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Abstract
High harmonic spectroscopy in solids is emerging as a new tool to investigate ultrafast electron dynamics in the presence of strong optical fields. However, the observed high harmonic spectra do not usually reflect the microscopic origin of high harmonic generation (HHG) because of nonlinear and/or linear propagation effects. Here, we systematically investigate the HHG in reflection and transmission from gallium arsenide exposed to intense mid-infrared optical pulses. In transmission geometry, we find that the properties of high harmonics are drastically changed by nonlinear effects during the propagation of even tens of micrometers. Especially, the nonlinear absorption and/or nonlinearly induced ellipticity of the drive pulses as well as a cascade nonlinear mixing significantly alter the high harmonic signals in the case of the transmission geometry, making an extraction of the microscopic electron dynamics of gallium arsenide difficult. On the contrary, in reflection geometry, we obtain HHG spectra that are free from propagation effects, opening a general approach for high harmonic spectroscopy.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High harmonic generation (HHG) in solids emerges as a result of a field-driven electron motion in the entire Brillouin zone on a time scale of few-femtoseconds to attoseconds [1–6]. The elementary process for the HHG, however, is not easy to be extracted because of propagation effects such as linear absorption of produced harmonics and phase mismatch [7] and nonlinear interactions through a medium of a mesoscopic propagation length [8]. Previously, experimental observations of HHG in reflection (abbreviated as RHHG) were limited to few-order harmonic generation [9–11]. Especially, second harmonic generation at the interface has been intensively used to probe surface reactions and dynamics, as reviewed in [12]. Recently, RHHG from a solid surface was considered theoretically [13] and demonstrated experimentally [14]. In [14], comparison of HHG in reflection and transmission was demonstrated to study propagation effects inside solids. It was shown that HHG in transmission from a 0.2-mm-thick MgO crystal has a spatially and spectrally broader profile than the case of the reflection geometry, which indicates self-focusing and self-phase modulation of drive pulses induced by propagation through the crystal.
In this study, RHHG and HHG in transmission (abbreviated as THHG) are demonstrated for a semiconductor gallium arsenide (GaAs) when exposed to an intense mid-infrared (MIR) optical pulse. Systematic investigations reveal that nonlinear propagation effects drastically change the property of high harmonics in THHG, making it difficult to extract the microscopic origin. It is shown that an RHHG configuration minimizes the influence of the linear and nonlinear propagations, and therefore, RHHG signals can be directly connected to the underlying electron dynamics.
2. Experiment
A KTiOAsO4 (KTA)-based optical parametric amplifier pumped by a Ti:sapphire chirped-pulse amplification system generates femtosecond MIR optical pulses (149 μJ, 120 fs, 3.5 μm, and 300 Hz) [15]. The output MIR pulses from the parametric amplifier are further compressed down to 60 fs (5 cycles) by spectral broadening in a 5 mm-thick anti-reflection-coated germanium plate followed by dispersion compensation in sapphire and fused silica plates. The linearly polarized MIR pulses are focused with a CaF2 lens (f = 300 mm, f-number of 32) at an angle of incidence of ∼ 5° onto the (001) surface (fourfold symmetry) of a GaAs sample, as shown in Fig. 1. All the spectra are measured with a fiber-coupled thermoelectric cooled spectrometer (Ocean Optics, QEPro). Note that the transmittance difference between the s- and p-polarizations on the front surface is ∼0.25%. The orientation dependence of the high harmonic spectrum is measured by rotating the polarization of the MIR pulses.
Fig. 1 HHG experiment using GaAs (001). (a) Experimental arrangement for HHG experiments in reflection and transmission geometries. HWP, zero-order half waveplate; L1 (f = 300 mm) and L2 (f = 50 mm), CaF2 lenses; M1, Aluminum concave mirror (radius of curvature = 100 mm). (b) Crystal structure of GaAs (001).
3. Result
Figure 2 shows the RHHG and THHG spectra for three GaAs samples of different thicknesses (45, 170, and 650 μm), where the MIR polarization is set along the [110] direction. These samples are prepared by conventional chemo-mechanical polishing with bromine-methanol solutions, which can reduce the thickness of a semiconductor wafer down to a few tens of micrometers. The peak intensity of the ∼20 μJ pulses at the sample position in the air is estimated to be 600 GW/cm2 (21 MV/cm), which corresponds to 430 GW/cm2 (10 MV/cm) inside the samples. The sharp peak around 1.4 eV near the 4th harmonic is due to the fluorescence generated by the recombination between an electron and a hole. While the 5th harmonic in RHHG is more than one order of magnitude less than that in THHG, the harmonics higher than the 7th order in RHHG exceed those in THHG. The most striking feature is that the spectral width of the high harmonics in RHHG is much narrower than that in THHG. This strongly indicates the onset of self-phase modulation (SPM) during the propagation of the MIR pulses in the samples, resulting in the spectral broadening of the MIR pulses [16] and high harmonics. We note that GaAs is a direct bandgap semiconductor, and therefore, the absorption lengths of the 5th and 7th harmonics and the higher harmonics are approximately several hundreds of nanometers and less than 100 nm, respectively [17]. Therefore, in THHG, the high harmonics are generated from the back surface of all the samples with the interaction length limited by the above mentioned absorption. Figure 3 demonstrates that the intensity dependences of the 7th and 9th harmonics in RHHG are much steeper than those in THHG. The saturation effect, which is found in the THHG signals, is not observed in RHHG, indicating that THHG is severely affected by propagation effects and that the RHHG signal can reflect a bulk response without suffering from propagation. In the case of THHG with a low electric field, the harmonic intensities are less dependent on the thickness of the sample. However, with high electric fields above 10 MV/cm in the air, it is clear that high harmonics from thicker samples saturate faster. The saturation effect is partially explained by the nonlinear absorption of the MIR pulses in the sample, which will be discussed later. Figures 4(a) and 4(b) show the measured orientation dependences of the 7th harmonics in both geometries without resolving their polarization, respectively, where the peak intensity at the sample position is 630 GW/cm2 in the air for all the measurements. The angular coordinate shows the angle between the field polarization and [100] direction of GaAs. All the orientation dependences show the fourfold symmetry, which is consistent with the crystal structure of GaAs (001). In THHG, the orientation dependence becomes more anisotropic for the thicker sample. In contrast, the orientation dependence in RHHG is less anisotropic than in THHG.
Fig. 2 Above-bandgap HHG spectra in RHHG (gray shaded area) and THHG for 45- (green), 170- (blue), and 650-μm-thick (red) samples. The MIR field directs along [110] (θ = 45°).
Fig. 3 Dependences of the spectrally integrated yield of the 7th and 9th harmonics on the peak intensity of the MIR pulses. The dashed lines show the scaling of the yield near the saturation area of THHG and the dotted ones the perturbative scaling. Note that a 9th harmonic is hardly observed with the 650-μm-thick GaAs sample in THHG. The MIR field is set along [100] (θ = 0°).
Fig. 4 Orientation dependence of the normalized yield of the 7th harmonic in (a) RHHG and (b) THHG. Note that the polarization of the harmonics is not resolved.
We investigate the effects of nonlinear propagation on the deformation of the fundamental MIR pulses in detail, and find that the different properties observed for RHHG and THHG can be qualitatively explained by the waveform deformation. Firstly, we measure the transmitted energy of the MIR pulses and find significant influence of the nonlinear absorption, which leads to a transmittance far below that calculated using Fresnel’s formula shown in Fig. 5(a). This can be the main reason why THHG cannot produce higher order harmonics than RHHG in Fig. 2. The nonlinear absorption of GaAs is experimentally found to be anisotropic with respect to the crystallographic orientation (see inset of Fig. 5(a)), which has been observed before in the two- and three-photon absorption regime [18] (four-photon or beyond in our case). The transmittances of the MIR pulses with their fields in the [110] (45°) and [100] (0°) directions are 22.8% and 23.8%, respectively, for the case of the 170-μm-thick sample, where the peak intensity at the sample position is 630 GW/cm2 in the air. This small angular variation of the fundamental MIR pulse’ transmittance can lead to an angular variation of 30% for the 7th-harmonic yield assuming a perturbative scaling of I7, which is consistent with the observed angular variation shown in Fig. 4(b). Secondary, we measure the polarization state of the transmitted MIR pulses as a function of the angle between the incident polarization direction and the [100] crystal axis. Figure 5 (b) shows the orientation dependence of the transmitted MIR pulses through a polarizer that is placed after the 650-μm-thick sample in the direction parallel or perpendicular to the incident polarization. At rotation angles of 20° and 70°, where the parallel component becomes minimum, we observe a significant increase of perpendicular components. We also obtained a maximum ellipticity of 0.36 at 20° by analyzing the measured transmittance through the polarizer, as shown in Fig. 5(c), which reveals the emergence of a strong nonlinear birefringence. This nonlinear birefringence is consistent with the phenomenon known as cross-polarized wave (XPW) generation due to anisotropic third-order nonlinearity, which is generally observed in cubic materials and most effective around 22.5° and 67.5° from the [100] axis [19]. GaAs (100) has a fourfold symmetry, and thus, the measured orientation dependence of the perpendicular component (blue curve in Fig. 5(b)) can be explained as XPW.
Fig. 5 (a) The transmittance of the MIR pulses is measured as a function of the peak intensity of the incident MIR pulses, and (inset) the angle between (b) Orientation dependence of the transmitted MIR pulses through a polarizer. The orange and blue curves represent the components parallel and perpendicular to the incident MIR field. (c) Ellipticity analysis at one of the angle where the perpendicular component becomes maximum as indicated in (b). The red circles represent the measured transmittance as a function of the polarizer rotation angle. (d) Ellipticity dependence of the normalized yield of the 7th harmonic.
We measure the 7th-harmonic yield in RHHG by changing the ellipticity of the MIR pulses to investigate the influence of the ellipticity of the drive field in THHG, as shown in Fig. 5(d). At an ellipticity of 0.36, the 7th-harmonic yield drops to ∼40%, which clearly explains the appearance of dips around 22.5° and 67.5° from the [100] axis for the case of the 650-μm-thick sample in Fig. 4(b). These measurements clearly represent the advantages of RHHG over THHG for extracting fundamental electronic dynamics in solid HHG because it is free from the above-mentioned nonlinear propagation effects.
We apply the RHHG and THHG methods to the different (011) surface of a 30-μm-thick GaAs sample that has a non-centrosymmetric axis and allows the generation of even-order harmonics, as shown in the inset of Fig. 6. Figure 6 shows the high harmonic spectra in RHHG (shaded area) and THHG (red curve) with the peak intensity of 570 GW/cm2 at the sample position in the air. The direction of the MIR field is set to maximize the yield of the even-order harmonics, as indicated by the red arrow in the inset of Fig. 6. In RHHG, all the even-order harmonics are much weaker than their two adjacent odd harmonics. We note that the sharp peak around 1.4 eV in RHHG is fluorescence because it is completely depolarized, which may mask a possibly weak 4th harmonic signal. Similar experimental results were reported for a few-layer thick gallium selenide (GaSe) sample for which the second harmonic generation was much weaker than third-harmonic generation [20,21]. Quantum mechanical calculations for a unit cell have also predicted that even-order harmonics are much weaker than odd-order harmonics for gapped graphene [22] and GaSe [23]. The latter also shows that the slope of the odd-order harmonics is much steeper than that of the even-order harmonics. This is because the even-order harmonics are exclusively produced from interband transitions, whereas the intraband current dominates the odd-order harmonics. Similarly, we observe that the intensity of the odd-order harmonics decreases much more steeply with increasing harmonic order (dashed line in Fig. 6) than the even-order harmonics (dotted line in Fig. 6). This is a clear demonstration that even- and odd-order harmonics originate from interband polarization and intraband current, respectively. In contrast to RHHG, the even-order harmonics in THHG are strongly enhanced. This drastic increase cannot be explained by SPM, nonlinear absorption, or nonlinear birefringence. Therefore, the result shown in Fig. 6 suggests that, in the case of THHG, the even-order harmonics are enhanced by cascade-like sum and/or difference frequency mixing processes between odd-order harmonics and fundamental MIR pulses. The fact that the intensity of the even-order harmonics becomes comparable with that of the adjacent odd harmonics supports the existence of cascade effects.
Fig. 6 Above-bandgap HHG spectra using GaAs (011). The inset shows the crystal structure of GaAs (011) and the direction of the MIR field. Note that the spectrally broad 4th harmonic and the sharp peak due to fluorescence are simultaneously observed in THHG.
4. Conclusion
In conclusion, we successfully produced high harmonics in GaAs, both in reflection and transmission geometries. In the case of RHHG, high harmonics were produced up to the 15th order (5.3 eV), which is well beyond the bandgap of GaAs. A systematic study of RHHG and THHG in GaAs (001) samples of different thicknesses revealed that anisotropic nonlinear propagation effects of the fundamental MIR pulses, such as SPM, nonlinear absorption, and nonlinear birefringence, significantly affected (i) the spectral profile of high harmonics, (ii) the intensity scaling of harmonic yields, and (iii) the orientation dependence of harmonic yields. Moreover, using GaAs (011) samples, we showed that the even-order harmonics in THHG were strongly enhanced by cascade frequency mixing processes. The RHHG has advantages over the THHG because it is free from linear and/or nonlinear propagation effects and applicable to any bulk sample with a polished surface, offering a new methodology for high harmonic spectroscopy in solids.
Funding
Japan Society for the Promotion of Science (JSPS) (KAKENHI JP18H01469, JP18H05250, JP17H04816); Advanced Leading Graduate Course for Photon Science (ALPS); Quantum Basic Research Coordinated Development Program by MEXT.
Acknowledgment
We thank Nariyuki Saito for his efforts in the HHG experiments in solids.
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