Publications (35)92.97 Total impact
 [Show abstract] [Hide abstract]
ABSTRACT: It seems to be selfevident that stable optical pulses cannot be considerably shorter than a single oscillation of the carrier field. From the mathematical point of view the solitary solutions of pulse propagation equations should loose stability or demonstrate some kind of singular behavior. Typically, an unphysical cusp develops at the soliton top, preventing the soliton from being too short. Consequently, the power spectrum of the limiting solution has a special behavior: the standard exponential decay is replaced by an algebraic one. We derive the shortest soliton and explicitly calculate its spectrum for the socalled short pulse equation. The latter applies to ultrashort solitons in transparent materials like fused silica that are relevant for optical fibers.  [Show abstract] [Hide abstract]
ABSTRACT: Rogue waves, by definition, are rare events of extreme amplitude. At the same time, they are surprisingly ubiquitous, in the sense that they can exist in a wide range of physical contexts and possess probability distributions that exhibit heavier tails than the normal Gaussian distribution. While many mechanisms have been demonstrated to explain the appearance of rogue waves in various specific systems, there is no known generic mechanism or general set of criteria shown to rule their appearance. Presupposing only the existence of a nonlinear Schrödingertype equation together with a concave dispersion profile around a zerodispersion wavelength, we demonstrate that solitons may experience acceleration and strong reshaping due to the interaction with continuum radiation, giving rise to extremevalue phenomena. The mechanism appears to be widely independent from interactions specific to the optical context, e.g., the Raman effect or other scattering processes that have no equivalent in other wavesupporting physical systems. In our system, a strong increase in the peak power may appear via reshaping while the pulse energy is nearly conserved. The conservative nature of the proposed reshapinginduced appearance of rogue waves makes this mechanism particularly robust.  [Show abstract] [Hide abstract]
ABSTRACT: We study the propagation of fewcycle optical solitons in nonlinear media with an anomalous, but otherwise arbitrary, dispersion and a cubic nonlinearity. Our approach does not derive from the slowly varying envelope approximation. The optical field is derived directly from Maxwell's equations under the assumption that generation of the third harmonic is a nonresonant process or at least cannot destroy the pulse prior to inevitable linear damping. The solitary wave solutions are obtained numerically up to nearly singlecycle duration using the spectral renormalization method originally developed for the envelope solitons. The theory explicitly distinguishes contributions between the essential physical effects such as higherorder dispersion, selfsteepening, and backscattering, as well as quantifies their influence on ultrashort optical solitons.  [Show abstract] [Hide abstract]
ABSTRACT: Rogue waves are solitary waves with extreme amplitudes, which appear to be a ubiquitous phenomenon in nonlinear wave propagation, with the requirement for a nonlinearity being their only unifying characteristics. While many mechanisms have been demonstrated to explain the appearance of rogue waves in a specific system, there is no known generic mechanism or general set of criteria shown to rule their appearance. Presupposing only the existence of a nonlinear Schr\"odingertype equation together with a concave dispersion profile around a zero dispersion wavelength we demonstrate that solitons may experience acceleration and strong reshaping due to the interaction with continuum radiation, giving rise to extremevalue phenomena. The mechanism is independent of the optical Raman effect. A strong increase of the peak power is accompanied by a mild increase of the pulse energy and carrier frequency, whereas the photon number of the soliton remains practically constant. This reshaping mechanism is particularly robust and may explain the appearance of rogue waves in a large class of systems.  [Show abstract] [Hide abstract]
ABSTRACT: We demonstrate that a generalized nonlinear Schrödinger equation (NSE), which includes dispersion of the intensitydependent group velocity, allows for exact solitary solutions. In the limit of a long pulse duration, these solutions naturally converge to a fundamental soliton of the standard NSE. In particular, the peak pulse intensity times squared pulse duration is constant. For short durations, this scaling gets violated and a cusp of the envelope may be formed. The limiting singular solution determines then the shortest possible pulse duration and the largest possible peak power. We obtain these parameters explicitly in terms of the parameters of the generalized NSE.  [Show abstract] [Hide abstract]
ABSTRACT: We demonstrate that ultrashort optical pulses propagating in a nonlinear dispersive medium are naturally described through incorporation of analytic signal for the electric field. To this end a secondorder nonlinear wave equation is first simplified using a unidirectional approximation. Then the analytic signal is introduced, and all nonresonant nonlinear terms are eliminated. The derived propagation equation accounts for arbitrary dispersion, resonant fourwave mixing processes, weak absorption, and arbitrary pulse duration. The model applies to the complex electric field and is independent of the slowly varying envelope approximation. Still the derived propagation equation posses universal structure of the generalized nonlinear Schrödinger equation (NSE). In particular, it can be solved numerically with only small changes of the standard splitstep solver or more complicated spectral algorithms for NSE. We present exemplary numerical solutions describing supercontinuum generation with an ultrashort optical pulse. 
Article: Manipulating light by light
[Show abstract] [Hide abstract]
ABSTRACT: We propose and verify a new concept of an alloptical transistor based on crossphase modulation between a signal pulse (SP) and a control pulse (CP). An effective interaction is achieved if the CP is temporally locked to the SP in an optical event horizon. This enables modification of carrier frequency, energy, and pulse duration of a SP by a considerably weaker CP.  [Show abstract] [Hide abstract]
ABSTRACT: We experimentally demonstrate a novel allopticaltransistor concept, allowing for cascadable switching of a stronger pulse with a significantly weaker control pulse. This concept employs crossphase modulation in the extended interactionzone of an optical event horizon.  [Show abstract] [Hide abstract]
ABSTRACT: A novel concept for an alloptical transistor is proposed and verified numerically. This concept relies on crossphase modulation between a signal and a control pulse. Other than previous approaches, the interaction length is extended by temporally locking control and the signal pulse in an optical event horizon, enabling continuous modification of the central wavelength, energy, and duration of a signal pulse by an up to sevenfold weaker control pulse. Moreover, if the signal pulse is a soliton it may maintain its solitonic properties during the switching process. The proposed alloptical switching concept fulfills all criteria for a useful optical transistor in [Nat. Photon. 4, 3 (2010)], in particular, fanout and cascadability, which have previously proven as the most difficult to meet. 
Article: Dissipative solitons
[Show abstract] [Hide abstract]
ABSTRACT: The present review summarizes experimental and theoretical work dealing with selforganized solitary localized structures (LSs) that are observed in spatially extended nonlinear dissipative systems otherwise exhibiting translational and rotational symmetry. Thereby we focus on those LSs that essentially behave like particles and that we call dissipative solitons (DSs). Such objects are also solutions of corresponding nonlinear evolution equations and it turns out that they are rather robust with respect to interaction with each other, with impurities, and with the boundary; alternatively they are generated or annihilated as a whole. By reviewing the experimental results it turns out that the richest variety of DS phenomena has been observed in electrical transport systems and optical devices. Nevertheless, DSs show up also in many other systems, among which nerve pulses in living beings are of uppermost importance in practice. In most of these systems DSs behave very similarly. The experimental results strongly suggest that phenomenon of DSs is universal. On the background of the experimental findings models for a theoretical understanding are discussed. It turns out that in a limited number of cases a straightforward quantitative description of DS patterns can be carried out. However, for the overwhelming number of systems only a qualitative approach has been successful so far. In the present review particular emphasis is laid on reactiondiffusion systems for which a kind of ‘normal form’ can be written down that defines a relatively large universality class comprising e.g. important electrical transport, chemical, and biological systems. For the other large class of DS carrying systems, namely optical devices, the variety of model equations is much larger and one is far away, even from a universal qualitative description. Because of this, and due to the existence of several extensive reviews on optical systems, their theoretical treatment has been mentioned only shortly. Finally, it is demonstrated that in terms of a singular perturbation approach the interaction of DSs and important aspects of their bifurcation behaviour, under certain conditions, can be described by rather simple equations. This is also true when deriving from the underlying field equations a set of ordinary differential equations containing the position coordinates of the individual DSs. Such equations represent a theoretical foundation of the experimentally observed particlelike behaviour of DSs. Though at present there is little real practical application of DSs and related patterns in an outlook we point out in which respects this might change in future. A systematic summary of a large amount of experimental and theoretical results on reactiondiffusion systems, being rather close to the subject of the present review, can also be found on the website http://www.unimuenster.de/Physik.AP/Purwins/ResearchSummary.  [Show abstract] [Hide abstract]
ABSTRACT: A Hamiltonian framework is developed for a sequence of ultrashort optical pulses propagating in a nonlinear dispersive medium. To this end a secondorder nonlinear wave equation for the electric field is transformed into a firstorder propagation equation for a suitably defined complex electric field. The Hamiltonian formulation is then introduced in terms of normal variables, i.e., classical complex fields referring to the quantum creation and annihilation operators. The derived zpropagated Hamiltonian accounts for forward and backward waves, arbitrary medium dispersion, and fourwave mixing processes. As a simple application we obtain integrals of motion for the pulse propagation. The integrals reflect timeaveraged fluxes of energy, momentum, and photons transferred by the pulse. Furthermore, pulses in the form of stationary nonlinear waves are considered. They yield extremal values of the momentum flux for a given energy flux. Simplified propagation equations are obtained by reduction of the Hamiltonian. In particular, the complex electric field reduces to an analytic signal for the unidirectional propagation. Solutions of the full bidirectional model are numerically compared to the predictions of the simplified equation for the analytic signal and to the socalled forward Maxwell equation. The numerics is effectively tested by examining the conservation laws.  [Show abstract] [Hide abstract]
ABSTRACT: Padé approximation is superior to Taylor expansion when functions contain poles. This is especially important for response functions in complex frequency domain, where singularities are present and intimately related to resonances and absorption. Therefore, we introduce a rational Padé approximant for the complex medium refractive index n(ω). The approximant is calculated using only local information of medium dispersion properties close to a carrier frequency ω0. In return it typically offers an accurate global representation of medium dispersion and absorption. Moreover, the fulfillment of the causality principle and the Kramers–Kronig relation can be established. In practice, our results are relevant if n(ω) is known only for ω≃ω0 whereas optical field is spectrally broad such that (i) the resonance absorption becomes important and (ii) a traditional polynomial dispersion operator diverges and induces huge errors. As an exemplary application we use the approximant to derive a nonlocal envelope model for ultrashort pulses. The model provides a natural bridge between the commonly used local envelope equations and the most general nonenvelope models operating directly with the electric field.  [Show abstract] [Hide abstract]
ABSTRACT: Propagation of short optical pulses in a onedimensional nonlinear medium is considered without the use of the slow envelope and unidirectional propagation approximations. The existence of uniformly moving solitary solutions is predicted for a Sellmeiertype dispersion function in the anomalous dispersion domain. A fourparametric family of such solutions is found that contains the classical envelope soliton in the limit of large pulse durations. In the opposite limit we get another family member, which, in contrast to the envelope soliton, strongly depends on the nonlinearity model and represents the shortest and the most intense pulse that can propagate in a stationary manner. 
Article: Formation and stabilisation of single current filaments inplanar dielectric barrier discharge
[Show abstract] [Hide abstract]
ABSTRACT: . In the experimental part we report on a typical bifurcation scenario of the current distribution in the discharge plane of a planar dielectric barrier discharge system. Increasing the amplitude of the sinusoidal driving voltage after breakdown a large number of dynamic solitary filaments is observed and the subsequent decrease of results in a pronounced hysteresis with decreasing number of filaments. In this way isolated single stationary filaments can be generated. In the theoretical part the latter are modeled by a reactiondriftdiffusion equation that is solved in three dimensional space numerically without any fitting procedure. As a result we obtain well defined stationary filaments of which size an shape essentially are independent of the initial conditions and having a width and an amplitude that agree with experiment rather well. On the basis of the numerical results we consider mechanisms of filament stabilisation. This includes the discussion of the well known surface charges as well as an additional focusing effect of volume charges.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the stability of the localized stationary solutions of a threecomponent reactiondiffusion system with one activator and two inhibitors. A change of the time constants of the inhibitors can lead to a destabilization of the stationary solution. The special case we are interested in is that the breathing mode becomes unstable first and the stationary dissipative soliton undergoes a bifurcation from a stationary to a "breathing" state. This situation is analyzed performing a twotimescale expansion in the vicinity of the bifurcation point thereby obtaining the corresponding amplitude equation. Also numerical simulations are carried out showing good agreement with the analytical predictions.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we present experimental results on dielectric barrier gasdischarge (DBD) systems in helium where the surface is prepared with some humidity. We observe welldefined solitary current filaments with selfpropelled motion. The trajectories of the filaments in the discharge plane resemble a random walk motion with memory. The mechanism leading to motion is attributed to the mutual interaction of gasdischarge and local humidity at the dielectric surface. The phenomenon is a new drift mechanism for filaments in DBD and may be important in applications being related to plasma surface treatment.  [Show abstract] [Hide abstract]
ABSTRACT: We report on pattern formation phenomena in the filamentary dielectric barrier discharge between plane glass electrodes. It is for the first time that a threedimensional (3D) selforganized glow pattern was both observed in an actual experiment and directly calculated in a full 3D discharge simulation in a quantitative manner. Specifically, we investigate the genesis of periodic patterns during the first breakdowns. Despite our simple driftdiffusion discharge model, the correspondence of experimental and numerical findings is surprisingly good.  [Show abstract] [Hide abstract]
ABSTRACT: Lagrangian variables are used to describe both surface and volume nonlinear oscillations of spatially bounded plasmas. Simple exact, or nonperturbative, solutions can often be obtained for complicated problems. Here, several recent analytical results on nonlinear standing waves in bounded systems are reviewed. These include temperature effects on nonlinear standing waves, oscillations of expanding multispecies plasma, and phase locking and transition to chaos. The analytical solutions are in good agreement with that from particleincell simulation. The exact solutions are also useful as starting point and verification for novel perturbation or numerical schemes, as well as stability analysis. They are of particular interest with respect to electromagnetically trapped nonneutral as well as freely expanding plasmas.  [Show abstract] [Hide abstract]
ABSTRACT: Electric breakdown and ionization fronts are considered theoretically in a sandwichlike dc discharge system consisting of two planeparallel electrodes and a gaseous gap in between. The key system feature is a highohmic cathode opposite to an ordinary metal anode. Such systems have received much attention from experimental studies because they naturally support current patterns. Using adiabatic description of electrons and twoscale expansion we demonstrate that in the lowcurrent Townsend mode the discharge is governed by a twocomponent reactiondiffusion system. The latter provides quantitative system description on the macroscopic time scale (i.e., much larger than the ion travel time). The breakdown appears as an instability of the uniform overvoltage state. A seed current fluctuation triggers a shocklike ionization front that propagates along the discharge plane with constant speed (typically approximately 10(4) cm/s). Depending on the cathode resistivity the front exhibits either monotonic or oscillatory behavior in space. Other breakdown features, such as damping transient oscillations of the global current, can also be found as solutions of the reactiondiffusion equations.  [Show abstract] [Hide abstract]
ABSTRACT: Pattern formation in a gas discharge operating on the lefthand branch of the Paschen curve (obstructed discharge) is investigated. Such a discharge (in contrast to that operated on the righthand branch) has a monotonic current–voltage characteristic and is believed to occupy the whole electrode area and to evolve continuously with current increase. This theoretical picture sharply contradicts a recent experiment (Nasuno S 2003 Chaos 13 1010–13) where both current constriction and current patterns have been observed for an obstructed dc discharge. We demonstrate experimentally in this paper that such a behaviour is indeed possible, but is always accompanied by current pulsation. We also suggest that the current pulsation, constriction and pattern formation are caused by local currentheating of the gas. This process explains the observed timescales, occurrence and multiplicity of current spots.
Publication Stats
576  Citations  
92.97  Total Impact Points  
Top Journals
 Physical Review E (8)
 Physical Review A (5)
 Plasma Physics Reports (3)
 Physics of Plasmas (2)
 Advances In Physics (1)
Institutions

20082014

Weierstrass Institute for Applied Analysis and Stochastics
Berlín, Berlin, Germany


19992006

Russian Academy of Sciences
 Division of Theoretical Physics
Moskva, Moscow, Russia
