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.

Figure 1

A sketch of possible motions of the plasma surface. (a) Single “hump” per half-laser period: two saddle points are responsible for the generation of the $n$th harmonic. (b) More complex plasma surface motion; the saddle points appear in pairs, two per hump. Interference between the different saddle points leads to modulations of the spectrum [7].

Figure 2

Spectra of the reflected radiation for the laser amplitudes $a_0=5,10,20$. The broken line marks the universal scaling $I\propto {\omega }^{-5/2}$.

Figure 3

Electron distribution function The helix represents the electron surface motion in the laser field. The reddish downward spikes stay for the surface relativistic motion towards the laser. These spikes are responsible for the zeptosecond pulse generation.

Figure 4

Zeptosecond pulse train: (a) temporal structure of the reflected radiation, (b) zeptosecond pulse train seen after spectral filtering, (c) one of the zeptosecond pulses zoomed; its FWHM duration is about 300 zs.