High-order harmonic generation in helium: A comparison study

We report a detailed study of high-order harmonic generation (HHG) in helium. When comparing predictions from a single-active-electron model with those from all-electron simulations, such as the attomesa code and $R$-matrix-with-time-dependence method, which can include different numbers of states in the close-coupling expansion, it seems imperative to generate absolute numbers for the HHG spectrum in a well-defined framework. While qualitative agreement in the overall frequency dependence of the spectrum, including the cutoff frequency predicted by a semiclassical model, can be achieved by many models in arbitrary units, only absolute numbers can be used for benchmark comparisons between different approaches.

Figure 1

Spectrum for a 15-cycle pulse with a central wavelength of 248.6 nm with a ${\sin }^2$ envelope of the electric field and peak intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The results from our SAE model are compared with predictions from attomesa and three RMT calculations with a different number of coupled states. Note that the RMT-1st and RMT-3st results are practically indistinguishable. The predicted cutoff energy from the three-step model is approximately equal to 26 eV, corresponding to the fifth harmonic. The crosses are the results for the peak values obtained by Guan et al. [13] in the acceleration form.

Figure 2

Spectrum for a 30-cycle pulse with a central wavelength of 248.6 nm with a ${\sin }^2$ envelope of the electric field and peak intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. Again, the RMT-1st and RMT-3st results are practically indistinguishable.

Figure 3

Spectrum for a 30-cycle pulse with a central wavelength of 248.6 nm with a ${sin}^2$ envelope of the electric field and peak intensity of $5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The predicted cutoff energy from the three-step model is approximately equal to 34 eV, corresponding to the seventh harmonic.

Figure 4

Dipole expectation value as a function of time for a 30-cycle pulse with a central wavelength of 248.6 nm with a ${\sin }^2$ envelope of the electric field and peak intensity of $5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ (same laser parameters as in Fig. 3).

Figure 5

Difference between the dipole expectation values obtained with the RMT-1st and RMT-6st models relative to the results obtained with RMT-3st as a function of time for the laser parameters used to produce Fig. 3 (30-cycle pulse with a central wavelength of 248.6 nm with a ${\sin }^2$ envelope of the electric field and peak intensity of $5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$).

Figure 6

Spectrum for a 60-cycle pulse with a central wavelength of 1064 nm with a ${\sin }^2$ envelope of the electric field and peak intensity of $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. The results from the SAE model are compared with predictions from three RMT calculations with a different number of coupled states. The predicted cutoff energy from the three-step model is approximately equal to 41 eV, corresponding to the 35th harmonic.

Figure 7

Spectral density as defined in Refs. [11, 12] for a 60-cycle pulse with a central wavelength of 1064 nm with a ${\sin }^2$ envelope of the electric field and peak intensity of $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ (the parameters of Fig. 6). The $y$ axis is logarithmic.

Figure 8

Dipole expectation value as a function of time for a 60-cycle pulse with a central wavelength of 1064 nm with a ${\sin }^2$ envelope of the electric field and peak intensity of $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ (the parameters used to produce Fig. 6).

Figure 9

Difference between the dipole expectation values obtained with the RMT-1st and RMT-6st models relative to the results obtained with RMT-3st as a function of time for the laser parameters used to produce Fig. 6 (central wavelength of 1064 nm with a ${\sin }^2$ envelope of the electric field and peak intensity of $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$).

Figure 10

Super-Gauss window function centered on the plateau with different widths to focus on the frequencies of the plateau in Fig. 6.

Figure 11

Reconstructed part of the dipole moment with the wider super-Gauss window function centered on the plateau of Fig. 6.

Figure 12

Reconstructed part of the dipole moment with the narrower super-Gauss window function centered on the plateau of Fig. 6.

Figure 13

Same as Fig. 6 for a peak intensity of $4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The predicted cutoff energy from the three-step model is approximately equal to 160 eV, corresponding to the 137th harmonic.

Figure 14

Spectral density as defined in Refs. [11, 12] for the laser parameters used to produce Fig. 13. The $y$ axis is logarithmic.