# Nonlinear TeraHertz Transmission by Liquid Water at 1 THz
## 3 Results and Discussion
The experiments in Panels b–c are performed with a field detector (EOS). thinner spacer the THz response is \"sharper\": the nonlinear water signal is evident over a narrower range around z = 0 when the thinner spacer is used (compare Figure 3c with Figure 2d). Routine z scan measurements are obtained either with an open or a partially closed aperture in front of the detector [23]. Often, z scan experiments measure only the transmitted intensity with, e.g., a bolometer or a diode, and not the field transmittance [26; 27; 28]. An open aperture intensity z scan allows to quantify only the nonlinear absorption (\(\alpha^{\text{NL}}\)) [26], while a partially closed aperture intensity z scan is sensitive both to the nonlinear absorption as well as the nonlinear refraction [26]. The nonlinear refractive index (\(n^{\text{NL}}\)) can be estimated by comparing two intensity-sensitive z scans, obtained either with open or partially closed aperture [26]. Here we propose to replace the closed aperture intensity z scan, with an open aperture field z scan. The transmitted field measured with, e.g., EOS detection, must depend on the nonlinear absorption and, through the change of the spot size induced by the Kerr lensing, also on the nonlinear refraction. The advantage of field-sensitive z scan is the superior dynamic range, which can exceed 60 dbm as we demonstrated recently for THz-TDS based on an amplified laser [29]. The nonlinear refractive index can be estimated as shown in Equation (7) [23]:  
\[n^{\text{NL}}=\Delta T\frac{\alpha}{\text{P}\times\left(1-\text{e}^{-\alpha \text{L}}\right)}\frac{\lambda}{1.8\times\left(1-\text{S}\right)^{0.25}}, \tag{7}\]  
where \(\Delta T\) is the modulation of a z scan estimated as the difference between minimum and maximum transmission, \(\lambda\) is the radiation wavelength, and S is the linear transmission of the aperture. This equation is strictly valid only when comparing intensity-resolved z scan experiments with different apertures, but we use it in the following to estimate tentatively the nonlinear refraction by comparing intensity- and field-resolved measurements. To first approximation we assume S = 0. For simplicity, we assume that the refraction-induced transmission change \(\Delta T\) (Equation (8)) corresponds to the difference between the nonlinear transmission detected with a field detector (which depends on both \(\alpha^{\text{NL}}\) as well as \(n^{\text{NL}}\)[38]) and the nonlinear transmission acquired with an intensity detector (which depends only on \(\alpha^{\text{NL}}\)[27]):  
\[\Delta T\approx\left(d_{E}(0)\right)^{2}-d_{I}(0)-1=0.0094\pm 0.0038, \tag{8}\]  
where the first term is squared to account for the relation between field and intensity. Thus, the nonlinear refraction of pure liquid water at \(\sim\)1 THz corresponds to \(n^{\text{NL}}_{\text{watt}}=\left(6.5\pm 2.6\right)\times 10^{-10}\text{cm}^{2}/ \text{W}\). Within the error bars, this value agrees again with the value \(n^{\text{NL}}_{\text{watt}}\approx 7\times 10^{-10}\text{cm}^{2}/\text{W}\) reported by [PERSON] et al. [23] for a water jet. This additional benchmark confirms that while some of the above approximations are crude, they provide correct estimation of the magnitude of both amplitude as well as phase nonlinearities displayed by liquid water at \(\sim\)1 THz. In the data analysis we neglected the contribution of multiple reflections coming from the cell windows. We note that when we measured the THz peak field (Figure 3), the reflections originating from the layer between the diamond windows are delayed. This can be estimated to twice the layer thickness divided by the speed of light, resulting in a delay of ca. 0.5 ps. As the near-infrared pulse used for the electro-optical sampling detection is shorter than 0.1 ps [32], the \(\sim\)0.5 ps delay moves the reflected field outside the detection window. However, when we measured the THz intensity with a power detector (Figure 2), the THz beams originating from multiple reflections are fully detected.