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Abstract
Since high-order harmonic generation (HHG) from atoms depends sensitively on the polarization of the driving laser field, the polarization gating (PG) technique was developed and applied successfully to generate isolated attosecond pulses from atomic gases. The situation is, however, different in solid-state systems as it has been demonstrated that due to collisions with neighboring atomic cores of the crystal lattice strong HHG can be generated even by elliptically- and circularly-polarized laser fields. Here we apply PG to solid-state systems and find that the conventional PG technique is inefficient for the generation of isolated ultrashort harmonic pulse bursts. In contrast, we demonstrate that a polarization-skewed laser pulse is able to confine the harmonic emission to a time window of less than one-tenth of the laser cycle. This method provides a novel way to control HHG and to generate isolated attosecond pulses in solids.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-order harmonic generation (HHG) from atoms which are driven by strong laser fields forms the foundation of attosecond science [1–3]. HHG is not only the main source for the generation of attosecond pulses, but is also an important tool to detect and analyze the ultrafast electron dynamics with attosecond time resolution. The fundamental mechanisms governing HHG from atoms are well explained by the semiclassical three-step model, i.e., ionization, acceleration, and recombination [4]. Based on this simple model, several approaches have been proposed to control the motion of electrons in order to obtain desired attosecond pulses and to explore the ultrafast dynamics. The first isolated attosecond pulse was generated using amplitude gating [5]. In this technique the amplitude difference between the strongest single and other half-cycles of a waveform-stabilized few-cycle driving field leads to a smooth spectrum in the cutoff range of HHG. Filtering out the continuous high-energy spectrum yields an isolated attosecond pulse [5–7]. Other powerful methods for generating isolated attosecond pulses, including the PG [8–10] and the double optical gating [11–15] techniques, are based on the ellipticity dependence of HHG from atoms. According to the three-step model, the electron can recombine with the parent ion only for the case of linearly polarized laser fields and already small deviations from linear polarization can result in a lateral displacement of the returning electron which substantially reduces the probability of recombination and the subsequent HHG [8–10]. Therefore, it is possible to generate broadband isolated attosecond pulses by combining two counter-rotating circularly-polarized laser pulses with a proper delay to obtain a temporal window of linear polarization in the overlapping region. These attosecond pulse generation techniques have strongly promoted time-resolved studies of the electron and nuclear dynamics on ultrafast time scales [16–19].
More recently it has been demonstrated that HHG can be generated from solids [20] which has sparked a tremendous interest in this field as, due to the huge number of available materials and material combinations, solids offer a broad range of design possibilities which can be used to increase the efficiency and intensity of HHG. Due to the huge application potential, e.g., in compact short-wavelength light sources and as tools for imaging the dynamics and structure in crystals, immediately after the first observation of HHG in a bulk crystal of ZnO was reported in 2011 [20] the underlying microscopic mechanisms that govern solid-state HHG have attracted significant attention and are still intensively explored [21–27]. The electron/hole dynamics in solids is much more complex than in atoms. Both the interband polarization and the intraband current contribute to HHG in solids [28]. When the driving laser field is weak, it has been found that the generation mechanism of the interband HHG is similar to that of atomic HHG and that a three-step model adapted to solid state system is able to explain experimental results well. However, for relatively high intensity driving fields, the three-step model for solids fails to describe numerous phenomena including the HHG driven by circularly- and elliptically-polarized laser fields [29–33]. In our previous work, we have demonstrated that in addition to the electron-hole recombination driven by the laser field, collisions with neighboring atomic cores can also lead to recombination of the electron with its associated hole, which is the origin of the anisotropy, the delocalization, and the elliptical dependence of HHG in MgO and ZnO [34]. Due to the discrepancies between the underlying mechanisms of atomic and solid HHG, it is not clear which parts of atomic attosecond technology can be directly applied to solids.
In this paper, we investigate possibilities to control the electron dynamics in solids on ultrafast time scales by intense light fields. We start by exploring the laser intensity dependence of solid-state HHG in linearly polarized laser fields to illustrate the collision and recombination dynamics in solids. This reveals that the strength of the two recombination channels in solids, i.e., the "laser-field driven electron-hole recombination" and the "neighboring-core collision driven electron-hole recombination", can be controlled via the laser intensity. Then we investigate how these two recombination processes change when driven by time-delayed counter-rotating circularly-polarized laser fields, i.e., by applying the PG technique. Finally, we propose a new method to generate an isolated intense emission burst by driving with a polarization-skewed laser pulse. This paper is organized as follows. In Sec. 2 we describe our theoretical approach, i.e., the semiconductor Bloch equations (SBE) for a two-band model. The results of our numerical simulations are presented and discussed in Sec. 3 and we conclude with a brief summary of our main results in Sec. 4.
2. Theoretical approach
For our simulations of solid-state HHG we use the semiconductor Bloch equations which include the coupled inter- and intraband dynamics and for a two-band model read [35,36]:
Here, $\varepsilon _{\mathbf {k}}^{e(h)}$ are the energies of electrons (holes), $T_{2}$ is the dephasing time, $\textbf {d}_{\mathbf {k}}$ is the dipole matrix element, and $\textbf {E}(t)$ denotes the electric field. According to the Bloch acceleration theorem [37], in the presence of a homogeneous electric field the electron’s crystal momentum in a given band changes according to $\textbf {K}(t)=\textbf {k}-\textbf {A}(t)$, where $\textbf {A}(t)$ is the vector potential of the electric field $-d\textbf {A}/dt=\textbf {E}(t)$ and $\textbf {k}=\textbf {K}(t_{0})$ is the initial momentum. As a result, the semiconductor Bloch equations can be transformed to and solved in a moving time-dependent frame $\textbf {K}(t)$ [28]:
Here, $\Omega (\mathbf {K}, t)=\mathbf {E}(t) \cdot \mathbf {d}[\mathbf {K}+\mathbf {A}(t)]/\hbar$ is the Rabi frequency, $S(\mathbf {K},t)=\hbar ^{-1}\int _{-\infty }^{t}\varepsilon _{\mathrm {g}}\left [\mathbf {K}+\mathbf {A}\left (t^{\prime }\right )\right ] d t^{\prime }$ is the classical action, and $\varepsilon _{g} [\mathbf {k}] = \varepsilon _{\mathbf {k}}^{c}-\varepsilon _{\mathbf {k}}^{v} = \varepsilon _{g} + \varepsilon _{\mathbf {k}}^{e} + \varepsilon _{\mathbf {k}}^{h}$ is the k-dependent transition energy between the valence and conduction bands.
We define the propagation direction of the incident laser field as the $z$-direction, which we assume to coincide with the $\Gamma - A$ of the ZnO crystal. In our simulations we set $k_z=0$ and use a two-dimensional hexagonal two-band model for the band structure of the ZnO crystal which is taken from Refs. [28,31] as
withThe subscript $m = c,v$ denotes the conduction and valance band, respectively. $a_x = 5.32 a.u.$ and $a_y = 6.14 a.u.$ are the lattice constants in the $x$- and $y$-directions. All other parameters $(t_m, t_m^{\prime }, q_m, p_m, u)$ can be found in Ref. [31] where a more detailed description is provided.
For the excitation conditions considered here, the HHG from ZnO is dominated by the interband current, which is given by
The HHG spectrum is obtained by performing a Fourier transform of the interband current [25,28].
3. Results and discussion
Before we apply the PG technique to solids, we first investigate the intensity dependence of HHG for pulses which are linearly polarized along the $\Gamma -M$-direction. The laser frequency is taken to be 0.0117 a.u. (wavelength $\lambda =3.9 \mu$m) and the dephasing time $T_2$ is set to 4fs according to Ref. [28]. Consistent with previous studies [31,34], two channels for HHG exist which can be distinguished due to their intensity dependence, as is visualized in Figs. 1(a) and 1(b). When the laser field amplitude is weak, the mechanism of interband HHG in solids is quite similar to the three-step model of the atomic HHG, see Fig. 1(a). In momentum space, the electron is excited to the conduction band mainly around the $\Gamma$ point and then the electron and its associated hole are accelerated by the laser field in their respective bands. In real space, they firstly move away from each other with a velocity determined by the gradient of the energy bands. After the direction of the electric field is reversed, the electron and hole begin to decelerate. When the electron and hole reach the $\Gamma$ point in momentum space again, their velocities are reduced to 0, and thereafter they begin to move towards each other until they reencounter and emit a harmonic photon with an energy which is given by the interband energy difference at the momentum where they recombine. Similar to atomic HHG, for this HHG channel the electron-hole pair can be driven back by the laser field and recombine together only if they are generated after the peak of the electric field. We denote this channel "laser-driven electron-hole recombination". It occurs mainly when the driving field is weak and the electron/hole cannot reach the boundary of the Brillouin zone. The high harmonic spectrum shown in Fig. 1(c) is dominated by this generation channel. The corresponding time-frequency analysis shown in Fig. 1(d) indicates that there are two emission bursts within one laser cycle. One of the typical characters of this HHG channel is that when the driving field amplitude is enhanced, the cutoff of the harmonic spectrum moves to higher energies as is indicated by the two processes shown in Fig. 1(a) where the black solid line corresponds to a stronger driving field. These arguments are also supported by comparing Figs. 1(c) and 1(d) with Fig. 1(e) and 1(f).
Fig. 1. Schematic illustration of the two different HHG channels in ZnO. (a) and (b) Schematic diagrams of the "laser-driven electron-hole recombination" (a) and the neighboring core "collision-driven electron-hole recombination" (b). The gray dashed lines corresponds to a weak field amplitude, whereas the black solid line shows the process for a stronger driving field. (c)-(h) HHG spectra (left column) and the corresponding time-frequency analyses (right column) generated by linearly-polarized pulses with three different field strengths, respectively. The FWHM of the pulses is 20$T_0$ where $T_0$ the duration of one laser cycle.
When the field strength is increased, electron-hole pairs generated before the peak of the electric field can reach the boundaries of the Brillouin zone where electron/hole collisions with neighboring atomic cores can occur. The backward scattering can lead to a reversal of the directions of the electron and hole so that they can move towards each other and recombine together without the need of reversal of the sign of the electric field. This HHG channel is called "collision-driven electron-hole recombination" [31,34]. In Figs. 1(e) and 1(f) with a driving field amplitude of $E_{0}=0.003$ a.u., both HHG channels coexist (as indicated by "a" and "b", respectively), and therefore the harmonic spectrum at the plateau becomes very complex. The emission times of the two HHG channels are different and, as a result, these two channels can be distinguished in the time-frequency analysis, see Fig. 1(f), where four emission bursts within one laser cycle can be clearly seen. In this case, the harmonic emission via the "collision-driven electron-hole recombination" channel is mainly located at the high energy part of the spectrum. Therefore, the emission intensity of the high energy cutoff part of the harmonic spectrum is higher than that in the plateau region, see Fig. 1(e). When the field strength is further increased, as shown in Figs. 1(g) and 1(h), the emission energy of the "collision-driven electron-hole recombination" is extended to the lower energy part and covers the whole plateau and cutoff region, see Figs. 1(b) and 1(h). Thus in this case, the "laser-driven electron-hole recombination" channel is suppressed and the "collision-driven electron-hole recombination" channel dominates the HHG. As a result, we can select a specific HHG channel by a proper choice of the intensity of the laser field. It should be noted here that it has been demonstrated that the "collision-driven electron-hole recombination" is the main source of HHG in circularly and elliptically polarized laser fields [31,34].
Owing to the similarity between the HHG from atoms and solids for weak laser fields, we investigate whether the PG technique which was developed for atomic HHG can also be applied successfully to solid-state HHG. However, the situation differs significantly when the laser field strength is high enough that "collision-driven electron-hole recombination" channel plays an essential role. In the following, we mainly focus on investigating and exploiting this point. The counter-rotating circularly polarized laser field we adopt is given by
Fig. 2. (a) The x- and y-components of a counter-rotating circularly-polarized field with $T_{d}=8 T_{0}$, $\tau _{p}=5{T_{0}}$, $E_{0}=0.003\mathrm {a.u.}$, and $\omega =0.0117\mathrm {a.u.}$, (b) the HHG spectrum and (c) the corresponding time-frequency analysis.
In atomic HHG few-cycle pulses are usually considered in the conventional PG scheme and the time delay $T_{d}$ is taken to be similar to the pulse duration of the circularly-polarized pulse $\tau _{p}$. Therefore we start to investigate HHG in ZnO with such PG pulse and set $T_{d}=\tau _{p}=2{T_{0}}$. The synthetic electric field and the x- and y-components are shown in Fig. 3(a1). In this case, the amplitude of the central linearly-polarized electric field is enhanced compared to the case of a larger time delay and therefore the harmonic emission via the "collision-driven electron-hole recombination" channel occurs both within and outside the polarization gate, see Fig. 3(a2). Figure 3(a3) show the temporal harmonic pulse derived from Fourier transforms of the spectra of the entire plateau region ranging from the 15th to 37th harmonics and demonstrates that multiple emission bursts are present.
Fig. 3. The time-frequency analysis (second row) and temporal pulse (third row) of HHG driven by different synthetic electric fields (first row). In (a1)-(a3) and (b1)-(b3) counter-rotating circularly-polarized pulses with $T_{d}=\tau _{p}=2T_{0}$ and $T_{d}=\tau _{p}=T_{0}$ are used, respectively. (c1)-(c3) arises for driving by a polarization-skewed electric field as described in the main text.
Decreasing the time delay to one laser cycle ($T_{d}=T_{0}$), the peak of the combined electric field is located at the central part where the electric field is linearly-polarized along the x-direction and the amplitude is high enough that the HHG via the "collision-driven electron-hole recombination" channel dominates the entire plateau region, see Fig. 3(b2). In this case, the electric with large ellipticity at the leading and ending parts is too weak to induce the "collision-driven electron-hole recombination” and therefore HHG is essentially suppressed in these parts. The time-frequency shown in Fig. 3(b2) indicates that the HHG via the "laser-driven electron-hole recombination" channel is negligible in this case. Comparing to the case with $T_{d}=\tau _{p}=2T_{0}$ (Fig. 3(a3)), the number of the harmonic pulse bursts is clearly reduced, whereas the intensity of the central harmonic pulse burst is enhanced by more than two orders of magnitude. However, quite high satellite peaks of about half the intensity of the central pulse still exist, see Fig. 3(b3).
Further investigations show that the "collision-driven electron-hole recombination" channel is very sensitive to small changes of the electric field component along the y-direction which affects the lateral motion of the electron and hole. Changing the phase of the electric field components along the y-direction by $\frac {\pi }{2}$ while all other parameters are unchanged compared to those used in Fig. 3(b1), i.e., considering
4. Conclusions
In conclusion, we investigate the intensity- and polarization-controlled HHG in ZnO. We found that the electron-hole recombination, which is the main source of the interband HHG in solids, can occur either due to the "laser-driven electron-hole recombination" or the neighboring core "collision-driven electron-hole recombination". These two generation channels can be selectively controlled by changing the intensity of the driving laser field. When the two channels co-exist, the conventional PG is inefficient since the neighboring core "collision-driven electron-hole recombination" enables HHG for driving with circularly-polarized laser fields.
To generate a strong isolated harmonic pulse, we propose a new scheme which adopts a polarization-skewed ultrashort laser field. Due to the design of such a pulse the central strongest near-linearly polarized half-cycle generates a broad continuum spectrum, while the recombination of the electron-hole pair created from the other half-cycles is suppressed due to the lateral displacement induced by the skewed polarization of the laser field. Our proposed method should be realizable in experiments on, e.g., ZnO and MgO and is expected to substantially advance the capabilities of attosecond technology and allow to generate ultrafast field bursts using solid-state HHG. For crystals with spatial asymmetry where Berry curvature plays an essential role or bilayer material [27], the application of this method needs further discussion.
Funding
National Natural Science Foundation of China (12074240); Natural Science Foundation of Hainan Province (122CXTD504, 123MS002); Deutsche Forschungsgemeinschaft (231447078 TRR 142 (subproject A10)); Sino-German Center for Research Promotion (M-0031).
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 12074240); Hainan Provincial Natural Science Foundation of China (Grant No.122CXTD504 and 123MS002); Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number 231447078 TRR 142 (subproject A10); Sino-German Mobility Programme (Grant No. M-0031).
Disclosures
The authors declare that there is no conflict of interest regarding the publication of this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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