# Interference-Induced Phenomena in High-Order Harmonic Generation from Bulk Solids
## 3 Results  
### General Properties of the HHG Spectra  
The HHG spectra that can be obtained as described above, show a weak dependence on the number of the conduction bands that we take into account. According to our experience, for moderate exciting field intensities, a four-band model (VB + 3 CBs) is sufficient in the sense that in adding more CBs, the results practically do not change. (Note that a peak field strength of \(F_{0}=1\) GV/m, together with a focal spot size of around 50 microns and additional pulse parameters given in the caption of Figure 3, means pulse energies of the order of 0.01 mJ). Additionally, in order to see the physical meaning of the relevant parameter ranges, it is worth providing the ratio of some important energies and frequencies. For a peak field strength of 1 GV/m (that corresponds to \(1.33\times 10^{11}\) W/cm\({}^{2}\) peak intensity) and a characteristic lattice constant of 5 A, the Bloch frequency \(\omega_{B}=aeF_{0}/\hbar\) has the same order of magnitude as \(\omega_{L}\) at \(\lambda_{L}=3\) microns. \(\hbar\omega_{L}\) is 7.7 times smaller than the minimal bandgap (3.2 eV), and more than 50 times smaller than the maximal bandgap (for both potentials). Considering \(aeF_{0}\) to be the characteristic laser-induced energy, we find that it has the order of magnitude of a few eV, which is considerably below the binding energy of both of the potentials--see Figure 2.  
As expected, the choice of the periodic potential \(U\) (as given by Equations (6) and (7)) also influences the resulting spectra. However, we found that the qualitative features of the HHG spectra to be discussed below, are general--that is, the same for both potentials. Therefore, apart from a single example (see the next subsection), we are not going to compare all the results for \(U^{(a)}\) and \(U^{(b)}\). Figure 3 shows representative HHG spectra calculated using \(U^{(a)}\). As can be seen, the usual structure of the high-order harmonic spectra can be identified in this Figure--the most dominant peak corresponds to the exciting frequency, and there is a plateau region with peaks of approximately the same heights, as well as a cutoff where the peaks disappear. A difference of several orders of magnitude can be observed between the maxima and minima of the spectra. Additionally, in this intensity regime, the cutoff scales linearly with the amplitude of the exciting field, in a qualitative agreement with experimental findings [17]. Quantitatively, however, we observe more harmonics than the ones seen in  
the experiment. For a more realistic model that takes e.g. propagation effects also into account, we can expect a better agreement. Furthermore, the increase of individual HH peaks as a function of the intensity of the excitation shows qualitative agreement with the experimental results--see Figure 4. Similarly to [17], the heights of the peaks follow a power law for low intensities, and we can also see a deviation from this law at higher intensities. Quantitatively, this change in behavior occurs slightly earlier than in the experimental data. Figure 4: The height of the seventh and 11 th high harmonic peak as a function of the peak field strength of the external field. (Please note the log–log scale.) Figure 3: Representative high-order harmonic spectra for different peak field strengths, which correspond to 0.083, 0.33, 0.75, and \(1.33\times 10^{11}\) W/cm\({}^{2}\) peak intensities. The band scheme of the target is derived using the potential \(U^{(a)}\). Parameters: \(\lambda_{L}=3\) μm and \(\tau=300\) fs. The frequency on the horizontal axis is given in units of \(\omega_{L}=2\pi c/\lambda_{L}\). To summarize, our model can reproduce most of the qualitative features of the experimental spectra, but since neither the propagation, nor the decay effects beyond the phenomenological ones were taken into account, no strong quantitative agreement can be found.