# Nonlinear TeraHertz Transmission by Liquid Water at 1 THz
## 3 Results and Discussion
In order to estimate the nonlinear THz intensity transmission by water (Equation (2)), we plot  
\[\text{d}_{\text{I}}(z)=2\times\frac{\text{I}_{\text{watt}}(z)}{\text{I}_{ \text{watt}}(z=22\text{ mm})}-\frac{\text{I}_{0}(z)}{\text{I}_{0}(z=22\text{ mm})} \tag{2}\]  
Figure 2: (**a**) Absorption coefficient of a single diamond window of the empty sample holder measured with linear THz time-domain spectroscopy. (**b**) The absorption can be fitted to a Drude peak with few free carriers, corresponding to less than one carrier per billion carbons. (**c**) Z scan measurements of the THz intensity transmitted by the empty diamond cell (black), and of the liquid cell filled with a 100-micron thick water layer (blue). The error bars are the standard error of the mean calculated from 300 measurements. The squares and circles are the experimental results, the solid lines are guides to the eye. (**d**) The nonlinear intensity transmission of the water layer is estimated as the difference of the intensity transmissions of filled and empty cell (d\({}_{\text{I}}\)). The solid red line is a Gaussian fit. The experiments reported in Panels c–d are performed with an intensity detector (bolometer). in Figure 2d. The factor two, roughly accounts for the fact that two diamond windows contribute to the nonlinear transmission of the empty sample holder, while only one diamond window contributes to the water-filled cell nonlinear transmission. The amplitude of d\({}_{\text{I}}\) at the THz focus can be estimated to \(1.026\pm 0.008\) with a Gaussian function (solid red line in Figure 2d). When the electromagnetic power (P) increases up to the nonlinear response regime, the absorption coefficient of a material can be written as Equation (3) [27]:  
\[\alpha(\text{P})=\alpha+\alpha^{\text{NL}}\times\text{P}, \tag{3}\]  
where \(\alpha\) is the linear absorption term, and \(\alpha^{\text{NL}}\) quantifies the nonlinear contribution that increases with P. When the sign of \(\alpha^{\text{NL}}\) is negative, the absorption decreases and the nonlinear response can be described as saturable absorption or bleaching. Open aperture z scan intensity measurements are insensitive to the nonlinear refraction [26], and can quantify the nonlinear absorption term (Equation (4)) as in [27]:  
\[\alpha^{\text{NL}}\approx(1-\text{d}_{\text{I}}(0))\frac{\alpha}{\text{P} \times(1-\text{e}^{-\alpha\text{L}})}, \tag{4}\]  
where L is the sample thickness. From the value d\({}_{\text{I}}(0)=1.026\pm 0.008\) obtained from Figure 2d, the THz peak power P = 61.4 MW/cm\({}^{2}\), the sample thickness, and the equilibrium absorption coefficient of water [33] (\(\alpha=197.4\) cm\({}^{-1}\) at 1 THz), we obtain \(\alpha^{\text{NL}}_{\text{wat}}=-87\pm 27\) cm/GW. Within the error bars, this value agrees with \(\alpha^{\text{NL}}_{\text{wat}}\approx-78\) cm/GW estimated from Figure 2a of [23]. To achieve the first approximation, in the data analysis we assumed that the nonlinear transmissions of both the two diamond windows of the empty cell, as well as of the empty and filled cell, are additive. Following these assumptions, we estimated a nonlinear absorption for liquid water that agrees with previous results. This good agreement confirms the validity of the approximations used. In the next step, we want to investigate the nonlinear response of the index of refraction (Equation (5)). In the simplest case:  
\[\text{n}(\text{P})=\text{n}+\text{n}^{\text{NL}}\times\text{P}, \tag{5}\]  
where n is the linear refraction term, and n\({}^{\text{NL}}\) (also called n\({}_{2}\) or Kerr term) quantifies the nonlinear contribution. Open aperture z scan measurements performed with an intensity detector (e.g., bolometer, Figure 2), are sensitive to absorption variations, but not to n\({}^{\text{NL}}\). The nonlinear refractive index is typically measured by comparing open and partially closed aperture z scan experiments, both measured with an intensity detector.