Polarization singularities of vectorial electromagnetic fields locate at the positions (such as points, lines, or surfaces) where properties of polarization ellipses are not defined. They are manifested as circular and linear polarization, for which respectively the semi-major axes and normal vectors of polarization ellipses become indefinite. First observed in conical diffraction in the 1830s, the field of polarization singularities has been systematically reshaped and deepened by many pioneers of wave optics. Together with other exotic phenomena such as non-Hermiticity and topology, polarization singularities have been introduced into the vibrant field of nanophotonics, rendering unprecedented flexibilities for manipulations of light-matter interactions at the nanoscale. Here we review the recent results on the generation and observation of polarization singularities in metaphotonics. We start with the discussion of polarization singularities in the Mie theory, where both electric and magnetic multipoles are explored from perspectives of local and global polarization properties. We then proceed with the discussion of various photonic-crystal structures, for which both near- and far-field patterns manifest diverse polarization singularities characterized by the integer Poincare or more general half-integer Hopf indices (topological charges). Next, we review the most recent studies of conversions from polarization to phase singularities in scalar wave optics, demonstrating how bound states in the continuum can be exploited to generate directly optical vortices of various charges. Throughout our paper, we discuss and highlight several fundamental concepts and demonstrate their close connections and special links to metaphotonics. We believe polarization singularities can provide novel perspectives for light-matter manipulation for both fundamental studies and their practical applications.
Using full-wave numerical simulations and analytical multipole expansions we investigated the properties of real-space polarization singularities that emerge in light scattering by subwavelength particles. We considered spherical and torus particles made of dielectric or perfect-electric-conductor material. We determined the topological indices and the trajectories of electric-field polarization singularities in both the near-field and far-field regions. In the far-field region, a total of four singularities are identified and the sum of their polarization topological indices is two, independe…
We describe the evolution of a paraxial electromagnetic wave characterizing by a non-uniform polarization distribution with singularities and propagating in a weakly anisotropic medium. Our approach is based on the Stokes vector evolution equation applied to a non-uniform initial polarization field. In the case of a homogeneous medium, this equation is integrated analytically. This yields a 3-dimensional distribution of the polarization parameters containing singularities, i.e. C-lines of circular polarization and L-surfaces of linear polarization. The general theory is applied to specific exa…
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, and the C lines can induce complex force and torque on a dielectric/magnetic particle. The force and torque manifest dramatic spatial variations with interesting symmetry proper…
We present a convenient method to generate vector beams of light having polarization singularities on their axis, via partial spin-to-orbital angular momentum conversion in a suitably patterned liquid crystal cell. The resulting polarization patterns exhibit a C-point on the beam axis and an L-line loop around it, and may have different geometrical structures such as \qo{lemon}, \qo{star}, and \qo{spiral}. Our generation method allows us to control the radius of L-line loop around the central C-point. Moreover, we investigate the free-air propagation of these fields across a Rayleigh range.
Topological singularities are ubiquitous in many areas of physics. Polarization singularities are locations at which an aspect of the polarization ellipse of light becomes undetermined or degenerate. At C points the orientation of the ellipse becomes degenerate and light's electric field vector describes a perfect circle in time. In 2D slices of 3D random fields the distribution in space of the C points is reminiscent of that of interacting particles. With near-field experiments we show that when light becomes truly 2D, this has severe consequences for the distribution of C points in space. Th…
The work is devoted to the self-action of laser beams propagating in the isotropic phase of a cholesteric liquid crystal near the transition temperature to the mesophase in a wide range of parameter values characterizing the nonlocality of the nonlinear optical response of the medium, defocusing in the medium, the power of incident laser radiation and diffraction in the liquid crystal. The origin of the primary singularities of both polarization components of radiation in a wide range of parameters of radiation and the medium is investigated for different values of the parameter characterizing…
Addressing subwavelength object and displacement is crucial in optical nanometrology. We show in this Letter that nano antennas with subwavelength structures can be addressed precisely by incident beams with singularity. This accurate feeding beyond the diffraction limit can lead to dynamic control of the unidirectional scattering in the far field. The combination of polarization discontinuity of the incoming singular beam, along with the rapid phase variation near the antenna leads to remarkable sensitivity of the far field scattering to displacement at deep subwavelength scale. This opens a …
Optical singularities play a major role in modern optics and are frequently deployed in structured light, super-resolution microscopy, and holography. While phase singularities are uniquely defined as locations of undefined phase, polarization singularities studied thus far are either partial, i.e., bright points of well-defined polarization, or unstable for small field perturbations. We demonstrate for the first time a complete, topologically protected polarization singularity; it is located in the 4D space spanned by the three spatial dimensions and the wavelength and is created in the focus…
The practical applications of chiral discrimination are usually limited by the weak chiral response of enantiomers and the high complexity of detection methods. Here, we propose to use the C lines (i.e., lines of polarization singularities) emerged in light scattering by a metal sphere to detect small chiral particles. Using full-wave numerical simulations and analytical multipole expansions, we determined the absorption dissymmetry of deep-subwavelength helices (i.e., chiral particles) at different positions on the C lines. We find that the absorption dissymmetry can be much larger than that …
Core concepts in singular optics, especially the polarization singularity, have rapidly penetrated the surging fields of topological and nonhermitian photonics. For open photonic structures with degeneracies in particular, the polarization singularity would inevitably encounter another sweeping concept of Berry phase. Several investigations have discussed, in an inexplicit way, the connections between both concepts, hinting at that nonzero topological charges for far-field polarizations on a loop is inextricably linked to its nontrivial Berry phase when degeneracies are enclosed. In this work,…
Structured optical fields embedded with polarization singularities (PSs) have attracted extensive attention due to their capability to retain topological invariance during propagation. Many advances in PSs research have been made over the past 20 years in the areas of mathematical description, generation and detection technologies, propagation dynamics, and applications. However, one of the most crucial and difficult tasks continues to be manipulating PSs with multiple degrees of freedom, especially in three-dimensional (3D) tailored optical fields. We propose and demonstrate the longitudinal …
Epsilon-near-zero and epsilon near-pole materials enable reflective systems supporting a class of symmetry-protected and accidental embedded eigenstates (EE) characterized by a diverging phase-resonance. Here we show that pairs of topologically protected scattering singularities necessarily emerge from EEs when a non-Hermitian parameter is introduced, lifting the degeneracy between oppositely charged singularities. The underlying topological charges are characterized by an integer winding number and appear as phase vortices of the complex reflection coefficient. By creating and annihilating th…
All-optical programming of polarization singularities in a photonic-crystal laser
Ecole Centrale de Lyon, CNRS, INSA Lyon, Université Claude Bernard Lyon 1, CPE Lyon, Institut des Nanotechnologies de Lyon (INL)（里昂中央理工学院，法国国家科学研究中心，里昂国立应用科学学院，里昂第一大学，里昂高等化工学院，里昂纳米技术研究所）；Department of Physics, NISM Institute, University of Namur（纳慕尔大学物理系，NISM研究所）；Centre for OptoElectronics and Biophotonics (COEB), School of Electrical and Electronic Engineering, Nanyang Technological University（南洋理工大学电气与电子工程学院光电与生物光子学中心）；CNRS-International-NTU-Thales Research Alliance (CINTRA)（法国国家科学研究中心-国际南洋理工大学-泰雷兹研究联盟）
A new classification of circular polarization C points in three-dimensional polarization ellipse fields is proposed. The classification type depends on the out-of-plane variation of the polarization ellipse axis, in particular, whether the ellipse axes are in the plane of circular polarization one or three times. A minimal set of parameters for this classification are derived, and discussed in the context of the familiar in-plane C point classification into lemon, star, and monstar types. This new geometric classification is related to the Möbius index of polarization singularities recently in…
A single paraxial beam reflection at a plane dielectric interface, configured appropriately, can lead to the formation of a polarization singularity in the inhomogeneously polarized output beam-field for any central angle of incidence. In this paper we derive the necessary condition to realize this effect. We explore the phase singularity characteristics associated to this polarization-singular field; and explore the dynamics of the singularities due to controlled variations of the input polarization. The simulation-generated exact field information lead to the exploration of the unique Goos-H…
In this article, we study the emergence of polarization singularities in the scattered fields of optical resonators excited by linearly polarized plane waves. First, we prove analytically that combinations of isotropic electric and magnetic dipoles can sustain L surfaces, and C lines that propagate from the near-field to the far field. Moreover, based on these analytical results, we derive anomalous scattering Kerker conditions trough singular optics arguments. Secondly, through exact full-field calculations, we demonstrate that high refractive index spherical resonators present such topologic…
Chiral optical effects are generally quantified along some specific incident directions of exciting waves (especially for extrinsic chiralities of achiral structures) or defined as direction-independent properties by averaging the responses among all structure orientations. Though of great significance for various applications, chirality extremization (maximized or minimized) with respect to incident directions or structure orientations have not been explored, especially in a systematic manner. In this study we examine the chiral responses of open photonic structures from perspectives of quasi…
A nonlinear self-focusing material can amplify random small-amplitude phase modulations present in an optical beam, leading to the formation of amplitude singularities commonly referred to as optical caustics. By imposing polarization structuring on the beam, we demonstrate the suppression of amplitude singularities caused by nonlinear self-phase modulation. Our results are the first to indicate that polarization-structured beams can suppress nonlinear caustic formation in a saturable self-focusing medium and add to the growing understanding of catastrophic self-focusing effects in beams conta…
A physical theory is presented for polarized light from an aspect of polarization singularity.This is carried out by analyzing the evolution equation of the Stokes parameters that is derived from the nonlinear Schrodinger type equation. The problem is explored from two aspects:The first is concerning the single mode Stokes parameter that is described by the propagation distance $ z $. The trajectory of the polarization singularities is simply given by the points satisfying a rule: $ S_3 (z) = +(-)S_0 $ and $ 0 $ ($ S_3 $ represents the third component of the Stokes parameters), which form a tu…
Speckle patterns produced by random optical fields with two (or more) widely different correlation lengths exhibit speckle spots that are themselves highly speckled. Using computer simulations and analytic theory we present results for the point singularities of speckled speckle fields: optical vortices in scalar (one polarization component) fields; C points in vector (two polarization component) fields. In single correlation length fields both types of singularities tend to be more{}-or{}-less uniformly distributed. In contrast, the singularity structure of speckled speckle is anomalous: for …
We present the results of a study of the Poynting vector field generic singularities at the resonant light scattering of a plane monochromatic linearly polarized electromagnetic wave by a subwavelength particle. We reveal the impact of the problem symmetry, the spatial dimension, and the energy conservation law on the properties of the singularities. We show that, in the cases when the problem symmetry results in the existence of an invariant plane for the Poynting vector field lines, a formation of a standing wave in the immediate vicinity of a singularity gives rise to a saddle-type singular…
Material characteristics and input-field specifics limit the controllability of nonlinear electromagnetic-field interactions. As these nonlinear interactions could be exploited to create strongly localized bright and dark waves, such as nonlinear surface polaritons, ameliorating this limitation is important. We present our approach to amelioration, which is based on a surface-polaritonic waveguide reconfiguration that enables excitation, propagation and coherent control of coupled dark rogue waves having orthogonal polarizations. Our control mechanism is achieved by finely tuning laser-field i…
For propagating and evanescent vector Bessel beams, we study the singularities of the Poynting vector (Poynting singularities), at which the energy flux density turns to zero. Poynting singularities include all the phase singularities and some of polarization ones (L- and C-points). We reveal the existence conditions and positions of singularities, which are located at cylindrical surfaces around the beam axis. We mark the special case of the evanescent Bessel beam in the form of cylindrical standing wave, that is singular at any spatial point.
Dynamics of polarization optical singularities chain reactions in generic elliptically polarized speckle fields created in photorefractive crystal LiNbO3 was investigated in details Induced speckle field develops in the tens of minutes scale due to photorefractive 'optical damage effect' induced by incident beam of He-Ne laser. It was shown that polarization singularities develop through topological chain reactions of developing speckle fields driven by photorefractive nonlinearities induced by incident laser beam. All optical singularities (C points, optical vortices, optical diabolos,) are d…
In order to explore electric-field-induced transformations of polarization singularities in the polarization-resolved angular (conoscopic) patterns emerging after deformed helix ferroelectric liquid crystal (DHFLC) cells with subwavelength helix pitch, we combine the transfer matrix formalism with the results for the effective dielectric tensor of biaxial FLCs evaluated using an improved technique of averaging over distorted helical structures. Within the framework of the transfer matrix method, we deduce a number of symmetry relations and show that the symmetry axis of L lines (curves of line…
Topological physics desires stable methods to measure the polarization singularities in optical vector fields. Here a periodic plasmonic metasurface is proposed to perform divergence computation of vectorial paraxial beams. We design such an optical device to compute spatial differentiation along two directions, parallel and perpendicular to the incident plane, simultaneously. The divergence operation is achieved by creating the constructive interference between two derivative results. We demonstrate that such optical computations provide a new direct pathway to elucidate specific polarization…
We present a set of paraxial light beams with cylindrical symmetry, smooth and localized transversal profile carrying finite power, that develop intensity singularities when they are focused in a linear medium, such as vacuum. They include beams with orbital angular momentum and with radial polarization, in which case they develop punctual phase and polarization singularities surrounded by infinitely bright rings, along with singular longitudinal fields. In practice, these effects are manifested in focal intensities and spot sizes, vortex bright ring intensities and radii, and strengths of the…
The fundamental polarization singularities of monochromatic light are normally associated with invariance under coordinated rotations: symmetry operations that rotate the spatial dependence of an electromagnetic field by an angle $θ$ and its polarization by a multiple $γθ$ of that angle. These symmetries are generated by mixed angular momenta of the form $J_γ= L + γS$ and they generally induce Möbius-strip topologies, with the coordination parameter $γ$ restricted to integer and half-integer values. In this work we construct beams of light that are invariant under coordinated rotations for arb…
The geometric phase is usually treated as a quantity modulo 2π, a convention carried over from early work on the subject. The results of a series of optical interference experiments involving polarization of light, done by the present author (reviewed in R.Bhandari, Phys. Rep. 281 (1997) p.1) question the usefulness of such a definition of the geometric phase in that it throws away useful and measurable information about the system, for example strengths of singularities giving rise to the geometric phase. Such singularities have been directly demonstrated by phase-shift measurement in interfe…
Optical singularities manifesting at the center of vector vortex beams are unstable, since their topological charge is higher than the lowest value permitted by Maxwell's equations. Inspired by conceptually similar phenomena occurring in the polarization pattern characterizing the skylight, we show how perturbations that break the symmetry of radially symmetric vector beams lead to the formation of a pair of fundamental and stable singularities, i.e. points of circular polarization. We prepare a superposition of a radial (or azimuthal) vector beam and a uniformly linearly polarized Gaussian be…
Bound states in the continuum (BICs), circularly polarized states (C points) and degenerate states are all of three types of singular points of polarization in the momentum space. For photonic crystal slabs (PhCSs) with linearly polarized far fields, BICs were found to be the centers of polarization vortices and attracted more attention in the previous studies. Here, we theoretically demonstrate that the far fields of PhCSs can exhibit remarkably diverse polarizations due to the robust existences of C points in the continuum. Only a pair of C points with identical handedness and opposite topol…
Electromagnetic multipoles have been broadly adopted as a fundamental language throughout photonics, of which general features such as radiation patterns and polarization distributions are generically known, while their singularities and topological properties have mostly slipped into oblivion. Here we map all the singularities of multipolar radiations of different orders, identify their indexes, and show explicitly the index sum over the entire momentum sphere is always $2$, consistent with the Poincaré-Hopf theorem. Upon those revealed properties, we can attribute the formation of bound stat…
Spin angular momentum, an elementary dynamical property of classical electromagnetic fields, plays an important role in spin-orbit and light-matter interactions, especially in near-field optics. The research on optical spins has led to the discovery of phenomena such as optical spin-momentum locking and photonic topological quasiparticles, as well as applications in high-precision detection and nanometrology. Here, we investigate spin-momentum relations in paraxial optical systems and show that the optical spin angular momentum contains transverse and longitudinal spin components simultaneousl…
Topology has appeared in different physical contexts. The most prominent application is topologically protected edge transport in condensed matter physics. The Chern number, the topological invariant of gapped Bloch Hamiltonians, is an important quantity in this field. Another example of topology, in polarization physics, are polarization singularities, called L lines and C points. By establishing a connection between these two theories, we develop a novel technique to visualize and potentially measure the Chern number: it can be expressed either as the winding of the polarization azimuth alon…
We study the angular structure of polarization of light transmitted through a nematic liquid crystal (NLC) cell by analyzing the polarization state as a function of the incidence angles and the polarization of the incident wave. The polarization resolved angular patterns emerging after the NLC cell illuminated by the convergent light beam are described in terms of the polarization singularities such as C-points (points of circular polarization) and L-lines (lines of linear polarization). For the homeotropically aligned cell, the Stokes polarimetry technique is used to measure the polarization …
Electromagnetic multipoles can be continuously mapped to tangent vectors on the momentum sphere, the topology of which guarantees the existence of isolated singularities. For pure (real or imaginary) vectors, those singularities correspond to zeros of tangent fields, which can be classified by integer Poincaré indices. Nevertheless, electromagnetic fields are generally complex vectors, a comprehensive characterization of which requires the introduction of line fields and line singularities categorized by half-integer Hopf indices. Here we explore complex vectorial electromagnetic multipoles fr…
We generalized the conventional concept of q-plate, allowing in its definition non linear functions of the azimuthal coordinate, and simulated the resulting fields of applying this kind of element to uniformly polarized input beams, both in the near (Fresnel diffraction) and the far field (Fraunhofer diffraction) approximations. In general terms, when working in the near field regime, the chosen function defines the output polarization structure for linearly polarized input beams, and the phase of the output field for circularly polarized input beams. In the far field regime, it is obtained th…
Fast electrons spectromicroscopies enable to measure quantitatively the optical response of excitations with unrivaled spatial resolution. However, due to their inherently scalar nature, electron waves cannot access to polarization-related quantities. In spite of promising attempts based on the conversion of concepts originating from singular optics (such as vortex beams), the definition of an optical polarization analogue for fast electrons has remained a dead letter. Here, we establish such an analogue as the dipole transition vector of the electron between two well-chosen singular wave stat…
Among the interesting outcomes of the study of the physical applications of spectral singularities in PT-symmetric optical systems is the discovery of CPA-lasers. These are devices that act both as a threshold laser and a coherent perfect absorber (CPA) for the same values of their physical parameters. Unlike a homogeneous slab that is made to act as a CPA, a slab CPA-laser would absorb the incident waves coming from the left and right of the device provided that they have appropriate intensity and phase contrasts. We provide a comprehensive study of one of the simplest experimentally accessib…
Symmetries and conservation laws of energy, linear momentum and angular momentum play a central role in physics, in particular in nonlinear optics. Recently, light fields with non trivial topology, such as polarization Möbius strips or torus-knot beams, have been unveiled. They cannot be associated to well-defined values of orbital and spin angular momenta (OAM and SAM), but are invariant under coordinated rotations, i.e. rotational symmetries that are generated by the generalized angular momentum (GAM) operator, a mixture of the OAM and SAM operators. The discovery of the GAM, which at varian…
Exploiting topological ideas has been a major theme in modern photonics, which provides unprecedented opportunities to design photonic devices with robustness against defects and flaws. While most previous works in topological photonics have focused on band theory, recent theoretical advances extend the topological concepts to the analysis of scattering matrices and suggest a topological route to complete polarization conversion (CPC), a unique photonic phenomenon without an electronic counterpart. Here, we report on the experimental observation of the topological effect in reflection matrices…
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 desc…
