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 cm2/W in the THz frequency range, which exceeds the values for the visible and NIR ranges by 6 orders of magnitude. To estimate n2, 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 Ipeak ∼ 1.1×1013 W/cm2 [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 n2. 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 n2 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 n2 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 (n2) 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 LiNbO3 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 0.5×108 W/cm2. 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.

 

Fig. 1 (a) The experimental setup for measuring the nonlinear refractive index (n2) of a liquid jet in the THz spectral range. Two parabolic mirrors (PM1 and PM2) with a focal length of 12.5 mm form the caustics area where the water jet (jet) is scanned along the z axis. The synchronization is performed using the mechanical modulator (M) located between the lens and the Golay cell (GC). The aperture (A) is moved from open to closed position to change the geometry of Z-scan from open to closed aperture. Insert - Geometrical position of the jet moved along the z axis relative to the THz radiation. The temporal waveform (b) and its spectrum (c) of the THz pulse generated by the TERA-AX system.

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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 n2 determination we use experimental data with the closed aperture (Fig. 2(b)).

 

Fig. 2 Z-scan curves for a 0.1 mm thick water jet measured with open (a) and closed (b) aperture for different THz radiation energy values of 4 nJ, 40 nJ and 400 nJ. Δ T = 0.013 is the differential of the Z-scan curve measured with the closed aperture of radius 1.5 mm.

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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 n2 of water according to the results of our measurements shown in Fig. 2(b):

n2=ΔT0.406Iin×2λ2πLα(1S)0.25
where ΔT = 0.013 (Fig. 2(b)) is the difference between the maximum and minimum transmission, S is the linear transmission of the aperture, Lα= 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 Iin 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 λ0 = 0.4 mm (ν0 = 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 n2 = 7×10−10 cm2/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]:

T(z)=+PT(ΔΦ0(t))dtS+Pi(t)dt
where Pi(t)=πw02I0(t)/2 is the instantaneous input power (within the sample), S=1exp(2ra2/wa2) is the aperture linear transmittance; the transmitted power through the aperture gives
PT(ΔΦ0(t))=c0N0π0raa|Ea(r,t)|2rdr
where
Ea(r,t)=E(z,r=0,t)exp(αL/2)×m=0+[iΔϕ0(z,t)]mm!wm0wmexp(r2wm2ikr22Rm+iQm)
and E(z, r = 0, t) = E0sin(2πν0t)w0/w(z), Δϕ0(z,t)=ΔΦ0(t)/(1+z2/z02), ΔΦ0(t) = kΔn0(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 ν0 = 0.75 THz (λ0= 0.4 mm), beam waist radius w0 = 0.5 mm, aperture radius ra = 1.5 mm, radius of the THz beam wa = 12.5 mm, intensity of the THz beam in caustic I0 = 0.5×108 W/cm2, nonlinear refractive index coefficient n2 = 7×10−10 cm2/W. This value of n2 was obtained in the experiment previously.

 

Fig. 3 Comparison of the experimental results of the closed aperture measurement of Z-scan method for the pulsed broadband THz radiation for the water jet 0.1 mm thick with an analytical Z-scan curve for monochromatic radiation with the wavelength of 0.4 mm. The analytical curve was calculated using Eq. (2).

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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 n2 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 ω0 is much smaller than the fundamental vibrational frequency resulting in absorption peak λ = 3 μm (ω ≈ 100 THz) [38]. For our experiment this condition is well satisfied, as ω0/2π is approximately 0.75 THz and ω/2π is 15.9 THz. This equation takes the form:

n¯2,ν=n¯2,ν(1)+n¯2,ν(2)=3a12m2ω4αT232n0π2q2N2kB2[n0,ν21]3932πNn0ω[n0,ν21]2
We evaluate this expression through the use of the following values: a1 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, αT = 0.2×10−3 (°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 H2O molecule equal to the weight of molecule (1×2 + 16) times amu (1.67×10−24). It results in N = 3.3×1022 in 1 cm3. We take the refractive index as n0,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 n2 for water in the low-frequency limit is n2 = 5×10−10 cm2/W.

4. Summary

In conclusion we have experimentally demonstrated the possibility of direct measurement of the nonlinear refractive index coefficient n2 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 n2 = 7×10−10 cm2/W, which is 6 orders of magnitude higher than for the visible and IR ranges [26–29] where n2 has the magnitude of 10−16 cm2/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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References

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  1. S. Baierl, M. Hohenleutner, T. Kampfrath, A. Zvezdin, A. Kimel, R. Huber, and R. Mikhaylovskiy, “Nonlinear spin control by terahertz-driven anisotropy fields,” Nature Photon. 10, 715–718 (2016).
    [Crossref]
  2. J. Dong, A. Locquet, M. Melis, and D. Citrin, “Global mapping of stratigraphy of an old-master painting using sparsity-based terahertz reflectometry,” Sci. Rep. 7, 15098 (2017).
    [Crossref] [PubMed]
  3. K. J. Kaltenecker, A. Tuniz, S. C. Fleming, A. Argyros, B. T. Kuhlmey, M. Walther, and B. M. Fischer, “Ultrabroadband perfect imaging in terahertz wire media using single-cycle pulses,” Optica 3, 458–464 (2016).
    [Crossref]
  4. T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nature Photon. 10, 371–379 (2016).
    [Crossref]
  5. Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
    [Crossref]
  6. P. Bowlan, J. Bowlan, S. A. Trugman, R. V. Aguilar, J. Qi, X. Liu, J. Furdyna, M. Dobrowolska, A. J. Taylor, D. A. Yarotski, and R. P. Prasankumar, “Probing and controlling terahertz-driven structural dynamics with surface sensitivity,” Optica 4, 383–387 (2017).
    [Crossref]
  7. O. Smolyanskaya, I. Schelkanova, M. Kulya, E. Odlyanitskiy, I. Goryachev, A. Tcypkin, Y. V. Grachev, Y. G. Toropova, and V. Tuchin, “Glycerol dehydration of native and diabetic animal tissues studied by thz-tds and nmr methods,” Biomed. Opt. Express 9, 1198–1215 (2018).
    [Crossref] [PubMed]
  8. X. C. Zhang, A. Shkurinov, and Y. Zhang, “Extreme terahertz science,” Nature Photon. 11, 16–18 (2017).
    [Crossref]
  9. M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2015).
    [Crossref] [PubMed]
  10. P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96, 187402 (2006).
    [Crossref]
  11. J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in ge, si, and gaas by terahertz pump–terahertz probe measurements,” Phys. Rev. B 81, 035201 (2010).
    [Crossref]
  12. M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Thz-pump/thz-probe spectroscopy of semiconductors at high field strengths,” J. Opt. Soc. Am. B 26, A29–A34 (2009).
    [Crossref]
  13. G. Sharma, I. Al-Naib, H. Hafez, R. Morandotti, D. Cooke, and T. Ozaki, “Carrier density dependence of the nonlinear absorption of intense thz radiation in gaas,” Opt. Express 20, 18016–18024 (2012).
    [Crossref] [PubMed]
  14. D. Turchinovich, J. M. Hvam, and M. C. Hoffmann, “Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor,” Phys. Rev. B 85, 201304 (2012).
    [Crossref]
  15. S. Li, G. Kumar, and T. E. Murphy, “Terahertz nonlinear conduction and absorption saturation in silicon waveguides,” Optica 2, 553–557 (2015).
    [Crossref]
  16. S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
    [Crossref] [PubMed]
  17. C. Vicario, M. Shalaby, and C. P. Hauri, “Subcycle extreme nonlinearities in gap induced by an ultrastrong terahertz field,” Phys. Rev. Lett. 118, 083901 (2017).
    [Crossref] [PubMed]
  18. S. Lin, S. Yu, and D. Talbayev, “Measurement of quadratic terahertz optical nonlinearities using second-harmonic lock-in detection,” Phys. Rev. Appl. 10, 044007 (2018).
    [Crossref]
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  20. K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
    [Crossref]
  21. A. Woldegeorgis, T. Kurihara, B. Beleites, J. Bossert, R. Grosse, G. G. Paulus, F. Ronneberger, and A. Gopal, “Thz induced nonlinear effects in materials at intensities above 26 gw/cm 2,” J. Infrared Millim. Terahertz Waves 39, 667–680 (2018).
    [Crossref]
  22. A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
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  23. A. V. Balakin, S. V. Garnov, V. A. Makarov, N. A. Kuzechkin, P. A. Obraztsov, P. M. Solyankin, A. P. Shkurinov, and Y. Zhu, “”terhune-like” transformation of the terahertz polarization ellipse “mutually induced” by three-wave joint propagation in liquid,” Opt. Lett. 43, 4406–4409 (2018).
    [Crossref] [PubMed]
  24. Q. Jin, Yiwen E, K. Williams, J. Dai, and X.-C. Zhang, “Observation of broadband terahertz wave generation from liquid water,” Appl. Phys. Lett. 111, 071103 (2017).
    [Crossref]
  25. Yiwen E, Q. Jin, A. Tcypkin, and X.-C. Zhang, “Terahertz wave generation from liquid water films via laser-induced breakdown,” Appl. Phys. Lett. 113, 181103 (2018).
    [Crossref]
  26. M. Paillette, “Recherches expérimentales sur les effets kerr induits par une onde lumineuse,” Ann. Phys.-Paris,  14, 671–712 (1969).
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  27. W. L. Smith, P. Liu, and N. Bloembergen, “Superbroadening in h 2 o and d 2 o by self-focused picosecond pulses from a yalg: Nd laser,” Phys. Rev. A 15, 2396–2403 (1977).
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  28. P. Ho and R. Alfano, “Optical kerr effect in liquids,” Phys. Rev. A 20, 2170 (1979).
    [Crossref]
  29. R. W. Boyd, Nonlinear optics (Elsevier, 2003).
  30. K. Yang, P. Richards, and Y. Shen, “Generation of far-infrared radiation by picosecond light pulses in linbo3,” Appl. Phys. Lett. 19, 320–323 (1971).
    [Crossref]
  31. C. Lombosi, G. Polonyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fulop, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).
    [Crossref]
  32. A. Watanabe, H. Saito, Y. Ishida, M. Nakamoto, and T. Yajima, “A new nozzle producing ultrathin liquid sheets for femtosecond pulse dye lasers,” Opt. Commun. 71, 301–304 (1989).
    [Crossref]
  33. D. A. Kislin, M. A. Knyazev, Y. A. Shpolyanskii, and S. A. Kozlov, “Self-action of nonparaxial few-cycle optical waves in dielectric media,” JETP Letters 107, 753–760 (2018).
    [Crossref]
  34. X. Zheng, R. Chen, G. Shi, J. Zhang, Z. Xu, X. Cheng, and T. Jiang, “Characterization of nonlinear properties of black phosphorus nanoplatelets with femtosecond pulsed z-scan measurements,” Opt. Lett. 40, 3480–3483 (2015).
    [Crossref] [PubMed]
  35. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
    [Crossref]
  36. M. Yin, H. Li, S. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single z-scan method,” Appl. Phys. B 70, 587–591 (2000).
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  37. Y. Cheung and S. Gayen, “Optical nonlinearities of tea studied by z-scan and four-wave mixing techniques,” J. Opt. Soc. Am. B 11, 636–643 (1994).
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  38. G. M. Hale and M. R. Querry, “Optical constants of water in the 200-nm to 200-μm wavelength region,” Appl. Opt. 12, 555–563 (1973).
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  39. P. Schatzberg, “Molecular diameter of water from solubility and diffusion measurements,” J. Phys. Chem. 71, 4569–4570 (1967).
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  40. G. Kell, “Precise representation of volume properties of water at one atmosphere,” JCED 12, 66–69 (1967).
  41. L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. Keiding, “Thz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
    [Crossref]

2018 (8)

Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
[Crossref]

O. Smolyanskaya, I. Schelkanova, M. Kulya, E. Odlyanitskiy, I. Goryachev, A. Tcypkin, Y. V. Grachev, Y. G. Toropova, and V. Tuchin, “Glycerol dehydration of native and diabetic animal tissues studied by thz-tds and nmr methods,” Biomed. Opt. Express 9, 1198–1215 (2018).
[Crossref] [PubMed]

A. Woldegeorgis, T. Kurihara, B. Beleites, J. Bossert, R. Grosse, G. G. Paulus, F. Ronneberger, and A. Gopal, “Thz induced nonlinear effects in materials at intensities above 26 gw/cm 2,” J. Infrared Millim. Terahertz Waves 39, 667–680 (2018).
[Crossref]

A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
[Crossref]

A. V. Balakin, S. V. Garnov, V. A. Makarov, N. A. Kuzechkin, P. A. Obraztsov, P. M. Solyankin, A. P. Shkurinov, and Y. Zhu, “”terhune-like” transformation of the terahertz polarization ellipse “mutually induced” by three-wave joint propagation in liquid,” Opt. Lett. 43, 4406–4409 (2018).
[Crossref] [PubMed]

S. Lin, S. Yu, and D. Talbayev, “Measurement of quadratic terahertz optical nonlinearities using second-harmonic lock-in detection,” Phys. Rev. Appl. 10, 044007 (2018).
[Crossref]

Yiwen E, Q. Jin, A. Tcypkin, and X.-C. Zhang, “Terahertz wave generation from liquid water films via laser-induced breakdown,” Appl. Phys. Lett. 113, 181103 (2018).
[Crossref]

D. A. Kislin, M. A. Knyazev, Y. A. Shpolyanskii, and S. A. Kozlov, “Self-action of nonparaxial few-cycle optical waves in dielectric media,” JETP Letters 107, 753–760 (2018).
[Crossref]

2017 (5)

Q. Jin, Yiwen E, K. Williams, J. Dai, and X.-C. Zhang, “Observation of broadband terahertz wave generation from liquid water,” Appl. Phys. Lett. 111, 071103 (2017).
[Crossref]

X. C. Zhang, A. Shkurinov, and Y. Zhang, “Extreme terahertz science,” Nature Photon. 11, 16–18 (2017).
[Crossref]

P. Bowlan, J. Bowlan, S. A. Trugman, R. V. Aguilar, J. Qi, X. Liu, J. Furdyna, M. Dobrowolska, A. J. Taylor, D. A. Yarotski, and R. P. Prasankumar, “Probing and controlling terahertz-driven structural dynamics with surface sensitivity,” Optica 4, 383–387 (2017).
[Crossref]

J. Dong, A. Locquet, M. Melis, and D. Citrin, “Global mapping of stratigraphy of an old-master painting using sparsity-based terahertz reflectometry,” Sci. Rep. 7, 15098 (2017).
[Crossref] [PubMed]

C. Vicario, M. Shalaby, and C. P. Hauri, “Subcycle extreme nonlinearities in gap induced by an ultrastrong terahertz field,” Phys. Rev. Lett. 118, 083901 (2017).
[Crossref] [PubMed]

2016 (4)

S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
[Crossref] [PubMed]

S. Baierl, M. Hohenleutner, T. Kampfrath, A. Zvezdin, A. Kimel, R. Huber, and R. Mikhaylovskiy, “Nonlinear spin control by terahertz-driven anisotropy fields,” Nature Photon. 10, 715–718 (2016).
[Crossref]

K. J. Kaltenecker, A. Tuniz, S. C. Fleming, A. Argyros, B. T. Kuhlmey, M. Walther, and B. M. Fischer, “Ultrabroadband perfect imaging in terahertz wire media using single-cycle pulses,” Optica 3, 458–464 (2016).
[Crossref]

T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nature Photon. 10, 371–379 (2016).
[Crossref]

2015 (5)

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2015).
[Crossref] [PubMed]

S. Li, G. Kumar, and T. E. Murphy, “Terahertz nonlinear conduction and absorption saturation in silicon waveguides,” Optica 2, 553–557 (2015).
[Crossref]

K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
[Crossref]

C. Lombosi, G. Polonyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fulop, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).
[Crossref]

X. Zheng, R. Chen, G. Shi, J. Zhang, Z. Xu, X. Cheng, and T. Jiang, “Characterization of nonlinear properties of black phosphorus nanoplatelets with femtosecond pulsed z-scan measurements,” Opt. Lett. 40, 3480–3483 (2015).
[Crossref] [PubMed]

2012 (2)

G. Sharma, I. Al-Naib, H. Hafez, R. Morandotti, D. Cooke, and T. Ozaki, “Carrier density dependence of the nonlinear absorption of intense thz radiation in gaas,” Opt. Express 20, 18016–18024 (2012).
[Crossref] [PubMed]

D. Turchinovich, J. M. Hvam, and M. C. Hoffmann, “Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor,” Phys. Rev. B 85, 201304 (2012).
[Crossref]

2010 (1)

J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in ge, si, and gaas by terahertz pump–terahertz probe measurements,” Phys. Rev. B 81, 035201 (2010).
[Crossref]

2009 (1)

2006 (1)

P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96, 187402 (2006).
[Crossref]

2000 (1)

M. Yin, H. Li, S. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single z-scan method,” Appl. Phys. B 70, 587–591 (2000).
[Crossref]

1995 (1)

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. Keiding, “Thz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
[Crossref]

1994 (1)

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

1989 (1)

A. Watanabe, H. Saito, Y. Ishida, M. Nakamoto, and T. Yajima, “A new nozzle producing ultrathin liquid sheets for femtosecond pulse dye lasers,” Opt. Commun. 71, 301–304 (1989).
[Crossref]

1979 (1)

P. Ho and R. Alfano, “Optical kerr effect in liquids,” Phys. Rev. A 20, 2170 (1979).
[Crossref]

1977 (1)

W. L. Smith, P. Liu, and N. Bloembergen, “Superbroadening in h 2 o and d 2 o by self-focused picosecond pulses from a yalg: Nd laser,” Phys. Rev. A 15, 2396–2403 (1977).
[Crossref]

1973 (1)

1971 (1)

K. Yang, P. Richards, and Y. Shen, “Generation of far-infrared radiation by picosecond light pulses in linbo3,” Appl. Phys. Lett. 19, 320–323 (1971).
[Crossref]

1969 (1)

M. Paillette, “Recherches expérimentales sur les effets kerr induits par une onde lumineuse,” Ann. Phys.-Paris,  14, 671–712 (1969).
[Crossref]

1967 (2)

P. Schatzberg, “Molecular diameter of water from solubility and diffusion measurements,” J. Phys. Chem. 71, 4569–4570 (1967).
[Crossref]

G. Kell, “Precise representation of volume properties of water at one atmosphere,” JCED 12, 66–69 (1967).

Aguilar, R. V.

Alfano, R.

P. Ho and R. Alfano, “Optical kerr effect in liquids,” Phys. Rev. A 20, 2170 (1979).
[Crossref]

Al-Naib, I.

Argyros, A.

Baierl, S.

S. Baierl, M. Hohenleutner, T. Kampfrath, A. Zvezdin, A. Kimel, R. Huber, and R. Mikhaylovskiy, “Nonlinear spin control by terahertz-driven anisotropy fields,” Nature Photon. 10, 715–718 (2016).
[Crossref]

S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
[Crossref] [PubMed]

Balakin, A. V.

Beleites, B.

A. Woldegeorgis, T. Kurihara, B. Beleites, J. Bossert, R. Grosse, G. G. Paulus, F. Ronneberger, and A. Gopal, “Thz induced nonlinear effects in materials at intensities above 26 gw/cm 2,” J. Infrared Millim. Terahertz Waves 39, 667–680 (2018).
[Crossref]

Bespalov, V.

A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
[Crossref]

Bespalov, V. G.

Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
[Crossref]

Bloembergen, N.

W. L. Smith, P. Liu, and N. Bloembergen, “Superbroadening in h 2 o and d 2 o by self-focused picosecond pulses from a yalg: Nd laser,” Phys. Rev. A 15, 2396–2403 (1977).
[Crossref]

Bossert, J.

A. Woldegeorgis, T. Kurihara, B. Beleites, J. Bossert, R. Grosse, G. G. Paulus, F. Ronneberger, and A. Gopal, “Thz induced nonlinear effects in materials at intensities above 26 gw/cm 2,” J. Infrared Millim. Terahertz Waves 39, 667–680 (2018).
[Crossref]

Bowlan, J.

Bowlan, P.

Boyd, R.

A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
[Crossref]

Boyd, R. W.

K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
[Crossref]

R. W. Boyd, Nonlinear optics (Elsevier, 2003).

Braun, L.

S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
[Crossref] [PubMed]

Chen, R.

Cheng, X.

Cheung, Y.

Citrin, D.

J. Dong, A. Locquet, M. Melis, and D. Citrin, “Global mapping of stratigraphy of an old-master painting using sparsity-based terahertz reflectometry,” Sci. Rep. 7, 15098 (2017).
[Crossref] [PubMed]

Cooke, D.

Dai, J.

Q. Jin, Yiwen E, K. Williams, J. Dai, and X.-C. Zhang, “Observation of broadband terahertz wave generation from liquid water,” Appl. Phys. Lett. 111, 071103 (2017).
[Crossref]

Do, T.-M.

S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
[Crossref] [PubMed]

Dobrowolska, M.

Dolgaleva, K.

K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
[Crossref]

Dong, J.

J. Dong, A. Locquet, M. Melis, and D. Citrin, “Global mapping of stratigraphy of an old-master painting using sparsity-based terahertz reflectometry,” Sci. Rep. 7, 15098 (2017).
[Crossref] [PubMed]

Drozdov, A.

A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
[Crossref]

Ducournau, G.

T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nature Photon. 10, 371–379 (2016).
[Crossref]

E, Yiwen

Yiwen E, Q. Jin, A. Tcypkin, and X.-C. Zhang, “Terahertz wave generation from liquid water films via laser-induced breakdown,” Appl. Phys. Lett. 113, 181103 (2018).
[Crossref]

Q. Jin, Yiwen E, K. Williams, J. Dai, and X.-C. Zhang, “Observation of broadband terahertz wave generation from liquid water,” Appl. Phys. Lett. 111, 071103 (2017).
[Crossref]

Elsaesser, T.

P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96, 187402 (2006).
[Crossref]

Fiebig, M.

S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
[Crossref] [PubMed]

Fischer, B. M.

Fleming, S. C.

Fulop, J. A.

C. Lombosi, G. Polonyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fulop, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).
[Crossref]

Furdyna, J.

Gaal, P.

P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96, 187402 (2006).
[Crossref]

Garnov, S. V.

Gayen, S.

Gopal, A.

A. Woldegeorgis, T. Kurihara, B. Beleites, J. Bossert, R. Grosse, G. G. Paulus, F. Ronneberger, and A. Gopal, “Thz induced nonlinear effects in materials at intensities above 26 gw/cm 2,” J. Infrared Millim. Terahertz Waves 39, 667–680 (2018).
[Crossref]

Goryachev, I.

Grachev, Y. V.

O. Smolyanskaya, I. Schelkanova, M. Kulya, E. Odlyanitskiy, I. Goryachev, A. Tcypkin, Y. V. Grachev, Y. G. Toropova, and V. Tuchin, “Glycerol dehydration of native and diabetic animal tissues studied by thz-tds and nmr methods,” Biomed. Opt. Express 9, 1198–1215 (2018).
[Crossref] [PubMed]

Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
[Crossref]

Grosse, R.

A. Woldegeorgis, T. Kurihara, B. Beleites, J. Bossert, R. Grosse, G. G. Paulus, F. Ronneberger, and A. Gopal, “Thz induced nonlinear effects in materials at intensities above 26 gw/cm 2,” J. Infrared Millim. Terahertz Waves 39, 667–680 (2018).
[Crossref]

Hafez, H.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
[Crossref]

Hale, G. M.

Han, P.

G. Kaur, P. Han, and X. Zhang, “Terahertz induced nonlinear effects in doped silicon observed by open-aperture z-scan,” in “Infrared Millimeter and Terahertz Waves (IRMMW-THz), 2010 35th International Conference on,” (IEEE, 2010), pp. 1–2.

Hauri, C. P.

C. Vicario, M. Shalaby, and C. P. Hauri, “Subcycle extreme nonlinearities in gap induced by an ultrastrong terahertz field,” Phys. Rev. Lett. 118, 083901 (2017).
[Crossref] [PubMed]

M. Shalaby and C. P. Hauri, “Demonstration of a low-frequency three-dimensional terahertz bullet with extreme brightness,” Nat. Commun. 6, 5976 (2015).
[Crossref] [PubMed]

Hebling, J.

C. Lombosi, G. Polonyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fulop, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).
[Crossref]

J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in ge, si, and gaas by terahertz pump–terahertz probe measurements,” Phys. Rev. B 81, 035201 (2010).
[Crossref]

M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Thz-pump/thz-probe spectroscopy of semiconductors at high field strengths,” J. Opt. Soc. Am. B 26, A29–A34 (2009).
[Crossref]

Hey, R.

P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96, 187402 (2006).
[Crossref]

Ho, P.

P. Ho and R. Alfano, “Optical kerr effect in liquids,” Phys. Rev. A 20, 2170 (1979).
[Crossref]

Hoffmann, M. C.

D. Turchinovich, J. M. Hvam, and M. C. Hoffmann, “Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor,” Phys. Rev. B 85, 201304 (2012).
[Crossref]

J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in ge, si, and gaas by terahertz pump–terahertz probe measurements,” Phys. Rev. B 81, 035201 (2010).
[Crossref]

M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Thz-pump/thz-probe spectroscopy of semiconductors at high field strengths,” J. Opt. Soc. Am. B 26, A29–A34 (2009).
[Crossref]

Hohenleutner, M.

S. Baierl, M. Hohenleutner, T. Kampfrath, A. Zvezdin, A. Kimel, R. Huber, and R. Mikhaylovskiy, “Nonlinear spin control by terahertz-driven anisotropy fields,” Nature Photon. 10, 715–718 (2016).
[Crossref]

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S. Baierl, M. Hohenleutner, T. Kampfrath, A. Zvezdin, A. Kimel, R. Huber, and R. Mikhaylovskiy, “Nonlinear spin control by terahertz-driven anisotropy fields,” Nature Photon. 10, 715–718 (2016).
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D. Turchinovich, J. M. Hvam, and M. C. Hoffmann, “Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor,” Phys. Rev. B 85, 201304 (2012).
[Crossref]

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J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in ge, si, and gaas by terahertz pump–terahertz probe measurements,” Phys. Rev. B 81, 035201 (2010).
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A. Watanabe, H. Saito, Y. Ishida, M. Nakamoto, and T. Yajima, “A new nozzle producing ultrathin liquid sheets for femtosecond pulse dye lasers,” Opt. Commun. 71, 301–304 (1989).
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Kampfrath, T.

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D. A. Kislin, M. A. Knyazev, Y. A. Shpolyanskii, and S. A. Kozlov, “Self-action of nonparaxial few-cycle optical waves in dielectric media,” JETP Letters 107, 753–760 (2018).
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A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
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Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
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A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
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S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
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M. Yin, H. Li, S. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single z-scan method,” Appl. Phys. B 70, 587–591 (2000).
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Lin, S.

S. Lin, S. Yu, and D. Talbayev, “Measurement of quadratic terahertz optical nonlinearities using second-harmonic lock-in detection,” Phys. Rev. Appl. 10, 044007 (2018).
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W. L. Smith, P. Liu, and N. Bloembergen, “Superbroadening in h 2 o and d 2 o by self-focused picosecond pulses from a yalg: Nd laser,” Phys. Rev. A 15, 2396–2403 (1977).
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Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
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Materikina, D. V.

K. Dolgaleva, D. V. Materikina, R. W. Boyd, and S. A. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
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C. Lombosi, G. Polonyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fulop, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).
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J. Dong, A. Locquet, M. Melis, and D. Citrin, “Global mapping of stratigraphy of an old-master painting using sparsity-based terahertz reflectometry,” Sci. Rep. 7, 15098 (2017).
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S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
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S. Baierl, M. Hohenleutner, T. Kampfrath, A. Zvezdin, A. Kimel, R. Huber, and R. Mikhaylovskiy, “Nonlinear spin control by terahertz-driven anisotropy fields,” Nature Photon. 10, 715–718 (2016).
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T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nature Photon. 10, 371–379 (2016).
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A. Watanabe, H. Saito, Y. Ishida, M. Nakamoto, and T. Yajima, “A new nozzle producing ultrathin liquid sheets for femtosecond pulse dye lasers,” Opt. Commun. 71, 301–304 (1989).
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J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in ge, si, and gaas by terahertz pump–terahertz probe measurements,” Phys. Rev. B 81, 035201 (2010).
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M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Thz-pump/thz-probe spectroscopy of semiconductors at high field strengths,” J. Opt. Soc. Am. B 26, A29–A34 (2009).
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C. Lombosi, G. Polonyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fulop, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).
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C. Lombosi, G. Polonyi, M. Mechler, Z. Ollmann, J. Hebling, and J. A. Fulop, “Nonlinear distortion of intense THz beams,” New J. Phys. 17, 083041 (2015).
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Putilin, S.

A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
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Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
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T. Nagatsuma, G. Ducournau, and C. C. Renaud, “Advances in terahertz communications accelerated by photonics,” Nature Photon. 10, 371–379 (2016).
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K. Yang, P. Richards, and Y. Shen, “Generation of far-infrared radiation by picosecond light pulses in linbo3,” Appl. Phys. Lett. 19, 320–323 (1971).
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A. Woldegeorgis, T. Kurihara, B. Beleites, J. Bossert, R. Grosse, G. G. Paulus, F. Ronneberger, and A. Gopal, “Thz induced nonlinear effects in materials at intensities above 26 gw/cm 2,” J. Infrared Millim. Terahertz Waves 39, 667–680 (2018).
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C. Vicario, M. Shalaby, and C. P. Hauri, “Subcycle extreme nonlinearities in gap induced by an ultrastrong terahertz field,” Phys. Rev. Lett. 118, 083901 (2017).
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K. Yang, P. Richards, and Y. Shen, “Generation of far-infrared radiation by picosecond light pulses in linbo3,” Appl. Phys. Lett. 19, 320–323 (1971).
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X. C. Zhang, A. Shkurinov, and Y. Zhang, “Extreme terahertz science,” Nature Photon. 11, 16–18 (2017).
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Shpolyanskii, Y. A.

D. A. Kislin, M. A. Knyazev, Y. A. Shpolyanskii, and S. A. Kozlov, “Self-action of nonparaxial few-cycle optical waves in dielectric media,” JETP Letters 107, 753–760 (2018).
[Crossref]

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W. L. Smith, P. Liu, and N. Bloembergen, “Superbroadening in h 2 o and d 2 o by self-focused picosecond pulses from a yalg: Nd laser,” Phys. Rev. A 15, 2396–2403 (1977).
[Crossref]

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Solyankin, P. M.

Talbayev, D.

S. Lin, S. Yu, and D. Talbayev, “Measurement of quadratic terahertz optical nonlinearities using second-harmonic lock-in detection,” Phys. Rev. Appl. 10, 044007 (2018).
[Crossref]

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M. Yin, H. Li, S. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single z-scan method,” Appl. Phys. B 70, 587–591 (2000).
[Crossref]

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Tcypkin, A.

O. Smolyanskaya, I. Schelkanova, M. Kulya, E. Odlyanitskiy, I. Goryachev, A. Tcypkin, Y. V. Grachev, Y. G. Toropova, and V. Tuchin, “Glycerol dehydration of native and diabetic animal tissues studied by thz-tds and nmr methods,” Biomed. Opt. Express 9, 1198–1215 (2018).
[Crossref] [PubMed]

A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
[Crossref]

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[Crossref]

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L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. Keiding, “Thz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330–333 (1995).
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Tsypkin, A. N.

Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
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Tuniz, A.

Turchinovich, D.

D. Turchinovich, J. M. Hvam, and M. C. Hoffmann, “Self-phase modulation of a single-cycle terahertz pulse by nonlinear free-carrier response in a semiconductor,” Phys. Rev. B 85, 201304 (2012).
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M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
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C. Vicario, M. Shalaby, and C. P. Hauri, “Subcycle extreme nonlinearities in gap induced by an ultrastrong terahertz field,” Phys. Rev. Lett. 118, 083901 (2017).
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Watanabe, A.

A. Watanabe, H. Saito, Y. Ishida, M. Nakamoto, and T. Yajima, “A new nozzle producing ultrathin liquid sheets for femtosecond pulse dye lasers,” Opt. Commun. 71, 301–304 (1989).
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M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990).
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Q. Jin, Yiwen E, K. Williams, J. Dai, and X.-C. Zhang, “Observation of broadband terahertz wave generation from liquid water,” Appl. Phys. Lett. 111, 071103 (2017).
[Crossref]

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P. Gaal, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Nonlinear terahertz response of n-type gaas,” Phys. Rev. Lett. 96, 187402 (2006).
[Crossref]

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A. Woldegeorgis, T. Kurihara, B. Beleites, J. Bossert, R. Grosse, G. G. Paulus, F. Ronneberger, and A. Gopal, “Thz induced nonlinear effects in materials at intensities above 26 gw/cm 2,” J. Infrared Millim. Terahertz Waves 39, 667–680 (2018).
[Crossref]

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S. Baierl, J. H. Mentink, M. Hohenleutner, L. Braun, T.-M. Do, C. Lange, A. Sell, M. Fiebig, G. Woltersdorf, T. Kampfrath, and R. Huber, “Terahertz-driven nonlinear spin response of antiferromagnetic nickel oxide,” Phys. Rev. Lett. 117, 197201 (2016).
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A. Watanabe, H. Saito, Y. Ishida, M. Nakamoto, and T. Yajima, “A new nozzle producing ultrathin liquid sheets for femtosecond pulse dye lasers,” Opt. Commun. 71, 301–304 (1989).
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K. Yang, P. Richards, and Y. Shen, “Generation of far-infrared radiation by picosecond light pulses in linbo3,” Appl. Phys. Lett. 19, 320–323 (1971).
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Yeh, K.-L.

J. Hebling, M. C. Hoffmann, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Observation of nonequilibrium carrier distribution in ge, si, and gaas by terahertz pump–terahertz probe measurements,” Phys. Rev. B 81, 035201 (2010).
[Crossref]

M. C. Hoffmann, J. Hebling, H. Y. Hwang, K.-L. Yeh, and K. A. Nelson, “Thz-pump/thz-probe spectroscopy of semiconductors at high field strengths,” J. Opt. Soc. Am. B 26, A29–A34 (2009).
[Crossref]

Yin, M.

M. Yin, H. Li, S. Tang, and W. Ji, “Determination of nonlinear absorption and refraction by single z-scan method,” Appl. Phys. B 70, 587–591 (2000).
[Crossref]

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S. Lin, S. Yu, and D. Talbayev, “Measurement of quadratic terahertz optical nonlinearities using second-harmonic lock-in detection,” Phys. Rev. Appl. 10, 044007 (2018).
[Crossref]

Zhang, J.

Zhang, X.

G. Kaur, P. Han, and X. Zhang, “Terahertz induced nonlinear effects in doped silicon observed by open-aperture z-scan,” in “Infrared Millimeter and Terahertz Waves (IRMMW-THz), 2010 35th International Conference on,” (IEEE, 2010), pp. 1–2.

Zhang, X. C.

X. C. Zhang, A. Shkurinov, and Y. Zhang, “Extreme terahertz science,” Nature Photon. 11, 16–18 (2017).
[Crossref]

Zhang, X.-C.

Y. V. Grachev, X. Liu, S. E. Putilin, A. N. Tsypkin, V. G. Bespalov, S. A. Kozlov, and X.-C. Zhang, “Wireless data transmission method using pulsed thz sliced spectral supercontinuum,” IEEE Photonics Technol. Lett. 30, 103–106 (2018).
[Crossref]

A. Tcypkin, S. Putilin, M. Kulya, M. Melnik, A. Drozdov, V. Bespalov, X.-C. Zhang, R. Boyd, and S. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline znse in the terahertz spectral range,” Bull. Russ. Acad. Sci.: Phys. 82, 1547–1549 (2018).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1 (a) The experimental setup for measuring the nonlinear refractive index (n2) of a liquid jet in the THz spectral range. Two parabolic mirrors (PM1 and PM2) with a focal length of 12.5 mm form the caustics area where the water jet (jet) is scanned along the z axis. The synchronization is performed using the mechanical modulator (M) located between the lens and the Golay cell (GC). The aperture (A) is moved from open to closed position to change the geometry of Z-scan from open to closed aperture. Insert - Geometrical position of the jet moved along the z axis relative to the THz radiation. The temporal waveform (b) and its spectrum (c) of the THz pulse generated by the TERA-AX system.
Fig. 2
Fig. 2 Z-scan curves for a 0.1 mm thick water jet measured with open (a) and closed (b) aperture for different THz radiation energy values of 4 nJ, 40 nJ and 400 nJ. Δ T = 0.013 is the differential of the Z-scan curve measured with the closed aperture of radius 1.5 mm.
Fig. 3
Fig. 3 Comparison of the experimental results of the closed aperture measurement of Z-scan method for the pulsed broadband THz radiation for the water jet 0.1 mm thick with an analytical Z-scan curve for monochromatic radiation with the wavelength of 0.4 mm. The analytical curve was calculated using Eq. (2).

Equations (5)

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n 2 = Δ T 0.406 I in × 2 λ 2 π L α ( 1 S ) 0.25
T ( z ) = + P T ( Δ Φ 0 ( t ) ) d t S + P i ( t ) d t
P T ( Δ Φ 0 ( t ) ) = c 0 N 0 π 0 r a a | E a ( r , t ) | 2 r d r
E a ( r , t ) = E ( z , r = 0 , t ) exp ( α L / 2 ) × m = 0 + [ i Δ ϕ 0 ( z , t ) ] m m ! w m 0 w m exp ( r 2 w m 2 i k r 2 2 R m + i Q m )
n ¯ 2 , ν = n ¯ 2 , ν ( 1 ) + n ¯ 2 , ν ( 2 ) = 3 a 1 2 m 2 ω 4 α T 2 32 n 0 π 2 q 2 N 2 k B 2 [ n 0 , ν 2 1 ] 3 9 32 π N n 0 ω [ n 0 , ν 2 1 ] 2

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