# Nonlinear TeraHertz Transmission by Liquid Water at 1 THz
## 2 Materials and Methods  
We performed experiments at the THz source TeraFERMI [32], located at the free electron laser FERMI at the Elettra synchrotron (Trieste, Italy). The source emitted almost single-cycle pulsed THz fields centered at \(\sim\)1 THz and at a repetition rate of 50 Hz. The liquid is enclosed in a static cell with 0.5 mm thick diamond windows and a clear aperture of \(\sim\)25 mm. The pure liquid water thickness is set to ca. 100 or 50 \(\upmu\)m with Kapton spacers. The temperature of the sample holder is stabilized to \(20\pm 0.05\)\({}^{\circ}\)C with a recirculating chiller. The diffusion limited thermalization time of the sample is approximately \(t\approx\frac{r^{2}}{D}\) with \(D\) diffusivity and \(r\) radius of a sphere of volume equal to the sample volume [33; 34]. Considering \(D=1.4\times 10^{-7}\)m\({}^{2}\)/s for liquid water and the sample volume, we estimate the thermalization time of the irradiated water to \(t\approx 0.3-1.7\) s. The repetition rate of the source is 50 Hz (0.02 s), thus up to \(15-85\) pulses can cumulatively heat up the water sample. Assuming that all the THz power is instantaneously converted into heat, we estimate the temperature increase of the liquid water sample to within \(+0.02\)\({}^{\circ}\)C and \(+0.04\)\({}^{\circ}\)C. This temperature increase is lower than the noise level of the recirculating chiller that sets the temperature of the static cell to \(20\pm 0.05\)\({}^{\circ}\)C. For this reason, thermal effects can be disregarded. The THz radiation is detected with an intensity detector (silicon bolometer) or via field-sensitive electro-optical sampling (EOS). EOS is performed with a 100 \(\upmu\)m thick ZnTe crystal. From the optical coefficients [35], the THz spot sizes measured with a camera (Ophir PyrocamIIIHR), and accounting for the reflection losses of the windows, we estimate the maximum THz electric field to be E\({}_{0}=215\) kV/cm at the focal position inside the diamond cell. As sketched in Figure 1, we performed z scan experiments on both the empty as well as the filled cell, and with either intensity (bolometer) or field (EOS) detectors. TeraFERMI is focused with a 4\" diameter off-axis parabolic mirror (OAPM) down to a spot size with full-width at half maximum (FWHM) of \(\sim\)1.1 mm. The sample position is controlled with a mechanical stage. When the sample holder is placed at the focal position (z = 0), the peak power (P) reaches the largest value P = 100% = 61.4 MW/cm\({}^{2}\). At z = 0, both the empty as well as the water-filled cell display nonlinear transmission (sketched in red in Figure 1a). When the sample is moved away from the focus to z > 22 mm, P decreases below 17% (\(<\)10 MW/cm\({}^{2}\), Figure 1b) and approaches the regime of linear transmission. In Figure 1b we show the THz spot size characterized with a THz camera replacing the sample, at the sample position. We note that in this case there is no sample displaying a nonlinear response. We characterized previously the optical properties of the diamond windows of the sample holder with linear, tabletop terahertz time-domain spectroscopy (THz-TDS). The details of this linear spectroscopy technique are reported elsewhere [36]. The diamond windows display a small absorption in the THz range (orange diamonds, Figure 2a). The frequency-dependent absorption coefficient \(\alpha(\omega)\) can be fitted to [37] the Drude model as shown in Figure 2b and Equation (1):  
\[\alpha(\omega)=\frac{\mathrm{eN_{c}\gamma^{2}\mu}}{\varepsilon_{0}\mathrm{n}c (\omega^{2}+\gamma^{2})} \tag{1}\]  
where e is the electric charge, N\({}_{c}\) is the concentration of free charge carriers, \(\gamma\) is the scattering rate, \(\mu\) is the carriers mobility, \(\varepsilon_{0}\) is the free space permittivity, n is the index of refraction, and c is the speed of light. From the fit we estimate N\({}_{c}\approx 1.13\times 10^{-14}\)cm\({}^{-3}\), which corresponds to ca. 6 free carriers in 10 billion carbon atoms of a diamond window.