High Kerr nonlinearity of water in THz spectral
ANTO N N. TC Y P K I N ,1,* MAKSIM V. MELNIK,1MA R IA O. ZHU KOVA ,1
IRINA O. VO R O N T S OVA,1SERGEY E. PUTILIN,1
SERGEI A. KO Z L OV,1AND XI-CH E N G ZH A N G 1,2
International Laboratory of Femtosecond Optics and Femtotechnologies, ITMO University, St. Petersburg
2The Institute of Optics, University of Rochester, Rochester, NY 14627, USA
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
/W in the THz frequency range, which exceeds
the values for the visible and NIR ranges by 6 orders of magnitude. To estimate
, 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
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Terahertz (THz) frequency range is ﬁnding more and more applications in diﬀerent ﬁelds
of fundamental research  and a variety of everyday applications such as nondestructive
spectroscopy  and imaging , communications [4, 5] and ultrafast control  as well as
biomedicine . Recent advances in research have brought high intensity broadband sources of
THz radiation into play . These perspectives have already been shown for highly intensive
THz pulses generated from organic crystals with peak intensity
Ipea k ∼
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 , absorption bleaching by means of pump-
probe technique [11,12], nonlinear free-carrier response [13, 14], ﬁeld-induced transparency ,
spin response , giant cross phase modulation and THz-induced spectral broadening of
femtosecond pulses , or quadratic THz optical nonlinearities by measuring the quadratic THz
Kerr eﬀect .
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
. In the ﬁrst report of such kind of measurements  silicon was tested by the use of the
open aperture Z-scan technique. Originally, it was predicted theoretically  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
Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10419
Received 2 Jan 2019; revised 13 Mar 2019; accepted 17 Mar 2019; published 1 Apr 2019
experimentally by the Z-scan method [21,22]. Also, estimates of the liquid nitrogen
made on the basis of a change in the rotation angle of the THz pulse polarization ellipse .
However, in these works only indirect estimations of n2were 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 (
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 . 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 . In our experiment, we use the parabolic mirror with
a focal length of 25 mm to collimate the THz radiation generated from LiNbO
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
. 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 . 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
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  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.
Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10420
Fig. 1. (a) The experimental setup for measuring the nonlinear refractive index (
) 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.
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
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 . 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 . 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
according to the results of our measurements shown in Fig. 2(b):
= 0.013 (Fig. 2(b)) is the diﬀerence between the maximum and minimum transmission,
is the linear transmission of the aperture,
] is the eﬀective interaction
Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10421
is the sample thickness,
is the absorption coeﬃcient (
= 100 cm
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
= 0.4 mm (
= 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.
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.
T = 0.013 is
the diﬀerential of the Z-scan curve measured with the closed aperture of radius 1.5 mm.
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 :
is the instantaneous input power (within the sample),
is the aperture linear transmittance; the transmitted power through the aperture gives
a|Ea(r,t)|2r dr (3)
and E(z,r=0,t)=E0si n(2πν0t)w0/w(z),∆φ0(z,t)=∆Φ0(t)/(1+z2/z2
The following values are used in these equations: absorption coeﬃcient
= 100 cm
= 0.1 mm, central frequency of the radiation
= 0.75 THz (
= 0.4 mm), beam waist
= 0.5 mm, aperture radius
= 1.5 mm, radius of the THz beam
= 12.5 mm,
intensity of the THz beam in caustic
, nonlinear refractive index coeﬃcient
n2= 7×10−10 cm2/W. This value of n2was 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.
Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10422
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).
3. Theoretical estimate of the nonlinear refractive index coefﬁcient
We estimate the nonlinear refractive index coeﬃcient
of liquid water through the use of a
recent theoretical treatment . 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 , which applies to the situation
where the THz frequency
is much smaller than the fundamental vibrational frequency resulting
in absorption peak
100 THz) . For our experiment this condition is well
is approximately 0.75 THz and
is 15.9 THz. This equation takes the
We evaluate this expression through the use of the following values:
is the lattice constant; for
our estimations in case of liquid we use the water molecule diameter 2.8
g is the reduced mass of the vibrational mode,
is the thermal
expansion coeﬃcient . The parameter
is the eﬀective charge of the chemical bond; for
simplicity, we take this quantity to be the electron charge.
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
O molecule equal to the weight of molecule (1
2 + 16) times amu (1.67
It results in
in 1 cm
. We take the refractive index as
= 2.3 which is the averaged
refractive index in 0.3–1.0 THz region . Using these values, we ﬁnd that the predicted value
of n2for water in the low-frequency limit is n2= 5×10−10 cm2/W.
In conclusion we have experimentally demonstrated the possibility of direct measurement of the
nonlinear refractive index coeﬃcient
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
Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10423
coeﬃcient of water calculated from the experiment is
/W, which is 6 orders of
magnitude higher than for the visible and IR ranges [26
has the magnitude of 10
/W. These results demonstrate the high cubic nonlinearity of water in the THz frequency
range and conﬁrm a recent theoretical prediction  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
Russian Foundation of Basic Research (RFBR) (19-02-00154). X.-C. Zhang acknowledges
support from U.S. Army Research Oﬃce (W911NF-17-1-0428). In addition, S.E. Putilin
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