Enhancing the Spin Hall Effect of Cylindrically Polarized Beams.

Two linked gear wheels in a micromachine can be simultaneously rotated in opposite directions by using a laser beam that has in its section areas the spin angular momentum (SAM) of the opposite sign. However, for instance, a cylindrical vector beam has zero SAM in the focus. We alter a cylindrical vector beam so as to generate areas in its focus where the SAM is of opposite signs. The first alteration is adding to the cylindrical vector beam a linearly polarized beam. Thus, we study superposition of two rotationally symmetric beams: those with cylindrical and linear polarization. We obtain an expression for the SAM and prove two of its properties. The first property is that changing superposition coefficients does not change the shape of the SAM density distribution, whereas the intensity changes. The second property is that maximal SAM density is achieved when both beams in the superposition have the same energy. The second perturbation is adding a spatial carrier frequency. We study the SAM density of a cylindrical vector beam with a spatial carrier frequency. Due to periodic modulation, upon propagation in space, such a beam is split into two beams, having left and right elliptic polarization. Thus, in the beam transverse section, areas with the spin of different signs are separated in space, which is a manifestation of the spin Hall effect. We demonstrate that such light beams can be generated by metasurfaces, with the transmittance depending periodically on one coordinate.

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.

Spin Hall Effect in the Paraxial Light Beams with Multiple Polarization Singularities.

Elements of micromachines can be driven by light, including structured light with phase and/or polarization singularities. We investigate a paraxial vectorial Gaussian beam with multiple polarization singularities residing on a circle. Such a beam is a superposition of a cylindrically polarized Laguerre-Gaussian beam with a linearly polarized Gaussian beam. We demonstrate that, despite linear polarization in the initial plane, on propagation in space, alternating areas are generated with a spin angular momentum (SAM) density of opposite sign, that manifest about the spin Hall effect. We derive that in each transverse plane, maximal SAM magnitude is on a certain-radius circle. We obtain an approximate expression for the distance to the transverse plane with the maximal SAM density. Besides, we define the singularities circle radius, for which the achievable SAM density is maximal. It turns out that in this case the energies of the Laguerre-Gaussian and of the Gaussian beams are equal. We obtain an expression for the orbital angular momentum density and find that it is equal to the SAM density, multiplied by -m/2 with m being the order of the Laguerre-Gaussian beam, equal to the number of the polarization singularities. We consider an analogy with plane waves and find that the spin Hall affect arises due to the different divergence between the linearly polarized Gaussian beam and cylindrically polarized Laguerre-Gaussian beam. Application areas of the obtained results are designing micromachines with optically driven elements.

Hall Effect at the Focus of an Optical Vortex with Linear Polarization.

The tight focusing of an optical vortex with an integer topological charge (TC) and linear polarization was considered. We showed that the longitudinal components of the spin angular momentum (SAM) (it was equal to zero) and orbital angular momentum (OAM) (it was equal to the product of the beam power and the TC) vectors averaged over the beam cross-section were separately preserved during the beam propagation. This conservation led to the spin and orbital Hall effects. The spin Hall effect was expressed in the fact that the areas with different signs of the SAM longitudinal component were separated from each other. The orbital Hall effect was marked by the separation of the regions with different rotation directions of the transverse energy flow (clockwise and counterclockwise). There were only four such local regions near the optical axis for any TC. We showed that the total energy flux crossing the focus plane was less than the total beam power since part of the power propagated along the focus surface, while the other part crossed the focus plane in the opposite direction. We also showed that the longitudinal component of the angular momentum (AM) vector was not equal to the sum of the SAM and the OAM. Moreover, there was no summand SAM in the expression for the density of the AM. These quantities were independent of each other. The distributions of the AM and the SAM longitudinal components characterized the orbital and spin Hall effects at the focus, respectively.

Double and Square Bessel-Gaussian Beams.

We obtain a transform that relates the standard Bessel-Gaussian (BG) beams with BG beams described by the Bessel function of a half-integer order and quadratic radial dependence in the argument. We also study square vortex BG beams, described by the square of the Bessel function, and the products of two vortex BG beams (double-BG beams), described by a product of two different integer-order Bessel functions. To describe the propagation of these beams in free space, we derive expressions as series of products of three Bessel functions. In addition, a vortex-free power-function BG beam of the mth order is obtained, which upon propagation in free space becomes a finite superposition of similar vortex-free power-function BG beams of the orders from 0 to m. Extending the set of finite-energy vortex beams with an orbital angular momentum is useful in searching for stable light beams for probing the turbulent atmosphere and for wireless optical communications. Such beams can be used in micromachines for controlling the movements of particles simultaneously along several light rings.

Spin Hall Effect of Double-Index Cylindrical Vector Beams in a Tight Focus.

We investigate the spin angular momentum (SAM) of double-index cylindrical vector beams in tight focus. Such a set of beams is a generalization of the conventional cylindrical vector beams since the polarization order is different for the different transverse field components. Based on the Richards-Wolf theory, we obtain an expression for the SAM distribution and show that if the polarization orders are of different parity, then the spin Hall effect occurs in the tight focus, which is there are alternating areas with positive and negative spin angular momentum, despite linear polarization of the initial field. We also analyze the orbital angular momentum spectrum of all the components of the focused light field and determine the overwhelming angular harmonics. Neglecting the weak harmonics, we predict the SAM distribution and demonstrate the ability to generate the focal distribution where the areas with the positive and negative spin angular momentum reside on a ring and are alternating in pairs, or separated in different semicircles. Application areas of the obtained results are designing micromachines with optically driven elements.

Tailoring the Topological Charge of a Superposition of Identical Parallel Laguerre-Gaussian Beams.

In optical computing machines, data can be transmitted by optical vortices, and the information can be encoded by their topological charges. Thus, some optical mechanisms are needed for performing simple arithmetic operations with the topological charges. Here, a superposition of several parallel identical Laguerre-Gaussian beams with single rings is studied. It is analytically and numerically shown that if the weighting coefficients of the superposition are real, then the total topological charge of the superposition is equal to the topological charge of each component in the initial plane and in the far field. We prove that the total topological charge of the superposition can be changed by the phase delay between the beams. In the numerical simulation, we demonstrate the incrementing and decrementing the topological charge. Potential application areas are in optical computing machines and optical data transmission.

Orbital angular momentum of paraxial propagation-invariant laser beams.

For propagation-invariant laser beams represented as a finite superposition of the Hermite-Gaussian beams with the same Gouy phase and with arbitrary weight coefficients, we obtain an analytical expression for the normalized orbital angular momentum (OAM). This expression is represented also as a finite sum of weight coefficients. We show that a certain choice of the weight coefficients allows obtaining the maximal OAM, which is equal to the maximal power of the Hermite polynomial in the sum. In this case, the superposition describes a single-ringed Laguerre-Gaussian beam with a topological charge equal to the maximal OAM and to the maximal power of the Hermite polynomial.

Propagation of a multi-vortex beam: far-field diffraction of a Gaussian beam from a multi-fork phase grating.

In this work, the far-field propagation of multi-vortex beams is investigated. We consider diffraction of a Gaussian wave from a spatial light modulator (SLM) in which a multi-fork grating is implemented on it at the waist plane of the Gaussian wave. In the first-order diffraction pattern a multi-vortex beam is produced, and we consider its evolution under propagation when different multi-fork gratings are implemented on the SLM. We consider two different schemes for the phase singularities of the implemented grating. A topological charge (TC) equal to l1 is considered at the center of the grating, and four similar phase singularities all having a TC equal to l2=l14 (or l2=-l14) are located on the corners of a square where the l1 singularity is located on the square center. Some cases with different values of l1, and consequently l2, are investigated. Experimental and simulation results show that if signs of the TCs at the corners and center of the square are the same, the radius of the central singularity on the first-order diffracted beam increases, and it convolves the other singularities. If their signs are opposite, the total TC value equals zero, and at the far-field, the light beam distribution becomes a Gaussian beam. For determining the TCs of the resulting far-field beams, we interfere experimentally and by simulation the resulting far-field beams with a plane wave and count the forked interference fringes. All the results are consistent.

Dividing the Topological Charge of a Laguerre-Gaussian Beam by 2 Using an Off-Axis Gaussian Beam.

In optical computing machines, many parameters of light beams can be used as data carriers. If the data are carried by optical vortices, the information can be encoded by the vortex topological charge (TC). Thus, some optical mechanisms are needed for performing typical arithmetic operations with topological charges. Here, we investigate the superposition of a single-ringed (zero-radial-index) Laguerre-Gaussian (LG) beam with an off-axis Gaussian beam in the waist plane. Analytically, we derive at which polar angles intensity nulls can be located and define orders of the optical vortices born around these nulls. We also reveal which of the vortices contribute to the total TC of the superposition and which are compensated for by the opposite-sign vortices. If the LG beam has a TC of m, TC of the superposition is analytically shown to equal [m/2] or [m/2] + 1, where [] means an integer part of the fractional number. Thus, we show that the integer division of the TC by two can be done by superposing the LG beam with an off-axis Gaussian beam. Potential application areas are in optical computing machines and optical data transmission.

Optical Polarization Sensor Based on a Metalens.

We investigated an optical microsensor of the polarization state of a laser light based on a metalens. In contrast to known polarization sensors based on metasurfaces that deflect different polarization types using various angles to the optical axis, the studied polarization sensor generated different patterns in the metalens focus to realize varied polarization states: left circular polarization generated a light ring in the focus, right circular polarization generated a circular focal spot, and linear polarization generated an elliptic spot with two sidelobes. Moreover, the tilt angle of the linear polarization matched the tilt angle of the elliptic focal spot. The simulation results were consistent with the theoretical predictions. A metalens with a diameter of several tens of microns was designed and fabricated in a thin amorphous silicon film with a thickness of 120 µm and a low aspect ratio, high numerical aperture, and short focal distance equal to a wavelength of 633 nm.

Ultra-Thin, Short-Focus, and High-Aperture Metalens for Generating and Detecting Laser Optical Vortices.

A combined high-aperture metalens in a thin silicon nitride film that consists of two tilted sectored metalenses is considered. Each sector of the metalens consists of a set of binary subwavelength gratings. The diameter of the metalens is 14 µm. Using a time-domain finite difference method, we show that the metalens can simultaneously detect optical vortices with two topological charges -1 and -2, almost over the entire spectrum of visible wavelengths. The metalens can distinguish several wavelengths that are focused at different points in the focal plane due to a 1-nm change in wavelength resulting in a focal spot shift of about 4 nm. When the metalens is illuminated by a Gaussian beam with left-handed circular polarization, two optical vortices with topological charges 1 and 2 are simultaneously formed 6-µm apart at the focal distance of 6 µm.

Spin-Orbital Conversion with the Tight Focus of an Axial Superposition of a High-Order Cylindrical Vector Beam and a Beam with Linear Polarization.

In this paper, spin-orbital conversion in the tight focus of an axial superposition of a high-order (order m) cylindrical vector beam and a beam with linear polarization is theoretically and numerically considered. Although such a beam does not have a spin angular momentum in the initial plane and the third projection of its Stokes vector is equal to zero, subwavelength local regions with a transverse vortex energy flow and with the non-zero third Stokes projection (the longitudinal component of the spin angular momentum) are formed in the focal plane for an odd number m. This means that such a beam with an odd m has regions of elliptical or circular polarization with alternating directions of rotation (clockwise and counterclockwise) in the focus. For an even m, the field is linearly polarized at every point of the focal plane, and the transverse energy flux is absent. These beams can be used to create a micromachine in which two microparticles in the form of gears are captured in the focus of the beam into neighboring local areas in which the energy flow rotates in different directions, and therefore, these gears will also rotate in different directions.

Spin-Orbital Conversion of a Strongly Focused Light Wave with High-Order Cylindrical-Circular Polarization.

We discuss interesting effects that occur when strongly focusing light with mth-order cylindrical-circular polarization. This type of hybrid polarization combines properties of the mth-order cylindrical polarization and circular polarization. Reluing on the Richards-Wolf formalism, we deduce analytical expressions that describe E- and H-vector components, intensity patterns, and projections of the Poynting vector and spin angular momentum (SAM) vector at the strong focus. The intensity of light in the strong focus is theoretically and numerically shown to have an even number of local maxima located along a closed contour centered at an on-axis point of zero intensity. We show that light generates 4m vortices of a transverse energy flow, with their centers located between the local intensity maxima. The transverse energy flow is also shown to change its handedness an even number of times proportional to the order of the optical vortex via a full circle around the optical axis. It is interesting that the longitudinal SAM projection changes its sign at the focus 4m times. The longitudinal SAM component is found to be positive, and the polarization vector is shown to rotate anticlockwise in the focal spot regions where the transverse energy flow rotates anticlockwise, and vice versa-the longitudinal SAM component is negative and the polarization vector rotates clockwise in the focal spot regions where the transverse energy flow rotates clockwise. This spatial separation at the focus of left and right circularly polarized light is a manifestation of the optical spin Hall effect. The results obtained in terms of controlling the intensity maxima allow the transverse mode analysis of laser beams in sensorial applications. For a demonstration of the proposed application, the metalens is calculated, which can be a prototype for an optical microsensor based on sharp focusing for measuring roughness.

Optical vortex beams with a symmetric and almost symmetric OAM spectrum.

We show both theoretically and numerically that if an optical vortex beam has a symmetric or almost symmetric angular harmonics spectrum [orbital angular momentum (OAM) spectrum], then the order of the central harmonic in the OAM spectrum equals the normalized-to-power OAM of the beam. This means that an optical vortex beam with a symmetric OAM spectrum has the same topological charge and the normalized-to-power OAM has an optical vortex with only one central angular harmonic. For light fields with a symmetric OAM spectrum, we give a general expression in the form of a series. We also study two examples of form-invariant (structurally stable) vortex beams with their topological charges being infinite, while the normalized-to-power OAM is approximately equal to the topological charge of the central angular harmonic, contributing the most to the OAM of the entire beam.

Minimal Focal Spot Size Measured Based on Intensity and Power Flow.

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

Converting an array of edge dislocations into a multi-vortex beam.

We theoretically show how, using a cylindrical lens, a Gaussian beam with a finite number of parallel zero-intensity lines (edge dislocations) is transformed into a vortex beam that carries orbital angular momentum (OAM) and topological charge (TC). Remarkably, while the original beam is assumed to carry a non-zero OAM and have no TC, the latter is shown to appear during free-space propagation. Considering two parallel center-symmetric zero-intensity lines located as an example, we look into the dynamics of generating two intensity nulls at the double focal length: with increasing distance between the vertical zero-intensity lines, two optical vortices are first generated on the horizontal axis before converging at the origin and then diverging along the vertical axis. Irrespective of the between-line distance, such an optical vortex has ${\rm TC} = - {2}$ at any distance from the optical axis, except for the original plane. With changing distance between the zero-intensity lines, the OAM that the beam carries is changing, taking positive and negative values, or a zero value at a certain between-line distance. We also show that if the number of zero-intensity lines is infinite, a vortex beam with finite OAM and infinite TC is generated.

Inversion of the axial projection of the spin angular momentum in the region of the backward energy flow in sharp focus.

We show theoretically and numerically that when strongly focusing a circularly polarized optical vortex, the longitudinal component of its spin angular momentum undergoes inversion. A left-handed circularly polarized input beam is found to convert in the focus and near the optical axis to a right-handed circularly polarized beam. Thanks to this effect taking place near the strong focus, where a reverse energy flow is known to occur, the spin angular momentum inversion discovered can be utilized to detect a reverse energy flow.

Orbital angular momentum and topological charge of a multi-vortex Gaussian beam.

We report on a theoretical and numerical study of a Gaussian beam modulated by several optical vortices (OV) that carry same-sign unity topological charge (TC) and are unevenly arranged on a circle. The TC of such a multi-vortex beam equals the sum of the TCs of all OVs. If the OVs are located evenly along an arbitrary-radius circle, a simple relationship for the normalized orbital angular momentum (OAM) is derived for such a beam. It is shown that in a multi-vortex beam, OAM normalized to power cannot exceed the number of constituent vortices and decreases with increasing distance from the optical axis to the vortex centers. We show that for the OVs to appear at the infinity of such a combined beam, an infinite-energy Gaussian beam is needed. On the contrary, the total TC is independent of said distance, remaining equal to the number of constituent vortices. We show that if TC is evaluated not along the whole circle encompassing the singularity centers, but along any part of this circle, such a quantity is also invariant and conserves on propagation. Besides, a multi-spiral phase plate is studied for the first time to our knowledge, and we obtained the TC and OAM of multi-vortices generated by this plate. When propagated through a random phase screen (diffuser) the TC is unchanged, while the OAM changes by less than 10% if the random phase delay on the diffuser does not exceed half wavelength. Such multi-vortices can be used for data transmission in the turbulent atmosphere.

Topological charge of asymmetric optical vortices.

We obtain theoretical relationships to define topological charge (TC) of vortex laser beams devoid of radial symmetry, namely asymmetric Laguerre-Gaussian (LG), asymmetric Bessel-Gaussian (BG), and asymmetric Kummer beams, as well as Hermite-Gaussian (HG) vortex beams. Although they are obtained as superposition of respective conventional LG, BG, and HG beams, these beams have the same TC equal to that of a single mode, n. At the same time, the normalized orbital angular momentum (OAM) that the beams carry is different, differently responding to the variation of the beam's asymmetry degree. However, whatever the asymmetry degree, TC of the beams remains unchanged and equals n. Although separate HG beam does not have OAM and TC, superposition of only two HG modes with adjacent numbers (n, n + 1) and a π/2-phase shift produces a modal beam whose TC is -(2n + 1). Theoretical findings are validated via numerical simulation.