Nonlinear Travelling-wave Equilibria for Free Electron Laser Applications

The class of large-amplitude travelling wave solutions to the nonlinear Vlasov-Maxwell equations are investigated in which the wave pattern is stationary in a frame of reference moving with the pondermotive phase velocity v = p w/(k + k0). Here, X0 = 27r/k 0 is the wavelength of the transverse helical wiggler field, and (wk) are the frequency and wavenumber of the saturated radiation field which is assumed to be monochromatic and circularly polarized. The conservation of (average) density, momentum and energy are imposed as additional exact constraint equations that connect the final (saturated) and initial states of the combined electron beam-radiation field-wiggler field system. These constraint equations reduce the generality of the nonlinear equilibrium BGK solutions, and allow estimates to be made of the saturated field amplitude in terms of initial properties of the beam-wiggler system. As a simple example that is analytically tractable, we consider the case where the initial distribution F0 (y) and the saturated untrapped distribution Fu (y') are prescribed by rectangular distribution functions centered around axial velocity vz = (k + k 0), assuming a moderate field amplitude with bT = ed T/mc2 k<1 and small fractional energy spread in the beam electrons. For a tenuous beam with w=~ kc 2 2 2 -1/2 and k ~ (1 + v /c) Y k , where y = (1 v /c ) , it is found that the saturated amplitude of the radiation field is given approximately by A W2 S= L p 1 + 10(1 + b21/2 c2k2 (l w 0 2 2 where b = eB /mc k, A Lmc is the characteristic half-width energy spread in the laborabory frame, and w2 = 47nb e 2 /m is the nonrelativistic plasma frequency-squared.