${\mathrm{N}}_2$ HOMO-1 orbital cross section revealed through high-order-harmonic generation

We measure multi-orbital contributions to high harmonic generation from aligned nitrogen. We show that the change in revival structure in the cutoff harmonics has a counterpart in the angular distribution when a lower-lying orbital contributes to the harmonic yield. This angular distribution is directly observed in the laboratory without any further deconvolution. Because of the high degree of alignment we are able to distinguish angular contributions of the highest occupied molecular orbital 1 (HOMO-1) orbital from angle-dependent spectroscopic features of the HOMO. In particular, we are able to make a direct comparison with the cross section of the HOMO-1 orbital in the extreme ultraviolet region.

Figure 1

Experimental setup; see text for details.

Figure 2

Raw spectrum in logarithmic scale of harmonics generated from ${\mathrm{N}}_2$ as a function of delay between the first pump and the probe and of the harmonic energy. The second pump is present at a delay of 8.36 ps with respect to the first pump. Strong features at times of alignment for all harmonics are visible. However, further features at times of anti-alignment before the full alignment revival (delay $\approx 16$ ps) become more visible as the harmonic energy increases.

Figure 3

Yield of harmonics H21, H23, H25, H31, H35, and H39 from ${\mathrm{N}}_2$, normalized to the respective isotropic values, as a function of delay between the first pump and the probe in 20 fs steps (black). In all cases the second pump is at a delay of 8.36 ps with respect to the first pump. Harmonics 21 and 23 show the same behavior as our calculations of $\langle {cos}^4\theta \rangle$, depicted in green (top panel). Harmonics 31 and higher show a peak at the anti-alignment dip (16.78 ps) in front of the full revival (17.04 ps). In particular, for harmonic 39 this peak becomes brighter than the full-revival peak. There are other smaller features in the delay scan of harmonic 39 that show a clear departure from the expected $\langle {cos}^4\theta \rangle$. Also shown is the fitting using the procedure outlined in Ref. [18] (red). The error bars are the standard deviation from 18 averages, resulting in a standard error of the mean that is $\sqrt{18}$ smaller than the standard deviation.

Figure 4

Measured harmonic yield of individual harmonics as a function of angle between pump and probe at the full revival of 17.04 ps ($\langle {cos}^2\theta \rangle \approx 0.67$). The experimental data were generated under the same conditions as presented in Fig. 3. Each harmonic yield is normalized to its corresponding isotropic value for an unaligned molecular ensemble. The red circles represent radial values of 1 and 3 units.

Figure 5

A comparison between the experimental angle scans (solid red line) at the full revival of 17.04 ps and theoretical calculations of the photoionization cross sections of HOMO (dotted line) and HOMO-1 (dashed line) from Ref. [43]. The cross section is normalized to 1, while the experimental data are normalized to the isotropic harmonic yield. The dotted line of the PICS of HOMO shows a strong minimum at $0^{\circ }$ and ${180}^{\circ }$ that is due to the Cooper minimum in nitrogen. Without better alignment, this feature cannot be fully resolved. However, the feature at ${90}^{\circ }$ can only be explained by the strong contribution from HOMO-1 to HHG (dashed line). As we then look at the experimental data as a function of harmonic order, the yield at ${90}^{\circ }$ gets stronger as we increase the harmonic order, compared with $0^{\circ }$ and ${180}^{\circ }$.