Theory of high harmonic generation in relativistic laser interaction with overdense plasma

High harmonic generation due to the interaction of a short ultra relativistic laser pulse with overdense plasma is studied analytically and numerically. On the basis of the ultra relativistic similarity theory we show that the high harmonic spectrum is universal, i.e. it does not depend on the interaction details. The spectrum includes the power law part $I_n\propto n^{-8/3}$ for $n<\sqrt{8\alpha}\gamma_{\max}^3$, followed by exponential decay. Here $\gamma_{\max}$ is the largest relativistic $\gamma$-factor of the plasma surface and $\alpha$ is the second derivative of the surface velocity at this moment. The high harmonic cutoff at $\propto \gamma_{\max}^3$ is parametrically larger than the $4 \gamma_{\max}^2$ predicted by the ``oscillating mirror'' model based on the Doppler effect. The cornerstone of our theory is the new physical phenomenon: spikes in the relativistic $\gamma$-factor of the plasma surface. These spikes define the high harmonic spectrum and lead to attosecond pulses in the reflected radiation.