The stratification of a wave packet in a nonlinear medium

The possibility of controlled large-scale stratification of the wave impulse in the nonlinear medium is discussed. The wide class of exact analytical solutions, describing, in the bounds of the nonlinear geometrical optics, the formation of the new intensity maxima, are found. The characteristics dimensions of the oscillations of the packet are much greater than the wave length.     

Complex geometrical optics of Kerr type nonlinear media

The paper generalizes paraxial complex geometrical optics (PCGO) for Gaussian beam (GB) propagation in nonlinear media of Kerr type. Ordinary differential equations for the beam amplitude and for complex curvature of the wave front are derived, which describe the evolution of axially symmetric GB in a Kerr type nonlinear medium. It is shown that PCGO readily provides the solutions of NLS equation obtained earlier from diffraction theory on the basis of the aberration-free approach. Besides reproducing classical results of self-focusing PCGO readily describes an influence of the initial curvature of the wave front on the beam evolution in a medium of Kerr type including a nonlinear graded-index fiber. The range of applicability of the PCGO theory is discussed as well which is helpful for avoiding nonphysical solutions.     

Light propagation in media with a highly nonlinear response: An analytical study

The problem of light propagation in highly nonlinear media is studied with the help of a recently introduced systematic approach to the analytical solution of equations of nonlinear optics [L.L. Tatarinova, M.E. Garcia, Exact solutions of the eikonal equations describing self-focusing in highly nonlinear geometrical optics, Phys. Rev. A 78 (2008) 021806(R)(1—4)]. Numerous particular cases of media exhibiting high-order nonlinear refractive indices are considered. We obtain analytical expressions for determining the self-focusing position and a new exact expression for calculating the filament intensity. The constructed solutions allowed us to revise a so-called self-focusing scaling law, i.e., the functional dependence of the self-focusing position on the initial light peak intensity. It was demonstrated that this dependence is governed by the form of the nonlinear refractive index and not by the laser beam shape at the boundary.     

On the change in electromagnetic wave polarization in a smooth one-dimensionally inhomogeneous medium

Propagation of an electromagnetic wave in a smooth one-dimensionally inhomogeneous isotropic medium is considered in the second approximation of geometrical optics. The polarization evolution is studied extensively. It is known that in the first (Rytov) approximation of geometrical optics, there is only the rotation of the plane of polarization (with no change in the polarization shape and sign) for rays with torsion. In the case considered, both the shape of polarization ellipse and the sign of polarization change proportionally to the integral of the squared ray curvature even for plane rays. The effect is of nonlocal geometrical nature and can be described in terms of the generalized geometrical phase incursion between two linear polarizations.     

Self-constriction of a wave packet in a non-linear medium

The possibility of a controlled deformation of a wave packet in the non-linear medium is discussed. The exact analytical solutions, describing, in the bounds of non-linear geometrical optics, the pulse envelope evolution and the non-linear modulation of such a packet, are found.     

Development of the model of HF radiowave propagation in the ionosphere

The model of HF radiowave propagation, based on a geometrical optics approximation, has been extended for the case of a complex geometrical optics. The radiowave propagation in the illuminated region without going to the caustic shadow region is considered. The radiowave propagation model has been generalized for the case of wideband HF signals in the ionosphere. A dynamic representation of LFM signals in the form of a wave packet sequence was used for this purpose. The radiowave propagation model was also adapted to a global self-consistent dynamic model of the thermosphere, ionosphere, and protonosphere, which made it possible to study specific features in the formation of HF radiowave ray paths during geomagnetic storms in a 3D inhomogeneous anisotropic medium.     

Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation

We obtain exact spatiotemporal periodic traveling wave solutions to the generalized (3+1)-dimensional nonlinear Schr?dinger equation with distributed coefficients. We utilize these solutions to construct analytical light bullet soliton solutions of nonlinear optics.     

The Non-Linear Geometric Optics of the Localized Wave Fields

The new approach to the self-action theory of intensive localized pulses, based on the hydrodynamical analogy in the non-linear geometrical optics, is proposed. The complex of phenomena of amplitude-phase non-stationary evolution of the intensive localized electromagnetic wave pulses in the dispersive medium is analysed in the framework of such approach. The wide classes of exact analytical solutions of the non-linear self-action equations, connected with such pulses, are constructed. The simple form of these solutions, represented with the well-known eigen-functions of the Laplace equation in special variables, permits to divide the pulse non-linear deformation qualitatively different effects. These solutions predict the large-scale pulse self-stratification and the origin of the quick intensity increase area during the non-linear evolution of the initially smooth distribution of the wave. The characteristic points of such evolution are represented by the singularities in the exact solutions of the non-linear geometrical optics. All results, describing the dynamics of the non-linear amplitude-phase re-building of the pulse, are represented in the simple algebraic form.     

Numerical implementation of complex geometrical optics

We propose a numerical scheme for calculation of the wave field, based on the geometrical-optics method generalized for complex values. The main advantage of the complex-value method is a possibility to take into account diffraction effects using only the ordinary differential equations of the geometrical optics. This allows one to significantly reduce the amount of computations and, hence, computation time. The efficiency of this algorithm is illustrated by two numerical examples that allow comparison with the known analytical solutions: the plane-wave field behind the caustic in the linearly inhomogeneous layer and the field of a Gaussian beam in a homogeneous medium. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 7, pp. 630–637, July, 2000.     

Transverse shift by total reflection of a circularly polarised light beam: theory and experimental proof

The Goos and Hänchen experiments of 1947 have shown that in total reflection the photons tunnel through the second medium (which we take to be the vacuum), whence a longitudinal shift x; Imbert's experiments of 1970 have shown that if the incident beam is circularly polarized there is also a transverse shift z, the sign of which depends on the helicity sign. We briefly explain this new phenomenon, first in terms of a generalized geometrical optics where the velocity and momentum of a spinning photon are non-collinear, then in terms of wave optics, using an appropriate class of solutions of Maxwell's equations.Presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

Electromagnetic light rays in local dielectrics

An approach to deal with the limit of geometrical optics of electromagnetic waves which propagate in moving nonlinear local dielectric media in the context of Maxwellian electrodynamics is here developed in order to apply to quite general material media. Fresnel equations for the light rays are generically found, and its solutions are intrinsically obtained. The multi-refringence problem is addressed, and no more than four monochromatic polarization modes are found to propagate there.     

Dynamics of solitons of the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients

We present three families of soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with distributed coefficients. We investigate the dynamics of these solitons in nonlinear optics with some selected parameters. Different shapes of bright solitons, a train of bright solitons and dark solitons are observed. The obtained results may raise the possibilities of relevant experiments and potential applications.     

Hodographic vortices

Vortices are screw phase dislocations associated with helicoidal wave-fronts. In nonlinear optics, vortices arise as singular solutions to the phase-intensity equations of geometric optics. They exist for a general class of nonlinear response functions. In this sense, vortices possess a universal character. Analysis of geometric optics equations on the hodograph plane leads to deformed vortex type solutions that are sensitive to the form of the nonlinearity. The case of a Kerr type nonlinear response is discussed as a specific example.     

WKB Analysis for Nonlinear Schrödinger Equations with Potential

We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besides subquadratic potentials, we consider initial phases which may be unbounded. For this, we construct solutions for some compressible Euler equations with unbounded source term and unbounded initial velocity.     

Optical solitons and optical rogons of generalized resonant dispersive nonlinear Schrödinger's equation with power law nonlinearity

This paper obtains solitons and singular periodic solutions to the generalized resonant dispersive nonlinear Schrödinger’ equation with power law nonlinearity. There are several integration tools that are adopted to extract these solutions. They are simplest equation method, functional variable method, sine–cosine function method, tanh function method and the G′/G-expansion method. These integration techniques reveal bright and singular solitons as well as the corresponding singular periodic solutions to the nonlinear evolution equation. These solitons solutions are important in the nonlinear fiber optics community as well as in the study of rogue waves.     

Soliton-like solutions for the nonlinear schr?dinger equation with variable quadratic hamiltonians

We construct one-soliton solutions for the nonlinear Schr¨odinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of the complete (super) integrability of generalized harmonic oscillators. The soliton-wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schr¨odinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painlev′e transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons and Jacobi elliptic and second Painlev′e transcendental solutions for several variable Hamiltonians that are important for research in nonlinear optics, plasma physics, and Bose–Einstein condensation. The Feshbach-resonance matter-wave-soliton management is briefly discussed from this new perspective.