The origins and spread of domestic horses from the Western Eurasian steppes.

Domestication of horses fundamentally transformed long-range mobility and warfare1. However, modern domesticated breeds do not descend from the earliest domestic horse lineage associated with archaeological evidence of bridling, milking and corralling2-4 at Botai, Central Asia around 3500 BC3. Other longstanding candidate regions for horse domestication, such as Iberia5 and Anatolia6, have also recently been challenged. Thus, the genetic, geographic and temporal origins of modern domestic horses have remained unknown. Here we pinpoint the Western Eurasian steppes, especially the lower Volga-Don region, as the homeland of modern domestic horses. Furthermore, we map the population changes accompanying domestication from 273 ancient horse genomes. This reveals that modern domestic horses ultimately replaced almost all other local populations as they expanded rapidly across Eurasia from about 2000 BC, synchronously with equestrian material culture, including Sintashta spoke-wheeled chariots. We find that equestrianism involved strong selection for critical locomotor and behavioural adaptations at the GSDMC and ZFPM1 genes. Our results reject the commonly held association7 between horseback riding and the massive expansion of Yamnaya steppe pastoralists into Europe around 3000 BC8,9 driving the spread of Indo-European languages10. This contrasts with the scenario in Asia where Indo-Iranian languages, chariots and horses spread together, following the early second millennium BC Sintashta culture11,12.

Orbital angular momentum of paraxial propagation-invariant laser beams.

For propagation-invariant laser beams represented as a finite superposition of the Hermite-Gaussian beams with the same Gouy phase and with arbitrary weight coefficients, we obtain an analytical expression for the normalized orbital angular momentum (OAM). This expression is represented also as a finite sum of weight coefficients. We show that a certain choice of the weight coefficients allows obtaining the maximal OAM, which is equal to the maximal power of the Hermite polynomial in the sum. In this case, the superposition describes a single-ringed Laguerre-Gaussian beam with a topological charge equal to the maximal OAM and to the maximal power of the Hermite polynomial.

Propagation of a multi-vortex beam: far-field diffraction of a Gaussian beam from a multi-fork phase grating.

In this work, the far-field propagation of multi-vortex beams is investigated. We consider diffraction of a Gaussian wave from a spatial light modulator (SLM) in which a multi-fork grating is implemented on it at the waist plane of the Gaussian wave. In the first-order diffraction pattern a multi-vortex beam is produced, and we consider its evolution under propagation when different multi-fork gratings are implemented on the SLM. We consider two different schemes for the phase singularities of the implemented grating. A topological charge (TC) equal to l1 is considered at the center of the grating, and four similar phase singularities all having a TC equal to l2=l14 (or l2=-l14) are located on the corners of a square where the l1 singularity is located on the square center. Some cases with different values of l1, and consequently l2, are investigated. Experimental and simulation results show that if signs of the TCs at the corners and center of the square are the same, the radius of the central singularity on the first-order diffracted beam increases, and it convolves the other singularities. If their signs are opposite, the total TC value equals zero, and at the far-field, the light beam distribution becomes a Gaussian beam. For determining the TCs of the resulting far-field beams, we interfere experimentally and by simulation the resulting far-field beams with a plane wave and count the forked interference fringes. All the results are consistent.

Optical vortex beams with a symmetric and almost symmetric OAM spectrum.

We show both theoretically and numerically that if an optical vortex beam has a symmetric or almost symmetric angular harmonics spectrum [orbital angular momentum (OAM) spectrum], then the order of the central harmonic in the OAM spectrum equals the normalized-to-power OAM of the beam. This means that an optical vortex beam with a symmetric OAM spectrum has the same topological charge and the normalized-to-power OAM has an optical vortex with only one central angular harmonic. For light fields with a symmetric OAM spectrum, we give a general expression in the form of a series. We also study two examples of form-invariant (structurally stable) vortex beams with their topological charges being infinite, while the normalized-to-power OAM is approximately equal to the topological charge of the central angular harmonic, contributing the most to the OAM of the entire beam.

Converting an array of edge dislocations into a multi-vortex beam.

We theoretically show how, using a cylindrical lens, a Gaussian beam with a finite number of parallel zero-intensity lines (edge dislocations) is transformed into a vortex beam that carries orbital angular momentum (OAM) and topological charge (TC). Remarkably, while the original beam is assumed to carry a non-zero OAM and have no TC, the latter is shown to appear during free-space propagation. Considering two parallel center-symmetric zero-intensity lines located as an example, we look into the dynamics of generating two intensity nulls at the double focal length: with increasing distance between the vertical zero-intensity lines, two optical vortices are first generated on the horizontal axis before converging at the origin and then diverging along the vertical axis. Irrespective of the between-line distance, such an optical vortex has ${\rm TC} = - {2}$ at any distance from the optical axis, except for the original plane. With changing distance between the zero-intensity lines, the OAM that the beam carries is changing, taking positive and negative values, or a zero value at a certain between-line distance. We also show that if the number of zero-intensity lines is infinite, a vortex beam with finite OAM and infinite TC is generated.

Topological charge of asymmetric optical vortices.

We obtain theoretical relationships to define topological charge (TC) of vortex laser beams devoid of radial symmetry, namely asymmetric Laguerre-Gaussian (LG), asymmetric Bessel-Gaussian (BG), and asymmetric Kummer beams, as well as Hermite-Gaussian (HG) vortex beams. Although they are obtained as superposition of respective conventional LG, BG, and HG beams, these beams have the same TC equal to that of a single mode, n. At the same time, the normalized orbital angular momentum (OAM) that the beams carry is different, differently responding to the variation of the beam's asymmetry degree. However, whatever the asymmetry degree, TC of the beams remains unchanged and equals n. Although separate HG beam does not have OAM and TC, superposition of only two HG modes with adjacent numbers (n, n + 1) and a π/2-phase shift produces a modal beam whose TC is -(2n + 1). Theoretical findings are validated via numerical simulation.

Topological charge of a linear combination of optical vortices: topological competition.

We theoretically show that optical vortices conserve the integer topological charge (TC) when passing through an arbitrary aperture or shifted from the optical axis of an arbitrary axisymmetric carrier beam. If the beam contains a finite number of off-axis optical vortices with same-sign different TC, the resulting TC of the beam is shown to equal the sum of all constituent TCs. If the beam is composed of an on-axis superposition of Laguerre-Gauss modes (n, 0), the resulting TC equals that of the mode with the highest TC. If the highest positive and negative TCs of the constituent modes are equal in magnitude, the "winning" TC is the one with the larger absolute value of the weight coefficient. If the constituent modes have the same weight coefficients, the resulting TC equals zero. If the beam is composed of two on-axis different-amplitude Gaussian vortices with different TC, the resulting TC equals that of the constituent vortex with the larger absolute value of the weight coefficient amplitude, irrespective of the correlation between the individual TCs. In the case of equal weight coefficients of both optical vortices, TC of the entire beam equals the greatest TC by absolute value. We have given this effect the name "topological competition of optical vortices".

Magnon Landau Levels and Spin Responses in Antiferromagnets.

We study gauge fields produced by gradients of the Dzyaloshinskii-Moriya interaction and propose a model of an AFM topological insulator of magnons. In the long wavelength limit, the Landau levels induced by the inhomogeneous Dzyaloshinskii-Moriya interaction exhibit relativistic physics described by the Klein-Gordon equation. The spin Nernst response due to the formation of magnonic Landau levels is compared to similar topological responses in skyrmion and vortex-antivortex crystal phases of AFM insulators. Our studies show that AFM insulators exhibit rich physics associated with topological magnon excitations.

Orbital angular momentum and topological charge of a multi-vortex Gaussian beam.

We report on a theoretical and numerical study of a Gaussian beam modulated by several optical vortices (OV) that carry same-sign unity topological charge (TC) and are unevenly arranged on a circle. The TC of such a multi-vortex beam equals the sum of the TCs of all OVs. If the OVs are located evenly along an arbitrary-radius circle, a simple relationship for the normalized orbital angular momentum (OAM) is derived for such a beam. It is shown that in a multi-vortex beam, OAM normalized to power cannot exceed the number of constituent vortices and decreases with increasing distance from the optical axis to the vortex centers. We show that for the OVs to appear at the infinity of such a combined beam, an infinite-energy Gaussian beam is needed. On the contrary, the total TC is independent of said distance, remaining equal to the number of constituent vortices. We show that if TC is evaluated not along the whole circle encompassing the singularity centers, but along any part of this circle, such a quantity is also invariant and conserves on propagation. Besides, a multi-spiral phase plate is studied for the first time to our knowledge, and we obtained the TC and OAM of multi-vortices generated by this plate. When propagated through a random phase screen (diffuser) the TC is unchanged, while the OAM changes by less than 10% if the random phase delay on the diffuser does not exceed half wavelength. Such multi-vortices can be used for data transmission in the turbulent atmosphere.

Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments.

Two simple and high-efficiency techniques for measuring the orbital angular momentum (OAM) of paraxial laser beams are proposed and studied numerically and experimentally. One technique relies on measuring the intensity in the Fresnel zone, followed by calculating the intensity that is numerically averaged over angle at discrete radii and deriving squared modules of the light field expansion coefficients via solving a linear set of equations. With the other technique, two intensity distributions are measured in the Fourier plane of a pair of cylindrical lenses positioned perpendicularly, before calculating the first-order moments of the measured intensities. The experimental error grows almost linearly from ~1% for small fractional OAM (up to 4) to ~10% for large fractional OAM (up to 34).

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.

Orbital angular momentum of an elliptic beam after an elliptic spiral phase plate.

We obtain a simple closed expression for the normalized orbital angular momentum (OAM) (OAM per unit power) of an arbitrary paraxial light beam with an elliptic shape, diffracted by an elliptic spiral phase plate (SPP), rotated by an arbitrary angle around the optical axis. Moreover, ellipticities of the beam and of the SPP can be different. It is shown that when an elliptic beam illuminates an elliptic SPP, the normalized OAM of the output beam is maximal (minimal) when both the beam and the SPP are oriented in the same (orthogonal) directions. The results can be used in optical trapping, e.g., for continuous change of the OAM transferred to a particle by rotating the SPP around the optical axis.

Astigmatic laser beams with a large orbital angular momentum.

We show that an elliptic Gaussian beam, focused by a cylindrical lens, can be represented as a linear combination of a countable number of only even angular harmonics with both positive and negative topological charge. For the orbital angular momentum (OAM) of the astigmatic Gaussian beam, an exact expression is obtained in a form of a converging series of the Legendre functions of the second kind. It is shown that at some conditions only the terms with the positive or negative topological charge are remained in this series. Using a hybrid numeric-experimental approach, we obtained the normalized OAM of the astigmatic beam, equal to 109, which is just 6% different from the exact OAM of 116, calculated by the equation. To generate such laser beams, there is no need in special optical elements such as spiral phase plates. The OAM of such beams can be adjusted by varying the waist radius of the Gaussian beam and the focal length of the cylindrical lens. The OAM of such beams can reach large values.

Tailoring polarization singularities in a Gaussian beam with locally linear polarization.

Here we theoretically study Gaussian beams with arbitrarily located polarization singularities (PSs). Under PSs, we mean here an isolated intensity null with radial, azimuthal, or radial-azimuthal polarization around it. An expression is obtained for the complex amplitude of such beams. We study in detail cases in which there is one off-axis PS, two opposite PSs, or more than two PSs located in the vertices of a regular polygon. If such a beam has one or two opposite PSs, these PSs are the centers of radial polarization. If there are three PSs, then one of them has radial polarization, and the other two have mixed radial-azimuthal polarization. If the beam has four PSs, then there are two PSs with radial polarization and two PSs with azimuthal polarization. When propagating in space, PSs are shown to appear in a discrete set of planes, in contrast to the phase singularities existing in any plane. If the beam has two PSs, their polarization is shown to transform from the radial in the initial plane to the azimuthal in the far field. The results can find application in optical communications by using non-uniform polarization.

Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy.

Using the Richards-Wolf formulas for an arbitrary circularly polarized optical vortex with an integer topological charge m, we obtain explicit expressions for all components of the electric and magnetic field strength vectors near the focus, as well as expressions for the intensity (energy density) and for the energy flux (components of the Poynting vector) in the focal plane of an aplanatic optical system. For m=2, from the obtained expressions it follows that the energy flux near the optical axis propagates in the reversed direction, rotating along a spiral around the optical axis. On the optical axis itself, the reversed flux is maximal and decays rapidly with the distance from the axis. For m=3, in contrast, the reversed energy flux in the focal plane is minimal (zero) on the optical axis and increases (until the first ring of the light intensity) as a squared distance from the axis.

Circularly polarized Hankel vortices.

We discuss vector Hankel beams with circular polarization. These beams appear as a generalization of a spherical wave with an embedded optical vortex with topological charge n. Explicit analytical relations to describe all six projections of the E- and H-field are derived. The relations are shown to satisfy Maxwell's equations. Hankel beams with clockwise and anticlockwise circular polarization are shown to have peculiar features while propagating in free space. Relations for the Poynting vector projections and the angular momentum in the far field are also obtained. It is shown that a Hankel beam with clockwise circular polarization has radial divergence (ratio between the radial and longitudinal projections of the Poynting vector) similar to that of the spherical wave, while the beam with the anticlockwise circular polarization has greater radial dependence. At n = 0, the circularly polarized Hankel beam has non-zero spin angular momentum. At n = 1, power flow of the Hankel beam with anticlockwise polarization consists of two parts: right-handed helical flow near the optical axis and left-handed helical flow in periphery. At n ≥2, power flow is directed along the right-handed helix regardless of the direction of the circular polarization. Power flow along the optical axis is the same for the Hankel beams of both circular polarizations, if they have the same topological charge.

Asymmetric Gaussian optical vortex.

We theoretically study a Gaussian optical beam with an embedded off-axis optical vortex. We also experimentally generate such an asymmetric Gaussian optical vortex by using an off-axis spiral phase plate. It is shown that depending on the shift distance the laser beam has the form of a crescent, which is rotated upon propagation. An analytical expression is obtained for the orbital angular momentum of such a beam, which appears to be fractional. When the shift increases, the greater the number of spirality of the phase plate or the "fork" hologram, the slower the momentum decreases. The experimental results are in qualitative agreement with the theory.

Astigmatic transforms of an optical vortex for measurement of its topological charge.

We obtain analytical expressions for the complex amplitudes of optical vortices deformed by astigmatic transforms, i.e., passed either through a cylindrical lens or through an inclined spherical lens. We also obtain similar analytical expressions describing propagation of an optical vortex generated when a Gaussian beam illuminates an inclined spiral phase plate (SPP) or when an elliptic Gaussian beam illuminates a SPP (not inclined). All these optical vortices with a topological charge (TC) n are described by the n-th order Hermite polynomial with a complex argument. It is shown that the argument is real only on a straight line in the transverse plane of the laser beam. There are n intensity nulls on this line. The treated here astigmatic transforms are used to determine the integer TC of optical vortices. We conduct a comparative experimental study of different astigmatic transforms and we show that the transform with a cylindrical lens is the best for determining the TC. Unlike other similar works, in this study we achieve transformation of n-degenerate intensity null of an optical vortex with the TC n=100 into n isolated first-order intensity nulls.

Optical trapping and moving of microparticles by using asymmetrical Laguerre-Gaussian beams.

We considered a generalization of the Laguerre-Gaussian (LG) laser beam family by using a complex shift of the beam complex amplitude in Cartesian coordinates. In this case, LG-beams lose their axial symmetry. The normalized orbital angular momentum is the sum of the beam topological charge and the term which is in square dependence on the asymmetry parameter. By optical trapping and moving the polystyrene microspheres in the focus of the asymmetric LG-beam, it is proven that the velocity of the microspheres increases with increasing the asymmetry parameter and constant topological charge.

Magnon Spin Nernst Effect in Antiferromagnets.

We predict that a temperature gradient can induce a magnon-mediated spin Hall response in an antiferromagnet with nontrivial magnon Berry curvature. We develop a linear response theory which gives a general condition for a Hall current to be well defined, even when the thermal Hall response is forbidden by symmetry. We apply our theory to a honeycomb lattice antiferromagnet and discuss a role of magnon edge states in a finite geometry.