Temperature-independent non-linear terahertz transmission by liquid water

On the microscopic scale, water molecules in the liquid phase make hydrogen bonds with tetrahedral structures that fluctuate on the picosecond (ps) timescale.¹ On the macroscopic scale, thermodynamic properties of liquid water, such as density and compressibility, are anomalous as they scale non-continuously.² However, it is unclear how the macroscopic behavior emerges from the microscopic properties.³ An inherently powerful spectroscopic tool to study liquid water is terahertz (THz) radiation because it can reveal the fluctuations of the water network on the picosecond time scale, which are of the same order of magnitude of the relaxation time constants.⁴ In fact, THz radiation between about 0.01 and 20 THz is strongly absorbed by the intermolecular collective modes of liquid water.^5–11 

Recently, several non-linear optical techniques in the THz range were applied to the study of liquid water. Optical-pump THz-probe experiments revealed the coupling between a dye and the solvating water,¹² the inhomogeneity in the bulk liquid,^13,14 and proton quantum effects.¹⁵ THz-Kerr experiments employ a THz pulse to reorient molecules and an optical probe pulse to detect the induced birefringence at visible or near-infrared frequencies, which is modulated by the Raman-active modes.¹⁶ These works revealed the anisotropic polarizability of hydrogen-bonded water molecules,¹⁷ the coupling between rotational and translational modes,¹⁸ and the effect of salts on the water structure.¹⁹ THz-pump THz-probe experiments are complementary to the aforementioned approaches because both the pump and the probe interact resonantly with infrared-active modes. While these latter, non-linear THz spectroscopy experiments reported signals with similar sizes at ∼1^20–24 and 12.3 THz,^25,26 i.e., a third-order nonlinear response with a magnitude of about χ³ ∼ 10⁻¹³ cm²/V² (see Table 1 in Ref. 26), the results have been rationalized in different ways.

Previously, we performed THz-pump THz-probe experiments at ∼12 THz^25,26 and z-scan measurements at ∼1 THz on water.²⁴ In Ref. 25, with the aid of molecular dynamics calculations and a dedicated model that includes all possible non-linear effects and every perturbation order, we proposed that the non-linear response of water in the THz range is due to the resonant reorientation of hydrogen-bonded water molecules. The Elsaesser group^22,23 performed THz-pump THz-probe experiments at ∼1 THz. They suggested that the THz pump is responsible for the irreversible ionization of water molecules and the subsequent generation of solvated electrons. The Kozlov group^20,21 performed z-scan measurements at ∼1 THz on water. Even though liquid water absorbs THz radiation^5–11 and resonant non-linear phenomena are typically stronger than non-resonant ones,²⁷ these authors^20,21 suggested that the non-linear refraction of water at ∼1 THz should originate from the non-resonant contribution by oscillations in the mid-infrared, at about 100 THz, 3 μm, or 3300 cm⁻¹. At these mid-infrared frequencies, liquid water absorbs radiation via the intramolecular O–H stretch mode,²⁸ rather than through the intermolecular modes involving hydrogen-bonded water molecules.^5–11 The non-resonant model proposed in Refs. 20 and 21 implies that the non-linear and frequency-dependent refraction is proportional to the thermal expansivity squared. For example, see Eq. (5) in Ref. 20 and Fig. 4 in Ref. 21. Thus, considering that the expansivity of water depends on temperature and becomes null at²⁹ 4 °C, one expects that the non-linear refraction by water at ∼1 THz should decrease upon cooling.

Here, we show that the non-linear transmission of intense radiation at ∼1 THz by water is the same at 21 and 4 °C, within the experimental uncertainty. This implies that the non-linear absorption coefficient of water is equal at these two temperatures. These results might be of help in understanding which molecular model should be preferred in the description of the non-linear response by liquid water in the THz range and suggest that models that are dependent weakly on temperature should be preferred.

We performed experiments at the beamline TeraFERMI in Trieste. The source and the available optical setups are detailed elsewhere.^30,31 In short, TeraFERMI generates intense and almost single-cycle THz fields lasting ∼1 ps with a typical spectrum centered at ∼1 THz [Fig. 1(a)]. The THz fields are detected via electro-optical sampling^32,33 in a 0.1 mm thick and [110]-oriented gallium phosphide (GaP) crystal with balanced photo-diodes. As the photo-voltage of each single unbalanced diode was not simultaneously detected, it was not possible to estimate the THz field amplitude from the electro-optical coefficients.³⁴ Due to the finite thickness of the detection crystal, reflections of the sampling and terahertz beams peak at electro-optical delays of about −2 and +2 ps, respectively. In order to avoid the spurious contribution of these reflections to the THz spectrum, we obtained Fig. 1(a) by detecting the THz field only between about −1 and +1 ps. This windowing induces the apparent frequency oscillations in the spectra of Fig. 1(a), with a period of about 1 THz. We will use only the value of the THz peak field amplitude in the following analysis. Please note that the spectrum in Fig. 1(a) is just an example of the typical source spectrum. It has been obtained under slightly different experimental conditions and cannot be used as a quantitative reference for the THz fields transmitted by water and shown in Fig. 1(c). A 2 in. off-axis parabolic mirror with an effective focal length of 3 in. focuses the THz field into a static cell with 0.5 mm thick diamond windows enclosing a 100 µm thick layer of pure water. The static cell is magnetically attached to a copper plate, whose temperature is stabilized to either 21 °C or 4 °C by a recirculating chiller, with an accuracy of ±0.1 °C. As shown in Fig. 1(b), the area of the THz spot size at the sample position was measured to be about 0.6 mm² full width at half maximum with a camera (Pyrocam IIIHR). A maximum THz power of ∼390 µW was measured with a pyro-electric based detector (Gentec THZ12D-3S-VP-INT-D0). The intensity lost by reflection at the first air/diamond interface, r_ad², can be estimated to r_ad² = ((1 − 2.4)/(1 + 2.4))² ∼ 17%, where r_ad is the Fresnel coefficient, 1 is the index of refraction of air, and ∼2.4 is the index of refraction of the diamond window.^35,36 Please note that the index of refraction of diamond is practically constant across the frequency³⁵ (∼1 THz) and temperature³⁶ (4 and 21 °C) ranges investigated here. The intensity lost by reflection at the second interface between diamond and liquid water is approximately r_dw² = ((2.4 − 2)/(2.4 + 2))² ∼ 0.8%, where we took the index of refraction of liquid water equal to^37,38 2, for simplicity. Thus, the THz intensity transmitted by the input diamond window is (1 − r_ad²)(1 − r_dw²) ∼ 82%, considering both air/diamond and diamond/water interfaces. As the maximum THz power available was about 390 µW, we estimate the power transmitted by the front diamond window of the static cell and hitting liquid water to 390 µW·82% ∼ 320 µW. This results in a maximum THz energy of 320 µW/50 Hz ∼ 6.4 µJ/pulse, fluence of 6.3 µJ/0.6 mm² ∼ 1 mJ/cm², peak power of 1 mJ/cm²/1 ps ∼ 1 GW/cm², and peak field amplitude³⁹ of (2376.7 Ω·1 GW/cm²)^0.5 ∼ 0.9 MV/cm. Please note that in the estimation of the peak power and field amplitude we assumed, for simplicity, that the temporal shape of the THz pulse is a square with a width of ∼1 ps. We controlled the THz intensity at the sample with two polarizers with a 10⁴:1 contrast ratio (infraspecs P01). Finally, we would like to point out that the non-linear response by the diamond windows (χ³ ∼ 10⁻¹⁷ cm²/V², see Table 4.1.2 in Ref. 27) is about four orders of magnitude smaller than in pure water (χ³ ∼ 10⁻¹³ cm²/V², see Table 1 in Ref. 26). Thus, to a first approximation, here, we neglect the non-linear contributions by the diamond windows.

The precise estimation of thermal phenomena requires solving intricate differential equations.⁴⁰ Here, for simplicity, we assumed that all the THz radiation is absorbed in a uniform layer of water as thick as one penetration depth and that the temperature decays by thermal diffusion. By using this approach, we were able to estimate the correct order of magnitudes of the thermalization timescales observed in previous experiments.^26,38 Thus, the thermalization time of the sample can be estimated to³⁸ τ ∼ L²/D ∼ 17 ms, where L = 50 µm is the penetration depth of liquid water at 1 THz and D = 1.5 × 10⁻⁷ m²/s is the thermal diffusivity. TeraFERMI emits radiation at a repetition rate of 50 Hz and the time delay between subsequent THz pulses is 20 ms. The thermal relaxation time (τ ∼ 17 ms) is comparable to the time delay between two subsequent pulses (20 ms), and we expect that the temperature buildup of the sample due to the THz pulse train is negligible. Thus, we can overestimate the maximum temperature increase of the sample due to the absorption of THz radiation from the energy of 2× subsequent pulses.²⁶ For simplicity, if we assume that two THz pulses are fully and instantaneously absorbed and transformed into heat in a water layer as thick as one penetration depth at 1 THz (L = 50 µm), we overestimate the maximum temperature increase of the water sample to 2·1 mJ/cm²/50 µm/(4.2 J/(cm³ °C)) ∼ 0.1 °C. As this is comparable to the accuracy of the recirculating chiller that stabilizes the sample within ±0.1 °C, we can neglect heating effects.

This estimation of the water temperature ignores the heat dissipated onto both diamond windows of the static cell, which reduces the heating of water even further.²⁶ Previously, we demonstrated that both non-linear refraction and absorption of liquid water at ∼1 THz could be estimated correctly from a static liquid cell.²⁴ The results from water in the static sample holder were identical to the ones obtained by others in a free-flowing liquid jet,^20,21 within the error bars. In addition, in that case, it was found that heating effects are irrelevant for liquid water at the 50 Hz repetition rate of TeraFERMI.²⁴ 

At equilibrium, the linear absorption coefficient of liquid water at THz frequencies depends strongly on temperature.^37,38,41,42 Between about 0.1 and 3 THz, the linear absorption increases upon heating and decreases upon cooling. Based on previous studies,^37,38,41,42 by decreasing the liquid temperature from 21 °C down to 4 °C, the absorption coefficient of liquid water at 1 THz is expected to drop by Δα₀ = (50 ± 9) cm¹. In Fig. 1, we show the THz spectrum [Fig. 1(a)], the THz spot size [Fig. 1(b)], and the THz fields transmitted by the diamond sample cell when filled with a 100 µm layer of water at two temperatures, 21 and 4 °C [Fig. 1(c)]. When we cool the liquid to 4 °C, the transmission increases, compared to that at 21 °C, as shown in the blue curve in Fig. 1(c).

At the electro-optical delay time at which the THz field reaches its maximum value [∼0 ps in Fig. 1(c)], all the frequency components of the THz beam add up constructively. Thus, the peak transmission of the THz field reveals the response of the sample that is averaged over the THz spectrum, which peaks at ∼1 THz as shown in Fig. 1(a). The advantage of using the THz peak field is that it can be detected quickly, with less time waited for the movement of the electro-optical sampling delay stage, thus allowing the signal-to-noise ratio to be improved. For these reasons, in the following, we discuss only the transmission of the THz peak field. In order to ensure the detection of the THz peak, we measured a few points around time zero in Fig. 1(c), i.e., over a temporal range of at least 0.5 ps with 0.07 ps long temporal steps of the electro-optical sampling delay. In this way, we were able to pinpoint and measure the value of the THz peak. Please note that resolving the full THz pulse is slow here because of the low repetition rate of the source used (50 Hz). In future experiments, we plan to compare these results with the ones obtained with laser-based high repetition rate systems that allow accurate frequency-resolved measurements. For these future experiments, we also plan to measure the complete fluence-dependent response, thus disentangling the absolute value of the nonlinear contribution to the THz transmission.

The ratio between the peak field transmitted by water at 21 °C and the peak field transmitted by water at 4 °C (TR) reveals the temperature-dependent absorption coefficient of liquid water. At equilibrium conditions (linear optics), the ratio between the transmitted THz peak fields at 21 and 4 °C is

where Δα₀ = α₀(21 °C) − α₀(4 °C) is the difference between the equilibrium absorption coefficients, d is the sample thickness, and the factor 2 accounts for the proportionality between the intensity and electric field squared. Experimentally, when the THz intensity (peak field) impinging upon the water sample is set to 0.3 GW/cm² (0.5 MV/cm), we find that the ratio between the peak field transmitted by water at 21 °C and the one transmitted at 4 °C is TR = (79.4 ± 0.5)%. By inserting TR = (79.4 ± 0.5)% and d = 100 µm in Eq. (1), we estimate an absorption change of Δα₀ = (46 ± 1.5) cm⁻¹ upon cooling water from 21 to 4 °C. This value agrees with the literature result at equilibrium,^37,38,41,42 Δα₀ = (50 ± 9) cm⁻¹. Please note that THz fields with amplitudes of ∼0.5 MV/cm do trigger sizable non-linear signals from water.^{20–22,24,26} Thus, this experimental observation implies that the non-linear response of water depends weakly on temperature. In order to investigate this further, we performed additional measurements at increasing THz fields as detailed in the next paragraphs.

When the intensity of the input radiation source is high enough, the absorption coefficient of any material—including liquid water—displays a non-linear response. At simplest, the absorption coefficient becomes intensity-dependent via the following equation:

where α₀ is the absorption coefficient of the material at equilibrium, I is the source intensity, and α_NL is the non-linear term. It has been demonstrated before^20,24,43 that liquid water at room temperature displays a non-linear absorption coefficient at ∼1 THz equal to α_NL ∼ −80 cm/GW. As shown in Fig. 2, the same value of the transmission ratio was detected at increasing THz intensities (I = 0.3, 0.7, 1 GW/cm²). This experimental finding indicates that the non-linear transmission and absorption by liquid water at 21 and 4 °C are the same, within the experimental uncertainty that amounts to about ±0.5% in the transmission ratio or ±1.5 cm⁻¹ in absorption.

In other words, we obtain the non-linear and intensity-dependent transmission ratio of the sample, TR(I), by combining Eqs. (1) and (2),

where Δα_NL = α_NL(21 °C) − α_NL(4 °C) is the difference between the non-linear THz absorption coefficients of water at the two temperatures. The transmission ratio described by Eq. (3) becomes independent of the THz intensity when the argument of the second exponential is negligible, i.e., when Δα_NL·I·d/2 ∼ 0. As demonstrated before,^20,21,25 the THz intensities used here are enough to induce large non-linear responses in water. Thus, the only condition by which Eq. (3) becomes independent of the THz intensity is that Δα_NL ∼ 0, implying α_NL(21 °C) ∼ α_NL(4 °C) within the noise level of this experiment, ∼1.5 cm⁻¹. Only when this is valid, it is possible to obtain a value of the TR that is independent of the THz intensity, as experimentally demonstrated in Fig. 2. In order to estimate the sensibility of these measurements, we assume, for simplicity, that the non-linear absorption coefficient of liquid water is zero at 4 °C, α_NL(4 °C) ∼ 0, and equal to the literature value at^20,24 21 °C, α_NL(21 °C) ∼ −80 cm/GW. In this case, the transmission ratio would vary a lot across the intensity range explored here: TR(I = 0.3 GW/cm²) ∼ +89% for the minimum THz intensity used, 0.3 GW/cm², and TR(I = 1 GW/cm²) ∼ +118% for the maximum one, 1 GW/cm². However, as shown in Fig. 2, we experimentally detect a “flat” TR(I) ∼ 79.5% within the experimental noise. Please note that the quantification of the change of the non-linear absorption vs temperature from the theory developed in Refs. 20 and 21 is made difficult by the modified Kramers–Kronig transformation of complex meromorphic functions.^44,45

In conclusion, we measured the THz transmission of a 100 µm thick water layer at two temperatures (21 and 4 °C) as a function of the intensity of the input THz radiation (0.3, 0.7, and 1 GW/cm²). For THz fields high enough to trigger a sizable non-linearity in water, we showed that the transmission is the same at 21 and 4 °C. This implies that the non-linear absorption of liquid water at ∼1 THz is equal at these temperatures, within the experimental error (±0.5% in TR and ±1.5 cm⁻¹ in absorption).

While several models have been proposed to explain the non-linear THz signals in water,^21,22,25,26 the experimental finding reported here suggests that we might prefer models predicting a weakly temperature-dependent non-linear THz response by liquid water. In particular, it is known that both the absorption of the librational band³⁷ and the generation of solvated electrons⁴⁶ depend weakly on temperature. Thus, it is possible that the large non-linear responses detected in water at THz frequencies are related either to tunneling ionization^22,23 or to the resonant excitation of librational motions.^25,26 However, as discussed before,²⁶ tunneling ionization cannot account for the non-linear response of water at 12.3 THz because the ponderomotive energy is about 1000× smaller at 12.3 THz than at 1 THz. For these reasons, the THz fields could be resonantly reorienting hydrogen-bonded water molecules in the liquid phase.²⁵ Among the different mechanisms proposed to date,^21,22,25,26 this dynamic process of THz-driven reorientation²⁵ might be the most compatible with all of the available experimental results.^20–26 

We acknowledge financial support from the Cluster of Excellence RESOLV (Grant No. EXC 2033–390677874) funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) and by the ERC Advanced Grant No. 695437 (THz Calorimetry). F.N. acknowledges funding by the DFG with Project No. 509442914. We acknowledge support by the Open Access Publication Funds of the Ruhr-Universität Bochum. These results are part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 805202-Project Teraqua). This project received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 801459-FP-RESOMUS. We thank CNR-IOM for the use of the MENLO C-Fiber780 laser. We are grateful to P. Di Pietro, A. Perucchi, and M. Havenith for support and discussions.