Phase properties of the cutoff high-order harmonics

The cutoff regime of high-order harmonic generation (HHG) by atoms in an intense laser field is studied numerically and analytically. We find that the cutoff regime is characterized by equal dephasing between the successive harmonics. The change of the harmonic phase locking when HHG evolves from the cutoff to the plateau regime determines the optimal bandwidth of the spectral region which should be used for attosecond pulse generation via the amplitude gating technique. The minimal pulse duration which can be obtained with this technique in argon without using dispersion elements is approximately 0.08–0.1 of the laser cycle for different intensities and frequencies of the fundamental. The cutoff regime is also characterized by a linear dependence of the harmonic phase on the fundamental intensity. The proportionality coefficient grows as the cube of the fundamental wavelength, thus this dependence becomes very important for the HHG by midinfrared fields. Moreover, for every high harmonic there is a range of laser intensities providing the generation in the cutoff regime and the atomic response magnitude in this regime can be greater than that in the plateau regime. Thus, the cutoff regime substantially contributes to the harmonic energy emitted under typical experimental conditions where the laser intensity varies in time and space.

Figure 1

The phase coefficient $\alpha$ for H17 generated in Xe as a function of laser intensity. The result obtained via Eq. (6) (blue line) and the approximation for the cutoff region (7) (thick red line) are presented. Lines with symbols show the results of two methods based on the numerical three-dimensional TDSE solution (the methods are described in the Appendix).

Figure 2

${\alpha }^{\left(1\right)}$ for the shortest trajectory contribution (solid blue) and the intensity (dotted black) of this contribution. The thick red line shows $\alpha$ for the cutoff harmonics given by Eq. (7). The results of method 1 (open circles) and method 2 (solid circles) calculated for H17 generated in Xe for (a) the laser wavelength 800 nm, (b) for H27 generated in Ar by the 800 nm laser field, and (c) for H57 generated in Ar by the 1067 nm laser field.

Figure 3

The intensities corresponding to the cutoff-plateau transition obtained theoretically (red line) and approximately (blue dashed curve) for different fundamental frequencies. The three-step (“simple-man”) and exact cutoff intensities (see text) are shown with light- and dark-green lines, respectively. The results are presented for (a) H17 generated by Xe atom and (b) H91 generated by He atom. The blue-filled area shows the region where the ionization is more than 10%.

Figure 4

Calculated attosecond pulses obtained by using all harmonics higher than $q_{\text{low}}$ shown in the legends. (a) Laser frequency is $1.5{\omega }_{\text{Ti:sapphire}}$ and intensity $4.6\times {10}^{14}$ W/${\mathrm{cm}}^2$, corresponding to cutoff at the $21\mathrm{st}$ harmonic. (b) Laser frequency is ${\omega }_{\text{Ti:sapphire}}$ and intensity $2.4\times {10}^{14}$ W/${\mathrm{cm}}^2$, corresponding to the cutoff at the $39\mathrm{th}$ harmonic.

Figure 5

Parameter $\beta$ (solid lines) and the single-attosecond-pulse duration (dashed lines) for certain values of $\beta$ for different laser intensities corresponding to 41.9 eV (red), 51.2 eV (green), and 60.5 eV (blue) cutoff harmonic energies.

Figure 6

Laser pulse intensity (solid blue line) given by Eq. (A1). After ${\tau }_f=3$ cycles turning on, the intensity increases linearly during $\tau =8$ cycles, and then turns off during 3 cycles. Dotted red line presents the square of the field.

Figure 7

Harmonic spectrum found via numerical three-dimensional TDSE solution for the model Xe atom. The laser wavelength is 800 nm and the laser intensity envelope is shown in Fig. 6. Red dashed lines show exact harmonic frequencies. One can see the shortest trajectory contribution (1), next-to-shortest trajectory contribution (2), and the contributions from other trajectories.

Figure 8

$\alpha$ maps calculated for different $\tau$ and $\gamma$ for H17 in Xe. The laser wavelength is 800 nm.