A self-guided Gaussian laser pulse propagating in a non-uniform plasma channel is unstable to a plasma wave perturbation co-moving with the laser with its group velocity. The plasma wave amplitude has a maximum radial profile along the laser propagation axis. As the plasma wave propagates through a non-uniform plasma channel, the plasma wave perturbation causes focusing in the part of the laser that propagates with the electron density trough and defocusing in the part moving with the crest. This yields an axial gradient in the intensity of the laser and produces a ponderomotive force on the electrons. The ponderomotive force drives the plasma wave, which in turn makes the modulational instability grow. The growth rate of the modulational instability becomes larger with the increase in non-uniformity of a shallow plasma channel. In a deep cavitated plasma channel the growth rate of the modulational instability increases with the laser pulse amplitude.

We study the generation of resonant third harmonic laser radiation in a density non-uniform rippled plasma channel. An introduction of plasma channel non-uniformity strongly enhances the self-focusing and compression of main laser pulse at lower powers. In a deeper plasma channel, self-focusing is less sensitive to laser amplitude variation but increases compression. Plasma density ripple ‘nq’ leading to resonant third harmonic generation when kq = 4ω2p/3meω0cγ0, where ‘ω’p is electron plasma frequency, ‘ω0’ is laser frequency, and ‘γ0’ is the electron Lorentz factor. Third harmonic is produced through the beating of ponderomotive force induced second harmonic density oscillations and the oscillatory velocity of electrons at main laser frequency. The self-focusing and compression of the fundamental pulse periodically enhances the intensity of the third-harmonic pulse at lower powers of main laser. In a deeper plasma channel, the third harmonic power is less effective by self-focusing and the compression of main laser, and increase with main laser pulse power.