Multitwist Möbius Strips and Twisted Ribbons in the Polarization of Paraxial Light Beams

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.

Nonlinear up-conversion of a polarization Möbius strip with half-integer optical angular momentum

Symmetries and conservation laws of energy, linear momentum, and angular momentum play a central role in nonlinear optics. Recently, paraxial light fields with nontrivial topology have been attracting a keen interest. Despite not being eigenstates of the orbital and spin angular momenta (OAM and SAM), they are eigenstates of the generalized angular momentum (GAM) operator-a mixture of the OAM and SAM operators with fractional eigenvalues. By driving high harmonic generation with a polarization Möbius strip carrying a half-integer GAM charge and implementing angular momentum characterization methods in the extreme ultraviolet range, we demonstrate the linear scaling of the GAM with the harmonic order, each harmonic carrying a precise half-integer GAM charge. Our work shows that beyond SAM and OAM, the GAM is, in some situations, an appropriate quantum number. It paves the way for finer manipulations and applications of light beams containing fractional-order polarization singularities.

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.

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.

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.

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.

Circularly polarized luminescence from organic micro-/nano-structures

Circularly polarized light exhibits promising applications in future displays and photonic technologies. Circularly polarized luminescence (CPL) from chiral luminophores is an ideal approach to directly generating circularly polarized light, in which the energy loss induced by the circularly polarized filters can be reduced. Among various chiral luminophores, organic micro-/nano-structures have attracted increasing attention owing to the high quantum efficiency and luminescence dissymmetry factor. Herein, the recent progress of CPL from organic micro-/nano-structures is summarized. Firstly, the design principles of CPL-active organic micro-/nano-structures are expounded from the construction of micro-/nano-structure and the introduction of chirality. Based on these design principles, several typical organic micro-/nano-structures with CPL activity are introduced in detail, including self-assembly of small molecules, self-assembly of π-conjugated polymers, and self-assembly on micro-/nanoscale architectures. Subsequently, we discuss the external stimuli that can regulate CPL performance, including solvents, pH value, metal ions, mechanical force, and temperature. We also summarize the applications of CPL-active materials in organic light-emitting diodes, optical information processing, and chemical and biological sensing. Finally, the current challenges and prospects in this emerging field are presented. It is expected that this review will provide a guide for the design of excellent CPL-active materials.

Digital Stokes polarimetry and its application to structured light: tutorial

Stokes polarimetry is a mature topic in optics, most commonly performed to extract the polarization structure of optical fields for a range of diverse applications. For historical reasons, most Stokes polarimetry approaches are based on static optical polarization components that must be manually adjusted, prohibiting automated, real-time analysis of fast changing fields. Here we provide a tutorial on performing Stokes polarimetry in an all-digital approach, exploiting a modern optical toolkit based on liquid-crystal-on-silicon spatial light modulators and digital micromirror devices. We explain in a tutorial fashion how to implement two digital approaches, based on these two devices, for extracting Stokes parameters in a fast, cheap, and dynamic manner. After outlining the core concepts, we demonstrate their applicability to the modern topic of structured light, and highlight some common experimental issues. In particular, we illustrate how digital Stokes polarimetry can be used to measure key optical parameters such as the state of polarization, degree of vectorness, and intra-modal phase of complex light fields.

Twisted ribbon carousels in random, three-dimensional optical fields

The instantaneous electric vector in a random three-dimensional optical field is shown to generate twisted ribbon carousels that spin about their axes. The ribbons can be right or left handed and can unwind and rewind in time, changing their handedness during an optical cycle. Analytical formulas describing this behavior are presented.

Hybrid order Poincaré spheres for Stokes singularities

Hybrid order Poincaré spheres to represent more general Stokes singularities are presented. Polarization singularities form a subset of Stokes singularities, and therefore induction of these spheres brings completeness. The conventional understanding of Poincaré beams as hybrid order Poincaré sphere beams is also expanded to include more beams. Construction and salient properties of these spheres are explained with illustrations to show their ability to represent more exotic Poincaré beams that have zero total helicity irrespective of their size. Pancharatnam-Berry geometric phase formulation using these new spheres is also possible.

Three-dimensional structure of polarization singularities of a light field near a dielectric spherical nanoparticle

We numerically study the structure of polarization singularity lines in a near-field of the sub-wavelength dielectric particle when it is irradiated by a monochromatic elliptically polarized plane wave. For the various values of the ellipticity degree of the incident radiation, we trace the C^T and L^T-lines on which the polarization ellipse turns into a circle or a line, respectively. The continuous variation of the isotropy parameters of the singularities is visualized along the lines. The main rules of C^T and L^T-lines deformation, emergence, and vanishing are revealed.

Polarization Möbius strips on elliptical paths in three-dimensional optical fields

Optical singularities in nonparaxial structured light are currently of special interest. Here we study polarization Möbius strips on elliptical paths surrounding lines of circular polarization in fully three-dimensional fields. We find that as the eccentricity, azimuthal orientation, or centering of the path changes, right-handed Möbius strips can change into left-handed ones, and vice versa, and that Möbius strips with one half twist can change into strips with three half twists, and vice versa. These transformations are shown to occur in a possibly unexpected way, not observed previously, that is universal for all two-component singularities.

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 Poincaré 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 Möbius strips.

Optical singularities and Möbius strip arrays in tailored non-paraxial light fields

A major current challenge in the field of structured light represents the development from three- (3d) to four-dimensional (4d) electric field structures, in which one exploits the transverse as well as longitudinal field components in 3d space. For this purpose, non-paraxial fields are required in order to be able to access visionary 3d topological structures as optical cones, ribbons and Möbius strips formed by 3d polarization states. We numerically demonstrate the customization of such complex topological structures by controlling generic polarization singularities in non-paraxial light fields. Our approach is based on tightly focusing tailored higher-order vector beams in combination with phase vortices. Besides demonstrating the appearance of cones and ribbons around the optical axis, we evince sculpting arrays of Möbius strips realized around off-axis generic singularities.

Polarization nano-tomography of tightly focused light landscapes by self-assembled monolayers

Recently, four-dimensional (4D) functional nano-materials have attracted considerable attention due to their impact in cutting-edge fields such as nano-(opto)electronics, -biotechnology or -biomedicine. Prominent optical functionalizations, representing the fourth dimension, require precisely tailored light fields for its optimal implementation. These fields need to be like-wise 4D, i.e., nano-structured in three-dimensional (3D) space while polarization embeds additional longitudinal components. Though a couple of approaches to realize 4D fields have been suggested, their breakthrough is impeded by a lack of appropriate analysis techniques. Combining molecular self-assembly, i.e., nano-chemistry, and nano-optics, we propose a polarization nano-tomography of respective fields using the functional material itself as a sensor. Our method allows a single-shot identification of non-paraxial light fields at nano-scale resolution without any data post-processing. We prove its functionality numerically and experimentally, elucidating its amplitude, phase and 3D polarization sensitivity. We analyze non-paraxial field properties, demonstrating our method’s capability and potential for next generation 4D materials.

The realisation of 4D light fields, where longitudinal polarisation represents the fourth dimension, has been limited by the lack of appropriate analysis techniques. Here, the authors use interaction with self-assembled monolayers of fluorescent molecules, which allow for identification of non-paraxial light fields based on a single image frame.

Photonic spin-controlled generation and transformation of 3D optical polarization topologies enabled by all-dielectric metasurfaces

Optical polarization topology is a spatially varying polarization structure, which usually exists around the polarization singularity. In three-dimensional (3D) space, optical polarization topologies mainly contain two fundamental structures, Möbius strip and twisted ribbon, depending on the parity of half-twist number. These spectacular topologies have been widely found in the existence of electric fields from multi-beam interference. Here, we propose and numerically demonstrate that, depending on the photonic spin state of light, an ultrathin all-dielectric metasurface can achieve efficient generation and transformation of two arbitrary 3D polarization topologies. The spin-controlled, tightly-focused Poincaré beams generated by the metasurface exhibit topologically stable 3D polarization topologies around the waist of the focal point. The preparation of such optical polarization topologies may have potential applications in compact complex beam engineering, optical signal multiplexing and optical fabrication of microstructures with nontrivial topology.

Generation of exotic optical polarization Möbius strips

We report a method of generating exotic optical polarization Möbius strips through tightly focusing an arbitrary vector beam. A heart-shaped Möbius strip, an "8"-shaped twin Möbius strip, and a circular Möbius strip with varying polarization twisting rate are demonstrated. The ability of tailoring three-dimensional optical polarization topologies may spur novel studies of optics and physics and find their applications in sensing, light coupling to nanostructures, light-matter interaction, and metamaterial fabrication.

Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities

Thirty years ago, Coullet et al. proposed that a special optical field exists in laser cavities bearing some analogy with the superfluid vortex. Since then, optical vortices have been widely studied, inspired by the hydrodynamics sharing similar mathematics. Akin to a fluid vortex with a central flow singularity, an optical vortex beam has a phase singularity with a certain topological charge, giving rise to a hollow intensity distribution. Such a beam with helical phase fronts and orbital angular momentum reveals a subtle connection between macroscopic physical optics and microscopic quantum optics. These amazing properties provide a new understanding of a wide range of optical and physical phenomena, including twisting photons, spin-orbital interactions, Bose-Einstein condensates, etc., while the associated technologies for manipulating optical vortices have become increasingly tunable and flexible. Hitherto, owing to these salient properties and optical manipulation technologies, tunable vortex beams have engendered tremendous advanced applications such as optical tweezers, high-order quantum entanglement, and nonlinear optics. This article reviews the recent progress in tunable vortex technologies along with their advanced applications.