Hybridizing traditional and next-generation reservoir computing to accurately and efficiently forecast dynamical systems.

Reservoir computers (RCs) are powerful machine learning architectures for time series prediction. Recently, next generation reservoir computers (NGRCs) have been introduced, offering distinct advantages over RCs, such as reduced computational expense and lower training data requirements. However, NGRCs have their own practical difficulties, including sensitivity to sampling time and type of nonlinearities in the data. Here, we introduce a hybrid RC-NGRC approach for time series forecasting of dynamical systems. We show that our hybrid approach can produce accurate short-term predictions and capture the long-term statistics of chaotic dynamical systems in situations where the RC and NGRC components alone are insufficient, e.g., due to constraints from limited computational resources, sub-optimal hyperparameters, sparsely sampled training data, etc. Under these conditions, we show for multiple model chaotic systems that the hybrid RC-NGRC method with a small reservoir can achieve prediction performance approaching that of a traditional RC with a much larger reservoir, illustrating that the hybrid approach can offer significant gains in computational efficiency over traditional RCs while simultaneously addressing some of the limitations of NGRCs. Our results suggest that the hybrid RC-NGRC approach may be particularly beneficial in cases when computational efficiency is a high priority and an NGRC alone is not adequate.

Positron Acceleration by Plasma Wakefields Driven by a Hollow Electron Beam.

A scheme for positron plasma wakefield acceleration using hollow or donut-shaped electron driver beams is studied. An annular-shaped, electron-free region forms around the hollow driver beam, creating a favorable region (longitudinal field is accelerating and transverse field is focusing) for positron acceleration. For Facility for Advanced Accelerator Experimental Tests (FACET)-like parameters, the hollow beam driver produces accelerating gradients on the order of 10 GV/m. The accelerating gradient increases linearly with the total charge in the driver beam. Simulations show acceleration of a 23-GeV positron beam to 35.4 GeV with a maximum energy spread of 0.4% and very small emittance over a plasma length of 140 cm is possible.

Absolute Instability near the Band Edge of Traveling-Wave Amplifiers.

Applying the Briggs-Bers "pole-pinch" criterion to the exact transcendental dispersion relation of a dielectric traveling wave tube (TWT), we find that there is no absolute instability regardless of the beam current. We extend this analysis to the circuit band edges of a linear beam TWT by approximating the circuit mode as a hyperbola in the frequency-wave-number (ω-k) plane and consider the weak coupling limit. For an operating mode whose group velocity is in the same direction as the beam mode, we find that the lower band edge is not subjected to absolute instability. At the upper band edge, we find a threshold beam current beyond which absolute instability is excited. The nonexistence of absolute instability in a linear beam TWT and the existence in a gyrotron TWT, both at the lower band edge, is contrasted. The general study given here is applicable to some contemporary TWTs such as metamaterial-based and advanced Smith-Purcell TWTs.

Direct measurement of the electron density of extended femtosecond laser pulse-induced filaments.

We present direct time- and space-resolved measurements of the electron density of femtosecond laser pulse-induced plasma filaments. The dominant nonlinearity responsible for extended atmospheric filaments is shown to be field-induced rotation of air molecules.

Interaction of an ultrashort laser pulse and relativistic electron beam in a corrugated plasma channel.

Copropagation of a laser pulse and a relativistic electron beam in a corrugated plasma channel has been proposed for the direct laser acceleration of electrons [Palastro, Phys. Rev. E 77, 036405 (2008)]. The corrugated plasma channel allows for the guiding of laser pulses composed of subluminal spatial harmonics. Phase matching between the electron beam and the spatial harmonics results in acceleration, but for high beam densities, the pulse energy can be rapidly depleted. This depletion may result in interaction times shorter than the waveguide length limited time or pulse length dephasing time. We present an analytic model and self-consistent simulations of the electron beam-laser pulse interaction. A linear dispersion relation is derived. The effect of the electron beam on the pulse after the occurrence of axial bunching is examined. Injection of axially modulated electron beams is also explored. In particular, we find that a properly phased electron beam can transfer energy to the laser pulse as an inverse process to acceleration.

Exact results for the Kuramoto model with a bimodal frequency distribution.

We analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed. The dynamics of this system has been the subject of long-standing interest. In 1984 Kuramoto proposed several conjectures about its behavior; ten years later, Crawford obtained the first analytical results by means of a local center manifold calculation. Nevertheless, many questions have remained open, especially about the possibility of global bifurcations. Here we derive the system's stability diagram for the special case where the bimodal distribution consists of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott and Antonsen, we show that in this case the infinite-dimensional problem reduces exactly to a flow in four dimensions. Depending on the parameters and initial conditions, the long-term dynamics evolves to one of three states: incoherence, where all the oscillators are desynchronized; partial synchrony, where a macroscopic group of phase-locked oscillators coexists with a sea of desynchronized ones; and a standing wave state, where two counter-rotating groups of phase-locked oscillators emerge. Analytical results are presented for the bifurcation boundaries between these states. Similar results are also obtained for the case in which the bimodal distribution is given by the sum of two Gaussians.

Pulse propagation and electron acceleration in a corrugated plasma channel.

A preformed plasma channel provides a guiding structure for laser pulses unbound by the intensity thresholds of standard waveguides. The recently realized corrugated plasma channel [Layer, Phys. Rev. Lett. 99, 035001 (2007)] allows for the guiding of laser pulses with subluminal spatial harmonics. These spatial harmonics can be phase matched to high energy electrons, making the corrugated plasma channel ideal for the acceleration of electrons. We present a simple analytic model of pulse propagation in a corrugated plasma channel and examine the laser-electron beam interaction. Simulations show accelerating gradients of several hundred MeV/cm for laser powers much lower than required by standard laser wakefield schemes.

External periodic driving of large systems of globally coupled phase oscillators.

Large systems of coupled oscillators subjected to a periodic external drive occur in many situations in physics and biology. Here the simple paradigmatic case of equal strength, all-to-all sine coupling of phase oscillators subject to a sinusoidal external drive, is considered. The stationary states and their stability are determined. Using the stability information and numerical experiments, parameter space phase diagrams showing when different types of system behavior apply are constructed, and the bifurcations marking transitions between different types of behavior are delineated. The analysis is supported by results of direct numerical simulation of an ensemble of oscillators.

Effect of pulse duration on resonant heating of laser-irradiated argon and deuterium clusters.

We study the effect of pulse duration on the heating of single van der Waals bound argon and deuterium clusters by a strong laser field using a two-dimensional (2D) electrostatic particle-in-cell (PIC) code in the range of laser-cluster parameters such that kinetic as well as hydrodynamic effects are active. Heating is dominated by a collisionless resonant absorption process that involves energetic electrons transiting through the cluster. A size-dependent intensity threshold defines the onset of this resonance [T. Taguchi, Physical Review Letters, 92, 20 (2004)]. It is seen that increasing the laser pulse duration lowers this intensity threshold and the energetic electrons take multiple laser periods to transit the cluster instead of one laser period. Our simulations also show that strong electron heating is accompanied by the generation of a high-energy peak in the ion energy distribution function. We also calculate the yield of thermonuclear fusion neutrons from exploding deuterium clusters using the PIC model with periodic boundary conditions that allows for the interaction of ions from neighboring clusters.

Wave interaction in relativistic harmonic gyro-traveling-wave devices.

In gyro-traveling-wave devices, several waves can be excited at different cyclotron harmonics simultaneously. This paper analyzes the interaction between three waves synchronous with gyrating electrons at different cyclotron harmonics in two relativistic gyro-amplifier configurations; viz., gyro-traveling-wave tubes and gyrotwystrons. Two types of nonlinear interactions are considered: (a) excitation of two waves at cyclotron harmonics by a wave excited at the fundamental resonance, and (b) excitation of a wave at the fundamental resonance and another wave at the third harmonic by a wave excited at the second cyclotron harmonic. The effect of the overlapping of electron cyclotron resonances on the performance of relativistic gyrodevices is investigated as well.

Parametric instability in the formation of plasma waveguides.

Plasma waveguides generated by focusing a moderate intensity laser into neutral gas with an axicon lens can be unstable to the generation of axial modulations in the channel parameters. A model is proposed in which the modulations are due to the nonlinear coupling between the axicon field and a scattered mode in the evolving channel. Good agreement is found with experimental measurements of these modulations.

Propagation of intense short laser pulses in a gas of atomic clusters.

We present a model and numerical simulations for the propagation of intense short laser pulses in gases of atomic clusters. As the pulse propagates through the clusters, they absorb energy, expand and explode. The clustered gas thus acts as a medium with time dependent effective dielectric constant. A self-consistent model for the cluster expansion and the laser pulse propagation is developed. Self-focusing of the laser pulse, coupling of laser energy to clusters and the evolution of the pulse spectrum are studied for a laser-cluster system with typical laboratory parameters.

Blowout bifurcations and the onset of magnetic activity in turbulent dynamos.

The transition to magnetic-field self-generation in a turbulent, electrically conducting fluid is shown to exhibit intermittent bursting characterized by distinct scaling laws. This behavior is predicted on the basis of prior analysis of a type of bifurcation (called a blowout bifurcation) occurring in chaotic systems with an invariant phase space submanifold. The predicted scalings are shown to be consistent with numerical solutions of the governing magnetohydrodynamic equations, and implications for recently implemented experimental programs are discussed.

Compressing and focusing a short laser pulse by a thin plasma lens.

We consider the possibility of using a thin plasma slab as an optical element to both focus and compress an intense laser pulse. By thin we mean that the focal length is larger than the lens thickness. We derive analytic formulas for the spot size and pulse length evolution of a short laser pulse propagating through a thin uniform plasma lens. The formulas are compared to simulation results from two types of particle-in-cell code. The simulations give a greater final spot size and a shorter focal length than the analytic formulas. The difference arises from spherical aberrations in the lens which lead to the generation of higher-order vacuum Gaussian modes. The simulations also show that Raman side scattering can develop. A thin lens experiment could provide unequivocal evidence of relativistic self-focusing.

Resonant self-trapping of high intensity Bessel beams in underdense plasmas.

We present a comprehensive report based on recent work [Phys. Rev. Lett. 84, 3085 (2000)] on resonant self-trapping and enhanced absorption of high power Bessel beams in underdense plasmas. The trapping resonance is strongly dependent on initial gas pressure, Bessel-beam geometry, and laser wavelength. Analytic estimates, and simulations using a one-dimensional Bessel-beam-plasma interaction code consistently explain the experimental observations. These results are for longer, moderate intensity pulses where the self-trapping channel is induced by laser-heated plasma thermal pressure. To explore the extension of this effect to ultrashort, intense pulsed Bessel beams, we perform propagation simulations using the code WAKE [Phys. Rev. E 53, R2068 (1996)]. We find that self-trapping can occur as a result of a plasma refractive index channel induced by the combined effects of relativistic motion of electrons and their ponderomotive expulsion.

Wakefield generation and GeV acceleration in tapered plasma channels.

To achieve multi-GeV electron energies in the laser wakefield accelerator (LWFA), it is necessary to propagate an intense laser pulse long distances in a plasma without disruption. One of the purposes of this paper is to evaluate the stability properties of intense laser pulses propagating extended distances (many tens of Rayleigh ranges) in plasma channels. A three-dimensional envelope equation for the laser field is derived that includes nonparaxial effects such as group velocity dispersion, as well as wakefield and relativistic nonlinearities. It is shown that in the broad beam, short pulse limit the nonlinear terms in the wave equation that lead to Raman and modulation instabilities cancel. This cancellation can result in pulse propagation over extended distances, limited only by dispersion. Since relativistic focusing is not effective for short pulses, the plasma channel provides the guiding necessary for long distance propagation. Long pulses (greater than several plasma wavelengths), on the other hand, experience substantial modification due to Raman and modulation instabilities. For both short and long pulses the seed for instability growth is inherently determined by the pulse shape and not by background noise. These results would indicate that the self-modulated LWFA is not the optimal configuration for achieving high energies. The standard LWFA, although having smaller accelerating fields, can provide acceleration for longer distances. It is shown that by increasing the plasma density as a function of distance, the phase velocity of the accelerating field behind the laser pulse can be made equal to the speed of light. Thus electron dephasing in the accelerating wakefield can be avoided and energy gain increased by spatially tapering the plasma channel. Depending on the tapering gradient, this luminous wakefield phase velocity is obtained several plasma wavelengths behind the laser pulse. Simulations of laser pulses propagating in a tapered plasma channel are presented. Experimental techniques for generating a tapered density in a capillary discharge are described and an example of a GeV channel guided standard LWFA is presented.

Traveling-wave tubes and backward-wave oscillators with weak external magnetic fields.

Recent development of plasma-assisted slow-wave oscillators [Goebel et al. IEEE Trans. Plasma Sci. 22, 547 (1994)], microwave sources that operate without guiding magnetic fields, has stimulated interest in the theoretical analysis of such tubes. In principle, in the absence of guiding magnetic fields, due to the space charge forces and the radial electric field of the wave, the electrons may propagate radially outward which increases electron coupling to the slow wave whose field is localized near the slow-wave structure (SWS). This increases the wave growth rate and efficiency, and hence allows one to shorten the interaction region. So the radial electron motion can be beneficial for operation if it does not lead to interception of electrons by the SWS. To avoid this interception a weak external magnetic field can be applied. The theory developed describes the effect of weak magnetic fields on the operation of traveling-wave tubes and backward-wave oscillators with electrons moving not only axially but also transversely. This theory allows one to estimate the magnetic field required for protecting the SWS from electron bombardment at different power levels. Theoretical predictions of the efficiency enhancement due to the weak magnetic field are confirmed in experiments.

Raman scattering of intense, short laser pulses in modulated plasmas.

We examine the exponentiation of the Raman forward scattering instability in modulated plasma channels computationally and analytically. An evolution equation for the complex phases of the Raman scattered waves treating the spatial localization and discrete nature of the channel modes is derived. Simulations with WAKE [P. Mora and T. M. Antonsen Jr., Phys. Plasmas 4, 217 (1997)] verify the theory in the linear growth regime and provide insight into the nonlinear stage of the instability when cascading and pump depletion play a role. We find that the exponentiation in modulated channels depends on two factors: the increase in coupling due to the increased plasma wavenumber in the high-density regions of the channel and a decreased coupling due to the reduced longitudinal spatial coherence. For the parameters considered, simulations show that the finite extent of the pump pulse is more significant in determining the exponentiation than phase mixing due to the transverse variation of the channel. Both the theory and simulations confirm that modulated channels allow for the stable guiding of longer pulses than nonmodulated channels.

Influence of the Weibel instability on the expansion of a plasma slab into a vacuum.

The development of the Weibel instability during the expansion of a thin plasma foil heated by an intense laser pulse is investigated, using both analytical models and relativistic particle-in-cell simulations. When the plasma has initially an anisotropic electron distribution, this electromagnetic instability develops from the beginning of the expansion. Then it contributes to suppress the anisotropy and eventually saturates. After the saturation, the strength of the magnetic field decreases because of the plasma expansion until it becomes too weak to maintain the distribution isotropic. For this time, the anisotropy rises as electrons give progressively their longitudinal energy to ions, so that a new instability can develop.

Effect of short ray trajectories on the scattering statistics of wave chaotic systems.

In many situations, the statistical properties of wave systems with chaotic classical limits are well described by random matrix theory. However, applications of random matrix theory to scattering problems require introduction of system-specific information into the statistical model, such as the introduction of the average scattering matrix in the Poisson kernel. Here, it is shown that the average impedance matrix, which also characterizes the system-specific properties, can be expressed in terms of classical trajectories that travel between ports and thus can be calculated semiclassically. Theoretical results are compared with numerical solutions for a model wave chaotic system.