Tunable Free-Electron
Stephen Vincent Benson
Accelerator Division
Continuous Electron Beam Accelerator Facility
Newport News, Virginia
1.1 Description of FEL Physics
The free-electron laser (FEL) uses a relativistic beam of electrons passing
through an undulating magnetic field (a wiggler) to produce stimulated emission
of electromagnetic radiation (Fig. 1). The quantum-mechanical description for
this device is based on stimulated emission of Bremsstrahlung [1]. The initial
and final states of the electron are continuum states so the emission wavelength
is not fixed by a transition between bound states. Although the initial description
by Madey was quantum mechanical, there was no dependence of the gain on
Planck's constant. This is a necessary but not sufficient condition for the exis-
tence of a classical theory for the laser. In fact, it was found that the device was
almost completely described by a classical theory [2].
The classical theory of FELs is an extension of the theory of the ubitron
developed by Phillips [3,4]. The ubitron is a nonrelativistic version of the FEL. It
was developed in a classified program between 1957 and 1964. It is a fast-wave
variant of the traveling-wave tube (TWT) amplifier and uses a transverse motion
of the electrons to couple a copropagating electromagnetic wave to the electron
beam. The classical formulation is therefore similar to the formulation for a
Tunable Lasers Handbook
Copyright 9 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
Stephen Vincent Benson
FIGURF ] A schematic of a FEL oscillator is shown. The electron beam is bent into a wiggler
using bending magnets (not shown). The electron beam wiggles along the optical axis of a cavity,
which is collinear with the axis of the wiggler. The wiggler shown consists of alternating North (light
gray) and South (dark gray) poles that alternately bend the beam left and right. The light is usually
coupled out of one of the mirrors. In an actual device, the electron beam and usually the mirrors are
in a vacuum chamber. The electron beam is shown as a continuous line, but in most devices it is a
pulsed beam.
TWT amplifier. It describes the interaction as a bunching of the electrons at a
wavelength near the resonant wavelength ~'0 defined by the relation
)~0 = 2h[~272
/ 2 ]
1 + 2~mc 2 ,
where h is the harmonic number for harmonic lasing, B is the rms magnetic field
in the wiggler, 13 is the speed of the electrons divided by the speed of light, ), is
the electron-beam relativistic energy divided by its rest mass
mc 2,
and ~v is the
wiggler wavelength. Equation (1) assumes that the electromagnetic wave is trav-
eling at the speed of light in a vacuum. Doria
et al.
have described the resonance
condition for a FEL in a waveguide for which the phase velocity is greater than c
[5]. We will assume here that the electromagnetic wave is traveling in a vacuum.
Figure 2 graphically shows how the resonance works. At the resonant wave-
length, one wavelength of the optical wave slips past the electron in the time that
the electron travels one wiggler period. At wavelengths near ~'0 the vector prod-
uct E.v is slowly varying so that there is a net exchange of energy between the
optical and electron beams. At exactly the resonant wavelength when the beam
current is low there is as much electron acceleration as deceleration so there is
no net gain. For wavelengths longer than ~'0 the interaction provides net gain for

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