Nonlinear traveling-wave equilibria for free-electron-laser applications

The class of large-amplitude traveling-wave solutions to the nonlinear Vlasov-Maxwell equations is investigated in which the wave pattern is stationary in a frame of reference moving with the pondermotive phase velocity $v_p=\frac{\omega }{(k+k_0)}$. Here, ${\lambda }_0=\frac{2\pi }{k_0}$ is the wavelength of the transverse helical wiggler field, and ($\omega ,k$) are, respectively, the frequency and wave number 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 Bernstein-Greene-Kruskal 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 $F_0\left(\gamma \right)$ and the saturated untrapped distribution $F_u\left({\gamma }^{\prime }\right)$ are prescribed by rectangular distribution functions centered around axial velocity $v_z=\frac{\omega }{(k+k_0)}$, assuming a moderate field amplitude with $b_T=\frac{e\delta {\hat{B}}_T}{m{c}^{2}k}<1\mathrm{}$ and small fractional energy spread in the beam electrons. For a tenuous beam with $\omega \simeq \mathrm{kc}$ and $k\simeq (1+\frac{v_p}{c}){\gamma }_p^2{k}_0$ where ${\gamma }_p={(1-\frac{v_p^2}{c^2})}^{-\frac{1}{2}}$, it is found that the saturated amplitude of the radiation field is given approximately by $\delta {\hat{B}}_T=\frac{{\Delta }_L}{10{(1+b_w^2)}^{\frac{1}{2}}}\frac{{\omega }_p^2}{c^2{k}_0^2}\left[1+\frac{v_p}{c}\right]{\hat{B}}_w$ where