L. Friedland

Autoresonant (nonstationary) excitation of pendulums, Plutinos, plasmas, and other nonlinear oscillators

A weakly driven pendulum cannot be strongly excited by a fixed frequency drive. The only way to strongly excite the pendulum is to use a drive whose frequency decreases with time. Feedback is often used to control the rate at which the frequency decreases. Feedback need not be employed, however; the drive frequency can simply be swept downwards. With this method, the drive strength must exceed a threshold proportional to the sweep rate raised to the 3/4 power. This threshold has been discovered only recently, and holds for a very broad class of driven nonlinear oscillators. The threshold may explain the abundance of 3:2 resonances and dearth of 2:1 resonances observed between the orbital periods of Neptune and the Plutinos (Pluto and many of the Kuiper Belt objects), and has been extensively investigated in the Diocotron system in pure-electron plasmas.

Electron beam dynamics in combined guide and pump magnetic fields for free electron laser applications

The propagation of a cold relativistic electron beam in a free electron laser with an axial guide magnetic field is considered. The possibility of several steady‐state helical trajectories for the electrons is shown, and the stability against perturbations and accessibility of such steady states is considered. Necessary and sufficient conditions for the stability are derived and indicate the importance of the transition region at the entrance of the laser. Possible modes of operation of the laser in different steady‐state regimes are suggested and illustrated by numerical examples.

Congruent reduction in geometric optics and mode conversion

Standard eikonal theory reduces, to N=1, the order of the system of equations underlying wave propagation in inhomogeneous plasmas. The condition for this remarkable reducibility is that only one eigenvalue of the unreduced N×N dispersion matrix D(k,x) vanishes at a time. If, in contrast, two or more eigenvalues of D become simultaneously small, the geometric optics reduction scheme becomes singular. These regions are associated with linear mode conversion and are described by higher‐order systems. A new reduction scheme is developed based on congruent transformations of D, and it is shown that, in degenerate regions, a partial reduction of order is still possible. The method comprises a constructive step‐by‐step procedure, which, in the most frequent (doubly degenerate) case, yields a second‐order system, describing the pairwise mode conversion problem in four‐dimensional plasmas. This N=2 case is considered in detail, and dimensionality arguments are used in studying the characteristic ordering of the e...

Electron cyclotron resonance heating of plasmas in tandem mirrors

The feasibility of heating tandem-mirror devices with microwaves near the electron cyclotron frequency or its harmonic is examined. Ray-tracing calculations are performed to find the optimal angle of wave launching for maximal absorption. It is found that the mirror plugs of present-day tandem-mirror devices can be heated effectively via the extraordinary mode, and that in the next generation of devices both the ordinary and the extraordinary mode of propagation can be used. Presently available gyrotrons can be used as sources of microwave energy in these experiments.

Migration Timescale Thresholds for Resonant Capture in the Plutino Problem

Dynamic autoresonance theory is applied to the problem of thresholds on migration timescales for capture into resonances in the planar-restricted three-body problem with m1 m2 m0 and slowly migrating masses m1, 2. The thresholds are found analytically, scale as (m2/m1)-4/3, and yield an order of magnitude longer timescales required for capture of m0 into 2 : 1 outer resonance as compared with 3 : 2 and other resonances. The difference is due to the rotation of the primary mass m1, affecting the 2 : 1 resonance only. This could explain the observed small abundance of Kuiper Belt objects in the 2 : 1 resonance and could define accurate bounds on the timescales involved in the early evolution of the solar system.

Phase-space solution of the linear mode-conversion problem☆

Abstract The problem of linear mode conversion in a weakly inhomogeneous medium is posed and solved by phase-space methods. The PDEs for the two coupled modes are transformed to a simple first-order ordinary differential equation by a canonical transformation, wherein the two dispersion functions become essentially a locally conjugate pair of coordinates.

Robust Autoresonant Excitation in the Plasma Beat-Wave Accelerator: A Theoretical Study

A modified version of the Plasma Beat-Wave Accelerator scheme is presented, which is based on autoresonant phase-locking of the nonlinear Langmuir wave to the slowly chirped beat frequency of the driving lasers via adiabatic passage through resonance. Compared to traditional approaches, the autoresonant scheme achieves larger accelerating electric fields for given laser intensity; the plasma wave excitation is more robust to variations in plasma density; it is largely insensitive to the details of the slow chirp rate; and the quality and uniformity of the resulting plasma wave for accelerator applications may be superior.

Autoresonant (nonstationary) excitation of a collective nonlinear mode

The autoresonant (nonlinear phase locking) manipulation of the diocotron mode in a non-neutral plasma is investigated. Autoresonance is a very general phenomenon in driven nonlinear oscillator and wave systems. By sweeping or chirping the drive frequency, autoresonance allows the amplitude of a nonlinear wave to be controlled without the use of feedback. The experimental results, including a novel scaling relation, are in excellent agreement with a simple theoretical model. These are the first controlled laboratory studies of autoresonance in a collective plasma system.

SPATIAL AUTORESONANCE CYCLOTRON ACCELERATOR

A mechanism of electron acceleration scheme utilizing the spatial autoresonance phenomenon in combined axial guide magnetic field and a traveling electromagnetic wave with adiabatically varying parameters is analyzed. The acceleration is achieved due to the self‐tendency of the particles to stay in the relativistic gyroresonance with the wave, despite the variation of systems parameters. As compared to other traveling wave cyclotron acceleration schemes, the spatial autoresonance accelerator does not require a precise initial frequency matching and a rigid tuning of the guide magnetic field for sustaining the resonance. The method is illustrated in the case of a microwave driver with a tapered axisymmetric waveguide geometry.

Renormalized geometric optics description of mode conversion in weakly inhomogeneous plasmas

Conventional mode conversion theory in inhomogeneous plasmas starts with a local dispersion relation, which serves as a base for constructing a differential equation describing the waves in nearly degenerate plasma regions, where the mode conversion can take place. It will be shown, however, that the usual geometric optics perturbation scheme, which in the zero order leads to the aforementioned dispersion relation, predicts either rapid variations of the local wave vector and amplitude of the wave, or large first‐order corrections to the amplitude in nearly degenerate plasma regions. A novel, general, renormalized perturbation scheme will be suggested in order to remove this singular behavior. The new method is formulated in terms of the conventional, general plasma dielectric tensor and yields two coupled, energy‐conserving differential equations describing the mode conversion. Simple asymptotic solutions of these equations exist if the mode coupling is localized and weak. The method is applied to the cl...