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.

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.

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.

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.

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.