Investigation of Fully Relativistic Effects Growth Rate for Free-Electron Laser in Completely Filled Waveguide

Farokhi, Bizhan1, Abdikian, Alireza2
1 Department of physics, Azad University of Arak, Arak, Iran
2 Physics Department, Bu-Ali Sina University, Hamadan, Iran
Abstract. A free electron laser with a helical wiggler field, cylindrical metallic waveguide, and axial guide field
operating in the collective regime infinitesimally above cutoff a transverse magnetic (TM) mode is considered. The
waveguide is completely filled with a relativistic electron beam. 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 3-dimensions. A
nonlinear wave equation for the 3-wave interaction is derived and employed to obtain a formula for spatial growth rate of
excited eigenmodes. It is found that the relativistic treatment of the electron oscillations in the wiggler field destroyed the
cyclotron resonance which appears in the non-relativistic case. Numerical analysis is conducted to study the growth rate,
radiation wavelength and required relativistic factor as function of axial magnetic field B0.
Keywords: Free Electron Laser, Helical Wiggler, Relativistic Electron Beam, Spatial Growth Rate.
PACS: 41.60.Cr, 52.35.Fp, 52.40.Fd
Introduction
Free-electron lasers are sources of coherent electromagnetic radiation with frequencies in the microwave,
infrared, visible, and ultraviolet. These can be operated efficiently at very high power levels and be continuously
tuned in frequency. The basic operating principle is stimulated scattering resulting from the passage of a relativistic
electron beam through a static magnetic wiggler. This instability process is caused by the interaction between the
transverse component of the electron motion in the wiggler field and the radiation field, leading to the axial
bunching of the electron beam. This axial bunching, in the collective regime, drives the space-charge waves. On the
other hand, interaction between the wiggler field and the space-charge waves amplifies the radiation field1.
Motivation for the present study is to develop a linear theory, in three dimensions, for Raman scattering in an
ubitron when the backscattered wave is an electromagnetic (EH) waveguide mode. This is a generalization of the
theory by Willett et al.2 in which the waveguide is completely filled with a relativistic electron beam. They have
treated the electron motion in the wiggler field nonrelativistically, and they have also imposed electrostatic
approximation on the space charge mode. On the other hand, Maraghechi et al.3 had some relativistic treatment by
assuming γ relativistic factor was constant. Finally, the backscattered EH waveguide mode was taken to be near
cutoff. In our analysis, electrons motion in the wiggler field is treated fully relativistically; therefore the wiggler is
not limited small amplitudes. This analysis is complicated by the first that space-charge and waveguide modes are
not, in general, purely transverse magnetic or purely transverse electric.
Three-Wave Decomposition
Consider a cylindrical metallic waveguide completely filled with relativistic electron beam in a uniform, static,
axial (guide) magnetic field. The beam is passing through a static, helical (wiggler) magnetic field, which is spatially
periodic along the guide axis. In the beam frame of reference, the wiggler field comprises a propagating
electromagnetic (pump) wave that undergoes stimulated Raman backscattering. This process is characterized by the
parametric decay of the pump wave(ω1,k1) into a forward- scattered space-charge wave (ω2,k2) and a backscattered
transverse TM waveguide mode (ω3,k3=0). The total beam frame electric field E, magnetic field B, electron fluid
velocity v, and electron density n will be written as an unperturbed part (subscript 0) plus a small perturbation that
we denote it with tilde. An analysis will be carried out in the beam frame based on the continuity equation, the coldfluid
momentum transfer equation, and the Maxwell’s equations in SI units. In the beam.

Numerical Results
This section describes the results of a numerical study of the lab-frame spatial growth rate ΔL based on the
formula derived in above (Eq. 15). The inner radius of the waveguide was taken to be R=3cm. Lab-frame value for
the unperturbed electron density, wiggler wavelength and wiggler amplitude were taken to be n0L=1012cm-3, λw=2cm
and Bw=1kG, respectively. In this stimulated scattering process, the growing modes were taken as the first
azimuthally symmetrical space-charge eigenmode of the beam- filled waveguide (p0ν= p01=2.405) and the first TM
waveguide eigenmode with azimuthally variation number l=1(p1μ= p11=3.832). Calculations were made with the
frequency of the TM mode held infinitesimally above cutoff and with the axial guide magnetic field B0ν=B0Lvaried
from .5 to 50 KG. Figure shows the growth rate ΔL in reciprocal centimeters (cm-1) as a function of axial magnetic
field B0 in kilogauss (KG) for group II in relativistic case and for nonrelativistic case.

Non relativistic Group II
FIGURE 1. Laboratory-frame special growth rate ΔL as a function of the axial magnetic field B0.
REFERENCES
1. H. Motz, J. Appl. Phys, 22, 527-535 (1951).
2. J. E. Willett, Y. Aktas, and H. Mehdian, Phys. Plasmas 2, 1311-1315 (1995).
3. B. Maraghechi, and S. Mirzanejhad, Phys. Plasmas 4, 2727-2735 (1997)
4. B. Farokhi and A. Abdikian, 13th International Congress on Plasma Physics, Kiev, May 22-26, 2006.
5. B. Farokhi and A. Abdikian, Phys Plasmas (to be published)

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