## 82 Fel Theory

8.2.1 Physical Mechanism

The FEL is conceptually quite simple, consisting only of an electron beam, a periodic pump field, and the radiation field. The most common pump field is a static periodic magnetic field called a wiggler, but any field capable of producing a transverse electron oscillation could in principle be used. A typical configuration of the FEL is illustrated in the simple schematic diagram of Fig. 8.2. The wiggler field is perpendicular to the FEL axis, so electrons injected into the wiggler along the axis begin to oscillate because of the v x B force. The radiation from the oscillating electrons combines with the wiggler field to produce a beat wave (referred to as the ponderomotive potential), which tends to axially bunch the electrons. The bunching caused by the ponderomotive potential provides the coherence that distinguishes the FEL from ordinary synchrotron light sources.

To show explicitly how the ponderomotive force arises and to see how the radiation frequency of the FEL is determined, we assume a particular form for the magnetic wiggler field:

Here, the wiggler period is Xw = 2^/kw, and ey is the unit vector in the y direction. The radiation fields are assumed to have amplitudes ER and BR and vary as cos(kz - wi), where the frequency and wavelength are related by the usual vacuum dispersion relation: w = ck = 2^c/X. Electrons injected into the wiggler with axial velocity v0 acquire a transverse oscillatory velocity (or wiggle velocity) whose amplitude can easily be shown from the Lorentz force equation to be given by eB

7o mkw where -e is the charge of an electron, 70 = (1 - v0/c2)-1/2 is the relativistic mass factor, c is the speed of light, and m is the rest mass of the electron. The vw x BR term in the force equation then produces the axial ponderomotive force, which can be shown to vary as sin[(k + kw)z - wi]. The argument of the sine function is just the relative phase between an electron's oscillatory motion and the radiation field, which is usually denoted by The phase velocity vph = w/(k + kw) of the ponderomotive potential must be approximately equal to the electron axial velocity in order for the electrons to remain in phase with the potential "wave" long enough to become strongly bunched. Thus we find that to synchronize the ponderomotive wave and the electrons, the frequency must satisfy the resonance condition w = /(1 - Pz), where pz = vz/c. In terms of the wavelength, this expression is 0 0