Free Electron Laser Equations Research Articles

On a generalized fractional intero-differential equation of volterra-type

The following fractional generalization of the free electron laser (FEL) equation is studied: . An analytical treatment of this fractional integral equation leads to a closed of form solution in terms of Kummer functions φ(a,b;x). The solution is also expressed in terms of incomplete gamma functions . Finally, Tau method approximation is used in the numerical evaluation of the results.

Storage ring free electron laser dynamics, with the inclusion of bunch lengthening and energy spread increasing effects

We develop a rate equation model of the Storage Ring Free Electron Laser dynamics, which includes bunch lengthening and energy spread increasing effects, due to the combined mechanism of potential well distortion and collective random excitations. The model is based on the ordinary storage ring Free Electron Laser equations, coupled to Haissinski-type equations, accounting for the previously quoted effects. The obtained results are shown to provide a reliable tool, yielding a transparent understanding of the interplay between Free Electron Laser and e-beam induced effects.

Transient analysis of free-electron lasers with discrete radiators

In the linear regime before saturation, we solve the one-dimensional, classical free-electron laser equations, maintaining the longitudinal discreteness of the electrons throughout the analysis. We then take the limit in which the beam of discrete electrons is approximated by a continuum fluid. In the continuum limit, we recover the Green function used by Wang and Yu in their treatment of self-amplified spontaneous emission (SASE). For a bunched electron beam, we discuss both incoherent and coherent SASE. We also discuss the field radiated from a bunch whose length is short compared to the radiation wavelength. {copyright} {ital 1999} {ital The American Physical Society}

A fractional integro-differential equation of volterra type

An analytical and numerical treatment of a fractional integro-differential equation has been considered recently by Boyadjiev et al. [1]. This paper deals with a fractional generalization of the Free Electron Laser (FEL) equation, and the solution is obtained by a method that combines the variation of parameters and successive approximations. The numerical values have been obtained employing the algebra system MAPLE V.

Analytical and numerical treatment of a multidimensional fractional FEL equation

The Free Electron Laser (FEL) integral equation is derived from the complete set of coupled Lorentz-Maxwell equations. Its rules the evolution of the field amplitude in different regions: small-signal, longitudinal and transverse mode dynamics as well. The goal of this paper is to derive generalizations of a multidimensional FEL equation in the context of fractional calculus. The considerations are based on the perturbative approach, yielding closed-form expressions for the solution, which are suitable for numerical treatment.

Analytical and numerical treatment of a fractional integro-differential equation of volterra-type

An analytical treatment of a fractional generalisation of the free electron laser (FEL) equation is given, whereby a closed form solution is obtained in terms of Kummer functions Φ( a, c; z). Subsequently, Tau method approximation of these functions facilitates the numerical evaluation of the solution of the FEL equation.

Analytical treatment of the high-gain free electron laser equation

Exact solutions are obtained for the monodimensional and higher dimensional Free Electron Laser high-gain equations. These equations, which belong to the class of integrodifferential Volterra equations are treated within the context of a perturbative approach, yielding suitable closed-form expressions for the relevant solutions. The proposed method allows to obtain a unified analytical formalism for the full FEL dynamics. Numerical results are given for the one-dimensional case where a suitable expression has been derived for the solution in terms of easily computable functions.

The high-gain free-electron laser equation: exact, perturbative, and approximated solutions

We use the Laplace-transform technique for a general analysis of the integral equation that describes the Free-Electron Laser (FEL) small-signal evolution. We include both pulse propagation effects and inhomogeneous broadening contributions. The solutions are obtained in exact form and by using perturbative expansions. It is also shown that these solutions can be approximated with elementary functions and that most of the FEL high-gain evolution can be handled with almost negligible computational effort. The analysis is extended to includee-beam pre-bunching effects, and a generalization of the Madey theorem is proved.

Emission of short free electron laser pulse by high charge electron bunches

A radiation packet generated by electron bunches with lengths less than the radiation wavelength traversing under a magnetic wiggler is analyzed. Radiation damping is taken into account in this beam-wave resonant interaction by using the high gain free electron laser equations. For these bunches, the electrons occupy only a fraction of the 2π phase space in the ponderomotive well as they enter the wiggler which corresponds to a pre-bunched configuration in a conventional long beam free electron laser. Grazing limit where the group velocity of the radiation packet matches the resonant longitudinal electron bunch velocity for maximum wave packet power is considered.

Development of coherence in the Raman free-electron laser

The free-electron laser equations in the Raman régime are treated analytically to demonstrate the initiation of the free-electron laser instability by noise. Allowance is made for plasma oscillations in the electron beam, using a connection between the autocorrelations of the bunching and of the energy. The effect of these oscillations on the spontaneous spectrum is calculated and the continuity between spontaneous and stimulated processes is demonstrated.

High power self-similar radiation pulse in a free electron laser

Using 1D short wavelength (8 μm) Compton free electron laser (FEL) equations with slippage, we explore propagation of a high power pulse down a tapered undulator FEL travelling wave amplifier. For an appropriate taper, a short pulse (∼300 fs FWHM) with regular features will propagate self-similarly as it grows in power, slipping through a much longer electron pulse. The power spectrum of the pulse is nearly Gaussian with no sidebands. The electron energy is depressed by ∼33%, but slippage causes the peak pulse intensity to be about the same as the electron beam power density, ∼10 TW/cm2.

Free-electron-laser model without the slowly-varying-envelope approximation.

A mathematical model is presented which describes the evolution of electromagnetic radiation in a free-electron laser (FEL), without any stringent assumptions regarding the envelope of the radiation pulse. The derived set of equations is nearly identical to the traditional set, which needed the assumption of a slowly varying radiation amplitude and phase. Although rapid variations in the radiation envelope do have influence on the dynamics of the electrons, ignoring this fact does not cause excessive errors. Consequently, it is concluded that the region of validity of the traditional FEL equations is much larger than has been realized sor far.

Designs for an infrared free-electron laser based on gyrotron-powered electromagnetic wigglers

The possibility of developing gyrotron-powered electromagnetic wiggler free-electron laser (FEL) oscillators is of considerable interest for reducing the electron energy required for operation in the infrared spectrum. This paper considers the design of a 10.6 /spl mu/m electromagnetic wiggler FEL experiment based on the University of California at Santa Barbara (UCSB) 6 MeV electrostatic accelerator. Both waveguide and quasioptical wiggler resonator configurations are considered. A prescription for optimizing the gain of a waveguide electromagnetic wiggler FEL is derived and the nonlinear regime is treated by relating the electromagnetic wiggler FEL equations-of-motion to the universal normalized FEL equations. Point designs are given showing accelerator, gyrotron, and output parameters for 10.6 /spl mu/m Proof-of-Principle FEE experiments based on both waveguide gyrotron and quasioptical gyrotron wigglers. Designs are given for both a 5- and a 2-A FEL electron beam. >

Superradiant free-electron lasers in the high-gain strong pump regime

Time-dependent high-gain free-electron laser equations are generalized to the strong pump regime where fast, 2π/kw, spatial oscillations can no longer be eliminated by spatial averages, and evolution of the radiation phase needs to be taken into account. The equations are solved numerically along the characteristics of the beam and the laser field for superradiant solutions in the strong pump regime using the parameters of the Electron Laser Facility (ELF) of the Lawrence Livermore National Laboratory, [Phys. Rev. Lett. 54, 889 (1985); Nucl. Instrum. Methods A 250, 144 (1986); 285, 217 (1989); Phys. Rev. A 35, 2184 (1987)]. With beam pulses short compared to the wiggler length and long compared to the cooperation length, numerical solutions show that the initial laser fields get amplified to the steady-state level intensity in only a few bucket distances in the slippage region along the beam characteristics. The underlying mechanism for this superradiant gain gradient is due to the amplification of the incoming fields by the electrons prebunched by the earlier outgoing fields. Contrary to the earlier results, no spikes on the laser amplitude is observed in the slippage region. On the other hand, the solutions do show spikes on the relative laser phase. However, it is not clear on the credibility and the significance of these spikes, whether in amplitude or in phase.

High gain strong pump free-electron lasers with fast 2π/k w spatial oscillations

In the high gain strong pump regime, the radiation field amplitude and phase can vary substantially over a wiggler period. Also, due to the periodic transverse motion, the longitudinal electron energy suffers large variations over the same distance. Because of these considerations, the spatial average that leads to the classical one-dimensional free-electron laser equations is not applicable here. Thus the rederived equations suitable for the high gain strong pump regime contain 2π/kw periodic terms. Comparisons with the Electron Laser Facility (ELF) experiments of the Lawrence Livermore National Laboratory [Phys. Rev. Lett. 54, 889 (1985); 57, 2172 (1986); Phys. Rev. A 35, 2184 (1987)] on the operational characteristics including the relative phase of the radiation field are made.

High gain weak pump free electron laser amplifier analysis

The one dimensional free electron laser equations are solved in the steady state case for high gain weak pump solutions. The quadratic dispersion relation obtained by series expansion is compatible with the classical cubic form by linearization. The functional dependence of the exponentiation length of the radiation field on the machine parameters can be used as a design guideline.

Wall shorting of space charge field in free electron lasers

The one-dimensional free electron laser equations with space charge field are extended to include the effect of the conducting waveguide wall. Green function solution of the two-dimensional Poisson equation indicates a wall shorting factor much less than unity which greatly attenuates the space charge field inside the waveguide. Physically, this attenuation is due to the action of the image charges.

Superradiant free electron laser with space charge field

Linearized time dependent free electron laser equations with space charge field are solved for the space charge effects on the superradiant solution which now has a cosine square modulation on the intensity along the electron beam.

Universal efficiency and gain computations for high-gain free-electron-laser amplifiers

The free-electron-laser (FEL) equations are reduced to a set of one-dimensional, normalized equations that allow a universal (dimensional) analysis. In universal parameters, numerical integration of the FEL equations indicates a relatively constant saturated ponderomotive wave amplitude independent of both the normalized wiggler potential amplitude and the injected signal level. The constant ponderomotive wave amplitude and an empirical fit for the universal saturation length as a function of normalized wiggler potential amplitude and gain permits unnormalized design calculations for saturated power and saturated length over a wide parameter range. Tapering is considered by deriving analytical expressions for the intrinsic efficiency and taper length. Design values for a high-gain, high-efficiency, tapered amplifier at 280 and 560 GHz are presented.

Self-Consistent Kinetic Description of the Free Electron Laser Instability in a Planar Magnetic Wiggler

The linearized Vlasov-Maxwell equations are used to investigate detailed free electron laser (FEL) stability properties for a tenuous relativistic electron beam propagating in the z direction through the planar wiggler magnetic field B0(x) = -Bw, cos k0zêx. Here, Bw = constant is the wiggler amplitude, and λ0 = 2π/k0 = constant is the wiggler wavelength. The theoretical model neglects longitudinal perturbations (δϕ = 0) and transverse spatial variations (∂/∂x = 0 = ∂/∂y). Moreover, the model is based on the Vlasov-Maxwell equations for the class of self-consistent beam distribution functions of the form fb(Z, p, t) = n,bδ(px) δ(Py) G(z, pz, t), where p = γmv is the mechanical momentum, and Py is the canonical momentum in the y direction. For low or moderate electron energy, there can be a sizable modulation of beam equilibrium properties by the wiggler field and a concomitant coupling of the kth Fourier component of the wave to the components k ± 2k0, k ± 4k0, ··· in the matrix dispersion equation. In the diagonal approximation, investigations of detailed stability behavior range from the regime of strong instability (monoenergetic electrons) to weak resonant growth (sufficiently large energy spread). In the limit of ultrarelativistic electrons and very low beam density, the kinetic dispersion relation is compared with the dispersion relation obtained from a linear analysis of the conventional Compton-regime FEL equations. Finally, assuming ultrarelativistic electrons and a sufficiently broad spectrum of amplifying waves, the quasi-linear kinetic equations appropriate to the planar wiggler configuration are presented.