Oscillating two-stream instability of a plasma wave in a plasma channel

A long wavelength Langmuir wave (ω₀, k₀), propagating through a parabolic plasma density channel, can decay into a low-frequency mode (ω,k&oarr;) and two short wavelength Langmuir wave sidebands (ν_1,2,k&oarr;_1,2 ), via two-stream instability where ω_1,2=ω∓ω₀ and k&oarr;_1,2=k&oarr;∓k&oarr;₀. Depending on the mode number n, the growth rate maximizes in the range γ_max≃0.1ω_pi-0.4ω_pi for the range of k from 0.1(ω_pi/c_s) to 0.2(ω_pi/c_x) for ν₀~ν_th where ν₀ and ν_th are the oscillatory and thermal velocities of electrons, ω_pi is the ion plasma frequency on the axis, and c_s is the sound speed. The growth rate increases with the width a of the plasma density channel. It decreases with the mode number. The instability may be relevant to laser based charged particle accelerators