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Abstract
The enhancement of even-order harmonics near the cut-off of high-order harmonic spectra from monolayer MoS2 has been experimentally observed recently by several groups. Here we demonstrate that this enhancement can be interpreted as a result of spectral interference between half-cycles with opposite polarity by adopting a fully quantum mechanical calculation. We found that, due to the energy modulation induced by Berry connections, only half-cycles with the same polarity can generate high-order harmonics near the cut-off frequency, thus the lack of destructive interference leads to the enhanced intensity of the corresponding even-order harmonics. The explanation is supported by the frequency shift of the measured harmonic peaks. Our finding revealed the role of inter-half-cycle interference in high-harmonic generation (HHG) from non-centrosymmetric materials.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
HHG from gas phase has long been investigated as a method to produce coherent broadband extreme ultraviolet (EUV) light and attosecond pulses [1,2]. Recently, HHG from solid phase has been reported from a variety of materials [3–15], and appears promising for a compact, highly efficient and tailorable attosecond EUV source [5,11,12,16–19]. Its ability to probe material electronic properties [20–23] and ultrafast electron dynamics [4,24,25] has also been proved. While progresses have been made to model the process in terms of interband transition and intraband electron motion [26–30], the underlying mechanics are still under debate.
One topic that attracts much attention is the generation of high-order harmonics from materials without centrosymmetry [8,24,31–35]. In centrosymmetric systems, the electron motions in positive and negative half-cycles of the laser field are identical. The harmonics generated in two adjacent half-cycles interfere with each other and the even-order harmonics are cancelled out. In non-centrosymmetric systems, as the electron motion varies in different half-cycles, the inter-half-cycle interference is not completely destructive, which is the microscopic origin of even-order harmonics. Since experiments [36] have shown that the macroscopic nonlinear propagation effects in bulk materials can strongly modify the even harmonic yield, the microscopic mechanism needs to be disentangled from bulk effects when studying the dynamics. To serve this purpose, measuring high-harmonics from two-dimensional materials is an effective approach. HHG from non-centrosymmetric two-dimensional materials has been reported in experiments [8,33]. Especially, Liu [8] and Yoshikawa [33] have found the enhancement of even harmonic intensity near the cut-off frequency in monolayer MoS2 and other transition metal dichalcogenides (TMDs). The phenomenon has been explained by inter-band resonance to the band nesting energy [33]. However, no work to our best knowledge has linked the even harmonic yield to the interference between different half-cycles, which, according to our analysis above, is of fundamental importance in the generation of even harmonics.
In this paper, we conduct experimental and theoretical study on HHG from monolayer MoS2 (1L-MoS2). The observed enhancement of even-order harmonics near the cut-off region can be well produced by the fully quantum mechanical calculation. By separating the calculated high-harmonic current in the time domain, we attribute the phenomenon to the spectral interference between harmonics generated from adjacent half-optical-cycles with opposite polarity. This explanation is supported by the frequency shift of the measured harmonic peaks. The underlying mechanism is revealed by saddle point analysis, which shows the modification of Berry connections of MoS2 to the interband electron dynamics. While inter- and intra-cycle interferences have been studied in HHG from centrosymmetric materials [25,37–39], our work emphasized the role of nontrivial inter-half-cycle interference in modulating the high-harmonic spectrum from non-centrosymmetric materials, which has long been overlooked in the study of HHG from solids.
2. Experiment and results
We excited 1L-MoS2 with an s-polarized mid-infrared driving laser (3.8 µm, 60 fs) [40] and measured the HHG spectra using a thermoelectrically cooled Si charge-coupled device (CCD) spectrometer. The sample is grown by chemical vapor deposition (CVD) and transferred to a barium fluoride (BaF2) substrate. A parabolic mirror with 100 mm focal length focused the laser beam to a spot of about 20 µm in radius. By observing the shape of MoS2 triangular flakes (average size 40 µm) in situ with a microscope, we make sure only one single crystal domain is irradiated at a time. Such method also allows us to identify the crystal orientation [41]. Figure 1(a) shows a typical harmonic spectrum obtained with driving pulse polarized along armchair direction (I = 1.7 TW/cm2). The result has been corrected for the grating efficiency, optics reflectivity, and quantum efficiency of the spectrometer. Both odd- and even-order harmonics are observed, with even-orders being overall weaker than odd-orders. As we look into the order dependence of harmonic yield (dotted line in Fig. 1(a)), a sharp increase in even harmonic intensity is seen between the 8th and 10th high-harmonic (HH8 and HH10). Such result is distinctively different from odd harmonics, which follows a typical gas-like pattern (a plateau extended to HH7 followed by a cut-off at around HH9). We also measured the harmonic yield as a function of driving laser intensity. The result can be found in Fig. 7(a) in Appendix A, which shows the consistency of our finding under different laser intensities.
Fig. 1. High-harmonic spectrum measured from monolayer MoS2 (I = 1.7 TW/cm2). Blue and orange dotted line shows the order-dependent intensity of the odd- and even-order harmonics, respectively. The driving pulse is polarized along armchair direction.
3. Theoretical simulation
To explain the anomalous intensity increase between HH8 and HH10, we did a theoretical stimulation adopting two-band (see Fig. 2(a)) semiconductor Bloch equations (SBEs) under length gauge [32,34,35]. The driving laser field adopted for the simulation is a Gaussian pulse with full width at half maximum (FWHM) duration τ = 60 fs and laser wavelength λ0 = 3.8 µm, which agrees with the experimental condition. The band structure and transition dipole matrix are derived from an empirical tight-binding model [42]. As the crystal is excited along the armchair direction, the selection rule restricts all the harmonics to be polarized in parallel with the incident light [43]. Thus the SBEs can be solved in one-dimension, namely, the K-K (or K’-K’) direction in the reciprocal space. The phenomenological dephasing time in the SBEs is set to 1 fs as used in Ref. [4,5,24,32]. More details of the simulation can be found in Appendix B. In Fig. 2(b), the calculated spectrum is shown in comparison with the experimental result. As the two spectra agree well with each other, we believe the calculation has captured the key effects in the experiment (the laser-intensity-dependence of the simulated harmonics can be found in Appendix A and offers additional support to the validity of the simulation). In Fig. 2(c), we compare the contributions from interband and intraband polarization to the total harmonic yield [26,44]. The spectrum is dominated by interband harmonics, which compliances with the previous experimental result found in ZnO [45].
Fig. 2. Theoretical simulation of HHG from monolayer MoS2. (a) The highest valence band and the lowest conduction band along K-K (K’-K’) direction in the reciprocal space. (b) Calculated spectrum (orange line, EL = 3.4 V/nm) in comparison with the experimental result (black line). (c) Interband and intraband contributions of the high-harmonics. The dephasing time is set as 1 fs.
4. Inter-half-cycle interference
As the validity of the simulation is confirmed, we do a time-frequency analysis of the calculated high-order harmonic current to gain more information on the HHG process. In Fig. 3, most signals are generated slightly after the zero points of the laser field, which indicates their origin from the short trajectory [25]. In our calculation, the long trajectory is suppressed since the inter-band dephasing time in the SBEs is set shorter than half of an optical cycle (see Appendix C for the effect of the dephasing time on the result). Meanwhile, the harmonic current generated in negative half-cycles extends to higher frequency than its counterpart in positive half-cycles, which creates a gap between around HH10 and HH11 where only negative half-cycles can generate such high frequency harmonics. The difference in the emitted harmonic frequency results from the modulation of Berry connections on the interband electron dynamics, which will be explained in detail in Section 5.
Fig. 3. Time-frequency analysis of the calculated high-order harmonic current. (a) Instantaneous electric field of the driving laser pulse. (b) Time-frequency diagram of the high-order harmonics calculated under EL=3.4 V/nm. ω0 is the frequency of the driving laser.
In Fig. 4, we artificially separate the contributions from different half-cycles in the time domain to see how the inter-half-cycle interference affects the intensity of different harmonic orders. Figure 4(a) shows the high-order harmonic current as a function of time. High-harmonic bursts appear just after (n/2 + 1/4) T0 (T0 is the optical cycle), which compliances with our time-frequency analysis. We use super Gaussian functions $f(t) = \exp [ - {(t - {t_n})^4}/\varDelta {t^4}]$ (blue and red line) to filter these harmonic bursts. The contributions from positive and negative half-cycles are transformed respectively into frequency domain and compared with their sum. Figure 4(b) shows the results of four near-cut-off harmonics. The harmonics generated from positive and negative half-cycles both decrease monotonically as a function of order. As HH10 lies in the frequency gap discussed above, the contribution from positive half-cycles decreases much faster than that from negative half-cycles. HH8 and HH10 come from the destructive interference between adjacent half-cycles (see Section 5 for the analytical description). The lack of destructive interference leads to anomalous intensity increase between HH8 and HH10. HH9 and HH11 come from the constructive interference, thus their intensity decreases normally with order. In the calculation, the time width Δt for the super Gaussian filter is 0.3T0, which allows us to separate the contribiutions from different half-cycles without inducing significant modulation on the harmonic current. The stability of the filtered harmonic spectra in Fig. 4(b) has been checked by changing Δt. As long as the aforementioned qualifications are met, the filtered harmonic spectra are relatively stable. Therefore, we have shown that the order-dependence of the harmonic intensity observed from experiments can be well explained by the inter-half-cycle interference.
Fig. 4. Spectral interference between harmonics generated from positive and negative half-cycles. (a) High-order harmonic current and the filter functions to separate the contributions from different half-cycles. (b) Contributions from positive half-cycles (red line), negative half-cycles (blue line) and their sum (black line).
Additional evidence of the inter-half-cycle interference can be found in the measured frequency shift of harmonic peaks. In Fig. 5(a), the frequency shift of each harmonic order, (ωs/sω0 -1), of the HHG spectrum shown in Fig. 1 is calculated, with ωs being the measured peak frequency of harmonic order s. The frequency shift of HH8 deviates strongly from other orders, which, we believe, results from the destructive interference between different half-cycles. Due to the nonadiabatic mechanism [46], the high harmonic frequencies contributed by positive and negative half-cycles are shifted by different values. Since HH8 comes from their destructive interference, the frequency shift of the total spectrum can be bigger than both half-cycles, which explains the result in Fig. 5(b).
Fig. 5. Frequency shift of the high-order harmonics. (a) Measured frequency shift of each harmonic order. (b-c) Zoom-in spectra of HH8 from the experiment (b) and simulation (c).
A demonstration of the above analysis can be found in Fig. 5(c), which presents the half-cycle-resolved spectra of HH8 in Fig. 4(b). The shift of central frequency of HH8 contributed by negative and positive half-cycles are 0.1% and -0.004%, respectively. Because of the destructive interference between half-cycles, the total spectral shape of HH8 in Fig. 5(c) is highly asymmetric. When we integrate the spectrum between 7.8 and 8.2 ω0, the central-frequency-shift is found to be -0.3%, which is bigger than both half-cycles. We are aware that the frequency shift of HH8 in Fig. 5(b) is not reproduced by the simulation. Especially, the total spectrum in Fig. 5(c) is red-shifted, while in the experiment is blue-shifted. This may result from the inaccuracy of harmonic yield calculated in different half-cycles. In Fig. 5(c), if the yield from the positive half-cycles decreases, the total spectrum will be shifted to the blue side. As our simulation is done in a two-band system without properly including the multiple-electron effect [47,48], such discrepancy in harmonic yield is expected.
5. Generation of even-order harmonics
In section 4, we have shown that the inter-half-cycle interference leads to the anomalous order-dependence of harmonic intensity. Next, we will explain the origin of different electron dynamics under positive and negative half-cycles of the laser field. Since the dominant role of the interband mechanism is confirmed in Section 3 (Fig. 2(c)), we limit the following discussion to interband dynamics. Therefore the corresponding electron motion can be seen as an analogy to the three-step model in gas [27]. The frequency of the high-harmonic photon emitted in the recombination step can be derived by applying the saddle point method to the two-band SBEs [34,49]:
Figure 6 presents a schematic diagram of the effect of such energy modulation –E(tr)Δd to HHG. As an example, the instantaneous laser field at the recombination time is set as ${\pm} $1.0 V/nm, which is the instantaneous laser field at the recombination time of tr = -0.2 T0 in Fig. 3. For the electron recombination in the negative half-cycle (Fig. 6(b)), the energy modulation leads to a maximum energy gap of 3.42 eV (10.5 $\hbar {\omega _0}$) between the conduction and valence band. In the positive half-cycle (Fig. 6(c)), the modulated maximum energy gap becomes 3.01 eV (9.5 $\hbar {\omega _0}$). Therefore, harmonic orders between 9.5 and 10.5 can only be generated in negative half-cycles, which explains the harmonic frequency difference found in Fig. 3. Note that the above analysis is only qualitative, as the specific electron trajectories for HHG are not known in this paper, and E(tr) is dependent on trajectories. In Fig. 7, HH10 is found brighter than HH8 at all the laser intensity measured, which indicates its predominant origin from negative half-cycles under the experimental conditions.
Fig. 6. Schematic diagram of the electron-hole dynamics of HHG from monolayer MoS2. (a) The excitation and recombination time of electron-hole pair. (b-c) Electron recombination in the reciprocal space under negative (b) and positive (c) laser field. The colored line shows the field-modulated band structure.
When the harmonics generated from two adjacent half-cycles of opposite polarity interfere with each other, the total high-order harmonic current is given by:
where $J(s)$ is the high-order harmonic current of order s, ${J_p}(s)$ the current generated in positive half-cycle, ${J_n}(s)$ in the negative half-cycle. $s{\omega _0}{T_0}/2$ stands for the phase difference results from the time interval between two adjacent half-cycles. The minus sign comes from the change in laser field direction. For odd-order harmonics, we have $- {e^{is{\omega _0}{T_0}/2}} = 1$, while for even-order harmonics, $- {e^{is{\omega _0}{T_0}/2}} ={-} 1$. This explains the constructive and destructive interference of odd- and even- order orders in Fig. 4(b). Since the frequency modulation in the two half-cycles is different, ${J_p}(s) \ne {J_n}(s)$. Therefore the incomplete destructive interference leads to the generation of even-order harmonics.By far, our calculation and analysis are limited to a two-band system. When higher energy bands are involved, the bandgap energy will increase, and the multiple-band interference will modify the intensity of even-order harmonics [24]. We did a three-band calculation to compare with the two-band case. Though the result harmonic spectrum is not exactly the same, the intensity minimum at HH8 can still be produced, which shows the effect of inter-half-cycle interference.
6. Conclusion
In conclusion, the anomalous intensity increases of even-order harmonics near the cut-off region of 1L-MoS2 observed in our experiments has been linked to the spectral interference between positive and negative half-cycles of the pump laser field. As the harmonic photon energy emitted by the electron-hole recombination is modulated by the external laser field at the time of recombination, some high-frequency harmonic photons can only be generated under a certain field polarity. Therefore, the even-order harmonics near the cut-off region are enhanced thanks to the lack of destructive interference. Our explanation is verified by a fully quantum mechanical calculation and further supported by the frequency shift of the observed harmonics. While being demonstrated in 1L-MoS2, we believe such mechanism exists generally in HHG from non-centrosymmetric materials. Our finding shows the effects of inter-half-cycle spectral interference on the HHG from asymmetric materials. The physical process we revealed may find applications in the measurement of ultrafast optical pulses or production of isolated attosecond bursts.
Appendix A: intensity-dependence of the harmonic yield
In Fig. 7, we compare the simulated laser-intensity-dependence of the harmonic yield with the experimental data (a). The two figures generally coincide with each other. Especially, in both the experiment and simulation, HH8 is found weaker than HH10 at all laser intensities, which provides additional support to the validity of our simulation. Higher yield of HH6 and HH10 in the simulation may result from the inaccuracy of the tight binding model [42]. In the experimental result, the intensity-dependence of HH8 follows a power law of I3.1, while a more complicated dependence is found in the calculation. This may come from the fact that the measured harmonic intensity is actually an integration of HHG spectra driven by different laser intensities at the focal spot, which evens out the perturbation in the intensity-dependence of harmonic yield.
Appendix B: theoretical model using semiconductor Bloch equations
The two-band SBEs we used for the calculation are written as [34,35]:
Here ${P_{c,v}}(k,t)$ is the interband polarization between the conduction and valence band, ${n_{c(v)}}$ the occupation of electrons (holes). ${d_{c,v}}(k)$ is the interband transition dipole, ${d_{c,v}}(k)$ $={-} ie\left\langle {{u_c}(k)} \right|{\nabla _k}|{{u_v}(k)} \rangle$, where ${u_c}(k)$ and ${u_v}(k)$ are the periodic parts of the Bloch wave functions. $ {\tilde{\varepsilon }_g}(k)= {\varepsilon _c}(k) - {\varepsilon _v}(k) + eE({t_r})[{A_{c,c}}(k) - {A_{v,v}}(k)]$, where ${\varepsilon _c}(k)$ and ${\varepsilon _v}(k)$ denotes the band energy while ${A_{c,c}}(k)$ and ${A_{v,v}}(k)$ are Berry connections. For any band indices m and n, ${A_{m,n}}(k) = i\left\langle {{u_m}(k)} \right|{\nabla _k}|{{u_n}(k)} \rangle $. By definition, we have ${d_{c,v}}(k) ={-} i{A_{c,v}}(k)$.
In Eqs. (3)–(4), the band energy ${\varepsilon _c}(k)$ and ${\varepsilon _v}(k)$, transition dipole ${d_{c,v}}(k)$, and Berry connections ${A_{c,c}}(k)$ and ${A_{v,v}}(k)$ are derived from an empirical tight-binding (TB) model of MoS2 [42] with the method demonstrated in Ref. [32]. The value of ${\varepsilon _c}(k)$ and ${\varepsilon _v}(k)$ has been shown Fig. 2(a). In Fig. 8, we plot the real and imaginary part of ${d_{c,v}}(k)$. As the twisted parallel transport (TPT) gauge is adopted [35], ${A_{c,c}}(k)$ and ${A_{v,v}}(k)$ are independent of k. Our calculation shows ${A_{c,c}} = $-2.42 a.u. and ${A_{v,v}} = $-2.39 a.u..
After solving the SBEs, one can obtain the inter- and intra-band current
Note that Eq. (1) in Section 5 can be derived from Eqs. (3) and (5) by applying saddle point condition ${\partial _t} = 0$. Details of the substantiation can be found in Ref. [34] and Ref. [49].
Appendix C: effect of the dephasing time on the inter-half-cycle interference
In Fig. 3, only the short trajectories are seen in the time-frequency spectrum due to the short dephasing time T2 = 1 fs. While this value has been commonly adopted when comparing the theoretical simulations with the experimental results [4,5,24,32], one may wonder what longer trajectories are like in HHG from such system, and how T2 will change the results of inter-half-cycle interference. In Fig. 9, we show the calculated HHG spectra and time-frequency analysis results by adopting longer dephasing times.
Fig. 9. Effect of dephasing time on high-harmonic generation. (a-b) High-harmonic spectra calculated with dephasing time T2 = 2 fs (a) and 12.7 fs (b). (c-d) Corresponding time-frequency distributions.
When the dephasing time T2 is increased to 2 fs, it changes the relative intensity between the harmonic yield from negative and positive half-cycles (Fig. 9(c)), which diminishes the destructive interference and increases the intensity of HH8 (Fig. 9(a)). We believe this is due to that in the negative half-cycles, the electron needs longer excursion time to reach the same field-modulated bandgap energy in the reciprocal space. Therefore, when T2 increases, it affect more intensely the harmonic yield from the negative half-cycles. In the real world, since higher energy bands might be involved, more trajectories can participate in the inter-half-cycle interference. As distinctively weaker intensity of even-order harmonics is found in the experimental result, we believe the real harmonic yield from the positive and negative half-cycles are close.
In Figs. 9(b) and (d), we set the dephasing time T2 to 12.7 fs in the calculation, which equals to one optical cycle. In such case, more complex trajectories appear, and the resultant harmonic peaks becomes split in the spectrum. The destructive interference of HH8 disappeared, which can be understood since the dipole phase of high-harmonics contributed by longer trajectories are more dependent on the waveform of the electric field [25]. Thus, the interference between half-cycles deviates from fully constructive or destructive. As better agreement with the experimental result is acquired with shorter dephasing time, we believe the short trajectories contribute to most high-harmonic signals in our experiment.
Funding
National Natural Science Foundation of China (11127901, 11134010, 61221064, 61405222); Strategic Priority Research Program of Chinese Academy of Sciences (XDB16000000); Scientific Instrument Developing Project of the Chinese Academy of Sciences (YJKYYQ20180023); Shanghai Municipal Development and Reform Commission.
Acknowledgement
We thank Prof. Chengpu Liu and Prof. Candong Liu for fruitful discussions.
Disclosures
The authors declare that they have no competing interests.
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