Search Results (19)

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. Full article

In this work, using a Richards-Wolf formalism, we derive explicit analytical relationships to describe vectors of the major and minor axes of polarization ellipses centered in the focal plane when focusing a cylindrical vector beam of integer order n. In these beams, the major axis of a polarization ellipse is found to lie in the focal plane, with the minor axis being perpendicular to the focal plane. This means that the polarization ellipse is perpendicular to the focal plane, with its polarization vector rotating either clockwise or anticlockwise and forming “photonic wheels”. Considering that the wave vector is also perpendicular to the focal plane, we conclude that the polarization ellipse and the wave vector are in the same plane, so that at some point these can coincide, which is uncharacteristic of transverse electromagnetic oscillations. In a cylindrical vector beam, the spin angular momentum vector lies in the focal plane, so when making a circle centered on the optical axis, at some sections, the handedness of the spin vector and circular motion are the same, being opposite elsewhere. This effect may be called an azimuthal transverse spin Hall effect, unlike the familiar longitudinal spin Hall effect found at the sharp focus. The longitudinal spin Hall effect occurs when opposite-sign longitudinal projections of the spin angular momentum vector are spatially separated in the focal plane. In this work, we show that for the latter, there are always an even number of spatially separated regions and that, when making an axis-centered circle, the major-axis vector of polarization ellipse forms a two-sided twisted surface with an even number of twists. Full article

In this paper, tight focusing of a superposition of a vortex laser beam with topological charge n with linear polarization and a plane wave with the same linear polarization directed along the horizontal axis is considered. Using the Richards–Wolf formalism, analytical expressions are obtained for the intensity distribution and longitudinal projection of the spin angular momentum in the focal plane. It is shown that for even and odd numbers n, the intensity and the spin angular momentum have different symmetries: for even n they are symmetric about both Cartesian axes, and for odd n they are symmetric only about the vertical axis. The intensity distribution has n local maxima at the focus, and it is nonzero on the optical axis for any n. The distribution of the longitudinal spin angular momentum (spin density) in the focal plane has (n + 2) subwavelength regions with a positive spin angular momentum and (n + 2) regions with a negative spin angular momentum, the centers of which alternately lie on a circle of a certain radius with a center on the optical axis. This spin distribution with different signs demonstrates the spin Hall effect at the focus. Negative and positive spins are mutually compensated, and the total spin is equal to zero at the focus. We have shown that by changing the topological charge of the optical vortex, it is possible to control the spin Hall effect at the focus, that is, to change the number of regions with spins of different signs. Full article

In this work, the tight focusing of vector beams with azimuthal polarization and beams with a V-line of polarization singularity (sector azimuthal polarization) was simulated numerically using the Richards–Wolf formulas. It was demonstrated that in a tight focus for these beams, there is no longitudinal component of the electric field. Previously, a similar effect was demonstrated for azimuthally polarized light only. The longitudinal component of the spin angular momentum for these beams was calculated, and the possibility of creating sector azimuthally polarized beams (beams with V-line singularities) using vector waveplates was demonstrated. Full article

The Richards–Wolf formulas not only adequately describe a light field at a tight focus, but also make it possible to describe a light field immediately behind an ideal spherical lens, that is, on a converging spherical wave front. Knowing all projections of light field strength vectors behind the lens, the longitudinal components of spin and orbital angular momenta (SAM and OAM) can be found. In this case, the longitudinal projection of the SAM immediately behind the lens either remains zero or decreases. This means that the Spin–Orbital Conversion (SOC) effect where part of the “spin goes into orbit” takes place immediately behind the lens. And the sum of longitudinal projections of SAM and OAM is preserved. As for the spin Hall effect, it does not form right behind the lens, but appears as focusing occurs. That is, there is no Hall effect immediately behind the lens, but it is maximum at the focus. This happens because two optical vortices with topological charges (TCs) 2 and −2 and with spins of different signs (with left and right circular polarization) are formed right behind the lens. However, the total spin is zero since amplitudes of these vortices are the same. The amplitude of optical vortices becomes different while focusing and at the focus itself, and therefore regions with spins of different signs (Hall effect) appear. A general form of initial light fields which longitudinal field component is zero at the focus was found. In this case, the SAM vector can only have a longitudinal component that is nonzero. The SAM vector elongated only along the optical axis at the focus is used in magnetization task. Full article

We have shown how the spin Hall effect is formed in a tight focus for two light fields with initial linear polarization. We have demonstrated that an even number of local subwavelength regions appear in which the sign of the longitudinal projection of the spin angular momentum (the third Stokes component) alternates. When an optical vortex with topological charge n and linear polarization passes through an ideal spherical lens, additional optical vortices with topological charges n + 2, n − 2, n + 1, and n − 1 with different amplitudes are formed in the converged beam. The first two of these vortices have left and right circular polarizations and the last two vortices have linear polarization. Since circularly polarized vortices have different amplitudes, their superposition will have elliptical polarization. The sign of this elliptical polarization (left or right) will change over the beam cross section with the change in the sign of the difference in the amplitudes of optical vortices with circular polarization. We also have shown that optical vortices with topological charges n + 2, n − 2 propagate in the opposite direction near the focal plane, and together with optical vortices with charges n + 1, n − 1, they form an azimuthal energy flow at the focus. Full article

In the framework of the Richards–Wolf formalism, the spin–orbit conversion upon tight focusing of an optical vortex with circular polarization is studied. We obtain exact formulas which show what part of the total (averaged over the beam cross-section) longitudinal spin angular momentum is transferred to the total longitudinal orbital angular momentum in the focus. It is shown that the maximum part of the total longitudinal angular momentum that can be transformed into the total longitudinal orbital angular momentum is equal to half the beam power, and this maximum is reached at the maximum numerical aperture equal to one. We prove that the part of the spin angular momentum that transforms into the orbital angular momentum does not depend on the optical vortex topological charge. It is also shown that by virtue of spin–orbital conversion upon focusing, the total longitudinal energy flux decreases and partially transforms into the whole transversal (azimuthal) energy flow in the focus. Moreover, the longitudinal energy flux decreases by exactly the same amount that the total longitudinal spin angular momentum decreases. Full article

We derive analytical formulae for the complex amplitudes of variants of generalized Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams. We reveal that, at particular values of parameters of the exact solution of the paraxial propagation equation, these generalized beams are converted into conventional elegant HG and LG beams. We also deduce variants of asymmetric HG and LG beams that are described by complex amplitudes in the form of Hermite and Laguerre polynomials whose argument is shifted into the complex plane. The asymmetric HG and LG beams are, respectively, shown to present the finite superposition of the generalized HG and LG beams. We also derive an explicit relationship for the complex amplitude of a generalized vortex HG beam, which is built as the finite superposition of generalized HG beams with phase shifts. Newly introduced asymmetric HG and LG beams show promise for the study of the propagation of beams carrying an orbital angular momentum through the turbulent atmosphere. One may reasonably believe that the asymmetric laser beams are more stable against turbulence when compared with the radially symmetric ones. Full article

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. Full article

In this paper, using the Richards–Wolf equations, the focusing of circularly polarized light with flat diffractive lenses is considered. It is shown that, as the numerical aperture (NA) of the lens increases, the size of the focal spot first decreases and then begins to grow. The minimum focal spot is observed at NA = 0.96 (FWHM = 0.55 λ). With a further increase in the numerical aperture of the lens, the growth of the longitudinal component leads to an increase in the size of the focal spot. When a flat diffractive lens is replaced by an aplanatic lens, the size of the focal spot decreases monotonically as the numerical aperture of the lens increases. In this case, the minimum focal spot will be FWHM = 0.58 λ and, with a larger numerical aperture, NA = 0.99. We also reveal that, at the focus of a circularly polarized laser beam, different radius circles are observed to be centered on the optical axis, where polarization vectors rotate oppositely (clockwise and anticlockwise). This phenomenon of radius-dependent ‘spin’ separation may be interpreted as a manifestation of the radial spin Hall effect at the focus. Full article

The family of Poincaré beams has three parameters, including two real-valued angular parameters, which specify a definite polarization state on the Poincaré sphere, and a third integer parameter n specifying the beam singularity order. We show theoretically and through a numerical simulation that, while being inseparable and not allowing for the separation of polarization and orbital degrees of freedom in the general case, the Poincaré beams display remarkable properties when tightly focused. We find that at n = 2, a reverse energy flow occurs near the optical axis, which is mathematically expressed as the negative projection of the Poynting vector. We also reveal that given certain parameters of the Poincaré beams, the energy flow rotates around the optical axis due to spin–orbital conversion. We also reveal a radial optical Hall effect that occurs at the tight focus of Poincaré beams, when the on-axis components of the spin angular momentum vector have different signs on certain different-radius circles centered at the focal spot center. Full article

In this paper, using a Richards–Wolf method, which describes the behavior of electromagnetic waves at the sharp focus, we show that high-order spin and orbital Hall effects take place at the focal plane of tightly focused laser beams. We reveal that four local subwavelength regions are formed at the focus of a linearly polarized optical vortex with unit topological charge, where the spin angular momentum behaves in a special way. Longitudinal projections of the spin angular momentum are oppositely directed in the adjacent regions. We conclude that this is because photons falling into the neighboring regions at the focus have the opposite spin. This newly observed phenomenon may be called a spin Hall effect of the 4-th order. We also show that tightly focusing the superposition of cylindrical vector beams of the m-th and zero-order produces 2m subwavelength regions in the focal plane, such that longitudinal projections of the orbital angular momentum are oppositely directed in the neighboring regions. This occurs because photons falling into the neighboring regions at the focus have the opposite signs of the on-axis projections of the orbital angular momentum. This phenomenon may be termed an orbital Hall effect of the 2m-th order. Full article

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. Full article

We have considered the tight focusing of light with linear polarization. Using the Richards–Wolf formalism, it is shown that before and after the focal plane, there are regions in which the polarization is circular (elliptical). When passing through the focal plane, the direction of rotation of the polarization vector is reversed. If before the focus in a certain area there was a left circular polarization, then directly in the focus in this area there will be a linear polarization, and after the focus in a similar area there will be a right circular polarization. This effect allows linearly polarized light to be used to rotate dielectric microparticles with little absorption around their center of mass. Full article

We show that when strongly focusing a linearly polarized optical vortex with the topological charge 2 (or −2) in the near-focus region, there occurs not only a reverse energy flow (where the projection of the Poynting vector is negative) but the right- (or left-) handed circular polarization of light as well. Notably, thanks to spin–orbital conversion, the on-axis polarization vector handedness is the same as that of the transverse energy flow, i.e., anticlockwise (clockwise). An absorbing spherical microparticle centered on the optical axis placed in the focus may be expected to rotate anticlockwise (clockwise) around its axis and its center of masses. We also show that in the case of sharp focusing of light with linear polarization (without an optical vortex) before and after focus, the light has an even number of local regions with left- and right-handed circular (elliptical) polarizations. Theoretical predictions are corroborated by the numerical simulation. Full article