Paraxial Vector Research Articles

Spin Hall effect of linearly polarized light passed through a metasurface

The density of the longitudinal component of the spin angular momentum (SAM) vector is calculated for a paraxial vector Gaussian beam with a periodic one-dimensional modulation. For the beam under consideration, the SAM in the initial plane is zero and the polarization is inhomogeneous and linear. When this beam propagates in free space, due to periodic modulation it is effectively divided into two beams with left-handed and right-handed elliptical polarization. That is, in the cross section of the beam, regions with spins of different signs are separated in space, which is a manifestation of the spin Hall effect. This beam can be formed using a metasurface whose transmission periodically depends on one coordinate.

Theory of paraxial optical skyrmions

Vector light beams, characterized by a spatially varying polarization, can exhibit localized structures reminiscent of the skyrmions familiar from the study of magnetic media. We present a theory of such skyrmions within paraxial optics, exploiting mathematical analogies with the study of superfluids, especially the A phase of superfluid He 3 . The key feature is the skyrmion field, which, together with the underlying skyrmion vector potential, determines the properties of the skyrmions and, more generally, the polarization structure of every paraxial vector beam. In addition to structures with integer skyrmion numbers, we find polarization patterns with non-integer skyrmion numbers; these seem to have no analogue in other fields of physics.

Spin angular momentum of Gaussian beams with several polarization singularities

We study a paraxial vector Gaussian beam with several polarization singularities located on a circle. Such a beam is superposition of a cylindrically polarized Laguerre-Gaussian beam and a linearly polarized Gaussian beam. It is found that although polarization in the initial plane is linear, alternating regions with the different-sign spin angular momentum density are generated upon free-space propagation, showing that a spin Hall effect arises. For an arbitrary transverse plane, it is shown that the spin angular momentum magnitude is maximal on a certain-radius circle. We obtain an approximate expression for the distance to the transverse plane where the spin angular momentum density is maximal. Besides, we derive an optimal radius of the singularity-containing circle in the initial plane for which the maximal spin angular momentum density can be achieved upon propagation. It is revealed that in this case, the energies of the Laguerre-Gaussian beam and the Gaussian beam are equal to each other. We also obtain an expression for the orbital angular momentum density and find it to be defined by the spin angular momentum density, multiplied by –m/2, with m being the upper index of the Laguerre-Gaussian beam, equal to the number of the polarization singularities. An analogy with plane waves reveals that the spin Hall effect arises due to different divergence rates of the linearly polarized Gaussian beam and the cylindrically polarized Laguerre-Gaussian beam.

Spin Hall Effect of Two-Index Paraxial Vector Propagation-Invariant Beams

We investigate a simple paraxial vector beam, which is a coaxial superposition of two single-ringed Laguerre–Gaussian (LG) beams, linearly polarized along the horizontal axis, with topological charges (TC) n and −n, and of two LG beams, linearly polarized along the vertical axis, with the TCs m and −m. In the initial plane, such a vector beam has zero spin angular momentum (SAM). Upon propagation in free space, such a propagation-invariant beam has still zero SAM at several distances from the waist plane (initial plane). However, we show that at all other distances, the SAM becomes nonzero. The intensity distribution in the cross-section of such a beam has 2m (if m > n) lobes, the maxima of which reside on a circle of a certain radius. The SAM distribution has also several lobes, from 2m till 2(m + n), the centers of which reside on a circle with a radius smaller than that of the maximal-intensity circle. The SAM sign alternates differently: one lobe has a positive SAM, while two neighbor lobes on the circle have a negative SAM, or two neighbor pairs of lobes can have a positive and negative SAM. When passing through a plane with zero SAM, positive and negative SAM lobes are swapped. The maximal SAM value is achieved at a distance smaller than or equal to the Rayleigh distance.

Spin Hall Effect in Paraxial Vectorial Light Beams with an Infinite Number of Polarization Singularities.

Elements of micromachines can be driven by light, including structured light with phase and/or polarization singularities. We investigate here a paraxial vector Gaussian beam with an infinite number of polarization singularities residing evenly on a straight line. The intensity distribution is derived analytically and the polarization singularities are shown to exist only in the initial plane and in the far field. The azimuthal angle of the polarization singularities is shown to increase in the far field by π/2. We obtain the longitudinal component of the spin angular momentum (SAM) density and show that it is independent of the azimuthal angle of the polarization singularities. Upon propagation in free space, an infinite number of C-points is generated, where polarization is circular. We show that the SAM density distribution has a shape of four spots, two with left and two with right elliptic polarization. The distance to the transverse plane with the maximal SAM density decreases with decreasing distance between the polarization singularities in the initial plane. Generating such alternating areas with positive and negative SAM density, despite linear polarization in the initial plane, manifests the optical spin Hall effect. Application areas of the obtained results include designing micromachines with optically driven elements.

Polarisation-spatial entanglement upon reflection across a dielectric

Polarisation-spatial entanglement in coherent reflected paraxial vector light fields across an infinite dielectric interface, for polarised incidence, is studied. Closed form expressions for the polarisation-spatial entanglement is obtained explicitly in the case of Hermite Gaussian and Laguerre Gaussian modes up to first order approximation, assuming arbitrary state of incident polarisation, angle of incidence, waist width, and mode indices. General observations regarding the polarisation-spatial entanglement which hold true irrespective of the aforementioned parameters are arrived at. It is observed that close to 1 ebit of polarisation-spatial entanglement can be obtained for angles near Brewster’s incidence. The feasibility of possible experimental measurement of the entanglement obtained in reflected light fields is discussed through the Laguerre Gaussian mode as an example.

Polarization states and Stokes vortices of dual Butterfly-Gauss vortex beams with uniform polarization in uniaxial crystals

We propose dual Butterfly-Gauss vortex beams with uniform polarization, which are composed of two finite-energy Butterfly-Gauss beams with vortex cores. Based on the paraxial vector theory, the analytical propagation expressions of the proposed beams in uniaxial crystals orthogonal to the optical axis are derived and used to investigate the evolutions of autofocusing behavior, polarization states and Stokes vortices. The results show that the autofocusing behavior and its autofocusing length can be affected by topological charge, and the input uniform polarization is disordered into non-uniform polarization owing to anisotropic diffraction. The symmetry and rotation of polarization states are also found for the case of vortex. The Stokes parameters on the Poincaré sphere can be manipulated by adjusting the refractive index ratio, azimuthal angle and phase retardation angle. Their Stokes vortices, e.g. S12 vortices, present the behavior of merging, separation and reversal. The presented work may be useful for manipulating polarization dynamics of catastrophe optical field in uniaxial crystal and has potential values in polarization-sensitive detection.

Polarization-spatial Gaussian entanglement in partially coherent light fields.

The problem of bipartite entanglement in partially coherent paraxial vector light fields is addressed. A generalized uncertainty principle suited for the polarization-spatial degrees of freedom is introduced. Partial transpose is implemented through the obtained generalized uncertainty principle. Partial transpose is shown to be necessary and sufficient in detecting entanglement for a class of partially coherent vector light fields which have a spatial part to be Gaussian. An experimental realization of the studied entangled states using classical optical interferometry is outlined.

Spin angular momentum flux density of non-uniformly polarized vortex beams with tunable polarization angles in uniaxial crystals

We propose the non-uniformly polarized vortex (NUPV) beams with the changing retardation angle and the tunable angle between polarization and radial direction, in which the radially polarized vortex (RPV) beam is a particular case. Based on the paraxial vector theory, the analytical propagation expressions of NUPV beams in uniaxial crystals orthogonal to the optical axis are derived and used to investigate the evolutions of spin angular momentum (SAM) flux densities, polarization states, and the Stokes parameters. The results show a more complex behavior of SAM such as rotation, creation and disappearance in cyclical ways can be found in anisotropic media, which is different from a monotonous rotation of SAM for hybridly polarized beams without a vortex. Besides, it is worth mentioning that the SAM of the NUPV beams also presents the more complex evolution in isotropic media rather than stable Gaussian-like structures, which indicate the more flexible polarization behavior of NUPV beams. Then the influence of refractive index ratio ne/no and topological charge number m on SAM is discussed. The results show the flexibility of polarization is strengthened by anisotropy or the embedded vortex. The polarization states and the Stokes parameters on the Poincaré spheres both verified the change in SAM. The presented work has potential values and broad prospects in polarization-sensitive sensing and detection.

Gaussian spatial-polarization entanglement in a folded Mach-Zehnder interferometer.

Gaussian spatial-polarization entanglement in a coherent vectorial paraxial light field is studied. Detection of spatial-polarization entanglement through fringe movement on rotation of a linear polarizer, with the light field passing through the polarizer, is outlined. The fringe movement is shown to be a sufficient condition for the detection of spatial-polarization entanglement in coherent paraxial vector light fields. Two Gaussian light fields with a small relative tilt but with significant spatial overlap and with orthogonal polarizations are shown to possess close to 1 ebit of spatial-polarization entanglement. Tunable Gaussian spatial-polarization entanglement is experimentally demonstrated in a folded Mach-Zehnder interferometer.

Analysis of characteristics of paraxial vector Gaussian beams affecting the formation of microstructures in an azopolymer

The paper considers paraxial Gaussian laser beams with different polarization states and their characteristics, such as the Poynting vector, gradient force, and force density, which are important for evaluating the effects of radiation on matter. An analysis of the considered characteristics of laser radiation from the point of view of the formation of photoinduced microrelief in an azopolymer film is carried out.

Vector properties of radially self-accelerating beams

We present a complete and consistent theory of vector radially self-accelerating beams (RSABs). We use this theory as a model for describing the properties of focused RSABs, and to show, in particular, that only circular polarised RSABs maintain their self-accelerating character upon focusing. Moreover, we also calculate the linear and angular momentum for paraxial vector RSABs, and discuss both their global and local properties.

Propagation property of the non-paraxial vector Lissajous singularity beams in free space

The analytic expressions for the free-space propagation of paraxial and non-paraxial vector Lissajous singularity beams are derived, and used to compare the propagation property of a Lissajous singularity carried by paraxial and non-paraxial vector beams in free space. It is found that the creation of a single Lissajous singularity, the creation and annihilation of pairs Lissajous singularities may take place for the both cases. However, after the annihilation of a pair of singularities, no Lissajous singularities appear in the output field for non-paraxial vector Lissajous singularity beams, which is different from the paraxial vector Lissajous singularity beams.

振幅比对矢量光束相干和非相干叠加合成Stokes奇点的影响

Based on the Stokes parameters,the complex Stokes scalar fields and the Fresnel diffraction integral method,the expressions for the composite Stokes singularities versus amplitude ratioηin two coherently and incoherently superimposed paraxial monochromatic vector optical beams were derived.The properties of composite Stokes singularities in coherently and incoherently superimposed vector optical beams with amplitude ratio propagating in free space were investigated,and the degree of polarization versus the amplitude ratio is analyzed.It is found that,during propagating in free space,the number,position and singularity index of composite Stokes singularities are affected by the amplitude ratio of two vector beams.In the coherent superposition the degree of polarization P=1.At the z=0plane,when the amplitude ratioη≠1,singularity indices of C-points,S23and S31singularities may be reversed.The creation of S23singularity with double singularity index may take place at the z0plane.In the incoherent superposition,the degree of polarization Pis less than 1and is variable withη,by varying the amplitude ratioηV-points may appear,which are unstable and split up into S23singularities and C-points with handedness reversal.With a further increase ofηS23singularities may annihilate.In addition,the sum of singularity indices of composite Stokes singularities is conserved in the process.

振幅比对矢量光束相干和非相干叠加合成Stokes奇点的影响

Based on the Stokes parameters,the complex Stokes scalar fields and the Fresnel diffraction integral method,the expressions for the composite Stokes singularities versus amplitude ratioηin two coherently and incoherently superimposed paraxial monochromatic vector optical beams were derived.The properties of composite Stokes singularities in coherently and incoherently superimposed vector optical beams with amplitude ratio propagating in free space were investigated,and the degree of polarization versus the amplitude ratio is analyzed.It is found that,during propagating in free space,the number,position and singularity index of composite Stokes singularities are affected by the amplitude ratio of two vector beams.In the coherent superposition the degree of polarization P=1.At the z=0plane,when the amplitude ratioη≠1,singularity indices of C-points,S23and S31singularities may be reversed.The creation of S23singularity with double singularity index may take place at the z0plane.In the incoherent superposition,the degree of polarization Pis less than 1and is variable withη,by varying the amplitude ratioηV-points may appear,which are unstable and split up into S23singularities and C-points with handedness reversal.With a further increase ofηS23singularities may annihilate.In addition,the sum of singularity indices of composite Stokes singularities is conserved in the process.

Vector erf-Gaussian beams: fractional optical vortices and asymmetric TE and TM modes

We have considered the paraxial vector erf-Gaussian beams with field distribution in the form of the error function that are shaped by the cone of plane waves with a fractional step of the azimuthal phase distribution modulated by the Gaussian envelope. We have revealed that the initial distributions of the transverse electric and transverse magnetic fields have a far from standard form but at the far diffraction field the field distributions recover nearly the symmetric form.

Diffraction–attenuation resistant beams: their higher-order versions and finite-aperture generations

Recently, a method for obtaining diffraction-attenuation resistant beams in absorbing media has been developed in terms of suitable superposition of ideal zero-order Bessel beams. In this work, we show that such beams keep their resistance to diffraction and absorption even when generated by finite apertures. Moreover, we shall extend the original method to allow a higher control over the transverse intensity profile of the beams. Although the method is developed for scalar fields, it can be applied to paraxial vector wave fields, as well. These new beams have many potential applications, such as in free-space optics, medical apparatus, remote sensing, and optical tweezers.

Topological and morphological transformations of developing singular paraxial vector light fields

The paper is devoted to establishment of the real-time topological and morphological dynamics of generic developing paraxial elliptic speckle fields generated and driven by the system ‘laser beam + photorefractive crystal LiNbO 3:Fe’. Generic space-time development of full gamut of polarization ellipse parameters (ellipticity, azimuth, morphology of C points, optical diabolos and handedness) and their combination in fixed beam cross-section was measured in details by the elaborated quick-action real-time Stokes-polarimetry. Whole field irreversible evolution is fulfilled through totality of random space/time C point pair nucleation/annihilation. The ‘life-story’ of C point and optical diabolo pairs is realized through ‘local topological/morphological transition’ with fully reversible scenario. It starts from smooth fragment of speckle field by formation of pre-nucleation local structure and finishes by after-annihilation local structure which decays to another smooth structure. Scenarios of star–monstar pair nucleation/annihilation and monstar ↔ lemon transformation were established. Measured statistics of C point and diabolo morphological forms was in excellent agreement with theory predictions. All allowed scenarios of diabolo pair ‘life-story’ started/finished as star–hyperbolic monstar–hyperbolic pair were measured. Evolution of polarization ellipses handedness is implemented through L contours movement and reconnection with a saddle as the catalyst. Reconnection of L contour peninsula leads to birth of closed L contour delimiting island of fixed handedness ellipses with/without C points. Elaborated approach and presented results start the dynamic singular optics of time-dependent vector light fields.

Propagation of flattened-Gaussian beams in uniaxial crystals

The propagation of flattened-Gaussian beams(FGBs) in uniaxial crystals is studied based on the paraxial vector theory of the propagation of beams in uniaxially anisotropic media. Analytical propagation equations are derived, which permit us to study the propagation properties of FGBs in unaixal crystals in a simple way. It is shown that the initial circular symmetry of FGBs cannot be kept up, and the state of polarization of FGBs varies during propagation due to the anisotropy of crystals. The spatial evolutions of ordinary and extraordinary components, as well as these in x and y directions in the crystals originating from input FG Bs are illustrated with detailed numerical examples.

Polarization singularities in paraxial vector fields: morphology and statistics

Polarization patterns in the transverse plane generically contain singularities: points of circular polarization (C points), lines of linear polarization (L lines), instantaneous zeros (disclinations) and component zeros. We investigate the geometry of ellipse fields at these singularities, using the Stokes parameters and others to characterize the singular geometry and morphology. Comparison is made with analogous structures on random surfaces, namely umbilic points and parabolic lines. The densities and correlations of the different types of polarization singularities are calculated in random polarization fields, and compared to the statistics of phase singularities and random surfaces.