Relativistic harmonic content of nonlinear electromagnetic waves inunderdense plasmas

The relativistic harmonic content of large-amplitude electromagnetic waves propagating in underdense plasmas is investigated. The steady-state harmonic content of nonlinear linearly polarized waves is calculated for both the very underdense (w_p/w ₀)≪1 and critical density (w_p/w₀)≃1 limits. For weak nonlinearities, eE₀/mcω₀<1, the nonlinear source term for the third harmonic is derived for arbitrary w_p/w₀. Arguments are given for extending these results for arbitrary wave amplitudes. It is also shown that the use of the variable x-ct and the quasi-static approximation leads to errors in both magnitude and sign when calculating the third harmonic. In the absence of damping or density gradients the third harmonic's amplitude is found to oscillate between zero and twice the steady-state value. Preliminary PIC simulation results are presented. The simulation results are in basic agreement with the uniform plasma predictions for the third-harmonic amplitude. However, the higher harmonics are orders of magnitude larger than expected and the presence of density ramps significantly modifies the results