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

A long wavelength Langmuir wave (/spl omega//sub 0/, k/sub 0/), propagating through a parabolic plasma density channel, can decay into a low-frequency mode (/spl omega/,k/spl I.oarr/) and two short wavelength Langmuir wave sidebands (/spl nu//sub 1,2/,k/spl I.oarr//sub 1,2/), via two-stream instability where /spl omega//sub 1,2/=/spl omega//spl mnplus//spl omega//sub 0/ and k/spl I.oarr//sub 1,2/=k/spl I.oarr//spl mnplus/k/spl I.oarr//sub 0/. Depending on the mode number n, the growth rate maximizes in the range /spl gamma//sub max//spl sime/0.1/spl omega//sub pi/-0.4/spl omega//sub pi/ for the range of k from 0.1(/spl omega//sub pi//c/sub s/) to 0.2(/spl omega//sub pi//c/sub x/) for /spl nu//sub 0//spl sim//spl nu//sub th/ where /spl nu//sub 0/ and /spl nu//sub th/ are the oscillatory and thermal velocities of electrons, /spl omega//sub pi/ is the ion plasma frequency on the axis, and c/sub 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.