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We investigate the conversion between orbital and spin components of the angular momentum of light beams in the second-harmonic generation process from the near-surface layer of the nonlinear isotropic medium without spatial dispersion at oblique incidence of the elliptically polarized fundamental beam. The conservation of the projection of both spin and orbital angular momenta onto the normal to the surface of the medium during the transformation of the incident wave into a reflected double frequency wave has been demonstrated.

We discuss analytically found expressions, which relate the values of two parameters characterizing the topological type of linear and circular polarization singularities in nonparaxial light fields to the values of the complex amplitude components of the electric field and their first spatial derivatives. The necessary conditions are described for the formation of light pulse at the doubled frequency in the bulk of an isotropic gyrotropic medium with a frequency dispersion of quadratic nonlinearity and they impose constraints on the spatial transverse structure of the non-uniformly polarized incident radiation pulse and its time structure. The electric field of a light beam reflected from the surface of an isotropic gyrotropic medium with spatial dispersion of quadratic nonlinearity is investigated in the case of an arbitrary structure and geometry of incidence of non-uniformly polarized fundamental light beams. The ranges of the parameters of an elliptically polarized Gaussian beam and a medium with local and nonlocal cubic nonlinearity are determined, at which the lines of circular polarization singularity appear in cross sections of self-focused beam.

Efficiency of convolutional artificial neural networks in the problem of finding nonlinearity parameters proportional to the local and non-local cubic dielectric susceptibilities of a medium and intensity of the incident radiation, fully describing the self-focusing character of elliptically polarized laser beams, is demonstrated. It is shown that realization of the predictive algorithm by the neural network can be improved by using complex structured light so that the error is lowered down to percent units.

We discuss key features of the conversion of spin and orbital angular momentum of electromagnetic waves in the process of second-harmonic generation from the surface of the isotropic medium at oblique incidence. Conservation of the projections of spin and orbital parts of angular momentum of interacting waves onto the normal to the surface is shown for an arbitrary case of polarization and mode structure of the incident light beam.

The interconversion and total conservation of spin and orbital components of angular momentum of light is analytically considered in various nonlinear optical processes in isotropic chiral medium, such as sum frequency and second harmonic generation on the surface and in the bulk of the medium. The nonlocality of nonlinear response and the symmetry break in its near surface layer were taken into account, and for each process the selection rules for interacting photons were derived.

We investigate the interconversion between spin components of the angular momentum (SAM) and orbital components of the angular momentum (OAM) of collinear elliptically polarized monochromatic fundamental beams with frequencies ω1 and ω2 in the five-wave mixing process in homogeneous isotropic nonracemic mixtures of chiral molecules with nonlinear fourth-order susceptibility. Fundamental beams have helicoidal wavefronts and arbitrary radial distributions of intensity. The discovered non-conservation of the total SAM and OAM projections on the direction of propagation of the interacting waves during the generation of radiation at the frequency 3ω1 − ω2 compensate each other, ensuring the conservation of the total angular momentum projection. A straightforward classical and quantum interpretation of the results is given.

Nonlinear optical mixing of light waves, carrying both orbital and spin angular momentum, can lead to conversion between spin and orbital angular momenta of the fundamental and signal photons. From this point of view, a number of classical problems of nonlinear optics of media with local and nonlocal nonlinear response were studied analytically and numerically in which the propagation of wide class of uniform and nonuniform elliptically polarized waves have been discussed.

We numerically study the structure of polarization singularity lines in a near-field of the sub-wavelength dielectric particle when it is irradiated by a monochromatic elliptically polarized plane wave. For the various values of the ellipticity degree of the incident radiation, we trace the C T and L T -lines on which the polarization ellipse turns into a circle or a line, respectively. The continuous variation of the isotropy parameters of the singularities is visualized along the lines. The main rules of C T and L T -lines deformation, emergence, and vanishing are revealed.

The work is devoted to the self-action of laser beams propagating in the isotropic phase of a cholesteric liquid crystal near the transition temperature to the mesophase in a wide range of parameter values characterizing the nonlocality of the nonlinear optical response of the medium, defocusing in the medium, the power of incident laser radiation and diffraction in the liquid crystal. The origin of the primary singularities of both polarization components of radiation in a wide range of parameters of radiation and the medium is investigated for different values of the parameter characterizing the nonlocality of nonlinear optical response in chiral medium. For each of the values, the visibility diagrams of the primary singularities of both polarization components are plotted depending on the parameters of the medium and radiation, and their asymmetry is found., 7 pages, 3 figures

The emerging and evolution of polarization singularities in various nonlinear optical processes in isotropic media was studied both analytically and numerically. The interacting light beams with non-uniform polarization were considered in parabolic approximation and the longitudinal components of their fields were taken into account. The generation of the nonlinear signal from the bulk and surface of the medium was shown to be caused by its chirality and nonlocal quadratic response.

Isotropy parameters of truly linear (${L}^{T}$) and truly circular (${C}^{T}$) polarization singularities in a three-dimensional monochromatic electromagnetic field are defined. It is demonstrated that these parameters characterize the distribution of polarization ellipses near the singular points in a more detailed way than a topological index. A numerical algorithm which tracks ${C}^{T}$ and ${L}^{T}$ lines and visualizes the changes of isotropy parameters along the lines is proposed. The algorithm is applied to visualize the polarization singularity lines and their properties in the vicinity of a metallic nanospheroid that is irradiated by a plane elliptically polarized wave. It is shown that both ${C}^{T}$ and ${L}^{T}$ lines appear close to the surface of the nanoparticle and they are three-dimensional closed loops or arcs that start and end at the surface of the nanospheroid. In most cases the singular lines of different types are topologically linked.

The interaction between spin and orbital components of the angular momentum of an elliptically polarized beam with helicoidal phase profile and arbitrary axially symmetric intensity profile in the third-harmonic generation process is investigated. The third-harmonic beam is generated in an isotropic medium with cubic nonlinearity within undepleted pump approximation. The conservation of the total projections of both the spin angular momentum and orbital angular momentum components in four-wave mixing is demonstrated for interacting waves.

We have numerically and analytically shown that polarization singularities can emerge when a homogeneously elliptically polarized light beam undergoes self-focusing in an isotropic third-order Kerr medium without frequency and spatial dispersion (fused silica, liquids, gases etc.) In the case of axially symmetric beam the emerging C -lines have the shape of circumference with the center at the beam’s axis and they are located in the separate transversal planes in the medium. If the axial symmetry of the incident beam is broken then the even number of C -points with opposite topological charges are nucleated in the medium. They exist in a certain propagation coordinate range and then they collide and annihilate each other.

We study the interaction between spin and orbital components of the angular momentum of electromagnetic waves in the sum-frequency generation process on a surface of the nonlinear isotropic chiral medium within undepleted pump approximation. Both bulk and near-surface responses of the medium are taken into account. Classical and quantum explanations of particular features of three-wave mixing on the surface of a nonlinear medium are presented.

A phenomenological description of nondegenerate and degenerate three-wave mixing in the bulk of an isotropic chiral medium is given for the wide class of fundamental light beams with nonuniform polarization distribution and multimode transverse structure. Special attention is paid to the classical and quantum interpretation of interconversion between spin and orbital parts of the angular momentum of the input and output beams.

The components of electric field of the third harmonic beam, generated in isotropic medium with cubic nonlinearity by a monochromatic light beam carrying polarization singularity of an arbitrary type, are found analytically. The relation between C-points characteristics in the fundamental and signal beams are determined, as well as the impact of the phase mismatch on the shape of the C-lines.

We used the high-order finite element method to numerically study the self-action of light in a nonliner metamaterial composed of gold helices. It is shown that such a metamaterial can selectively reflect the waves with certain circular polarization. The impact of the nonlinear optical response of gold on the value of the transmission coefficient and the frequency range, in which this selective reflection occurs, is studied.

The emerging of polarization singularity lines (C-lines), accompanying the self-focusing of an initially elliptically polarized Gaussian beam in an isotropic medium with various mechanisms of cubic nonlinearity, is studied numerically. Lines of circular polarization can appear in planes perpendicular to the beam axis as circles centered at the axis in both sub-threshold and super-threshold regimes of nonlinear self-action. The range of beam power values, for which the C-lines are formed, rapidly expands as the initial polarization of the beam gets closer to circular. The radius of the first formed C-line and the distance from the C-line to the border of the medium are decreasing if the beam power increases.

Sum-frequency generation in the case of coaxial interaction of elliptically polarized Gaussian beams of fundamental radiation in isotropic chiral medium was studied. The system of three equations of nonlinear diffraction was solved numerically without assumption of undepleting fundamental (``pump'') waves. It has been shown that only for small (less than 0.1) values of nonlinear coupling coefficient of the interacting waves their propagation can be described within the framework of such approximation with precision of about tenths of percent. However, the results of the undepleting pump approximation are remarkably different from those obtained in numerical solution of system of nonlinear equations for greater values of nonlinear coupling coefficient. In the first turn, this difference appears in the amplitude and polarization of the sum frequency beam. When the nonlinear coupling coefficient is of the order of unity, its further increase does not lead to the growth of the conversion efficiency, which remains of the order of 0.01. At the same time, there appear notable changes of intensity and polarization of light in transverse sections of the beams at fundamental frequencies.

Sum topological charges of the circularly polarized components of a beam, generated in the bulk of an isotropic chiral medium by elliptically polarized fundamental beams with arbitrary transverse structure, are found. The impact of the ellipticity degrees of the polarization ellipses of the fundamental beams and the angle between the major axes of the ellipses on the sum topological charges is discussed.

We have analysed the conditions for the appearance of polarisation singularities in the second-harmonic beam cross section arising in the case of reflection of a uniformly elliptically polarised Gaussian beam at the fundamental frequency from the surface of an isotropic gyrotropic medium. It is shown that there are elliptical polarisation states of the incident light at which the cross section of the second-harmonic reflected beam contains either one or two C lines and either two, or one, or none L lines [the loci of the points where the propagating radiation is circularly (C) or linearly (L) polarised].The formulas determining the conditions for the occurrence of L and C lines and specifying their orientation in the plane of the cross-section of the second-harmonic beam are obtained.

We have investigated the spatial distribution of the electric field vector in a light wave interacting with a planar metamaterial with strong dispersive and nonlinear optical properties. Its basic element consists of two spatially separated silver strips. It is shown that polarization singularities of the light field (lines) appear in the vicinity of these strips. The shapes and the locations of the lines are determined by the frequency and intensity of elliptically polarized monochromatic light incident on the metamaterial.

Expressions for the electric field at a sum frequency generated by a collinear elliptically polarized Gaussian beam and circularly polarized Laguerre-Gaussian beam in an isotropic chiral nonlinear medium are obtained in quadratures. The amount and locations of $C$ points in the cross section of a signal beam at a sum frequency are shown to be dependent on frequency and diffraction lengths ratios of fundamental beams and on the ellipticity degree of the Gaussian beam's polarization ellipse. Possible values of total topological charges of the emergent $C$ points are determined by the topological charge of the Laguerre-Gaussian beam and remain constant while the radiation propagates in nonlinear media. In case of nonzero total topological charge $C$ lines form helical structures, the parameters of which depend on the wave-vector mismatch. Otherwise, $C$ lines form a loop. As the wave-vector mismatch grows the loop undergoes deformation and breaks up, creating new $C$ lines.

Considering sum-frequency generation in an isotropic chiral nonlinear medium, we analyze the transfer of the spin angular momentum of fundamental elliptically polarized Gaussian light beams to the signal beam, which appears as the superposition of two Laguerre–Gaussian modes with both spin and orbital angular momentum. Only for the circular polarization of the fundamental radiation is its angular momentum fully transferred to the sum-frequency beam; otherwise, part of it can be transferred to the medium. Its value, as well as the ratio of spin and orbital contributions in the signal beam, depends on the fundamental frequency ratio and the polarization of the incident beams. Higher energy conversion efficiency in sum-frequency generation does not always correspond to higher angular momentum conversion efficiency.

The second harmonic generation in the bulk of an isotropic chiral liquid with spatial or temporal nonlocality of its quadratic response is studied analytically. The fundamental pulsed beam has Gaussian time envelope and carries a polarization singularity at its axis. The influence of the topological charge and the handedness of the polarization singularity on the polarization state of the signal beam is revealed.

When proceeding to the oblique incidence from thenormal incidence geometry in surface secondharmonic generation, the symmetry of the problemand, thus, of the reflected signal beam breaks. Inthis case there can appear C-points in thetransversal section of the signal beam. Using theanalytical formulae obtained in previous works, westudied the conditions of appearance and thebehaviour of the C-points in reflected signal beam(in particular case of SHG), and also the way of theline-to-point singularity transformation.

For the first time we have obtained the analytical expressions describing the spatial distribution of the polarization of a second harmonic beam's light field reflected from the surface of an isotropic gyrotropic medium in the case when a normally incident fundamental beam contains a polarization singularity of an arbitrary type. The contribution of bulk and surface responses of the nonlinear medium in the formation of the lines of circular polarization in the second harmonic beam were analyzed. The relation between topological characteristics and polarization states of the singularities in the incident and reflected beams was established.

The interaction of polarization singularities (C-points) of a light field in collinearly propagating beams in an isotropic gyrotropic medium with nonlocal nonlinear cubic optical response was studied numerically. The nonlinear self-action significantly affects the transversal polarization and intensity distribution in the propagating beams, while the nonlocality can change the relative weights or the polarization states of the intense peaks in the transversal section. The sum of the topological charges of all of the singularities in the propagating field remains unchanged. The scenario of their interaction is, in general, the same, as in the linear medium, but depending on the nonlinearity character it can either be quickened or slowed down by the increase of the intensity of the incident waves. In the case of very large values of the initial intensity the interaction between the C-points (pairwise creation and annihilation) ceases due to the strong separate self-focusing of circularly polarized components of the field.