High Harmonic Generation and the Role of Atomic Orbital Wave Functions

High harmonic spectra were recorded from different rare-gas atoms under identical experimental conditions. It is shown that although each atom’s spectrum is different, the differences are due almost entirely to the orbital influence in the recombination step. The amplitude of the continuum electron wave packet versus kinetic energy is derived from these data and is shown to be largely independent of the atom, in agreement with models of tunnel ionization. We compare the measurements with calculations in both the length gauge and the velocity gauge and show that the two gauges imply a different de Broglie wavelength.

Figure 1

High harmonic signals recorded for three atoms under identical experimental conditions. Also shown are the calculated orbital shapes for He ($1s$), Ne ($2p$), and Ar ($3p$).

Figure 2

Length gauge transition dipole matrix elements $⟨{\psi }_0|x|k⟩$ between each atom’s HOMO, ${\psi }_0$, and a plane wave of momentum $k$, plotted against emitted xuv frequency $\Omega$ in multiples of the driving laser frequency. In (a), the field-free electron dispersion relation is used, $\Omega =k^2/2$. In (b), the ionization potential, $I_p$, has been subtracted from the emitted xuv frequency, $\Omega$, to determine the corresponding plane wave momentum, $k$: $\Omega =k^2/2+I_p$.

Figure 3

The recorded HHG spectra were divided by the transition dipole in the length gauge for each atomic orbital to yield the amplitude of the continuum electron wave packet as it returns to the atom. We plot ${\Omega }^2|a(\omega )|=\sqrt{S(\Omega )}/|D_{\mathrm{len}}(k)|$ and utilize the dispersion relation $\Omega =k^2/2$. Because the laser intensity was well above the saturation intensity for argon, argon’s higher harmonics fall off rapidly. The experimental data have been corrected for detector response and have been vertically scaled because of differing gas densities and ionization probabilities. The horizontal axis is the electron kinetic energy $\omega$ just before it interacts with the atomic potential, i.e., $\omega =\Omega -I_p$.

Figure 4

Experimental high harmonic spectra from Fig. 1 divided by the transition dipole matrix elements calculated using both the length (solid lines) and the velocity gauge (dashed lines). In the lower group, we plot ${\Omega }^2|a(\omega )|=\sqrt{S(\Omega )}/|D_{\mathrm{vel}}(k)|$ and utilize the dispersion relation $\Omega =k^2/2+I_p$. The length gauge curves are repeated from Fig. 3 to show that both gauges give similar results. The two gauges have been vertically displaced so that they can be compared.