Exactly unitary discrete representations of the metaplectic transform for linear-time algorithms.

The metaplectic transform (MT), a generalization of the Fourier transform sometimes called the linear canonical transform, is a tool used ubiquitously in modern optics, for example, when calculating the transformations of light beams in paraxial optical systems. The MT is also an essential ingredient of the geometrical-optics modeling of caustics that we recently proposed. In particular, this application relies on the near-identity MT (NIMT); however, the NIMT approximation used so far is not exactly unitary and leads to numerical instability. Here, we develop a discrete MT that is exactly unitary, and approximate it to obtain a discrete NIMT that is also unitary and can be computed in linear time. We prove that the discrete NIMT converges to the discrete MT when iterated, thereby allowing the NIMT to compute MTs that are not necessarily near-identity. We then demonstrate the new algorithms with a series of examples.

Theory of the Tertiary Instability and the Dimits Shift from Reduced Drift-Wave Models.

Tertiary modes in electrostatic drift-wave turbulence are localized near extrema of the zonal velocity U(x) with respect to the radial coordinate x. We argue that these modes can be described as quantum harmonic oscillators with complex frequencies, so their spectrum can be readily calculated. The corresponding growth rate γ_{TI} is derived within the modified Hasegawa-Wakatani model. We show that γ_{TI} equals the primary-instability growth rate plus a term that depends on the local U^{''}; hence, the instability threshold is shifted compared to that in homogeneous turbulence. This provides a generic explanation of the well-known yet elusive Dimits shift, which we find explicitly in the Terry-Horton limit. Linearly unstable tertiary modes either saturate due to the evolution of the zonal density or generate radially propagating structures when the shear |U^{'}| is sufficiently weakened by viscosity. The Dimits regime ends when such structures are generated continuously.

Pseudo-differential representation of the metaplectic transform and its application to fast algorithms.

The metaplectic transform (MT), also known as the linear canonical transform, is a unitary integral mapping that is widely used in signal processing and can be viewed as a generalization of the Fourier transform. For a given function $ \psi $ψ on an $ {N} $N-dimensional continuous space $ {\textbf q} $q, the MT of $ \psi $ψ is parameterized by a rotation (or more generally, a linear symplectic transformation) of the $ 2{N} $2N-dimensional phase space $ ({\textbf q},{\textbf p}) $(q,p), where $ {\textbf p} $p is the wavevector space dual to $ {\textbf q} $q. Here, we derive a pseudo-differential form of the MT. For small-angle rotations, or near-identity transformations of the phase space, it readily yields asymptotic differential representations of the MT, which are easy to compute numerically. Rotations by larger angles are implemented as successive applications of $ {K} \gg 1 $K≫1 small-angle MTs. The algorithm complexity scales as $ {O}({K}{{N}^3}{{N}_p}) $O(KN3Np), where $ {{N}_p} $Np is the number of grid points. Here, we present a numerical implementation of this algorithm and discuss how to mitigate the associated numerical instabilities.

Ladder Climbing and Autoresonant Acceleration of Plasma Waves.

When the background density in a bounded plasma is modulated in time, discrete modes become coupled. Interestingly, for appropriately chosen modulations, the average plasmon energy might be made to grow in a ladderlike manner, achieving upconversion or downconversion of the plasmon energy. This reversible process is identified as a classical analog of the effect known as quantum ladder climbing, so that the efficiency and the rate of this process can be written immediately by analogy to a quantum particle in a box. In the limit of a densely spaced spectrum, ladder climbing transforms into continuous autoresonance; plasmons may then be manipulated by chirped background modulations much like electrons are autoresonantly manipulated by chirped fields. By formulating the wave dynamics within a universal Lagrangian framework, similar ladder climbing and autoresonance effects are predicted to be achievable with general linear waves in both plasma and other media.

Nonlinear amplification and decay of phase-mixed waves in compressing plasma.

Through particle-in-cell simulations, we show that plasma waves carrying trapped electrons can be amplified manyfold via compressing plasma perpendicularly to the wave vector. These simulations are the first ab initio demonstration of the conservation of nonlinear action for such waves, which contains a term independent of the field amplitude. In agreement with the theory, the maximum of amplification gain is determined by the total initial energy of the trapped-particle average motion but otherwise is insensitive to the particle distribution. Further compression destroys the wave; electrons are then untrapped at suprathermal energies and form a residual beam. As compression continues, the bump-on-tail instability is triggered each time one of the discrete modes comes in resonance with this beam. Hence, periodic bursts of the electrostatic energy are produced until a wide quasilinear plateau is formed.

Negative-mass instability in nonlinear plasma waves.

The negative-mass instability, previously found in ion traps, appears as a distinct regime of the sideband instability in nonlinear plasma waves with trapped particles. As the bounce frequency of these particles decreases with the bounce action, bunching can occur if the action distribution is inverted in trapping islands. In contrast to existing theories that also infer instabilities from the anharmonicity of bounce oscillations, spatial periodicity of the islands turns out to be unimportant, and the particle distribution can be unstable even if it is flat at the resonance. An analytical model is proposed that describes both single traps and periodic nonlinear waves and concisely generalizes the conventional description of the sideband instability in plasma waves. The theoretical results are supported by particle-in-cell simulations carried out for a regime accentuating the negative-mass instability.

Spin Hall effect of radiofrequency waves in magnetized plasmas.

In inhomogeneous media, electromagnetic-wave rays deviate from the trajectories predicted by the leading-order geometrical optics. This effect, called the spin Hall effect of light, is typically neglected in ray-tracing codes used for modeling waves in plasmas. Here, we demonstrate that the spin Hall effect can be significant for radiofrequency waves in toroidal magnetized plasmas whose parameters are in the ballpark of those used in fusion experiments. For example, an electron-cyclotron wave beam can deviate by as large as 10 wavelengths (∼0.1 m) relative to the lowest-order ray trajectory in the poloidal direction. We calculate this displacement using gauge-invariant ray equations of extended geometrical optics, and we also compare our theoretical predictions with full-wave simulations.

Nonlinear dispersion of stationary waves in collisionless plasmas.

A nonlinear dispersion of a general stationary wave in collisionless plasma is obtained in a nondifferential form expressed in terms of a single-particle oscillation-center Hamiltonian. For electrostatic oscillations in nonmagnetized plasma, considered as a paradigmatic example, the linear dielectric function is generalized, and the trapped particle contribution to the wave frequency shift Δω is found analytically as a function of the wave amplitude a. Smooth distributions yield Δω ∼ a(1/2), as usual. However, beamlike distributions of trapped electrons result in different power laws, or even a logarithmic nonlinearity, which are derived as asymptotic limits of the same dispersion relation.

Steepest-descent algorithm for simulating plasma-wave caustics via metaplectic geometrical optics.

The design and optimization of radiofrequency-wave systems for fusion applications is often performed using ray-tracing codes, which rely on the geometrical-optics (GO) approximation. However, GO fails at wave cutoffs and caustics. To accurately model the wave behavior in these regions, more advanced and computationally expensive "full-wave" simulations are typically used, but this is not strictly necessary. A new generalized formulation called metaplectic geometrical optics (MGO) has been proposed that reinstates GO near caustics. The MGO framework yields an integral representation of the wavefield that must be evaluated numerically in general. We present an algorithm for computing these integrals using Gauss-Freud quadrature along the steepest-descent contours. Benchmarking is performed on the standard Airy problem, for which the exact solution is known analytically. The numerical MGO solution provided by the new algorithm agrees remarkably well with the exact solution and significantly improves on previously derived analytical approximations of the MGO integral.

Controlling hot electrons by wave amplification and decay in compressing plasma.

Through particle-in-cell simulations, it is demonstrated that a part of the mechanical energy of compressing plasma can be controllably transferred to hot electrons by preseeding the plasma with a Langmuir wave that is compressed together with the medium. Initially, a wave is undamped, so it is amplified under compression due to plasmon conservation. Later, as the phase velocity also changes under compression, Landau damping can be induced at a predetermined instant of time. Then the wave energy is transferred to hot electrons, shaping the particle distribution over a controllable velocity interval, which is wider than that in stationary plasma. For multiple excited modes, the transition between the adiabatic amplification and the damping occurs at different moments; thus, individual modes can deposit their energy independently, each at its own prescribed time.

Dressed-particle approach in the nonrelativistic classical limit.

For a nonrelativistic classical particle undergoing arbitrary oscillations in external fields, the generalized effective potential Psi is derived through calculating the nonlinear eigenfrequencies of the particle-field system. Specifically, the ponderomotive potential is extended to a nonlinear oscillator, resulting in multiple branches near the primary resonance. For a pair of particle natural frequencies in a beat resonance, Psi scales linearly with the internal actions and is analogous to the dipole potential for a two-level quantum system. Thus cold quantum particles and highly excited quasiclassical objects permit uniform manipulation tools, particularly, one-way walls.

Positive and negative effective mass of classical particles in oscillatory and static fields.

A classical particle oscillating in an arbitrary high-frequency or static field effectively exhibits a modified rest mass m(eff) derived from the particle averaged Lagrangian. Relativistic ponderomotive and diamagnetic forces, as well as magnetic drifts, are obtained from the m(eff) dependence on the guiding center location and velocity. The effective mass is not necessarily positive and can result in backward acceleration when an additional perturbation force is applied. As an example, adiabatic dynamics with m||>0 and m||<0 is demonstrated for a wave-driven particle along a dc magnetic field, m|| being the effective longitudinal mass derived from m(eff). Multiple energy states are realized in this case, yielding up to three branches of m|| for a given magnetic moment and parallel velocity.

Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation.

Inhomogeneous drift-wave turbulence can be modeled as an effective plasma where drift waves act as quantumlike particles and the zonal-flow velocity serves as a collective field through which they interact. This effective plasma can be described by a Wigner-Moyal equation (WME), which generalizes the quasilinear wave-kinetic equation (WKE) to the full-wave regime, i.e., resolves the wavelength scale. Unlike waves governed by manifestly quantumlike equations, whose WMEs can be borrowed from quantum mechanics and are commonly known, drift waves have Hamiltonians very different from those of conventional quantum particles. This causes unusual phase-space dynamics that is typically not captured by the WKE. We demonstrate how to correctly model this dynamics with the WME instead. Specifically, we report full-wave phase-space simulations of the zonal-flow formation (zonostrophic instability), deterioration (tertiary instability), and the so-called predator-prey oscillations. We also show how the WME facilitates analysis of these phenomena, namely, (i) we show that full-wave effects critically affect the zonostrophic instability, particularly its nonlinear stage and saturation; (ii) we derive the tertiary-instability growth rate; and (iii) we demonstrate that, with full-wave effects retained, the predator-prey oscillations do not require zonal-flow collisional damping, contrary to previous studies. We also show how the famous Rayleigh-Kuo criterion, which has been missing in wave-kinetic theories of drift-wave turbulence, emerges from the WME.

Kinetic simulations of ladder climbing by electron plasma waves.

The energy of plasma waves can be moved up and down the spectrum using chirped modulations of plasma parameters, which can be driven by external fields. Depending on whether the wave spectrum is discrete (bounded plasma) or continuous (boundless plasma), this phenomenon is called ladder climbing (LC) or autoresonant acceleration of plasmons. It was first proposed by Barth et al. [Phys. Rev. Lett. 115, 075001 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.075001] based on a linear fluid model. In this paper, LC of electron plasma waves is investigated using fully nonlinear Vlasov-Poisson simulations of collisionless bounded plasma. It is shown that, in agreement with the basic theory, plasmons survive substantial transformations of the spectrum and are destroyed only when their wave numbers become large enough to trigger Landau damping. Since nonlinear effects decrease the damping rate, LC is even more efficient when practiced on structures like quasiperiodic Bernstein-Greene-Kruskal (BGK) waves rather than on Langmuir waves per se.

Nonadiabatic tunneling in ponderomotive barriers.

Localized regions of intense large-scale radiofrequency field are known to act like effective ("ponderomotive") potential barriers, which scatter particles elastically and in the direction determined by the particle initial velocity rather than phase. In smaller-scale fields, transmission through a ponderomotive barrier is probabilistic and resembles tunneling of a quantum particle through a static potential. We derive asymptotic expressions for the phase-averaged transmission coefficient T as a function of the particle energy E0. We show that, unlike for a truly quantum particle, T(E0) is of algebraic form and has a threshold, below which transmission does not occur. We also find a threshold in E0, above which all particles are transmitted regardless of their initial phase.

Ponderomotive ratchet in a uniform magnetic field.

We show how a ratchet effect, generally used in systems with periodic potentials, can also be practiced on charged particles by an ac field alone, in a background magnetic field near the cyclotron resonance. The effect relies entirely on the spatial inhomogeneity of the high-frequency drive, which produces a deterministic asymmetric ponderomotive barrier for undamped particles. Such a barrier can reflect particles incident from one side while transmitting those incident from the opposite side, hence acting somewhat like a Maxwell demon. The necessary fields are perhaps most easily realized in a plasma, though the effect is more general.

Motion of charged particles near magnetic-field discontinuities.

Charged particles near discontinuities in magnetic fields, so-called "boundary particles," can be constrained to remain near the discontinuity, even an arbitrarily fractured discontinuity, as the particle drifts along the fractured boundary. These particles are shown to exhibit new and interesting effects along broken and branching surfaces, including the wetting of fractured surfaces.

Relativistic electron acceleration in focused laser fields after above-threshold ionization.

Electrons produced as a result of above-threshold ionization of high-Z atoms can be accelerated by currently producible laser pulses up to GeV energies, as shown recently by Hu and Starace [Phys. Rev. Lett. 88, 245003 (2002)]. To describe electron acceleration by general focused laser fields, we employ an analytical model based on a Hamiltonian, fully relativistic, ponderomotive approach. Though the above-threshold ionization represents an abrupt process compared to laser oscillations, the ponderomotive approach can still adequately predict the resulting energy gain if the proper initial conditions are introduced for the particle drift following the ionization event. Analytical expressions for electron energy gain are derived and the applicability conditions of the ponderomotive formulation are studied both analytically and numerically. The theoretical predictions are supported by numerical computations.

Negative effective mass of wave-driven classical particles in dielectric media.

For a classical particle undergoing nonlinear interaction with a wave in dielectric medium, a perturbation theory is developed, showing that the particle motion can be described in terms of an effective parallel mass which can become negative. A relativistic particle interacting with a circularly polarized wave and a static magnetic field is studied as an example. For the three stationary orbits corresponding to the same velocity parallel to the magnetic field, the conditions are found under which all these equilibria are centerlike, or neutrally stable. It is shown that a negative parallel mass is realized in the vicinity of the intermediate-energy equilibrium and can lead to a plasma collective instability.