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Abstract
The effect of driving wavelengths on high harmonic generation (HHG) have long been a fundamental research topic. However, despite of abundant efforts, the investigation of wavelength scaling of HHG in solids is still confined within the scope of theoretical predictions. In this work, we for the first time to the best of our knowledge, experimentally reveal wavelength scaling of HHG yields and cutoff energy in three typical solid media (namely pristine crystals GaSe, CdTe and polycrystalline ZnSe), driven in a broad mid-infrared (MIR) range from 4.0 to 8.7 µm. It is revealed that when the driving wavelength is shorter than 6.5-7.0 µm, HHG yields decrease monotonously with the MIR driving wavelengths, while they rise abruptly by 1-3 orders of magnitude driven at longer wavelength and exhibit a crest at 7.5 µm. In addition, the cutoff energies are found independent on driving wavelengths across the broad MIR pump spectral range. We propose that the interband mechanism dominates the HHG process when the driving wavelength is shorter than 6.5-7.0 µm, and as the driving wavelength increases, intraband contribution leads to an abrupt rise of the HHG yields, which is verified by the HHG polarization measurement driven at 3.0 and 7.0 µm. This work not only experimentally demonstrate the wavelength scaling of HHG in solids, but more importantly blazes the trail for optimizing the HHG performance by choosing a driving wavelength and provides experimental method to distinguish the interband and intraband dynamics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-order harmonic generation (HHG) is a canonical nonperturbative nonlinear optical phenomenon generated from the intense laser interacting with matters, which provides a unique pathway towards extreme ultraviolet sources and attosecond photonics [1]. On the other hand, in solid materials with complex electronic band structures, HHG also provides a fascinating avenue for investigation of electron dynamics and nonadiabatic tunneling dynamics [2–4]. Since the first observation of solid-state HHG in a zinc oxide (ZnO) crystal in 2011 [5], a large number of experimental demonstrations of HHG in various solid-state media, such as semiconductors [6–10], meta-structures [11–13], 2D materials [14,15] and nano thin-films [16–19] have been conducted. Moreover, the wavelengths of driving lasers for HHG in solids have been extended from near-infrared to mid-infrared (MIR) regions owing to the merits of longer driving wavelengths such as larger material damage intensity, and easier access of harmonics without vacuum apparatus [20–23]. Many researches have been attended to explore the wavelength scaling of HHG yields and cutoff energies in solids in a broad spectra range of driving lasers [24–31], however, wavelength scaling of HHG in solids is still a controversial topic for its complex radiation process. It is generally accepted that solid-state HHG is dominated by two channels: interband polarization between the valence and conduction bands, and nonlinear intraband current in the individual bands [24,25]. In 2017, Wang et al. measured the HHG spectrum of ZnO at driving wavelengths from 1.8 to 3.6 µm and found that the HHG yield exponentially decreases with driving wavelengths. Through theoretical investigation, further extending into the longer driving wavelength till 10 µm, the simulated interband and intraband yields both decrease monotonically with a fitted slope of λ−(15.3 ± 0.8) and λ−(7.0 ± 0.9) respectively [26]. Subsequently, it is theoretically predicted that harmonic yield in solid follows a scaling of λ−4, which is attributed to the wave-packet spreading (λ−3) and the energy distribution effect (λ−1) due to the increase of the cutoff [27]. Unlike the monotonical scaling, Vampa et al. theoretically proposed a complex wavelength dependence that the HHG spectral intensity decreases monotonically at MIR wavelengths and then increases at longer wavelength region [24,25]. Recently, Yue et al. theoretically characterized the yields of even-order harmonics orders, and predicted that HHG strengths fluctuated in a broad range of driving wavelengths [28]. On the other hand, the wavelength scaling of HHG cutoff energies have also been theoretically investigated. In 2015, Wu et al. investigated the high-harmonic radiation basing on a 1D single-electron model, revealing the cutoff scales approximately linearly with the driving wavelength [29]. In 2017, Tancogne-Dejean et al. studied the wavelength scaling of cutoff energy of solid through ab initio simulations cover wavelengths from 0.8 to 3.6 µm. It predicted that the cutoff energy is driving-wavelength independent [30]. In the same year, another theoretical study demonstrated that cutoff energies of interband HHG is linearly dependent on driving wavelengths, whereas for the intraband contribution, cutoff energies of HHG are wavelength independent [31]. It is highly desired to understand the wavelength scaling of HHG in solids and have a clear guidance on choosing the driving laser wavelength for a broad range of HHG experiments, such as strong-field optoelectronics [32,33], tailor of temporal structures of HHG in solids [34,35], and ultrafast dynamics investigations of nan-films [18,36]. However, till now, the dependences of harmonic yields and cutoff energies on driving wavelengths in solids are not explicated, and there is no directly experimental evidence in the broad mid-IR driving wavelength range.
In this work, the wavelength scaling of HHG yields and cutoff energies are symmetrically investigated in pristine crystals cadmium telluride (CdTe), gallium selenide (GaSe) and polycrystal zinc selenide (ZnSe) in a broad range of MIR driving wavelengths spanning from 4 to 8.7 µm, pumped by a two-stage tunable optical parametric amplifier (OPA). It is found that when the driving wavelength is shorter than 6.5 -7.0 µm, the HHG yields decrease monotonously with the MIR driving wavelength, while they rise up abruptly driven at longer wavelength and exhibit a crest at 7.5 µm. Strikingly, the HHG yield could be enhanced by 1-3 orders of magnitude by tuning to the optimal driving wavelength compared to that of adjacent pump wavelengths. The variations of harmonic intensities can be summarized as a two regions characteristic. We propose that the interband trajectory dominates the HHG process when the driving wavelength is shorter than 6.5-7.0 µm, and as the driving wavelength increases, intraband contribution leads to an abrupt rise of the HHG yields driven at 7.5 µm. This is verified by the HHG polarization measurement by scanning the MIR pump polarization, driven at 3.0 and 7.0 µm. In addition, the HHG cutoff energies is found independent on driving wavelengths across the broad MIR spectral range for the three crystals. Our work not only reveals wavelength scaling of HHG yields and cutoff energies in solid media for the first to the best of knowledge, more importantly, it provides the direct gauge on choosing the MIR driving wavelength for either optimizing the HHG yields and cutoff energies.
2. Experimental setup and parameters
Pristine crystals CdTe, GaSe and polycrystal ZnSe are chosen as targeted materials where sufficient harmonic orders and yields could be generated driven at MIR wavelengths, which is beneficial for investigating wavelength scaling in a broad driving wavelength range. The schematic diagram of experimental setup is shown in Fig. 1(a).
Fig. 1. (a) The experimental schematic of MIR pulse generation and HHG measurement based on a two-stage MIR OPA system. HWP, half-wave plate; TFP, thin film polarizer; YAG, yttrium aluminum garnet; LPF, long pass filter (1100 nm); DM1, dichroic mirror (HT@1135-1600 nm, HR@1030 nm); DM2, dichroic mirror (HT@4-11µm, HR@1030 nm); LGS, LiGaS2 crystal; Ge, germanium window 3-12 µm AR coated; BD, beam dump. The output pulses in a broad MIR spectral range spanning from 4 to 8.7 µm are shown in the inset of Fig. 1(a). (b)-(d) The normalized spectra of MIR pulses at central wavelengths of 5.5, 6.5 and 7.5 µm, respectively. The insets are the measured MIR beam profiles before focusing at central wavelengths of 5.5, 6.5 and 7.5 µm, respectively. (e)-(g) The measured (black) and retrieved (red) interferometric autocorrelation (IAC) traces of the MIR pulses centered at 5.5, 6.5 and 7.5 µm. (h)-(j) The retrieved temporal profiles of the MIR pulses at 5.5, 6.5 and 7.5 µm and the retrieved pulse widths are 200, 185 and 194 fs respectively.
In this experiment, a commercial Yb-doped regenerative amplifier emitting at a central wavelength of 1030 nm, with a pulse width of 250 fs, at a repetition rate of 5 kHz, serves as the pump source of the tunable MIR femtosecond laser. The signal pulse is obtained through self-phase modulation in bulk YAG crystal and corresponding spectrum component is tunable by adjusting the laser intensity. The tunable MIR pulses with the spectral range tunable from 4.0 to 8.7 µm are generated through a two-stage MIR OPA system based on KTiOAsO4 (KTA, Type I) or LiGaS2 (LGS, Type I) crystals. More details of the MIR system could be found in [37]. The MIR pulses are focused on pristine crystals CdTe ((111) plane), GaSe ((100) plane) and polycrystal ZnSe, with a thickness of 1.0, 1.0, and 3.0 mm, respectively, by a ZnSe lens with a focal length of 100.0 mm. All HHG spectra are measured in a transmission geometry. The generated HHG spectra are directly collected by a spectrometer (Ocean Optics USB 2000 Pro). To study the wavelength scaling of HHG in solids throughout the broad MIR driving wavelength range, careful characterizations of the MIR spectra, beam diameters, and pulse widths are conducted, to generate a constant driving intensity for all the pump wavelengths. The spectra of MIR pulses are measured by scanning-grating monochromator with a liquid-nitrogen-cooled MCT detector. Figures. 1(b-d) present the measured spectra of three typical wavelengths at 5.5, 6.5, and 7.5 µm. The MIR beam profiles are characterized with an infrared CCD camera (Dataray, WinCamD-IR-BB-7.5 system). The corresponding MIR beam profiles of 5.5, 6.5 and 7.5 µm wavelengths before focusing are shown in the insets, respectively. The beam radius of the focal spot is calculated by equation: $2{\omega _0} = 4{M^2}\; \lambda f/\pi D$, where ${\omega _0} $ is the beam radius of the focal spot, ${M^2}$ is beam quality parameter, $\lambda $ is wavelength, f is focal length of lens and D is beam diameter at lens. The temporal profiles of MIR pulses are measured by a home-built IAC setup and reconstructed through “evolutionary phase retrieval from IAC (EPRIAC)” algorithm [38]. In the Taylor expansion of the reconstructed spectral phase, dispersion terms up to 7th order are considered. The retrieved pulse widths through EPRIAC algorithm are 200, 185 and 194 fs at 5.5, 6.5 and 7.5 µm respectively, as shown in Figs. 1(h)–1(j). In Table 1, laser parameters of MIR pulses with 8 wavelengths ranging from 4.0 to 8.7 µm throughout the broad spectral range for investigating wavelength scaling of HHG in solids are summarized, and the laser intensity is kept at 40 GW/cm2 for all the driving wavelengths.
Table 1. The laser parameters of HHG in solids driven in a broad MIR spectral range from 4.0 to 8.7 µm at a fixed intensity of 40 GW/cm2.
As the high-harmonic flux depends on the emitting area, we calculate the correction factors to evaluate the error caused by different emitting areas through the ratio of powers $({{p_1}/{p_i}} )$ = $({{I_1}/{I_i}} )\; $* $({{r_1}^2/{r_i}^2} )$ = $({{r_1}^2/{r_i}^2} )$, where ${p_1},$ ${I_1}$ and ${r_1}$ is the peak power, intensity and beam radius at 6.5 µm, respectively and ${p_i},$ ${I_i}$ and ${r_i}$ are the parameters at other wavelength conditions. This criterion implies a range of correction factors between 0.5-2.25, as shown in Table 1. In addition, the influences of nonlinear dispersion and free-carrier absorption on HHG have been considered. We calculate the linear dispersion of 1.0 mm CdTe, 1.0 mm GaSe, and 3.0 mm ZnSe in the MIR wavelength region, as plotted in Supplement 1. It could be seen that the linear dispersion is small. In the nonlinear region, the spectra before and after CdTe crystal which has the biggest nonlinear refractive index (n2 ∼ 5 × 10<συπ>−17</sup> m2/W) at the same intensity of the experiment are plotted in Supplement 1. It is revealed that there is no noticeable spectral broadening after propagating through the CdTe crystal, which implies that the nonlinear accumulation such as self-phase modulation is small or negligible. In addition, the transmission of MIR driving laser is measured, which is not decreased significantly due to the excitation of HHG.
3. Results and discussion
3.1 Wavelength scaling of harmonic yields and cutoff energies in solids
Figures. 2(a-d) present the HHG spectra generated from CdTe at typical wavelengths of 5.5, 6.5, 7.0 and 8.0 µm, and the laser intensity is fixed at 40 GW/cm2, for all the driving wavelengths. It is worth noting that all the harmonic photon energies are above the bandgap (1.44 eV) as marked by dashed lines for each wavelength. As shown in Fig. 2(a), driven at 5.5 µm, the measured harmonics intensities of 7th to 11th orders are of the magnitude of 2.4-7.2 × 10−7. As the pump wavelength is increased to 6.5 µm, the harmonics intensities of 8th to 12th orders drop by one order of magnitude to 1.5-6.0 × 10−8, as depicted in Fig. 2(b). Strikingly, unlike the monotonic wavelength scaling in gas-HHG, when the pump wavelength further increases, measured intensities of 9th to 14th and 11th to 15th harmonics driven at 7.0 and 8.0 µm rebound to 5.0 × 10−8−1.4 × 10−7 and 1.8-4.3 × 10−7, as shown in Figs. 2(c) and 2(d), respectively. All the graphs have been normalized to the maximum of all harmonics (5th harmonics of ZnSe at 4.0 µm).
Fig. 2. (a)-(d) The normalized HHG spectra of CdTe at typical mid-infrared pump wavelengths of 5.5, 6.5, 7.0 and 8.0 µm. The orders of harmonics and cutoff are marked in each graph. The black and blue dashed lines indicate the first and second plateaus, respectively. The laser intensity is fixed at 40 GW/cm2. (e) The variations of harmonic intensities with pump wavelengths in a broad wavelength range from 4.0 to 8.7 µm, with harmonic orders 5-15. (f) The measured harmonics cutoff energies versus pump wavelengths.
Figure. 2(e) summarizes the wavelength scaling of HHG yields in CdTe, with the harmonics intensities of 5th to 15th orders extracted from the measured HHG spectra. The HHG spectra at wavelengths of 4.0, 6.0, 7.5 and 8.7 µm are provided in Supplement 1 (a-d). Throughout the broad range of MIR pump wavelengths, the HHG yields drop as the pump wavelength increases from 4.0 to 6.5 µm, and rise up with the wavelength till 8 µm. The HHG yields decrease again as the pump wavelength increases further. The wavelength scaling of harmonics cutoff energy in CdTe is characterized too, as presented in Fig. 2(f). The minimum cutoff photon energy is 2.23 eV, which occurs at driving wavelength of 4.0 µm and corresponding harmonic order is 7, as shown in Supplement 1. The cutoff energy is significantly smaller than those of other driving wavelengths, as harmonics are confined in the first plateau. As the driving wavelength increasing, the harmonic spectra of CdTe show a characteristic of two-plateaus, as presented in Figs. 2(a-d) and Supplement 1. The cutoff energy of HHG in CdTe exhibits a crest of 4.25 eV corresponding to the 24th order, driven at 7.0 µm. However, there is no clear wavelength-dependence but only fluctuations of cutoff energies and transitions from two plateaus observed. Moreover, HHG in GaSe which is a common crystal for HHG in solids with 6-fold symmetry and bandgap of 2.1 eV is characterized driven in the MIR wavelengths from 4.0 to 8.7 µm. Figures. 3(a-d) depict the HHG spectra of GaSe driven at wavelengths of 6.0, 6.5, 7.5 and 8.7 µm, respectively, with a fixed pump intensity of 40 GW/cm2. The HHG spectra of GaSe at wavelengths of 4.0, 5.5, 7.0 and 8.0 µm are provided in Supplement 1. Only harmonics below the bandgap are measured. As presented in Figs. 3(a) and (b), the harmonic intensities decrease from the magnitude of 10−6 to 10−7, when the driving wavelength increases from 6.0 to 6.5 µm. As the pump wavelength further increases, the harmonic intensities rebound to a magnitude of 10−5 driven at 7.5 µm, and then quickly drop to a magnitude of 1.7-5.7 × 10−8 driven at 8.7 µm. The wavelength scaling of harmonics cutoff energy in GaSe again shows no wavelength-dependence, but fluctuation in a small range of cutoff energy from 1.7 to 2.05 eV, as shown in Fig. 3(f).
Fig. 3. (a)-(d) The normalized HHG spectra of GaSe at typical mid-infrared pump wavelengths of 6.0, 6.5, 7.5 and 8.7 µm. The orders of harmonics and cutoff are marked in each graph. The laser intensity is fixed at 40 GW/cm2. (e) The variations of harmonic intensities with pump wavelengths in a broad wavelength range from 4.0 to 8.7 µm, with harmonic orders 5-10. (f) The measured harmonics cutoff energies versus pump wavelengths.
The wavelength scaling of HHG in polycrystalline ZnSe with the bandgap of 2.1 eV is investigated, as shown in Fig. 4. Only harmonics below the bandgap are measured and compared. The harmonic yield decreases by 2-orders of magnitude as the driving wavelength increases from 5.5 to 6.5 µm. The HHG intensity quickly rebounds by 3-orders of magnitude when the driving wavelength is increased to 7.5 µm. As the driving wavelength keeps increasing to 8.0 µm, the harmonics intensity drops abruptly. Like the CdTe and GaSe crystal, the wavelength scaling of harmonics cutoff energy in polycrystalline ZnSe exhibits clear fluctuation in the range of 1.9 to 2.5 eV, but no wavelength dependence observed, as presented in Fig. 4(f). The HHG spectra of ZnSe at wavelengths of 4.0, 6.0, 7.0 and 8.7 µm are provided in Supplement 1.
Fig. 4. (a)-(d) The normalized HHG spectra of polycrystalline ZnSe at typical mid-infrared pump wavelengths of 5.5, 6.5, 7.5 and 8.0 µm. The orders of harmonics and cutoff are marked in each graph. The laser intensity is fixed at 40 GW/cm2. (e) The variations of harmonic intensities with pump wavelengths in a broad wavelength range from 4.0 to 8.7 µm, with harmonic orders 5-12. (f) The measured harmonics cutoff energies versus pump wavelengths.
The wavelength scaling of harmonic intensity can be divided into two regions. When the driving wavelength is shorter than 6.5-7.0 µm, the HHG trajectory may be dominated by interband mechanism, which locates at the left region of black dashed line, as shown in Figs. 2(e), 3(e) and 4(e). The intensities of HHG decreases monotonically with the increase of pump wavelengths, the similar trend was firstly observed by the theoretical work of Vapma et al. [25]. At a fixed laser intensity, the longer the driving wavelength, the more photons are needed in the interband process, which results in the harmonic efficiency decreases with increasing of driving wavelengths. In the second region, when the driving wavelength is longer than 7.5 µm, the intraband process becomes dominant, which locates on the right side of black dashed line in Figs. 2(e), 3(e) and 4(e). Intraband emission presents a fluctuated characteristic with pump wavelengths. The intraband HHG exhibit a complex trend that firstly increases and then decreases with increasing the pump wavelengths, as depicted in Figs. 2(e), 3(e) and 4(e). The reason for this fluctuated characteristic is still not clear, and worth investigating in future. We experimentally distinguish the interband and intraband mechanisms in solids through their different wavelength dependences, which provides a direct measurement of the interband and intraband dynamics. The wavelength scaling of cutoff energy of HHG in solids is another topic without clear conclusion. Several studies claim that cutoff energy of solid-state HHG linearly scales with driving wavelengths, whereas some think that it is independent on driving wavelengths [29–31]. In this work, we demonstrate that the measured cutoff energy of solid-state HHG shows no wavelength dependence for harmonics both below and above the bandgap of materials.
3.2 HHG trajectory dependence of the harmonic polarization
It has been recently revealed that the HHG mechanism is closely related to harmonic polarization [39]. As depicted in Fig. 5(a), to test the HHG trajectories driven at wavelengths from the two regions, the polarization angle φ of the high harmonic emission is measured at two wavelengths, namely 3.0 and 7.0 µm, through rotating the pump polarization θ. A CdTe crystal with 6-fold symmetry is excited and a polarizer working in a wavelength range of 450 to 750 nm is used, thus polarizations of harmonics with the 4th to 5th orders and 11th to 14th orders are measured at 3.0 and 7.0 µm, respectively. As shown in Fig. 5(b), when a driving wavelength of 3.0 µm is chosen, it is observed that the odd harmonics (the 5th order) keep along with the pump polarization, whereas the polarization direction of the even order (the 4th order) switches from parallel to perpendicular with respect to the pump when the pump polarization changes from θ = 0° to 30°. This response is similar to the results observed by the previous works in MoS2 [36] and GaSe [34] and in accordance with symmetry requirements of the crystals with broken inversion symmetry. Moreover, it shows a fixed polarization for the perpendicular excitation (θ ≈ ± 30°), which demonstrate that the HHG emission is dominated by the interband contribution [39]. When the driving wavelength is increased to 7.0 µm, polarizations of even order harmonics (12th and 14th order) are fixed along the crystal mirror planes for the parallel excitation (θ ≈ 0°), which again agrees well with the theoretically predicted polarization of the intraband contribution [39]. Compared to the low odd-order harmonic (the 5th order) with no noticeable modulation, the higher odd-order harmonics (the 11th and 13th orders) exhibit stronger modulation due to the electron-hole trajectories of higher-order harmonics are highly deflected by the periodic crystal potentials, as shown in Fig. 5(c). A similar modulation was observed in previous studies of HHG in MoSe2 [39] and MgO [9,40]. In addition, CdTe has smaller bandgap energy (1.44 eV) compared to those of transition-metal dichalcogenides WS2 (2.01 eV) and dielectrics MgO (7.77 eV), which suggests that stronger effects of nonparabolic band in CdTe amplify the deflection. The polarization analysis provides an experimental validation to interband and intraband dynamics in two regions, where interband mechanism dominates the HHG in short-wave mid-IR wavelengths, whereas the intraband emission is the dominated mechanism at long-wave mid-IR wavelengths.
Fig. 5. Analysis of interband-intraband dynamics of HHG of CdTe. (a) Schematic diagram of the HHG polarization measurement. The yellow and pink lines indicate laser polarization and harmonics polarization respectively. The gray dotted line represents the mirror plane of crystal. The MIR polarization angle θ and HHG polarization angle φ are referenced with respect to the crystal mirror plane. (b) HHG polarization angle φ as a function of θ at pump wavelength of 3.0 µm. The 5th harmonics follows the MIR polarization direction. The polarization direction of 4th harmonics switches from parallel to perpendicular with respect to the pump polarization. (c) and (d) Polarization angle φ as a function of θ at pump wavelength of 7.0 µm. The odd-order harmonics shows a noticeable modulation.
4. Conclusion
In conclusion, we experimentally investigate the wavelength scaling of cutoff energy and intensity of solid-state HHG driven at wavelengths from 4.0 to 8.7 µm in CdTe, GaSe and ZnSe bulk materials. The variations of harmonic intensity can be summarized as a two regions characteristic. When the driving wavelength is shorter than 6.5-7.0 µm, HHG in solids is dominated by interband mechanism, where the harmonic efficiency decreases with increasing driving wavelengths. In the second region, when the driving wavelength is longer than 7.5 µm, the HHG process is dominated by intraband mechanism, in which the harmonic intensity fluctuates with pump wavelengths. The interband and intrabnd dynamics in two regions are supported by the polarization analysis. Moreover, the cutoff energy is experimentally observed to be wavelength independent. Our work is useful not only for understanding the ultrafast dynamics of HHG in solids, but more importantly, providing a guidance in choosing MIR driving wavelengths for strong-field experiments in solid-state material systems.
Funding
National Natural Science Foundation of China (12175157, 12274230, 62075144); Outstanding Youth Science and Technology Talents Program of Sichuan (2022JDJQ0031); China Postdoctoral Science Foundation (BX20220311).
Acknowledgments
The authors would like to truly thank Dr. Zhong Guan, Jiaqi Liu, Cheng Jin, and Xuebin Bian very much for the helpful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Supplemental document
See Supplement 1 for supporting content.
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