11institutetext: Division of Informatics, Imaging and Data Sciences, School of
Health Sciences, University of Manchester22institutetext: Division of Cancer
Sciences, University of Manchester
22email<EMAIL_ADDRESS>
# HEMIT: H&E to Multiplex-immunohistochemistry Image Translation with Dual-
Branch Pix2pix Generator

Chang Bian 11 Beth Philips 22 Tim Cootes 11 Martin Fergie 11

###### Abstract

Computational analysis of multiplexed immunofluorescence histology data is
emerging as an important method for understanding the tumour micro-environment
in cancer. This work presents HEMIT, a dataset designed for translating
Hematoxylin and Eosin (H&E) sections to multiplex-immunohistochemistry (mIHC)
images, featuring DAPI, CD3, and panCK markers. Distinctively, HEMIT’s mIHC
images are multi-component and cellular-level aligned with H&E, enriching
supervised stain translation tasks. To our knowledge, HEMIT is the first
publicly available cellular-level aligned dataset that enables H&E to multi-
target mIHC image translation. This dataset provides the computer vision
community with a valuable resource to develop novel computational methods
which have the potential to gain new insights from H&E slide archives.

We also propose a new dual-branch generator architecture, using residual
Convolutional Neural Networks (CNNs) and Swin Transformers which achieves
better translation outcomes than other popular algorithms. When evaluated on
HEMIT, it outperforms pix2pixHD, pix2pix, U-Net, and ResNet, achieving the
highest overall score on key metrics including the Structural Similarity Index
Measure (SSIM), Pearson correlation score (R), and Peak signal-to-noise Ratio
(PSNR). Additionally, downstream analysis has been used to further validate
the quality of the generated mIHC images. These results set a new benchmark in
the field of stain translation tasks.

###### Keywords:

HEMIT dataset Hematoxylin and Eosin (H&E) Multiplex-immunohistochemistry
(mIHC) stain translation Swin Transformers Downstream analysis.

##  1 Introduction

Multiplex immunohistochemistry/immunofluorescence (mIHC/IF) technologies have
revolutionized the study of the tumor microenvironment (TME), providing
critical insights for cancer research and immunotherapy [25]. These
technologies enable the detection of multiple markers in a single tissue
slice, revealing intricate spatial interactions. However, the complexity and
cost of mIHC limits its accessibility [24]. Deep learning offers a promising
solution to these challenges, with recent advancements in stain translation
and virtual staining allowing for better exploitation of existing H&E slides
[3, 10, 21, 6, 27].

Capitalizing on the ease and cost-efficiency of producing H&E images, this
paper aims to develop an image-to-image translation method that can convert
H&E images into their corresponding mIHC counterparts.

Image-to-image translation algorithms are well-established across a range of
image analysis domains. Specifically, pix2pix [9] employs a conditional
generative adversarial network (cGAN) approach for paired images, adeptly
generating high-fidelity translations. Building upon this, pix2pixHD [26] has
demonstrated significant outcomes with high-resolution paired images. Notably,
a recent study [14] introduced a multi-scale loss term, improving the
translation performance for HER2 images. Furthermore, other methods [5, 8, 12,
13, 11, 19] have been designed specifically for image-to-image translation
tasks, underscoring the breadth and depth of research in this domain.

High-quality datasets are essential for effective supervised image
translation, yet the realm of pathological image translation lacks
comprehensive resources. The BCI dataset [14] is proposed for H&E to HER2 IHC
image translation. However, it is limited in use as staining is performed on
consecutive tissue sections so there is no cell-to-cell mapping across stains.
Further, it is limited to one target stain (HER2) unlike mIHC where multiple
stains can be predicted simultaneously. In response, we present HEMIT: A
dataset for H&E to mIHC Translation. This is the first publicly-available
cellular-level aligned dataset for stain translation. Concurrently, we propose
a specialized method tailored for this task which we compare to various state-
of-the-art image translation algorithms. This work makes the following
contributions:

  1. 1.

Introduction of HEMIT: a paired dataset for H&E to mIHC image translation. To
the best of our knowledge, HEMIT is the first publicly available cellular-
level aligned dataset that enables H&E to multi-target mIHC image translation.

  2. 2.

Development of a SwinTransformer-CNN-based dual-branch pix2pix strategy to
convert H&E images into mIHC versions. Our methodology assimilates both global
information and spatial details, culminating in superior outcomes, setting
benchmark results for the proposed dataset.

  3. 3.

A thorough empirical evaluation was conducted on the newly introduced HEMIT
dataset, setting a benchmark for future investigations within the research
community. Further downstream analysis, employing Qupath [1], substantiated
the high quality of the images produced. This establishes a solid foundation
for subsequent studies and applications, encouraging a deeper exploration into
the dataset’s potential.

##  2 HEMIT Dataset

We present HEMIT: A cellular-level aligned dataset for H&E to mIHC Image
Translation. A schematic of our dataset’s construction pipeline is provided in
fig. 1a. Full dataset will be released upon acceptance of this paper.

###  2.1 Data Collection

HEMIT’s raw data is sourced from ImmunoAIzer [2] which we have adapted to make
it suitable for the computer vision community. Notably, HEMIT distinguishes
itself from other datasets: the H&E and mIHC slide pairs are derived from the
identical tissue section, not consecutive slides. The tissue was first stained
with mIHC protocols and then bleached before the H&E staining. This feature
leads to higher fidelity and alignment between the matched image pairs and
better translation outcomes [4].

In the mIHC images, three pivotal cell type identification markers are
incorporated: DAPI to signify cell nuclei, pan-cytokeratin (panCK) to
highlight tumor regions, and CD3 to pinpoint T cells—all of which are integral
to TME analysis. These specific markers serve as the foundation for our H&E to
mIHC stain translation benchmark. We have collated a selection of publicly
available datasets that leverage H&E images for predicting IHC expressions. A
comparative overview of their characteristics is presented in table 1.

Table 1: Summary of publicly available datasets Datasets | Staining Types | Sectioning Approach | IHC/mIHC Markers | Ground Truths
---|---|---|---|---
HEMIT | H&E & mIHC | same slide | DAPI, CD3, panCK | cellular level
BCI [14] | H&E & IHC | consecutive slides | HER2 | structural level
HEROHE [7] | H&E | H&E only | Clinical HER2 status | slide level

###  2.2 Data Preprocessing

Despite both H&E and mIHC staining being performed on the same tissue slide,
the re-staining and scanning processes mean that the captured images do not
align perfectly. We employ a 2-step registration process to ensure cellular-
wise alignment of the image pairs which is crucial for optimal training
performance. Implementation details of the registration process are given in
the supplementary material.

Upon concluding the registration, a margin of 50 pixels from each edge is
trimmed to account for rotational transformations. Subsequently, the block
pairs are cropped into $1024\times 1024$ patches, maintaining a 50% overlap.
Color normalization [18] is then applied to all H&E patches to mitigate stain
variations. Visualizations of the final registered image pairs are shown in
fig. 1b. This process resulted in 5292 matched image patch pairs. This
processed data is distributed into three partitions: _training_ , _validation_
, and _testing_. The division yields $3717$ patches designated for training,
$630$ for validation, and $945$ for the testing phase. Importantly, to prevent
data leakage of our evaluation, patches across these subsets are derived from
unique patient samples, precluding potential data overlap.

---
(a) Processing pipeline of the HEMIT dataset.
(b) Visualization of the registered image pairs, along with constitute
channels.
Figure 1: Overview of dataset processing and visualization in the HEMIT
dataset: (a) Processing pipeline. (b) Visualization of registered image pairs.

##  3 Proposed Method

###  3.1 Architecture

Our dual-branch architecture is based on the original pix2pix [9] model, and
we focus on the Swin Transformer-based branch. The overall framework is shown
in fig. 2.

We devised a dual-branch generator architecture. The main branch with residual
blocks of the generator is to extract spatial nuances from the input H&E
images, and an auxiliary branch powered by Swin Transformers [16] is
incorporated to integrate information across multiple scales. The features
extracted by each stage of the Swin Transformer branch are fused with the
feature maps of the CNN branch by the Feature Map Fusion (FMF) module. This
architecture caters to the multi-scale nature of pathological images. This
design captures information from the structure of individual cells to the
interaction of cell clusters, vital for interpreting complex tissue
environments.

The auxiliary branch, operating in parallel with the main branch, leverages
Swin Transformer modules [16] to adeptly capture multi-scale features.
Notably, the Swin Transformer introduces both Window-based Multi-head Self
Attention (W-MSA) and Shifted Window-based MSA (SW-MSA). Prior to each MSA
module and MLP, a Layer Normalization (LN) is applied. $\hat{s}_{l}$ and
$s_{l}$ denote the outputs of W-MSA and MLP respectively, at the $l^{th}$
layer. These are computed as:

$\displaystyle\hat{s}_{l}$
$\displaystyle=\text{W-MSA}(\text{LN}(s_{l-1}))+s_{l-1};s_{l}=\text{MLP}(\text{LN}(\hat{s}_{l}))+\hat{s}_{l},$
(1) $\displaystyle\hat{s}_{l+1}$ $\displaystyle=\text{SW-
MSA}(\text{LN}(\hat{s}_{l}))+s_{l};s_{l+1}=\text{MLP}(\text{LN}(\hat{s}_{l+1}))+\hat{s}_{l+1}$

The FMF module integrates features from the two branches by utilizing Gated
Cross Attention, which emphasizes spatially significant regions. The Swin
Transformer output $S_{i}$ is aligned with the CNN feature map $F_{i}$ in both
dimensions and resolution to form $S^{\prime}_{i}$. A gating layer then
processes $F_{i}$ through convolution and sigmoid activation to produce a
gating map $G$, pinpointing areas of importance within $F_{i}$. The model then
selects the top $k$ pixels in $G$, indicative of the most informative regions.
Subsets $F_{i,k}$ and $S^{\prime}_{i,k}$ are extracted from $F_{i}$ and
$S^{\prime}_{i}$ correspondingly for cross attention operation in equation 2.
Focusing on these $k$ points allows the cross-attention to exchange
information primarily in regions crucial for the model’s performance, thereby
improving efficiency. The choice of $k$ is based on empirical analysis,
details in the supplementary materials.

$A=\text{Attention}(F_{i,k},S^{\prime}_{i,k},S^{\prime}_{i,k})$ (2)

where:

  * •

$F_{i,k}$ serves as the query, representing the top k elements from the CNN
feature maps.

  * •

$S^{\prime}_{i,k}$ serves as the key and value, derived from the corresponding
elements in the Swin Transformer output.

The adversarial loss is adopted from the pix2pix framework:

$\mathcal{L}_{cGAN}(G,D)=\mathbb{E}_{x,y}[\log
D(x,y)]+\mathbb{E}_{x,z}[\log(1-D(x,G(x,z)))].$ (3)

The terms $x$, $y$, and $z$ denote the input H&E image, the ground truth mIHC
image, and random noise, respectively. Additionally, the L1 loss from pix2pix
is employed to preserve structural consistency:

$\mathcal{L}_{1}=\mathbb{E}_{x,y,z}[\|y-G(x,z)\|_{1}].$ (4)

Thus, our overarching objective function becomes:

$G^{*}=\text{arg}\ \underset{G}{\text{min}}\ \underset{D}{\text{max}}\
\mathcal{L}_{cGAN}(G,D)+\lambda\mathcal{L}_{1}(G),$ (5)

where $\lambda$ balances the contributions of the adversarial loss and the
$L_{1}$ loss.

Figure 2: Overall structure of the proposed generator.

##  4 Experiments

We used the Adam optimizer over 100 epochs, with an initial learning rate of
0.00003 for 50 epochs, and then linearly decayed to zero over the remaining
epochs, on an NVIDIA Tesla V100 16GB GPU.

###  4.1 Benchmark Results

Following established benchmarks in image-to-image translation [14, 3, 6, 15],
we adopt the Structural Similarity Index Measure (SSIM), Pearson correlation
score (R), and Peak Signal to Noise Ratio (PSNR) as evaluation metrics to
gauge the quality of the synthesized images.

Our method demonstrated superior performance on the HEMIT dataset, achieving
the highest scores in SSIM (0.875), Pearson correlation (0.746), and PSNR
(29.886), as shown in table 2. This indicates its effectiveness in image
translation quality compared to other models.

Table 2: Comparison of evaluation metrics across different methods. | SSIM | R | PSNR (dB)
---|---|---|---
Methods | DAPI | CD3 | panCK | Average | DAPI | CD3 | panCK | Average | DAPI | CD3 | panCK | Average
U-Net | 0.791 | 0.907 | 0.901 | 0.866 | 0.659 | 0.005 | 0.949 | 0.538 | 27.929 | 24.733 | 33.773 | 28.695
ResNet | 0.790 | 0.907 | 0.898 | 0.865 | 0.677 | 0.005 | 0.949 | 0.544 | 27.531 | 24.733 | 33.212 | 28.268
pix2pix_UNet | 0.775 | 0.879 | 0.903 | 0.852 | 0.652 | 0.455 | 0.943 | 0.683 | 27.691 | 25.659 | 34.306 | 29.219
pix2pix_ResNet | 0.723 | 0.906 | 0.914 | 0.848 | 0.668 | 0.553 | 0.946 | 0.723 | 27.152 | 26.225 | 34.359 | 29.189
pix2pixHD | 0.786 | 0.900 | 0.888 | 0.858 | 0.721 | 0.530 | 0.943 | 0.731 | 28.156 | 26.281 | 34.033 | 29.471
ours | 0.815 | 0.898 | 0.913 | 0.875 | 0.716 | 0.571 | 0.951 | 0.746 | 28.610 | 26.349 | 34.875 | 29.886

Visually, as seen in fig. 3, while U-Net and ResNet scored well in SSIM and
PSNR, they fell short in Pearson correlation due to insufficient emphasis on
less dominant markers, such as CD3. This highlights the importance of Pearson
correlation for evaluating multi-stain translations and demonstrates that GAN-
based frameworks significantly improve the realism of image translations,
particularly for low-expression markers.

Figure 3: Visualization of different methods on HEMIT dataset. DAPI is shown
in blue, panCK in red and CD3 in green.

###  4.2 Downstream Analysis

Our evaluation, incorporating downstream analysis with Qupath [1, 17],
compared the fidelity of our generated images and other methods against real
images. We analyzed 15 image blocks, using Stardist [22] for cell detection
and the Otsu method [20] for marker positivity, to validate the translation
accuracy by comparing the positive cell proportions.

Mean absolute error ratios of positive cell proportions are shown in fig. 4A.
Our method shows considerable improvement over the existing state-of-the-art
methods, especially on the challenging CD3 marker. Further comparative results
of our method are shown in fig. 4B-D. C and D showcase the close
correspondence between real and generated images, evidenced by the tight
distribution around the identity line in scatter plots and marginal deviations
in Bland-Altman plots.

Figure 4: Downstream Analysis Results: (A) Comparative MAE Ratios for CD3 and
PanCK; (B) shows boxplots of cell counts for CD3, panCK, and total of the
generated images of our method and real images; (C) and (D) provide scatter
plots and Bland-Altman plots for CD3 and panCK positive cell proportions of
the generated images of our method and the real images.

##  5 Conclusion

In this study, we introduced HEMIT, a pioneering dataset tailored for
pathology image translation tasks. HEMIT is specifically designed to bridge
the gap between H&E stained sections and their mIHC counterparts, showcasing
multiple markers including DAPI, CD3, and panCK. Notably, the mIHC images in
our dataset are multi-component, providing a cellular-level registration with
the associated H&E images. This unique alignment presents an invaluable
opportunity for supervised stain translation research.

Complementing our dataset contribution, we proposed a dual-branch generator
architecture optimized for pathology image analysis. Through extensive
experimentation, we contrasted our approach with several SOTA algorithms. The
empirical outcomes with downstream analysis underscore the efficacy of our
method, establishing a new benchmark for the H&E to mIHC translation domain.
We hypothesise that the significant performance improvement is due to the
spatial context provided by the Swin Transformer where the tissue compartment
can inform the cell protein expression. This is in line with other works [23]
which has demonstrated that tissue context can inform cell classification
leading to improved accuracy.

This work provides an opportunity for further exploration in the use of
predicting biomarkers from H&E pathology images to accelerate research in
biomarker prediction.

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Also at ]International Laser Centre, Ilkovicova 3, 84104 Bratislava, Slovakia

# Mid-infrared Kerr index evaluation via cross-phase modulation with a near-
infrared probe beam

D. Lorenc  [  Z. Alpichshev  Institute of Science and Technology Austria, Am
Campus 1, 3400 Klosterneuburg, Austria

###### Abstract

We propose a simple method to measure nonlinear Kerr refractive index in mid-
infrared frequency range that avoids using sophisticated infrared detectors.
Our approach is based on using a near-infrared probe beam which interacts with
a mid-IR beam via wavelength-non-degenerate cross-phase modulation (XPM). By
carefully measuring XPM-induced spectral modifications in the probe beam and
comparing the experimental data with simulation results we extract the value
for the non-degenerate Kerr index. Finally, in order to obtain the value of
degenerate mid-IR Kerr index we use the well-established two-band formalism of
Sheik-Bahae et al., which is shown to become particularly simple in the limit
of low frequencies. The proposed technique is complementary to the
conventional techniques such as z-scan and has the advantage of not requiring
any mid-infrared detectors.

††preprint: AIP/123-QED

Nonlinear optical frequency conversion has gone a long way in the past decades
from the initial observation of faint second-harmonic generation in the 60’s
Bloembergen  _et al._ (1968) to the present day when nonlinear-optics-based
sources of coherent broadband radiation have become a staple of a modern
optics lab Cerullo and Silvestri (2003). Given the importance, there is a
continual effort to improve the efficiency of such devices and, as one
possible direction, it was demonstrated that many of the relevant nonlinear
frequency-conversion phenomena, - be it high-harmonic generation Li  _et al._
(2020); Chen  _et al._ (2014) or the production of strong THz-range pulses
through optical rectification Fedorov and Tzortzakis (2020), - become
particularly efficient when the frequencies of the primary (fundamental)
pumping beams happen to lie in the mid-infrared domain. Naturally, the
progress within this approach is contingent upon detailed characterization of
the optical properties of nonlinear materials in the infrared range.

Kerr effect (KE) is a nonlinear optical phenomenon wherein the refractive
index of a material is changing as a response to the application of an
external electric field. Unlike the closely related Pockels effect, KE is
proportional to the square of the field and hence, does not require broken
inversion symmetry in the host medium (Boyd, 2008). In fact, it is virtually
ubiquitous and manifests itself prominently in many different contexts, being
responsible for a great number of phenomena such as self- and cross-phase
modulation of the beams, Kerr lensing, self-focusing, optical soliton
formation, optical switching, passive mode-locking Boyd (2008), etc., and as
such has to be taken into account when designing any practical nonlinear
optical application.

The magnitude of KE is determined by the so-called nonlinear Kerr index
$n_{2}$. On the conceptual level the problem of measuring $n_{2}$ is long
solved, thanks to advent of techniques such as z-scan Chapple  _et al._
(1997), I-scan Taheri  _et al._ (1996), four-wave mixing Samoc  _et al._
(1998); Pigeon  _et al._ (2016), nearly degenerate three wave mixing Adair,
Chase, and Payne (1989), two-beam coupling Kang, Krauss, and Wise (1997) and a
number of interferometric techniques (e.g. Jansonas  _et al._ (2022)).
However, in practice it may still pose considerable challenges, especially
when it comes to the mid- and deep-IR wavelengths where experiments require
specialized components (most notably detectors) and often are not as
straightforward as they are with visible or near-IR radiation. As a result,
the amount of information on mid-IR Kerr refractive indices is comparatively
scarce and even in the case of some standard materials, mid-IR Kerr index has
been characterized only recently for a limited number of wavelength values
Jansonas  _et al._ (2022); Ensley and Bambha (2019); Patwardhan  _et al._
(2021).

In this Letter we propose to go around this issue by dropping the challenging
part of mid-IR detection altogether and use a near-IR probe to gauge the
changes to optical properties of a material as a response to mid-IR radiation.
To this end we study the non-degenerate Kerr effect-mediated interaction
between near- and mid-IR pulses manifested as cross-phase modulation (XPM)
Boyd (2008) between the two. Specifically, we observe the mid-IR pump-induced
spectral changes in the probe pulse from which we extract the value for non-
degenerate Kerr index $n_{2}(\Omega,\omega)$. We then use the well-established
two band-theory of Sheik-Bahae et al Hutchings  _et al._ (1992) to reconstruct
from $n_{2}(\Omega,\omega)$ the mid-IR Kerr index. The convenience of the
proposed methods is especially apparent in the limit where both pump and probe
photon energies are significantly less than the band gap. In this regime the
two-band model predicts that relationship between the degenerate and non-
degenerate Kerr indices becomes particularly simple and universal, the only
material-specific property being the band-gap width that enters as a simple
scaling factor for frequencies.

Figure 1: The setup to investigate XPM-induced spectral shift. Intensity of
the probe is controlled via a half-wave plate (HWP) and Glan-Taylor polarizer
(GT); the intensity of the mid-IR pump coming from an OPA is tuned by a pair
of wire-grid polarizers (WP). Pump and probe beams are joined by a
beamsplitter (BS) and focused onto the sample with a f=100mm lens. After the
sample the probe beam is re-collimated with a f=30mm lens and after passing
through monochromator detected by an avalanche photodiode detector. B) A
cartoon demonstrating cross-phase modulation (XPM) in the ideal case of no
walk-off between pulses. The probe beam frequency is red- or blue-shifted
depending on the sign of the Kerr index and instantaneous time derivative of
pump intensity (black) (difference in pump- and probe pulse durations is
exaggerated for the purpose of illustration). C) Experimental XPM-induced
spectral shift of a $\lambda_{0}=$1028nm probe by a
$\lambda_{\mathrm{pump}}=$450 nm pump in a polycrystalline ZnSe sample as a
function of pump-probe delay at peak pump intensity $I_{0}=2.2\times
10^{14}$W/cm2. D) XPM with non-matching pump and probe pulse group velocities
$v^{\mathrm{MIR}}_{g}$ and $v^{\mathrm{NIR}}_{g}$ respectively
($v^{\mathrm{MIR}}_{g}<v^{\mathrm{NIR}}_{g}$). E) XPM-induced spectral changes
for the situation laid out in panel D. Evidently, in the limit of large group
velocity mismatch $\Delta v_{\mathrm{g}}$, the temporal separation between
extremal shifts (blue and red) is determined by $\Delta v_{\mathrm{g}}$ rather
than original pump pulse-width.

In our study we take ZnSe as a model system due to its relatively large band
gap, well-studied linear- and nonlinear optical properties Patwardhan  _et
al._ (2021); Weber (2018) and low absorption in both near-IR and mid-IR
spectral ranges. The sketch of the experimental setup is shown Fig. 1A.
Tunable-frequency mid-IR pulses are generated by an optical parametric
amplifier (OPA; Light Conversion Orpheus-HE) pumped by a femtosecond laser
system (Light Conversion Pharos) producing a train of pulses with a repetition
rate of 3kHz; central wavelength $\lambda_{0}$=1028nm; pulse duration of
$\tau_{\mathrm{FWHM}}$=270fs; and 2mJ/pulse. A small fraction (5%) of the main
beam from the amplifier is split off and used as a near-IR probe while the
main part pumps the OPA to produce $\lambda$=4.5$\mu$m pulses used as a mid-IR
pump. Pump and probe pulses are spatially and temporally overlapped inside a
$d$=1.25mm-thick, polycrystalline ZnSe sample (Crystran Ltd). After the
sample, the XPM-affected probe beam is spectrally analyzed with a
monochromator (Horiba H10) connected to an avalanche photo-diode (Becker&Hickl
APM-400-P-078). To increase the signal-to-noise ratio, the signal from the
detector is first passed through a boxcar integrator (SRS SR250) before being
analyzed in a lock-in amplifier (SRS SR830).

In the experiment we record the spectrum of the probe beam at the exit from
the sample for every value of pump-probe delay as shown in Fig. 1C. In the
range where pump and probe overlap temporally the spectrum of the probe is
visibly perturbed by the pump. As can be seen in Fig.1C and Fig.2 the spectral
shift is changing sign depending on time delay between pump and probe pulses.
Qualitatively the character of the spectral change is in line with what one
expects from XPM, which can be intuitively understood as shown in Fig.1B. As a
result of Kerr effect, the refractive index $n_{0}$ of a medium acquires a
nonlinear correction $\delta n$ proportional to the intensity $I$ of the pump:
$\delta n=n_{2}I$, where $n_{2}$ is the nonlinear (Kerr) refractive index.
Now, on the one hand, the total refractive index $n=n_{0}+\delta n$ determines
the total optical phase accumulated by the probe pulse with wavelength
$\lambda=2\pi c/\omega$ as it propagates through the sample of thickness $L$:
$\phi=2\pi nL/\lambda$. On the other, time derivative of the phase determines
the probe frequency $\omega=\dot{\phi}$. It is then clear that since time-
dependent pump intensity $I(t)$ modifies $\phi(t)$, the frequency
$\delta\omega$ acquires a correction proportional to the derivative:

$\delta\omega=2\pi\frac{L}{\lambda}n_{2}\frac{dI(t)}{dt}$

Therefore analysing the value of XPM-induced spectral shift and knowing the
properties of the pump pulse such as its duration and peak intensity, one can
in principle expect to be able to extract the value for nonlinear Kerr index
$n_{2}$ of the medium.

Needless to say, this simple analysis is only valid for the artificial case of
a monochromatic probe and a steadily growing pump intensity $I(t)$. In the
more realistic case of pulsed pump and probe, a more involved analysis is
necessary. The most important issue in a real experiment is that unlike the
situation depicted in Fig.1B, the duration of pump and probe pulses are
similar, which means that different parts of the latter experience different
spectral shifts. Another problem can be seen upon inspecting Fig.1C or Fig.2.
Here one can see that the points of maximum spectral shifts are separated by
about $\delta\tau\approx 0.7$ps. This is considerably more than the pulsewidth
of of the pump pulse ($\tau_{\mathrm{pu}}\approx 270$fs), which according to
the naive cartoon in Fig.1B should set the value for $\delta\tau$. This
discrepancy comes from the fact that due to large wavelength difference, the
mid-IR pump and near-IR probe pulses propagate through the sample with
significantly different group velocities. Then the interaction between the
pulses occurs more like the Fig.1D
($v_{g}^{\mathrm{NIR}}>v_{g}^{\mathrm{MIR}}$). Here, in the generic situation
when the two pulses meet in the middle of a sufficiently thick sample, each
temporal segment of the probe (gray) experiences the entire pump (red-blue)
pulse, therefore net spectral shift in the former integrates to zero. The
spectral shift becomes non-zero only when the delay between the pulses is such
that they overlap near one of the sample edges. The maximum spectral shifts
then is achieved when the probe pulse center coincides with one of the slopes
of the pump pulse. The separation between these points is consequently
$\delta\tau\approx L/|v_{g}^{\mathrm{NIR}}-v_{g}^{\mathrm{MIR}}|$ (for
$\delta\tau\gg\tau_{\mathrm{pu}}$).

To accurately take these effects into account we simulate the interaction
between pulses by means of solving a system of coupled generalized nonlinear
Schroedinger equations (CGNLSE) for frequency-nondegenerate fields Agrawal
(2013) (see Supplementary Material for details). In order to simplify the
calculation we ignore all absorption effects, which is justified since
$\hbar\left(\omega+\Omega\right)<\Delta_{\mathrm{gap}}$, where
$\Delta_{\mathrm{gap}}$ is the band-gap of ZnSe and $\omega$ and $\Omega$ are
probe and pump frequencies respectively.

Figure 2:  Blue dots: XPM-induced spectral shift of the central wavelength of
the probe pulse as a function of pump-probe delay at peak pump intensity
I${}_{0}=2.1\times 10^{13}$W/cm2; solid red curve: the result of the
corresponding CGNLSE simulation.

In Fig.2 we plot the central wavelength of the probe as a function of delay
between the pump and probe pulses (blue dots) taken with peak pump pulse
intensity of $I_{0}=2.1\times 10^{13}$W/cm2. This relatively moderate value
was chosen to avoid various heating- and multi-photon-absorption-related
phenomena not accounted for in our calculation. The solid red line in Fig.2 is
the result of a CGNLSE simulation with $n_{2}\approx 0.65\times 10^{-18}$
m2/W. Given the excellent agreement, we take it as the measured value for the
non-degenerate nonlinear Kerr index of (polycrystalline) ZnSe.

Figure 3:  Real part of nondegenerate Kerr index $n_{2}$($\omega,\Omega$)
calculated for 2-band model through dispersive analysis of Sheik-Bahae et al.
In the shaded regions, the relative error of the approximate expression in
eq.1 does not exceed 2% (darker green) and 10% (light green); The red spot
marks the point that corresponds to the conditions of the present experiment
($\omega/\Delta\approx 0.45$, $\Omega/\Delta\approx 0.10$; with
${\Delta}=2.82$eV for ZnSe).

In the last step we need to relate the value for the non-degenerate mid-
IR/near-IR Kerr index $n_{2}(\omega,\Omega)$ to the degenerate mid-IR value
$n_{2}(\Omega,\Omega)$. To this end we note that generally speaking, in the
limit $\omega^{2},\Omega^{2}\ll{\Delta^{2}_{\mathrm{gap}}}$ Kerr index must
have the form:

$n_{2}(\omega,\Omega)\propto
1+\Gamma\times\left(\frac{\omega^{2}+\Omega^{2}}{\Delta_{\mathrm{gap}}^{2}}\right)$
(1)

with $\Gamma$ being some numeric factor on the order of unity whose exact
value depends on the microscopic details of a given material. The form of eq.1
is dictated by the fact that on the one hand, $n_{2}(\omega,\Omega)$, – being
an observable quantity, – must be an even function of both $\omega$ and
$\Omega$, and on the other, it must be symmetric against
$\omega\leftrightarrow\Omega$ owing to Kleinman symmetry which holds in the
low-frequency limit Boyd (1999). Having said that $\Gamma$ is not expected to
differ significantly from unity, we can also calculate its value for the
specific case of two-band model Sheik-Bahae  _et al._ (1991) which is known to
adequately capture nonlinear properties of ZnSe Hutchings  _et al._ (1992).
Fitting the expression for $n_{2}(\omega,\Omega)$ (see Supplementary Material)
around the origin with eq.1 gives $\Gamma=1.35$. The accuracy of the low-
frequency expansion in eq.1 can be seen in Fig.3 where we plot
$n_{2}(\omega,\Omega)$, as calculated according to the two band model, and
mark the experimental conditions of the present work with a red spot. As is
seen, the quadratic approximation provides satisfactory agreement in a broad
region around the origin which includes the wavelengths used in the present
work. Using eq.1 and the value for non-degenerate $n_{2}$ we obtain a value
for degenerate nonlinear Kerr index of polycrystalline ZnSe
$n^{\mathrm{poly}}_{2}\approx(0.5\pm 0.1)\times 10^{-18}$ m2/W at
$\lambda$=4.5$\mu$m; the main source of error here comes from the
determination of the pump beam intensity (see Supplementary Material). This
number is similar in magnitude, albeit slightly less than the values obtained
previously for single- Patwardhan  _et al._ (2021) and poly-crystalline ZnSe
samples Jansonas  _et al._ (2022); Ensley and Bambha (2019); Werner  _et al._
(2019).

Lastly, one might be interested in the dependence of the magnitude of cross-
phase modulation on the polarizations of pump and probe beams. The brute force
approach would be to redo the spectral analysis above for all possible
polarization configuration. Such an experiment is certainly doable, albeit
time consuming. However, as we show below, it is in fact not necessary when
one is not after absolute values for susceptibilities, but is only interested
in the ratios between the different components of the nonlinear susceptibility
tensor $\chi^{(3)}_{\alpha\beta\gamma\delta}$. Since XPM was established above
as the main interaction channel between pump and probe pulses, one can take it
for granted. Then we note that the wavelength of the probe $\lambda=1028$nm
lies in the range where the sensitivity of a Si photodiode-based detector has
a strong wavelength dependence (see inset in Fig.5A for the responsivity of
Thorlabs PDA100A2 detector used here). Therefore, any pump-induced changes in
the probe spectrum will be detected by it.

Figure 4:  Signal at the Si-diode photodetector detecting probe beam after it
was modified by the pump through XPM for collinear polarization orientation of
pump and probe beams; Inset: sensitivity of the Si-based photodiode used in
this work as a function of wavelength Si_ ; red vertical line shows the probe
beam wavelength; B) Blue dots: peak-to-peak ($A_{\mathrm{pp}}$) amplitude of
the signal at the detector as a function of angle $\theta$ between pump and
probe polarizations; red line: fit to $a+b\cos(2\theta)$

In Fig.5A we show the transient signal at 1028nm obtained in the standard
transmission-geometry pump-probe configuration with collinear pulses having
collinear polarizations (see Supplementary Material for details). As expected
for XPM, the transient has a bi-polar character, qualitatively similar to
Fig.2. The peak-to-peak amplitude of the signal $A_{pp}$ in Fig.5A can
therefore be used as a measure of XPM, and consequently Kerr index $n_{2}$. In
Fig.5B we plot this quantity as a function of the angle $\theta$ between pump
and probe polarizations (blue dots). This simple angular dependence of the
Kerr index $n_{2}(\theta)$ in a polycrystalline sample is natural and can be
used to find relations between the various components of the susceptibility
tensor. Indeed, when expressed in terms of crystalline nonlinear
susceptibility $\chi_{\alpha\beta\gamma\delta}$, the effective nonlinear
susceptibility of a polycrystalline sample $\chi_{\mathrm{eff}}(\theta)$ must
have the following form (see Supplementary Material):

$n_{2}(\theta)\propto\chi_{\mathrm{eff}}(\theta)=\frac{1}{2}\left(\chi_{xxxx}+\chi_{xxyy}\right)+\\\
+\frac{\cos(2\theta)}{4}\left(\chi_{xxxx}-\chi_{xxyy}+\chi_{xyyx}+\chi_{xyxy}\right)$
(2)

The red solid curve in Fig.5B is a fit
$\chi_{\mathrm{eff}}(\theta)=a+b\cos(2\theta)$ producing an experimental value
for $b/a\approx 1/3$ that can be used to obtain the ratio between the
quantities in the parentheses. For this we note that for a cubic crystal
structure of ZnSe $\chi_{xxxx}\approx\chi_{xxyy}+\chi_{xyyx}+\chi_{xyxy}$
Burns and Bloembergen (1971), and find that for ZnSe at 4.5$\mu$m pump and
1.03$\mu$m probe, $\chi_{xxyy}\approx\chi_{xxxx}/2$ and
$\chi_{xyxy}=\chi_{xyyx}\approx\chi_{xxxx}/4$.

In conclusion, we introduce an alternative technique to measure the nonlinear
Kerr index in the mid-infrared range by studying the effects of cross-phase
modulation on a secondary near-infrared probe beam. Employing this near-IR
beam as a probe allows circumventing the necessity for sophisticated IR
detectors necessary for conventional methods such as z-scan. In order to
relate the measured non-degenerate and the sought-after degenerate mid-IR Kerr
indices, we analyze the frequency dependence of nonlinear refractive index and
with the help of nonlinear dispersive analysis of Sheik-Bahae et al.,
establish a general expression for the frequency dependence of non-degenerate
$n_{2}(\omega,\Omega)$ that is not limited to a specific material. In the
proof-of-principle experiment we measured the Kerr index for a polycrystalline
ZnSe sample to find $n_{2}\approx(0.5\pm 0.1)\times 10^{-18}$ m2/W at
4.5$\mu$m.
See the supplementary material for supporting content.

###### Acknowledgements.

The work was supported by IST Austria. The authors would like to gratefully
acknowledge the help and assistance of Prof. John M. Dudley.

## Author Declarations

Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
D. Lorenc: Investigation (equal); Methodology (equal); Visualization (equal);
Writing (equal). Z. Alpichshev: Investigation (equal); Methodology (equal);
Visualization (equal); Writing (equal).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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CGNLSE modeling. The XPM was modeled using the usual set of coupled nonlinear
Schroedinger equations Agrawal (2013):

$\frac{\partial A_{1}}{\partial z}+\frac{1}{v_{g1}}\frac{\partial
A_{1}}{\partial z}+i\frac{\beta_{21}}{2}\frac{\partial^{2}A_{1}}{\partial
z^{2}}=i\gamma_{1}(\left|A_{1}\right|^{2}+2\left|A_{2}\right|^{2})A_{1}$ (3)

$\frac{\partial A_{2}}{\partial z}+\frac{1}{v_{g2}}\frac{\partial
A_{2}}{\partial z}+i\frac{\beta_{22}}{2}\frac{\partial^{2}A_{2}}{\partial
z^{2}}=i\gamma_{2}(\left|A_{2}\right|^{2}+2\left|A_{1}\right|^{2})A_{2}$ (4)

with $A_{x}$ being the field amplitudes, $v_{gx}$ corresponding group
velocities and $\beta_{2x}$ and $\gamma_{x}$ being GVD coefficients and
nonlinear parameters respectively. Here we ignore all absorptive effects,
justification being that on the one hand the sample is relatively thin to
ignore linear absorption, and that
$\Omega_{\mathrm{pump}}+\omega_{\mathrm{probe}}<\Delta_{\mathrm{gap}}$ so we
can also ignore two-photon absorption. The set was integrated by using the
split-step Fourrier method.

In the course of analyzing the beam interaction for various parameter values,
we have also observed that for the conditions of the present experiment, group
velocity dispersion (GVD) has no observable effect on the simulation outcome
when corresponding coefficients in CGNLSE equations above were taken at their
typical values reported in literature (as in e.g. J. Connolly et al., Proc.
SPIE 181, 141 (1979)). Therefore it is even possible to simply put GVD
coefficients $\beta_{2i}$ equal to zero under these conditions.

Full expression for $n_{2}(\omega,\Omega)$. The approach closely follows the
treatment by Sheik-Bahae et.al.Hutchings  _et al._ (1992). The two-band model
expression for absorption (with Raman terms included) is used to recover the
real Kerr index $n_{2}$ by means of extended Kramers-Kronig analysis Sheik-
Bahae  _et al._ (1991). For reference we write down here the full expression
for $n_{2}(\Omega,\omega)$ calculated in this approximation (two-band model,
two-photon absortion and Raman):

$\displaystyle
n^{\mathrm{2PA+Raman}}_{2}(\omega,\Omega)\propto\frac{\Delta^{9/2}}{\omega^{4}\Omega^{4}}\cdot\left\\{{\left(\Delta-\omega-\Omega\right)^{3/2}}\left({\omega}+{\Omega}\right)^{2}+\right.$
$\displaystyle\left.+\left(\Delta+\omega-\Omega\right)^{3/2}\left({\Omega}-{\omega}\right)^{2}+\right.$
$\displaystyle\left.+\left(\Delta-\omega+\Omega\right)^{3/2}\left({\omega}-{\Omega}\right)^{2}+\right.$
$\displaystyle\left.+\left(\Delta+\Omega+\omega\right)^{3/2}\left(-{\Omega}-{\omega}\right)^{2}-\right.$
$\displaystyle\left.-2\Omega^{2}(\Delta-\Omega)^{3/2}\left[1+\frac{\omega^{2}}{\Omega^{2}}-\frac{3\omega^{2}}{\Omega(\Delta-\Omega)}+\frac{3}{8}\frac{\omega^{2}}{(\Delta-\Omega)^{2}}\right]-\right.$
$\displaystyle\left.-2\Omega^{2}(\Delta+\Omega)^{3/2}\left[1+\frac{\omega^{2}}{\Omega^{2}}+\frac{3\omega^{2}}{\Omega(\Delta+\Omega)}+\frac{3}{8}\frac{\omega^{2}}{(\Delta+\Omega)^{2}}\right]-\right.$
$\displaystyle\left.-2\omega^{2}(\Delta-\omega)^{3/2}\left[1+\frac{\Omega^{2}}{\omega^{2}}-\frac{3\Omega^{2}}{\omega(\Delta-\omega)}+\frac{3}{8}\frac{\Omega^{2}}{(\Delta-\omega)^{2}}\right]-\right.$
$\displaystyle\left.-2\omega^{2}(\Delta+\omega)^{3/2}\left[1+\frac{\Omega^{2}}{\omega^{2}}+\frac{3\Omega^{2}}{\omega(\Delta+\omega)}+\frac{3}{8}\frac{\Omega^{2}}{(\Delta+\omega)^{2}}\right]+\right.$
$\displaystyle\left.+4\Delta^{3/2}\left(\omega^{2}+\Omega^{2}\right)+9\frac{\omega^{2}\Omega^{2}}{\Delta^{1/2}}\right\\}$
(5)

where both $\omega$ and $\Omega$ are complex variables. In the degenerate
limit $\omega$ = $\Omega$ the above expression can be checked to coincide with
the functional dependence of $n_{2}(\omega,\Omega)$ given in Hutchings  _et
al._ (1992).

Pump-probe setup. The beam delivery side remained the same as in the
spectrally-resolved experiment, however, unlike the previous case we chop the
4.5$\mu$m pump beam. The probe beam interacts with pump throughout the sample
and is sampled by a Si-diode-based photodetector (Thorlabs PDA100A2). To
increase the signal-to-noise ratio, the signal is pre-processed through a
boxcar integrator (SRS SR250) before being analyzed as usual by a lock-in
amplifier (SRS SR830) synchronized with the optical chopper modulating the
pump intensity.

Polarization dependence of third-order susceptibility in polycrystalline ZnSe.
In order to derive an expression for the polarization dependence of
$\chi^{(3)}$ and consequently $n_{2}$ in a polycrystalline sample, we will
first write down the expression for the third-order susceptibility
$\chi^{(3)}_{\alpha\alpha\beta\beta}(\omega;\omega,\Omega,-\Omega)$ in a
single crystal sample for arbitrary polarization orientations of the pump and
probe. Here $\omega$ ($\Omega$) is the frequency of the probe (pump) beams
while $\alpha$ ($\beta$) mark the orientation of the polarization of the probe
(pump) beams with respect to the optical axis of the crystal. By using the
standard transformation properties of the susceptibility tensor, it is
straightforward to write:

$\displaystyle\chi^{(3)}_{\alpha\alpha\beta\beta}=\left[\cos^{2}(\alpha)\cos^{2}(\beta)+\sin^{2}(\alpha)\sin^{2}(\beta)\right]\cdot\chi_{xxxx}$
$\displaystyle+\left[\cos^{2}(\alpha)\sin^{2}(\beta)+\sin^{2}(\alpha)\cos^{2}(\beta)\right]\cdot\chi_{xxyy}+$
$\displaystyle+2\sin(\alpha)\cos(\alpha)\sin(\beta)\cos(\beta)\cdot\left[\chi_{xyxy}+\chi_{xyyx}\right]+$
$\displaystyle+\left[\chi_{xxxy}+\chi_{xxyx}\right]\cdot\sin(\beta)\cos(\beta)+\left[\chi_{xyxx}+\chi_{yxxx}\right]\cdot\sin(\alpha)\cos(\alpha)$

here we used the fact that the material in question (ZnSe) has cubic symmetry,
therefore $\chi_{xxxx}=\chi_{yyyy}$, $\chi_{xyxy}=\chi_{yxyx}$, etc. Now to
find the susceptibility $\chi_{\mathrm{eff}}(\theta)$ of a polycrystalline
sample, where $\theta=\beta-\alpha$, we assume that the single-crystal domains
are oriented with a random distribution across the sample, therefore

$\displaystyle\chi_{\mathrm{eff}}(\theta)=\int\frac{d\alpha}{2\pi}\,\,\chi^{(3)}_{\alpha\alpha(\alpha+\theta)(\alpha+\theta)}=$
$\displaystyle=\frac{1}{2}\left(\chi_{xxxx}+\chi_{xxyy}\right)+\frac{\cos(2\theta)}{4}\left(\chi_{xxxx}-\chi_{xxyy}+\chi_{xyyx}+\chi_{xyxy}\right)$

reproducing eq.2 of the main text.

Intensity measurements. The actual intensities as given in the text and used
in the CGNLSE simulation were calculated by measuring:

1.) the respective beam waist diameters using the knife-edge technique:

Figure 5: Transverse scan of the 1030 nm probe beam (left) and the 4500 nm
pump beam

with the corresponding beam-waist diameters of 24$\pm$ 1.4$\mu$m and 63$\pm$
6.1$\mu$m for the 1030 nm probe beam and 4500 nm pump beam respectively. Note
that the knife-edge scan as performed is a 1D technique and hence we estimate
an additional $\pm$10$\%$ may have arisen as a result of the pump beam
ellipticity.

2.) the actual power. Beam power was measured by means of an OPHIR VEGA
powermeter with a 3A-P-V1 thermal head and an estimated uncertainty of
$\pm$10$\%$. Hence we estimate the total uncertainty of the intensity
measurements on the level of $\pm$30$\%$