Control and enhancement of multiband high harmonic generation by synthesized laser fields

We start by studying the nonlinearity that underlies the enhancement mechanism by resolving the HHG yield and efficiency for various fundamental and bichromatic synthesized fields. The basic characteristics of the measured spectra can be seen in the representative experimental results shown in figure 2. Three key features are common to all the bichromatic measurements. The first, and most striking feature is the dramatic enhancement of the HHG yield for the bichromatic field compared with the single colour field. For the example shown in figure 2(a), the energy in the bichromatic field is about 20% higher than the single colour field, yet it produces an order of magnitude increase in the total HHG yield. Second, we observe an increase in spectral density due to the non-commensurate nature of the fundamental frequencies of the fields. The appearance of 'intermediate' harmonic orders, also observed in gas phase HHG, originate from the non-periodic nature of the field [24, 28]. Third, a high energy extension of the HHG cutoff ranging from 18–25 eV appears for each realization of the bichromatic field.

Zoom In Zoom Out Reset image size

The multi-plateau structures in the spectra can be understood by examining the band structure of MgO, as shown in figure 3. For interband HHG, the bandgap energy, and thereby the emitted XUV energy following electron–hole recombination, can be mapped to a corresponding point in the crystal momentum space. For the lower energy conduction band, shown in blue, the maximum bandgap energy is ∼18 eV. As shown in figure 2(a), below a specific intensity of the single colour field (∼0.7–0.8 × 10¹³ W cm⁻²), only the lower conduction band is populated and the 18 eV cutoff is observed. At intensities greater than ∼0.8 × 10¹³ W cm⁻² (figure 2(b)), excitation to the higher energy conduction band is induced and an additional plateau forms, with the cutoff energy extended from ∼18 eV to >25 eV. Previous studies have established that the extended cutoffs originate from multiple conduction bands [4, 10]. The higher energy cutoff corresponds to recombination at the maximum bandgap energy of the higher energy conduction band, shown in red in figure 3(b). In contrast to gas phase HHG, where the cutoff typically scales continuously with the field intensity, the increase in cutoff energy is discrete in solid state systems [29, 30].

Zoom In Zoom Out Reset image size

The cutoff extension appears for all the bichromatic fields. Although the power in the 800 nm field is modest, the peak intensity increase upon addition of the second field is substantial, leading to a sub-cycle nonlinear enhancement as discussed in section 1. Upon constructive interference of the fields, the maximum attainable peak intensity I_peak when the pulses are overlapped in time can be estimated as:

$$\begin{equation}{I}_{\text{peak}}={I}_{0}(1+\alpha +2\sqrt{\alpha }),\end{equation} \tag{ 1 }$$

where I₀ is the peak intensity of the stronger (1300 nm) pulse, and α = I₁/I₀ is the relative intensity of the two pulses, where I₁ is the intensity of the weaker (800 nm) pulse. If the pulse widths are equal, which can be assumed in these experiments, then α is simply the power ratio between the two pulses. From equation (1) and table 2, we see that all synthesized peak intensities exceed the ∼0.7–0.8 × 10¹³ W cm⁻² threshold for observing the higher energy plateau, which accounts for its appearance in all of the bichromatic spectra. The cutoff extension is dependent on the peak intensity and thus can be induced even when the combined two colour power is lower than the required power for the single colour field.

Table 2. Peak intensities of the synthesized fields in table 1, normalized in units of 10¹³ W cm⁻² yellow, purple, and cyan highlights correspond respectively to the field combinations with the five highest, five lowest, and four median peak intensities.

Similarly, the HHG yield for the bichromatic field is markedly higher than for a single colour field with the same total energy. Interference of the two fields drastically modulates the pulse envelope and concentrates the pulse energy in a smaller number of cycles with larger field amplitudes [26]. Thus the field amplitude, and consequently the peak intensity, within the modulated bichromatic pulse (shown by the red curve in figure 1(b)) can be considerably larger than a single colour pulse at the same power (blue transparent curve in figure 1(b)). Previous studies in the gas phase showed the ability of the bichromatic field to enhance the tunnelling probability, leading to large HHG enhancements spanning the XUV and soft x-ray regions [24, 25]. In this work, due to the exponential dependence of ionization on the laser field strength, the bichromatic field leads to higher interband currents (and consequently, higher HHG yield) compared with the single colour case, where the energy is spread over more cycles with lower field amplitudes.

We can now summarize the HHG yield and efficiency for the complete set of bichromatic spectral measurements. Figures 4(a)–(d) show the measured spectra for single colour and three sets of bichromatic synthesized fields. For each subfigure, we show spectra for five powers of the 1300 nm pulses at a single fixed power of the 800 nm pulse. All the subfigures share the same scale, allowing their direct comparison. Even at the lowest added power of the 800 nm pulse (figure 4(b)), the increase in spectral density from the intermediate harmonic orders is clearly visible, along with an increase in HHG amplitudes across the complete spectrum. The HHG flux progressively increases with each corresponding increase in the 800 nm power. In figure 5, we show the spectrally integrated HHG yields as a function of the 1300 nm pulse power. This figure summarizes the nonlinear scaling of the HHG for the complete set of synthesized fields. In particular, we can extract the yield enhancement (ratio between the bichromatic and single colour yields) for each constant value of the 1300 nm pulse power. This yield enhancement ranges from one order of magnitude, for larger 1300 nm powers, to a full two orders of magnitude for lower 1300 nm powers.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The enhanced HHG flux driven by the synthesized field was calibrated by measuring the XUV-induced current from an Si XUV photodiode (AXUV100G). For the maximum applied input power (164 mW in total power in both pulses), and after taking the known XUV transmission of the two Al/Mg/Al filters into account, we estimate a total XUV power of 0.1 nW, or 0.1 pJ/pulse at the 1 kHz repetition rate of the laser. This corresponds to a conversion efficiency of 6 × 10⁻¹⁰. Considering that the interaction length in solid state materials is highly limited due to XUV absorption, this is impressive considering the very low effective volume for HHG emission in the crystal.

Despite the high peak intensities used, we observed no damage to the MgO crystal during the duration of the measurements. This may indicate that the modulated pulse envelope created by the synthesized field is more favourable for limiting crystal damage, allowing the increase of the peak intensity while minimizing the incident pulse power on free standing crystals.

In the following stage, we obtain a deeper insight into the underlying mechanism and resolve the HHG yields for different spectral ranges. We consider three plateau regions of equal widths in energy. Plateaus 1 and 2 span between 14–17 eV and 21–24 eV respectively. Prior studies have shown that these regions are dominated by interband HHG between the two distinct conduction bands and the valence band (see figure 3 and references [4, 31]. A third plateau, which we term as plateau C, spans between 11–14 eV and was associated with an intraband process [32]. In figure 6 we plot the integrated HHG yield within each plateau region as a function of the 1300 nm field power for various values of the 800 nm field power. Plateaus 1 and 2 display very similar scaling characteristics, their comparison requires deeper analysis that will be presented in the next section. In contrast, the yield in plateau C scales far more weakly with power, as seen by the flatter curves compared with plateaus 1 and 2. This observation suggests that an alternate mechanism is responsible for driving the HHG at these lower energies.

Zoom In Zoom Out Reset image size

For each plateau, adding the 800 nm field results in progressively higher signal yields and a decrease in the exponential scaling factor. The strongest scaling occurs for the single colour fields, as seen by the slope of the lines in figure 6(a). Each subfigure can be matched to a column of synthesized peak intensity values in table 2. In the single colour case (figure 6(a)), the 1300 nm pulse power increases by 72%, from 70 mW to 120 mW, which corresponds to peak intensities between 0.58–1 × 10¹³ W cm⁻². In figure 6(b), with 6 mW of 800 nm pulse power, the bichromatic peak intensity ranges from 0.97–1.50 × 10¹³ W cm⁻². Compared with the single colour case, these start from a higher intensity value and have a smaller relative increase of 55%. This trend continues as the 800 nm pulse power is further increased. Therefore, for low fundamental intensities, the synthesized field has a larger influence on the nonlinear efficiency of the HHG process. In comparison, for higher single colour intensities, the nonlinear enhancement is lower upon addition of the second field, which infers the observed scaling trends from figures 6(a)–(g). Saturation of the interband currents with increasing peak intensity may also affect the scaling dependence.

A deeper look into the underlying mechanism requires a complete spectral analysis of the enhancement efficiency. In gas phase HHG, the strong field approximation provides a straightforward mapping between each XUV energy and its ionization time and probability. When driven by a bichromatic field, the ionization probability is enhanced throughout the entire ionization window. When HHG is dominated by the microscopic response, large signal enhancements are seen across the entire XUV spectral range [24, 25]. In contrast, solid state HHG offers more complex dynamics and therefore multiple mechanisms of spectral enhancement. In solids, the tunnel ionization yield, the excitation to higher bands, and the dynamics of the recombination step are controlled by the synthesized field. These combined effects give rise to enhancements with a pronounced dependence on the XUV energy.

In the following step, we perform a spectrally resolved study of the enhancement process. For a synthesized field with powers P₀ of the 1300 pulse and P₁ of the 800 nm pulse, we first extract the peak height H for each harmonic peak N in the spectrum. The enhancement factor F_enh for each harmonic energy ℏω_N is thus ${F}_{\text{enh}}=\frac{H({P}_{0},{P}_{1},\hslash {\omega }_{N})}{H({P}_{0},6\enspace \mathrm{m}\mathrm{W},\hslash {\omega }_{N})}$. Stated differently, F_enh  compares the bichromatic spectral peaks for driving field powers (P₀, P₁) with the spectral peaks in figure 4(b) for the same power P₀ and the lowest power of the 800 nm pulse. This analysis resolves the signal enhancement by the synthesized fields at ∼0.6 eV spectral intervals. The non-commensurate nature of the synthesized field is the key to achieving this resolution, owing to the increase in spectral density from the 'intermediate' harmonic orders. This provides additional spectral information that is not obtainable in single colour or two colour commensurate experiments [4, 33].

In general, increasing the power of the 800 nm pulse increases the enhancement factor as expected. However, the most striking result is the sharp rise in enhancement starting in the vicinity of 15–17 eV and extending through to the cutoff of the high energy plateau >25 eV. The 15–17 eV spectral range corresponds to plateau 1 energies where HHG is dominated by interband recombination between the lower conduction band and the valence band. The effect is strongest for synthesized fields with higher values of α. In contrast, the lowest F_enh are seen around 11–15 eV, overlapping with plateau C. In this region there is also a weak dependence of F_enh on the 1300 nm power for all bichromatic field combinations.

From gas phase HHG results, we expect large enhancements when α and the ionization energy I_p are large, thereby providing a large relative increase in the sub-cycle ionization probability. In MgO, tunnel ionization generally occurs near the Γ-point where the bandgap is at a minimum. If the enhancement was entirely due to an increase in the tunnel ionization rate, then both intraband and interband HHG would benefit from the increase in electron current in the first conduction band, leading to similar F_enh values for both processes. Therefore, we conclude that additional enhancement mechanisms are involved. Recent studies show that the most prominent HHG emission in MgO occurs via enhanced constructive interference of electron trajectories at the recombination step [4]. The interference is induced at the critical points in the band gaps, where the gradient of the bandgap energy with respect to k approaches zero, giving rise to spectral caustics [34]. From the lower energy conduction band shown in figure 3, the expected HHG energy of the caustic is ∼17 eV. In the bichromatic data, F_enh indeed rises with increasing energy once recombination becomes the dominant mechanism as revealed by the presence of the spectral caustic in plateau 1.

We can interpret the low F_enh in plateau C in the following way. For low peak intensities, electrons are driven at lower crystal momenta, where the intraband process can be significant. This is observed in the intregrated yield measurements shown in figures 6(a) and (b), where the signal from plateau C approaches, or can even exceed that of plateau 1 for low power and peak intensity of the bichromatic fields. A similar situation was described in references [32, 35] in cases where interband HHG could be suppressed. However, for larger intensities, electron current is driven more strongly to the Brillouin zone edge by the synthesized field, where additional processes contribute. The subsequent dynamics at higher intensities, which include recombination to the valence band and transitions to higher energy bands, depend sensitively on the band gap energy and sub-cycle field structure [36].

The enhancement in plateau 2 displays a more complex behaviour. For lower powers of the 800 nm pulses (figures 7(a) and (b)), F_enh is slightly lower for plateau 2 compared with plateau 1, whereas for higher 800 nm pulse powers (figures 7(d) and (e)) it is slightly higher. Clearly these effects cannot be explained by a straightforward increase in the interband tunnel ionization probability.

Zoom In Zoom Out Reset image size

The role of sub-cycle field-driven dynamics in multiband systems have been at the focus of a number of theoretical studies. The electron wavepacket's crystal momentum |k| is dependent on the vector potential of the laser field. Fields with a larger vector potential are more effective at accelerating the wavepacket to the edge of the Brillouin zone, where tunnel ionization between conduction bands is most likely to occur. However, for a single colour field, the field strength is zero where the vector potential is maximal. This creates unfavourable conditions for multiband HHG, and may account for the low HHG yield from higher energy plateaus [30, 36]. A multicolour synthesized field can overcome this limitation by tailoring the ionization times and probabilities at different sub-cycle points within the field. For multiband systems, such control results in larger interband currents and increased HHG signal yield, particularly for higher energy plateaus [13, 16]. Our measurements can thus serve as a benchmark for probing the temporal dynamics of multiband HHG emission. In addition, we note that if the peak intensity is sufficiently strong (see table 2), one must consider excitation to higher conduction bands, the dynamical band structure, and its effect on the ionization and recombination probabilities [37, 38].

In section 3.3, we discussed the importance of the sub-cycle electric field structure and its influence on the HHG yield. To this point, we have considered the scaling of the HHG with respect to the power in the synthesized field. What can we learn by looking at additional field parameters, such as the peak intensity or peak vector potential? In a multicolour field, the power level and the peak intensity can be independently controlled. For example, in tables 1 and 2 we see that the highest, lowest, and median combined powers of the synthesized field do not correspond precisely with the highest, lowest, and median combined peak intensities. A similar comparison can be made with the peak vector potential (table 3). What is the dependence of the HHG yield on these parameters?

Table 3. Peak vector potentials of the synthesized fields in table 1. The values are normalized in units of the vector potential of the highest intensity single colour field (1300 nm central wavelength, intensity = 10¹³ W cm⁻²), corresponding to a vector potential of 3.8 × 10⁻⁷ V s cm⁻¹. Yellow, purple, and cyan highlights correspond respectively to the peak vector potentials with the five highest, five lowest, and four median values.

In figure 8, the integrated HHG signals from each of the three plateaus and all 30 synthesized fields are combined onto a single graph. The signals are plotted as a function of the peak intensity (in units of 10¹³ W cm⁻²) for each respective bichromatic synthesized field according to equation (1). Figure 8 allows a direct comparison of the enhancement mechanism, induced by the various combinations of the synthesized fields. If the HHG process scales straightforwardly with the peak intensity, then different field combinations with similar intensities would lead to similar HHG signal yields. Somewhat surprisingly, this simple dependence is observed for the yield from plateau C—there is almost no spread in the yield for different synthesized fields at any value of the peak intensity. Indeed, for intraband HHG, the electrons sample the nonlinearity of the conduction band where the signal is largely determined by the magnitude of the current that is tunnel ionized from the valence band, dictated by the strength of the electric field.

Zoom In Zoom Out Reset image size

For plateaus 1 and 2, in which interband HHG is dominant, the observation is significantly different. As an example, we can consider peak intensities in the range of 1.9 × 10¹³ W cm⁻². The dark blue dashed oval in figure 8 highlights the integrated HHG yield within plateau 1 for five different combinations of the two fields. Although the power in the 1300 nm pulse is different for each of the five synthesized fields, there is only a ±2% variation in the peak intensities. Nevertheless, the measured signal yield varies by more than a factor of five. Similar observations can be made for the signal yield in plateau 2. However, if we inspect the peak vector potentials for these fields (normalized to that of a 1300 nm pulse with intensity of 10¹³ W cm⁻², see table 3), we find that the largest value (1.0(1300 nm) + 0.22(800 nm) = 1.22) corresponds to the largest HHG yield, and the smallest value (0.76(1300 nm) + 0.37(800 nm) = 1.14) corresponds to the smallest HHG yield. Due to the inverse frequency dependence of the vector potential, reducing α while keeping the peak intensity fixed, increases the vector potential of the synthesized fields. Indeed, figure 8 shows the signal yields are well ordered with respect to the peak vector potentials at almost any value of the peak intensity. The synthesized fields with the highest 1300 nm pulse power (darker shaded symbols) consistently produce the highest HHG yields in plateau 1 and plateau 2. For decreasing 1300 nm pulse power (progressively lighter shaded symbols), the HHG yield also tends to decrease monotonically.

Interference between the fields highly modulates the temporal envelope of the synthesized pulse. A more complete analysis must go beyond consideration of the peak intensity or vector potential, and study the multiple bands dynamics induced by the complete, multicycle synthesized pulse. Such studies have only been carried out theoretically [13, 15, 16]. Nevertheless, using these simple scaling relations we can experimentally resolve the spectral dependence of the peak intensity and vector potential in our data. These observations directly connect the sub-cycle field structure and the electronic dynamics that give rise to HHG in multiband systems.