Nonlinear equilibrium and stability analysis of rippled, partially neutralized, magnetically focused electron beams

In the first part of this work a higher-order solution of the anharmonic oscillator equation describing the nonlinear rippled equilibrium state of magnetically focused, partially neutralized electron beams is given. Thus, using the method of harmonic balance, we derive a ripple-amplitude solution of the form where ϕ=ω₁t+β₀, ω₁being the nonlinear proper frequency and β₀a phase shift depending on the initial conditions. In the second part of the work we carry out a stability analysis of the nonlinear equilibrium state found in the first part with respect to long- and short-wavelength surface space-charge perturbations. In the framework of a local approximation the wave equation for the rippled beam is found to be a Hill type of equation which contains harmonic terms up to cos 3k_sz (k_s is the wavenumber of the ripple). This equation is solved by a resonant-mode coupling method; coupling of fast—fast, slow—slow and slow—fast waves is considered. The growth rates and band widths for different possible wave couplings are derived and compared.