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.

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.

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.

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.

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.

Tight focusing of a cylindrical vector beam by a hyperbolic secant gradient index lens.

In this Letter, we investigate the tight focusing of a second-order cylindrical vector beam by a hyperbolic secant gradient index lens with a thickness of 10 µm, a radius of 9.43 µm, and a refractive index on the axis of 3.47 (silicon). It is shown that the lens forms the reverse energy flow near its shadow surface. Moreover, it was obtained that the spherical hole in the center of the shadow plane with a diameter of 0.3 µm allows us to localize the direct energy flow inside the lens material and with the reverse energy flow in an area of free space.

Reverse and toroidal flux of light fields with both phase and polarization higher-order singularities in the sharp focus area.

Based on the Richards-Wolf formalism, we obtain for the first time a set of explicit analytical expressions that completely describe a light field with a double higher-order singularity (phase and polarization), as well as distributions of its intensity and energy flux near the focus. A light field with the double singularity is an optical vortex with a topological charge m and with nth-order cylindrical polarization (azimuthal or radial). From the theory developed, rather general predictions follow. 1) For any singularity orders m and n, the intensity distribution near the focus has a symmetry of order 2(n - 1), while the longitudinal component of the Poynting vector has always an axially symmetric distribution. 2) If n = m + 2, there is a reverse energy flux on the optical axis near the focus, which is comparable in magnitude with the forward flux. 3) If m ≠0, forward and reverse energy fluxes rotate along a spiral around the optical axis, whereas at m = 0 the energy flux is irrotational. 4) For any values of m and n, there is a toroidal energy flux in the focal area near the dark rings in the distribution of the longitudinal component of the Poynting vector.

Subwavelength micropolarizer in a gold film for visible light.

We have designed and fabricated a 100 µm×100 µm four-sector binary subwavelength reflecting polarization microconverter in a gold film. Using finite-difference time-domain-aided numerical simulations and experiments, the micropolarizer was shown to convert an incident linearly polarized Gaussian beam of wavelength 532 nm into an azimuthally polarized beam. Conditions for generating on-axis regions of nonzero intensity when using propagating optical vortices with different initial polarization were deduced. By putting a spiral phase plate into an azimuthally polarized beam, the intensity pattern was shown to change from diffraction rings to a central peak.

Tight focus of light using micropolarizer and microlens.

Using a binary microlens of diameter 14 µm and focal length 532 nm (NA=0.997) in resist, we focus a 633 nm laser beam into a near-circular focal spot with dimensions (0.35 ± 0.02)λ and (0.38 ± 0.02)λ (λ is incident wavelength) at full width half-maximum intensity. The area of the focal spot is 0.105λ(2). The incident light is a mixture of linearly and radially polarized beams generated by reflecting a linearly polarized Gaussian beam at a 100 µm × 100 µm four-sector subwavelength diffractive optical microelement with a gold coating. The focusing of a linearly polarized laser beam (the other conditions being the same) is found to produce an elliptical focal spot measuring (0.40 ± 0.02)λ and (0.50 ± 0.02)λ. To our knowledge, this is the first implementation of subwavelength focusing of light using a pair of micro-optic elements (a binary microlens and a micropolarizer).

Photonic nanojets generated using square-profile microsteps.

We have shown experimentally that square-profile microsteps on a silica substrate, with square sides of 0.4, 0.5, 0.6, and 0.8 µm and height of 500 nm, illuminated through the substrate by a linearly polarized laser beam of wavelength λ=633 nm, produce, near the surface, enhanced-intensity regions (termed photonic nanojects), with their intensity being six times higher than that of the incident light and their respective full width at half-maximum diameters being 0.44λ, 0.43λ, 0.39λ, and 0.47λ, which is below the diffraction limit of 0.51λ. It is worth noting that when the step side is smaller than the wavelength, the focus is found within the step; otherwise the focus is outside the step, which is similar to an optical candle.

Curved laser microjet in near field.

With the use of the finite-difference time-domain-based simulation and a scanning near-field optical microscope that has a metal cantilever tip, the diffraction of a linearly polarized plane wave of wavelength λ by a glass corner step of height 2λ is shown to generate a low divergence laser jet of a root-parabolic form: over a distance of 4.7λ on the optical axis, the beam path is shifted by 2.1λ. The curved laser jet of the FWHM length depth of focus=9.5λ has the diameter FWHM=1.94λ over the distance 5.5λ, and the intensity maximum is 5 times higher than the incident wave intensity. The discrepancy between the analytical and the experimental results amounts to 11%.

Analysis of the shape of a subwavelength focal spot for the linearly polarized light.

By decomposing a linearly polarized light field in terms of plane waves, the elliptic intensity distribution across the focal spot is shown to be determined by the E-vector's longitudinal component. Considering that the Poynting vector's projection onto the optical axis (power flux) is independent of the E-vector's longitudinal component, the power flux cross section has a circular form. Using a near-field scanning optical microscope (NSOM) with a small-aperture metal tip, we show that a glass zone plate (ZP) having a focal length of one wavelength focuses a linearly polarized Gaussian beam into a weak ellipse with the Cartesian axis diameters FWHM(x)=(0.44±0.02)λ and FWHM(y)=(0.52±0.02)λ and the (depth of focus) DOF=(0.75±0.02)λ, where λ is the incident wavelength. The comparison of the experimental and simulation results suggests that NSOM with a hollow pyramidal aluminum-coated tip (with 70° apex and 100 nm diameter aperture) measures the transverse intensity, rather than the power flux or the total intensity. The conclusion that the small-aperture metal tip measures the transverse intensity can be inferred from the Bethe-Bouwkamp theory.

Diffraction of a Gaussian beam by a logarithmic axicon.

We derive an explicit analytical relationship to describe the axial light intensity when a Gaussian beam is diffracted by the logarithmic axicon (LA). An evaluation formula for the effective radius of the diffraction pattern that we deduce shows the said radius to be in inverse proportion to the LA "force" parameter. The finite-difference time-domain-based simulation has shown that using the LA makes it possible to go beyond the diffraction limit: in the LA vicinity, the FWHM of the light beam can be as small as one fifth of the illumination wavelength.