# Nonlinear TeraHertz Transmission by Liquid Water at 1 THz
## 3 Results and Discussion
The most intense THz pulse, with intensity P\({}_{0}\), is the one transmitted at every interface. The first and most intense reflected beam originates from two reflections at the air-diamond interface when the empty cell is measured. We estimate the intensity of the first reflection with Fresnel coefficients as:  
\[\text{d}_{\text{I}}(z)=2\times\frac{\text{I}_{\text{watt}}(z)}{\text{I}_{ \text{watt}}(z=22\text{ mm})}-\frac{\text{I}_{0}(z)}{\text{I}_{0}(z=22\text{ mm})} \tag{2}\]0where \(n_{dia}\approx 1.8\) (\(n_{air}\approx 1\)) is the index of refraction of the diamond window (air) at ca. 1 THz, and the second power accounts for the relation between field amplitude and intensity. The intensity of the second reflected beam, which originates from four reflections at the air-diamond interface, is even smaller (P\({}_{2}\approx 4\times 10^{-5}\times\)P\({}_{0}\)). Equation (9) indicates that by neglecting the first multiple reflection we induce a systematic error of less than 1%. We previously estimated the nonlinear absorption of liquid water to \(\alpha_{\text{water}}^{\text{NL}}=-87\pm 27\text{ cm}/\text{GW}\) and the nonlinear refraction to \(n_{\text{water}}^{\text{NL}}=(6.5\pm 2.6)\times 10^{-10}\text{cm}^{2}/\text{W}\). Both terms have statistical errors within 30% and 40%. Thus, we can neglect the \(<1\%\) systematic error induced by the multiple reflections. Finally, we note that when the cell is filled, liquid water strongly absorbs THz radiation [33] and the effect of multiple reflections is further reduced. The ab initio molecular dynamics (MD) calculations with static (DC) fields by [PERSON] et al. [14] suggest that the absorption at \(\sim\)1 THz by liquid water should decrease for increasing DC field strengths. This could be consistent with the experimental observation that the nonlinear absorption coefficient of water is negative (\(\alpha_{\text{water}}^{\text{NL}}=-87\pm 27\text{ cm}/\text{GW}\)), i.e., water is more transparent and absorbs less radiation for increasing field strength. However, it is important to note that the THz source used here is not DC [39; 40], but emits almost single-cycle pulses centered at \(\sim\)1 THz. New experiments and theory are needed to clarify the origin of both the absorptive (\(\alpha_{\text{water}}^{\text{NL}}\)) as well as the refractive (\(n_{\text{water}}^{\text{NL}}\)) nonlinearities detected here in bulk liquid water. We speculate that these large anisotropies might stem from the local asymmetry in the strength of the hydrogen bonds [25] and/or be rooted in the resonant enhancement of molecular reorientations [24]. In conclusion, we have shown that the full nonlinear THz transmission properties of liquid water can be deduced from measurements performed in a static cell with windows. Field-resolved detection allows estimation of both the real as well as the imaginary nonlinear optical coefficients, even when only open aperture z-scan transmission measurements are performed. Given the small sample volume required, this technique can be readily applied to assess the transient solvation dynamics even of complex solutes in aqueous environments, and could be useful to other research fields. Extensions of this novel experimental approach could also help to clarify the recent, puzzling, and mounting evidence of temporally delayed THz responses by liquid water [24; 41; 42; 43]. Conceptualization, F.N. ; methodology, F.N. and E.M.A. ; software, N.A., P.D.P., A.P. ; validation, F.N., C.Y.M., N.A., T.O., D.D.M., P.D.P., A.P. ; formal analysis, F.N. ; investigation, F.N., C.Y.M., N.A., E.M.A., T.O., D.D.M., P.D.P., A.P. ; resources, all authors; data curation, all authors; writing--original draft preparation, F.N., E.M.A., P.D.P., A.P., and M.H. ; writing--review and editing, F.N. and E.M.A. ; visualization, F.N. ; supervision, F.N., A.P., and M.H. ; project administration, F.N., A.P., and M.H. ; funding acquisition, M.H. All authors have read and agreed to the published version of the manuscript. M.H. acknowledges funding by the ERC Advanced Grant 695437.