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# Second-order artifact overwhelming the third-order signal in 2D Raman-THz
spectroscopy of non-centrosymmetric materials

Seyyed Jabbar Mousavi1,2,∗, Megan F. Biggs3, Jeremy A. Johnson3, Peter Hamm1
and Andrey Shalit1,∗ 1Department of Chemistry, University of Zurich,
Winterthurerstrasse 190, CH-8057 Zurich, Switzerland; email:
<EMAIL_ADDRESS>2Institute of Applied Physics, University of Bern,
Sidlerstrasse 5, 3012 Bern, Switzerland; email<EMAIL_ADDRESS>3Department of Chemistry and Biochemistry, Brigham Young University, Provo,
Utah 84602, USA

###### Abstract

Through comprehensive data analysis, we demonstrate that a second-order
artifact contributes significantly to the measured signal in 2D Raman-THz
spectroscopy of non-centrosymmetric materials. We confirm that by measuring
the 2D Raman-THz response of a beta barium borate (BBO) crystal, a well-known
${\chi}^{(2)}$ active sample. More importantly, we show that this strong
second-order artifact can be effectively suppressed by implementing a special
detection scheme. By doing so, we successfully isolate the desired third-order
nonlinear response. Our experimentally recovered 2D Raman-THz signal for an
x-cut BBO crystal reveals a distinct cross-peak feature, whose frequency
position suggests the presence of phonon-phonon coupling within this crystal.
Our study emphasizes the critical significance of considering second-order
nonlinear responses when conducting 2D THz experiments involving non-
centrosymmetric samples.

##  I Introduction

Two-dimensional (2D) THz spectroscopy intends to measure the third-order
nonlinear signal, giving useful insights into the dynamics and coupling of
low-frequency modes in the THz frequency range in condensed-phase molecular
systems.Kuehn et al. (2009); Woerner et al. (2013); Elsaesser et al. (2019);
Houver et al. (2019); Woerner et al. (2021) Recently, the application of this
technique was further expanded to explore the correlation and dynamics of
rotational degrees of freedom in gas-phase molecules, revealing a
multidimensional rotational spectroscopy.Fleischer et al. (2012); Lu et al.
(2016, 2019) In two-pulse 2D THz experiment on non-centrosymmetric materials,
${\chi}^{(2)}$ nonlinear susceptibility can also contribute to the nonlinear
signal.Reimann et al. (2021) Various second-order contributions, with
characteristic spectral features, originating from second harmonic generation
(SHG), and optical rectification (OR) processes have been previously reported
in 2D THz spectra of non-centrosymmetric materials.Woerner et al. (2013);
Elsaesser et al. (2019); Somma et al. (2014); Lu et al. (2017)

For example, Elsaesser and co-workers studied intersubband transitions in
GaAs/AlGaAs semiconductor quantum wells using two-pulse 2D THz
spectroscopy.Kuehn et al. (2011a, b) In addition to the third-order pump-probe
and photon echo signals, they also observed second-order contributions to the
2D THz spectrum. The observed ${\chi}^{(2)}$ features were assigned to
difference-frequency mixing in the bulk GaAs substrate. Nelson and co-workers
investigated nonlinear responses of magnons and their correlations in an
antiferromagnetic crystal, YFeO3. They also observed that the second-order
signals, including SHG and THz rectification (TR) peaks, originated from sum-
and difference-frequency mixing of the magnons, respectively, contribute to
the measured 2D THz spectra.Lu et al. (2017) In a very recent 2D THz
experiment on ZnTe, it has been shown that second-order contributions, such as
rectified nonlinearities and second harmonic signals arising from the electro-
optic sampling in the detection crystal, known as artifacts of the detection
scheme, can also contaminate 2D spectra collected in a collinear geometry.Liu
and Ankit (2024)

Hybrid spectroscopies with two complementary pulse sequences of 2D Raman-THz-
THz (2D RTT) Savolainen et al. (2013); Shalit et al. (2017); Ciardi et al.
(2019); Shalit et al. (2021); Mousavi et al. (2022) and 2D THz-THz-Raman (2D
TTR) Finneran et al. (2016, 2017); Mead et al. (2020); Knighton et al. (2019);
Lin et al. (2022) are alternative approaches for performing 2D experiments in
the THz frequency range. In this work, we present a new observation on how
second-order artifacts can dominate the third-order signal in 2D Raman-THz
experiments involving ${\chi}^{(2)}$ active samples. These artifacts are
effectively linear (i.e., 1D) responses and overwhelm the measured nonlinear
signal, potentially leading to misleading interpretations.

We will first present our measured 2D Raman-THz data for an x-cut BBO
nonlinear crystal. By analyzing the measured data, we confirm that the
observed features originated from the second-order contribution rather than
the third-order response of the sample. We also introduce the origin of this
contribution and explain why the differential chopping detection scheme is
insufficient to isolate the third-order signal in such cases. Finally, with
the goal of recovering the real 2D Raman-THz response of the BBO crystal, we
measure the x-cut BBO signal using a special detection scheme based on the
work of Johnson et al.Johnson et al. (2014) and Brunner et al.Brunner et al.
(2014). Our data provide experimental evidence suggesting the presence of
phonon-phonon coupling within the BBO crystal.

##  II  Methods

The experimental setup for 2D Raman-THz spectroscopy has been described in
detail before.Savolainen et al. (2013) Briefly, the output of an amplified
Ti:sapphire laser system, operating at a center wavelength of 800 nm, pulse
duration of 110 fs, and repetition rate of 5 kHz, is divided into Raman and
THz branches using a beamsplitter. The beam used for the THz branch is further
split into two beams: one for THz generation and the other for THz detection
(gated beam). All three laser beams have parallel polarizations, aligned
parallel to the crystallographic z-axis of BBO as shown in Fig. 1a. Broadband
THz pulses are generated via optical rectification in a 200 $\mu$m-thick (110)
GaP crystal. Before reaching the GaP crystal, the THz generation beam is
modulated using a mechanical chopper operating at a frequency of one-quarter
of the laser repetition rate (1.25 kHz). The generated THz pulses are first
focused onto the sample and then directed towards the detection crystal (a 200
$\mu$m-thick GaP crystal) using two custom-made aluminum elliptical mirrors.
The Raman pump pulses with energies up to 40 $\mu$J are first modulated at
half of the laser repetition rate (2.5 kHz) and subsequently focused onto the
sample. The temporal delays $t_{1}$, the delay between the Raman pump and
generated THz field, and $t_{2}$, the delay between THz generation and
detection, are controlled by two step-scan motors. 2D Raman-THz-THz (RTT)
pulse sequence is shown schematically in Fig. 1a. The emitted signal (red
dotted line shown in Fig. 1a) is focused onto the detection crystal and
measured using conventional electro-optic sampling (EOS) scheme (see Fig. 1b).
To prevent water vapor absorption, the entire THz section of the experimental
setup, including the THz generation and detection components, is enclosed
within a nitrogen-purged box.

Figure 1: a) Schematic of the 2D Raman-THz spectroscopy with RTT pulse
sequence. The polarization of the Raman and THz pulses is shown in the
circles. b) Electro-optic sampling (EOS) detection scheme. c) Angle of the
quarter-wave plate at $45^{\circ}$ in conventional EOS (left panel), and two
small opposite angles around zero in small bias detection scheme (right
panel).

We also employed an alternative experimental approach, known as the small bias
detection scheme,Johnson et al. (2014); Brunner et al. (2014); Krauspe et al.
(2020) to effectively isolate the very weak third-order nonlinear signal from
a strong second-order contribution caused by imperfect balancing in the 2D
Raman-THz experiment on the x-cut BBO crystal. As previously mentioned, in the
conventional EOS detection scheme (see Fig. 1b), the angle of the QWP is
typically set to $45^{\circ}$ (see Fig. 1c-left panel), converting the
linearly polarized probe beam into circular polarization which is subsequently
split by the Wollaston prism into two s and p polarization components with
equal intensity. In the small bias detection scheme, this angle is set to a
small angle close to $0^{\circ}$, which effectively reduces the optical bias
from the usual setting.Johnson et al. (2014) In this configuration, a
significant portion of the probe beam intensity, whose polarization is
parallel to the probe beam polarization ($I_{\parallel}$), is directed towards
one of the photodiodes. Balance detection can then be achieved by adjusting a
variable attenuator located in front of that photodiode, thereby attenuating
the main polarization component. The bias detection scheme has the same effect
on increasing the sensitivity of the THz detection as the Brewster windows
scheme that we had introduced previously,Ahmed et al. (2014) but as we will
see, introduces an additional tuning parameter that allows one to suppress the
second-order artifact.

##  III Results and Discussion

###  III.1 2D Raman-THz spectroscopy of x-cut BBO crystal

Figure 2a shows the 2D time domain response measured using the RTT pulse
sequence and detected by conventional EOS, for a 100 $\mu m$ thick x-cut BBO
crystal. As can be seen, a long-lived oscillatory feature along both t1 and t2
axes dominates the measured signal in the time domain. Figure 2b shows the
absolute value of the 2D Fourier transformation of the data in Fig. 2a. Three
main peaks with frequencies of 2.18 THz, 2.85 THz, and 3.5 THz are observed
along the f1 frequency axis. The observed features appear to climb up and
align along the diagonal line, accompanied by a splitting of these features
that follows the spectral shape of the instrument response function (IRF), as
discussed in Ref. Ciardi et al., 2019. It has been previously shown that the
convolution of molecular response with the IRF strongly affects the 2D Raman-
THz response in terms of spectral shape, intensity, and frequency
position.Ciardi et al. (2019) IRF has two nodal lines, one on the diagonal and
the other along the $f_{1}$ axis, suppressing diagonal peaks and those along
the $f_{1}$ axis with $f_{2}=0$. In the next section, we will prove that the
observed features in Fig. 2b are ${\chi}^{(2)}$ artifacts originating from
imperfect balancing in the conventional EOS detection scheme.

Figure 2: a) 2D time domain response measured using the RTT pulse sequence and
detected by conventional EOS, for the x-cut BBO crystal. b) Absolute value of
the 2D Fourier transformation for the data shown in panel (a).

###  III.2 THz signal at arbitrary quarter-wave plate angle

In the conventional electro-optic sampling detection scheme, where the
quarter-wave plate is oriented at $\theta=45^{\circ}$, the intensities of the
two orthogonal polarization components at the balanced photodiodes are:Lee
(2009)

$\displaystyle I_{\perp}=\frac{I_{0}}{2}\ (1+\sin(\phi))$ (1) $\displaystyle
I_{\parallel}=\frac{I_{0}}{2}\ (1-\sin(\phi))$ (2)

where $I_{0}$ is the probe beam intensity, and $\phi$ is the phase retardation
experienced by the probe beam due to the THz electric field. As a result, the
normalized EOS signal, which is obtained as the intensity difference between
two balanced photodiodes is:Lee (2009)

$\displaystyle I_{s}=\frac{I_{\perp}-I_{\parallel}}{I_{0}}=\sin(\phi).$ (3)

However, when a quarter-wave plate is oriented at an arbitrary angle $\theta$,
the intensities of the two orthogonal polarization components are given
byBrunner et al. (2014)

$\displaystyle\begin{split}I_{\perp}(\phi,\theta)&=\frac{I_{0}}{2}\times\\{1+\sin(2\theta)\sin(\phi)\\\
&\quad-\cos^{2}(2\theta)\cos(\phi)\\}\end{split}$ (4)
$\displaystyle\begin{split}I_{\parallel}(\phi,\theta)&=\frac{I_{0}}{2}\times\\{1-\sin(2\theta)\sin(\phi)\\\
&\quad+\cos^{2}(2\theta)\cos(\phi)\\}.\end{split}$ (5)

Here for simplicity, we neglect the small contributions from phase
retardations due to the strain-induced birefringence of the detection crystal
and the scattering background. To achieve balanced detection in the small bias
detection scheme, the main polarization component ($I_{\parallel}$) needs to
be attenuated by a factor $T\ll 1$:

$\displaystyle T=\frac{\sin^{2}(2\theta)}{1+\cos^{2}(2\theta)}$ (6)

in which case we have ($I_{\perp}=TI_{\parallel}$) when $\phi=0$. The
normalized balanced difference signal then is:

$\displaystyle\begin{split}I_{s}(\phi,\theta)&=\frac{I_{\perp}(\phi,\theta)-TI_{\parallel}(\phi,\theta)}{I_{\perp}(0,\theta)+TI_{\parallel}(0,\theta)}\\\
&=\frac{1}{\sin^{2}(2\theta)}\\{(1+T)(\sin(2\theta)\sin(\phi)\\\
&\quad-\cos^{2}(2\theta)\cos(\phi))-T+1\\}.\end{split}$ (7)

For assuming a small THz-induced phase retardation $(\phi\ll 1)$, we can
Taylor expand $\sin(\phi)$ and $\cos(\phi)$ up to second order, simplifying
equation 7:

$\displaystyle\begin{split}I_{s}(\phi,\theta)&=\frac{1}{\sin^{2}(2\theta)}\\{(1+T)(\sin(2\theta)\phi\\\
&\quad-\cos^{2}(2\theta)(1-\frac{\phi^{2}}{2}))-T+1\\}\end{split}$ (8)

In addition to balanced detection, we employ a differential chopping detection
scheme in order to isolate the nonlinear signal from other undesired signals
arising from individual pulses, such as scattering effects and optical
rectification. The implementation of the two-chopper scheme enables us to
acquire four distinct measurements, denoted as $I_{t,r}$, where the subscripts
$t$ and $r$ represent the states of the THz and Raman choppers, respectively
(1 or 0):

$\displaystyle I_{1,1}=THz_{On}Raman_{On}$ $\displaystyle
I_{1,0}=THz_{On}Raman_{Off}$ $\displaystyle I_{0,1}=THz_{Off}Raman_{On}$
$\displaystyle I_{0,0}=THz_{Off}Raman_{Off}$

The measured 2D Raman-THz signal is then calculated according to:

$\displaystyle S=\left((I_{1,1}-I_{1,0})-(I_{0,1}-I_{0,0})\right)$ (9)

Here, we first provide a mathematical demonstration of how second-order
contributions overwhelms the signal in non-centrosymmetric materials, and
subsequently validate it experimentally. Using equation 8 for the normalized
balanced difference signal in EOS, the four individual signals measured with
our two-chopper scheme can be written as:

$\displaystyle I_{0,0}(0,\theta)$
$\displaystyle=\frac{1}{\sin^{2}(2\theta)}\\{(1+T)(-\cos^{2}(2\theta))-T+1\\}$
(10)
$\displaystyle\begin{split}I_{1,0}(\phi_{THz},\theta)&=\frac{1}{\sin^{2}(2\theta)}\\{(1+T)(\sin(2\theta)\phi_{THz}\\\
&\quad-\cos^{2}(2\theta)(1-\frac{\phi_{THz}^{2}}{2}))-T+1\\}\end{split}$ (11)
$\displaystyle\begin{split}I_{0,1}(\phi_{Raman},\theta)&=\frac{1}{\sin^{2}(2\theta)}\\{(1+T)(\sin(2\theta)\phi_{Raman}\\\
&\quad-\cos^{2}(2\theta)(1-\frac{\phi_{Raman}^{2}}{2}))-T+1\\}\end{split}$
(12)
$\displaystyle\begin{split}I_{1,1}(\phi_{THzRaman},\theta)&=\frac{1}{\sin^{2}(2\theta)}\\{(1+T)(\sin(2\theta)\phi_{THzRaman}\\\
&\quad-\cos^{2}(2\theta)(1-\frac{\phi_{THzRaman}^{2}}{2}))\\\
&\quad-T+1\\}\end{split}$ (13)

where $\phi_{THz}$ and $\phi_{Raman}$ are phase retardations due to the THz
excitation pulse transmitted through the sample, and the Raman-induced
${\chi}^{(2)}$ process in the sample (BBO crystal), respectively. The term
$\phi_{THzRaman}$ represents the sum of the phase retardations induced by THz
($\phi_{THz}$) and Raman ($\phi_{THz}$) pulses, along with the contribution
from the desired third-order nonlinear signal $\phi_{3rdNL}$ (${\chi}^{(3)}$
response):

$\displaystyle\phi_{THzRaman}=\phi_{THz}+\phi_{Raman}+\phi_{3rdNL}$ (14)

Using the above definitions for the individual signals, the measured 2D Raman-
THz signal can be calculated as follows:

$\displaystyle\begin{split}S&=\frac{1}{\sin^{2}(2\theta)}\\{(1+T)(\sin(2\theta)\phi_{3rdNL}\\\
&\quad+\cos^{2}(2\theta)(\phi_{THz}\phi_{Raman}+\phi_{THz}\phi_{3rdNL}\\\
&\quad+\phi_{Raman}\phi_{3rdNL}+\frac{(\phi_{3rdNL})^{2}}{2})\\}\end{split}$
(15)

Theoretically, when the angle of the quarter-wave plate is set to
$45^{\circ}$, the $\cos^{2}(2\theta)$-terms become zero and the signal should
be:

$\displaystyle S=\frac{1+T}{\sin(2\theta)}\phi_{3rdNL}$ (16)

However, in practice, there is always some leakage due to the imperfections in
the probe beam polarization and the polarizing optics. As we show in this
paper, this issue becomes particularly crucial when trying to measure the
extremely weak ${\chi}^{(3)}$ signals of non-centrosymmetric materials
(${\chi}^{(2)}$ active samples). When the Raman-induced THz field
$\phi_{Raman}$ is present, the $\phi_{THz}\phi_{Raman}$ term (called as
product field) is much larger than any of other terms in the equation 15.
Therefore, signal $S$ is dominated by the product field:

$\displaystyle S\propto\phi_{THz}\phi_{Raman}$ (17)

regardless of the angle $\theta$ of the quarter-wave plate, making it
impossible for the conventional EOS detection scheme to isolate the weak
${\chi}^{(3)}$ signal from the strong ${\chi}^{(2)}$ artifact.

Figure 3: 2D plots in the time domain for a) the THz field transmitted through
the 100 $\mu m$ thick x-cut BBO crystal, denoted as $\phi_{THz}$, and b)
Raman-induced THz field, labeled as $\phi_{Raman}$. The inset in Fig. 3b shows
the Fourier transformation of 1D cut along t1 axis for $\phi_{Raman}$.
Multiplication of the transmitted THz field through the 100 $\mu m$ thick
x-cut BBO crystal and Raman-induced THz field, called as product field
($\phi_{THz}\phi_{Raman}$), in c) the time and d) frequency domains.

Figures 3a and 3b displays the 2D plots in the time domain for the THz field
transmitted through a 100 $\mu m$ thick x-cut BBO crystal, $\phi_{THz}$, and
Raman-induced THz field, $\phi_{Raman}$. Both are plotted in a 2D
representation, with the delay times as they are given by the experimental
configuration, but they are 1D signals. The inset in Fig. 3b shows the Fourier
transformation of 1D cut along t1 axis for the data shown in Fig. 3b. This
figure demonstrates that when the Raman pulse hits the BBO sample, it
generates a THz field, called as Raman-induced THz field, $\phi_{Raman}$, with
three distinct frequency components at 2.18 THz, 2.85 THz, and 3.5 THz. In
order to experimentally validate the theory discussed above, we multiply the
transmitted THz field through the BBO with that induced by Raman pulse,
$\phi_{THz}\phi_{Raman}$, as shown in Fig. 3c. Fig. 3d presents the absolute
value of the 2D Fourier transformation of the data in Fig. 3c. By comparing
Fig. 2b and Fig. 3d, one can clearly see that the measured signal of the BBO
crystal using conventional EOS is dominated by the product field
$\phi_{THz}\phi_{Raman}$ term, revealing a close agreement between the two
signals in both time and frequency domains. Thus, this comparison provides
conclusive evidence that the measured data for BBO is not a real
${\chi}^{(3)}$ signal; rather, it is a ${\chi}^{(2)}$ artifact due to
imperfect balancing. It is worth noting that the same signal remains even by
minimizing its size when trying to adjust $\theta$ as close as possible to
$45^{\circ}$. Therefore, in 2D Raman-THz experiments involving non-
centrosymmetric materials like BBO, where ${\chi}^{(2)}$ is nonzero, relying
solely on the differential chopping scheme is insufficient to recover the
desired third-order nonlinear signal.

###  III.3 Measurements at $\pm\theta$ around $\theta=0^{\circ}$

In the previous section, we demonstrated both theoretically and experimentally
that the measured signal in 2D Raman-THz experiments involving non-
centrosymmetric samples is overwhelmingly dominated by second-order artifact.
Here, to recover the extremely weak 3rd-order 2D Raman-THz response of the BBO
crystal, we use the small bias detection schemeJohnson et al. (2014); Brunner
et al. (2014); Krauspe et al. (2020). In this scheme, the balanced difference
signal is measured at opposite optical biases $\pm\theta$ where $\theta$ is a
small angle of the quarter-wave plate close to $0^{\circ}$, as shown in Fig.
1c, right panel. A difference signal was calculated by subtracting these two
consecutive measurements at each delay time. In that difference signal, the
sign of $\sin(2\theta)$ coefficient flips, while the sign of
$\cos^{2}(2\theta)$ coefficient remains unchanged (see equation 15). This is
not the case for two measurements taken around $45^{\circ}$ in the
conventional EOS detection scheme. The real third-order nonlinear signal
$S^{(3)}$ can thus be successfully recovered:

$\displaystyle
S^{(3)}=S(\phi,\theta)-S(\phi,-\theta)=\frac{2(1+T)}{\sin(2\theta)}\phi_{3rdNL}.$
(18)

Figure 4a presents the 2D difference signal obtained by subtracting the time
domain 2D signals measured at positive and negative optical biases,
$\theta$=±$2^{\circ}$. Figure 4b compares an 1D cut along t1 axis at
t${}_{2}=0$ for the 2D difference signal (dotted blue line), with the signal
that is dominated by the product field $\phi_{THz}\phi_{Raman}$ measured at
positive optical bias (in red). It is important to note that these two data
sets were extracted from the same measurement, enabling a meaningful
comparison. The first observation is that the 2D difference signal
(${\chi}^{(3)}$ signal) is significantly weaker, by a factor of 20, compared
to the ${\chi}^{(2)}$ artifact measured at opposite optical biases (see Fig.
4b).

Figure 4: a) Time-domain 2D Raman-THz data (2D difference signal) for the 100
$\mu m$ thick x-cut BBO crystal obtained through the subtraction of the
measured 2D signals at opposite optical biases. b) Comparison of 1D cuts along
t1 axis for the ${\chi}^{(2)}$ artifact measured at a positive optical bias
(in red), and the ${\chi}^{(3)}$ signal (in blue), which is obtained through
the subtraction of the measured signals at opposite optical biases. c) Fourier
transformation of the 1D time domain cuts shown in panel (b), which highlights
a distinct $\frac{\pi}{2}$ phase shift between the ${\chi}^{(3)}$ difference
signal and the ${\chi}^{(2)}$ artifact.

More importantly, this clearly shows that the difference signal is shifted in
phase compared to the ${\chi}^{(2)}$ artifact measured at positive optical
bias. It is worth mentioning that imperfection in subtraction can never lead
to a phase shift. To see the observed phase shift better we show in Fig. 4c
the real part of the Fourier transformation of the 1D time domain cuts. A
$\frac{\pi}{2}$ phase shift is clearly seen in this figure. The observed
$\frac{\pi}{2}$ phase shift between the two time-domain signals indicates that
they are generated through nonlinearities of different parity namely
${\chi}^{(2)}$ vs ${\chi}^{(3)}$, because each successive order of nonlinear
interaction introduces a $\frac{\pi}{2}$ phase shift due to the complex nature
of the nonlinear polarization.Boyd (2008)

The 2D Fourier transformation of the time domain data (Fig. 4a) is shown in
Fig. 5b. The resulting 2D difference spectrum looks significantly different
from the artifact 2D data shown in Fig. 2b. Specifically, it reveals a single
round peak that is sharper and lacks any above-diagonal features. In order to
assign this single peak, we measured both polarized Raman and THz transmission
spectra of x-cut BBO, which are shown in Fig. 5a and Fig. 5c, respectively.

Figure 5: a) Polarized Raman spectrum of x-cut BBO. b) Absolute value 2D
spectrum of the 2D difference signal, ${\chi}^{(3)}$ signal, shown in Fig. 4a.
c) THz transmission spectrum of x-cut BBO (sample oriented at $60^{\circ}$
with respect to the polarization direction of the incident THz beam, since
otherwise the absorbance of the THz modes would be too strong).

As can be seen from Fig. 5a, the polarized Raman spectrum of BBO reveals one
strong peak at around 3.7 THz, which corresponds to the $A_{1}$ phonon mode of
the BBO crystal and has been previously reported in the literatureNey et al.
(1998). The THz transmission spectrum of BBO in Fig. 5c exhibits three
distinct absorption features, a band at 2.15 THz which is assigned to $A_{1}$
mode, and two bands at 2.50 THz, and 2.85 THz which correspond to closely
placed $E$ and $A_{1}$ modes, in good agreement with the literature.Liu et al.
(2008); Liu and Zhang (2009); Ney et al. (1998) The position of the peak along
the f1 frequency axis (f1$\approx$ 3.65 THz) in the 2D spectrum (Fig. 5b),
coincides with $A_{1}$ phonon mode observed in the polarized Raman spectrum.
The peak position along the f2 frequency axis (f2$\approx$ 2.3 THz) falls
close to the position of the $A_{1}$ phonon mode observed at 2.15 THz in the
THz transmission spectrum. Considering that the $A_{1}$ mode excited by the
Raman pulse can only be coupled to the modes with the same symmetry this 2D
spectrum suggests the presence of an anharmonic coupling between these two
$A_{1}$ phonon modes within the BBO crystal. The small difference in the peak
position along f2 axis and the frequency of the $A_{1}$ phonon mode observed
in the THz transmission spectrum is due to the fact that the convolution of
the sample response with the IRF can shift the peak position along f2 towards
higher frequencies, as discussed in our previous work.Mousavi et al. (2022)

##  IV Conclusion

In conclusion, we have presented the 2D Raman-THz response of the x-cut BBO
crystal, revealing a long-lived oscillatory feature along t1 axis in the time-
domain data. Our comprehensive data analysis demonstrated that the observed
signal is dominated by a significant contribution from a second-order artifact
due to imperfect balancing. Ignoring these second-order contributions can lead
to misleading interpretations of the measured data. We showed both
theoretically and experimentally that the third-order nonlinear response of
the x-cut BBO crystal can be successfully recovered, effectively suppressing
the strong artifact induced by the ${\chi}^{(2)}$ effect, through the
application of the small bias detection schemeBrunner et al. (2014); Johnson
et al. (2014); Krauspe et al. (2020). By implementing a bias detection scheme,
we measured the signal at two small opposite optical biases around the zero
angle of the quarter-wave plate. The resulting 2D difference signal, obtained
by subtracting these two consecutive measurements, revealed a single round
cross peak feature at f${}_{1}\approx$ 3.65 THz and f${}_{2}\approx$ 2.3 THz.
These frequencies are in agreement with the frequency of two $A_{1}$ phonon
modes of the BBO crystal, which are observed in the Raman and THz transmission
spectra, respectively. Moreover, the observed $\frac{\pi}{2}$ phase shift
between the 2D difference signal and the 2D signal measured at positive
optical bias provides strong evidence that the 2D difference is indeed a
${\chi}^{(3)}$ signal. Therefore, our experimental findings suggest the
presence of phonon-phonon coupling within the BBO crystal. Overall, the
experimental approach presented here in signal recovery proves to be effective
in disentangling the true third-order nonlinear response from the dominant
artifact contribution arising from the ${\chi}^{(2)}$ process. This systematic
approach can be applied to any other 2D THz spectroscopy experiments involving
non-centrosymmetric samples, ensuring accurate data analysis and
interpretation. This result paves the way for further expanding the
applicability of 2D Raman-THz spectroscopy, thus facilitating the exploration
of anharmonic phonon couplings in nonlinear crystals.

Acknowledgement: The work has been supported by the Swiss National Science
Foundation (SNF) through the National Center of Competence and Research (NCCR)
MUST as well by the MaxWater network of the Max Planck Society.

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