Macroscopic aspects of relativistic x-ray-assisted high-order-harmonic generation

A theoretical model is developed describing high-order-harmonic generation (HHG) from a gas of multiply charged ions driven by a laser field of relativistic intensity. Macroscopic propagation of harmonics is investigated in a relativistic HHG setup where the relativistic drift is suppressed by means of x-ray field assistance of the driving laser field. The possibility of phase-matched emission of the harmonics is shown. The laser field geometry is optimized to maximize the HHG yield with the corresponding phase-matching schemes. Crucial issues determining the macroscopic HHG yield are discussed in detail.

Figure 1

Geometry of the medium and detector including the definitions of the coordinates. The dashed lines denote the divergence angle of the harmonic radiation. The box on the right schematically shows the measured angular distribution.

Figure 2

Geometry of the HHG process for a collinear alignment of the x-ray and laser field. The copropagating x-ray field (orange) has a frequency above the ionization energy to achieve drift compensation. The weak IR field (brown) is employed to accomplish phase matching.

Figure 3

(a) Single-atom emission probability for $E_0=2.5\mathrm{a}.\mathrm{u}.$, $I_{\mathrm{p},\mathrm{x}}=8\mathrm{a}.\mathrm{u}.$, ${\omega }_{\mathrm{x}}=14\mathrm{a}.\mathrm{u}.$ and $E_{\mathrm{x}}=0.65\mathrm{a}.\mathrm{u}.$ (black) and a conventional laser field ($E_{\mathrm{x}}=0$) with $E_0=2.5\mathrm{a}.\mathrm{u}.$ and $I_{\mathrm{p},\mathrm{t}}=4.8\mathrm{a}.\mathrm{u}.$ (dashed black) and the same configuration in the DA (gray). For the second configuration, $I_{\mathrm{p},\mathrm{t}}$ is chosen such that the average tunnel-ionization rate is the same as the single-photon ionization rate in the case before. (b) Separate HHG yields of the three contributing quasiclassical trajectories for the discussed setup [blue line in (a)]. The dotted red contribution (long trajectory) is suppressed because the drift for the trajectory is not completely compensated (as discussed in Sec. 3b2). The short trajectory contributions [solid lines in blue (gray thick line) and orange (gray thin line)] are nearly identical and separate only at the cutoff region where the saddle point approximation breaks down.

Figure 4

(a) We show the ionization phase saddle points of the 50 keV trajectory for different x-ray frequencies, which is the same as ionization time in units of radian. The dashed line indicates the value chosen in Fig. 3. The dotted and solid lines belong to the long and short trajectories, respectively. (b) Displays the initial momentum direction for different initial energies ${\omega }_{\mathrm{x}}-I_{\mathrm{p},\mathrm{x}}$ (as indicated in a.u. next to the respective arrow) needed for the emission of a 50 keV photon. The upper and lower branch correspond to the short uphill and downhill trajectories, respectively, where the color indication coincides with the one in (a). Note that the ionization phases ${\eta }_2$ are different for the uphill and downhill trajectories. $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{z}}$ are the propagation and polarization directions of the laser, respectively. (c) Differential ionization rate depending on the initial energy ${\omega }_{\mathrm{x}}-I_{\mathrm{p},\mathrm{x}}$ from Eq. (41). The considered direction is in the initial momentum direction determined by the saddle-point equations. From (b) we see that the $z$ component is approximately $p_{\mathrm{z},\mathrm{c}}=2.9\mathrm{a}.\mathrm{u}.$ and the $x$ component depends on ${\omega }_{\mathrm{x}}-I_{\mathrm{p},\mathrm{x}}$.

Figure 5

(a) Real part of spectral component of the locally emitted HHG field at 48.6 keV at different positions along the propagation direction. The blue dashed line is for HHG without the quasiphase-matching scheme. The red line is for the case of adding the weak counterpropagating field to achieve QPM. (b) The macroscopically emitted spectral photon number via Eq. (9) is displayed for the QPM scenario.