Helical Wiggler Free Electron Laser Research Articles

Effects of electron beam temperature on radiation growth rate in a realistic helical wiggler with ion-channel guiding

We study the effects of electron beam parameters such as electron beam temperature and Lorentz relativistic factor (or electron energy) on growth rate in a realistic (three-dimensional) helical wiggler free-electron laser with ion-channel guiding. A generalized dispersion relation is derived by considering the interaction between the radiation fields and electron beam when the effects of electron beam self-fields are taken into account. The growth rate is then calculated, and it is found that, for group I orbits, there is an optimum value for electron beam density at which the growth rate goes to its maximum. Also, there is an optimum value for Lorentz relativistic factor. The effect of beam temperature is also investigated, and it is shown that the growth rate decreases considerably as beam temperature increases.

Steady-state electron trajectories in a free electron laser with ion-channel and axial magnetic field in the presence of self-fields

A theory of self-fields in a one-dimensional helical wiggler free electron laser with ion-channel guiding and axial magnetic field is presented. The steady-state orbits under the influence of self-field are derived and discussed. The Φ function that determines the rate of change of axial velocity with energy is derived. The numerical results show the effects of self-fields and the two electron-beams guiding devices (ion-channel and axial magnetic field) on the trajectories when used separately and simultaneously. The study shows that new unstable orbits, in the first part of the group I and II orbits, are found. A detailed stability analysis of orbits is presented.

The Gain Equation of a Helical Wiggler Free Electron Laser with Ion-Channel Guiding and/or an Axial Magnetic Field

Electron motion in the combined ion-channel, helical wiggler and axial magnetic fields is analyzed in the absence of a radiation field. Detailed analysis of the gain equation in a free-electron laser is presented. Numerical calculations are made to illustrate the effects of the two electron-beam guiding devices on the gain when applied separately and simultaneously.

Adorable reversed field configuration with self-fields effect in a fel

The dynamical behavior of relativistic electron governed by the combination of a realistic helical-wiggler free-electron laser (FEL) with a uniform axial guide magnetic field is investigated by the consideration of the effect of the relativistic electrons self-fields. The electron beam is assumed to have uniform density. In Raman regime, a three-dimensional Hamiltonian approach is derived in detail. The consideration of the additional scalar potential Φs represents the basic feature of the analysis. The approach recognized the two usual constants of motion: one concerns the total energy while the other is the canonical axial angular momentum \(\hat P_{z'}\). After some tedious algebra, the dynamical variables problems are solved analytically to study stable and unstable fixed point. The additional scalar potential Φs changes the nature of groups, in group II orbits reversed field configuration near \(\hat \rho _0 = 0\) converted to a simple group II. At the time of the variation of e the energetic interaction zones are discussed. The stability zones of fixed points that allow an excellent interaction between the electron and the existing fields are limited. To validate our model, we apply it to the well-known experience of Conde and Bekefi [Phys. Lett. 67, 3082 (1991)] and get some encouraging results.

Self-field effects on electron dynamics in a three-dimensional helical wiggler free-electron laser with axial magnetic field

An analytic linear theory of the electron dynamics in a three-dimensional helical wiggler free electron laser (FEL) with axial magnetic field is presented. Orbits are obtained by perturbing the steady state-trajectories in order to determine the characteristic frequencies Ω± of the FEL. The effect of the self-fields on electron dynamics is studied and modified steady-state orbits and their stabilities have been analysed considering variation of electron energy and density. Among the features encountered is that in both group-I and group-II, one of the characteristic frequencies may have either signs affecting then the stability of the motion, while in group-II operation a repulsion of the frequencies at a pseudocrossing leads to highly perturbed trajectories when the wiggler frequency is approximately half the cyclotron frequency. Self-fields effects can significantly impair the stability of the electron orbits. For group-I orbits, they are more important for higher wiggler frequencies and lower beam energies. For group-II orbits, they remain less important for higher wiggler frequencies and lower beam energies before reaching the inversion zone, then they behave as for group-I orbits. It should be remarked that self-fields shift the inversion zone towards higher cyclotron frequencies the thing that is obtained by either decreasing the wiggler frequency or increasing the beam energy. It is shown that the axial velocity-induced self-magnetic field has a diamagnetic effect for both groups orbits, while the wiggler-induced self-magnetic field has a diamagnetic effect for group-I orbits and a paramagnetic effect for group-II orbits. The paramagnetic and diamagnetic effects are more important for higher beam energies and densities.

Unified model of electron beam shot noise and coherent spontaneous emission in the helical wiggler free electron laser

A numerical algorithm for simulating electron beam shot noise in free electron lasers (FELs) is presented. Shot noise is a source of spontaneous emission that may be amplified in the self-amplified spontaneous emission regime of operation. This regime is of great importance to XUV and x-ray FELs where the spontaneous emission is currently the only effective source available for amplification. The algorithm uses a quasiuniform phase-space distribution of appropriately charge weighted macroparticles. The statistical properties of the macroparticles are derived directly from the temporal Poisson statistical properties of the real electron distribution. Unlike previous algorithms, ours does not rely upon any averaging over a resonant radiation period time scale and so more correctly describes the underlying physics. The algorithm also allows shot noise to be modeled self-consistently in unaveraged FEL models which are able to describe subwavelength phenomena such as coherent spontaneous emission (CSE). The algorithm is used in the unaveraged 1D FEL numerical simulation code FemFel and demonstrates spontaneous emission due to shot noise and CSE in both rectangular and Gaussian electron pulse current profiles. The preliminary results show good qualitative and quantitative agreement with theory.

Classical harmonic oscillator approach of a helical-wiggler free-electron laserwith axial guide field

Electron beam dynamics in a helical-wiggler free-electron laser (FEL) with a uniform axial guide magnetic field are studied using a three-dimensional Hamiltonian approach. The basic feature of the analysis is the definition of a rotational variable, [Formula: see text], that plays the primordial role in lowering to the half the dimension of the quadratic Hamiltonian as a system of two uncoupled oscillators with definite frequencies and amplitudes. It is through applying this variable in the vicinity of a fixed point that the Heisenberg picture of the dynamics of the particles comes to light, leading thus to the association of the steady-state ideal helical trajectories with arbitrary trajectories. The approach recognized the usual two constants of motion, one being the total energy while the other is the canonical axial angular momentum, Pz'. If the value of the latter is such that a fixed point exists, the Hamiltonian is expanded about the fixed point up to second order. The so-obtained oscillator characteristic frequencies allowed one to study the different modes of propagation and to identify, and then avoid the problematic operating conditions of the FEL concerned. On the other hand, the amplitudes of the oscillations, which do depend on the frequencies, are fortunately found to be constants of motion and then controlled by the boundary conditions (initial conditions). PACS Nos.: 52.40-w, 52.60+h, 42.55.Tb, 52.75Ms

Classical harmonic oscillator approach of a helical-wiggler free-electron laserwith axial guide field

Electron beam dynamics in a helical-wiggler free-electron laser (FEL) with a uniform axial guide magnetic field are studied using a three-dimensional Hamiltonian approach. The basic feature of the analysis is the definition of a rotational variable, , that plays the primordial role in lowering to the half the dimension of the quadratic Hamiltonian as a system of two uncoupled oscillators with definite frequencies and amplitudes. It is through applying this variable in the vicinity of a fixed point that the Heisenberg picture of the dynamics of the particles comes to light, leading thus to the association of the steady-state ideal helical trajectories with arbitrary trajectories. The approach recognized the usual two constants of motion, one being the total energy while the other is the canonical axial angular momentum, Pz'. If the value of the latter is such that a fixed point exists, the Hamiltonian is expanded about the fixed point up to second order. The so-obtained oscillator characteristic frequencie...

Dispersion relation and growth in a free-electron laser with ion-channel guiding

A theory of a helical wiggler free-electron laser with ion-channel guiding is presented. Electron motion has been analyzed using single particle dynamics and regimes of orbit stability have been discussed. With the help of these trajectories, source currents have been obtained and the linear dispersion equation showing coupling of electromagnetic and space-charge waves by the wiggler field has been set up for a monoenergetic electron beam. Growth rates have been obtained for different ion-channel frequencies. The variation of resonant frequencies and peak growth rates with ion-channel frequency has been illustrated. Substantial enhancement in peak growth rate is obtained as the ion-channel frequency approaches the wiggler frequency.

Asymptotic electron trajectories and an adiabatic invariant for a helical-wiggler free electron laser with weak self-fields

The dynamics of a relativistic electron in the field configuration consisting of a constant-amplitude helical-wiggler magnetic field, a uniform axial magnetic field, and the equilibrium self-fields is described by a near-integrable three-degree-of-freedom Hamiltonian system. The system is solved asymptotically for small ε by the method of averaging, where ε measures the strength of the self-fields. Because the Hamiltonian does not depend on one of the coordinates, it immediately reduces to a two-degree-of-freedom system. For ε=0, this reduced system is integrable, but is not in standard form. The action-angle transformation to standard form is derived explicitly in terms of elliptic functions, thus enabling the application of the averaging procedure. For almost all regular electron trajectories the solution is explicitly derived in asymptotic form and an adiabatic invariant is constructed, both results are in a form that remains uniformly valid over the time interval for electrons to transit the laser. The analytical results are verified by numerical calculations for an example problem.

Electron trajectories and gain in free electron laser with ion channel guiding

A detailed analysis of electron trajectories and gain in a helical wiggler free electron laser (FEL) with ion channel focusing using single particle dynamics is presented. Conditions for stability of electron orbit have been obtained and stable regimes have been identified. Gain has been evaluated in the low-gain-per-pass limit with the help of the self-consistent pendulum equation. It is seen that the presence of ion channel leads to significant gain enhancement under appropriate conditions.

Identifying chaotic electron trajectories in a helical-wiggler free-electron laser.

When taking into account the equilibrium self-fields of the beam, it is confirmed that the motion of an electron in a helical wiggler with guide field may be chaotic. There is evidence of chaos from numerical calculations of nonzero Lyapunov exponents using different approaches of Benettin's method which are described and compared. Very accurate Poincar\'e maps are also performed.

Dynamics of the equilibrium electrons in a helical-wiggler free-electron laser with reversed guide field.

The dynamical behavior of a relativistic electron governed by the combination of a helical-wiggler magnetic field with a reversed guide magnetic field is investigated by generating Poincar\'e surface-of-section maps. The Hamiltonian description is derived in detail. It is found that the helical orbits are always stable. Computer simulation shows that the separatrix of the closed and open curves in phase portrait in this case is quite different from the one in the case of a positive guide field. An interesting effect numerically predicated is that increasing the self-field narrows the spread of longitudinal momentum. No chaos appears for the experimental parameters of Conde and Bekefi [Phys. Rev. Lett. 67, 3082 (1991)], although the self-fields destroy the integrability of the motion.

Self-field-induced Poincaré maps of the equilibrium electrons in a helical-wiggler free-electron laser with axial guide field

This paper presents a technique of generating Poincaré maps of the equilibrium electrons in a helical-wiggler free-electron laser with axial guide field, and investigates the influence of the electrostatic and magnetic self-fields on Poincaré maps by numerically integrating the equations of motion. Upon improving the accuracy of the iterations, we find that the self-field-induced chaotic behavior is not so serious as was indicated before. On the contrary, a new interesting result is observed: some unbounded curves in Poincaré maps are folded and closed with the increase of self-fields for group II orbits.

Macroclump model of the nonlinear evolution of the sideband instability in a helical wiggler free-electron laser

The nonlinear evolution of a helical wiggler, free-electron laser is investigated within the framework of a macroclump model for the trapped electrons. The model describes the nonlinear evolution of a right-circularly polarized electromagnetic wave with frequency ωs and wave number ks, and slowly varying amplitude âs(z,t) and phase δs(z,t) (eikonal approximation). The model further assumes that the trapped electrons can be treated as tightly bunched macroclumps that interact coherently with the radiation field. The analysis is carried out in the ponderomotive frame, which leads to a substantial simplification in both the analytical and numerical studies. As a first application, the nonlinear evolution of the primary signal is examined when ∂/∂l′=0 (no spatial variation of the wave amplitude and phase). The evolution equations are reduced to quadrature, and the maximum excursion of the wave amplitude âs,max is calculated analytically. Subsequently, the nonlinear evolution of the sideband instability is investigated, making use of the equations describing the self-consistent evolution of the wave amplitude âs and phase δs, which vary slowly with both space and time, together with the macroclump orbit equation. In the present analysis, the sideband signals are treated as perturbations (not necessarily small) about a constant-amplitude (â0s =const) primary electromagnetic wave with slowly varying phase δ0s(z′). The coupled orbit and field equations are investigated analytically and numerically over a wide range of system parameters to determine detailed scaling properties of the sideband instability. The results of the present analysis suggest that free-electron lasers operating with system parameters corresponding to the strong-pump regime [(Ω′B/Γ0)6/4≫1] are least vulnerable to the sideband instability. Moreover, the nonlinear evolution of the sideband instability is investigated numerically for system parameters corresponding to the Los Alamos free-electron laser experiment [Warren et al., IEEE J. Quantum Electron. QE-21, 882 (1985)]. In several aspects, the numerical results are found to be in good qualitative agreement with the experimental results.

Influence of the trapped-electron distribution on the sideband instability in a helical wiggler free-electron laser

Use is made of the Vlasov–Maxwell equations to investigate detailed properties of the sideband instability for a helical wiggler free-electron laser with wiggler wavelength λw=2π/k0=const and normalized wiggler amplitude aw=eB̂w/mc2k0=const. The model describes the nonlinear evolution of a right circularly polarized primary electromagnetic wave with frequency ωs, wave number ks, and slowly varying amplitude âs(z,t) and phase δs(z,t) (eikonal approximation). The coupled Vlasov and field-evolution equations are analyzed in the ponderomotive frame (‘‘primed’’ variables) moving with velocity vp=ωs/(ks+k0) relative to the laboratory. Detailed properties of the sideband instability are investigated for small-amplitude perturbations about a quasi-steady state characterized by an equilibrium electron distribution f0(γ′0) and a primary electromagnetic wave with constant amplitude â0s=const (independent of z′ and t′) and slowly varying phase δ0s(z′). A formal dispersion relation is derived for perturbations about a general equilibrium distribution f0(γ0) that may include both trapped and untrapped electrons. For the case where only trapped electrons are present, the dispersion relation is reduced to a simple analytical form. Detailed properties of the sideband instability are investigated for the case where the trapped electrons uniformly populate the ponderomotive potential up to an energy γ′M≤γ̂+, where γ̂′+ is the energy at the separatrix. Analysis of the dispersion relation shows that the maximum energy of the trapped-electron population (γM) significantly affects detailed stability properties in the strong-pump and intermediate-pump regimes.

Two-dimensional Vlasov treatment of free-electron laser sidebands

The Kroll–Morton–Rosenbluth equations [IEEE J. Quantum Electron. QE-17, 1436 (1981)] for a helical-wiggler free-electron laser are generalized to treat an electron beam with a prescribed radial density profile and an equilibrium distribution function that is an arbitrary function of the longitudinal action J. The principal approximation is the assumption that betatron frequencies of beam particles are low compared with typical synchrotron frequencies. Vlasov equilibria for finite-amplitude primary waves with time-varying phase are calculated for several distribution functions. Using these equilibria, radial eigenvalue equations for the frequency and growth rate of small-amplitude sidebands are derived and solved numerically. The radial mode structure is found to have no appreciable effect on sideband growth when the beam radius is large compared with [2ks min(Ω0, dφ0/dz)]−1/2, where ks and φ0 are the wavenumber and phase of the primary wave and Ω0 is the maximum synchrotron ‘‘frequency’’ in z of trapped electrons. In these effectively one-dimensional cases, the dispersion relation depends only on the distribution function and on a dimensionless density parameter η̄=kwa2wω2b/(c2γ3rΩ30i), where kw is the wiggler wavenumber, aw=eAw/(mc2) is the dimensionless wiggler vector potential, ωb is the maximum plasma frequency of the beam, and γr is the Lorentz factor for resonant particles. Both the upper and lower sidebands for a deeply trapped distribution (J≊0) have a maximum growth rate of (31/2/2)(η̄2/2)1/3 for η̄≪1 and (31/2/2)(η̄/2)1/3 for η̄≫1, and distributions with a spread in J invariably show slower sideband growth. For beams with a smaller radius, the radial density variation causes a further reduction in the peak sideband growth rate and narrows the spectrum of unstable modes.

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.

Influence of untrapped electrons on the sideband instability in a helical wiggler free electron laser

The detailed influence of an untrapped-electron population on the sideband instability in a helical wiggler free electron laser is investigated for small-amplitude perturbations about a constant-amplitude (â0s =const) primary electromagnetic wave with slowly varying equilibrium phase δ0s. A simple model is adopted in which all of the trapped electrons are deeply trapped, and the equilibrium motion of the untrapped electrons (assumed monoenergetic) is only weakly modulated by the ponderomotive potential. The theoretical model is based on the single-particle orbit equations together with Maxwell’s equations and appropriate statistical averages. Moreover, the stability analysis is carried out in the ponderomotive frame, which leads to a substantial simplification in deriving the dispersion relation. Detailed stability properties are investigated over a wide range of dimensionless pump strength ΩB /Γb ck0 and fraction of untrapped electrons fu =n̂u /n̂b. When both trapped and untrapped electrons) are present, there are generally two types of unstable modes, referred to as the sideband mode, and the untrapped-electron mode. For fu =0, only the sideband instability is present. As fu is increased, the growth rate of the sideband instability decreases, whereas the growth rate of the untrapped-electron mode increases until only the untrapped-electron mode is unstable for fu =1. It is found that the characteristic maximum growth rate of the most unstable mode varies by only a small amount over the entire range of fu from fu =0 (no untrapped electrons to fu =1 (no trapped electrons). The present analysis suggests that the linear and nonlinear evolution of the beam electrons and radiation field may be substantially modified by the present of an untrapped-electron component when fu ≳0.2.

Single-particle analysis of the free-electron laser sideband instability for primary electromagnetic wave with constant phase and slowly varying phase

Use is made of the single-particle orbit equations together with Maxwell’s equations and appropriate statistical averages to investigate detailed properties of the sideband instability for a helical-wiggler free-electron laser with wiggler wavelength λ0=2π/k0=const and normalized wiggler amplitude aw=eB̂w/mc2k0 =const. The model describes the nonlinear evolution of a right-circularly polarized primary electromagnetic wave with frequency ωs, wavenumber ks, and slowly varying amplitude âs(z,t) and phase δs(z,t) (eikonal approximation). The orbit and wave equations are analyzed in the ponderomotive frame (‘‘primed’’ variables) moving with velocity vp=ωs/(ks+k0) relative to the laboratory. Detailed properties of the sideband instability are investigated for small-amplitude perturbations about a quasisteady equilibrium state characterized by â0s =const (independent of z′ and t′). Two cases are treated. The first case assumes constant equilibrium wave phase δ0s =const, which requires (for self-consistency) both untrapped- and trapped-electron populations satisfying 〈∑j exp[ik′pzj0(t′) +iδ0s]/γ′j〉=0. Here kp=(ks+k0)/γp is the wavenumber of the ponderomotive potential, z′j0(t′) is the equilibrium orbit, and γjmc2 is the electron energy. The second case assumes that all of the electrons are deeply trapped, which requires a slow spatial variation of the equilibrium wave phase, ∂δ0s/∂z′=2Γ0(Γ0ck0/ΩB)2 k′p≠0. The resulting dispersion relations and detailed stability properties are found to be quite different in the two cases. Both the weak-pump and strong-pump regimes are considered.