# Polarization Dependent Excitation and High Harmonic Generation from Intense Mid-IR Laser Pulses in ZnO

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

_{2}lens onto the samples oriented such that the laser propagation direction was along the c-axis. The generated signals were collected in a transmission geometry and refocused onto the entrance slit of a spectrometer using a f = 5 cm CaF

_{2}lens. The mid-IR pulse duration and laser spot size in the plane of the ZnO sample were determined using a frequency resolved optical gating (FROG) [25] and a knife-edge technique, respectively. The laser intensity was then estimated from the experimentally determined pulse energy, duration and spot size. All experiments were carried out at room temperature and in an ambient atmosphere.

## 3. PL and HHG Emission from ZnO Thin Films and Bulk

## 4. Ellipticity Dependence of Strong Field Light Absorption

## 5. Ellipticity Dependence of HHG

^{2}, are shown in Figure 3a as a function of the laser ellipticity. All harmonic orders are most efficiently generated for linearly polarized laser pulses and vanish for circularly polarization. However, the drop of the harmonic intensities with increasing ellipticity depends on the harmonic order. In order to describe the dependence of the HHG process on the laser ellipticity, a super-Gaussian fit-function $\mathrm{exp}\left[-{\left(\epsilon /{\omega}_{n}\right)}^{2{k}_{n}}\right]$ was used, in accordance with the theory work published in reference [22]. The waist, determined by the fit parameter ${\omega}_{n}$, is used for a quantitative characterization of the sensitivity of the HHG process on the laser ellipticity for different harmonic orders n. Figure 3b shows the fit parameters ${\omega}_{n}$ and ${k}_{n}$, determined from the fitting procedure of the experimental results measured in the ZnO bulk (red) and thin film (blue) samples as a function of the harmonic order n. Furthermore, fit parameter values—calculated using theoretical models by Liu et al. (black) [22] and Zhang et al. (green) [23]—using a Gaussian fit function $({k}_{n}=1)$ are shown in the upper panel of Figure 3b. Both theoretical simulations are based on the density matrix equation and the results for the above band gap higher harmonics are interpreted in a frame of the semiclassical saddle-point analysis. The vertical dashed line depicts the ZnO bandgap at 3.2 eV. As it follows from the Figure 3b (upper panel), the ellipticity dependence of ${\omega}_{n}$ for below and above bandgap higher harmonics varies from 0.48 to 0.23 for the 5th and the 15th harmonic, respectively. The below band gap harmonics are much less sensitive to the laser ellipticity, corresponding to a larger ${\omega}_{n}$, than the above band gap harmonics. Qualitatively similar observations have been previously made experimentally [10] as well as theoretically [22]. Best fitting was achieved for a parameter ${k}_{n}$ > 1 value for all harmonics (see Figure 3b, lower panel). In combination with a larger coefficient of determination (R

^{2}), this motivated the use of the super-Gaussian fit function. Please note that the Gaussian curve waist is invariant under changes of the parameter ${k}_{n}.$ Therefore the additional fit parameter ${k}_{n}$has no influence on the comparison with the work of Liu et al. (as shown in the upper panel of Figure 3b), in which ${k}_{n}$ = 1 was used. Comparison of the sensitivity of the HHG-process in the bulk (red) and the thin film samples (blue) on the laser ellipticity reveal an identical behavior for harmonics below the 9th order. However, for higher harmonic orders the tendency varies: The HHG process in the bulk sample becomes more sensitive to the laser ellipticity with increasing harmonic order. In contrast, in the film sample, the generation of the 13th order is most sensitive on the laser ellipticity, compared to the lower and higher orders. The experimental results measured for the bulk sample are in very good agreement with numerical simulations published in reference [22]. The polycrystalline structure, a different impurity level and an associated modification of the electronic structure may explain the small deviations between the theory and experimental values obtained from the thin film. The significant effect of the electronic structure on the ellipticity dependence of HHG in ZnO was demonstrated by Liu et al. [22] In that work it is shown that small modifications of the material band structure qualitatively allows us to reproduce the measured ellipticity dependence of HHG in the thin film sample.

^{2}corresponding to 0.15 V/Å field strength and a laser ellipticity of 0.5. Here, the Bloch acceleration theorem $\overrightarrow{k}\left(t\right)=\frac{e}{\u045b}{{\displaystyle \int}}_{{t}_{0}}^{t}{\overrightarrow{E}}_{L}\left({t}^{\prime}\right)d{t}^{\prime}$ was used to determine the electron quasi momentum. Furthermore, it was assumed that the electron is generated at the Γ-point at a time ${t}_{0}$, when the electric field reaches its maximum value. Higher harmonics are only generated when the curvature of the electronic structure is not constant, i.e., when the band is non-parabolic. The band structure curvature of ZnO varies stronger for momentum values $\left|\overrightarrow{k}\right|$ with a large distance to the Γ-point. As shown in Figure 4b, an elliptically polarized laser field leads to a turn of the electron trajectory. Larger k-values, which are linked to an inhomogeneous electron movement, are thus less likely reached. Furthermore, for higher laser ellipticity an increasing part of the electron trajectory undergoes a homogeneous movement during a half cycle of the laser field, i.e., $\u2206{E}_{g}{}^{-1}|{}_{\overrightarrow{k}=\overrightarrow{k}\left(t\right)}=constant,$ resulting in a less efficient HHG. This effect is more pronounced for short wavelength lasers, since the laser field-based momentum transfer (quiver momentum) is directly proportional to the laser wavelength. Thus, the intraband generation mechanism is strongly affected by the laser ellipticity for short driving wavelengths, which is in good agreement with our findings, as displayed in Figure 5.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Photoluminescence (PL), defect emission and high harmonic generation (HHG) in ZnO. (

**a**) Schematic illustration of recombination processes upon strong field tunneling excitation of an electron-hole pair. Upper panel: PL and defect emission as a result of electron-hole recombination from conduction band (purple arrow) and defect states (green arrow). Lower panel: Harmonic radiation is generated by a field driven nonlinear intraband motion of the electron as well as electron-hole recombination. (

**b**) Emission from a thin film and bulk sample irradiated with linear polarized femtosecond laser pulses with a central wavelength at 3.85 µm. Upper panel: Normalized photoluminescence and defect emission spectra from the thin film (blue) and bulk (red) sample. Lower panel: High harmonic spectrum, on top of the PL and defect emission, generated in a thin film (blue) and bulk sample (red).

**Figure 2.**Photoluminescence (PL) and high harmonic emission from a ZnO thin film irradiated with femtosecond laser pulses at 3.8 µm as a function of the laser ellipticity.

**Figure 3.**Dependence of HHG in ZnO on the laser ellipticity for a laser intensity of 0.39 TW/cm

^{2}. (

**a**) Spectrally integrated emission of the 5th up to the 17th harmonic from the ZnO thin film sample. (

**b**) Upper panel: Comparison of the experimentally determined sensitivity of the detected harmonic orders n on the laser ellipticity, described by ${\omega}_{n}$, with theoretically determined values from Liu et al. [22] and Zhang et al. [23]. Lower panel: Fit parameter ${k}_{n}$ as a function of the detected harmonic order n from the bulk (red) thin film sample (blue).

**Figure 4.**Effect of the electronic structure symmetry properties on the ellipticity dependence of the HHG process. (

**a**) Ellipticity dependence of the 11th harmonic as a function of the laser polarization major axis rotation around the ZnO crystal c-axis. (

**b**) Electron momentum-dependent band gap. The high symmetry points Γ, K, K’ and M are labelled. The area from the Γ-point to the green dashed circle highlights the condition ${E}_{g}$($\overrightarrow{k}$) < 5.5 eV. This is the region where experimentally detectable harmonics are generated. The black dashed and red dotted lines indicate the electron trajectory upon strong field interaction when a 3.1 and 3.9 µm laser and an ellipticity of 0.5 is used, respectively.

**Figure 5.**Sensitivity of HHG from a ZnO thin film on the laser ellipticity as a function of the harmonic order and the emission frequency for five laser wavelengths. The laser intensity in the experiment was fixed to 0.39 TW/cm

^{2}. The inset depicts simulations performed by Liu et al. [22].

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**MDPI and ACS Style**

Hollinger, R.; Herrmann, P.; Korolev, V.; Zapf, M.; Shumakova, V.; Röder, R.; Uschmann, I.; Pugžlys, A.; Baltuška, A.; Zürch, M.;
et al. Polarization Dependent Excitation and High Harmonic Generation from Intense Mid-IR Laser Pulses in ZnO. *Nanomaterials* **2021**, *11*, 4.
https://doi.org/10.3390/nano11010004

**AMA Style**

Hollinger R, Herrmann P, Korolev V, Zapf M, Shumakova V, Röder R, Uschmann I, Pugžlys A, Baltuška A, Zürch M,
et al. Polarization Dependent Excitation and High Harmonic Generation from Intense Mid-IR Laser Pulses in ZnO. *Nanomaterials*. 2021; 11(1):4.
https://doi.org/10.3390/nano11010004

**Chicago/Turabian Style**

Hollinger, Richard, Paul Herrmann, Viacheslav Korolev, Maximilian Zapf, Valentina Shumakova, Robert Röder, Ingo Uschmann, Audrius Pugžlys, Andrius Baltuška, Michael Zürch,
and et al. 2021. "Polarization Dependent Excitation and High Harmonic Generation from Intense Mid-IR Laser Pulses in ZnO" *Nanomaterials* 11, no. 1: 4.
https://doi.org/10.3390/nano11010004