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Letter

Estimations of Low-Inertia Cubic Nonlinearity

Featured by Electro-Optical Crystals in the THz Range

Maria Zhukova * , Maksim Melnik, Irina Vorontsova , Anton Tcypkin and Sergei Kozlov

Laboratory of Femtosecond Optics and Femtotechnologies, ITMO University, 49 Kronverksky Pr.,

197101 St. Petersburg, Russia; mmelnik@itmo.ru (M.M.); iovorontsova@itmo.ru (I.V.); tsypkinan@itmo.ru (A.T.);

kozlov@mail.ifmo.ru (S.K.)

*Correspondence: mozhukova@itmo.ru

Received: 30 September 2020; Accepted: 26 October 2020; Published: 28 October 2020

Abstract:

Despite the growing interest in nonlinear devices and components for light by light

control in the terahertz range, there is a shortage of such materials and media used for

these purposes. Here,

we present the calculated values of low-inertia nonlinear refractive index

coefﬁcient for electro-optical crystals used in THz time-domain spectroscopy systems such as

ZnSe, ZnTe, CdTe, GaP, and LiNbO3.

The medium parameters affecting the cubic nonlinearity of

the vibrational nature increase in the range of 0.5–1 THz have been determined. Comparison of

theoretical calculations with known experimental results conﬁrm the theoretical model as well

as our analysis of media parameter inﬂuence on the cubic nonlinearity. In terms of applications,

results obtained open up new perspectives for studying various materials in the THz frequency range.

Keywords:

terahertz nonlinearity; kerr nonlinearity; electro-optical crystal; high field terahertz radiation

1. Introduction

At present, the development of THz technologies is advancing by leaps and bounds [

1

].

The amount of materials used for generating, detecting, modulating, and controlling THz radiation

is constantly expanding with the new ones, such as metamaterials and two-dimensional materials.

While the ﬁeld working on such material creation is currently at its development stage, there are a lot

of examples of media that proved themselves in the sphere,

i.e., crystals, glasses, liquids, and gases.

Nonlinear media are of particular interest among the materials mentioned. For example, nonlinear

electro-optical crystals are widely employed in THz technology as generators and detectors [

2

].

Aiming at ﬁnding some new applications for them, it is necessary to study the features of their

nonlinear responses in the THz range.

Owing to the recent developments in the ﬁeld of coherent THz radiation source creation,

pulsed sources featuring peak intensity of about 10

13

W/cm

2

have appeared [

3

]. Such large intensity

values furnish insights into the observation of not only linear [

4

], but also nonlinear phenomena [

5

]

of various materials in the THz region of the spectrum. As mentioned above, electro-optical crystals

used for THz radiation detection and generation [

6

] are the ﬁrst choice for the study of the nonlinear

response, since they are widely used in modern THz technology.

In this paper, we present the calculated values of the low-inertia nonlinear refractive index

coefﬁcients in the range of 0.5–1 THz for electro-optical crystals used in THz time-domain

spectroscopy systems such as

ZnSe, ZnTe, CdTe, GaP, and LiNbO3. The

results presented correspond

to the experimental estimations performed by other research teams so far. The accuracy is around an

order of magnitude. The subsequent analysis revealed the parameters inﬂuencing the

n2

coefﬁcient

value increase in the THz range. To prove the analysis validity, the results of our research for water are

then given. They conﬁrm the stated relationship between the media parameters and the

n2

coefﬁcient

Photonics 2020,7, 98; doi:10.3390/photonics7040098 www.mdpi.com/journal/photonics

Photonics 2020,7, 98 2 of 7

magnitude. The analysis performed is a reference point for experimental teams to choose of the

materials featuring high n2for the research.

2. Methods

As widely known, thermal nonlinearity is considered the largest one conventionally. However,

THz pulses feature the duration of 1–2 ps and the thermal nonlinearity contribution is small as it has

high-inertia nature. It has been shown both analytically [

7

,

8

] and experimentally [

9

] that regarding THz

frequency range, the nonlinearity of the vibrational nature contributes to the overall cubic (third-order)

nonlinearity the most, which occurs owing to its low inertia (of the order of or less than 1 ps).

The theoretical approach proposed in [

7

] for calculating the low-inertia nonlinear refractive index

coefﬁcient (

n2

) in the terahertz frequency range far from the fundamental resonance (Equation (55)

in [7]) is used for the evaluations carried out in this work. The formula has the following form:

n2,THz [CGS] = 3a2

lm2ω4

0α2

T

32n0π2q2N2k2

Bhn2

0,ν−1i3−9

32πNn0¯hωhn2

0,ν−1i2, (1)

where

n0,ν=q1+n2

0−n2

el

describes the vibrational contribution to the low-frequency refractive

index,

n0

denotes the linear refractive index in THz range (0.5–1.0 THz),

nel

is the linear refractive

index in the non-resonant electron contribution range (i.e., 800 nm),

ω0

is the fundamental vibration

frequency,

al

is the lattice constant,

m

describes the reduced mass of the vibrational mode for

AxBy:m= (mA×mB)/(mA+mB),αT

is the coefﬁcient of thermal expansion, and

S

denotes the

relative density. Equation (1) is given in the CGS system of units. To convert this value to the

corresponding one in SI, the following relation can be used: n2,T Hz[S I] = 4.2 ×10−7n2,T Hz[CGS]/n0.

This approach is making use of the fact that the dominant nonlinearity mechanism in the frequency

range mentioned features a vibrational nature (i.e., vibrations of lattice ions for crystals being the case)

and is based on a classical anharmonic oscillator model. The approach is valid for the case when the

radiation frequency is considered less than the one corresponding to the resonance of the medium,

a single vibrational resonance is inherent in the medium, or one of them is visibly dominant.

3. Results

Earlier, the

n2

coefﬁcient in the terahertz frequency range for crystalline quartz (SiO

2

) was

estimated by using the theoretical approach described [

7

]. Crystalline quartz has high transparency in

the visible and IR spectral ranges, as well as in the THz frequency range (starting from 100

µ

m),

and, therefore, it is used as a material for optical components (transparency windows, lenses)

in terahertz technology [

10

] actively. Additionally, the material mentioned is anisotropic and has

birefringence in the THz spectral region [

5

]. It was shown in [

7

] that the vibrational contribution in the

THz frequency range can exceed the electronic one signiﬁcantly and the value of

n2

in the THz range

is several orders of magnitude higher than in the NIR.

Active development of THz technologies and the appearance of high-intensity THz sources

gave impetus to the study of the nonlinear properties of various materials common for

the

THz range. For instance,

the vibrational nonlinearity in the THz frequency range featured

by electro-optical crystals used for the detection and generation of THz radiation, such as

ZnSe, ZnTe, CdTe, GaP, and LiNbO3,

is worth consideration. The parameters used for the calculations

performed are presented in Table 1.

The low-inertia

n2

calculation results for the media mentioned, as well as the values for SiO

2

provided for comparison, are illustrated in Figure 1(for LiNbO

3

, the mean

n2,THz

value for

a

and

c

axis

is shown). Table 2represents the low-inertia nonlinear refractive index coefﬁcients in the NIR (n2, IR).

Photonics 2020,7, 98 3 of 7

Table 1. Parameters for n2,T Hz calculations.

Crystal ω0ω0/2πn0nel

alcm mαTS

cm−1THz ×10−8cm ×10−23 ×10−6◦C−1

ZnSe 292 [11] 8.7 2.97 [12] 2.5 [13] 5.67 [14] 5.92 4.56 [15] 5.27 [16]

ZnTe 253 [17] 7.6 3.1 [18] 2.85 [13] 6.1 [19] 7.2 8.21 [20] 6.34 [16]

CdTe 141 [21] 4.2 3.23 [22] 2.95 [23] 6.48 [19] 9.98 5.0 [16] 6.20 [16]

GaP 367 [24] 11 3.31 [25] 3.18 [23] 5.45 [26] 3.6 5.3 [16] 4.13 [16]

LiNbO3(a) 187 [27] 5.6 5.15 [18] 2.28 [28,29] 5.15 [30] 15 14.8 [16] 4.64 [16]

LiNbO3(c) 147 [27] 4.4 6.7 [31] 2.2 [28,29] 13.9 [30] 14 4.1 [16] 4.64 [16]

Figure 1.

The results of

n2,THz

estimations (see also Table 2) for comparison with other results),

* calculated value from [7].

Table 2. The nonlinear refractive index coefﬁcients in the NIR and the THz frequency range.

Crystal

n2,T Hz ,cm2/W n2, IR ,cm 2/W

in THz Range in NIR Range

(Calculations)

ZnSe 1×10−13 3.8 ×10−14 [32]

ZnTe 3×10−14 1.3 ×10−12 [33]

CdTe 2×10−13 3.4 ×10−13 [34]

GaP 1×10−14 6.5 ×10−14 [35]

LiNbO3(a) 7×10−11 1.7 ×10−15 [36]

LiNbO3(c) 5×10−11 1.31 ×10−15 [36]

For some crystals, the low-inertia coefﬁcient of the nonlinear refractive index in the THz frequency

range exceeds the corresponding value in the NIR range. It can be related to the fact that the vibrational

contribution to the refractive index of the media is quite large.

The experimental conﬁrmation is needed to prove the analytical model validity.

However, there are

just few experimental works conducted so far. For instance,

in the paper [37],

authors using the measurement of the angular dependence of the Kerr signal and the theoretical

Photonics 2020,7, 98 4 of 7

analysis of the experiment determined the nonzero tensor elements of the third-order response

function for GaP and estimate its

n2

parameter to be 1.2

×

10

−13

cm

2

/W. For lithium niobate crystal,

experimental measurements based on the change of the transmitted THz pulse shape [

38

] give the

n2

value of 5.4

×

10

−12

cm

2

/W. For the THz range results, the values taken from other sources correlate

with our estimations within about an order of magnitude, which is a tolerable error and a standard

phenomenon for such a small

n2

value. However, it should be mentioned that these experimental

results were indirectly estimated.

4. Discussion

Regarding the media parameters affecting the value of

n2,THz

, an increase in this coefﬁcient can

be inherent in media featuring a higher coefﬁcient of thermal expansion, a larger difference between

linear refractive index in the range with non-resonant electronic contribution and linear refractive

index in the THz range considered, a larger fundamental frequency of vibrations, and a smaller value

of the numerical density of vibrations. The latter is ensured by an increase in the total mass of atoms

and a decrease in the relative density of the medium (see Figure 2a). It is also important that the

vibrational contribution to the low-frequency refractive index be larger (see Figure 2b). This depends

on the values of the linear refractive index in the THz frequency range and in the range containing the

non-resonant electronic contribution (NIR) and on difference between them.

Figure 2.

The nonlinear refractive index coefﬁcient dependence on the medium parameters for different

crystals: (

a

)

m

describes the reduced mass of the vibrational mode and (

b

)

n0,ν

describes the vibrational

contribution to the low-frequency refractive index (see Equation (1) description).

Moreover, we have tested this theoretical approach experimentally earlier for water [

9

]. Obviously,

the structure featured by liquids is different from the one of crystals, so it seems quite logical to doubt

the idea to implement this theoretical model in case of water. Here, the anharmonic oscillator vibration

model is considered. Crystals being the case, the vibrations mentioned are the ones of lattice ions,

whereas vibrations of a molecule as a separate structure are addressed regarding liquids. Both oscillation

types have the same nature; therefore, the approach is valid for liquids as well.

The z-scan method was used to conduct the measurements [

39

]. The technique essence consists of

the induced narrowing and broadening of an intense spherical light beam when a nonlinear medium

moves along its propagation axis and passes through the focus. The nonlinear medium then acts as a

thin lens and leads to a minimal change in the distribution of the beam ﬁeld in the far ﬁeld when placed

in or near the focus. The resulting characteristic z-scan curve represents the peak and valley of the

nonlinear medium transmission. The magnitude of the difference between the maximal and minimal

values allows to calculate the nonlinear refractive index coefﬁcient. Earlier we have shown that this

technique is applicable for THz frequency range featuring a very broad spectrum with the correct ratio

of the crystal thickness to the spatial size of the pulse [

40

]. Regarding water as the nonlinear medium,

we obtained theoretically a value of

n2

= 5

×

10

−10

cm

2

/W and experimentally

n2

= 7

±

5

×

10

−10

Photonics 2020,7, 98 5 of 7

cm

2

/W. These values have a very good correspondence and show that for THz, the frequency range

coefﬁcient of nonlinear refractive index coefﬁcient of water is 6 orders of magnitude higher than in

the NIR frequency range. Additionally, one can see that the

n2

value featured by water in THz range

is in the mean several orders of magnitude higher than in the case of crystals. Based on the above

given analysis of the medium parameters’ contribution to cubic nonlinearity, the explanation comes

from the water characteristics directly. Water manifests higher fundamental vibrational frequency

(100 THz), a larger difference between linear refractive index in the range with non-resonant electronic

contribution (1.33) and linear refractive index in the THz range (2.3), and a smaller value of the

numerical density of vibrations (3.3

×

10

22

). Furthermore, water exhibits a 1000 times greater thermal

expansion coefﬁcient (0.2 ×10−3◦C−1), which can also add to the overall n2value.

5. Conclusions

The work presents the analytically calculated values of the low-inertia nonlinear refractive index

coefﬁcient for common electro-optical crystals used in THz spectroscopy systems and their comparison

with the experimental data published before. The estimations conducted conﬁrm that the contribution

to the media nonlinearity associated with lattice ion vibrations is quite large in the THz spectral range.

The value for LiNbO

3

in the range of 0.5–1 THz exceeds the corresponding value in the NIR range.

The relationship between media parameters and

n2

is presented and analyzed in terms of vibration

related parameters. Our work [

9

] experimentally proves the validity of the theoretical approach

used and justiﬁes the conclusions regarding the medium parameters that determine the

n2

coefﬁcient

magnitude. Current work in the context of common THz crystals gives deeper insight into the materials

featuring high n2for experimental teams to base their choice of the media to investigate on.

The results presented can be used as a summary useful for a wide range of future studies

devoted to the search and choice of the media to be employed for different manipulation devices,

all-optical switching for routing, modulation and control of THz signals, i.e., [

41

], as well as

high-harmonic generation [

42

] in the THz range. The information provided in this article serves

as a platform for material development and study, i.e., 2D materials, metamaterials [

43

], and liquid

media [9], which have a potential to be used in nonlinear THz devices.

Author Contributions:

Conceptualization, M.Z., A.N. and S.K.; formal analysis, M.M.; investigation, M.Z. and I.V.;

data curation, M.Z.; writing—original draft preparation, M.Z. and I.V.; writing—review and editing, A.T. and S.K.;

visualization, M.Z.; supervision, S.K. All authors have read and agreed to the published version

of the manuscript.

Funding: This research was funded by RFBR project 19-02-00154.

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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