In complex scalar fields, singularities of the phase (optical vortices, wavefront dislocations) are lines in space, or points in the plane, where the wave amplitude vanishes. Phase singularities are illustrated by zeros in edge diffraction and amphidromies in the heights of the tides. In complex vector waves, there are two sorts of polarization singularity. The polarization is purely circular on lines in space or points in the plane (C singularities); these singularities have index +/- 1/2. The polarization is purely linear on lines in space for general vector fields, and surfaces in space or lines in the plane for transverse fields (L singularities); these singularities have index +/- 1. Polarization singularities (C points and L lines) are illustrated in the pattern of tidal currents.

This paper is a review and extension of recent work by Berry and Dennis (Proc. Roy. Soc. Lond. A456, pp. 2059-2079, 2000; A457, pp. 141-155, 2001), where the geometric structure of phase singularities (wave dislocations) in waves is studied, particularly for singularities in isotropic random wavefields. The anisotropy ellipse of a generic dislocation is defined, and I derive an angular momentum rule for its phase. Random wavefields are discussed, and statistical results for density, anisotropy ellipse eccentricity, and planar correlation functions are stated. The properties of the correlation functions are compared to analogous features from ionic structure theory, and are discussed in those terms. The results are given explicitly for four particular spectra: monochromatic waves propagating in the plane, monochromatic waves propagating in space, a speckle pattern in the transverse plane of a paraxial beam, and the Planck spectrum for blackbody radiation.

Vortex trajectories typically have one or more attached stationary point trajectories. We call colloquially such a trajectory collection a bundle, and we study the permissible topologies and geometries of the individual trajectories as well as those of the bundle as a whole. Changes in bundle structure under changes in foliation are also studied.

Currently, optical vortices (OVs) which are areas of circular motion of light flux in an electromagnetic wave, are being actively studied. Optical vortices can be divided into longitudinal OVs, where the axis of circular flow coincides with the direction wave propagation, and transversal OVs, where the axis is perpendicular to the wave propagation direction. Airy rings in a focal plane of a lens may serve as an example of transversal OVs. For both longitudinal and transversal OVs the OV axis is the line where the field amplitude is equal to zero. The phase is assumed to be uncertain, or singular, along the OV axis. We report a detailed study of a structure of a transversal OV created in an interference field of two two-dimensional (2D) Gaussian waves and the topological reactions which can happen in the interference field. A physical reason, which determines the direction of light flow circulation is discussed.

Structurally stable laser beams with phase singularities that are rotating under propagation (so called spiral beams) have been investigated in various aspects. Some integral invariants of general laser beams and an optical analog of the Steiner theorem in mechanics are presented. Similarity and distinction of spiral beams for different rotation behavior are shown. A usage of spiral beams shaped like a predetermined planar curve applying for construction of phase focusing element is discussed.

The interaction of a vortex and an edge phase dislocation nested in a Gaussian beam by its free space propagation is studied experimentally and theoretically in the paraxial regime. This interaction induces the nucleation of additional vortices of both topological charges. Their positions and number depend on which of the phase dislocations is shifted from the center of the host beam. The interaction of noncoaxial light beams containing these phase dislocations is also reported.

It is considered a construction possibility of high order paraxial singular beams from nonparaxial modes in a paraxial approximation. Laguerre-Gaussian beams cannot form from nonparaxial Legendre-Bessel beams by means of a relatively simple way over the whole optical axis on conventional approximate boundary conditions. Nevertheless, this operation may be realized at the expense of a new passage to the limit: a set of nonparaxial singular beams is unambiguously transformed into a set of guided modes of an optical fiber near its input. It is presented two practical significant cases.

In a free space, it may be created any singular beam field distributions but far from all those fields are generic beams, i.e. beams to be steady state with respect to small perturbations of beam parameters or a distance from an optical transparence or a hologram. In the given paper it is considered those explicit solutions of Maxwell's equations that are unstable for signs variations of a spin or a topological charge of singular beams in a free space. Those beam modes change their intrinsic structure on a wave propagation. So, the sum of E- and H-types of linearly polarized non-paraxial optical vortices form nongeneric beam modes that have a pure screw dislocation far from a focal plane. Near a beam waist, the pure screw dislocation transforms into a pure edge dislocation and recoveries again its initial structure on its further propagation. Due to a sign variation both a vortex spin and a topological charge, an edge dislocation sense rotates at 90 angular degrees -- the optical Magnus effect 'in a void.'

On the basis of spin continuity equation we study the propagation of light in locally isotropic inhomogeneous media. We have obtained refined value of linear birefringence of order ((lambda) /(alpha) )² in layered media. We have studied spin transformations in perturbed weakly guiding fibers and proved that spin dynamics in fibers is governed by Heisenberg equation for a spin-1/2 particle.

We analyze the properties of light beams carrying phase singularities, or optical vortices. The transformations of topological charge during free-space propagation of a light wave were presented in a theory and experiment. We report theoretical calculations of orbital angular momentum density (AMD) in a of combined beam, which is a superposition of LG₀¹, LG₁⁰ and LG₀⁰ modes with different relative phase shifts and intensity ratio. We show that the total orbital angular momentum in a combined beam is constant during the propagation, in spite of any variation in the absolute value of AMD.

It is shown that, when Laguerre-Gaussian light beam with non- zero azimuthal index is partially reflected at plane interface of two isotropic transparent media, a specific deformation of the beam occurs in this process. This deformation gives rise a transverse shift of the center of gravity of the reflected beam. When the beam is incident from a denser medium at an angle that is close to the critical angle for total reflection, the magnitude of the predicted shift, like the magnitude of the well-known lateral or Goos-Hanchen shift, can be many wavelengths.

The problems to ensure an effective functioning of the controlled optical systems at the long paths in the turbulent atmosphere are presented. It is shown that the conventional techniques of adaptive optics do not allow us to suppress the optical field distortions under the conditions of the strong turbulence, when the intensity distribution acquire the pronounced speckle-structure and the wave front gains the topological defects (dislocations). Dynamical and statistical dislocation characteristics typical for the turbulent atmosphere are considered. Novel theoretical and experimental approaches to the dislocation problem are proposed and prospect of its practical solution are estimated.

Process of speckle-field forming when a light beam is distorted by optical inhomogeneities is considered, spatial ray dynamics is studied, and also regularities of birth of the wave-front dislocations are revealed. The energy-stream lines are constructed for the Gaussian beam passed through an optical inhomogeneity on the basis of the solution of the parabolic equation for the complex amplitude of the monochromatic wave field. Simulating of the ray trajectories is accompanied by the study of the complete set of the critical points. The angular divergence of radiation is calculated as an integral characteristic of the wave front. The ray spatial dynamics shows the basic features of transformation of the first smooth wave front into the singular one. It is established that this process is accompanied by the energy redistribution in the space manifested as a local beam focusing. The intensity distribution in the plane transversal to the beam propagation direction transforms according the rays' behavior. The discovered ray local focusing is the forerunner of subsequent beginnings of the pair screw wave front dislocations in the points of zero intensity. When the beam propagates through the medium having the refractive index inhomogeneities the dislocations can annihilate and then born again and the intensity acquires the speckle-structure.

The interferometric technique of analysis of polarization singular skeleton (s contours and c points) of vector field is elaborated. It was shown that complete characteristics of c points and s contours may be reconstructed from interferometric data. The examples of elaborated interferometric technique application to the polarization speckle-field analysis are presented.

In the article, it is consisted of wave fields near a focal plane of a strongly focused beam. The theoretical and experimental analysis of exact solution of Maxwell's for optical vortex modes beyond the paraxial barrier are given, and a phase singularities spectrum of wave fields is drawn. The asymmetry of electrical and magnetic fields, and also the losses of phase synchronism between them is observed, which expressed as distinction of topological phases. Experimental results are well according with the theoretical ones.

It is developed the theory of diffraction of a singular light beam with axial optical vortex on a half-plane screen, which cuts off the vortex zero-amplitude center ('severe screening'). It is shown that such diffraction features differ strongly from well known diffraction of a light beam with smooth wave front. It was founded that the beam singular properties are restored on some distance behind the screen. It is happened through the complicated space dynamics due to generation of secondary vortices. The main features of severe screened singular beam 'self-treatment' are established.

It is presented numerical simulations for Optical Vortex (OV) diffraction on the basis of Rayleigh-Sommerfeld solution of an edge diffraction problem and experiments that establish the effect of surviving and restoring of OV in a diffracted beam. Dependence of singular point position on screening investigated.

The theoretical analysis and experimental results of this report illustrate the possibility of describing of diffraction phenomena using the objects and methods, which were developed in singular optics. The aperture-diffracted field can be represented as superposition of real physically existing waves: an ordinary wave component with an amplitude half that of the incident wave, and a wave component possessing a singularity. The investigation of the singular wave component as informative part of diffraction field is an important new trend. It was shown that a system of hidden dislocation lines composes the skeleton and represents the diffraction field topology. The analysis of topology of the diffraction field in case of plane wave diffraction on a slit was performed. The proposed experimental proof ground can be useful for studying the system of linear edge dislocations space evolution and properties, such as nucleation and annihilation, edge and mixed edge-screw transition.

It is studied structurally stable nonparaxial vector singular beams -- generic beams -- near an optical fiber input as solutions of Maxwell's equations. The beam behavior in a free space or a homogeneous isotropic medium is considered. Any variation of a beam Raylein length is turn out to change no number and form of phase and polarization singularities but to displace their coordinates. It is found that a field structure both of eigen modes in a free space and an optical fiber is much about the same. Nevertheless there exist an essential distinguished feature of generic fiber modes -- a spin-orbit interaction in a vortex field, that defines a polarization correction to a scalar propagation constant. It is the fact that enable us to divide fiber vortex sensors into two big groups: thermodynamics value devices and inertial navigation sensors.

Properties of vortical fields in an elliptic weakly guiding optical fiber are considered. The eigenfunctions and the spectrum of polarization corrections to the scalar propagation constant in the case of relatively large and small values of a fiber ellipticity are obtained by means of the spin-orbit interaction operator method. It is discussed the conversion process of a spin and orbit angular momenta on a vortex propagation.

A mechanical bent of an optical fiber induces perturbations of geometrical and material fiber parameters. Due to these factors there arise both wave field distortions and variations of a mode propagation constant. In the given paper, polarization corrections to propagation constants are shown to be mean values of the spin-orbit operator of a singular beam in a bent optical fiber. The propagation process of eigen modes in those fibers is analyzed.

In the paper the vortex pulse properties on the pulse propagation in the optical fibers with the step and parabolic refraction index are discussed. The pulse with an initial frequency deviation (modulated on the frequency at the fiber input) and without it is investigated in the linear regime. The dependence of the pulse dynamics from its transversal structure and the initial frequency deviations are shown. The frequency chirp of the vortex pulses with the initial frequency deviation and without it is considered.

A beam propagation through astigmatic square-law waveguides (n(x,y) equals K_x²X² + K_y²Y²) is investigated by theoretical and experimental means. It is shown that if certain condition between astigmatic value, length of waveguide, wave-front curvature and Gaussian parameters takes place then a Hermite-Gaussian beams transforms into Laguerre-Gaussian beams without residual astigmatism.

We study the scattering properties of optical vortex solitons generated in a self-defocusing nonlinear Kerr medium, analyzing both linear and nonlinear wave scattering. In the linear case, we find a splitting of a plane-wave front at the optical vortex proportional to the vortex circulation, similar to what occurs in the scattered wave of electrons for the Aharonov-Bohm effect. For larger wave amplitudes, we study analytically and numerically the scattering of a dark-soliton stripe, a nonlinear analog of a small-amplitude wavepacket, by a vortex and observe a significant asymmetry of the scattered wave. Subsequently, a wavefront splitting of the scattered wave develops into transverse modulation (snake) instability of the stripe that generates two trains of vortices with opposite charges.

Paper presents the results of semianalytical study of features and stability of weakly nonparaxial spatial (transversely two- dimensional) optical soliton in a transparent bulk medium with a cubic (Kerr-law) nonlinearity. We have developed a variant of perturbation theory for sufficiently wide solitons where a small parameter of nonparaxiality is the ratio of the light linear wavelength to the soliton width. By solution of approximate governing equation for electric field transverse components with the Townes mode with linear polarization as an initial iteration, we have found solitons with cylindrically asymmetric field distribution and elliptical polarization changing over the transverse section. The nonparaxial soliton stability was confirmed and weakly damping 'internal modes' of the soliton have been found.

We show the dynamics of evolution of screw phase-dislocations existing in the wave front of Gaussian beam with nested multiple-charged vortices that propagate in quadratic nonlinear crystals under conditions for seeded second-harmonic generation. The number of existing vortices is shown to depend on the input light and material conditions, including the topological charge, width and intensity of the pump and seed signals, as well as on the propagation length inside the crystal.

The stimulated Brillouin scattering (SBS) of focused laser beams with screw dislocation of phase front (Gauss-Laguerre vortex modes) has been numerically explored with allowing for transient processes, non-steady-state conditions, hypersound noise and an effect of Stokes field on laser field, i.e. SBS saturation. Calculations as well as an analysis on the basis of perturbation theory under near-threshold SBS conditions have shown that phase conjugation of vortex beams does not take place as a phenomenon because of failure of selection of Stokes mode conjugated to laser one. It occurs since the SBS gain coefficient does not depend on the laser field phase under the near-steady-state conditions, and the Stokes field does not 'feel' how the laser phase helix is twisted. Therefore (for example, in the case of a doughnut mode) the Stokes beam consists of a random superposition of a few modes including conjugated one. For rather wide class of vortex beams (for example, in the case of a laser mode with two rings), a novel phenomenon takes place that can be called as phase transformation at SBS. Its essence is in the fact that in Stokes beam a mode with regular phase front is selected which is orthogonal to laser mode at all. In the near- threshold SBS regime, this selected mode is a principal Gaussian mode. Calculations show that at SBS of vortex laser beam the generation of vortex hypersonic wave takes place in the SBS-medium. Its topological charge coincides with the laser beam charge in view of vortex-free structure of the Stokes beam.

Linear and nonlinear processes of generation and transformation of optical vortices in crystals were investigated. New universal methods for production of Bessel light beams with optical vortices of the first, second and higher order by means of uniaxial and biaxial crystals were proposed. Light beams with optical vortices of topological charge +/- 1 and +/- 2 are experimentally obtained using KTP and HIO₃ (iodic acid) biaxial crystals. We studied type II second harmonic generation by Bessel beams with optical vortices in nonlinear crystals. Results of investigation of the processes of Bessel light vortex order doubling, transfer of vortex to the second harmonic radiation, and annihilation of optical vortices with the opposite signa are presented.

The circularly distributed speckle pattern, as well as interference fringe structure in profile of the beam of stimulated Raman scattering (SRS) I Stokes component, pumped by multimode radiation with nearly four-fold azimuthal symmetry, have been observed. The SRS was excited near the threshold of generation by nanosecond pulses of laser radiation at wavelength 530 nm. The profile of output pump beam had a uniform intensity distribution, whereas the SRS beam profile showed kaleidoscopic change from shot to shot, while the energies of input pulses were kept stable. The interference fringes showed a number of points where the fringes originated or vanished. Such behavior, which is the vortex signature, allows to suppose that SRS waves, generated from quantum noise, carry screw dislocations.

In the paper the pulse field component equations for guided pulse in an optical fiber are obtained. The linear and nonlinear regimes on the pulse propagation are considered. It is shown that the equation systems for the pulse field component envelopes depend on mode structure both for the nonlinear and linear regimes. The topological charge and its sign determine the dynamics of the pulse component envelopes and phase of the vortex pulse as well as of a nonsingular pulse.

In the paper the guided vortex pulse is studied in the nonlinear regime in an optical fiber. The formation of the vector vortex soliton on the pulse propagation is considered. It is shown that the vortex pulse with the singularity on the longitudinal axis can generate the vortex soliton in the form of the ring. The vortex soliton and the nonsingular soliton have different dynamics on the propagation along the given fiber.

In this paper, we present theoretical and experimental results of our investigations on high resolution interference microscopy. We were able to measure phase variations with a spatial resolution in the order of 10 nm. For this purpose, we have modified a classical interference microscope based on a Mach-Zehnder interferometer to obtain ultra-high resolution. The optical fields were generated by periodic optical microstructures. The measured optical phase distribution in the near-field compares very well with the results of rigorous diffraction theory.

It is considered the construction problems of the fiber- optical sensors of physical values with phase singularities. The signal detection is carried out by the Mach-Zander interferometer with a single-mode fiber as the signal arm and the free propagating singular beam as the reference signal. The sensitivity of such sensor up to 160 rad/( degree(s)C(DOT)m) was achieved.

An optical device is proposed and realized for spatial separation into different channels of the Laguerre-Gaussian modes with even and odd mode indices. It was shown that a Gaussian beam diffracted by special computer-synthesized hologram attains in the first order of diffraction a single off-axis optical vortex, which may be represented as superposition of such modes. The proposed separator is able to create 'pure' axial vortex in one output channel and vortices- free beam in another channel. The obtained result is interpreted as a possibility to separate photons with zero and nonzero orbital angular momentum.

E-beam lithography was used to fabricate an 8-channel phase DOE capable of selecting angular harmonics with the number +/- 1, +/- 2, +/- 3, and +/- 4. It has experimentally been demonstrated that this filter can successfully extract angular harmonics from laser beams composed of Gauss-Laguerre modes.

An efficiency of phase correction a wave, propagating through a turbulent atmosphere, is considered. It is shown, that the account of the hidden phase, when using least-square method, allow to increase the efficiency of phase correction, when branch points are present in the phase of a turbulence- distorted optical field.

A geometrical optics consideration of the torque exerted on a particle by a light beam with an angular momentum is presented. The beam is modelled as a bundle of rays directed normally to the wave front and carrying the power proportional to the local value of the light intensity. A numerical code is developed to calculate the ray propagation through the refracting and absorbing particle of arbitrary convex shape, which is applied to evaluation of the beam angular momentum transformation during the light interaction with the particle. Calculated values of the mechanical reaction experienced by the particle are used for the estimation of its motion in the optical levitation trap. The possibility of regular and predictable particle behavior, which allows studying complicated patterns of light scattering by means of a single fixed detector, is shown.

A simple Laguerre-Gaussian (LG) -- Hermite-Gaussian (HG) mode converter structure is described that contains only flat mirrors and, possibly, transversely uniform phase correctors. Its action is grounded on the superposition of the initial beam with proper auxiliary beam, which is generated automatically by the corrector itself. In the case of lowest LG (HG) modes such auxiliary beam can be obtained by the initial beam reflection from the properly oriented flat mirror and uniform phase shift. The converter scheme has the interferometer-like structure, where the initial beam is splitted into two arms by the light magnitude divider. Within the first arm the beam retains its initial structure while the second arm allows preparing the auxiliary beam. Then two partial beams are re-coupled; in order to avoid irrelevant discrepancies between them proper shape-correcting and phase- shifting provisions should been made. The advantage of this converter system is the absence of focusing elements, simplicity of assemblage and practical arrangement. Instead, this method inevitably leads to energy losses and has rather specialized character: in general, for conversion of different modes different converter structures are necessary. Possible applications of such mode converters as interferometer devices are discussed.

A domain grating with dislocations in the process of magneto- optical diffraction performs the hologram function of an optical vortex and transforms a fundamental Gauss beam into a wave transmitting an angular impulse momentum. The conditions forming domains structures with the given location of a stripe domain in a plane were determined. The magnetic film parameters causing the formation of Y-type domains and clusters of V-shaped domains were obtained.

The new fiber vortex sensor for supersensitive temperature measurements requires using special ways of a computer processing. In the given paper, it is revealed and studied one from the possible ways of an image conversion of an interference helix into an intensity field with an edge dislocation. A helix rotation transfers into an edge dislocation rotation. An experimentally observed phase shift is recorded as an orientation angle of the edge dislocation axis. It is estimated experimental errors.

Mechanical action of optical fields with phase singularities (optical vortices) on laser cooled atoms is theoretically investigated. Due to small sizes and sufficiently low velocities of cold atoms it is intriguing to understand how they may be used to shed a light on some modern problems of singular optics. It is shown that cold atoms are suitable tools for learning of optical singularity evolution, its birth and transformations due to interactions between themselves and with other physical objects. Applications to guiding and trapping of atoms are discussed also.