• W.M. Fawley, J.-L. Vay
  LBNL, Berkeley, California

Numerical electromagnetic simulation of some systems containing charged particles with highly relativistic directed motion can by sped up by orders of magnitude by choice of the proper Lorentz-boosted frame*. A particularly good application for boosted frame calculation is short wavelength FEL simulation. In the optimal boost frame (i.e., the ponderomotive rest frame), the red-shifted FEL radiation and blue-shifted undulator field have identical wavelengths and the number of required time-steps for fully electromagnetic simulation (relative to the laboratory frame) decreases by a factor of gamma squared. We have adapted the WARP code** to apply this method to several FEL problems including coherent spontaneous emission from prebunched e-beams, strong exponential gain in a single pass amplifier configuration, and FEL emission from e^- beams in undulators with multiple harmonic components. We discuss our results and compare with those obtained using the "standard" FEL simulation approach which applies the eikonal and wiggler-period-averaging approximations.

* J.-L. Vay, Phys. Rev. Lett. 98, 130405 (2007).
** D.P. Grote, A. Friedman, J.-L. Vay, and I. Haber, AIP Conf. Proc. 749, 55 (2005).

• S.D. Webb, V. Litvinenko, G. Wang
  BNL, Upton, Long Island, New York

In this paper, we present an analytical three-dimensional theory of free electron lasers. Under several assumptions, we arrive at an integral equation similar to earlier work carried out by Ching, Kim and Xie, but using a formulation better suited for the initial value problem of Coherent Electron Cooling. We use this model in later papers to obtain analytical results for gain guiding, as well as to develop a complete model of Coherent Electron Cooling.

• S.D. Webb, V. Litvinenko
  BNL, Upton, Long Island, New York

We present analytical results for the one-dimensional dispersion relation for high-gain FELs. Using kappa-n distributions, we obtain analytical relations between the dispersion relations for various order kappa distributions. Since an exact solution exists for the kappa-1 (Lorentzian) distribution, this provides some insight into the number of modes on the way to the Gaussian distribution.

• G. Wang, V. Litvinenko
  BNL, Upton, Long Island, New York
• S.D. Webb
  Stony Brook University, Stony Brook

We solve linearized Vlasov-Maxwell FEL equations for a 3-D perturbation in an infinite electron beam with Lorentzian energy distributions using paraxial approximation. We present analytical solutions for various initial perturbations and discuss the effect of optical guiding in such system.

• F. Jafari Bahman, B. Maraghechi
  AUT, Tehran

Three-Dimensional simulation of Free-Electron Laser amplifiers at the presence of helical wiggler and ion-channel has been reported. The electromagnetic field is assumed to express in terms of the TE modes of a cylindrical waveguide in the absence of the electron beam. The final form of dynamical equations for the evolution of the slowly varying amplitude and wavenumber of TE mode is obtained by substitution of the vector potentials in to Maxwell’s equations. A cold, uniform, axisymmetric electron beam with a flat-top density profile has been considered for modeling the initial injection of the electron beam. The three-dimensional Lorentz force equation in the presence of a realistic helical magnetostatic wiggler, ion-channel electrostatic field and electromagnetic fields describes the electron dynamics. A set of coupled nonlinear first order differential equations is derived and solved numerically by Runge-Kutta method. The 10th-order Gussian quaderature technique is used for calculation of averages in the field equations. Finally, evolution of the radiation power and growth rate of the TE11 mode is shown.

• A. Raghavi
  pnum, Mashhad
• N. Mahdizadeh
  PPRC, Tehran

A combination of two planar magnetic wigglers with orthogonal fields and a shared electron beam is proposed for energy transfer between two different electromagnetic waves. It is shown that one of the wigglers can acts as an IFEL accelerator by extracting energy from a seed wave while simultaneously another wiggler works as a FEL and amplifies its corresponding resonant frequency. The equation of motion in the small signal gain (SSG) regime for this FEL-IFEL structure is studied. It is shown that the bunching process occurs for the electron beam in two different scales, corresponding to two different ponderomotive waves. It is concluded finally that, in principle, it is possible to use a FEL-IFEL scheme for energy exchange between two electromagnetic waves and retain an electron beam in resonance with two different electromagnetic waves simultaneously.

• V.A. Goryashko
  NASU/IRE, Kharkov

Using a time-dependent approach the analysis and optimization of a planar FEL-amplifier with an axial magnetic field and an irregular waveguide is performed. By applying methods of nonlinear dynamics three-dimensional equations of motion and the excitation equation are partly integrated in an analytical way. As a result, a self-consistent reduced model of the FEL is built in special phase space. The reduced model is the generalization of the Colson-Bonifacio model and takes into account the electrons’ intricate dynamics and intramode scattering. The reduced model and concepts of evolutionary computation are used to find optimal waveguide profiles. The numerical simulation of the original non-simplified model is performed to check the effectiveness of found optimal profiles. The FEL parameters are chosen to be close to the parameters of the experiment*, in which a sheet electron beam with the moderate thickness interacts with the TE01 mode of a rectangular waveguide. The results strongly indicate that one can improve the efficiency by a factor of five or six if the FEL operates in the magnetoresonance regime and if the irregular waveguide with the optimized profile is used.

*S. Cheng et al. IEEE Trans. Plasma Sci. 1996, vol. 24, p. 750

• V. Petrillo
  Universita' degli Studi di Milano, Milano
• M. Ferrario
  INFN/LNF, Frascati (Roma)
• A.R. Rossi
  Istituto Nazionale di Fisica Nucleare, Milano

We study both analytically and numerically the effects of a laser pulse on the longitudinal phase space of an electron beam in the stage of extraction from the cathode. We show how the interaction can produce modulations in the longitudinal momentum distribution.

• V. Petrillo, C. Maroli
  Universita' degli Studi di Milano, Milano
• M. Ferrario
  INFN/LNF, Frascati (Roma)
• L. Giannessi
  ENEA C.R. Frascati, Frascati (Roma)

We have written and numerically solved a set of 1-d FEL equations for electrons and radiation without the need of the slowly varying envelope approximation (SVEA). The equations, which take into account both forward and backward waves, have been applied to the case of a very short beam, as long as few wavelenghts, and to the case of long beams with short density modulations.

• J.T. Donohue
  CENBG, Gradignan
• J. Gardelle
  CESTA, Le Barp

Several years ago Andrews and Brau * presented a two-dimensional (2-D) theory for the production of coherent Smith-Purcell radiation by an initially continuous beam. An essential component of their analysis was the dispersion relation for a lamellar grating (i.e., rectangular profile) relating frequency and axial wave number k. Both simulations and an experiment performed at CESTA ** using a wide beam have confirmed the validity of their approach. However, all gratings are three-dimensional objects, and one may ask what modifications of the theory might be necessary. We present here our solution to the problem, which assumes a progressive wave in the direction of the grooves, with wave number q. A surprisingly simple modification of the Andrews and Brau 2-D dispersion relation is found. We have extensively tested our theory, both with simulations using the 3-D PIC code "MAGIC", and with measurements of the properties of the surface wave on the CESTA grating made using a network analyzer. Extremely good agreement is found, both with and without sidewalls on the grating.

* H. L. Andrews and C. A. Brau, Phys. Rev. ST Accel. Beams 7, 070701 (2004).
** J. Gardelle, L. Courtois, P. Modin and J.T. Donohue, Phys. Rev. ST Accel. Beams 12, 110701 (2009).

• J. Wu, J.J. Welch
  SLAC, Menlo Park, California
• R.A. Bosch
  UW-Madison/SRC, Madison, Wisconsin
• B. Jia
  FEL/Duke University, Durham, North Carolina
• A.A. Lutman
  DEEI, Trieste

A single-pass high-gain x-ray free electron laser (FEL) calls for a high quality electron bunch. In particular, for a seeded FEL amplifier and for a harmonic generation FEL, the electron bunch initial energy profile uniformity and peak current uniformity are crucial for generating an FEL with a narrow bandwidth. After the acceleration, compression, and transportation, the electron bunch energy profile entering the undulator can acquire temporal non-uniformity both in energy and local density. We study the effects of the electron bunch initial energy profile non-uniformity and local density variation on the FEL performance. Intrinsically, for a harmonic generation FEL, the harmonic generation starts with an electron bunch having energy modulation as well as density bunching at the previous stage FEL wavelength and its harmonics. Its effect on the harmonic generation FEL in the radiator is then studied.

• Z. Huang, Y.T. Ding, J. Wu
  SLAC, Menlo Park, California

Frequency-chirped FELs are useful to generate a large photon bandwidth or a shorter x-ray pulse duration. In this paper, we present a three-dimensional analysis of a high-gain FEL driven by the energy-chirped electron beam. We show that the FEL eigenmode equation is the same for a frequency-chirped FEL as for an undulator-tapered FEL. We study the transverse effects of such FELs including mode properties and transverse coherence. Comparison with numerical simulations are also discussed.

• L.T. Campbell, B.W.J. McNeil
  USTRAT/SUPA, Glasgow

An unaveraged 3D model of the FEL has been developed which can model variably polarised undulators. The radiation field polarisation is self-consistently driven by the electron dynamics and is completely variable. This paper describes both physical model and computational code.

• J. Henderson, B.W.J. McNeil
  USTRAT/SUPA, Glasgow
• D.J. Dunning
  STFC/DL/ASTeC, Daresbury, Warrington, Cheshire

The method of Echo Enabled Harmonic Generation is a possible method of achieving coherent short wavelengths in an FEL amplifier. In this paper the effects of noise variations is some of the important parameters affecting the stability of the final harmonic bunching of the electron beam are investigated numerically.

• O.A. Shevchenko, N. Vinokurov
  BINP SB RAS, Novosibirsk

The new approach for the SASE radiation properties calculation was proposed recently. It is based on the use of BBGKY chain of equations, adapted for FEL. In fact, it is the only known logically correct way to describe the SASE phenomenon. The two-time correlation function is necessary for calculation of averaged SASE spectrum. The solution of the correlation function equation for linear stage of SASE process is obtained.

• N. Mahdizadeh
  Islamic Azad University, Sabzevar Branch, Sabzevar
• F.M. Aghamir
  University of Tehran, Tehran
• A. Raghavi
  pnum, Mashhad
• S. Saviz
  PPRC, Tehran

The effects of temperature on the growth rate of a two-stream free electron laser (TSFEL) with planar wiggler magnetic pump have been investigated. The dispersion equation has been derived through the use of continuity, momentum transfer, and Maxwell's equations. In the analysis, only the longitudinal component of the pressure tensor is considered in the electron equation of motion. The characteristics of the dispersion relation along with the growth rate are analyzed numerically. The results show that the growth rate in this system (TSFEL) is relatively higher than the conventional FEL; finally, we compare our results with other cases, like without beam temperature, and conventional FEL.