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High Kerr nonlinearity of water in THz spectral

range

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

1

International Laboratory of Femtosecond Optics and Femtotechnologies, ITMO University, St. Petersburg

197101, Russia

2The Institute of Optics, University of Rochester, Rochester, NY 14627, USA

*tsypkinan@corp.ifmo.ru

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

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

Ipea k ∼

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

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

Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10419

#354766

https://doi.org/10.1364/OE.27.010419

© 2019

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

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 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 (

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 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.

Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10420

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.

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)).

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):

n2=

∆T

0.406Iin ×√2λ

2πLα(1−S)0.25 (1)

where

∆T

= 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

Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10421

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.

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 [35]:

T(z)=+∞

−∞ PT(∆Φ0(t))dt

S+∞

−∞ Pi(t)dt

(2)

where

Pi

(t)=

πw2

0I0(t)/2

is the instantaneous input power (within the sample),

S

=1-

exp(−

2

r2

a/w2

a)

is the aperture linear transmittance; the transmitted power through the aperture gives

PT(∆Φ0(t)) =c0N0πra

0

a|Ea(r,t)|2r dr (3)

where

Ea(r,t)=E(z,r=0,t)exp(−αL/2) ×

+∞

m=0

[i∆φ0(z,t)]m

m!

wm0

wm

exp(−r2

w2

m−ikr 2

2Rm

+iQm)(4)

and E(z,r=0,t)=E0si n(2πν0t)w0/w(z),∆φ0(z,t)=∆Φ0(t)/(1+z2/z2

0),∆Φ0(t)=k∆n0(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

×

10

8

W/cm

2

, 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

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

µ

m (

ω≈

100 THz) [38]. For our experiment this condition is well

satisﬁed, as

ω0/

2

π

is approximately 0.75 THz and

ω/

2

π

is 15.9 THz. This equation takes the

form:

¯n2,ν =¯n(1)

2,ν

+¯n(2)

2,ν

=

3a2

1m2ω4α2

T

32n0π2q2N2k2

Bn2

0,ν −13−9

32πNn0}ωn2

0,ν −12(5)

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

(

℃

)

−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

n0

= 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 n2for 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

Vol. 27, No. 8 | 15 Apr 2019 | OPTICS EXPRESS 10423

coeﬃcient of water calculated from the experiment is

n2

= 7

×

10

−10

cm

2

/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

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). X.-C. Zhang acknowledges

support from U.S. Army Research Oﬃce (W911NF-17-1-0428). In addition, S.E. Putilin

acknowledges support from the Government of the Russian Federation (project 3.9041.2017/7.8).

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