Pump depletion in four-photon collinear laser frequency multiplication in plasma

It was shown recently that exactly resonant four-photon scattering of collinear laser pulses in

plasma can be enabled by mildly-relativistic nonlinear renormalization of the classical dispersion law

which does not allow the resonant collinear four-photon scattering. It remained unclear, however,

if the collinear four-photon scattering may produce a significant depletion of the pump pulses,

because a noticeable pump depletion would change the nonlinear renormalization of laser frequencies

by values much greater than the width of the four-photon resonance. Here, we present the first

example of the collinear four-photon scattering maintaining the exact resonance throughout seed

amplification up to the complete pump depletion. It may enable the all-optical nearly doubling

of frequency of mildly-relativistic-intense laser pulses in the collinear geometry free of challenges

associated with the transverse slippage between the pulses.