Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases

A two-dimensional axisymmetric model of the propagation of intense femtosecond laser pulses through dispersion-free transparent media is described. The effects of diffraction, nonlinear Kerr effect (instantaneous and retarded) and multiphoton ionisation are included. Numerical results concerning air and other gases are discussed. In particular, time self-compression of femtosecond pulses is predicted. Stable self-guided pulses are simulated, in agreement with recent experimental observations. Received: 19 June 1998 / Received in final form: 14 January 1999     

Travelling pulses in reaction-diffusion systems under global constraints

The influence of global inhibition on travelling pulses both in infinite and in finite periodic systems is investigated. Analytic investigation of the modified piecewise linear Rinzel-Keller-Model (RKM) shows a shrinking and slowing down of pulses due to global inhibition. Analytic results obtained are confirmed by numerical simulations. The investigation is completed by numerical simulations of the modified Oregonator model for the light-sensitive BZR. Received: 25 September 1997 / Revised: 9 January 1998 / Accepted: 9 January 1998     

Solitary waves in the Madelung's fluid: Connection between the nonlinear Schrödinger equation and the Korteweg-de Vries equation

An investigation to deepen the connection between the family of nonlinear Schr?dinger equations and the one of Korteweg-de Vries equations is carried out within the context of the Madelung's fluid picture. In particular, under suitable hypothesis for the current velocity, it is proven that the cubic nonlinear Schr?dinger equation, whose solution is a complex wave function, can be put in correspondence with the standard Korteweg-de Vries equation, is such a way that the soliton solutions of the latter are the squared modulus of the envelope soliton solution of the former. Under suitable physical hypothesis for the current velocity, this correspondence allows us to find envelope soliton solutions of the cubic nonlinear Schr?dinger equation, starting from the soliton solutions of the associated Korteweg-de Vries equation. In particular, in the case of constant current velocities, the solitary waves have the amplitude independent of the envelope velocity (which coincides with the constant current velocity). They are bright or dark envelope solitons and have a phase linearly depending both on space and on time coordinates. In the case of an arbitrarily large stationary-profile perturbation of the current velocity, envelope solitons are grey or dark and they relate the velocity u₀ with the amplitude; in fact, they exist for a limited range of velocities and have a phase nonlinearly depending on the combined variable x-u_{0 s} (s being a time-like variable). This novel method in solving the nonlinear Schr?dinger equation starting from the Korteweg-de Vries equation give new insights and represents an alternative key of reading of the dark/grey envelope solitons based on the fluid language. Moreover, a comparison between the solutions found in the present paper and the ones already known in literature is also presented. Received 20 February 2002 and Received in final form 22 April 2002 Published online 6 June 2002     

Invariable-profile wavefunctions and Brittingham's focus wave modes

A family of invariable-profile wavefunctions is constructed. The relations found describe both transient and steady-state waves. The Gauss and Bessel-Gauss focused waves of order m can be obtained from these steady-state waves via Bateman's transformation. Received: 14 May 1998 / Accepted: 22 August 1998     

Nonlinear propagation of ultra-low-frequency electromagnetic modes in a magnetized dusty plasma

A theoretical investigation has been made of nonlinear propagation of ultra-low-frequency electromagnetic waves in a magnetized two fluid (negatively charged dust and positively charged ion fluids) dusty plasma. These are modified Alfvén waves for small value of and are modified magnetosonic waves for large , where is the angle between the directions of the external magnetic field and the wave propagation. A nonlinear evolution equation for the wave magnetic field, which is known as Korteweg de Vries (K-dV) equation and which admits a stationary solitary wave solution, is derived by the reductive perturbation method. The effects of external magnetic field and dust characteristics on the amplitude and the width of these solitary structures are examined. The implications of these results to some space and astrophysical plasma systems, especially to planetary ring-systems, are briefly mentioned. Received 8 July 1999 and Received in final form 11 October 1999     

Finite-amplitude electrostatic waves im magneto-active plasmas

Summary Weakly nonlinear dispersive longitudinal waves in an infinite homogeneous collisionless plasma in the presence of an external constant and uniform magnetic field are considered. Under specific physical assumptions and for an arbitrary three-dimensional envelope modulation of a plane wave, a purely differential system is derived. Taking into account the effect of wave-wave and wave-particle interaction, the evolution of the modulation is described by a modified nonlinear Schr?dinger equation, coupled to the space perturbation charge densities. The generation of a static mode is described. As a particular case the electron waves are discussed and some special solutions, resorting to the theory of the perturbed solitions.     

Rarefactive ion-acoustic electrostatic solitary structures in nonthermal plasmas

An investigation has been made of ion-acoustic solitary waves in an unmagnetized nonthermal plasma whose constituents are an inertial ion fluid and nonthermally distributed electrons. The properties of stationary solitary structures are briefly studied by the pseudo-potential approach, which is valid for arbitrary amplitude waves, and by the reductive perturbation method which is valid for small but finite amplitude limit. The time evolution of both compressive and rarefactive solitary waves, which are found to coexist in this nonthermal plasma model, is also examined by solving numerically the full set of fluid equations. The temporal behaviour of positive (compressive) solitary waves is found to be typical, i.e., the positive initial disturbance breaks up into a series of solitary waves with the largest in front. However, the behaviour of negative (rarefactive) solitary waves is quite different. These waves appear to be unstable and produce positive solitary waves at a later time. The relevancy of this investigation to observations in the magnetosphere of density depressions is briefly pointed out. Received 12 October 1999     

Nonlinear interactions between drift waves and zonal flows

Phase coherent interactions between drift waves and zonal flows are considered. For this purpose, mode coupling equations are derived by using a two-fluid model and the guiding center drifts. The equations are then Fourier analyzed to deduce the nonlinear dispersion relations. The latter depict the excitation of zonal flows due to the ponderomotive forces of drift waves. The flute-like zonal flows with insignificant density fluctuations have faster growth rates than those which have a finite wavelength along the magnetic field direction. The relevance of our investigation to drift wave driven zonal flows in computer simulations and laboratory plasmas is discussed. Received 5 April 2002 Published online 28 June 2002     

Electromagnetically induced transparency in a non-uniform plasma

Electromagnetically induced transparency in a non-uniform plasma is investigated. We discuss the possible frequency ranges in which a weak beam below cutoff can propagate transparently in this plasma with the presence of a strong high frequency field. We also analyze the effect of driving intensity on the frequency range of the probe. Received 3 March 2000 and Received in final form 21 April 2000     

Generation of zonal flows by interchange modes in a plasma

The generation of zonal flows by flute-like interchange modes in a nonuniform magnetoplasma is considered. The guiding center particle drifts are then used to derive a system of coupled mode equations. The latter are Fourier analyzed to obtain a nonlinear dispersion relation, which exhibits the excitation of zonal flows by the ponderomotive force of the interchange modes. The growth rate of the parametrically driven zonal flows is obtained. Received 26 July 2002 Published online 24 September 2002 RID="a" ID="a"e-mail: ps@tp4.ruhr-uni-bochum.de     

Properties of pulsed entangled two-photon fields

The dependence of one- and two-photon characteristics of pulsed entangled two-photon fields generated in spontaneous parametric down-conversion on the pump-pulse properties (shape of the pump-pulse spectrum and its internal structure) is examined. It is shown that entangled two-photon fields with defined properties can be generated. A general relation between the spectra of the down-converted fields is established. As a special case the interference of two partially overlapping pulsed two-photon fields is studied. The fourth-order interference pattern of entangled two-photon fields is investigated in the polarization analog of the Hong-Ou-Mandel interferometer. Received: 23 December 1998 / Received in final form: 14 April 1999     

Convective amplification of parametrically generated acoustic wave in a piezoelectric semiconductor plasma

Using hydrodynamic model of semiconductor plasmas and coupled-mode theory of interacting waves, we have analytically investigated parametric interaction in a magnetised piezoelectric semiconductor plasma in non-relativistic domain. The temperature dependence of momentum transfer collision frequency of electrons due to their heating by the pump is assumed to induce nonlinearity in the medium. We have derived a dispersion relation which finally gives four unstable acoustic modes; two forward amplifying modes and two backscattered attenuating modes. We have also obtained an expression for the critical pump amplitude ( ) at and around which gains and phase velocities of amplifying acoustic modes become least dependent on the pump amplitude and static magnetic field . The required can be readily obtained from a pulsed 10.6 μm CO₂ laser. The magnetic field is found to shift the critical point towards lower pump amplitudes. Received 5 September 2000 and Received in final form 5 March 2001     

Scaling of hysteresis in a multi-dimensional all-optical bistable system

We analyse the hysteresis enlargements of an optical bistable system involving three dynamical variables. We investigate, both experimentally and numerically, the local dynamics of the up- and down-switching process versus the sweeping frequency of the control parameter. In particular, we delineate the domain of validity of the scaling law predicted for one-dimensional systems. At high sweeping frequency, we show the appearance of another asymptotic scaling low in . Thereafter, we analyse the global evolution of the hysteresis loop induced by these processes. At low frequency, a scaling law is retrieved, whereas at high frequency, the dynamical behaviour is shown to strongly depend on the particular shape of the bistability curve. Received: 14 September 1998 / Received in final form: 15 February 1999     

Ionization waves: from stability to chaos and turbulence

The spatio-temporal dynamics of self-excited ionization waves in a neon glow discharge is experimentally investigated. Various mechanisms leading to ionization waves chaos and turbulence are identified: subharmonics cascade, Ruelle-Takens-Newhouse scenario, and spatio-temporal intermittency. The dynamical states involved in the transition scenarios from stability to chaotic regimes are characterized through both temporal and spatio-temporal analysis by means of the Biorthogonal Decomposition (BD). Received 25 October 2001     

Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation

There has been considerable interest in the study on the variable-coefficient nonlinear evolution equations in recent years, since they can describe the real situations in many fields of physical and engineering sciences. In this paper, a generalized variable-coefficient KdV (GvcKdV) equation with the external-force and perturbed/dissipative terms is investigated, which can describe the various real situations, including large-amplitude internal waves, blood vessels, Bose-Einstein condensates, rods and positons. The Painlevé analysis leads to the explicit constraint on the variable coefficients for such a equation to pass the Painlevé test. An auto-B?cklund transformation is provided by use of the truncated Painlevé expansion and symbolic computation. Via the given auto-B?cklund transformation, three families of analytic solutions are obtained, including the solitonic and periodic solutions.     

Envelope solitons induced by high-order effects of light-plasma interaction

The nonlinear coupling between light beams and non-resonant ion density perturbations in a plasma is considered, taking into account the relativistic particle mass increase and the light beam ponderomotive force. A pair of equations comprising a nonlinear Schr?dinger equation for light beams and a driven (by the light beam pressure) ion-acoustic wave response is derived. It is shown that the stationary solutions of our nonlinear equations can be represented in the form of a bright and dark/gray soliton for the one-dimensional problem. We also present numerical results which exhibit that our bright soliton solutions are stable exclusively for the values of the parameters compatible with our theory. Received 24 July 2002 Published online 31 October 2002 RID="a" ID="a"Permanent address: Dipartimento di Scienze Fisiche, Universitá Federico II and INFN, Complesso Universitario di M.S. Angelo, Via Cintia, 80126 Napoli, Italy e-mail: renato.fedele@na.infn.it RID="b" ID="b"Permanent address: Dipartimento di Fisica Generale, Universitá di Torino, Via Pietro Giuria 1, 10125 Torino, Italy RID="c" ID="c"Permanent address: Institute of Physics, Georgian Academy of Sciences, Tbilisi 380077, Georgia     

Modelling of Lévy walk kinetics of charged particles in edge electrostatic turbulence in tokamaks

We model and discuss the possible types of motion that charged particles may undergo in a stationary and spatially periodic electrostatic potential and a homogeneous magnetic field. The model is considered to be the simplest approximation of more complex phenomena of plasma edge turbulence in tokamaks. Therein, low frequency turbulence appears in the plasma edge, resulting in a fluctuation of the electron density, and also in the generation of a turbulent electrostatic field. Typical parameters of this turbulent electrostatic field are an electrical potential amplitude of 10–100 V and wave numbers k≈10³ m^-1. In our model, we consider these regimes, together with a homogeneous magnetic field with a magnitude of 1 T. We investigate the dynamics of singly-ionized carbon ions – a typical plasma impurity – with kinetic energies on the order of 10 eV. Besides the obvious Larmor and drift motions, a motion of random-walk and of Lévy walk character appear therein. All of these types of motion can play an important role in the modelling of the anomalous diffusion of particles from the plasma edge turbulence region. The dynamics mentioned will cause an inevitable escape of energetic particles and thus of power loss from the thermonuclear reactor. Moreover, Lévy walk kinetics represents a very interesting kind of kinetics, currently of great interest, which was previously not so often discussed.     

Far-field fluorescence microscopy with repetitive excitation

We introduce the concept of repeatedly exciting an excited state of a photostable fluorescent entity to generate a nonlinear fluorescence signal which is solely based on the linear susceptibility of the molecule. The excitation cycle between the fluorescent state and another state prolongs the average lifetime of , with emphasis on those molecules that are in the center of the focus. The photons emitted by the long-lived molecules in the center are recorded by a temporal filter and constitute fluorescence that depends nonlinearly on the excitation intensity. Theoretical analysis reveals that this concept can provide three-dimensional imaging and improve the spatial resolution in far-field fluorescence microscopy. We show that despite the presence of diffraction the effective focal waist can in principle be narrowed down to the molecular scale at the expense of signal. Received: 3 December 1998     

Modulational instability in asymmetric coupled wave functions

The evolution of the amplitude of two nonlinearly interacting waves is considered, via a set of coupled nonlinear Schr?dinger-type equations. The dynamical profile is determined by the wave dispersion laws (i.e. the group velocities and the group velocity dispersion terms) and the nonlinearity and coupling coefficients, on which no assumption is made. A generalized dispersion relation is obtained, relating the frequency and wave-number of a small perturbation around a coupled monochromatic (Stokes') wave solution. Explicitly stability criteria are obtained. The analysis reveals a number of possibilities. Two (individually) stable systems may be destabilized due to coupling. Unstable systems may, when coupled, present an enhanced instability growth rate, for an extended wave number range of values. Distinct unstable wavenumber windows may arise simultaneously.     

Acousto-optic modulation in magnetised diffusive semiconductors

The modulation of an intense electromagnetic beam induced by the acousto-optic (AO) effect has been analysed in a transversely magnetised semiconductor-plasma medium. The effect of carrier diffusion on the threshold field and gain profile of the modulated wave has been extremely investigated using coupled mode theory. The origin of the AO interaction is assumed to lie in the induced nonlinear diffusion current density of the medium. By considering the modulation process as a four wave parametric interaction an expression for effective third-order AO susceptibility describing the phenomena has been deduced. The modulation is greatly modified by propagation characteristics such as dispersion and diffraction due to dielectric relaxation of the acoustic mode. The threshold pump field and the steady state growth rates are estimated from the effective third-order polarisation in the plasma medium. Analytical estimation reveals that in the presence of enhanced diffusion due to excess charge carriers the modulated beam can be effectively amplified in a dispersionless acoustic wave regime. The presence of an external dc magnetic field is found to be favourable for the onset of diffusion induced modulational amplification of the modulated wave in heavily doped regime. Received 5 November 2001