Employing the Richards-Wolf formalism that adequately describes an electromagnetic field near the sharp focus of an ideal spherical lens, we demonstrate that certain light fields (linearly polarized optical vortex, cylindrical vector fields of an arbitrary order) have a reverse canonical energy flow in the focus plane. When the numerical aperture is 0.95, maximal magnitude of the reverse energy flow amounts to nearly 0.7% of the maximal magnitude of the direct energy flow. The distribution of the reverse canonical flow in the focus plane can have the shape of concentric rings or only arcs of the concentric rings. For certain light fields, for instance, for an azimuthally polarized light field, the longitudinal component of the canonical energy flow vector coincides with the longitudinal component of the Poynting vector. It is shown that a circularly polarized optical vortex does not have the reverse flow at the focus.

Abstract Besides scalar optical vortices that have a topological charge (TC), helical wave front, and carry an orbital angular momentum (OAM) that can be transferred to particles and rotate them along circular trajectories, polarization optical vortices are also known, whose polarization state in the beam section changes with the azimuthal angle. Such vortices are polarization singularities that are described by indices, similar to the TC. However, polarization OAM for polarization vortices still has not been considered, although laser beams with inhomogeneous polarization can perform spiral mass transport in polarization-sensitive media. In this work, we consider two analogues of the OAM for vector fields. One OAM analog is the polarization orientation azimuthally changing velocity (OACV), whereas the other OAM analog is the polarization ellipticity azimuthally changing velocity. For instance, the normalized OACV is equal to the order of a cylindrical vector beam and also equals the order of a Poincaré beam.

In this work, we theoretically and numerically analyze helical Ince–Gaussian (hIG) modes, hIG p , q ( x , y , ε ). We derive explicit analytical relationships to describe the ε -dependence (where ε is the ellipticity parameter) of the orbital angular momentum (OAM) of hIG p , q ( x , y , ε ) modes at p =2,3,4,5. The derivation procedure relies on expansions of the hIG modes in terms of Hermite–Gaussian modes. It is shown that in the general case, the OAM is an even function of ε and exhibits no monotonic behavior with ε varying from zero to plus and minus infinity.

Subject of study. Laser vector vortex beams and their tight focusing are investigated, specifically the spin-orbit transformation effect at the focus. Aim of study. The aim is to theoretically analyze and numerically simulate the features of the spin-orbit effect that arise during tight focusing of optical vortices with circular polarization. Method. Theoretical and numerical investigations were performed using the Debye-integral-based Richards-Wolf method. Main results. The transverse and longitudinal components of the Poynting vector (energy flow), spin angular momentum (SAM), and orbital angular momentum (OAM), averaged over the beam cross-section in the focal plane, were calculated. Practical significance. For the first time, this study shows that, contrary to common assumptions, during the spin-orbit transformation, a portion of the longitudinal SAM does not transform into longitudinal OAM. The total SAM is conserved during focusing and only redistributed-a part of the longitudinal component is converted into a transverse (azimuthal) component. The generation of OAM at the focus is attributed to a circularly polarized beam producing two optical vortices at the focus-a transverse vortex with a topological charge of 2 and a longitudinal vortex with a charge of 1. These vortices generate azimuthal energy flow in the focal plane. (c) 2026 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved.

We show that when an optical vortex with a topological charge n shifts from the optical axis of a Gaussian beam, the orbital angular momentum decreases, whereas the topological charge remains unchanged. We experimentally confirmed this phenomenon with the aid of a q-plate array generated in a liquid crystal cell filled with a frustrated chiral nematic. We also theoretically proved that the topological charge of an optical vortex is conserved when the optical vortex is displaced from the axis.

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.

We investigate the spin angular momentum of a superposition of two vector light beams with rotational symmetry. One beam is cylindrically polarized and another is linearly polarized. Radial form of these beams can be arbitrary (Laguerre-Gaussian, Bessel-Gaussian, or some other). For such a superposition, an analytical expression is derived for the spin angular momentum and two its properties are obtained. At first, we found that altering the coefficients does not affect the form of the spin angular momentum distribution, while the intensity distribution is changed. At second, we found that maximal spin angular momentum is generated if both beams are of equal power.

Рассматривается проблема выбора комплектов технологических баз при проектировании технологических процессов изготовления корпусных деталей на многофункциональных обрабатывающих центрах с ЧПУ. В результате проведенного анализа был выбран граф связи поверхностей детали для представления ее линейных размеров, допусков ориентации и местоположения, а также биения. Кроме того, для выбора комплекта баз установлено, что необходимо учитывать «нереальные» базы, погрешность базирования, площадь и устойчивость главной базы, унификацию поверхностей базирования, соответствие типовым схемам базирования, а также вспомогательное время на установку и снятие заготовки. Таким образом, с учетом рассмотренных критериев разработана методика выбора оптимального комплекта технологических баз при разработке маршрутного технологического процесса механической обработки корпусных деталей на обрабатывающих центрах с ЧПУ. The article considers the problems of choosing technological losses in the design of technological processes for manufacturing body parts on multifunctional CNC machining centers. A graph of the connections of the surfaces of the part to describe the dimensions and requirements are presented. Basing error, «unrealistic» bases, area and stability of the main base, unification of the biasing surfaces, compliance with standard basing schemes, as well as time for installation and removal of the workpiece, are described. Method of choosing the optimal technological bases in the development of the technological process of mechanical processing of machine body parts, are given.

The Richards-Wolf equations not only adequately describe a light field distribution at the sharp focus, but are also able to describe a light field distribution just behind an ideal spherical lens, i.e. on a converging spherical wavefront. Knowing all projections of light field strength vectors behind the lens, longitudinal components of the spin angular momentum and orbital angular momentum (SAM and OAM) can be derived. In this case, the longitudinal projection of the SAM just behind the lens either remains zero or decreases. This means that the spin-orbital conversion (SOC), where part of the “spin transfers orbit”, occurs just behind the ideal spherical lens. Notably, the sum of the longitudinal projections of SAM and OAM is conserved. Regarding the spin Hall effect, it is revealed that rather than forming just behind the lens, it appears as focusing occurs. Thus, we find that while just behind the lens there is no Hall effect, it becomes maximally pronounced in the focal plane. It is because just behind the ideal spherical lens, two optical vortices with topological charges (TCs) –2 and 2 and opposite-sign spins (with right and left circular polarization) are generated. However, the total spin is equal to zero because the two vortices have the same amplitudes. The amplitudes of the optical vortices become different in the course of focusing and in the focal plane and, therefore, areas with opposite-sign spins (Hall effect) are formed. We also present a general form of the incident light fields whose longitudinal component is zero in the focal plane. In this case, the SAM vector can only have the longitudinal non-zero component. The notion of the SAM vector elongated only along the optical axis in the focal plane is applied for solving magnetization problems.

We study the spin angular momentum of a superposition of two vector light beams radial symmetry, one has cylindrical polarization and another – linear. Both beams can have an arbitrary radial shape. An analytical expression is obtained for the spin angular momentum and two its properties are proven. The first one is that changing weight coefficients of the superposition does not changes the shape of the spin angular momentum density distribution, whereas the intensity shape can change. The second property is that the maximal spin angular momentum density is achieved when both constituent beams in the superposition have equal energy.

A metalens for detection a polarization ellipticity of an incident beam and the wavelength is considered in this work. The metalens is constructed of blocks of diffraction gratings with a height of 140 nm and a period of 220 nm. It works like a polarizer, which depends on one transverse coordinate, and a focuser. This metalens is capable of both separating linearly polarized radiation into two focal spots with circular polarizations of different signs, and detecting the direction and the ellipticity of the polarization. The metalens operates over a wide range of wavelengths from 0.55 to 0.837 μm, it can be used to estimate of the incident radiation wavelength in the range from 0.64 to 0.837 μm. This is possible due to the almost linear displacement of the focal spot in the transverse plane depending on the light wavelength.

Paraxial beam modes, which propagate in space and focus without changing their transverse intensity pattern, are of great value for multiplexing transmitted data in optical communications, both in waveguides and in free space. The best-known paraxial modes are the Hermite-Gaussian and Laguerre-Gaussian beams. Here, we derive explicit analytical expressions for Ince-Gaussian (IG) beams for several first values of the indices p = 3, 4, 5, and 6. In total, we obtain expressions for the amplitudes of 24 IG beams. These formulae are written as superpositions of the Laguerre-Gaussian (LG) or Hermite-Gaussian (HG) beams, with the superposition coefficients explicitly depending on the ellipticity parameter. Due to simultaneous representation of the IG modes via the LG and HG modes, it is easy to obtain the IG modes in the limiting cases wherein the ellipticity parameter is zero or approaches infinity. The explicit dependence of the obtained expressions for the IG modes on the ellipticity parameter makes it possible to change the intensity pattern at the beam cross-section by continuously varying the parameter values. For the first time, the intensity distributions of the IG beams are obtained for negative values of the ellipticity parameter. The obtained expressions could facilitate a theoretical analysis of properties of the IG modes and could find practical applications in the numerical simulation or generation of such beams with a liquid-crystal spatial light modulator.

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.

We obtain and investigate Bessel-Bessel-Gaussian vortex beams (BBG beams) with the complex amplitude being equal to a product of the Gaussian function with two Bessel functions, whose arguments are expressed as complicated radicals including the cylindrical coordinates and a free parameter that defines the shape of the intensity distribution. If this parameter is small, the intensity has the shape of an inhomogeneous ring. For larger values of this parameter, the intensity has the shape of two arcs or 'crescents', oriented by their concave sides to each other. The complex amplitude of such beams is derived in explicit form for an arbitrary distance from the waist. We demonstrate that the BBG beams rotate upon propagation anomalously fast: at a distance much shorter than the Rayleigh length, the intensity distribution is already rotated by almost 45 degrees, whereas typically, the rotation angle of vortex Gaussian beams is equal to the Gouy phase. It is also shown that the parameter of the BBG beam allows controlling its topological charge (TC): when the parameter value is positive and increases, the beam TC also increases stepwise by an even number. Besides, we study two other similar vortex BBG beams: either with four local intensity maxima, lying on the Cartesian coordinates axes, or with one intensity maximum with a crescent shape, whose center is on the horizontal axis. The derived three new families of asymmetric vortex laser beams, whose complex amplitude is described by explicit analytical expressions at an arbitrary distance from the waist, extend the variety of laser beams that can be used for manipulating and rotating microparticles, free space data transmission, and in quantum informatics.

We analyze the tight focusing of a generalized Poincaré beam using a Richards–Wolf formalism. Conventional Poincaré beams are superpositions of two Laguerre–Gaussian beams with orthogonal polarization, while the generalized Poincaré beams are composed of two arbitrary optical vortices with rotationally symmetric amplitudes. Analytical relationships for projections of the electric field in the focal plane are derived. Using the superposition of a right-handed circularly polarized plane wave and an optical vortex with a topological charge of −1 as an example, relationships for the intensity distribution and the longitudinal projection of the spin angular momentum vector are deduced. It is theoretically and numerically shown that the original beam has a topological charge of −1/2 and a C-point of circular polarization, and it is generated at the focal plane center, producing an on-axis C-line with a singularity index of −1/2 (a star). Furthermore, when making a full circle of some radius around the optical axis, the major axis vector of polarization ellipse is theoretically and numerically shown to form a one-sided polarization (Möbius) strip of order −3/2, which has three half-twists and a single ‘patching’ in which two oppositely directed vectors of the major axis of polarization ellipse occur close to each other.

It is known that in the cross-section of a high-order cylindrical vector beam (CVB), polarization is locally linear. The higher the beam order, the higher the number of full rotations of the vector of local linear polarization when passing along a contour around the optical axis. It is also known that both in the input and in the focal planes, the CVB has neither the spin angular momentum (SAM), nor the orbital angular momentum (OAM). We demonstrate here that near the focal plane of the CVB (before and after the focus), an even number of local subwavelength areas is generated, where the polarization vector in each point is rotating. In addition, in the neighboring areas, polarization vectors are rotating in different directions, so that the longitudinal component of SAM vectors in these neighboring areas is of the opposite sign. In addition, after the beam passes the focus, the rotation direction of the polarization vector in each point of the beam cross-section is changed to the opposite one. Such spatial separation of the left and right rotation of the polarization vectors manifests so that the optical spin Hall effect takes place.