1. Introduction

When measuring the absorption of electromagnetic radiation by a sample in transmission geometry, the optimal material thickness is often on the order of one penetration depth, i.e., the thickness of the sample should be comparable to the inverse of its absorption coefficient over the probed frequency range [1]. However, when measuring refraction properties, the amount of temporal delay accumulated by the transmitted field is proportional to the thickness of the sample: the thicker the sample, the larger the acquired phase. These different optimal thicknesses constrain the simultaneous detection of refractive and absorptive properties. The situation is particularly difficult when the material under investigation is “black” or highly absorptive, which limits its thickness and reduces the precision of measurement of the real part of the index of refraction [1].

This work focuses on one important substance in particular: liquid water and some of its diluted salt solutions. Water absorbs strongly at far-infrared [2–9] and terahertz [10–21] (THz) frequencies via the inter-molecular modes of hydrogen-bonded (HB) water molecules [22–32]. As an example, in Fig. 1 of Ref [33], the absorption coefficient of water roughly increases with THz frequency and reaches several thousand inverse centimeters, corresponding to penetration depths of a few microns. At a temperature of 20 °C and pressure of 1 atmosphere [34], liquid water displays three main spectroscopic features with center frequencies at approximately 0.02 THz (20 GHz), 6 THz, and 20 THz, respectively. At low frequencies, the “microwave” band at 20 GHz can be empirically fit to a simple Debye function. However, this fit fails at frequencies higher than ∼0.2 THz [35] and the microscopic origin of this feature is debated, i.e., it could originate either from molecular reorientations of hundreds of HB waters [36–39], percolation of bifurcated HB [40,41] or diffusion of charge defects [42], stochastic frequency modulation [43], or jumps between local energy minima involving few molecules [44]. The broad peak at ca. 6 THz is associated with restricted translational or intermolecular stretching modes wherein the water molecules move against each other along the directions of their HB [22,23,45,46]. The feature at ∼20 THz, which is asymmetrical because of a stronger absorption at frequencies between about 10 and 20 THz, is associated with restricted molecular rotations or librations, wherein the HB water molecules are involved in rocking or wagging motions that mainly displace the positions of the hydrogen atoms [47].

 

Fig. 1. Transmission by a 0.5 mm thick pure water layer enclosed in a static liquid cell with diamond windows. a) When the terahertz spectrometer is purged with a constant flow of nitrogen gas, the relative humidity is close to 10% rH, and the transmitted THz field is shown with the black curve. When there is no purge, the humidity is 50% rH, and the THz field is shown in blue. The difference between unpurged and purged (blue minus black curve) is shown in the inset with the red dots. b) The intensity spectrum of the THz field transmitted with ∼10% and 50% rH is shown with black circles and blue squares, respectively. The inset shows the phase of the fields between 0.25 THz and 0.67 THz. The gray curve refers to the right axis and quantifies the absorption coefficient of water vapor [114–117].

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The dissolution of salts affects the dielectric function of liquid water at THz frequencies and makes the microscopic understanding of the underlying molecular dynamics more difficult. A list of debated topics include the strength and the extent of the interaction of different salts with the surrounding, hydration water molecules [48–51]; possible super-additive cooperativity effects whereby some water molecules are affected by concerted ionic interactions [52–54]; the conductivity of charged ions in solution that limits electro-chemical processes [55]; and the amounts of the different kinds of ion pairs, i.e., how the thermodynamic activity depends on the specific salt concentration [56–58]. To help address some of these issues, more precise spectroscopic tools are needed.

Typical terahertz time-domain spectroscopy (THz-TDS) covers the spectral range from 0.1 to 3 THz [59–63]. THz-TDS on liquid water is routinely performed in the transmission geometry on samples with a thickness of about 100 microns [64–81], corresponding to ∼2x penetration depths at 1 THz. These relatively thin samples make the precise measurement of the index of refraction of aqueous solutions a difficult task. To overcome this limitation and allow the damped radiation transmitted by an opaque body to be measured, the THz detection could be enhanced by either increasing the repetition rate [82–88], and/or the total intensity of the input THz radiation. To my knowledge, only two previous works have probed the transmission of thick water layers with THz-TDS. Nazarov et al.[74] studied 0.5 mm thick and highly concentrated aqueous solutions containing both bovine serum albumin (BSA) proteins (≥ 30 mg/mL) and glucose (≥ 1.5 M). We have demonstrated before [89] that intense THz-TDS can be used as a high-precision, fast, and contactless probe sensitive to very diluted, colloidal aqueous solutions of gold nanoparticles (<0.5% weight). Here, this idea is extended to the study of the dielectric function of dilute aqueous solutions and demonstrated for a typical salt, sodium chloride. Most previous reports of successful THz-TDS measurements on NaCl solutions are at concentrations of about one Molar (1 M) [55,90–98]. For example, one of the highest precision measurements to date [16] did not report a detectable difference between pure water and 0.1 M NaCl at THz frequencies. Here, I show that it is possible to quantify the tiny change in the index of refraction of an aqueous solution upon dissolving ten times less solute, 0.01 M (10 mM) NaCl, at frequencies across the 300 GHz band and spanning the frequency range between 0.25 THz and 0.67 THz.

2. Methods

2.1 Source and samples

The input laser source is an amplified titanium sapphire (Ti:Sa) laser that emits ∼90 fs long pulses centered at 790 nm with a total output power of 7 W at 1 kHz repetition rate (Coherent). The intense and almost single-cycle THz pulses are generated by tilted-front optical rectification [99–102] in a congruent lithium niobate (cLN) prism (MolTech). The generated THz power measured by a detector (Gentec THZ12D-3S-VP-D0) amounted to ∼0.5 mW at the sample position, corresponding to an energy per pulse $\textrm{W} = 0.5\textrm{mW}/1\textrm{kHz} = 0.5\,\mathrm{\mu}\textrm{J}/\textrm{pulse}$. The sample is placed away from the focus, where the radius of the THz beam estimated with an iris amounts to $\textrm{b}\sim 10\textrm{mm}$. At this position, assuming for simplicity a square pulse, the order of magnitude of the THz peak field amounts to ${\approx} 10\textrm{kV}/\textrm{cm}$. This field value was estimated from the equation [103] $\sqrt {2\textrm{Z}\frac{\textrm{W}}{{\textrm{A} \cdot \mathrm{\tau }}}} $, where $\textrm{Z} = 376.7\,\mathrm{\Omega }$ is the vacuum impedance, $\textrm{W} = 0.5\,\mathrm{\mu}\textrm{J}/\textrm{pulse}$ the energy per pulse, $\textrm{A} = \mathrm{\pi } \cdot {\textrm{b}^2}$ the beam area, and $\mathrm{\tau } \approx 1\textrm{ps}$ the pulse duration.

The sodium chloride reagent with ≥99% purity was bought from Sigma-Aldricht (S9888) and used as is. The salt solutions were prepared by dissolving appropriate amounts of weighted NaCl powder in ultra-pure milli-Q water. The samples were enclosed in a static liquid cell with 0.5 mm thick diamond windows (Diamond Materials) and a 0.5 mm thick Teflon spacer that determines the amount of liquid under investigation. The sample holder is cleaned and assembled at the beginning and not demounted between measurements to ensure that the same thickness of liquid is measured. Before measuring a sample, the liquid cell was flushed with pure water and dried by blowing nitrogen gas inside the mounted sample holder, three times. To improve the repeatability of the salt solutions measurements, the samples were measured with increasing concentrations, i.e., 10 mM NaCl was measured first, then 100 mM NaCl, and finally 1 M NaCl. The static holder is magnetically attached to a copper plate whose temperature is stabilized to $20 \pm 0.05^\circ \textrm{C}$ by a recirculating chiller. The THz fields transmitted by the thick aqueous layers are detected by electro-optical sampling in a 1 mm thick zinc telluride crystal (ZnTe). The voltage produced by balanced photodiodes (Thorlabs) is averaged with a boxcar (Zurich Instruments) before acquisition as a function of the temporal delay set by an electronically controlled mechanical stage (PI). One scan of a THz field takes about 20 seconds. The setup is enclosed in a box and purged with nitrogen to control the relative humidity (rH). The THz fields are detected over a delay range of 11.9 ps (-5.2 ps to +6.7 ps, where 0 marks the arrival time of the peak field transmitted by pure water) in steps of 0.1 ps.

When a train of electromagnetic pulses interacts with a material, multiple nonlinear processes can appear depending on the magnitude of the driving fields and the repetition rate. From the fastest to the slowest processes, the temporal hierarchy of the nonlinearities encompasses optical, acoustic, and thermal effects [104]. Typically, nonlinear optical processes require high input intensity and last up to a few picoseconds. Here THz fields up to ${\sim} 10\textrm{kV}/\textrm{cm}$ were used, which are much smaller than the ones needed to drive liquid water into the non-linear response regime [104–111]. Thus, such high-order optical processes are ignored in this work. Acoustic phenomena are determined by the speed of sound and the sample dimensions [112]. In liquid water these acoustic effects happen on the microsecond timescale [112] and can be ignored at the inter-pulse delay of the laser used here ($1/1\textrm{kHz} = 1\textrm{ms}$), i.e., the next laser pulse interacts with a material characterized by “acoustic and density equilibrium.” Thermal effects are determined by the energy density of the absorbed radiation and by the time it takes for a sample to change temperature. The timescale of the temperature change is quantified by the thermalization timescale, ${t_{th}}\sim {l^2}/D$, with l sample length and D thermal diffusivity. For a static liquid water sample excited by radiation at ${\sim} 1\textrm{THz}$, the sample length can be taken equal to the penetration depth, $l\sim 50\,\mathrm{\mu}\textrm{m}$, while $D \approx 1.4 \cdot {10^{ - 7}}{\textrm{m}^2}/\textrm{s}$ is approximately constant at temperatures comprised between 5 °C and 50 °C [113]. The order of magnitude of the thermalization time of liquid water enclosed in a sample holder is ${t_{th}}\sim 18\textrm{ms}$. Thermal effects are expected to be present for our static water samples irradiated by a THz pulse train at 1 kHz, i.e., I expect that about 18 subsequent pulses, each one separated from the next by 1 ms, are contributing to the average heating of the sample determined by ${t_{th}}$. The amount of temperature change induced by one single pulse depends on the energy density of the absorbed radiation and can be estimated to $\Delta \textrm{T} = \frac{\textrm{W}}{{\textrm{A} \cdot {l} \cdot {\textrm{C}_\textrm{p}}}} \approx 8 \cdot {10^{ - 6}}\mathrm{^\circ{C}}$, where $\textrm{W}$, $\textrm{A}$, and ${l}$ were defined above and ${\textrm{C}_\textrm{p}} \approx 4.2\frac{\textrm{J}}{{\textrm{c}{\textrm{m}^3} \cdot \mathrm{^\circ{C}}}}$ is the specific heat or volumetric heat capacity of pure water. The temperature increase of the sample induced by a sequence of 18 subsequent pulses amounts to ca. 0.0001 °C, which is more than two orders of magnitude smaller than the temperature stability of the samples studied here (0.05 °C). For these reasons, thermal effects are irrelevant to the data presented here. This simple numerical estimate of the temperature change induced in liquid water by a train of THz pulses was validated before by experimental results [104].

The THz fields transmitted by a 0.5 mm thick layer of pure liquid water are shown in Fig. 1 for two different humidity levels inside the spectrometer. The black curve in Fig. 1(a) is the THz field transmitted at lower values, $\textrm{rH} = 10 \pm 1\%$, while the blue trace is obtained for higher humidity, $\textrm{rH} = 50 \pm 1\%$. When there is more water vapor in the spectrometer, the THz field is delayed in time, and its amplitude is slightly reduced, i.e., the peak of the THz blue trace in Fig. 1(a) arrives at a later time and has a somewhat smaller amplitude with respect to the THz black trace. The long-lived oscillations present for higher humidity are due to the sharp absorption lines of the water vapor molecules. The blue and black traces are each the average of 15 subsequent scans wherein the full THz field was recorded, amounting to a total acquisition time of about 10 minutes. A direct way to visualize the phase gained by the THz field in the presence of water vapor is to plot the difference between the THz field transmitted at $\textrm{rH}\sim 50\%$ and the one at $\textrm{rH}\sim 10\%$, which is shown with the red trace in the inset of Fig. 1(a). A temporal shift of the THz pulse shows up as a negative signal before time zero and a positive bump after time zero; see the red dots in Fig. 1(a). Figure 1(b) displays the magnitude squared or intensity spectrum (main panel) and the phase (inset panel) of the oscillating fields shown in Fig. 1(a), which were obtained by Fourier transformation (FT). Blue refers to $\textrm{rH}\sim 50\%$ and black to $\textrm{rH}\sim 10\%$. For higher humidity, the spectrum is shaped according to the sharp absorption lines of water vapor [114–117], which is shown as the gray curve that refers to the right axis in Fig. 1(b). As expected, the temporal delay acquired by the THz pulse for higher humidity shows up in the larger phase value shown in the inset of Fig. 1(b). Please note that the phase is shown only over the best frequency range of operation of the spectrometer used here, 0.25 THz to 0.67 THz.

Typical THz-TDS experiments are routinely performed with humidity levels below about 10%. However, unusual aspects can be noted when probing the transmission of highly absorbing and thick materials. Water vapor plays a lesser role here than in standard THz-TDS experiments. The spectrum transmitted by the aqueous layer includes prominent frequency components below ∼0.5 THz, for which the absorption by water vapor is negligible (gray curve in Fig. 1(b)). However, the vapor content affects the phase of the transmitted field, see the inset of Fig. 1(b). Thus, a key parameter to perform reliable measurement is that the amount of water vapor remains stable during the acquisition time. This opens the door to the possibility of performing similar experiments without the necessity of nitrogen purging, but either by controlling or stabilizing the humidity with a dehumidifier, or by recording the free varying humidity at the sample position and sorting the data in post processing.

The dynamic range (DR) of the spectrum transmitted by a liquid water sample is roughly equal to 70 dB, see the left axis on Fig. 1(b). This is not the DR that is often reported in the literature without a sample. Such a quantity cannot be reported here because, when the water layer is removed, the THz intensity reaching the electro-optical sensing detector is too high, over-rotating and saturating the response [118].

2.2 Data analysis

Figure 2 displays a generic three-layer system made of homogenous and isotropic materials. By ignoring the first terms originating from multiple reflections (E_out,R2, E_out,R3, and E_out,R4 in Fig. 2), the transmission of a single pulsed field is

 

Fig. 2. Cartoon of pulsed electromagnetic radiation transmitted by a multi-layer system (2,3,4) embedded in two semi-infinite media (1 and 5). All materials are isotropic, homogeneous, and optically thick. Typically, mediums 1 and 5 are nitrogen-purged air, 2 and 4 are static cell windows, and 3 is the sample. The inbound field (E_in) is coming from medium 1, is s-polarized, and impinges at the interface between medium 1 and 2 with an angle θ₁ with respect to the direction normal to the sample surface. The electromagnetic pulse that reaches the interface between medium 2 and 3 depends on the Fresnel transmission coefficient between medium 1 and 2 and on the phase acquired by traversing medium 2, φ₂. By considering all the interfaces and media traversed, it is possible to write an equation for the outbound field emitted by the multi-layer system, E_out. An example of secondary reflections originating from layer 2, layer 3, and layer 4 are sketched with the dashed lines and marked as E_out,R2, E_out,R3, and E_out,R4. All the angles are exaggerated for display purposes.

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For an aqueous solution enclosed in a static sample holder cell that is embedded in a controlled environment, medium 2 and medium 4 (Fig. 2) are two windows of the holder that often have similar thicknesses (${\textrm{d}_2} \approx {\textrm{d}_4}$) and composition (polyethylene, Teflon, quartz, diamond, etc.), and mediums 1 and 5 are the same (air, dry air, nitrogen gas, vacuum, etc.). Assuming for simplicity that an oscillating electric field of arbitrary polarization is propagating normally to the interface of the sample, i.e., that all the angles $\mathrm{\theta }$ are equal to zero, and Eq. (1) simplifies to

THz-TDS measurements require the detection of two THz traces: one for a reference sample and one for the sample under investigation. The reference sample used here is pure liquid water at $20 \pm 0.05\mathrm{^\circ{C}}$. This work aims to detect the tiny differences in the THz transmission of water upon subtle changes such as the dissolution of chemicals at low concentrations. In this case, it is possible to write two equations like Eq. (2), one for the THz transmitted by the aqueous solution under investigation ($\textrm{s} = \textrm{sol}$) and one for the reference, pure water ($\textrm{s} = \textrm{wat}$). The ratio between the complex and frequency-dependent THz field transmitted by the sample and the THz field transmitted by pure water is:

A simple analytical solution of Eq. (3) can be found by assuming that the pre-factor that multiplies the exponential functions, $\frac{{{{\mathbf n}_{\textrm{sol}}}{{({{{\mathbf n}_{\textrm{wat}}} + {{\mathbf n}_{\textrm{win}}}} )}^2}}}{{{{\mathbf n}_{\textrm{wat}}}{{({{{\mathbf n}_{\textrm{sol}}} + {{\mathbf n}_{\textrm{win}}}} )}^2}}}$, is real, i.e., that $\frac{{{{\mathbf n}_{\textrm{sol}}}{{({{{\mathbf n}_{\textrm{wat}}} + {{\mathbf n}_{\textrm{win}}}} )}^2}}}{{{{\mathbf n}_{\textrm{wat}}}{{({{{\mathbf n}_{\textrm{sol}}} + {{\mathbf n}_{\textrm{win}}}} )}^2}}} \approx \frac{{{\textrm{n}_{\textrm{sol}}}{{({{\textrm{n}_{\textrm{wat}}} + {\textrm{n}_{\textrm{win}}}} )}^2}}}{{{\textrm{n}_{\textrm{wat}}}{{({{\textrm{n}_{\textrm{sol}}} + {\textrm{n}_{\textrm{win}}}} )}^2}}}$. Separating the argument ($\textrm{arg}\left( {\frac{{{\textrm{E}_{\textrm{sol}}}(\omega )}}{{{\textrm{E}_{\textrm{wat}}}(\omega )}}} \right)$) and the modulus ($\left|{\frac{{{\textrm{E}_{\textrm{sol}}}(\omega )}}{{{\textrm{E}_{\textrm{wat}}}(\omega )}}} \right|$) of the complex and frequency-dependent ratio of the FT of the THz fields transmitted by sample and reference, $\frac{{{\textrm{E}_{\textrm{sol}}}(\omega )}}{{{\textrm{E}_{\textrm{wat}}}(\omega )}}$, the following equations determining the variation of the index of refraction and extinction coefficient of the sample (aqueous solution) with respect to the reference (pure water) are obtained:

The following relation is obtained by expressing Eq. (5) in terms of the absorption coefficient ($\mathrm{\alpha } = 2\mathrm{\omega k}/\textrm{c}$) and for $\frac{{{\textrm{n}_{\textrm{wat}}}{{({{\textrm{n}_{\textrm{sol}}} + {\textrm{n}_{\textrm{win}}}} )}^2}}}{{{\textrm{n}_{\textrm{sol}}}{{({{\textrm{n}_{\textrm{wat}}} + {\textrm{n}_{\textrm{win}}}} )}^2}}} \approx 1$:

In summary, performing THz-TDS measurements of a thick aqueous layer in a static liquid cell consists in recording the two THz fields transmitted by the sample (diluted solution) and by the reference (pure water), performing the FT on them, taking the ratio between these two FT, calculating the frequency-dependent phase ($\textrm{arg}\left( {\frac{{{\textrm{E}_{\textrm{sol}}}(\omega )}}{{{\textrm{E}_{\textrm{wat}}}(\omega )}}} \right)$) and magnitude ($\left|{\frac{{{\textrm{E}_{\textrm{sol}}}({\omega } )}}{{{\textrm{E}_{\textrm{wat}}}({\omega } )}}} \right|$) of this ratio, and applying Eq. (4) and Eq. (6) to calculate the variation of the refractive index and the absorption coefficient, respectively. Please note that Eq. (6) is the Beer-Lambert law, and it was obtained by assuming that the transmission is not influenced by the index of refraction of sample, reference, and windows used.

To derive Eq. (4) and Eq. (6), I assumed the propagation of an electromagnetic pulse through optically thick materials and that the pre-factor before the exponentials in Eq. (3) is real with a value close to one. Estimates of the limits of validity of these approximations follow.

When a pulsed field propagates through, the system in Fig. 2 is “optically thick” if multiple reflections can be neglected. This condition is met when the thickness of each slab is large enough that the reflections originating from their interfaces are delayed beyond the acquisition window or if the amplitude of the reflections is too small to be detected, i.e., if the Fresnel coefficient is close to zero and/or the sample absorption is high enough. When dealing with phase stable and pulsed THz radiation, as is the case for THz-TDS, the arrival time of the first single-cycle pulse that is transmitted by a multi-layer compound (E_out in Fig. 2) can be separated in time from the subsequent pulses originating from multiple reflections (E_out,R2, E_out,R3, and E_out,R4 in Fig. 2). In the simplest case, when electromagnetic radiation has propagated through a medium with index of refraction $\textrm{n}$ and thickness $\textrm{d}$, it acquires a temporal delay of $t = \textrm{d} \cdot \textrm{n}/\textrm{c}$, where $\textrm{c}$ is the speed of light. Thus, the additional delay accumulated by the first reflection originating in the second layer (E_out,R2) with respect to the main transmission E_out amounts to $2 \cdot {\textrm{d}_2} \cdot {\textrm{n}_2}/\textrm{c}\sim 7.9\textrm{ps}$, with ${\textrm{d}_2} = 0.5\textrm{mm}$ and ${\textrm{n}_2} = 2.37$ for the front window made of polycrystalline diamond [119–123]. This reflection is pushed beyond the observed time range because the electro-optical detection time extends only to +6.7 ps. Even if this reflected signal would be detected, it would show up in the FT at frequencies centered at about $1/7.9\textrm{ps}\sim 0.13\textrm{THz}$, which is below the detection window (0.25-0.67 THz). As the absorption and extinction coefficients of diamond are practically zero in the THz range [119–123] (${{n}_{\textrm{win}}} = 2.37$), the amplitude of the first reflection originating from the front window (E_out,R2) is determined by the Fresnel reflection coefficients between layers 2 (front diamond window) and 3 (water), ${\textrm{r}_{23}}$, and between 2 (diamond) and 1 (nitrogen gas, rH∼10%), ${\textrm{r}_{21}}$. To estimate the order of magnitude, I assume for simplicity s-polarized radiation, normal incidence, a nitrogen gas environment (${{\mathbf n}_\textrm{e}} \approx 1$ at THz frequencies) and take ${{\mathbf n}_{\textrm{wat}}} \approx 2.39$ between 0.2 THz and 0.7 THz. With these approximations, the magnitude of the first reflection is $|{{\textrm{E}_{\textrm{out},\textrm{R}2}}} |= |{{\textrm{r}_{23}}{\textrm{r}_{21}}} |= \left|{\frac{{{ n_\textrm{win}} - {\mathbf{n}_{\textrm{wat}}}}}{{{ n_\textrm{win}} + {\mathbf{n}_{\textrm{wat}}}}}\frac{{{n_\textrm{win}} - 1}}{{{n_\textrm{win}} + 1}}} \right|\approx 0.0017 \approx 1/585$, i.e., about $585$ times smaller than the first outbound pulse, E_out. The amplitude of this reflection is so much smaller than the main transmitted pulse that no trace of it can be seen by inspecting, e.g., the THz fields in Fig. 1(a). In summary, the first reflection from the front window can be ignored because it is pushed to longer delay times and lower frequencies, away from the detected time and frequency ranges, and because it is very weak in amplitude. The same estimate holds for the reflection originating from the back diamond window, ${\textrm{E}_{\textrm{out},\textrm{R}4}}$. The secondary reflection originating from the sample, ${\textrm{E}_{\textrm{out},\textrm{R}3}}$, is further, strongly diminished by traversing the highly absorbing aqueous layer two more times. Please note that, if the source emits continuous radiation (CW) as in standard Fourier-transform infrared (FTIR) spectroscopy, the multiple reflections cannot be separated in time and Fabry-Pérot, fringe, or etalon effects appear in the transmitted intensity, which can be used to perform high precision measurements [124].

To obtain Eq. (4) and Eq. (6), it was assumed that $\frac{{{{\mathbf n}_{\textrm{sol}}}{{({{{\mathbf n}_{\textrm{wat}}} + {{\mathbf n}_\textrm{win}}} )}^2}}}{{{{\mathbf n}_{\textrm{wat}}}{{({{{\mathbf n}_{\textrm{sol}}} + {{\mathbf n}_\textrm{win}}} )}^2}}} \approx 1$. Here, I perform a simple numerical estimate of the systematic error stemming from this approximation for three cases, i.e., when the dielectric function of the diluted aqueous solution is 5% larger than the one of pure water (${{\mathbf n}_{\textrm{sol}}} = 1.05 \cdot {{\mathbf n}_{\textrm{wat}}}$), when it increases by 0.5% (${{\mathbf n}_{\textrm{sol}}} = 1.005 \cdot {{\mathbf n}_{\textrm{wat}}}$), and when it is only 0.05% larger (${{\mathbf n}_{\textrm{sol}}} = 1.0005 \cdot {{\mathbf n}_{\textrm{wat}}}$). In other words, I assume ${{\mathbf n}_{\textrm{sol}}} = 1.05 \cdot {{n}_{\textrm{wat}}}$ and calculate the phase and modulus of $\frac{{{{\mathbf n}_{\textrm{sol}}}{{({{{\mathbf n}_{\textrm{wat}}} + {{\mathbf n}_\textrm{win}}} )}^2}}}{{{{\mathbf n}_{\textrm{wat}}}{{({{{\mathbf n}_{\textrm{sol}}} + {{\mathbf n}_\textrm{win}}} )}^2}}}$. The value of $\frac{\textrm{c}}{{\mathrm{\omega d}}}\textrm{arg}\left( {\frac{{{{\mathbf n}_{\textrm{sol}}}{{({{{\mathbf n}_{\textrm{wat}}} + {{\mathbf n}_\textrm{win}}} )}^2}}}{{{{\mathbf n}_{\textrm{wat}}}{{({{{\mathbf n}_{\textrm{sol}}} + {{\mathbf n}_\textrm{win}}} )}^2}}}} \right)$ quantifies the systematic error ($\mathrm{\delta }{n_{5\%}}$) in the estimate of the refraction $\Delta n(\omega )$ obtained via Eq. (4), and $- \frac{2}{\textrm{d}}\ln \left( {\left|{\frac{{{{\mathbf n}_{\textrm{sol}}}{{({{{\mathbf n}_{\textrm{wat}}} + {{\mathbf n}_\textrm{win}}} )}^2}}}{{{\mathbf{n}_{\textrm{wat}}}{{({{{\mathbf n}_{\textrm{sol}}} + {{\mathbf n}_\textrm{win}}} )}^2}}}} \right|} \right)$ quantifies the systematic error ($\mathrm{\delta }{\alpha _{5\%}}$) to the absorption coefficient $\Delta \alpha (\omega )$ calculated from Eq. (6). Table 1 displays the literature values of the index of refraction and absorption coefficient of pure liquid water at equilibrium and at the temperature of 20 °C between 0.1 and 1 THz, as obtained by averaging different previous reports in the literature [2––20], and the estimated systematic errors when the dielectric function of the water mixture increases by 5%, 0.5%, and 0.05% with respect to neat water.

Table 1. The index of refraction (${{\boldsymbol n}_{{wat}}}({\boldsymbol \omega } )$) and the absorption coefficient (${{\boldsymbol \alpha }_{{wat}}}({\boldsymbol \omega } )$) of liquid water [2––20] between 0.1 THz and 1 THz. The table lists the systematic errors of the estimated changes of absorption and refraction obtained via Eq. (4) and (Eq. (6)) when measuring a dilute solution with a dielectric function that is identical to the one of pure water within 5%, 0.5%, or 0.05%. See text for details

View Table

3. Results

The THz-TDS experimental results obtained for the water solutions containing 1 M, 0.1 M, and 0.01 M NaCl are shown in Fig. 3. The black THz field in Fig. 3(a) is the reference THz field transmitted by a 0.5 mm thick layer of pure water sandwiched between diamond windows and stabilized to $20 \pm 0.05\mathrm{^\circ{C}}$. The orange trace in Fig. 3(a) is the THz field transmitted by a 1000 mM NaCl solution. As expected, the THz field is delayed in time and reduced in amplitude. To highlight these effects, the difference between the THz field transmitted by the salt solution and the one transmitted by neat water is shown with the red curve in the inset of Fig. 3(a). This difference curve is negative before time zero and positive afterwards, which is the hallmark of the phase shift of the THz field, i.e., the transmitted THz field acquires a larger phase in the salt solution than in pure water. The same descriptions are valid for Fig. 3(b), where the THz field transmitted by a 100 mM NaCl solution is green, and Fig. 3(c), where the THz field transmitted by a 10 mM NaCl solution is purple. While it is difficult to see the difference between the green and black curves in Fig. 3(b), and between the purple and black curves in Fig. 3(c), a clear signal can be found in the difference shown in red in the corresponding inset. As expected, the overall shape of the difference signal is similar for different concentrations, but it becomes weaker for smaller salt concentrations.

 

Fig. 3. Terahertz radiation transmitted by aqueous solutions of sodium chloride. a) The black curve is the THz field transmitted by a 0.5 mm thick layer of pure water enclosed in a static cell with polycrystalline diamond windows. The orange curve is the THz transmission of a 1 M NaCl aqueous solution. Both the black and orange fields result from an average of 30 THz traces recorded subsequently. The red dots in the inset correspond to the difference between the THz field transmitted by water and the salt, i.e., the difference between the orange and black curves in the main panel. b) Same as panel a), but the green curve is the transmission by 100 mM NaCl dissolved in water. In this case, each THz trace was averaged over two independent sample preparations and 60 THz traces. c) Same as a) and b), for a diluted solution of 10 mM NaCl. Each THz trace was averaged over seven independent sample preparations and 210 THz traces. d) Difference between the absorption coefficient of the salt solution (1 M, orange; 0.1 M green; 0.01 M purple) and the one of neat water estimated from Eq. (6). e) Change of the refraction upon dissolution of salt at different concentrations, from Eq. (4). The statistical error bars are taken equal to twice the value of the standard error of the mean and are indicated in all panels with the same color of the corresponding curve. In panels d) and e) there are additional overlayed error bars displayed in black, which are the systematic errors summarized in Table 1.

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A single THz trace is recorded by step scanning the mechanical delay of the electro-optical sampling beam. For the 1 M NaCl salt solution (Fig. 3(a)), the water reference and the solution sample were recorded 30 times each, resulting in a total acquisition time of about 20 minutes. Two independent samples were measured for the 100 mM NaCl solution (Fig. 3(b)), and each THz trace averaged over 60 scans for a total acquisition time of ca. 40 minutes. For the 10 mM NaCl sample (Fig. 3(c)), I measured seven sample preparations for a total of 210 THz traces, resulting in a measurement time of less than two and a half hours. Starting from each data set, the statistical error bars are estimated to 2x the value of the standard error of the mean (s.e.m.) and are shown in all the panels of Fig. 3 with the same color as the corresponding curve, e.g., see the thick red vertical bars in the inset of Fig. 3(c) or the purple ones in Fig. 3(d) and Fig. 3(e). Some error bars are too small to be visible. For example, in the inset of Fig. 3(c), the statistical error bars become clearly visible at the smallest concentration of salt dissolved in water, 10 mM NaCl. Smaller error bars can be seen for 100 mM NaCl (inset of Fig. 3(b)), while they are hidden by the size of the red dots in the inset of Fig. 3(a) (1 M NaCl). Please note that the s.e.m. scales with the square root of the number of measurements, and thus the error bars are about $\sqrt 7 \sim 2.6$x times smaller for the 10 mM salt solution with respect to the 1 M NaCl sample.

From the THz fields shown in Fig. 3 and Eq. (6), it is possible to estimate the change in the THz absorption coefficient induced by dissolving sodium chloride in water. The results are displayed in Fig. 3(d). Similarly, from Eq. (4), it is possible to obtain the change to the real part of the index of refraction upon salt solvation, which is displayed in Fig. 3(e). As expected, the magnitude of the change in the dielectric function is roughly proportional to the concentration of sodium chloride. Figure 3(d) shows that the order of magnitude of the increase of the absorption coefficient is ca. 10 cm^-1 for 1 M NaCl, 1 cm^-1 for 0.1 M NaCl, and 0.1 cm^-1 for 0.01 M NaCl. The index of refraction increases by about 0.1 for 1 M NaCl, 0.01 for 0.1 M NaCl, and 0.001 for 0.01 M NaCl, see Fig. 3(e). To my knowledge, such precision in the estimate of the real part of the index of refraction of a diluted aqueous solution is unprecedented at frequencies between 0.25 THz and 0.67 THz. This is the first experimental measurement of a change in the index of refraction of water of ∼10⁻³, which is as low as $\Delta n = ({0.7 \pm 0.2} )\cdot {10^{ - 3}}$ at the probe frequency of 0.58 THz for the 10 mM NaCl solution (see the purple trace in Fig. 3(e)).

In addition to the statistical error, there is a systematic error stemming from the approximations used to obtain Eq. (4) and Eq. (6) from the geometrical model of Fig. 2. The dielectric function of the salt solutions is roughly 5%, 0.5%, and 0.05% larger than that of pure water for 1 M, 0.1 M, and 0.01 M NaCl, respectively. Thus, the systematic errors listed in Table 1 are relevant and can be displayed as the thin vertical black bars in Fig. 3(d) and Fig. 3(e). By inspecting Fig. 3(d) and Fig. 3(e), it is evident that the systematic errors are much larger than the statistical ones for the 1 M solution, comparable for 0.1 M sodium chloride, and are much smaller at the lowest concentration of salt studied here, 10 mM NaCl. As a rule of thumb, considering the noise level of this THz spectrometer, it is possible to state that the systematic errors can be neglected when the dielectric function of water changes by less than about 0.1% upon dissolution of a solute, and Eq. (4) and Eq. (6) are valid.

4. Conclusions

In summary, this article reported the changes of the dielectric function of liquid water at low terahertz frequencies (0.25-0.67 THz) upon dissolution of sodium chloride (1 M, 0.1 M, 0.01 M NaCl), detected via intense terahertz time-domain spectroscopy. A generic geometrical model with a four-layered system was derived and adapted for a static sample holder; simple analytical equations were obtained, and systematic and statistical errors were discussed. With the aid of a bright terahertz source, which is often used to perform nonlinear experiments [125–129], this article demonstrated the possibility of detecting changes as low as ∼10⁻³ in the linear index of refraction of an opaque medium, liquid water, subsequent to subtle modifications such as the dissolution of modest amounts of salt (10 mM NaCl). This novel spectroscopic approach could be directly used to investigate systematically the intricate dynamics of ions in water [48–58], or of other colloids that can be dispersed in a liquid only at low concentrations, like nanoparticles [89].

This technique is not restricted to liquids and could be applied to address different materials. The details found in the Methods section can be extended to other optically thick and opaque samples, and to different experimental configurations like beam geometry or THz polarization. For example, only the reflection of biological tissues is often studied with THz-TDS [130,131]. This includes composite, layered tissues like the cornea [132,133] or the skin [134,135]. With the approach introduced here, the THz field transmitted by tissues as thick as about one millimeter could be detected to high precision. This transmission depends on the dielectric response of all the sample components, not only of the outermost layer that typically dominates the sample reflectivity. In materials science research, considering the absorption and refraction of electromagnetic radiation by metals [136,137], I estimate that it could be possible to measure the THz transmission of ∼100 nanometers thick bulk gold, with a precision comparable to the one reported here for the salt solutions. In turn, probing the precise conductivity of thin metal slabs could be helpful in assessing the quality of each single electrode used in an electrochemical device [138], even when the morphology of the electrode makes the use of electrical contacts challenging [139–142].