Structural features of self-action of laser radiation in the electromagnetic induced transparency mode

Structural features of the laser radiation self-focusing dynamics in the electromagnetic induced transparency (EIT) band are studied for an atomic system with a Λ-type energy level diagram. Effective nonlinearity of an EIT medium is manifested primarily as nonlinear dispersion (dependence of the group velocity on the wave amplitude). Qualitative analysis of the dynamics of self-action of laser pulses, which is confirmed by numerical simulation, shows that nonlinear evolution of a 3D wave packet follows the scenario of self-focusing, which serves as the background on which the envelope profile turnover and the formation of a shock wave occur at an advanced rate.     

Self-action dynamics of ultrashort electromagnetic pulses

The self-action dynamics of three-dimensional wave packets whose width is on the order of the carrier frequency is studied under fairly general assumptions concerning the dispersion properties of the medium. The condition for the wave field collapse is determined. Self-action regimes in a dispersion-free medium and in media with predominance of anomalous or normal group velocity dispersions are numerically investigated. It is shown that, for extremely short pulses, nonlinearity leads not only to the self-compression of the wave field but also to a “turn-over” of the longitudinal profile. In a dispersionless medium, the formation of a shock front within the pulse leads to the nonlinear dissipation of linearly polarized radiation and to self-focusing stabilization. For circularly polarized radiation, the wave collapse is accompanied by the formation of an envelope shock wave.     

Self-focusing of laser radiation in cluster plasma

The self-focusing of laser radiation in plasma with ionized gaseous clusters is studied both analytically and numerically. An electrodynamic model is proposed for cluster plasma in a field of ultrashort laser pulse. The radiation self-action dynamics are studied using the equation for wave-field envelope with allowance for the electronic nonlinearity of the expanded plasma bunches and the group-velocity dispersion in a nanodispersive medium. It is shown that, for a laser power exceeding the self-focusing critical power, the wave-field self-compression occurs in a medium with dispersion of any type (normal, anomalous, or combined). Due to the strong dependence of the characteristic nonlinear field on the size of ionized cluster, the corresponding processes develop faster than in a homogeneous medium and give rise to the ultrashort pulses.     

Self-focusing of wave packets and envelope shock formation in nonlinear dispersive media

The self-action of three-dimensional wave packets is analyzed analytically and numerically under the conditions of competing diffraction, cubic nonlinearity, and nonlinear dispersion (dependence of group velocity on wave amplitude). A qualitative analysis of pulse evolution is performed by the moment method to find a sufficient condition for self-focusing. Self-action effects in an electromagnetically induced transparency medium (without cubic nonlinearity) are analyzed numerically. It is shown that the self-focusing of a wave packet is accompanied by self-steepening of the longitudinal profile and envelope shock formation. The possibility of envelope shock formation is also demonstrated for self-focusing wave packets propagating in a normally dispersive medium.     

Self-action of laser radiation under the conditions of electromagnetically induced transparency

The laser-radiation self-action dynamics in the electromagnetically induced transparency band were studied analytically and numerically for an atomic gas with the Λ-type energy-level scheme. The self-consistent system of equations describing the spatiotemporal evolution of a finite-amplitude wave packet in the field of a uniform pump wave was derived. The self-action in the presence of competing diffraction, nonlinear dispersion, and radiation absorption in the system was qualitatively analyzed; in particular, the conditions for self-focusing of a probe beam were determined. The results were confirmed by the numerical simulation of wave-packet spatiotemporal evolution.     

The structural features of wave collapse in medium with normal group velocity dispersion

The dynamic characteristics of self-action in three-dimensional wave packets described by the nonlinear Schrödinger equation with a hyperbolic space operator were studied analytically and numerically. The class of the initial wave field distributions for which self-focusing effects predominated over dispersion spreading and caused the arising of wave collapses was considered. The collapse of tubular wave packets was shown to be accompanied by packet shape changes during its contraction to the axis of the system. The nonlinear stabilization of collapses resulted in wave field fragmentation in the longitudinal direction followed by the expansion of the bunches thus formed along the axis. The dynamics of collapses was numerically studied taking into account medium nonlinearity saturation and nonlinear dissipation.     

Self-Action of Nonparaxial Few-Cycle Optical Waves in Dielectric Media

A mathematical model of evolution of the space–time spectra of nonparaxial few-cycle optical waves in isotropic dielectric media with an arbitrary dispersion of the refractive index and the inertialess third-order nonlinearity has been discussed. It has been shown that, at the self-focusing of a wave single-cycle at the input to a nonlinear medium into an optical filament with transverse dimensions comparable with the central radiation wavelength, the strength of the increased longitudinal component of its electric field can become larger than the initial longitudinal component by a factor of 7 and can be 18% of the transverse component of the input wave field. Errors of the calculations of the self-action of radiation with superwide time and spatial spectra within simplified mathematical models have been estimated.     

X and Y waves in the spatiotemporal Kerr dynamics of a self-guided light beam

The nonlinear stage of development of the spatiotemporal instability of the monochromatic Townes beam in a medium with self-focusing nonlinearity and normal dispersion is studied by analytical and numerical means. Small perturbations to the self-guided light beam are found to grow into two giant, splitting Y pulses featuring shock fronts on opposite sides. Each shocking pulse amplifies a co-propagating X wave, or dispersion- and diffraction-free linear wave mode of the medium, with super-broad spectrum.     

Dynamics of the spatial spectrum of a self-focusing optical wave in a nonlinear medium

A new nonlinear equation for the dynamics of the spatial spectrum of a self-focusing monochromatic wave in a medium with cubic nonlinearity is derived in the nonparaxial approximation. The formation of optical beams with cross section on the order of a wavelength is considered. Backward self-reflection is found to be the fundamental cause for the limitation of optical self-focusing     

Asymptotic theory of plane soliton self-focusing in two-dimensional wave media

An asymptotic method is developed to describe a long-term evolution of unstable quasi-plane solitary waves in the Kadomtsev-Petviashvili model for two-dimensional wave media with positive dispersion. An approximate equation is derived for the parameters of soliton transversal modulation and a general solution of this equation is found in an explicit form. It is shown that the development of periodic soliton modulation, in an unstable region, leads to saturation and formation of a two-dimensional stationary wave. This process is accompanied by the radiation of a small-amplitude plane soliton. In a stable region, an amplitude of the modulation is permanently decreasing due to radiation of quasi-harmonic wave packets. The multiperiodic regime of plane soliton self-focusing is also investigated.     

Modulation instability in systems of coupled waves with nonlinearity dispersion

Conditions for the appearance of modulation instability of an optical wave packet formed by one or two unidirectional strongly interacting waves are studied taking into account the influence of the temporal dispersion of a nonlinear response of the medium on this process. The regions of values of the frequency, input radiation intensity, and dispersion parameters in which the wave perturbations are increased are found. It is shown that the initial conditions for excitation of optical fibers of this type significantly affect the dynamics of the modulation instability.     

Lie-optics, geometrical phase and nonlinear dynamics of self-focusing and soliton evolution in a plasma

It is useful to state propagation laws for a self-focusing laser beam or a soliton in group-theoretical form to be called Lie-optical form for being able to predict self-focusing dynamics conveniently and amongst other things, the geometrical phase. It is shown that the propagation of the gaussian laser beam is governed by a rotation group in a non-absorbing medium and by the Lorentz group in an absorbing medium if the additional symmetry of paraxial propagation is imposed on the laser beam. This latter symmetry, however, needs care in its implementation because the electromagnetic wave of the laser sees a different refractive index profile than the laboratory observer in this approximation. It is explained how to estimate this non-Taylor paraxial power series approximation. The group theoretical laws so-stated are used to predict the geometrical or Berry phase of the laser beam by a technique developed by one of us elsewhere. The group-theoretical Lie-optic (or ABCD) laws are also useful in predicting the laser behavior in a more complex optical arrangement like in a laser cavity etc. The nonlinear dynamical consequences of these laws for long distance (or time) predictions are also dealt with. Ergodic dynamics of an ensemble of laser beams on the torus during absorptionless self-focusing is discussed in this context. From the point of view of new physics concepts, we introduce a stroboscopic invariant torus and a stroboscopic generating function in classical mechanics that is useful for long-distance predictions of absorptionless self-focusing.     

Formation of elliptically polarized ring-shaped electric field structures on the self-focusing of light in an isotropic medium featuring a spatially disperse nonlinearity

The possibility is demonstrated that the self-focusing of an elliptically polarized beam in a medium featuring a spatially disperse cubic nonlinearity can give rise to several radially symmetric ring-shaped regions of the same sense of rotation (right or left) of the electric-field vector in the cross-section of the beam. The formation dynamics and propagation features of such electric field structures are studied for various parameters of the incident radiation and nonlinear medium.     

Nonsteady-state self-interaction in a medium with striction nonlinearity

An analytic and numerical investigation is made of the self-focusing of a wave beam allowing for the inertia of the nonlinear response of the medium described by an acoustic type of equation. Some characteristics of the dynamics of self-interaction of the wave fields are analyzed in the paraxial optics approximation and the self-similar structures and space-time instability of a plane wave are considered. The stages of instability buildup, structure formation, and the establishment of a steady state are studied numerically.     

Peculiarities of the Self-Action of Inclined Wave Beams Incident on a Discrete System of Optical Fibers

Based on a discrete nonlinear Schrödinger equation (DNSE), we studied analytically and numerically the peculiarities of the self-action of one-dimensional quasi-optic wave beams injected into a spatially inhomogeneous medium consisting of a set of equidistant mutually coupled optical fibers. A variational approach allowing the prediction of the global evolution of localized fields with the initially plane phase front was developed. The self-consistent equations are obtained for the main parameters of such beams (the position of the center of mass, the effective width, and linear and quadratic phase-front corrections) in the aberrationless approximation. The case of radiation incident on a periodic system of nonlinear optical fibers at an angle to the axis oriented along them is analyzed in detail. It is shown that for the radiation power exceeding a critical value, the self-focusing of the wave field is observed, which is accompanied by the shift of the intensity maximum followed by the concentration of the main part of radiation only in one of the structural elements of the array under study. In this case, the beams propagate along paths considerably different from linear and the direction of their propagation changes compared to the initial direction. Asymptotic expressions are found that allow us to estimate the self-focusing length and to determine quite accurately the final position of a point with the maximum field amplitude after radiation trapping a channel. The results of the qualitative study of the possible self-channeling regimes for wave beams in a system of weakly coupled optical fibers in the aberrationless approximation are compared with the results of direct numerical simulations within the DNSE framework.     

Self-action of intense electromagnetic radiation in an electron-positron vacuum

An analysis is done of the effects of self-action of intense coherent electromagnetic radiation in an electron-positron vacuum that is in homogeneous electric and magnetic fields. A modified version of the Heisenberg-Euler theory, in which the Lagrangian incorporates terms with field derivatives, is used to take into account vacuum dispersion. The nonphysical branch of the solutions of the dispersion equation is excluded by a transition to a quasioptical equation for the slowly varying field envelope, an equation that describes the propagation of radiation with allowance for diffraction, spatial-temporal dispersion, and vacuum nonlinearity. The existence of dark solitons (with an intensity gap) in the vacuum is shown to be present. Finally, self-focusing of radiation in a vacuum is demonstrated and the critical self-focusing power is determined. Zh. éksp. Teor. Fiz. 113, 513–520 (February 1998)     

Wideband modulation instability at combination frequencies in wave beams

We study theoretically wideband modulation instability at combination frequencies in media having cubic nonlinearity of self-focusing type along with the higher-order defocusing nonlinearity. It is assumed that in a medium with a purely cubic nonlinearity, the medium dispersion does not permit modulation instability. In this case, a collapse of the wave field exists if the beam power is higher than the critical power of self-focusing. The higher-order nonlinearity limits the field at the nonlinear focus, and the instability at combination frequencies becomes possible. It turns out that the field at the nonlinear focus increases with increasing excess of the beam power over the critical power of self-focusing. The obtained values of the nonlinear dielectric permittivity are used for determination of the growth rates of instability at combination frequencies. These growth rates ensure an increase in the combination fields from noise levels up to values comparable with the field of the high-power beam. Such an increase takes place if the beam power is severalfold higher than the critical one. The developed theory can be used for explanation of spectrum superbroadening during self-focusing of sufficiently short laser pulses and high-harmonic generation. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 6, pp. 522–532, June 2007.     

All-optical limiting using nonlinear directional couplers composed of a self-focusing and a self-defocusing waveguide

We numerically demonstrate the all-optical limiting features in a mismatched nonlinear directional coupler (NLDC) composed of a self-focusing and a self-defocusing waveguide for both continuous wave and pulse cases. The working conditions required are analyzed. To obtain the limiting feature, the propagation constant of the self-focusing waveguide should not be larger than that of the self-defocusing waveguide. Cascaded asymmetric NLDCs are investigated to improve the limiting characteristics. The limiting threshold and the limiting output power can be adjusted by varying the coupler length or the ratio of the nonlinearity coefficients of the self-defocusing and self-focusing waveguides. Analytical solutions are presented in the case of a continuous wave. For the pulse case, numerical solutions show that the top part of the output pulse, if it exceeds the limiting threshold, will be tailored, while the rising and falling edges of the output pulse are almost the same as the input pulse. There is almost no pulse breakup. The influence of the second order dispersion and the intermodal dispersion on the limiting characteristics are analyzed. PACS 42.79.Gn; 42.79.Ta     

Multiple fragmentation of wave packets in a nonlinear medium with normal dispersion of the group velocity

The features of the dynamics of the self-ffects of the wave field are considered for the case when the dynamics is described by the nonlinear Schrodinger equation (NSE) with a hyperbolic spatial operator. An analytical investigation of the characteristic regimes of the self-effects is carried out; these regimes are due to the spatial competition of the self-focusing compression of the wave packet in one direction and its defocusing in another. The initial distributions of the wave field are analyzed with the goal of using them in a numerical modeling for illustration of the features of the self-effects. It is shown that three stages can be discerned in the evolution of the localized distributions: self-focusing filamentation of the wave field in the transverse direction, and compression and subsequent fragmentation in the defocusing (longitudinal) direction. The strongest non-uniformities are excited in the self-similar collapse of the wave field to hyperbolas (to the characteristic curves of the hyperbolic operator of the NSE). The stabilization of the development of the fragmentation instability is discussed separately.     

Catastrophic collapse of ultrashort pulses

We investigate theoretically the self-focusing dynamics of an ultrashort laser pulse both near and above the threshold at which the pulse effectively undergoes catastrophic collapse. We find that, as a result of space-time focusing and self-steepening, an "optical shock" wave forms inside the medium that gives rise to a broad blueshifted pedestal in the transmitted pulse spectrum. Our results are in good agreement with the primary features observed in experiments and thus provide a theoretical understanding for the underlying process that gives rise to "white-light" generation.