Polarization singularities of focused, radially polarized fields

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.

Dynamics of Topological Polarization Singularity in Momentum Space

The polarization singularity in momentum space has recently been discovered as a new class of topological signatures of Bloch modes in photonic crystal slabs concerning the far-field radiations, beyond its near-field description with widely explored topological band theory. Bound states in the continuum (BICs) in photonic crystal slabs are demonstrated as vortex eigenpolarization singularities in momentum space and the circular polarization points (C points) are also obtained based on BICs, opening up more possibilities for exotic light scattering and various topological phenomena of singular optics. Here, focusing on the nondegenerate bands, we report the generation to annihilation of two pairs of C points in momentum space in the photonic crystal slabs with inversion symmetry but broken up-down mirror symmetry. Interestingly, as the C points evolve with the structure parameter, we find two merging processes of C points, where an accidental at-Γ BIC and unidirectional radiative resonances with leaky channels of drastically different radiative lifetime emerge. The whole evolution is governed by the global charge conservation and the sum of topological charges equals to zero. Our findings suggest a novel recipe for dynamic generation and manipulation of various polarization singularities in momentum space and might shed new light to control the resonant and topological properties of light-matter interactions.

Energy transfer of the tightly focused hybridly polarized vector optical fields with elliptic symmetry in free space

We theoretically and experimentally present hybridly polarized vector optical fields (HP-VOFs) with elliptic symmetry in an elliptic coordinate system. Compared with the traditional cylindrical HP-VOFs, there is an additional degree of freedom for this new kind of vector optical field, which is the interval between the two foci in the elliptic coordinate system. Except for discussing the singularities of the HP-VOFs, we concentrate on studying the energy transfer of the tightly focused HP-VOFs with elliptic symmetry in free space. We summarize the rules of the energy transfer and introduce a reference optical field to explain them. We hope these results can provide a new way to flexibly modulate tightly focused fields, which may be applied in realms such as optical machining, optical trapping, and information transmission.

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.

Circularly Polarized States Spawning from Bound States in the Continuum

Bound states in the continuum in periodic photonic systems like photonic crystal slabs are proved to be accompanied by vortex polarization singularities on the photonic bands in the momentum space. The winding structures of polarization states not only widen the field of topological physics but also show great potential that such systems could be applied in polarization manipulating. In this Letter, we report the phenomenon that by in-plane inversion (C_{2}) symmetry breaking, pairs of circularly polarized states could spawn from the eliminated bound states in the continuum. Along with the appearance of the circularly polarized states as the two poles of the Poincaré sphere together with linearly polarized states covering the equator, full coverage on the Poincaré sphere could be realized. As an application, ellipticity modulation of linear polarization is demonstrated in the visible frequency range. This phenomenon provides a new degree of freedom in modulating polarization. The C points could also find applications in light-matter interactions. Further studying and manipulating the reported polarization singularities may lead to novel phenomena and physics in radiation modulating and topological photonics.

Evolution properties of the radially polarized multi-Gaussian Schell-model beam in uniaxial crystals

The evolution properties of the normalized intensity distribution, the spectral degree of coherence (SDOC), and the spectral degree of polarization (SDOP) of the radially polarized multi-Gaussian Schell-model (MGSM) beam in uniaxial crystals are illustrated. Numerical results show that the intensity distribution of the radially polarized MGSM beam gradually evolves from a doughnut shape into an elliptical symmetric flattop shape and retains its elliptical flattop shape on further propagation in anisotropic crystals. The evolution behavior of the SDOC and SDOP for the radially polarized MGSM beam is quite different from that of the linearly polarized one. In addition, the influences of the spatial coherence length δ₀, beam index M, and the ratio of the extraordinary refractive index to the ordinary refractive index n_e/n_o of the uniaxial crystals on the evolution properties of the normalized intensity distribution, the SDOC, and the SDOP of the radially polarized MGSM beam are discussed in detail.

Sculpting complex polarization singularity networks

Polarization singularities in vectorial light fields have become an important tool for different cutting-edge applications such as information processing with integer information units. However, even though vectorial singularities naturally form configurations of multiple singular points, up to now only rather simple, mostly cylindrical vector beams including single central singularities, have been considered. Here we demonstrate the customization of extended singularity networks embedded in a class of complex polarization structures based on general Ince-Gaussian modes, namely, Ince-Gaussian vector modes. Contributing to fundamental singular optics, our investigations evince the conservation of tailored singularity arrangements upon 3D propagation, whereby respective modes enlarge the range of stable vectorial fields, paving the way to information technologies with a significantly enhanced number of degrees of freedom.

Observation of Polarization Vortices in Momentum Space

The vortex, a fundamental topological excitation featuring the in-plane winding of a vector field, is important in various areas such as fluid dynamics, liquid crystals, and superconductors. Although commonly existing in nature, vortices were observed exclusively in real space. Here, we experimentally observed momentum-space vortices as the winding of far-field polarization vectors in the first Brillouin zone of periodic plasmonic structures. Using homemade polarization-resolved momentum-space imaging spectroscopy, we mapped out the dispersion, lifetime, and polarization of all radiative states at the visible wavelengths. The momentum-space vortices were experimentally identified by their winding patterns in the polarization-resolved isofrequency contours and their diverging radiative quality factors. Such polarization vortices can exist robustly on any periodic systems of vectorial fields, while they are not captured by the existing topological band theory developed for scalar fields. Our work provides a new way for designing high-Q plasmonic resonances, generating vector beams, and studying topological photonics in the momentum space.

Tailored intensity landscapes by tight focusing of singular vector beams

Vector beams are of major importance to tailor tightly focused fields by creating an additional z-polarization component. Till now, mainly focusing properties of fundamental vector beams have been investigated, whereas the knowledge of focused higher-order singular vector fields is still missing. We fill this gap by numerical analysis of these fields, applying their attractive characteristics as including a spatially adjustable amount of radial and azimuthal components. We demonstrate the realization of three-dimensional polarization structures whose total intensity resembles dark stars and bright flowers. Further, we tailor these focal intensity landscapes by modulating the order of incident vector fields. This in turn allows shaping the focus of a light field for specific applications as e.g. advanced microscopy.

Topological events on the lines of circular polarization in nonparaxial vector optical fields

In nonparaxial vector optical fields, the following topological events are shown to occur in apparent violation of charge conservation: as one translates the observation plane along a line of circular polarization (a C line), the points on the line (C points) are seen to change not only the signs of their topological charges, but also their handedness, and, at turning points on the line, paired C points with the same topological charge and opposite handedness are seen to nucleate. These counter-intuitive events cannot occur in paraxial fields.

Time-reversal constraint limits unidirectional photon emission in slow-light photonic crystals

Photonic crystal waveguides are known to support C-points-point-like polarization singularities with local chirality. Such points can couple with dipole-like emitters to produce highly directional emission, from which spin-photon entanglers can be built. Much is made of the promise of using slow-light modes to enhance this light-matter coupling. Here we explore the transition from travelling to standing waves for two different photonic crystal waveguide designs. We find that time-reversal symmetry and the reciprocal nature of light places constraints on using C-points in the slow-light regime. We observe two distinctly different mechanisms through which this condition is satisfied in the two waveguides. In the waveguide designs, we consider a modest group velocity of vg≈c/10 is found to be the optimum for slow-light coupling to the C-points.This article is part of the themed issue 'Unifying physics and technology in light of Maxwell's equations'.

Vector Hermite-Gaussian correlated Schell-model beam

A new kind of partially coherent vector beam named vector Hermite-Gaussian correlated Schell-model (HGCSM) beam is introduced as a natural extension of recently introduced scalar HGCSM beam. The realizability and beam conditions for a vector HGCSM beam with uniform state of polarization (SOP) or non-uniform SOP are derived, respectively. Furthermore, analytical formulae for a vector HGCSM beam propagating in free space are derived, and the propagation properties of a vector HGCSM beam with uniform SOP or non-uniform SOP in free space are studied and analyzed in detail. We find that the behaviors of a vector HGCSM beam on propagation are quite different from those of a conventional vector partially coherent beam with uniform SOP or non-uniform SOP, and modulating the structures of the correlation functions cannot only modulate the intensity distribution, but also the state of polarization, the degree of polarization and the polarization singularities of a partially coherent vector beam on propagation. Furthermore, we report experimental generation of a radially polarized HGCSM beam for the first time. Our results provide a novel way for polarization modulation.

Cycle of phase, coherence and polarization singularities in Young's three-pinhole experiment

It is now well-established that a variety of singularities can be characterized and observed in optical wavefields. It is also known that these phase singularities, polarization singularities and coherence singularities are physically related, but the exact nature of their relationship is still somewhat unclear. We show how a Young-type three-pinhole interference experiment can be used to create a continuous cycle of transformations between classes of singularities, often accompanied by topological reactions in which different singularities are created and annihilated. This arrangement serves to clarify the relationships between the different singularity types, and provides a simple tool for further exploration.

Exceptional polarization structures near the C-lines in diffracted near fields

We study the polarization structures in the vicinity of C-lines in the near fields diffracted from a pair of small holes. We find that, when the incident light is circularly polarized, both the true C-lines and the structures near them are controlled only by the longitudinal component. Furthermore, we find that all the existing singular lines of circular polarization have the winding number ±1, which is very different than the usual numbers ±1/2, and the structure of major axes of the polarization ellipses surrounding these lines are shown to form structures different than the Möbius strip type. All these features prove to be stable upon small changes of shapes or positions of the apertures. However, C-lines with a unit winding number split into two C-lines of half-winding numbers when the incident light is elliptically polarized light.

Correlation singularities in a partially coherent electromagnetic beam with initially radial polarization

We have investigated the correlation singularities, coherence vortices of two-point correlation function in a partially coherent vector beam with initially radial polarization, i.e., partially coherent radially polarized (PCRP) beam. It is found that these singularities generally occur during free space propagation. Analytical formulae for characterizing the dynamics of the correlation singularities on propagation are derived. The influence of the spatial coherence length of the beam on the evolution properties of the correlation singularities and the conditions for creation and annihilation of the correlation singularities during propagation have been studied in detail based on the derived formulae. Some interesting results are illustrated. These correlation singularities have implication for interference experiments with a PCRP beam.

Polarization tailored novel vector beams based on conical refraction

Coherent vector beams with involved states of polarization (SOP) are widespread in the literature, having applications in laser processing, super-resolution imaging and particle trapping. We report novel vector beams obtained by transforming a Gaussian beam passing through a biaxial crystal, by means of the conical refraction phenomenon. We analyze both experimentally and theoretically the SOP of the different vector beams generated and demonstrate that the SOP of the input beam can be used to control both the shape and the SOP of the transformed beam. We also identify polarization singularities of such beams for the first time and demonstrate their control by the SOP of the input beam.

Observer-dependent sign inversions of polarization singularities

We describe observer-dependent sign inversions of the topological charges of vector field polarization singularities: C points (points of circular polarization), L points (points of linear polarization), and two virtually unknown singularities we call γ(C) and α(L) points. In all cases, the sign of the charge seen by an observer can change as she changes the direction from which she views the singularity. Analytic formulas are given for all C and all L point sign inversions.

Optical Möbius strips and twisted ribbon cloaks

Optical Möbius strips that surround points of circular polarization, C points, in a generic three-dimensional optical field are cloaked by lines of twisted ribbons attached to the C points. When cloaking occurs, the observable signed twist index that counts the number of half-twists (one or three), and also measures the handedness (right or left), of a generic Möbius strip is determined by the twisted ribbon cloaks. Although some cloaks can be detached, they can never all be removed.

Electric and magnetic polarization singularities of first-order Laguerre-Gaussian beams diffracted at a half-plane screen

Based on the vector Fresnel diffraction integrals, analytical expressions for the electric and magnetic components of first-order Laguerre-Gaussian beams diffracted at a half-plane screen are derived and used to study the electric and magnetic polarization singularities in the diffraction field for both two- and three-dimensional (2D and 3D) cases. It is shown that there exist 2D and 3D electric and magnetic polarization singularities in the diffraction field, which do not coincide each other in general. By suitably varying the waist width ratio, off-axis displacement parameter, amplitude ratio, or propagation distance, the motion, pair-creation, and annihilation of circular polarization singularities, and the motion of linear polarization singularities take place in 2D and 3D electric and magnetic fields. The V point, at which two circular polarization singularities with the same topological charge but opposite handedness collide, appears in the 2D electric field under certain conditions in the diffraction field and free-space propagation. A comparison with the free-space propagation is also made.

Polarization singularities in superposition of vector beams

We present a systematic study of the superposition of two vector Laguerre-Gaussian (LG) beams. Propagation depended field distribution obtained from the superposition of two vector LG beams has many interesting features of intensity and polarization. Characteristic inhomogeneous polarization distribution of the vector LG beam appears in the form of azimuthally modulated intensity and polarization distributions in the superposition of the beams. We found that the array of polarization singular points, whose number depends upon the azimuthal indices of the two beams, evolves during propagation of the field. The position and number of C-points generated in the field were analyzed using Stokes singularity relations. Novel intensity and polarization patterns obtained from the superposition of two vector LG beams may find applications in the field of molecular imaging, optical manipulation, atom optics, and optical lattices.

Manifestation of the Gouy phase in strongly focused, radially polarized beams

The Gouy phase, sometimes called the focal phase anomaly, is the curious effect that in the vicinity of its focus a diffracted field, compared to a non-diffracted, converging spherical wave of the same frequency, undergoes a rapid phase change by an amount of π. We theoretically investigate the phase behavior and the polarization ellipse of a strongly focused, radially polarized beam. We find that the significant variation of the state of polarization in the focal region, is a manifestation of the different Gouy phases that the two electric field components undergo.

Controlling the polarization singularities of the focused azimuthally polarized beams

We mainly investigate the polarization singularities of the focused azimuthally polarized (AP) beams modulated by spiral phase and sector obstacles. The results reveal that either the spiral phase or sector obstacle can convert the central V-point to C-points, C-point dipoles, or even double V-points under certain conditions. The conversion can be selectively controlled by appropriately setting the topological charge of the spiral phase and the sector angle of the obstacle. These results may have implications for the researches on polarization, focal field manipulation, or even angular momentum of the focused cylindrically polarized beams.

Ordinary polarization singularities in three-dimensional optical fields

In generic three-dimensional optical fields the canonical point polarization singularities are points of circular polarization, C points on C lines, and points of linear polarization, L points on L lines. These special points are surrounded by a sea of ordinary points. In planes oriented normal to the principle axes of the polarization ellipse at the point, every ordinary point is also a singularity, here an ordinary polarization singularity, or O point. Interactions between O points, between O points and C points, and between O points and L points are described that highlight the fact that a consistent description of optical fields containing C and L lines must include O points.

Optical Möbius strips, twisted ribbons, and the index theorem

The twist numbers of circular optical Möbius strips and twisted ribbons are shown to obey the index theorem under rotation of the plane of observation, and under change in the radius of the path.

Composite polarization singularities in superimposed Laguerre-Gaussian beams beyond the paraxial approximation

The closed-form expression for the free-space propagation of superimposed Laguerre-Gaussian beams beyond the paraxial approximation is derived, and the composite polarization singularities formed by the transverse and longitudinal electric-field components are studied in detail. It is shown that there exist composite C-points and L-lines in vector nonparaxial fields. By suitably varying a control parameter, such as the off-axis distance, relative phase, or amplitude ratio, the motion, creation, and annihilation of composite C-points may appear, and in the process the sum of topological charge remains unchanged. The shift, deformation, combination, and disappearance of composite L-lines may take place. The topological relationship holds true. The results are compared with the previous work.

Electromagnetic spin-orbit interactions via scattering of subwavelength apertures

Circularly polarized electric fields incident on subwavelength apertures produce near-field phase singularities with phase vorticity +/-1 depending on the polarization handedness. These near-field phase singularities combine with those associated with orbital angular momentum and result in polarization-dependent transmission. We produce arbitrary phase vorticity in the longitudinal component of scattered electric fields by varying the incident beam and aperture configuration.

Polarization singularities of Gaussian vortex beams diffracted at a half-plane screen beyond the paraxial approximation

On the basis of the vector Rayleigh-Sommerfeld diffraction integrals, the analytical expression for Gaussian vortex beams diffracted at a half-plane screen beyond the paraxial approximation is derived and used to study the polarization singularities formed by the transverse and longitudinal electric field components in the diffracted field. It is shown that there exist C-points and L-lines that depend on the off-axis parameters in the x and y directions, waist width, wavelength, and topological charge of diffracted Gaussian vortex beams, as well as on the position parameter. By a suitable variation of the off-axis and position parameters, the creation, motion, and annihilation of polarization singularities may take place, and the topological relationship holds true. Therefore, the nonparaxial beam diffraction and propagation provide a method for generating polarization singularities.

Spectral Stokes singularities of stochastic electromagnetic beams

The spectral Stokes singularities are introduced to describe the polarization singularities of stochastic electromagnetic beams. The motion and annihilation, as well as the changes in the polarization of spectral Stokes singularities, take place in the free-space propagation. The results are illustrated by numerical examples.

Evolution of singularities in a partially coherent vortex beam

We study the evolution of phase singularities and coherence singularities in a Laguerre-Gauss beam that is rendered partially coherent by letting it pass through a spatial light modulator. The original beam has an on-axis minumum of intensity--a phase singularity--that transforms into a maximum of the far-field intensity. In contrast, although the original beam has no coherence singularities, such singularities are found to develop as the beam propagates. This disappearance of one kind of singularity and the gradual appearance of another is illustrated with numerical examples.

Observation of polarization singularities at the nanoscale

With a phase-sensitive near-field microscope we measure independently the two in-plane electric field components of light propagating through a 2D photonic crystal waveguide and the phase difference between them. Consequently, we are able to reconstruct the electric vector field distribution with subwavelength resolution. In the complex field distribution we observe both time-dependent and time-independent polarization singularities and determine the topology of the surrounding electric field.

Partially correlated sources with complete polarization

We show that the correlation matrix of any electromagnetic planar source endowed with complete polarization must be of a universal form, no matter how the field components are correlated across the source plane. Such form entails that both the realizability conditions and the search of vector modes of the source reduce to scalar problems. We shall also briefly dwell on how propagation affects the polarization properties of the field and, in particular, we discuss cases in which propagated light remains completely polarized.

Observation of singularities in multiply scattered microwave fields

Speckle patterns of arbitrary resolution are obtained by applying the sampling theorem to measurements of two orthogonal components of the microwave field transmitted through multiply scattering samples. Core structures of phase singularities, phase critical points, and polarization singularities are explored. We find that equiphase lines connect phase singularities with opposite topological signs except for the bifurcation lines, which run through a phase saddle point, in agreement with predictions by Freund [Phys. Rev. E25, 2348 (1995)]. We observe hyperbolic equiphase lines near phase saddle points and elliptical equiphase lines around phase extrema. Polarization singularities of the vector field with the three morphologies predicted are observed.