and molecules. Here, we report a study of optical force and torque on small particles induced by the optical polarization singularities of a gold cylinder. We show that the scattering of the cylinder generates both electric and magnetic C lines (i.e., lines of polarization singularities) in the near fields. The intrinsic spin density of the C lines can induce complex optical torque on a dielectric/magnetic particle, and the near-field evolutions of the C lines are accompanied by a gradient force on the particle. The force and torque manifest dramatic spatial variations, providing rich degrees of freedom for near-field optical manipulations. The study, for the first time to our knowledge, uncovers the effect of optical polarization singularities on light-induced force and torque on small particles. The results contribute to the understanding of chiral light-matter interactions and can find applications in on-chip optical manipulations and optical sensing.

Observing polarization patterns in the collective motion of nanomechanical arrays

In recent years, nanomechanics has evolved into a mature field, and it has now reached a stage which enables the fabrication and study of ever more elaborate devices. This has led to the emergence of arrays of coupled nanomechanical resonators as a promising field of research serving as model systems to study collective dynamical phenomena such as synchronization or topological transport. From a general point of view, the arrays investigated so far can be effectively treated as scalar fields on a lattice. Moving to a scenario where the vector character of the fields becomes important would unlock a whole host of conceptually interesting additional phenomena, including the physics of polarization patterns in wave fields and their associated topology. Here we introduce a new platform, a two-dimensional array of coupled nanomechanical pillar resonators, whose orthogonal vibration directions encode a mechanical polarization degree of freedom. We demonstrate direct optical imaging of the collective dynamics, enabling us to analyze the emerging polarization patterns, follow their evolution with drive frequency, and identify topological polarization singularities.

Coupled nanomechanical resonator arrays serve as model systems to study collective dynamical phenomena. Doster et al. introduce a two-dimensional array of pillar resonators encoding a mechanical polarization degree of freedom for analyzing polarization patterns and identifying topological singularities.

Writing and reading with the longitudinal component of light using carbazole-containing azopolymer thin films

It is well known that azobenzene-containing polymers (azopolymers) are sensitive to the polarization orientation of the illuminating radiation, with the resulting photoisomerization inducing material transfer at both the meso- and macroscale. As a result, azopolymers are efficient and versatile photonic materials, for example, they are used for the fabrication of linear diffraction gratings, including subwavelength gratings, microlens arrays, and spectral filters. Here we propose to use carbazole-containing azopolymer thin films to directly visualize the longitudinal component of the incident laser beam, a crucial task for the realization of 3D structured light yet remaining experimentally challenging. We demonstrate the approach on both scalar and vectorial states of structured light, including higher-order and hybrid cylindrical vector beams. In addition to detection, our results confirm that carbazole-containing azopolymers are a powerful tool material engineering with the longitudinal component of the electric field, particularly to fabricate microstructures with unusual morphologies that differentiate from the total intensity distribution of the writing laser beam.

Poincaré sphere analogue for optical vortex knots

We propose a Poincaré sphere (PS) analogue for optical vortex knots. The states on the PS analogue represent the light fields containing knotted vortex lines in three-dimensional space. The state changes on the latitude and longitude lines lead to the spatial rotation and scale change of the optical vortex knots, respectively. Furthermore, we experimentally generate and observe these PS analogue states. These results provide new insights for the evolution and control of singular beams, and can be further extended to polarization topology.

Particle-like topologies in light

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 [Formula: see text] 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.

Möbius Strip Microlasers: A Testbed for Non-Euclidean Photonics

We report on experiments with Möbius strip microlasers, which were fabricated with high optical quality by direct laser writing. A Möbius strip, i.e., a band with a half twist, exhibits the fascinating property that it has a single nonorientable surface and a single boundary. We provide evidence that, in contrast to conventional ring or disk resonators, a Möbius strip cavity cannot sustain whispering gallery modes (WGM). Comparison between experiments and 3D finite difference time domain (FDTD) simulations reveals that the resonances are localized on periodic geodesics.

Design of Metasurface with Nanoslits on Elliptical Curves for Generation of Dual-Channel Vector Beams

The manipulations of nanoscale multi-channel vector beams (VBs) by metasurfaces hold potential applications in various important fields. In this paper, the metasurface with two sets of nanoslits arranged on elliptic curves was proposed to generate the dual-channel focused vector beams (FVBs). Each set of nanoslits was composed of the in-phase and the out-of-phase groups of nanoslits to introduce the constructive interference and destructive interference of the output light field of the nanoslits, focusing the converted spin component and eliminating the incident spin component at the focal point. The two sets of nanoslits for the channels at the two focal points were interleaved on the same ellipses, and by setting their parameters independently, the FVBs in the two channels are generated under illumination of linearly polarized light, while their orders and polarization states of FVBs were controlled independently. The generation of the FVBs with the designed metasurfaces was demonstrated by the finite-difference time domain (FDTD) simulations and by the experimental verifications. The work in this paper is of great significance for the generation of miniaturized multi-channel VBs and for broadening the applications of metasurfaces.

Multidimensional phase singularities in nanophotonics

[Figure: see text].

Supertoroidal light pulses as electromagnetic skyrmions propagating in free space

Topological complex transient electromagnetic fields give access to nontrivial light-matter interactions and provide additional degrees of freedom for information transfer. An important example of such electromagnetic excitations are space-time non-separable single-cycle pulses of toroidal topology, the exact solutions of Maxwell's equations described by Hellwarth and Nouchi in 1996 and recently observed experimentally. Here we introduce an extended family of electromagnetic excitation, the supertoroidal electromagnetic pulses, in which the Hellwarth-Nouchi pulse is just the simplest member. The supertoroidal pulses exhibit skyrmionic structure of the electromagnetic fields, multiple singularities in the Poynting vector maps and fractal-like distributions of energy backflow. They are of interest for transient light-matter interactions, ultrafast optics, spectroscopy, and toroidal electrodynamics.

Shaping light in 3d space by counter-propagation

We extend the established transverse customization of light, in particular, amplitude, phase, and polarization modulation of the light field, and its analysis by the third, longitudinal spatial dimension, enabling the visualization of longitudinal structures in sub-wavelength (nm) range. To achieve this high-precision and three-dimensional beam shaping and detection, we propose an approach based on precise variation of indices in the superposition of higher-order Laguerre-Gaussian beams and cylindrical vector beams in a counter-propagation scheme. The superposition is analyzed experimentally by digital, holographic counter-propagation leading to stable, reversible and precise scanning of the light volume. Our findings show tailored amplitude, phase and polarization structures, adaptable in 3D space by mode indices, including sub-wavelength structural changes upon propagation, which will be of interest for advanced material machining and optical trapping.

Minimal Focal Spot Size Measured Based on Intensity and Power Flow

It is shown, theoretically and numerically, that the distributions of the longitudinal energy flow for tightly focused light with circular and linear polarization are the same, and that the spot has circular symmetry. It is also shown that the longitudinal energy flows are equal for optical vortices with unit topological charge and with radial or azimuthal polarization. The focal spot has a minimum diameter (all other characteristics being equal), which is measured based on the intensity of an optical vortex with azimuthal polarization. The diameter of the focal spot calculated from the energy flow for light with circular or linear polarization is slightly larger (by a fraction of a percentage). The magnitude of the diameter based on the intensity plays a role in the interaction of light with matter, and the magnitude of the diameter based on the energy flux affects the resolution in optical microscopy which is crucial in sensorial applications.

Generation of transversely oriented optical polarization Möbius strips

We report a time-reversal method based on the Richards-Wolf vectorial diffraction theory to generate a prescribed polarization topology on a defined trajectory within areas of relatively high intensity. An example is given to generate transversely oriented optical Möbius strips that wander around an axis perpendicular to the beam propagation direction. A number of sets of dipole antennae are purposefully positioned on a defined trajectory in the y = 0 plane and the radiation fields are collected by one high-NA objective lens. By sending the complex conjugate of the radiation fields in a time-reversed manner, the focal fields are calculated and the optical polarization topology on the trajectory can be tailored to form prescribed Möbius strips. The ability to control optical polarization topologies may find applications in nanofabrication, quantum communication, and light-matter interaction.

Efficient calculation of highly focused electromagnetic Schell-model beams

The calculation of the propagation of partially coherent and partially polarized optical beams involves using 4D Fourier Transforms. This poses a major drawback, taking into account memory and computational capabilities of nowadays computers. In this paper we propose an efficient calculation procedure for retrieving the irradiance of electromagnetic Schell-model highly focused beams. We take advantage of the separability of such beams to compute the cross-spectral density matrix by using only 2D Fourier Transforms. In particular, the number of operations depends only on the number of pixels of the input beam, independently on the coherence properties. To provide more insight, we analyze the behavior of a beam without a known analytical solution. Finally, the numerical complexity and computation time is analyzed and compared with some other algorithms.