Estimations of Low-Inertia Cubic Nonlinearity Featured by Electro-Optical Crystals in the THz Range

Despite the growing interest in nonlinear devices and components for light by light control in the terahertz range, there is a shortage of such materials and media used for these purposes. Here, we present the calculated values of low-inertia nonlinear refractive index coefficient for electro-optical crystals used in THz time-domain spectroscopy systems such as ZnSe, ZnTe, CdTe, GaP, and LiNbO3. The medium parameters affecting the cubic nonlinearity of the vibrational nature increase in the range of 0.5–1 THz have been determined. Comparison of theoretical calculations with known experimental results confirm the theoretical model as well as our analysis of media parameter influence on the cubic nonlinearity. In terms of applications, results obtained open up new perspectives for studying various materials in the THz frequency range.


Introduction
At present, the development of THz technologies is advancing by leaps and bounds [1]. The amount of materials used for generating, detecting, modulating, and controlling THz radiation is constantly expanding with the new ones, such as metamaterials and two-dimensional materials. While the field working on such material creation is currently at its development stage, there are a lot of examples of media that proved themselves in the sphere, i.e., crystals, glasses, liquids, and gases. Nonlinear media are of particular interest among the materials mentioned. For example, nonlinear electro-optical crystals are widely employed in THz technology as generators and detectors [2]. Aiming at finding some new applications for them, it is necessary to study the features of their nonlinear responses in the THz range.
Owing to the recent developments in the field of coherent THz radiation source creation, pulsed sources featuring peak intensity of about 10 13 W/cm 2 have appeared [3]. Such large intensity values furnish insights into the observation of not only linear [4], but also nonlinear phenomena [5] of various materials in the THz region of the spectrum. As mentioned above, electro-optical crystals used for THz radiation detection and generation [6] are the first choice for the study of the nonlinear response, since they are widely used in modern THz technology.
In this paper, we present the calculated values of the low-inertia nonlinear refractive index coefficients in the range of 0.5-1 THz for electro-optical crystals used in THz time-domain spectroscopy systems such as ZnSe, ZnTe, CdTe, GaP, and LiNbO 3 . The results presented correspond to the experimental estimations performed by other research teams so far. The accuracy is around an order of magnitude. The subsequent analysis revealed the parameters influencing the n 2 coefficient value increase in the THz range. To prove the analysis validity, the results of our research for water are then given. They confirm the stated relationship between the media parameters and the n 2 coefficient magnitude. The analysis performed is a reference point for experimental teams to choose of the materials featuring high n 2 for the research.

Methods
As widely known, thermal nonlinearity is considered the largest one conventionally. However, THz pulses feature the duration of 1-2 ps and the thermal nonlinearity contribution is small as it has high-inertia nature. It has been shown both analytically [7,8] and experimentally [9] that regarding THz frequency range, the nonlinearity of the vibrational nature contributes to the overall cubic (third-order) nonlinearity the most, which occurs owing to its low inertia (of the order of or less than 1 ps).
The theoretical approach proposed in [7] for calculating the low-inertia nonlinear refractive index coefficient (n 2 ) in the terahertz frequency range far from the fundamental resonance (Equation (55) in [7]) is used for the evaluations carried out in this work. The formula has the following form: where n 0,ν = 1 + n 2 0 − n 2 el describes the vibrational contribution to the low-frequency refractive index, n 0 denotes the linear refractive index in THz range (0.5-1.0 THz), n el is the linear refractive index in the non-resonant electron contribution range (i.e., 800 nm), ω 0 is the fundamental vibration frequency, a l is the lattice constant, m describes the reduced mass of the vibrational mode for , α T is the coefficient of thermal expansion, and S denotes the relative density. Equation (1) is given in the CGS system of units. To convert this value to the corresponding one in SI, the following relation can be used: This approach is making use of the fact that the dominant nonlinearity mechanism in the frequency range mentioned features a vibrational nature (i.e., vibrations of lattice ions for crystals being the case) and is based on a classical anharmonic oscillator model. The approach is valid for the case when the radiation frequency is considered less than the one corresponding to the resonance of the medium, a single vibrational resonance is inherent in the medium, or one of them is visibly dominant.

Results
Earlier, the n 2 coefficient in the terahertz frequency range for crystalline quartz (SiO 2 ) was estimated by using the theoretical approach described [7]. Crystalline quartz has high transparency in the visible and IR spectral ranges, as well as in the THz frequency range (starting from 100 µm), and, therefore, it is used as a material for optical components (transparency windows, lenses) in terahertz technology [10] actively. Additionally, the material mentioned is anisotropic and has birefringence in the THz spectral region [5]. It was shown in [7] that the vibrational contribution in the THz frequency range can exceed the electronic one significantly and the value of n 2 in the THz range is several orders of magnitude higher than in the NIR.
Active development of THz technologies and the appearance of high-intensity THz sources gave impetus to the study of the nonlinear properties of various materials common for the THz range. For instance, the vibrational nonlinearity in the THz frequency range featured by electro-optical crystals used for the detection and generation of THz radiation, such as ZnSe, ZnTe, CdTe, GaP, and LiNbO 3 , is worth consideration. The parameters used for the calculations performed are presented in Table 1.
The low-inertia n 2 calculation results for the media mentioned, as well as the values for SiO 2 provided for comparison, are illustrated in Figure 1 (for LiNbO 3 , the mean n 2,THz value for a and c axis is shown). Table 2 represents the low-inertia nonlinear refractive index coefficients in the NIR (n 2,IR ).     [36] For some crystals, the low-inertia coefficient of the nonlinear refractive index in the THz frequency range exceeds the corresponding value in the NIR range. It can be related to the fact that the vibrational contribution to the refractive index of the media is quite large.
The experimental confirmation is needed to prove the analytical model validity. However, there are just few experimental works conducted so far. For instance, in the paper [37], authors using the measurement of the angular dependence of the Kerr signal and the theoretical analysis of the experiment determined the nonzero tensor elements of the third-order response function for GaP and estimate its n 2 parameter to be 1.2 × 10 −13 cm 2 /W. For lithium niobate crystal, experimental measurements based on the change of the transmitted THz pulse shape [38] give the n 2 value of 5.4 × 10 −12 cm 2 /W. For the THz range results, the values taken from other sources correlate with our estimations within about an order of magnitude, which is a tolerable error and a standard phenomenon for such a small n 2 value. However, it should be mentioned that these experimental results were indirectly estimated.

Discussion
Regarding the media parameters affecting the value of n 2,THz , an increase in this coefficient can be inherent in media featuring a higher coefficient of thermal expansion, a larger difference between linear refractive index in the range with non-resonant electronic contribution and linear refractive index in the THz range considered, a larger fundamental frequency of vibrations, and a smaller value of the numerical density of vibrations. The latter is ensured by an increase in the total mass of atoms and a decrease in the relative density of the medium (see Figure 2a). It is also important that the vibrational contribution to the low-frequency refractive index be larger (see Figure 2b). This depends on the values of the linear refractive index in the THz frequency range and in the range containing the non-resonant electronic contribution (NIR) and on difference between them. Moreover, we have tested this theoretical approach experimentally earlier for water [9]. Obviously, the structure featured by liquids is different from the one of crystals, so it seems quite logical to doubt the idea to implement this theoretical model in case of water. Here, the anharmonic oscillator vibration model is considered. Crystals being the case, the vibrations mentioned are the ones of lattice ions, whereas vibrations of a molecule as a separate structure are addressed regarding liquids. Both oscillation types have the same nature; therefore, the approach is valid for liquids as well.
The z-scan method was used to conduct the measurements [39]. The technique essence consists of the induced narrowing and broadening of an intense spherical light beam when a nonlinear medium moves along its propagation axis and passes through the focus. The nonlinear medium then acts as a thin lens and leads to a minimal change in the distribution of the beam field in the far field when placed in or near the focus. The resulting characteristic z-scan curve represents the peak and valley of the nonlinear medium transmission. The magnitude of the difference between the maximal and minimal values allows to calculate the nonlinear refractive index coefficient. Earlier we have shown that this technique is applicable for THz frequency range featuring a very broad spectrum with the correct ratio of the crystal thickness to the spatial size of the pulse [40]. Regarding water as the nonlinear medium, we obtained theoretically a value of n 2 = 5×10 −10 cm 2 /W and experimentally n 2 = 7 ± 5 × 10 −10 cm 2 /W. These values have a very good correspondence and show that for THz, the frequency range coefficient of nonlinear refractive index coefficient of water is 6 orders of magnitude higher than in the NIR frequency range. Additionally, one can see that the n 2 value featured by water in THz range is in the mean several orders of magnitude higher than in the case of crystals. Based on the above given analysis of the medium parameters' contribution to cubic nonlinearity, the explanation comes from the water characteristics directly. Water manifests higher fundamental vibrational frequency (100 THz), a larger difference between linear refractive index in the range with non-resonant electronic contribution (1.33) and linear refractive index in the THz range (2.3), and a smaller value of the numerical density of vibrations (3.3 × 10 22 ). Furthermore, water exhibits a 1000 times greater thermal expansion coefficient (0.2 × 10 −3 • C −1 ), which can also add to the overall n 2 value.

Conclusions
The work presents the analytically calculated values of the low-inertia nonlinear refractive index coefficient for common electro-optical crystals used in THz spectroscopy systems and their comparison with the experimental data published before. The estimations conducted confirm that the contribution to the media nonlinearity associated with lattice ion vibrations is quite large in the THz spectral range. The value for LiNbO 3 in the range of 0.5-1 THz exceeds the corresponding value in the NIR range. The relationship between media parameters and n 2 is presented and analyzed in terms of vibration related parameters. Our work [9] experimentally proves the validity of the theoretical approach used and justifies the conclusions regarding the medium parameters that determine the n 2 coefficient magnitude. Current work in the context of common THz crystals gives deeper insight into the materials featuring high n 2 for experimental teams to base their choice of the media to investigate on.
The results presented can be used as a summary useful for a wide range of future studies devoted to the search and choice of the media to be employed for different manipulation devices, all-optical switching for routing, modulation and control of THz signals, i.e., [41], as well as high-harmonic generation [42] in the THz range. The information provided in this article serves as a platform for material development and study, i.e., 2D materials, metamaterials [43], and liquid media [9], which have a potential to be used in nonlinear THz devices. Funding: This research was funded by RFBR project 19-02-00154.

Conflicts of Interest:
The authors declare no conflict of interest.