# Cross-phase modulation in the two dimensional spectroscopy

Mao-Rui Cai  Graduate School of China Academy of Engineering Physics, No. 10
Xibeiwang East Road, Haidian District, Beijing 100193, China  Xue Zhang
Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East
Road, Haidian District, Beijing 100193, China  Zi-Qian Cheng  Graduate School
of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian
District, Beijing 100193, China  Teng-Fei Yan<EMAIL_ADDRESS>School of
Microelectronics, shanghai university, Shanghai 200444, China  Hui Dong
<EMAIL_ADDRESS>Graduate School of China Academy of Engineering Physics,
No. 10 Xibeiwang East Road, Haidian District, Beijing 100193, China

###### Abstract

Developing from the transient absorption (TA) spectroscopy, the two
dimensional (2D) spectroscopy with pump-probe geometry has emerged as a
versatile approach for alleviating the difficulty on implementing the 2D
spectroscopy with other geometries. However, the presence of cross-phase
modulation (XPM) in TA spectroscopy introduces significant spectral
distortions, particularly when the pump and probe pulses overlap. We
demonstrate that this phenomenon is extended to the 2D spectroscopy with pump-
probe geometry and the XPM is induced by the interference of the two pump
pulse. We present the oscillatory behavior of XPM in the 2D spectrum and its
displacement with respect to the waiting time delay through both experimental
measurements and numerical simulations. Additionally, we explore the influence
of probe pulse chirp on XPM and discover that by compressing the chirp, the
impact of XPM on the desired signal can be reduced.

##  I Introduction

Two dimensional (2D) spectroscopy [1, 2, 3, 4] is a powerful tool with growing
attention for studying the couplings and dynamics of systems in both condensed
[5, 6, 7, 8] and gaseous phases [9, 10, 11, 12]. Among the various 2D
spectroscopy beam geometries [13, 7, 8], the pump-probe geometry [13, 14, 15]
developed from the transient absorption (TA) spectroscopy [16, 17, 18, 19, 20]
has emerged as a widely adopted approach due to its ease of implementation. In
this technique, the pump beam is typically modulated by a pulse shaper [15,
14, 21] which is utilized to generate two identical pump pulses with interval
$\tau$ and to compensate for the chirp of the pump pulses [22]. As for the
probe arm, the chirp of the probe pulse is often compressed with an extra
compressor [23] or even left uncompressed, especially when the super-continuum
is applied for probe [15, 24]. Similar to TA spectroscopy, the chirp of the
probe pulse would introduce distortions [25, 19, 26] to the spectrum in the 2D
case [5, 27, 28]. This occurs because various frequency components of the
probe pulse reach the investigated sample at different times [29]. Taking
inspiration from the chirp correction scheme used in TA spectroscopy [25, 19,
26], it has been proposed that these distortions in the 2D spectroscopy could
be post-corrected [27, 28] by characterizing the chirp of the probe pulse.

Despite the distortion caused by the chirp of the probe pulse, there is
another significant artifact induced by the intense pump pulse in the TA
spectroscopy known as the cross-phase modulation (XPM) [29, 30, 31, 32, 33].
XPM universally exists in the TA spectroscopy when the pump and probe pulses
overlap temporally at the sample, particularly in the system of condensed
phase. Moreover, it get stronger as the chirp of the probe pulse increases
[29]. As an extension of the TA spectroscopy, the 2D spectroscopy with pump-
probe geometry also gets affected by the XPM [34] especially when dynamics
within several tens of femtoseconds is investigated. However, a comprehensive
theoretical understanding of the XPM in the 2D spectroscopy with pump-probe
geometry and its response to variations in the chirp of the probe pulse remain
elusive.

In this paper, we present a theoretical derivation of the XPM in the 2D
spectroscopy with pump-probe geometry. Our results demonstrate that XPM
manifests itself at the central frequency of the pump pulse on the excitation
axis and at the central frequency of the probe pulse on the detection axis.
Additionally, we observe oscillations and shifts of the XPM along the
detection axis, which are supported by experimental data using a sapphire
window as the sample and by our numerical simulations. Furthermore, we
investigate the effect of probe pulse chirp on the behavior of XPM through the
experiments and simulations. We demonstrated that by reducing the chirp of the
probe pulse, the influence of XPM on the desired signal is mitigated.

##  II theory

###  II.1 General two dimensional spectroscopy

In typical two dimensional spectroscopy experiments, the third-order optical
response [1, 2] of a sample is represented by the induced third-order
polarization as

$P^{(3)}\left(t\right)=\int_{0}^{\infty}dt_{3}\int_{0}^{\infty}dt_{2}\int_{0}^{\infty}dt_{1}R^{(3)}\left(t_{1},t_{2},t_{3}\right)E\left(t-t_{3}\right)E\left(t-t_{3}-t_{2}\right)E\left(t-t_{3}-t_{2}-t_{1}\right).$
(1)

where $R^{(3)}\left(t_{1},t_{2},t_{3}\right)$ is the third-order response
function of the sample and $E\left(t\right)$ is the external electric field.
As Fig. 1 shows, the external field typically consists of three pulses
arriving sequentially at time $t=0$, $t=\tau$, and $t=\tau+T$ [21, 35, 5, 36]
for 2D setup. As a result, both the external electric field and the induced
third-order polarization are functions of $\tau$, $T$ and $t$. Specifically,
we denote the induced third-order polarization as
$P^{(3)}\left(\tau,T,t\right)$, and the electric field as

$E\left(\tau,T,t\right)=E_{1}\left(t\right)+E_{2}\left(\tau,t\right)+E_{3}\left(\tau,T,t\right)+\mathrm{c.c.},$
(2)

$\displaystyle
E_{1}\left(t\right)=A_{1}\left(t\right)\exp\left\\{i\left(\vec{k}_{1}\cdot\vec{r}-\Omega_{1}t+\phi_{1}\right)\right\\},$
(3) $\displaystyle
E_{2}\left(\tau,t\right)=A_{2}\left(t-\tau\right)\exp\left\\{i\left[\vec{k}_{2}\cdot\vec{r}-\Omega_{2}\left(t-\tau\right)+\phi_{2}\right]\right\\},$
(4) $\displaystyle
E_{3}\left(\tau,T,t\right)=A_{3}\left(t-\tau-T\right)\exp\left\\{i\left[\vec{k}_{3}\cdot\vec{r}-\Omega_{3}\left(t-\tau-T\right)+\phi_{3}\right]\right\\}.$
(5)

where $A_{m}$, $\vec{k}_{m}$, $\Omega_{m}$, and $\phi_{m}$ $(m=1,2,3)$ are
respectively the envelopes, wave vectors, central frequencies, and initial
phases of the pulses, and $\vec{r}$ is the spatial location of the sample
molecule. The possible dispersion of the pulses is encoded into the envelopes.
For a transparent sample with length $L$, the induced polarization yields
signal [1] field

$E_{s}\left(\tau,T,t\right)=\frac{2\pi
i}{n}\frac{\Omega_{s}L}{c}P^{(3)}\left(\tau,T,t\right)=i\eta\Omega_{s}P^{(3)}\left(\tau,T,t\right),$
(6)

along several phase matching directions
$\vec{k}_{s}=\pm\vec{k}_{\alpha}\pm\vec{k}_{\beta}\pm\vec{k}_{\gamma}$
($\alpha,\beta,\gamma=1,2,3$), where $\eta=2\pi L/cn$, $n$ is the linear
refractive index of the sample, $c$ is the speed of light, and $\Omega_{s}$ is
the central frequency of the signal field.

In general, the two dimensional (2D) spectrum
$E_{s}\left(\omega_{\tau},T,\omega_{t}\right)$ is obtained by Fourier
transforming $E_{s}\left(\tau,T,t\right)$ with respect to $\tau$ and $t$. In
some cases, the time domain signal $E_{s}\left(\tau,T,t\right)$ is collected
by scanning both $\tau$ and $t$ [9, 10, 7, 8]. Alternatively, it is popular to
heterodynely detect [1, 35, 5] the frequency domain signal
$E_{s}\left(\tau,T,\omega_{t}\right)$ via a spectrometer and with a local
oscillator $E_{\mathrm{LO}}\left(t\right)$. In this technique, the
differential optical signal $S\left(\tau,T,\omega_{t}\right)$ is obtained as

$S\left(\tau,T,\omega_{t}\right)=\ln\frac{I_{s}\left(\tau,T,\omega_{t}\right)}{I_{\mathrm{LO}}\left(\omega_{t}\right)}\simeq\frac{I_{s}\left(\tau,T,\omega_{t}\right)}{I_{\mathrm{LO}}\left(\omega_{t}\right)}-1\simeq\frac{2\mathrm{Re}\left[E_{s}\left(\tau,T,\omega_{t}\right)E_{\mathrm{LO}}^{*}\left(\omega_{t}\right)\right]}{\left|E_{\mathrm{LO}}\left(\omega_{t}\right)\right|^{2}}=2\mathrm{Re}\left[\frac{E_{s}\left(\tau,T,\omega_{t}\right)}{E_{\mathrm{LO}}\left(\omega_{t}\right)}\right],$
(7)

where
$I_{\mathrm{LO}}\left(\omega_{t}\right)=\left|E_{\mathrm{LO}}\left(\omega_{t}\right)\right|^{2}$
and
$I_{s}\left(\tau,T,\omega_{t}\right)=\left|E_{s}\left(\tau,T,\omega_{t}\right)+E_{\mathrm{LO}}\left(\omega_{t}\right)\right|^{2}\simeq
2\mathrm{Re}\left[E_{s}\left(\tau,T,\omega_{t}\right)E_{\mathrm{LO}}^{*}\left(\omega_{t}\right)\right]+\left|E_{\mathrm{LO}}\left(\omega_{t}\right)\right|^{2}$,
because the signal field is usually much weaker than the local oscillator,
i.e.,
$\left|E_{s}\left(\tau,T,\omega_{t}\right)\right|\ll\left|E_{\mathrm{LO}}\left(\omega_{t}\right)\right|$.
The 2D spectrum $S\left(\omega_{\tau},T,\omega_{t}\right)$ is then obtained by
scanning and Fourier transforming with respect to only $\tau$,

$S\left(\omega_{\tau},T,\omega_{t}\right)=\mathcal{F}\left[S\left(\tau,T,\omega_{t}\right)\right].$
(8)

###  II.2 Two dimensional spectroscopy with pump-probe geometry

There are several experimental beam geometries in the 2D spectroscopy based on
the direction arrangement of the three pulses, such as collinear geometry [10,
7, 8] and non-collinear BOXCARS geometry [5, 35]. Besides, there is a
partially collinear geometry known as the pump-probe geometry [13, 14, 15]
where the first two pulses are collinear $\vec{k}_{1}=\vec{k}_{2}$ and the
third pulse serves also as the local oscillator. Such a beam geometry is
identical with the TA (pump-probe) spectroscopy geometry [16, 17, 18, 19, 20].
The first two pulses are often denoted as the pump pulses and the third pulse
as the probe pulse. The pump-probe geometry eases the difficulty of
experimentally implementing the 2D spectroscopy compared with the other
geometries, because the heterodyne detection is naturally satisfied by
measuring the spectrum of probe pulse, i.e., $\vec{k}_{s}=\vec{k}_{3}$.
However, signal fields that are spatially separated in the BOXCARS geometry
may get mixed [14, 23] in the pump-probe geometry. Specifically, signal fields
$E_{\alpha^{\prime},-\alpha^{\prime},3}\left(\tau,T,\omega_{t}\right)$,
$E_{1,-2,3}\left(\tau,T,\omega_{t}\right)$, and
$E_{-1,2,3}\left(\tau,T,\omega_{t}\right)$ are all emit along $\vec{k}_{3}$ in
the pump-probe geometry. Here, $\alpha^{\prime}=\pm 1,\pm 2,\pm 3$ and the
subscripts of the signal field $E_{\alpha,\beta,\gamma}$ denote that the
signal is induced by incident fields $E_{\alpha}$, $E_{\beta}$, and
$E_{\gamma}$ (or their conjugation if the corresponding subscripts are
negative). Among all these mixed signals,
$E_{-1,2,3}\left(\tau,T,\omega_{t}\right)$ and
$E_{1,-2,3}\left(\tau,T,\omega_{t}\right)$ are typically required for the
study of single-quantum coherence in the excitation time $\tau$. These two
signals are often denoted as the rephasing and non-rephasing signal [1, 5],
respectively.

Figure 1: Typical pulse sequence of the two dimensional spectroscopy, where
three pulses arrive at the sample at time $t=0$, $t=\tau$, and $t=\tau+T$
sequentially.

To sort out only $E_{-1,2,3}\left(\tau,T,\omega_{t}\right)$ and
$E_{1,-2,3}\left(\tau,T,\omega_{t}\right)$ in the total signal field, the
technique of phase cycling [14, 23, 11] is applied. According to the
discussion in Sec. II.1, the signal field
$E_{s}\left(\tau,T,\omega_{t}\right)$ from the sample depends on the initial
phase $\phi_{\alpha}$ of the incident fields. However, in the 2D spectroscopy
with pump-probe geometry, the phase of the third pulse $\phi_{3}$ is
eliminated because the third pulse serves as the local oscillator in the
heterodyne detection [14]. Thus, we denote the phase-depend signal field as
$E_{s}\left(\tau,T,\omega_{t};\phi_{1},\phi_{2}\right)$ and its components

$E_{\alpha,\beta,3}\left(\tau,T,\omega_{t};\phi_{1},\phi_{2}\right)=E_{\alpha,\beta,3}\left(\tau,T,\omega_{t}\right)\exp\left\\{i\left[\mathrm{sgn}\left(\alpha\right)\times\phi_{\left|\alpha\right|}+\mathrm{sgn}\left(\beta\right)\times\phi_{\left|\beta\right|}\right]\right\\},$
(9)

where $\alpha,\beta=\pm 1,\pm 2,\pm 3$,
$\mathrm{sgn}\left(x\right)=x/\left|x\right|$ is the sign function, and the
phase free term $E_{\alpha,\beta,3}\left(\tau,T,\omega_{t}\right)$ denotes the
signal field with initial phases $\phi_{1},\phi_{2}=0$ of the first two
pulses. By conducting two experiments with phase arrangements a)
$\phi_{1}=\pi$ and $\phi_{2}=0$; b) $\phi_{1}=\pi$ and $\phi_{2}=\pi$, the
phase cycled signal is given as

$S_{\mathrm{cyc}}\left(\tau,T,\omega_{t}\right)=\ln\frac{I\left(\tau,T,\omega_{t};\pi,0\right)}{I\left(\tau,T,\omega_{t};\pi,\pi\right)}\simeq\frac{I\left(\tau,T,\omega_{t};\pi,0\right)}{I\left(\tau,T,\omega_{t};\pi,\pi\right)}-1\simeq\frac{I\left(\tau,T,\omega_{t};\pi,0\right)-I\left(\tau,T,\omega_{t};\pi,\pi\right)}{I\left(\tau,T,\omega_{t};\pi,\pi\right)},$
(10)

with

$I\left(\tau,T,\omega_{t};\phi_{1},\phi_{2}\right)=\left|E_{s}\left(\tau,T,\omega_{t};\phi_{1},\phi_{2}\right)+E_{3}(\omega_{t})\right|^{2}\simeq\left|E_{3}(\omega_{t})\right|^{2}+2\mathrm{Re}\left[E_{s}\left(\tau,T,\omega_{t};\phi_{1},\phi_{2}\right)E_{3}^{*}(\omega_{t})\right],$
(11)

$I_{3}\left(\omega_{t}\right)=\left|E_{3}(\omega_{t})\right|^{2}$, and
$\left|E_{s}\left(\tau,T,\omega_{t};\phi_{1},\phi_{2}\right)\right|\ll\left|E_{3}(\omega_{t})\right|$.
Then, the phase cycled signal is expressed as

$S_{\mathrm{cyc}}\left(\tau,T,\omega_{t}\right)=2\mathrm{Re}\left[\frac{E_{s}\left(\tau,T,\omega_{t};\pi,0\right)-E_{s}\left(\tau,T,\omega_{t};\pi,\pi\right)}{E_{3}\left(\omega_{t}\right)}\right],$
(12)

which includes only the signals $E_{-1,2,3}$ and $E_{1,-2,3}$, while the parts
$E_{\alpha^{\prime},-\alpha^{\prime},3}$ are eliminated because they are not
affected by the initial phases of the pump pulses. The 2D spectrum
$S_{\mathrm{cyc}}\left(\omega_{\tau},T,\omega_{t}\right)$ is obtained by
firstly scanning $\tau$ to collect a series of
$S_{\mathrm{cyc}}\left(\tau,T,\omega_{t}\right)$ and then Fourier transforming
$S_{\mathrm{cyc}}\left(\tau,T,\omega_{t}\right)$ with respect to $\tau$. The
rephasing and non-rephasing parts can further be separated with additional
phase cycling [23].

###  II.3 Cross-phase modulation in the two dimensional spectroscopy with
pump-probe geometry

In most spectroscopy experiments, the frequencies of the incident electric
fields are typically chosen to resonate with (or nearly resonate with) one of
the transition frequencies of the investigated sample, thereby changing the
absorption rate [1, 2] (imaginary part of the electric susceptibility) of the
sample accordingly. However, in liquid sample where the solution (i.e., the
investigated sample) is dissolved in a specific solvent, the electric fields
also induce the non-resonant electronic response (ER) of the solvent, altering
the refractive index [37, 34] (real part of the electric susceptibility) of
the solvent through Kerr effect. The ER is approximately instantaneous [37,
38] (with a time scale of $0.1\sim 1\>\mathrm{fs}$) compared with the resonant
response of the solution and the duration of the incident pulses (with a time
scale of $10\sim 100\>\mathrm{fs}$). Therefore, the response function [37, 29,
34] is approximately written as

$R_{\mathrm{ER}}^{(3)}\left(t_{1},t_{2},t_{3}\right)=\sigma_{e}\delta\left(t_{1}\right)\delta\left(t_{2}\right)\delta\left(t_{3}\right).$
(13)

Substituting Eq. (13) into Eq. (1), the induced polarization of the electronic
response is given as
$P^{\mathrm{ER}}\left(\tau,T,t\right)=\sigma_{e}\left[E\left(\tau,T,t\right)\right]^{3}$,
and the corresponding signal field is

$E_{s}^{\mathrm{ER}}\left(\tau,T,t\right)=i\sigma_{e}\eta\Omega_{s}\left[E\left(\tau,T,t\right)\right]^{3}.$
(14)

Among all the heterodynely detected signal components in the 2D spectroscopy
with pump-probe geometry, $E_{3,-3,3}^{\mathrm{ER}}\left(\tau,T,t\right)$ and
$E_{-3,3,3}^{\mathrm{ER}}\left(\tau,T,t\right)$ are treated as the self-phase
modulation (SPM) [39] which describes the spectral and temporal modulation of
the probe pulse induced by itself. The other components are cross-phase
modulation (XPM) [29, 30, 31, 32, 33], describing the spectral and temporal
modulation of the probe pulse induced by the first pump pulse only
[$E_{1,-1,3}^{\mathrm{ER}}\left(\tau,T,t\right)$ and
$E_{-1,1,3}^{\mathrm{ER}}\left(\tau,T,t\right)$], the second pump pulse only
[$E_{2,-2,3}^{\mathrm{ER}}\left(\tau,T,t\right)$ and
$E_{-2,2,3}^{\mathrm{ER}}\left(\tau,T,t\right)$], or the interference of the
two pump pulses [$E_{-1,2,3}^{\mathrm{ER}}\left(\tau,T,t\right)$ and
$E_{1,-2,3}^{\mathrm{ER}}\left(\tau,T,t\right)$] where

$\displaystyle E_{-1,2,3}^{\mathrm{ER}}\left(\tau,T,t\right)$
$\displaystyle=i\sigma_{e}\eta\Omega_{s}E_{1}^{*}\left(t\right)E_{2}\left(t-\tau\right)E_{3}\left(t-\tau-T\right),$
(15) $\displaystyle E_{1,-2,3}^{\mathrm{ER}}\left(\tau,T,t\right)$
$\displaystyle=i\sigma_{e}\eta\Omega_{s}E_{1}\left(t\right)E_{2}^{*}\left(t-\tau\right)E_{3}\left(t-\tau-T\right).$
(16)

Via the phase cycling technique [14, 23, 11] described in Sec. II.2, all the
modulations of the probe pulse induced by a single pulse
$E_{\alpha^{\prime},-\alpha^{\prime},3}^{\mathrm{ER}}$ are eliminated.
Whereas, the modulations from the interference of the two pump pulses remain,

$S_{\mathrm{XPM}}\left(\tau,T,\omega_{t}\right)=-4\mathrm{Re}\left[\frac{E_{-1,2,3}^{\mathrm{ER}}\left(\tau,T,\omega_{t}\right)+E_{1,-2,3}^{\mathrm{ER}}\left(\tau,T,\omega_{t}\right)}{E_{3}\left(\omega_{t}\right)}\right],$
(17)

where $E_{-1,2,3}^{\mathrm{ER}}\left(\tau,T,\omega_{t}\right)$ and
$E_{1,-2,3}^{\mathrm{ER}}\left(\tau,T,\omega_{t}\right)$ are XPM signal fields
when the initial phases of the two pump pulses $\phi_{1},\phi_{2}=0$.

To give more explicit expressions of the XPM signal
$S_{\mathrm{XPM}}\left(\tau,T,\omega_{t}\right)$, we assume that the two pump
pulses are Fourier transform limited Gaussian pulses without chirp, i.e.,

$\displaystyle
E_{1}\left(t\right)=a_{1}\exp\left\\{-\frac{t^{2}}{2\tau_{1}^{2}}\right\\}\exp\left\\{-i\Omega_{1}t\right\\},$
(18) $\displaystyle
E_{2}\left(\tau,t\right)=a_{2}\exp\left\\{-\frac{\left(t-\tau\right)^{2}}{2\tau_{2}^{2}}\right\\}\exp\left\\{-i\Omega_{2}\left(t-\tau\right)\right\\},$
(19)

and the probe pulse is a Gaussian pulse with linear chirp [29, 28]

$E_{3}\left(\tau,T,t\right)=a_{3}\exp\left\\{-\frac{\left(t-\tau-T\right)^{2}}{2\tau_{3}^{2}}\right\\}\exp\left\\{-i\left[\Omega_{3}\left(t-\tau-T\right)+\beta_{3}\left(t-\tau-T\right)^{2}\right]\right\\},$
(20)

where $a_{\alpha}$ and $\tau_{\alpha}$ are the amplitude and duration of pulse
$E_{\alpha}$, and $\beta_{3}$ is the chirp rate of the probe pulse. The
spatial phase factors and initial phases are neglected here. We further
simplify our model by assuming $a_{1}=a_{2}$, $\tau_{1}=\tau_{2}$, and
$\Omega_{1}=\Omega_{2}$, which is accessible with the pulse shaper [15, 14,
21] on the pump arm. With these incident pulses in Eq. (18-20), we obtain the
phase cycled XPM signal as

$\displaystyle S_{\mathrm{XPM}}\left(\tau,T,\omega_{t}\right)$
$\displaystyle=\frac{8}{A^{1/4}}\sigma_{e}\eta\Omega_{3}\tau_{1}\left|a_{1}\right|^{2}\cos\left(\Omega_{1}\tau\right)\exp\left\\{-\frac{\tau^{2}}{4\tau_{1}^{2}}\right\\}$
(21)
$\displaystyle\times\exp\left\\{\frac{\left(\tau_{1}^{2}+2\tau_{3,0}^{2}\right)B-4\beta_{gdd}\tau_{3,0}^{2}\left(\Omega_{3}-\omega_{t}\right)t_{d}\left(\omega_{t}\right)}{A}\right\\}$
$\displaystyle\times\sin\left\\{\frac{2\beta_{gdd}B+2\tau_{3,0}^{2}\left(\Omega_{3}-\omega_{t}\right)\left(\tau_{1}^{2}+2\tau_{3,0}^{2}\right)t_{d}\left(\omega_{t}\right)}{A}+\phi\right\\},$

where $A=\left(\tau_{1}^{2}+2\tau_{3,0}^{2}\right)^{2}+4\beta_{gdd}^{2}$,
$B=\left[t_{d}\left(\omega_{t}\right)\right]^{2}+\tau_{3,0}^{4}\left(\Omega_{3}-\omega_{t}\right)^{2}$,
$t_{d}\left(\omega_{t}\right)=\tau/2+T-\beta_{gdd}\left(\Omega_{3}-\omega_{t}\right)$,
and
$\phi=\arctan\left[2\beta_{gdd}/\left(\tau_{1}^{2}+2\tau_{3,0}^{2}\right)\right]$.
The parameters $\beta_{gdd}$ and $\tau_{3,0}$ are the group delay velocity
(GDD) and Fourier limited duration of the probe pulse, respectively. These two
parameters are connected to $\tau_{3}$ and $\beta_{3}$ with relation
$\tau_{3}^{2}=\left(\tau_{3,0}^{4}+\beta_{gdd}^{2}\right)/\tau_{3,0}^{2}$ and
$\beta_{3}=0.5\beta_{gdd}/\left(\tau_{3,0}^{4}+\beta_{gdd}^{2}\right)$ [28].

The cosine term $\cos\left(\Omega_{1}\tau\right)$ and the Gaussian term with
variable $\left(\Omega_{3}-\omega_{t}\right)$ in Eq. (21) indicate that the
XPM signal on the 2D spectrum covers around
$\left(\omega_{\tau}=\Omega_{1},\omega_{t}=\Omega_{3}\right)$, i.e., typically
the resonant frequencies of the sample solution. We remark that the XPM signal
are oscillating along $\omega_{t}$ govern by the sine term in Eq. (21), and
the oscillation will move along $\omega_{t}$ as $T$ changes. We will show such
features with experiments and simulations in the following sections.

##  III Experiment and simulation

###  III.1 Experimental Setup

Our experimental setup is presented in Fig. 2. Laser pulses (800 nm, 7.15 W,
30.8 fs, at a 1 kHz repetition rate, and horizontally polarized with respect
to the table surface) delivered by the Ti:Sapphire amplifier laser system are
lead into an optical parametric amplifier (OPA) to produce pulses with desired
central wavelength at 680 nm. The laser beam is then lead into a commercial
integrated 2D spectroscopy system. Specifically, the laser beam after the OPA
is divided by a wedged window into two beams, i.e., the pump and probe beams,
and the power of the probe beam is controlled by a half-wave plate before the
wedged window. To introduce time delays between the pump and probe pulses, the
pump beam is directed into a pulse shaper and a delay stage before hitting the
sample. The pulse shaper [15, 14, 21] consists of an acousto-optic modulator
(AOM), two parabolic mirrors, and two gratings with 4-f geometry. The AOM,
synchronized with our laser system, modulates the first-order diffraction of
the pump pulses with customized mask through photoelastic effect. Typically,
the mask is designed such that one pump pulse is divided into two identical
transform limited pump pulses with interval $\tau$. The time delay $T$ between
the probe and the second pump pulse is controlled by the delay stage. The pump
and probe beams are aligned in parallel and focused onto the sample using a
parabolic mirror. After the sample, the pump beam is blocked by a beam trap,
and the probe beam, passing through a Glan-Taylor prism and a lens
($f=300\>\mathrm{mm}$), is lead into a spectrometer with the spectrum detected
by a charge coupled device (CCD) camera. The Glan-Taylor prism is used to
guarantee that only horizontally polarized component of the probe pulse is
measured. Furthermore, a half-wave plate and a polarizer in the pump arm
before the sample are utilized to ensure that only horizontally polarized
component pump pulses interact with the sample.

In our 2D spectroscopy experiment for detecting the XPM, a sapphire window (3
mm) serves as the sample. Additionally, to study the influence of the GDD of
the probe pulse to the XPM on the 2D spectrum, two experiments are conducted.
In our first experiment, the probe beam passes freely before the sample, while
in the second experiment, an additional sapphire window (SW, 3 mm thick) is
placed on probe arm to alter the $\beta_{gdd}$ of the probe pulse.

We remark that the phases of the pump pulses are also controlled by the AOM,
making phase cycling available in this system. Moreover, instead of the two-
frame cycling $(\phi_{1}=\pi,\phi_{2}=0)$ and $(\phi_{1}=\pi,\phi_{2}=\pi)$
introduced in Sec. II.2, we actually utilize a four-frame cycling [14]
$(\phi_{1}=\pi,\phi_{2}=0)$, $(\phi_{1}=\pi,\phi_{2}=\pi)$,
$(\phi_{1}=0,\phi_{2}=\pi)$, and $(\phi_{1}=0,\phi_{2}=0)$, which further
eliminates the scatter from the pump pulses.

Figure 2: Schematic experimental setup of the 2D spectroscopy with pump-probe
geometry in our experiment. Amplifier: Ti:Sapphire amplifier laser system;
OPA: optical parametric amplifier; HWP: half-wave plate; WW: wedged window;
PM: parabolic mirror; GR: optical grating; AOM: acousto-optic modulator; DS:
delay stage; PL: polarizer; SW: sapphire window (3 mm) used to altering the
GDD of the probe pulse; BT: beam trap; GTP: Glan-Taylor prism; Spectrometer:
spectrometer; CCD: charge coupled device camera. The system enclosed by a dash
box is an integrated 2D spectroscopy system. In our 2D spectroscopy experiment
for detecting the XPM, a sapphire window (3 mm) serves as the sample.

###  III.2 Pulse parameters

We presented in this section the parameters of the pulses applied in our
experiments. The spectrum of the probe pulse is shown in Fig. 3(a), with the
central frequency $\Omega_{3}/2\pi=438.09\>\mathrm{THz}$ and the bandwidth
$\sigma_{3,0}/2\pi=11.74\>\mathrm{THz}$ evaluated by the Gaussian fitting of
the spectrum. The bandwidth $\sigma_{3,0}$ corresponds to a Fourier limited
duration of the probe pulse, $\tau_{3,0}=1/\sigma_{3,0}=13.56\>\mathrm{fs}$.
The auto-correlation functions (ACF) of the probe pulses $E_{3}^{e_{1}}$ and
$E_{3}^{e_{2}}$ are given in Fig. 3(b) and (c). Here, the superscripts $e_{1}$
and $e_{2}$ are used to denote experiments without and with the additional SW
on the probe arm, respectively. The duration of the ACFs are
$\tau_{\mathrm{ACF}}^{e_{1}}=47.25\>\mathrm{fs}$ and
$\tau_{\mathrm{ACF}}^{e_{2}}=66.50\>\mathrm{fs}$, and the duration of the
electric fields are directly given by the ACFs as
$\tau_{3}^{e_{1}}=\tau_{\mathrm{ACF}}^{e_{1}}=47.25\>\mathrm{fs}$ and
$\tau_{3}^{e_{2}}=\tau_{\mathrm{ACF}}^{e_{2}}=66.50\>\mathrm{fs}$. Such
duration of the electric fields corresponds to the typical positive GDDs
$\beta_{gdd}^{e_{1}}=613.76\>\mathrm{fs}^{2}$ and
$\beta_{gdd}^{e_{2}}=882.79\>\mathrm{fs}^{2}$ according to the relation
$\beta_{gdd}^{2}=\tau_{3,0}^{2}\left(\tau_{3}^{2}-\tau_{3,0}^{2}\right)$. The
spectrum is collected by the spectrometer and CCD camera, and the ACFs are
measured by an auto-correlator.

Figure 3: (a) The spectrum and (b, c) the auto-correlation functions (ACFs) of
the probe pulses. The black solid lines are experimental results and the red
dash lines are Gaussian fittings. The Gaussian fitting in (a) suggests that
the probe spectrum centered at $\Omega_{3}/2\pi=438.09\>\mathrm{THz}$ with the
bandwidth $\sigma_{3,0}/2\pi=11.74\>\mathrm{THz}$. Such a bandwidth
corresponds to a Fourier limited duration
$\tau_{3,0}=1/\sigma_{3,0}=13.56\>\mathrm{fs}$ of the probe pulse. The ACFs in
(b) and (c) relate to the probe pulses without and with the additional SW,
respectively. Via the Gaussian fitting, the duration of the ACFs are
$\tau_{\mathrm{ACF}}^{e_{1}}=47.25\>\mathrm{fs}$ for (b) and
$\tau_{\mathrm{ACF}}^{e_{2}}=66.50\>\mathrm{fs}$ for (c) corresponding to the
duration $\tau_{3}^{e_{1}}=\tau_{\mathrm{ACF}}^{e_{1}}=47.25\>\mathrm{fs}$ and
$\tau_{3}^{e_{2}}=\tau_{\mathrm{ACF}}^{e_{2}}=66.50\>\mathrm{fs}$ for the
probe pulses.

The parameters of the pump pulses are not measured with the spectrometer,
instead we evaluate the parameters of the pump pulses by simulating the 2D
correlation spectra of the XPM (referred to as 2DCS-XPM) and Gaussian fitting
their projection traces on the axis of $\omega_{\tau}$. Firstly, the central
frequencies of the pump pulses are determined by projecting the experimental
results of the 2DCS-XPM onto the axis of $\omega_{\tau}$. Then the simulations
are conducted by scanning the duration of the pump pulses to obtain the best
duration, with which the simulated projection trace fits best with the
experiment (See Supplementary for details). The central frequencies and best
duration of the pump pulses are $\Omega_{1}^{e_{1}}/2\pi=449.59\>\mathrm{THz}$
and $\tau_{1}^{e_{1}}=48.85\>\mathrm{fs}$ for experiment without the SW and
$\Omega_{1}^{e_{2}}/2\pi=449.19\>\mathrm{THz}$ and
$\tau_{1}^{e_{2}}=50.48\>\mathrm{fs}$ for experiment with the SW. These
parameters are utilized in Sec. III.4 for numerical simulation.

###  III.3 XPM in 2D spectroscopy experiments

We have measured the 2DCS-XPM using a sapphire window as the sample. The time
delay $T$ is scanned from -100 fs to 30 fs with a step size of 10 fs. The
results from $T=-20\;\mathrm{fs}$ to $T=20\;\mathrm{fs}$ with 65% contours are
shown in Fig. 4, where the spectra in the first row are measured without the
SW and those in the second row are measured with the SW on the probe arm.

Figure 4: 2D correlation spectra of the XPM by using a sapphire window as the
sample with the corresponding $T$ denoted on. The spectra in the first row are
measured when the SW is absent on the probe arm. The spectra in the second row
are measured with the the SW placed. Those 2D spectra are obtained by Fourier
transform the time domain signal
$S_{\mathrm{XPM}}\left(\tau,T,\omega_{t}\right)$ with respect to $\tau$ and
then taking the absolute value of the transform results. To stress the signal
from the background and from the noise, 65% contours are shown here.

In Fig. 4, up to three peaks along $\omega_{t}$ are observable. Those peaks
correspond to the oscillation of the
$S_{\mathrm{XPM}}\left(\tau,T,\omega_{t}\right)$ introduced by the sine term
in Eq. (21). The additional peaks, which are also predicted by the sine term,
are not visible here constrained by the signal-to-noise ratio. We remark that
the intensities of the 2DCS-XPM presented in Fig. 4 are stronger when $T$ is
negative than the positive ones. This discrepancy arises due to the greater
likelihood of the probe pulse overlapping with the interference of the two
pump pulses under negative $T$ conditions.

Another feature of the 2DCS-XPM is the dynamic shift along $\omega_{t}$ from
higher frequency (shorter wavelength) to smaller frequency (longer wavelength)
regions as the delay time $T$ increases. We emphasize such a tendency by
projecting the 2DCS-XPM onto the axis of $\omega_{t}$, and the instances from
$T=-20\;\mathrm{fs}$ to $T=20\;\mathrm{fs}$ are presented in Fig. 5(a) for
experiment without the SW and Fig. 5(b) for experiment with the SW. After the
projection, one particular peak near $\Omega_{3}$ of the projection traces,
e.g., the one enclosed by red square in Fig. 5(a) or blue square in Fig. 5(b),
is fitted with the quadratic polynomial to obtain a central frequency of the
peak. Fig. 5(c) and (d) illustrates the shifts of the central frequencies as
$T$ changes from -100 fs to 30 fs with the red dots (for experiment without
the SW) and the blue triangles (for experiment with the SW), respectively. The
shifts are nearly linear about $T$ in both experiments and the slopes of the
linear fitting lines are $-0.205\>\mathrm{THz}/\mathrm{fs}$ for experiment
without the SW and $-0.153\>\mathrm{THz}/\mathrm{fs}$ for experiment with the
SW. The slope characterizes the speed of the peak displacement, and thus the
result shows that the XPM moves faster when the probe pulse is less chirped.

Figure 5: (a, b) Projection traces of the 2DCS-XPM onto the axis of
$\omega_{t}$ and (c, d) the dynamic shifts of the XPM along the axis of
$\omega_{t}$ as $T$ changes. The traces in (a) and the shifts in (c)
correspond to the 2DCS-XPM without the SW and those in (b) and (d) correspond
to the 2DCS-XPM with the SW on the probe arm. The displacement of the traces
in (a) and (b) are indicated by the gray dash lines, representing the central
frequencies of one specific peak, e.g., the one enclosed by red square in (a)
or blue square in (b) when $T=-20\;\mathrm{fs}$. The central frequencies are
obtained by fitting the peaks using the quadratic polynomial (green dash
lines). The shifts of the central frequencies as functions of $T$ are
quantified in (c) by the red dots and in (d) by the blue solid triangles.
Their corresponding linear fitting lines have slopes of
$-0.205\>\mathrm{THz}/\mathrm{fs}$ and $-0.153\>\mathrm{THz}/\mathrm{fs}$,
respectively. Additionally, the simulation results are depicted in (c) by the
pink circles and in (d) by the light-blue hollow triangles, with the slopes of
their linear fitting lines being $-0.231\>\mathrm{THz}/\mathrm{fs}$ and
$-0.165\>\mathrm{THz}/\mathrm{fs}$, respectively.

###  III.4 Simulation

To verify the mechanism of the XPM on the 2D spectroscopy, we simulate the
2DCS-XPM using Eq. (21) and with the pulse parameters in Sec. III.2. The
simulation results from $T=-20\;\mathrm{fs}$ to $T=20\;\mathrm{fs}$ with 65%
contours are taken as the instances in Fig. 6, where three distinct peaks are
observed as in the experimental results in Fig. 4. Also, as in Fig. 4,
simulations in the first row in Fig. 6 correspond to experiment without the
SW, and those in the second row correspond to experiment with the SW.

Figure 6: Simulated 2DCS-XPM with the corresponding $T$ denoted on. The
spectra in the first row are simulated for the experiment without the SW, and
the parameters are $\Omega_{3}/2\pi=438.09\>\mathrm{THz}$,
$\beta_{gdd}^{e_{1}}=613.76\>\mathrm{fs}^{2}$ for the probe pulse, and
$\Omega_{\tau}^{e_{1}}/2\pi=449.59\>\mathrm{THz}$,
$\tau_{1}^{e_{1}}=48.85\>\mathrm{fs}$ for the pump pulses. The spectra in the
second row are simulated for the experiment with the SW, and the parameters
are $\Omega_{3}/2\pi=438.09\>\mathrm{THz}$,
$\beta_{gdd}^{e_{2}}=882.79\>\mathrm{fs}^{2}$ for the probe pulse, and
$\Omega_{1}^{e_{2}}/2\pi=449.19\>\mathrm{THz}$,
$\tau_{1}^{e_{2}}=50.48\>\mathrm{fs}$ for the pump pulses. 65% contours are
shown here.

We also illustrate the dynamic shifts of the XPM on the simulated 2D spectra
by projecting them onto the axis of $\omega_{t}$ and then fitting the peak
near $\Omega_{3}$ to obtain a central frequency. The shifts of the central
frequencies in our simulations are depicted in Fig. 5(c) and (d), alongside
the corresponding experimental results. In Fig. 5(c) and (d), the
displacements of the simulations (represented by the hollow marks) meet well
with the experimental results (represented by the solid marks) for both
experiments without and with the SW. The slope of the linear fittings of the
simulations are $-0.231\>\mathrm{THz}/\mathrm{fs}$ and
$-0.165\>\mathrm{THz}/\mathrm{fs}$, close to the experimental results
$-0.205\>\mathrm{THz}/\mathrm{fs}$ and $-0.153\>\mathrm{THz}/\mathrm{fs}$.
Both experiments and simulations prove that as the chirp of the probe pulse
gets larger, the XPM moves slower with respect to delay time $T$ and will
cover the desired signal for a wider range of $T$.

The speed of the XPM displacement as a function of the GDD of the probe pulse
$\beta_{gdd}$ is presented in Fig. 7, where the slopes for both positive and
negative $\beta_{gdd}$ are simulated (red solid marks) and then fitted (red
dash lines) with reciprocal function. Unlike in the positive $\beta_{gdd}$
region where the slope is negative and the XPM moves from higher frequency
(shorter wavelength) to smaller frequency (longer wavelength), in the negative
$\beta_{gdd}$ region, slope is positive and the XPM moves from smaller
frequency (longer wavelength) to larger frequency (shorter wavelength).
Whereas, the reciprocal relation between the slope and the $\beta_{gdd}$
retains. Such a reciprocal relation moves the XPM away from the frequency
region of interest much faster with respect to $T$ as $\beta_{gdd}$ decreases.
Hence, suppressing the chirp of the probe pulse also suppresses the range of
$T$ within which the desired signal is affected by the XPM. Moreover, the
intensity of the XPM at $T=0\>\mathrm{fs}$ is significantly reduced by
suppressing $\beta_{gdd}$, as depicted by the black solid line in Fig. 7. With
these two features, one is able to minimize the influence of the XPM on the 2D
spectroscopy by suppressing the chirp of the probe pulse.

Figure 7: Simulated intensities and moving speed of the XPM on the 2D spectra
as functions of the GDD of the probe pulse. In the simulation, the central
frequency and the Fourier limited duration of the probe pulse are
$\Omega_{3}/2\pi=438.09\>\mathrm{THz}$ and $\tau_{3,0}=13.56\>\mathrm{fs}$,
and the central frequency and the Fourier limited duration of the pump pulse
are $\Omega_{1}/2\pi=449.46\>\mathrm{THz}$ and $\tau_{1}=50\>\mathrm{fs}$. The
maximum intensity of the XPM within the region
$\left\\{\left(\omega_{\tau},\omega_{t}\right)|444.46\>\mathrm{THz}\leq\omega_{\tau}/2\pi\leq
454.46\>\mathrm{THz},433.09\>\mathrm{THz}\leq\omega_{t}/2\pi\leq
443.09\>\mathrm{THz}\right\\}$ at $T=0\>\mathrm{fs}$ is illustrated by the
black solid line. The moving speed, i.e., the slope, of the XPM with respect
to $T$ is represented by the red solid marks, which are then fitted with
reciprocal function by the red dash lines.

##  IV Conclusion

We have successfully demonstrated the presence of XPM induced by the
interference of the two pump pulses in 2D spectroscopy with pump-probe
geometry. This XPM phenomenon predominantly occurs at
$\left(\omega_{\tau}=\Omega_{1},\omega_{t}=\Omega_{3}\right)$, which typically
corresponds to the resonant frequencies of the target sample. Consequently,
the XPM effect can significantly overlap with the desired signal on the 2D
spectra, especially when studying liquid samples due to the substantial XPM
introduced by the solvent.

We have present theoretically and experimentally the pattern and the
displacements of the XPM on the 2D spectrum. We observe that the XPM
oscillates along $\omega_{t}$, while the observable oscillations (or peaks)
are limited by the signal-to-noise ratio. In our experiments, three peaks are
observed. As for the displacements of the XPM, we show that for positive
chirped probe pulse, the XPM on the 2D spectra moves from higher frequency
(shorter wavelength) to smaller frequency (longer wavelength) regions as $T$
increases. The direction of such a displacement is reversed for negative
chirped probe pulse. Additionally, we determine that the speed of the
displacement with respect to $T$ is inversely related to the GDD of the probe
pulse $\beta_{gdd}$. When the $\beta_{gdd}$ gets suppressed, the XPM moves
faster and thus will disturb the measurement of the desired signal for a
narrow range of $T$. Moreover, suppressing the $\beta_{gdd}$ of the probe
pulse reduces the intensity of the XPM.

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