## Abstract

The values of the nonlinear refractive index coefficient for various materials in the terahertz frequency range exceed the ones in both visible and NIR ranges by several orders of magnitude. This allows to create nonlinear switches, modulators, systems requiring lower control energies in the terahertz frequency range. We report the direct measurement of the nonlinear refractive index coefficient of liquid water by using the Z-scan method with broadband pulsed THz beam. Our experimental result shows that nonlinear refractive index coefficient in water is positive and can be as large as 7×10^{−10} cm^{2}/W in the THz frequency range, which exceeds the values for the visible and NIR ranges by 6 orders of magnitude. To estimate *n*_{2}, we use the theoretical model that takes into account ionic vibrational contribution to the third-order susceptibility. We show that the origins of the nonlinearity observed are the anharmonicity of molecular vibrations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Terahertz (THz) frequency range is finding more and more applications in different fields of fundamental research [1] and a variety of everyday applications such as nondestructive spectroscopy [2] and imaging [3], communications [4, 5] and ultrafast control [6] as well as biomedicine [7]. Recent advances in research have brought high intensity broadband sources of THz radiation into play [8]. These perspectives have already been shown for highly intensive THz pulses generated from organic crystals with peak intensity *I _{peak}* ∼ 1.1×10

^{13}W/cm

^{2}[9]. The exploration of matter nonlinearities in THz frequency range can open new directions for devices and systems development, components and instrumentation operation.

Despite the growing interest to this issue, the observations of nonlinearities in THz frequency range were carried out without using direct measurements of material properties in most cases. The latter ones can be nonlinearity and dispersion in the wave propagation, nonlinear optical response caused by intense ultrashort THz pulses [10], absorption bleaching by means of pump-probe technique [11,12], nonlinear free-carrier response [13,14], field-induced transparency [15], spin response [16], giant cross phase modulation and THz-induced spectral broadening of femtosecond pulses [17], or quadratic THz optical nonlinearities by measuring the quadratic THz Kerr effect [18].

The most important parameter characterizing the nonlinearity of the material response in the field of intense waves is the coefficient of its nonlinear refractive index, usually denoted as *n*_{2}. In the first report of such kind of measurements [19] silicon was tested by the use of the open aperture Z-scan technique. Originally, it was predicted theoretically [20] that the nonlinear refractive index coefficient for various materials in the THz frequency range exceeds the ones in both visible and NIR ranges by several orders of magnitude. Then this fact was proven experimentally by the Z-scan method [21, 22]. Also, estimates of the liquid nitrogen *n*_{2} were made on the basis of a change in the rotation angle of the THz pulse polarization ellipse [23]. However, in these works only indirect estimations of *n*_{2} were made.

Until recently, the study of water in the THz frequency range has been considered impossible due to its large absorption. The broadband THz wave generation from a water film [24,25] was experimentally demonstrated and opened a new field of interest. In this article, we present the direct measurement of water nonlinear refractive index coefficient for the broadband pulsed THz radiation with the conventional Z-scan method. Since the Z-scan method can be utilized for plane-parallel samples only, we use flat water jet. We demonstrate that the nonlinear refractive index coefficient exceeds its visible and NIR ranges’ values by 6 orders of magnitude [26–29].

## 2. Experiment and analytical model

Figure 1(a) illustrates an experimental setup for measuring the nonlinear refractive index (*n*_{2}) of a flat liquid jet with THz pulses. We use TERA-AX (Avesta Project) as a source of THz radiation. In this system generation of the THz radiation is based on the optical rectification of femtosecond pulses in a lithium niobate crystal [30]. The generator TERA-AX is pumped using a femtosecond laser system (duration 30 fs, pulse energy 2.2 mJ, repetition rate 1 kHz, central wavelength 800 nm). The THz pulse energy is 400 nJ, the pulse duration is 1 ps (Fig. 1(b)) and the spectrum width 0.1–2.5 THz (Fig. 1(c)). The measurement of the THz electric field was held by the conventional electro-optical detection system. The THz radiation intensity is controlled by reducing the femtosecond pump beam intensity. The pump intensity variation during the THz generation leads to a change in the divergence of the terahertz beam and also influences the terahertz central position [31]. In our experiment, we use the parabolic mirror with a focal length of 25 mm to collimate the THz radiation generated from LiNbO_{3} crystal. Next, when adjusting the experimental setup, we ensure that the THz beam of 25.4 mm in diameter obtained at the output of the TERA-AX is collimated at all femtosecond pump energies and it’s optical axis passes through the center of the PM1 parabolic mirror. Pulsed THz radiation is focused and collimated by two parabolic mirrors (PM1 and PM2) with a focal length of 12.5 mm. The spatial size of the THz radiation at the output of the generator is 25.4 mm. Caustic diameter is 1 mm (FWHM). In order to get a higher intensity in the waist, we use a short-focus parabolic mirror with a large NA. Such a geometry allows us to get the radiation peak intensity in the caustic of the THz beam 0.5×10^{8} W/cm^{2}. Flat water jet (jet) is moved along the caustic area from −4 mm to 4 mm using a motorized linear translator, the restriction on the displacement is determined by the jet width and the focusing geometry of the THz radiation (see Fig. 1(a) insert). The polarization of the THz radiation is vertical. In this experiment we used distilled water which does not contain any substances as medium. The water jet has a thickness of 0.1 mm and is oriented along the normal to the incident radiation. The jet is obtained using the nozzle which combines the compressed-tube nozzle and two razor blades [32]. This design forms a flat water surface with a laminar flow. The optical path of the THz pulse passes through the center of the jet area of a constant thickness. Due to the use of the pump, water is released under the pressure. The hydroaccumulator in the system of water supply allows to reduce the pulsations associated with the operation of the water pump significantly. The THz radiation is collimated by the parabolic mirror PM2 and focused by the lens (L) on the Golay cell (GC). For the closed aperture geometry the aperture (A) is moved in the beam (closed position). The synchronization is performed using the mechanical modulator (M) located between the lens and the Golay cell. When the jet is moved along the *z* axis through the focal region of the THz radiation, the average power of the latter is measured using open and closed aperture.

Despite the fact the experimental setup implies nonparaxial radiation, as was shown in [33] for pulses from a small number of oscillations, the differences between the paraxial and nonparaxial modes are negligible. Figure 2 shows Z-scan curves for the water jet measured with open Fig. 2(a) and closed 2(b) aperture for different values of the THz radiation energy. Each line is averaged over 50 measurements. Figure 2(a) shows the water bleaching by around 2% which is caused by the THz radiation pump energy growth by 2 orders. For *n*_{2} determination we use experimental data with the closed aperture (Fig. 2(b)).

Usually, the Z-scan technique is strictly valid only for quasi-monochromatic radiation. However, it is also widely used in the case of femtosecond pulsed radiation that possesses a broad spectrum [34]. As seen from Fig. 2(b), moving the jet along the z axis leads to a noticeable change of the measured intensity of the THz beam, which is a distinguishing feature of Z-scan curves obtained by the known method [35]. It is caused by different divergences of the radiation at various positions of the water jet in the caustic, where a nonlinear Kerr lens is induced by the THz radiation field. In view of this we use standard formulas [35–37] to evaluate n_{2} of water according to the results of our measurements shown in Fig. 2(b):

*T*= 0.013 (Fig. 2(b)) is the difference between the maximum and minimum transmission,

*S*is the linear transmission of the aperture,

*L*=

_{α}*α*

^{−1}[1-exp(−

*αL*)] is the effective interaction length,

*L*is the sample thickness,

*α*is the absorption coefficient (

*α*= 100 cm

^{−1}),

*λ*is the wavelength, and

*I*is the input radiation intensity. The linear transmission of the aperture is 2%, which allows to maximize the sensitivity of the measurement method but reduces the signal-to-noise ratio. The radiation wavelength was chosen to be

_{in}*λ*

_{0}= 0.4 mm (

*ν*

_{0}= 0.75 THz). It corresponds to the maximum of the generation spectrum of the THz radiation (see Fig. 1(b)). The result of the evaluation calculated by Eq. (1) gives the value

*n*

_{2}= 7×10

^{−10}cm

^{2}/W.

To illustrate the correctness of using Eq. (1) for the calculation of the nonlinear refractive index coefficient in the case of the broadband THz radiation, we compare the experimental data with the analytical Z-scan curve for monochromatic radiation (Fig. 3) calculated by the equation [35]:

*E*(

*z*,

*r*= 0,

*t*) =

*E*

_{0}

*sin*(2

*πν*

_{0}

*t*)

*w*

_{0}/

*w*(

*z*), $\mathrm{\Delta}{\varphi}_{0}(z,t)=\mathrm{\Delta}{\mathrm{\Phi}}_{0}(t)/\left(1+{z}^{2}/{z}_{0}^{2}\right)$, ΔΦ

_{0}(

*t*) =

*k*Δ

*n*

_{0}(

*t*)

*L*.

The following values are used in these equations: absorption coefficient *α* = 100 cm^{−1}, sample length *L* = 0.1 mm, central frequency of the radiation *ν*_{0} = 0.75 THz (*λ*_{0}= 0.4 mm), beam waist radius *w*_{0} = 0.5 mm, aperture radius *r _{a}* = 1.5 mm, radius of the THz beam

*w*= 12.5 mm, intensity of the THz beam in caustic

_{a}*I*

_{0}= 0.5×10

^{8}W/cm

^{2}, nonlinear refractive index coefficient

*n*

_{2}= 7×10

^{−10}cm

^{2}/W. This value of

*n*

_{2}was obtained in the experiment previously.

As can be seen, the experimental Z-scan curve for broadband THz radiation agrees with the analytical Z-scan curve for the monochromatic radiation well.

## 3. Theoretical estimate of the nonlinear refractive index coefficient

We estimate the nonlinear refractive index coefficient *n*_{2} of liquid water through the use of a recent theoretical treatment [20]. This treatment ascribes the THz nonlinearities in media to a vibrational response that is orders of magnitude larger than typical electronic responses. This model assumes that the nature of the nonlinearity of the water refractive index in the experiment is not caused by the thermal expansion of the substance (as well as by its density change). This expansion process is inertial. The initial cause of low-inertia nonlinearity of the refractive index measured and the subsequent inertial thermal expansion of the substance is the anharmonicity of molecular vibrations. We make use of Eq. (55) of reference [20], which applies to the situation where the THz frequency *ω*_{0} is much smaller than the fundamental vibrational frequency resulting in absorption peak *λ* = 3 *μ*m (*ω* ≈ 100 THz) [38]. For our experiment this condition is well satisfied, as *ω*_{0}/2*π* is approximately 0.75 THz and *ω*/2*π* is 15.9 THz. This equation takes the form:

*a*

_{1}is the lattice constant; for our estimations in case of liquid we use the water molecule diameter 2.8×10

^{−8}cm [39],

*m*= 1.6×10

^{−24}g is the reduced mass of the vibrational mode,

*α*= 0.2×10

_{T}^{−3}(°C)

^{−1}is the thermal expansion coefficient [40]. The parameter

*q*is the effective charge of the chemical bond; for simplicity, we take this quantity to be the electron charge.

*N*is the number density of vibrational units. We calculate this value as the ratio between specific gravity of water equal to 1 and the total mass of H

_{2}O molecule equal to the weight of molecule (1×2 + 16) times amu (1.67×10

^{−24}). It results in

*N*= 3.3×10

^{22}in 1 cm

^{3}. We take the refractive index as

*n*

_{0,ν}= 2.3 which is the averaged refractive index in 0.3–1.0 THz region [41]. Using these values, we find that the predicted value of

*n*

_{2}for water in the low-frequency limit is

*n*

_{2}= 5×10

^{−10}cm

^{2}/W.

## 4. Summary

In conclusion we have experimentally demonstrated the possibility of direct measurement of the nonlinear refractive index coefficient *n*_{2} of water in the THz frequency range. Z-scan curves obtained for broadband THz radiation experimentally are in good agreement with the analytical model of the method for monochromatic radiation. The value of the nonlinear refractive index coefficient of water calculated from the experiment is *n*_{2} = 7×10^{−10} cm^{2}/W, which is 6 orders of magnitude higher than for the visible and IR ranges [26–29] where *n*_{2} has the magnitude of 10^{−16} cm^{2}/W. These results demonstrate the high cubic nonlinearity of water in the THz frequency range and confirm a recent theoretical prediction [20] that the ionic vibrational contribution to the third-order susceptibility renders THz nonlinearities much larger than typical optical-frequency nonlinearities. Therefore, in terms of applications, our demonstration opens up new perspectives for studying various materials in the THz frequency range. Nonlinear optics, in its turn, finds applications in the creation of light modulators, transistors, switchers and others in this spectral range.

## Funding

Russian Foundation of Basic Research (RFBR) (19-02-00154); Ministry of Science and Higher Education of the Russian Federation (project 08-08); Government of the Russian Federation (project 3.9041.2017/7.8); U.S. Army Research Office (W911NF-17-1-0428).

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