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; email@example.com (M.M.); firstname.lastname@example.org (I.V.); email@example.com (A.T.);
Received: 30 September 2020; Accepted: 26 October 2020; Published: 28 October 2020
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.
terahertz nonlinearity; kerr nonlinearity; electro-optical crystal; high field terahertz radiation
At present, the development of THz technologies is advancing by leaps and bounds [
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 [
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
have appeared [
]. Such large intensity
values furnish insights into the observation of not only linear [
], but also nonlinear phenomena [
of various materials in the THz region of the spectrum. As mentioned above, electro-optical crystals
used for THz radiation detection and generation [
] 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
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
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.
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 [
] and experimentally [
] 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 [
] for calculating the low-inertia nonlinear refractive index
) in the terahertz frequency range far from the fundamental resonance (Equation (55)
in ) is used for the evaluations carried out in this work. The formula has the following form:
n2,THz [CGS] = 3a2
describes the vibrational contribution to the low-frequency refractive
denotes the linear refractive index in THz range (0.5–1.0 THz),
is the linear refractive
index in the non-resonant electron contribution range (i.e., 800 nm),
is the fundamental vibration
is the lattice constant,
describes the reduced mass of the vibrational mode for
is the coefﬁcient of thermal expansion, and
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.
coefﬁcient in the terahertz frequency range for crystalline quartz (SiO
estimated by using the theoretical approach described [
]. Crystalline quartz has high transparency in
the visible and IR spectral ranges, as well as in the THz frequency range (starting from 100
and, therefore, it is used as a material for optical components (transparency windows, lenses)
in terahertz technology [
] actively. Additionally, the material mentioned is anisotropic and has
birefringence in the THz spectral region [
]. It was shown in [
] that the vibrational contribution in the
THz frequency range can exceed the electronic one signiﬁcantly and the value of
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
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.
calculation results for the media mentioned, as well as the values for SiO
provided for comparison, are illustrated in Figure 1(for LiNbO
, the mean
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.
cm−1THz ×10−8cm ×10−23 ×10−6◦C−1
ZnSe 292  8.7 2.97  2.5  5.67  5.92 4.56  5.27 
ZnTe 253  7.6 3.1  2.85  6.1  7.2 8.21  6.34 
CdTe 141  4.2 3.23  2.95  6.48  9.98 5.0  6.20 
GaP 367  11 3.31  3.18  5.45  3.6 5.3  4.13 
LiNbO3(a) 187  5.6 5.15  2.28 [28,29] 5.15  15 14.8  4.64 
LiNbO3(c) 147  4.4 6.7  2.2 [28,29] 13.9  14 4.1  4.64 
The results of
estimations (see also Table 2) for comparison with other results),
* calculated value from .
Table 2. The nonlinear refractive index coefﬁcients in the NIR and the THz frequency range.
n2,T Hz ,cm2/W n2, IR ,cm 2/W
in THz Range in NIR Range
ZnSe 1×10−13 3.8 ×10−14 
ZnTe 3×10−14 1.3 ×10−12 
CdTe 2×10−13 3.4 ×10−13 
GaP 1×10−14 6.5 ×10−14 
LiNbO3(a) 7×10−11 1.7 ×10−15 
LiNbO3(c) 5×10−11 1.31 ×10−15 
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 ,
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
parameter to be 1.2
/W. For lithium niobate crystal,
experimental measurements based on the change of the transmitted THz pulse shape [
] give the
value of 5.4
/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
value. However, it should be mentioned that these experimental
results were indirectly estimated.
Regarding the media parameters affecting the value of
, 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.
The nonlinear refractive index coefﬁcient dependence on the medium parameters for different
describes the reduced mass of the vibrational mode and (
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 [
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 [
]. 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 [
]. Regarding water as the nonlinear medium,
we obtained theoretically a value of
/W and experimentally
Photonics 2020,7, 98 5 of 7
/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
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
). Furthermore, water exhibits a 1000 times greater thermal
expansion coefﬁcient (0.2 ×10−3◦C−1), which can also add to the overall n2value.
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
in the range of 0.5–1 THz exceeds the corresponding value in the NIR range.
The relationship between media parameters and
is presented and analyzed in terms of vibration
related parameters. Our work [
] experimentally proves the validity of the theoretical approach
used and justiﬁes the conclusions regarding the medium parameters that determine the
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., [
], as well as
high-harmonic generation [
] in the THz range. The information provided in this article serves
as a platform for material development and study, i.e., 2D materials, metamaterials [
], and liquid
media , which have a potential to be used in nonlinear THz devices.
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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