Relativistic Effects in a Waveguide Completely Filled By Energetic Electron Beam for Free-Electron Laser

Farokhi, Bizhan1, Abdikian, Alireza2
1 Department of physics, Azad University of Arak, Arak, Iran
2 Physics Department, Bu-Ali Sina University, Hamadan, Iran
Abstract. In this paper an analysis of the high frequency eigenmodes of a cylindrical metallic waveguide completely
filled with a one dimensional helical wiggler and relativistic magnetized plasma is presented. A relativistic nonlinear
wave equation is derived in a form which includes the coupling of EH and HE modes due to the finite axial magnetic
field. Numerical analyses for four families of electrostatic and electromagnetic are shown that the dispersion relations in
relativistic case dependent on γ in low frequency that is ignored in previous work. In drawn figures, it is shown that
difference between relativistic and non-relativistic cases.
Keywords: Lorantz relativistic factor, High frequency eigenmodes,Relativistic nonlinear wave equation, Helical wiggler.
PACS: 41.60.Cr, 52.35.Fp, 52.38.Kd, 52.40.Fd
Introduction
The plasma waveguide (a cylindrical waveguide filled completely or partially with plasma) is an essential
component in many plasma electronics devices. It is used also for transporting charged particles and
electromagnetic energy, and as well as in investigating the properties of plasma and measuring its
parameters. The plasma medium may be magnetized by the application of a uniform static axial magnetic field.
The free electron laser (FEL) is a coherent radiation source in which a relativistic electron beam passes through
a wiggler magnetic field which is static and spatially periodic along the beam axis. For operation over the entire
electromagnetic spectrum, the beam may be enclosed in a waveguide and guided by an axial magnetic field1.
Any application of the plasma waveguide presupposes a thorough knowledge of the properties of its
eigenmodes. Analysis of the eigenmodes of a plasma waveguide is equivalent to the analysis for a
charged-particle-beam wave guide carried out in the beam frame with the beam self-fields neglected. Four
families of high-frequency eigenmodes of a magnetized plasma wave guide are known, namely EH and HE wave
guide modes, cyclotron modes, and space-charge modes. This paper presents a study of the relativistic parametric
excitation of space charge wave, cyclotron wave, EH and HE modes in a completely filled waveguide. This is a
generalization of some part of the theory by Uddholm et al2. in which they have investigated a study of
nonrelativistic parametric excitation of a space charge wave and transverse waveguide modes in a beam filled
waveguide with an axial magnetic field of arbitrary magnitude. In the lab frame, an analysis will be presented that is
based on Maxwell’s equations and the cold fluid equations in the beam frame using SI units with the wiggler field
which is a time-independent, spatially periodic, helical magnetic field.
The Basic Differential Equation
Imagine a cylindrical metallic waveguide of radius R, with a finite axial magnetic field B0z. The waveguide
contains a relativistic electron beam which completely fills it. The beam passes through a static, helical magnetic
field which is spatially periodic along the guide axis. The dynamics of the electron beam will be described.

Numerical Results
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, w λ =2cm and BW=1 KG,
respectively. Figure 1 shows the Correction factor α as a function of normalized cyclotron frequency w / ck 0 Ω . In
the relativistic case, due to wiggler field there are two groups of orbits for electron trajectory. For Group II, note that
in the low value of w / ck 0 Ω and in the sufficiently high w / ck 0 Ω , the Correction factor α is near zero (it will be as
constant as nonrelativistic case).

FIGURE 1. Correction factor α as a function of normalized w / ck 0 Ω .
Figure (2) illustrates ω of electromagnetic mode EH01 as a function of normalized wavenumber w k / k . The
dash curve is nonrelativistic case which is located between Group I and Group II.

FIGURE 2. Dispersion relation of electromagnetic mode EH01 as a function of normalized wavenumber w k / k .
REFERENCES
1. P. Freund, R. H. Jackson, D. E. Pershing, and J. M. Taccetti, Phys. Plasmas 1, 1046-1052 (1994).
2. P. Uddholm, J. E. Willet, and S. Bilikmen, J.Phys. D Appl. Phys., 24, 1278-1287, (1991).
3. H. P. Freund and T. M. Antenson, Jr., “Principal of Free-Electron Laser”, Chapman and Hall, London, 1996, pp. 42-70.
4. B. Farokhi and A. Abdikian , Phys. Plasmas (to be published).

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