In this work, we are the first, to our knowledge, to demonstrate both analytically and numerically that in the cross-section of a 2D paraxial accelerating Airy beam propagating on a parabolic trajectory, there are areas where a canonical energy backflow occurs. These areas arise in the far sidelobes of the Airy beam characterized by a subwavelength local period (superoscillation areas). At an arbitrary propagation distance from the initial plane, there is a threshold transverse coordinate, such that at all smaller negative values of the transverse coordinate, the longitudinal component of the canonical energy flow is negative. For a nonparaxial Airy beam, we derive an explicit analytical expression for the canonical energy flow near the initial plane. The energy backflow is revealed to occur near the intensity null, where a phase jump of π takes place. Near the intensity null, the phase derivative with respect to the longitudinal coordinate takes large negative values, leading to the longitudinal wavevector projection becoming larger than the wavenumber of light and directed oppositely. The maximum energy backflow is found to be about one-fifth of the direct maximum energy flow. Results of the numerical simulation for the paraxial and non-paraxial Airy beams are shown to agree with the theoretical prediction.

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.

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.

Exact analytical expressions for the density of the longitudinal projection of the orbital angular momentum (OAM) vector in the sharp focus plane are obtained in this work. We derive expressions of four light fields with uniform and non-uniform polarizations: an optical vortex with elliptical polarization, a superposition of a cylindrical vector beam and a beam with linear polarization, an optical vortex with cylindrical polarization, and a beam with non-uniform elliptical polarization. All OAM densities for these four fields depend on the polarization state of the initial light field. For two light fields with hybrid polarization, the OAM density at the focus changes sign with a change in both the azimuthal and radial coordinates. It is known that in the case of paraxial optical vortices with elliptical polarization, the OAM density does not depend on the polarization state, and is completely determined by the optical vortex topological charge. Therefore, in the paraxial case, the azimuthal orbital energy flux always rotates in one direction, determined by the sign of the optical vortex topological charge. However, in the case of non-paraxial light fields, the OAM density at the focus is shown in this work to depend on the polarization state. The energy flow can rotate in different directions at different distances from the optical 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.

In this work we propose a simple optical method for non-contact measuring of small shifts or thin film thicknesses. The sensor consists of a laser light source, spiral amplitude zone plate with a high numerical aperture, and a CCD camera connected to a computer. The zone plate produces rotating beams. By measuring the rotation angle of this beams a small shift along the optical axis could be measured.

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.

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.

Polarization of a higher-order cylindrical vector beam (CVB) is known to be locally linear. The higher the beam order, the larger number of full circles the local linear polarization vector makes around the optical axis. It is also known that the CVB with radially symmetric amplitude has zero spin angular momentum (SAM) and zero orbital angular momentum (OAM) both in the initial plane and in the focal plane (because in both Cartesian components of the vector field, the angular derivative of phase is zero). We show here that near the focal plane of the CVB (i.e. before and beyond the focus), an even number of local subwavelength areas with rotating polarization vectors are generated. In addition, in the neighboring areas, the polarization vectors are rotating in the opposite directions. Thus, the longitudinal components of the SAM vector in such neighboring areas are of different sign. After passing through the focal plane, the rotation direction of the polarization vector at each point of the beam cross-section changes to the opposite one. Such a spatial separation of the left and right rotation of the polarization vectors is a manifestation of the optical spin Hall effect.

It is shown in this work that, with strong focusing of a beam with optical vortex and circular polarization, three energy flows take place in the focal plane: direct longitudinal, reverse longitudinal and azimuthal transverse flows. Calculations are made analytically using the Richards–Wolf formalism and by numerical simulation. Moreover, the energy rotation at different lengths from the optical axis occurs in different directions. Therefore, the focal plane intersects along the optical axis only part of the initial beam energy per unit time. The same energy part (other things being equal) intersects the focal plane along the positive direction of the optical axis when an optical vortex with cylindrical polarization is focused. The difference is that, if an optical vortex is present, then the transverse energy flux at the focus rotates around the optical axis. If an optical vortex is not present (a beam with only cylindrical polarization), then the average transverse flow in the focal plane is zero, though, in some regions in the focal plane, the flow is directed towards the optical axis and, in other regions, away from it. This behavior of the transverse energy flow at the focus (flow direction towards the optical axis and away from the optical axis) of a cylindrical vector beam can be deemed another kind of Hall effect.

In this work, a metalens for detecting an incident field with a fractional topological charge ranging from – 2 to 0 is proposed. The metalens is based on a spiral zone plate with a topological charge of 1.5. A change in the topological charge of the incident beam is numerically shown to lead to an off-axis shift of the focal spot from the center, with the intensity maximum value also changing. This results in a 6.9-fold change in the on-axis intensity while the topological charge of the incident beam changes from –0.6 to –1.5. The on-axis intensity at the focus is also shown to be affected by the rotation of the fractional vortex beam. This makes it possible to use the proposed metalens for measuring the angle of rotation of the incident beam in the range from 0 to 110°.

Readers will learn in which ways light can be "confined" within a subwavelength region smaller than half a wavelength. Strictly within the focal spot, all degrees of freedom of light interact and manifest themselves in a dramatic way. The size and shape of the focal spot and the magnitude of side-lobes depend on the polarization state alongside phase and amplitude distributions of a light beam. Readers will learn techniques in which inhomogeneously (i.e., azimuthally and radially) polarized optical beams can be focused. In sharp focus, exotic phenomena can occur, including the negative propagation of light and a toroidal optical flow. Throughout the book, the numerical simulation is performed using the rigorous solution of Maxwell's equations based on a Finite-Difference Time-Domain (FDTD) approach, which makes the results of modeling highly reliable. The photonic components, including optical metasurfaces, discussed in the book have been implemented using state-of-the-art techniques of electron beam writing and reactive ion-beam etching of microrelief. Two chapters are concerned with photonics hot spots, which deal with the control of light by means of optical metasurfaces and the generation of an energy backflow in the region of sharp focus of a laser beam. Another hot topic is diffractive polarization converters implemented as subwavelength diffraction gratings to convert polarization of light. By way of illustration, such converters are shown to perform linear-to-radial or linear-to-azimuthal polarization conversion. The book describes advanced photonic components fabricated by the authors to perform sharp focusing of light, including binary zone plates, binary axicons, a planar photonic crystal lens, diffraction polarization converters, and metalenses. This book is a must-have for individuals and institutions studying cutting edge optics.

A metalens for detection an incident field with initially a fractional topological charge in the range from –2 to 0 is considered in this work. The metalens is constructed utilizing a spiral zone plate with a topological charge of –1.5. A change in the topological charge of the focused incident beam is shown by simulation to lead to a displacement of its focal spot from the center on the optical axis and to a change in the intensity maximum value, which results in the change in the intensity on the optical axis by 6.9, the change from –0.6 to –1.5 of the topological charge of the incident beam was considered. The intensity at the focus on the optical axis is also affected by the rotation of the beam with a fractional topological charge. This makes it possible using the metalens to measure the tilt angle of the incident beam in the range from 0° to 110°.

We investigate the common topological charge of a superposition of parallel identical vortex beams with an arbitrary transverse shape, either Laguerre–Gaussian beams or Bessel–Gaussian beams or some other vortex beams with rotationally symmetric intensity distribution. It is known that if all the beams in the superposition have the same phase then the common topological charge of the whole superposition equals the topological charge of each constituent beam n. We show that if the beams are located on a circle and their phases increase linearly along this circle so that the phase delay between the neighbor beams on the circle is 2πp/N with N being the number of beams and p being an integer number, then the common topological charge of the superposition is equal to n + p.

In this paper, the light propagation through an ideal spherical lens is theoretically studied applying the Richards-Wolf theory. It is shown that the spin-to-orbital conversion happens just after the lens. At the same time there is no the spin Hall effect just after the lens. It is maximum in the focal plane.