Relativistic Doppler Effect: Universal Spectra and Zeptosecond Pulses

We report on a numerical observation of the train of zeptosecond pulses produced by the reflection of a relativistically intense femtosecond laser pulse from the oscillating boundary of an overdense plasma because of the Doppler effect. These pulses promise to become unique experimental and technological tools since their length is of the order of the Bohr radius and the intensity is extremely high $\propto {10}^{19}\mathrm{W}/{\mathrm{c}\mathrm{m}}^2$. We present the physical mechanism, analytical theory, and direct particle-in-cell simulations. We show that the harmonic spectrum is universal: the intensity of $n$th harmonic scales as $1/n^p$ for $n<4{\gamma }^2$, where $\gamma$ is the largest $\gamma$ factor of the electron fluid boundary, and $p=3$ and $p=5/2$ for the broadband and quasimonochromatic laser pulses, respectively.