Helical Wiggler Magnetic Field Research Articles

Resonant Second Harmonic Generation of a Millimeter Wave in a Plasma Filled Waveguide

A plasma filled waveguide, immersed in a helical wiggler magnetic field, is suitable for resonant second harmonic generation of a millimeter wave. The wiggler provides the additional momentum required for the generation of harmonic photons. The millimeter wave induces oscillatory electron velocity that couples with the wiggler to produce density oscillations. These oscillations in conjunction with the oscillatory velocity produce a nonlinear current driving the resonant second harmonic. The wiggler introduces an azimuthal assymetry in the second harmonic field. The power conversion efficiency scales as the square of wiggler amplitude, electron density and oscillatory electron velocity.

Space-charge waves in the wiggler field of a Raman free-electron laser

A waveguide filled with a relativistic electron beam that passes through a helical wiggler magnetic field and a uniform axial magnetic field is considered. The propagation of two types of space-charge waves in this device is analyzed. An electrostatic approximation is employed that is based on Gauss's law and the requirements that the magnetic field of the wave and the curl of the electric field of the wave both be zero in the electron-beam reference frame. These equations transformed into the laboratory reference frame are shown to be more accurate than a system of equations that includes Gauss's law in conventional form. A dispersion relation is derived with the combined effects of the wiggler and axial magnetic fields and the waveguide boundary included. Some numerical results are presented for both plasma and cyclotron types of space-charge waves.

Free electron laser in a partially filled waveguide

A linear theory for a free electron laser with a one-dimensional helical wiggler and axial magnetic field in the collective regime is presented. The configuration consists of a cylindrical waveguide with arbitrary ratio of electron beam radius a to waveguide inner radius R. Parametric decay of the wiggler pump wave, in the beam frame, into a space-charge wave and an electric–magnetic (EH) waveguide mode is analyzed in three dimensions. A nonlinear wave equation for the three-wave interaction is derived and employed to obtain a formula for the spatial growth rate of the excited eigenmodes. It was found that the relativistic treatment of the electron oscillations in the wiggler field destroys the cyclotron resonance which appears in the nonrelativistic case. Nevertheless, appreciable amplification was found. Numerical analysis is conducted to study the growth rate, radiation wavelength, and required relativistic factor as functions of axial magnetic field B0 and radius ratio a/R. The suitable value for a/R was found to be around 0.65 for our choice of parameters.

Beam space-charge modes in a free-electron laser

Propagation of space-charge waves in a Raman free-electron laser is analyzed. The device under consideration consists of a cylindrical metallic waveguide filled with a relativistic electron beam which passes through a helical wiggler magnetic field and a uniform axial magnetic field. A solution of the fluid equations and Gauss’s law is developed in the form of truncated Fourier and Fourier–Bessel series. The dispersion relation is derived and the combined effects of the waveguide boundary and the wiggler and axial magnetic fields on cyclotron-like waves are studied numerically. In the limit of infinite waveguide radius, the dispersion relation yields a well-known expression in which the combined wiggler and axial fields determine an effective plasma frequency. In the limit of zero wiggler field, it reduces to the dispersion relation for axisymmetric space-charge waves which, in the beam frame, are purely electrostatic.

Conversion of a whistler wave into a controllable helical wiggler magnetic field

Plasma in the presence of a static magnetic field supports a whistler wave. It is shown that when the static magnetic field is switched off, the energy of the whistler wave is converted into the energy of a helical wiggler magnetic field.

Soliton solutions for free-electron-laser applications.

A new class of nonlinear traveling-wave solutions of the relativistic cold-fluid model are presented for a system consisting of a relativistic cold electron beam propagating through a helical wiggler magnetic field. The solutions, which are in the form of isolated soliton pulses of coupled electromagnetic and plasma waves, are obtained numerically. They represent possible nonlinear saturated states of the free-electron-laser instability and may also have useful applications in particle acceleration schemes.

Nonlinear kinetic theory of the free-electron laser.

A kinetic analysis of the nonlinear evolution of the free-electron-laser (FEL) instability is presented. The governing equations are the coupled Vlasov-Maxwell equations, which are investigated for a system consisting of a relativistic electron beam propagating through a helical wiggler magnetic field. Assuming that a single cavity mode of the electromagnetic field takes part in the lasing, a general nonlinear solution of the Vlasov equation is obtained in the resolvent formalism. Use of this solution in the wave equation provides a nonlinear description of the FEL. The saturation properties of the FEL are discussed by numerical and analytical solutions of this equation.

Self-field-induced chaoticity in the electron orbits in a helical-wiggler free-electron laser with axial guide field

The motion of a relativistic electron is analyzed in the field configuration consisting of a constant-amplitude helical wiggler magnetic field, a uniform axial magnetic field, and the equilibrium self-electric and self-magnetic fields produced by the non-neutral electron beam. By generating Poincaré surface-of-section maps, it is shown that the equilibrium self-fields destroy the integrability of the motion, and consequently part of phase space becomes chaotic. In particular, the Group I and Group II orbits can be fully chaotic if the self-fields are sufficiently strong. The threshold value of the self-field parameter ε=ω2pb/4Ω2c for the onset of beam chaoticity is determined numerically for parameter regimes corresponding to moderately high beam current (and density). It is found that the characteristic time scale for self-field-induced changes in the electron orbit is of the order of the time required for the beam to transit one wiggler period. An analysis of the first-order, self-field-induced resonances is carried out, and the resonance conditions and scaling relations for the resonance width are derived. The analytical estimates are in good qualitative agreement with the numerical simulations.

Two-dimensional solutions of the Dirac equation for the motion of an electron in a helical magnetic field

Quantum theory of a free electron in a helical wiggler magnetic field has been studied. In the previous studies the transverse momentum operators in the Dirac hamiltonian were ignored. In this note, the 2-D solutions of the Dirac equation, based on a hamiltonian which includes one of the transverse momentum operators ( p x ≠ 0, p y = 0) and the wiggler field as an external potential is derived. It is shown that there are solutions in terms of the Mathieu functions of fractional order.

Kinetic analysis of the sideband instability in a helical wiggler free-electron laser for electrons trapped near the bottom of the ponderomotive potential.

A kinetic formalism based on the Vlasov-Maxwell equations is used to investigate properties of the sideband instability for a tenuous, relativistic electron beam propagating through a constant-amplitude helical wiggler magnetic field (wavelength lambda/sub 0/ -- 2..pi../k/sub 0/ and normalized amplitude a/sub w/ -- eB-circumflex/sub w//mc/sup 2/k/sub 0/). The analysis is carried out for perturbations about an equilibrium Bernstein-Greene-Kruskal state in which the distribution of beam electrons G/sub s/(..gamma..') and the wiggler magnetic field coexist in quasisteady equilibrium with a finite-amplitude, circularly polarized, primary electromagnetic wave (..omega../sub s/,k/sub s/) with normalized amplitude a/sub s/ -- eB-circumflex/sub s//mc/sup 2/k/sub s/ and constant equilibrium wave phase.

Nonlinear traveling waves in a helical wiggler free-electron laser.

Nonlinear electromagnetic wave propagation is investigated in a system consisting of a relativistic, cold-electron beam propagating through a helical wiggler magnetic field. By transforming to wiggler coordinates and introducing the traveling-wave ansatz, a set of three, coupled, nonlinear ordinary differential equations is derived which describes nonlinear wave propagation in the system. In the Compton-regime limit, the number of differential equations reduces to two. These equations are particularly useful for studying nonlinear saturated states of the free-electron-laser instability, corresponding to values of the dimensionless phase speed (\ensuremath{\beta}=u/c) that are less than the dimensionless beam speed (${\ensuremath{\beta}}_{b}$=${V}_{0}$/c).

Electron orbits in free-electron lasers with helical wiggler and axial guide magnetic fields

An analytic theory of nonsteady-state trajectories in a fully three-dimensional, helical wiggler magnetic field is developed. In contrast to the well-known class of helical steady-state orbits, these orbits are not axicentered. Two limiting cases are considered. In the case of trajectories which are axis-encircling, we obtain a class of orbits by perturbation about the steady-state limit. We also consider the opposite regime in which particle trajectories are not axis-encircling, and derive a guiding-center theory for the limiting case in which the particles remain far from the axis of symmetry. In both limits we obtain trajectories which consist of periodic motions corresponding to both the fast wiggler and Larmor periods, as well as a slow motion corresponding to betatron oscillations arising from the transverse inhomogeneity in the wiggler field.

Compton and Raman free electron laser stability properties for a cold electron beam propagating through a helical magnetic wiggler

This paper gives an extensive characterization of the range of validity of the Compton and Raman approximations to the exact free electron laser dispersion relation for a cold, relativistic electron beam propagating through a constantamplitude helical wiggler magnetic field. The electron beam is treated as infinite in transverse extent. Specific properties of the exact and approximate dispersion relations are investigated analytically and numerically. In particular, a detailed numerical analysis is carried out to determine the range of validity of the Compton approximation.

Quasilinear stabilization of the free electron laser instability for a relativistic electron beam propagating through a transverse helical wiggler magnetic field

A quasilinear model is developed that describes the nonlinear evolution and stabilization of the free electron laser instability in circumstances where a broad spectrum of waves is excited. The relativistic electron beam propagates perpendicular to a helical wiggler magnetic field B0=−B̂ cos k0 z êx−B̂ sin k0 z êy, and the analysis is based on the Vlasov–Maxwell equations assuming ∂/∂x=0=∂/∂ y and a sufficiently tenuous beam that the Compton-regime approximation is valid (δφ≂0). Coupled kinetic equations are derived that describe the evolution of the average distribution function G0( pz,t) and spectral energy density ℰk(t) in the amplifying electromagnetic field perturbations. A thorough exposition of the theoretical model and general quasilinear formalism is presented, and the stabilization process is examined in detail for weak resonant instability with small temporal growth rate γk satisfying ‖γk/ωk‖≪1 and ‖γk/k Δvz‖≪1. Assuming that the beam electrons have small fractional momentum spread (Δ pz/p0≪1), the process of quasilinear stabilization by plateau formation in the resonant region of velocity space (ωk−kvz=0) is investigated, including estimates of the saturated field energy, efficiency of radiation generation, etc.

Instability of relativistic-electron helical trajectories in combined uniform axial and helical wiggler magnetic fields

The stability is investigated of helical trajectories of relativistic electrons in combined axial guide and helical wiggler magnetic fields appropriate for studies of free-electron lasers without using the frequently invoked assumption of evaluating the wiggler field on the laser axis. This is the first analysis of such trajectories which is free of this approximation. It is found that the helical trajectories are unstable with the possible exception of one value of the orbital radius. The instabilities are of two types corresponding to different properties of the eigenvalues of the matrix associated with the linearization of the equations of motion. These properties, in turn, depend upon the relative signs and magnitudes of the physical parameters that occur in the problem. At the exceptional radial value the trajectories can be either unstable or linearly stable depending upon the values of these parameters. For the linearly stable situations it is not known whether the trajectories are nonlinearly stable or unstable. We also prove that the trajectories of one of the instability types are of a particular kind called conditionally stable. This type of unstable trajectory has the property that it is approached by some (but not all) solutions of the equations of motion as time approaches positive or negative infinity. This type of instability is found to be more severe for larger values of the orbital radius. Even so, it still appears to be significant for small radii, of the order of $\frac{1}{100}$ of the wiggler wavelength.

Free electron laser instability for a relativistic annular electron beam in a helical wiggler field

A free electron laser instability is investigated for a relativistic annular electron beam propagating through a helical wiggler magnetic field. It is assumed that the beam is thin, with radial thickness (2a) much smaller than the beam radius (R0), and that ν/γb≪1, where ν is Budker’s parameter. The stability analysis is carried out within the framework of the linearized Vlasov–Maxwell equations for perturbations with general azimuthal harmonic number l and radial mode number s, including the important influence of (a) finite beam geometry in the radial direction, (b) positioning of the beam radius relative to the outer conducting wall (R0/Rc), and (c) finite wiggler amplitude (δB). All of these effects are shown to have an important influence on stability behavior. Moreover, the maximum coupling between the transverse and longitudinal modes increases substantially with increasing radial mode number s. It is also found that the transverse magnetic (TM) mode has slightly larger growth rate than the transverse electric (TE) mode.

Free Electron Laser Instability for a Relativistic Electron Beam in a Helical Wiggler Field

The free electron laser instability is investigated for a relativistic annular or solid electron beam propagating through a helical wiggler magnetic field. It is assumed that ?/?b<<l, where ? is Budker's parameter. In this regard, the stability analysis utilizes the vacuum transverse electric and transverse magnetic waveguide modes as a convenient basis to represent the electromagnetic field perturbations. Stability properties are investigated including the important influence of (a) finite beam geometry in the radial direction, (b) positioning of the beam radius relative to the outer conducting wall radius (Rc) and (c) finite wiggler amplitude. All of these effects are shown to have an important influence on detailed stability behavior.