Relativistic harmonic content of nonlinear electromagnetic waves in underdense 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