Phase Properties of Laser High-Order Harmonics Generated on Plasma Mirrors

As a high-intensity laser-pulse reflects on a plasma mirror, high-order harmonics of the incident frequency can be generated in the reflected beam. We present a numerical study of the phase properties of these individual harmonics, and demonstrate experimentally that they can be coherently controlled through the phase of the driving laser field. The harmonic intrinsic phase, resulting from the generation process, is directly related to the coherent sub-laser-cycle dynamics of plasma electrons, and thus constitutes a new experimental probe of these dynamics.

Figure 1

Emission times of CWE and ROM attosecond pulses (superposition of harmonics 4 to 10 for CWE, and 11 to 35 for ROM), for a plasma of maximum density $110n_c$ ($n_c$ critical density at the laser frequency), driven by a 35-cycle ${cos}^2(t)$ laser pulse at 45° incidence. In (a), ions are kept fixed, and the exponential density gradient scale length is $\lambda /100$ ($\lambda /30$) for CWE (ROM). (b) The chirp parameter of individual CWE harmonics obtained from PIC simulations in this case. In (c) ions are allowed to move, and the initial density gradient is infinitely steep. (d) The fields of two individual CWE attosecond pulses in the train in this case.

Figure 2

(a) Measured spectral width of harmonic 13 generated by CWE ($a_0=0.25$ for FTL pulses), as a function of the dimensionless chirp parameter $\xi$ of the laser pulse. The images show HHG spectra obtained around harmonic 13 for the two different chirps indicated on the graph. The spectral width of harmonic 18 generated by ROM ($a_0=2$) with a FTL laser pulse is shown for comparison. (b) Diameter of the CWE harmonic beam in the far field, generated on a plastic target (orders $\approx 13$ to 15 filtered using a 100 nm thick Al filter), as a function of the distance between target surface and best focus ($a_0=0.6$ at best focus). The two upper images show the beam patterns for two different positions.

Figure 3

(a)–(b) $x-p_x$ phase-space densities of plasma electrons at two different times, corresponding, respectively, to (a) the frequency upshift of the reflected light by relativistic outgoing Brunel electrons, and (b) to the penetration of Brunel electrons in the density gradient. In the lower graphs, electrons occupying the areas in the white dashed boxes have been selected. (c) Trajectories and (d) velocity $v$ (along specular direction) of Brunel electrons responsible for ROM harmonics (fixed ions with imposed gradient of $L=\lambda /30$). (e) Trajectories of Brunel electrons triggering plasma oscillations (same conditions with $L=\lambda /60$). In each case, two sets of trajectories are shown, corresponding to two laser intensities. The upper panels show the amplitude of the magnetic field $B$ of the corresponding attosecond pulses.