Influence of finite radial geometry on the generation of coherent radiation by a relativistic electron beam in a longitudinal magnetic wiggler

The influence of finite radial geometry on the longitudinal wiggler free-electron-laser instability is investigated for TE-mode perturbations about a uniform-density electron beam with radius ${\stackrel{^}{R}}_{b}$. The equilibrium and stability analysis is carried out for a thin, tenuous electron beam propagating down the axis of a multiple-mirror (undulator) magnetic field ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})\ensuremath{\simeq}{B}_{0}[1+(\frac{\ensuremath{\delta}B}{{B}_{0}})sin({k}_{0}z)]{\stackrel{\ensuremath{\rightarrow}}{\mathrm{e}}}_{z}$, where ${\ensuremath{\lambda}}_{0}=\frac{2\ensuremath{\pi}}{{k}_{0}}=\mathrm{const}$ is the wiggler wavelength. It is assumed that ${k}_{0}^{2}{\stackrel{^}{R}}_{b}^{2}\ensuremath{\ll}1$, and that perturbations are about the self-consistent Vlasov equilibrium ${f}_{b}^{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}})=(\frac{{\stackrel{^}{n}}_{b}}{\ensuremath{\pi}})\ensuremath{\delta}({p}_{\ensuremath{\perp}}^{2}\ensuremath{-}{2\ensuremath{\gamma}}_{b}m{\ensuremath{\omega}}_{b}{P}_{\ensuremath{\theta}}\ensuremath{-}{2\ensuremath{\gamma}}_{b}m{\stackrel{^}{T}}_{\ensuremath{\perp}b})\ensuremath{\delta}({p}_{z}\ensuremath{-}{\ensuremath{\gamma}}_{b}m{V}_{b})$, where ${p}_{\ensuremath{\perp}}^{2}={p}_{r}^{2}+{p}_{\ensuremath{\theta}}^{2}$, ${P}_{\ensuremath{\theta}}$ is the canonical angular momentum, and ${\stackrel{^}{n}}_{b}$, ${\ensuremath{\gamma}}_{b}$, ${\ensuremath{\omega}}_{b}$, ${\stackrel{^}{T}}_{\ensuremath{\perp}b}$, and ${V}_{b}$ are positive constants. For $\frac{\ensuremath{\delta}B}{{B}_{0}}\ensuremath{\ll}1$ and slow beam rotation (${\ensuremath{\omega}}_{b}\ensuremath{\ll}{\ensuremath{\omega}}_{\mathrm{cb}}=\frac{e{B}_{0}}{{\ensuremath{\gamma}}_{b}\mathrm{mc}}$), the equilibrium density is uniform (${\stackrel{^}{n}}_{b}$) out to the beam radius ${\stackrel{^}{R}}_{b}={(\frac{2{\stackrel{^}{T}}_{\ensuremath{\perp}b}}{{\ensuremath{\gamma}}_{b}m{\ensuremath{\omega}}_{b}{\ensuremath{\omega}}_{\mathrm{cb}}})}^{\frac{1}{2}}$. Detailed free-electron-laser stability properties are investigated for the case where the amplifying radiation field has TE-mode polarization with perturbed field components ($\ensuremath{\delta}{\stackrel{^}{E}}_{\ensuremath{\theta}}$,$\ensuremath{\delta}{\stackrel{^}{B}}_{r}$,$\ensuremath{\delta}{\stackrel{^}{B}}_{z}$). The matrix dispersion equation is analyzed in the diagonal approximation, and it is shown that the positioning of the beam radius (${\stackrel{^}{R}}_{b}$) relative to the conducting wall radius (${R}_{c}$) can have a large influence on the growth rate and detailed stability properties. Analytic and numerical studies show that the growth rate increases as $\frac{{\stackrel{^}{R}}_{b}}{{R}_{c}}$ is increased.