# Physics-guided adversarial networks for artificial digital image correlation
data generation

David Melching  German Aerospace Center (DLR), Institute of Materials
Research, Linder Hoehe, 51147 Cologne, Germany.  Corresponding author:
<EMAIL_ADDRESS>Erik Schultheis  German Aerospace Center (DLR),
Institute of Materials Research, Linder Hoehe, 51147 Cologne, Germany.  Eric
Breitbarth  German Aerospace Center (DLR), Institute of Materials Research,
Linder Hoehe, 51147 Cologne, Germany.

(March 28, 2023)

###### Abstract

Digital image correlation (DIC) has become a valuable tool in the evaluation
of mechanical experiments, particularly fatigue crack growth experiments. The
evaluation requires accurate information of the crack path and crack tip
position, which is difficult to obtain due to inherent noise and artefacts.
Machine learning models have been extremely successful in recognizing this
relevant information given labelled DIC displacement data. For the training of
robust models, which generalize well, big data is needed. However, data is
typically scarce in the field of material science and engineering because
experiments are expensive and time-consuming. We present a method to generate
synthetic DIC displacement data using generative adversarial networks with a
physics-guided discriminator. To decide whether data samples are real or fake,
this discriminator additionally receives the derived von Mises equivalent
strain. We show that this physics-guided approach leads to improved results in
terms of visual quality of samples, sliced Wasserstein distance, and geometry
score.

Keywords— physics-guided neural networks, generative adversarial networks,
digital image correlation, fatigue crack growth

##  1 Introduction

Digital image correlation (DIC) is a contact-less optical measurement
technique [1]. It computes full-field surface displacements by tracking a
previously applied stochastic random pattern and comparing current images with
a reference image. However, DIC data is subject to inherent noise and
artefacts due to influences such as pattern quality, sensor noise, air
movement, etc. [2].

In recent years, DIC has become an important tool to accompany and evaluate
mechanical experiments [3], especially fatigue crack growth (FCG) experiments.
FCG experiments are of significant importance to determine the lifetime and
damage tolerance of critical engineering structures and components that are
subjected to non-constant loads [4]. The DIC data serves as the basis for the
subsequent mechanical evaluation of fracture mechanical quantities like the
stress intensity factors [5] and $J$-integral [6]. For the evaluation, the
spatial location of the crack and especially the exact crack tip position is
crucial. However, due to the inherent noise of DIC, this information can be
difficult to obtain. To solve this problem in an automated way, specially
designed convolutional neural networks have been successfully applied [7, 8,
9].

For these powerful data-driven models to work reliably, they need a diverse
set of labelled training. However, data is sparse in our domain, experiments
are expensive, and manual labelling is extremely tedious and time-consuming.
To address the problem of insufficient data, Strohmann et al. [8] added
artificial training data in the form of finite element simulations. However,
simulations are idealized and lack the characteristic DIC noise. In this case,
it is difficult to imitate the DIC characteristics by simply adding noise to
the data.

In the field of data-driven modeling, generative adversarial networks (GANs)
have proven to be powerful data generators. GANs are a generative,
unsupervised approach to machine learning based on deep neural networks
trained using an adversarial process. Recently, deep convolutional GANs (DC-
GANs) produced state-of-the-art results in many generative and translative
computer vision tasks such as image generation [10, 11, 12], style transfer
[13], image-to-image translation [14, 15], text-to-image translation [16], or
semantic image synthesis [17].

However, training deep neural networks typically requires large amounts of
data, which are often not available in the scientific domain. For example, it
is not possible to test an entire aircraft just to generate training data.
Without sufficient data, deep learning models are often unreliable and poorly
generalize to domains not covered by the training data. To overcome this
problem, efforts have been made to integrate fundamental physical laws and
domain knowledge into machine learning models. Karpatne et al. [18] describe
theory-guided data science as an emerging paradigm aiming to use scientific
knowledge to improve the effectiveness of data science models. They presented
several approaches for integrating domain knowledge in data science. Daw et
al. [19] proposed a physics-guided neural network (PGNN) by adding a physics-
based term to the loss function of a neural network to encourage physically
consistent results. Karniadakis et al.[20, 21] coined the term physics-
informed neural networks (PINNs) - a deep learning framework which enables the
combination of data-driven and mathematical models described by partial
differential equations. Yang et al. [22] combined this approach with GANs by
adding a physics-based loss to the generator and, recently, Daw et al. [23]
introduced a physics-informed discriminator GAN, which physically supervises
the discriminator instead.

In this work, we generate synthetic DIC displacement data using GANs. Our GAN
framework is based on the classical convolutional DC-GAN architecture [24]. We
incorporate mechanical knowledge by using a physics-guided discriminator. In
addition to the generated displacement data, this discriminator receives the
derived equivalent strain according to von Mises [25] as an additional input
feature. This mechanically motivated physical feature guides the adversarial
training process to physically consistent generated data. The synthetically
generated data samples can be used to increase data variation of a given DIC
dataset. Although this synthetic data is not labelled, it has the potential to
also improve supervised machine learning tasks, e.g. by using it for
unsupervised pre-training [26] or label propagation [27]. We use DIC data from
FCG experiments of a middle tension specimen manufactured from an aluminium-
based alloy (see Section 2.2) to train several GANs. To demonstrate the merits
of the physics-guided method, we compare the results of the physics-guided
approach to a classical one in terms of visual quality of generated samples
(see Section 3.2) and distance of fake data distributions to the real training
data. The latter is quantified using the sliced Wasserstein distance (SWD)
[28] and the geometry score (GS) [29] (see Section 2.4).

##  2 Methodology

Generative Adversarial Networks (GANs) are generative machine learning models
learned using an adversarial training process [30]. In this framework, two
neural networks - the generator $G$ and the discriminator $D$ \- contest
against each other in a zero-sum game. Given a training dataset characterized
by a distribution $p_{\text{data}}$, the generator aims to produce new data
following $p_{\text{data}}$ while the task of the discriminator is to
distinguish generated data samples from actual training data samples.

Given a noise vector $z$ sampled from a prior, e.g. the standard normal
distribution, the generator outputs data samples $G(z)$, called fake samples,
trying to follow the training data distribution $p_{\text{data}}$. Given a
real or fake sample, the discriminator is supposed to decide whether it is
real or fake by predicting the probability of it belonging to the training
dataset.

Both models $G$ and $D$ are trained simultaneously contesting against each
other in a two-player zero-sum minimax game with the value function $V(G,D)$:

$\min_{G}\max_{D}V(G,D):=\mathbb{E}_{x\sim
p_{\text{data}}}\left[\log(D(x))\right]+\mathbb{E}_{z}\left[\log(D(G(z)))\right]$
(1)

This means $D$ is trained to minimize the loss

$L_{D}=-\log(D(x))-\log(D(G(z))),$ (2)

whereas $G$ is trained to minimize the loss

$L_{G}=-\log(D(G(z))).$ (3)

As the discriminator gets better at identifying fake samples $G(z)$, the
generator has to improve on generating samples which are more similar to the
real training samples $x\sim p_{\text{data}}$. We refer to [30] for further
details of the training algorithm.

###  2.1 Digital image correlation

DIC is a contact-less, optical method to obtain full-field displacements and
strains. It is widely applied in science and engineering to quantify
deformation processes. In experimental mechanics, it is used to monitor and
evaluate fatigue crack growth (FCG) experiments [31] by determining fracture
mechanical parameters like stress intensity factors (SIFs) [5] or J-integral
[6]. Essentially, DIC measurements are based on the comparison of a current
image with a reference image using tracking and image registration techniques.
The cross correlation method requires a random speckle pattern on the sample
surface. Various external and internal influences such as illumination, air
movement, vibrations, facet size and spacing, camera settings, sensor noise,
pattern quality, etc. lead to inherent noise in the DIC data. Our goal is to
generate realistic DIC data synthetically with GANs that incorporate these
types of uncertainties.

###  2.2 Training data

As training dataset, we use planar displacement fields $u=(u_{x},u_{y})$
obtained during FCG experiments of the aluminium alloy AA2024-T3 using a
commercial GOM Aramis 12M 3D-DIC system. Details on the general experimental
setup can be found in [8]. For the dataset, we use one FCG experiment
performed on a middle tension (MT) specimen (width $w=160$ mm, thickness $t=2$
mm) at a maximal stress of $\sigma_{\text{max}}=47$ MPa and $R=0.3$ with 20
load cycles per second. Every 0.5 mm of crack growth (measured by direct
current potential drop), 3 images (at minimal load
$F_{\text{min}}=R\cdot\sigma_{F_{max}}$, mean load
$F_{\text{min}}+(F_{\text{max}}-F_{\text{min}})/2$ and maximum load
$F_{\text{max}}$) were acquired. From the resulting DIC dataset, we take the
planar displacements $u_{x}$ and $u_{y}$ of the specimen and linearly
interpolate the displacements from an area of $70\times 70\,\text{mm}^{2}$ of
the right-hand side of the specimen on an equidistant $256\times 256$ pixel
grid. This procedure results in 838 training data samples of shape $2\times
256\times 256$, where the first dimension stands for the $x$\- and $y$\-
displacements. Each of the two channels is normalized to $[-1,1]$ by the min-
max-scaling and shift

$u_{\rm scaled}=2\cdot\frac{u-u_{\rm min}}{u_{\rm max}-u_{\rm min}}-1$ (4)

such that the minimum and maximum are mapped to $-1$ and $1$, respectively.

###  2.3 Physics-guided DIC-GAN

We aim to generate artificial DIC displacement data using deep convolutional
GANs. For this, we mainly follow the architectural guidelines from [24].
However, in order to reduce checkerboard artefacts, we choose nearest-neighbor
upsampling instead of transposed convolutions in the generator [32].

Figure 1: Physics-guided DIC-GAN framework: A deep convolutional generator
creates fake DIC displacement data from noise. The discriminator gets the
derived corresponding von Mises equivalent strain as an additional feature to
decide whether samples are real or fake.

Generator. The input of the generator network is a $n$-dimensional vector $z$
randomly sampled from a standard normal distribution. For definiteness, we
choose $n=5$ throughout all our training experiments. First, the random vector
$z$ passes a fully-connected layer with $8\cdot 8\cdot 512=32768$ neurons,
batch-normalization [33], and ReLU activation [34]. The output of this layer
is then reshaped into 512 features of size $8\times 8$. After that, these
features are successively doubled in size using the base block (upsampling
$\rightarrow$ batch normalization $\rightarrow$ ReLU $\rightarrow$
convolution) four times. The final block ends with a $\tanh$ activation
instead of ReLU. Therefore, in accordance with the training data, the
generator outputs fake samples with pixel values between $-1$ and $1$.

Discriminator. For the discriminator, we implemented the following two
approaches:

  1. 1.

Classical: The discriminator gets real and fake pairs of $x$\- and $y$\-
displacement fields $(u_{x},u_{y})$ and predicts a (pseudo-)probability of the
sample being real. We refer to this approach as classical DIC-GAN.

  2. 2.

Physics-guided: In addition to the displacement fields, the corresponding von
Mises equivalent strain $\varepsilon_{\rm vm}$ is calculated based on the
generated and real displacement fields and the discriminator gets the triple
$(u_{x},u_{y},\varepsilon_{\rm vm})$ as input in order to decide whether it is
fake or real. For small strains, the von Mises equivalent strain is defined as
the scalar quantity

$\varepsilon_{\rm vm}=\sqrt{\frac{2}{3}\varepsilon_{\rm dev}:\varepsilon_{\rm
dev}},\qquad\text{where }\varepsilon_{\rm dev}=\varepsilon-\frac{1}{3}{\rm
tr}(\varepsilon)$ (5)

denotes the deviatoric part of the three-dimensional strain tensor

$\varepsilon=\begin{pmatrix}\varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz}\\\
\varepsilon_{xy}&\varepsilon_{yy}&\varepsilon_{yz}\\\
\varepsilon_{xz}&\varepsilon_{yz}&\varepsilon_{zz}\\\ \end{pmatrix}$ (6)

In case of plane stress, $\varepsilon_{xz}=\varepsilon_{yz}=0$ and
$\varepsilon_{zz}=-\nu(\varepsilon_{xx}-\varepsilon_{yy})$. Assuming volume
constancy with a Poisson ratio of $\nu=1/2$, Formula (5) simplifies to

$\varepsilon_{\rm
vm}=\frac{2}{\sqrt{3}}\sqrt{\varepsilon_{xx}^{2}+\varepsilon_{yy}^{2}+\varepsilon_{xy}^{2}+\varepsilon_{xx}\varepsilon_{yy}},$
(7)

We use Formula (7) for the physics-guided discriminator. Therfore, we
numerically approximate the strains using finite differences, e.g.

$\varepsilon_{xy}(x,y)=\frac{\partial}{\partial
y}u_{x}(x,y)=\frac{u_{x}(x,y+h)-u_{x}(x,y)}{h}+\mathcal{O}(h).$ (8)

To guarantee differentiability, the square-root function is smoothed by using
$\sqrt{\cdot+\delta}$ with $\delta\ll 1$. We refer to this approach as
physics-guided DIC-GAN (see Figure 1).

In both cases, we choose the same model architecture for the discriminator.
The input of size $2\times 256\times 256$, or $3\times 256\times 256$ in case
of the physics-guided discriminator, is successively downsampled to the size
$1\times 32\times 32$ using three blocks of strided convolutions, batch
normalization, and LeakyReLU activation [35], where
$\text{LeakyReLU}(t)=\max(\alpha t,t)$ with $\alpha=0.2$. The extracted
features are then flattened and pass the last fully-connected layer with one
output neuron and sigmoid activation. The output is a number between 0 and 1
and is interpreted as the probability of the sample being real.

###  2.4 Evaluation of GANs

In classical supervised learning, a model is trained by minimizing a specific
loss (e.g. mean squared error), which quantitatively compares model
predictions with the expected target. After training, models can be evaluated
and compared by calculating the loss (and accuracy) for independent labeled
test data. GAN generators, however, are trained in an adversarial fashion
using a second model (the discriminator) to classify generated data as real or
fake. Both models are trained simultaneously to maintain an equilibrium.
Therefore, there is no natural objective function to evaluate GAN generators
quantitatively. Instead GANs are evaluated by assessing the quality and
variation of generated data. This is typically achieved by visual inspection
of generated samples or by calculating the inception score (IS) [36] and
Fréchet inception distance (FID) [37]. However, in case of DIC data, several
domain experts would be needed to objectively grade the visual quality of
generated samples. Moreover, quantitative metrics like IS or FID can only be
employed for natural images since they use image classification networks like
Inception [38], which are pre-trained on ImageNet [39]. Therefore, in addition
to a visual examination of generated samples in Section 3.2, we use metrics
which are independent of the data type and do not use any pre-trained models.
More precisely, we use the following two metrics:

Sliced Wasserstein distance. In mathematics, the Wasserstein distance is a
natural distance function between two distributions. Intuitively, it can be
viewed as the minimal cost of transforming one of the distributions into the
other. In case of image-like datasets $X=\\{X_{n}\\}_{n=1,\dots N}$ and
$Y=\\{Y_{n}\\}_{n=1,\dots N}$ with same number of samples $N$ and image sizes
$c\times h\times w$, where $c$ is the number of channels and $h$ and $w$
denote the height and width of images, respectively, the (quadratic)
Wasserstein distance is given by,

$W(X,Y)^{2}=\min_{\pi}\sum_{i,j,k,n}|X_{n}(i,j,k)-Y_{\pi(n)}(i,j,k)|^{2},$ (9)

where the minimum is taken over all permutations $\pi$ of the set $\\{1,\dots
N\\}$ [28]. Due to the high dimensionality of images and the large number of
samples, the exact computation of the Wasserstein distance is computationally
infeasible. This is because the number of permutations scales exponentially
with the number of samples $N$. Therefore, instead of (9), we use the sliced
Wasserstein distance (SWD) introduced in [28] as an approximation, which is
amendable for efficient numerical computation. The main idea of slicing is to
map the high dimensional image data from $\mathbb{R}^{c\times h\times w}$ onto
one-dimensional slices. On these slices, the Wasserstein distance can be
calculated in loglinear time by using the ordered structure of one-dimensional
Euclidean space. The sliced Wasserstein distance is defined as,

$\tilde{W}(X,Y)^{2}=\int_{\theta\in\Omega}\min_{\pi_{\theta}}\sum_{n=1}^{N}|\langle
X_{n}-Y_{\pi_{\theta}(n)},\theta\rangle|^{2}d\theta,$ (10)

where $\Omega=\\{\theta\in\mathbb{R}^{c\times h\times w}:\|\theta\|=1\\}$
denotes the unit sphere. We refer to [28, 10] and Section 3.3 below for
further details.

Geometry score. Introduced by Khrulkov & Oseledets [29], the geometry score
(GS) allows to quantify the performance of GANs trained on datasets of
arbitrary nature. It measures the similarity between the real dataset $X_{\rm
real}$ and a generated one $X_{\rm fake}$ by comparing topological properties
of the underlying low-dimensional manifolds [40]. The detailed quantitative
characterization of the underlying manifold of a given dataset $X$ is usually
very hard. The core idea of [29] is to choose random subsets $L\subset X$
called landmarks and to build a family of simplicial complexes, parametrized
by a non-negative, time-like persistance parameter $\alpha$. For small
$\alpha$, the complexes consist of a disjoint union of points. Increasing
$\alpha$ adds more and more simplicies finally leading to one single connected
blob. For each value of $\alpha$, topological properties of the corresponding
simplicial complex, namely the number of one-dimensional holes in terms of
homology, $\beta_{1}(\alpha)$, are calculated (see, e.g., [41]). From this,
the authors propose to compute Relative Living Times (RLTs) for every number
of holes that was observed [29]. For each non-negative number $i$, the RLT is
the amount of the time when exactly $i$ holes were present relative to the
overall time $\alpha_{\text{max}}$ after which everything is connected. More
precisely,

$\textrm{RLT}(i,X,L)=\frac{\mu\left(\\{\alpha\in[0,\alpha_{\text{max}}]:\beta_{1}(\alpha)=i\\}\right)}{\alpha_{\text{max}}},$
(11)

where $\mu$ denotes the standard Lebesgue measure. Since the RLTs depend on
the choice of landmarks $L$, we choose a collection of $n$ random sets of
landmarks $L_{j}$ and define the Mean Relative Living Times (MRLTs) as

$\textrm{MRLT}(i,X)=\frac{1}{n}\sum_{j=1}^{n}\textrm{RLT}(i,X,L_{j}).$ (12)

The MRLT is a discrete probability distribution over the non-negative
integers. It can be interpreted as the probability of the manifold having
exactly $i$ one-dimensional holes (on average). The $L^{2}$-distance between
the MRLT distributions of $X_{\rm real}$ and $X_{\rm fake}$ defines a measure
of topological similarity between the real dataset and the generated one,
called geometry score (GS):

$\textrm{GS}(X_{\rm fake},X_{\rm real})=\sum_{i=0}^{i_{\rm
max}-1}\left|\textrm{MRLT}(i,X_{\rm fake})-\textrm{MRLT}(i,X_{\rm
real})\right|^{2},$ (13)

where $i_{\rm max}$ is an upper bound on the number of holes. We refer to [29]
for further theoretical details and to Section 3.4 for the choice of
hyperparameters and results in our case.

##  3 Results and discussion

In order to demonstrate the effectiveness of the method and to compare the
classical and the physics-guided discriminator approach, we trained 10
randomly initialized classical and physics-guided DIC-GANs each for 100
epochs. Moreover, we trained two classical and physics-guided DIC-GANs each
for 1000 epochs in order to compare both architectures after long training
runs. The training setup is described in Section 3.1 below. The trained models
are evaluated qualitatively and quantitatively by using the following
criteria:

  * •

Visual inspection of samples (Section 3.2)

  * •

Sliced Wasserstein distances (Section 3.3)

  * •

Geometry scores (Section 3.4)

A summary of the results can be seen in Table 1. In short, the physics-guided
DIC-GAN approach leads to visually better results after 100 epochs and overall
to measurably better results. For a detailed discussion, we refer to the
sections below.

Model | Epoch | Visual quality |  GS $\times 10^{3}$ |  SWD $\times 10^{3}$
---|---|---|---|---
Classical | 100 | low | 183.49 | 151.70
Physics-guided | 100 | medium | 157.58 | 82.96
Classical | 1000 | high* | 23.83 | 83.19
Physics-guided | 1000 | high* | 1.93 | 61.61
Table 1: Subjective visual quality, calculated geometry score (GS), and sliced
Wasserstein distance (SWD) (lower is better) for different DIC-GAN model
architectures and training lengths. (*) apart from garbage samples (see Figure
4)

###  3.1 Training procedure

Before training, the filters of the convolution layers in both generator and
discriminator network are initialised randomly from a normal distribution with
zero mean and a standard deviation $0.02$. In contrast, the weights of the
batch normalization layers are initialised from mean $1$ and standard
deviation of $0.02$, whereas the biases are initialised with zeros.

For training, we choose the Adam optimizer [42] with a learning rate of
$0.002$, momentum parameters of $\beta_{1}=0.5,\,\beta_{2}=0.999$, and a batch
size of $8$. We noticed that occasionally models suffer from mode collapse
during training. This means that the generator always outputs the same or
visibly similar fake data samples and stops to learn. This problem is well-
known and still part of active research. Popular strategies to overcome
convergence issues of GANs regularize or perturb the discriminator [43, 44] or
by using a more sophisticated loss function [45]. In our case, if mode
collapse happened, we restarted the training and discarded the collapsed
model. All neural networks and training loops were implemented using PyTorch
[46]. The hardware for the training was an NVIDIA RTX8000 graphics card.

###  3.2 Visual evaluation

We begin with a visual inspection of the generated data and compare real
training data to representative samples generated by the classical DIC-GAN and
the physics-guided DIC-GAN. We refer to fake samples generated by the
classical or physics-guided DIC-GAN generators as classical or physics-guided
DIC-GAN samples, respectively.

Figure 2 shows real DIC data obtained during FCG experiments as described in
Section 2.2. The figure contains planar displacements and von Mises equivalent
strains of 9 data samples. Images belong together in the sense that the $x-$
displacement of the first sample is located at the top left of the left
column. The corresponding $y-$ displacement is located at the same position in
the middle column, and the corresponding calculated equivalent strain is
located at the same position in the right column. Here, the crack path as well
as the characteristic crack tip field is clearly visible.

Figure 2: Real samples of DIC displacement data (left and middle) and
corresponding von Mises equivalent strains (right).

In Figure 3, we see fake samples after 100 epochs of GAN training. We can
often identify the initial crack on the left edge and the crack path. Whereas
most generated displacements are visually close to real displacements,
significant differences are revealed in the von Mises strains, which are
calculated afterwards. Especially classical DIC-GAN samples contain
inconsistencies between $x-$ and $y-$ displacements and visual artefacts. This
leads to large-scale vortexes and small-scale noise in the von Mises strains
①. Although far from being perfect, physics-guided DIC-GAN samples contain
significantly less of these artefacts and inconsistencies and visually capture
the inherent noise of the DIC system much better than classical DIC-GAN
samples. Nevertheless, most fake samples are still visually distinguishable
from real samples. In order to make sure the models are fully converged, we
also performed some longer training runs.

Figure 3: Visual comparison of generated classical and physics-guided DIC-GAN
samples after 100 epochs of training. Classical DIC-GAN samples show a larger
noise level in the von Mises strains compared to phyiscs-guided DIC-GAN
samples ①.

Figure 4 shows fake samples after 1000 epochs. At this stage, the models are
well converged and the visual difference between classical and physics-guided
DIC-GAN samples is mainly disappeared. In general, the fake samples of both
models show much better visual quality and less artefacts and inconsistencies
compared to fake samples of generators trained for only 100 epochs ③. However,
few samples suffer from severe inconsistencies and are qualitatively inferior
②. We refer to these failures as garbage samples. Apart from these outliers,
the vast majority of samples (of both models) are visually indistinguishable
from real samples. Nevertheless, domain experts may notice that the
characteristic crack tip field still seems unphysical in the fake samples
especially when compared to real samples with long cracks.

Figure 4: Visual comparison of generated classical and physics-guided DIC-GAN
samples after 1000 epochs of training. Both models seem to produce mostly good
samples ③ but also few garbage samples ②.

###  3.3 Sliced Wasserstein distances

For a thorough comparison of GANs, one needs to inspect a large number of fake
samples. Doing this manually, would be very tedious and subjective. Instead,
one should compare the results using meaningful, quantitative metrics.

For this, we follow [10] and calculate the sliced Wasserstein distances (SWD)
introduced in Section 2.4 between fake data samples and real data samples on
various scales. These scales are introduced by building a 5-level Laplacian
pyramid [47] with resolutions $16\times 16$, $32\times 32$, $64\times 64$,
$128\times 128$, $256\times 256$. Each pyramid level corresponds to a specific
spatial resolution. For each level, we compute the SWD between the training
dataset and a generated fake dataset of the same size. More precisely, the
SWDs are calculated between datasets of random $7\times 7$ patches of the
pyramid samples. The patches are pre-processed by normalizing each channel
(i.e. $x$ and $y$ displacement) to mean 0 and standard deviation 1. To reduce
uncertainty, we average the SWDs of ten runs with randomly sampled fake data.
Since there are less unique patches for low resolutions, we adapt the number
of random patches depending on the pyramid level. For the five resolutions,
$16\times 16$, $32\times 32$, $64\times 64$, $128\times 128$, and $256\times
256$, we use 128, 256, 512, 1024, 2048 patches, respectively. The integral in
Equation (10) is approximated by choosing 512 random slices and average the
results. We implemented a GPU-enabled version of the code from [10] using the
PyTorch [46] framework.

At least intuitively, a small SWD shows that the fake and real samples are
similar. At low resolution (e.g. $16\times 16$) only large-scale features like
the crack length are visible and a small SWD would indicate that the variation
of crack lengths in the fake dataset is similar to the training dataset. At
high resolution (e.g. $256\times 256$) very fine-grained structures like the
inherent DIC noise is encoded in the patches.

Figure 5 shows the calculated SWDs of classical and physics-guided DIC-GAN
samples after 100 epochs of training. In order to estimate uncertainty, we
trained 10 randomly initialized models each with the classical and the
physics-guided DIC-GAN architecture. The main observation is that for all
resolutions, physics-guided samples are closer to the training data than
classical DIC-GAN samples. This indicates that physics-guided DIC-GAN samples
are better in quality and variation. Especially for the high resolution
$256\times 256$, the SWDs show a large gap and confirm our visual observation
of artefacts and unphysical noise as seen in the classical DIC-GAN samples in
Figure 3. Nevertheless, the results can be significantly different for each
trained generator. This fact is reflected in the large error bars of the SWDs.

Figure 5: Comparison of SWDs between classical DIC-GAN (left) and physics-
guided DIC-GAN (right) trained for 100 epochs. The boxplot intervals range
from the minimal to the maximal SWDs. The box includes ranges from the 25% to
the 75% quantile and shows the median.

The results after 1000 training epochs are displayed in Figure 6. Here, we
used 2 training runs for each GAN architecture. As expected, the distances are
all smaller than after 100 epochs. In contrast to the results after 100 epochs
(cf. Figure 5), both GAN architectures are closer together. However, the
physics-guided DIC-GAN samples have significantly smaller SWDs for the fine
resolution $256\times 256$ and the low resolutions $16\times 16$ and $32\times
32$. This suggests that after 1000 epochs of training the physics-guided
samples are still closer to the real samples in terms of quality and
variation. Nevertheless, the few garbage samples seen in Figure 4 ② could
influence the SWDs especially at smaller pyramid levels.

Figure 6: Comparison of SWDs between classical DIC-GAN (left) and physics-
guided DIC-GAN (right) trained for 1000 epochs. The boxplot intervals range
from the minimal to the maximal SWDs. The box includes ranges from the 25% to
the 75% quantile and shows the median.

###  3.4 Geometry scores

To compare the geometry score (GS) introduced in Section 2.4 of different
trained GANs, we generated fake datasets with the same number of samples
$N=838$ as the training dataset. To calculate the MRLTs of the real and fake
datasets, we mainly follow the recommendations in [29]. We set $i_{\rm
max}=100$ and use $n=1000$ random landmarks. The number of samples in each
landmark is $64$. The maximal persistance time $\alpha_{\rm max}$ is
proportional to the maximal pairwise Euclidean distance between samples in
each landmark, i.e. for $j=1,\dots,n$:

$\alpha_{\rm max}^{j}=\gamma\max({\rm
dist}(L_{j},L_{j})),\quad\gamma=\frac{1}{128}/\frac{N}{5000}.$ (14)

We used the implementation from [29] to calculate the MRLTs.

Figure 7 shows the distributions of MRLTs after 100 epochs of training. The
error band originates from the uncertainty induced by the random landmarks
and, even more so, from the 10 different models trained for each GAN
architecture. This results in large variations of calculated MRLTs.
Nevertheless, on average the phyiscs-guided DIC-GAN distribution is closer to
the MRLTs of the real data distribution than the classical DIC-GAN
distribution. This observation is quantitatively reflected in a smaller mean
GS of the phyiscs-guided models (see Table 1). However, both fake data
distributions are still far away from the real data distribution and the GSs
are large.

Figure 7: Comparison of MRLT distributions between the real dataset and fake
datasets generated by classical and physics-guided DIC-GAN after 100 epochs of
training.

In Figure 8, we see the MRLT distributions after 1000 epochs of training. Both
GAN results are much closer to the real data than after 100 epochs and the
phyiscs-guided DIC-GAN MRLTs almost coincide with the real data MRLTs. This
accordance is shown in the calculated GSs in Table 1 as well.

Figure 8: Comparison of MRLT distributions between the real dataset and fake
datasets generated by classical and physics-guided DIC-GAN after 1000 epochs
of training.

##  4 Conclusion

We introduced a machine learning framework to generate synthetic full-field
DIC displacements by learning the underlying data distribution from a
sufficiently large experimental dataset. The training data was obtained during
fatigue crack growth experiments of the aluminium alloy AA2024-T3. In contrast
to finite element simulations, our method is able to produce realistic data
with inherent DIC noise. However, data-driven methods typically lack an
understanding of physics and tend to overfit on the known training data.

Our approach is based on deep convolutional generative adversarial networks.
The main novelty compared to the classical approach is a physics-guided
discriminator. This discriminator, in addition to the generated $x$\- and
$y$\- displacement fields, gets also the derived von Mises equivalent strain
as input. This enables the discriminator to detect physical inconsistencies in
the generated fake samples more easily, thus enhancing the training process.

In order to evaluate trained generator models on an objective basis, we used
two quantitative metrics. First, the sliced Wasserstein distance (SWD) between
real and fake samples and, secondly, the geometry score (GS) approximating the
topological distance between a generated data manifold and the training data
manifold.

We observed superior performance of the physics-guided DIC-GAN compared to the
classical DIC-GAN approach. This result was observed by visual evaluation of
generated samples and confirmed by lower SWDs and GSs of the physics-guided
models. Both, SWD and GS, proved themselves to be valuable evaluation metrics.
They are useful to identify mode collapse and to select the best trained
models. Our findings confirm that hybrid models, which combine data-driven
methods with physical domain knowledge, can lead to more powerful generative
models and faster training.

The visual inspection revealed a varying sample quality. Especially the
converged models after 1000 epochs of training, apart from mostly good
samples, produce few garbage samples. Although the number of these garbage
samples is model-dependent, we were not able to avoid their occurrence
completely. Moreover, we still face the issue of (local) non-convergence and
mode collapse. To overcome these issues, one could try to stabilize training
using suitable regularization techniques [48, 44].

Another open problem concerns the control of boundary conditions like the
crack path and external force. In contrast to finite element simulations, with
our approach it is not possible to control them. This challenge could be
tackled by using a conditional GAN framework instead [49] and is part of
current research.

##  5 Acknowledgements

We acknowledge the financial support of the DLR-Directorate Aeronautics.

##  6 Data availability

The code and training data will be publicly available on Github
(https://github.com/dlr-wf) and Zenodo
(https://doi.org/10.5281/zenodo.7737880).

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May 2023

††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work††thanks: Corresponding author:
<EMAIL_ADDRESS>
# Optical contrast analysis of $\alpha$-RuCl3 nanoflakes on oxidized silicon
wafers

Tatyana V. Ivanova  Institute of Photonics and Quantum Sciences, SUPA, Heriot-
Watt University, Edinburgh EH14 4AS, UK  Daniel Andres-Penares  Institute of
Photonics and Quantum Sciences, SUPA, Heriot-Watt University, Edinburgh EH14
4AS, UK  Yiping Wang  Department of Physics, Boston College, Chestnut Hill, MA
02467, USA  Jiaqiang Yan  Materials Science and Technology Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, USA  Daniel Forbes  Institute
of Photonics and Quantum Sciences, SUPA, Heriot-Watt University, Edinburgh
EH14 4AS, UK  Servet Ozdemir  Institute of Photonics and Quantum Sciences,
SUPA, Heriot-Watt University, Edinburgh EH14 4AS, UK  Kenneth S. Burch
Department of Physics, Boston College, Chestnut Hill, MA 02467, USA  Brian D.
Gerardot  Institute of Photonics and Quantum Sciences, SUPA, Heriot-Watt
University, Edinburgh EH14 4AS, UK  Mauro Brotons-Gisbert  Institute of
Photonics and Quantum Sciences, SUPA, Heriot-Watt University, Edinburgh EH14
4AS, UK

###### Abstract

$\alpha$-RuCl3, a narrow-band Mott insulator with large work function, offers
intriguing potential as a quantum material or as a charge acceptor for
electrical contacts in van der Waals devices. In this work, we perform a
systematic study of the optical reflection contrast of $\alpha$-RuCl3
nanoflakes on oxidized silicon wafers and estimate the accuracy of this
imaging technique to assess the crystal thickness. Via spectroscopic micro-
ellipsometry measurements, we characterize the wavelength-dependent complex
refractive index of $\alpha$-RuCl3 nanoflakes of varying thickness in the
visible and near-infrared. Building on these results, we simulate the optical
contrast of $\alpha$-RuCl3 nanoflakes with thicknesses below $100$ nm on
SiO2/Si substrates under different illumination conditions. We compare the
simulated optical contrast with experimental values extracted from optical
microscopy images and obtain good agreement. Finally, we show that optical
contrast imaging allows us to retrieve the thickness of the RuCl3 nanoflakes
exfoliated on an oxidized silicon substrate with a mean deviation of $-0.2$ nm
for thicknesses below 100 nm with a standard deviation of only 1 nm. Our
results demonstrate that optical contrast can be used as a non-invasive, fast,
and reliable technique to estimate the $\alpha$-RuCl3 thickness.

##  I Introduction

Van der Waals materials offer an unprecedented possibility to design and
fabricate two-dimensional (2D) heterostructure devices due to a broad palette
of atomically thin crystals with unique optical, electric, and magnetic
properties. Among the different van der Waals layered crystals, the narrow-
band Mott insulator $\alpha$-RuCl3 has emerged as a new building block for
achieving local charge control in 2D van der Waals heterostructures. Its
narrow electronic bands and large work function make $\alpha$-RuCl3 (hereafter
RuCl3) an excellent atomic crystalline charge acceptor, enabling modulation
doping of several atomically thin crystals even when just a single RuCl3 layer
is employed [1]. Recently, RuCl3 has been used to demonstrate low-resistance
Ohmic contact for p-type WSe2 at low temperature and low carrier densities [2,
3], overcoming the well-known challenges associated with achieving low-
resistance electrical contacts to monolayer transition-metal dichalcogenides
semiconductors [4, 5]. Furthermore, the unique electronic structure of RuCl3
itself has attracted a lot of attention due its complex magnetic interactions,
including access to a Kitaev quantum spin liquid phase [6, 7, 8, 9]. Combined,
these properties position 2D RuCl3 as a versatile building block in the van
der Waals platform, with the potential to incorporate novel magnetic states
[10] and the resulting topological excitations into 2D devices [1].

Similar to other 2D materials, high quality RuCl3 nanoflakes are usually
obtained by mechanical exfoliation from bulk crystals on oxidized silicon
wafers [11], which typically results in stochastic nanoflakes with randomly
varying thicknesses, lateral dimensions, and spatial locations on the
substrate. In this context, an experimental technique that allows a fast, non-
destructive, and accurate estimation of the thickness of the RuCl3 nanoflakes
would represent a valuable asset in the fabrication of 2D heterostructures
incorporating this material. Although optical microscopy is routinely employed
as initial step in the thickness estimation of mechanically exfoliated RuCl3
[12, 3], to our knowledge, a systematic study of the optical reflection
contrast of RuCl3 nanoflakes on oxidized silicon wafers is still missing.

Here, we perform spectroscopic ellipsometry measurements at room temperature
on mechanically exfoliated RuCl3 nanoflakes of different thickness for photon
wavelengths in the visible and near infrared range. Our measurements allow us
to estimate the wavelength-dependent complex refractive index of RuCl3
nanoflakes in this spectral range, confirming their thickness-independent
optical properties. Building on these results, we exploit the estimated
complex refractive index to calculate the optical contrast of RuCl3 nanoflakes
between $\sim 2-100$ nm on SiO2/Si substrates under different illumination
conditions. We compare the calculated optical contrast with the thickness-
dependent experimental optical contrast extracted from optical microscopy
images, showing good agreement between the simulated and experimental values.
The good agreement of the calculated and experimental optical contrast allows
us to predict the monochromatic illumination wavelengths that maximise the
optical contrast response of RuCl3 nanoflakes below 10 nm for two standard
thicknesses (90 and 295 nm) of the SiO2 layer on a Si substrate. Finally, we
show that for RuCl3 nanoflakes in the range $2-100$ nm imaged with our
calibrated standard optical microscope, the crystal thickness estimated
exclusively from the optical contrast analysis shows only a mean deviation of
$-0.2$ nm from the value measured by a combination of atomic force microscopy
(AFM) and spectroscopic ellipsometry, with a standard deviation close to the
thickness of a single RuCl3 monolayer. Our results provide valuable
information about the optical properties of 2D RuCl3 flakes in the visible and
near infrared, which are crucial to exploit this material in 2D nanodevices.
Moreover, we show that optical contrast can be used as a non-invasive, fast,
and reliable technique to estimate RuCl3 crystal thicknesses.

##  II Complex refractive index of RuCl3 in the visible and near-infrared

We begin by performing spectroscopic micro-ellipsometry measurements at room
temperature on RuCl3 nanoflakes with thicknesses between $\sim$3 and 40 nm for
photon wavelengths in the visible and near infrared ($\sim$1.4 - 3.1 eV). The
RuCl3 nanoflakes were obtained by mechanical exfoliation from bulk crystals
directly on Si substrates with a top SiO2 layer with a nominal thickness of
295 nm. The thickness of each nanoflake was measured by AFM. The spectroscopic
micro-ellipsometry measurements were carried out using an Accurion EP4 imaging
ellipsometer with a spatial resolution of $\sim 1$ $\mu m$. Figure 1(a) shows
a sketch of the experimental measurement setup. The ellipsometric $\Delta$ and
$\Psi$ angles of the multilayer system (RuCl3/SiO2/Si) were measured as a
function of the energy and the angle of incidence (AOI) of the illumination
light. Figure 1(b) shows the $\Delta$ (red) and $\Psi$ (blue) angles measured
under AOIs of 45∘ (filled circles) and 50∘ (empty circles) for RuCl3
nanoflakes with thicknesses of 2.9, 6.9, and 36.5 nm. We note that these
measurements were repeated for different in-plane rotations of the RuCl3
samples, leading to imperceptible differences on the measured $\Delta$ and
$\Psi$ angles, as expected from the in-plane isotropic optical response of the
material.

Figure 1: (a) Schematic representation of the experimental spectroscopic
ellipsometry setup. (b) $\Delta$ (red) and $\Psi$ (blue) ellipsometric angles
measured under AOIs of 45∘ (filled circles) and 50∘ (empty circles) for RuCl3
nanoflakes with thicknesses of 2.9 (bottom), 6.9 (middle) and 36.5 nm (top).
The solid lines represent the fit of the experimental data using a Mueller
matrix formalism. (c) Estimated complex refractive index of RuCl3 obtained
from a simultaneous fitting of the $\Delta$ and $\Psi$ angles shown in (b) for
different crystal thicknesses.

For a quantitative analysis of the optical properties of the RuCl3 nanoflakes,
it is necessary to fit the measured ellipsometry angles with an appropriate
multilayer model. With the aim of minimizing substrate-induced uncertainties
in the determination of the refractive index of RuCl3 nanoflakes,
ellipsometric angles from the bare SiO2/Si substrate were also measured
simultaneously on spatial positions a few microns away from each RuCl3
nanoflake using the same experimental conditions. The thickness of the SiO2
oxide layer corresponding to each spatial location was estimated by fitting
the experimental substrate data to a multilayer model using the Mueller matrix
formalism, where we employed reported refractive indices for Si and SiO2 [13].
Next, the same Mueller matrix formalism was employed to estimate the energy-
dependent complex refractive index of RuCl3 nanoflakes with different
thicknesses. Figure 1(c) shows the estimated complex refractive index of RuCl3
obtained from a simultaneous fitting of the $\Delta$ and $\Psi$ angles shown
in Fig. 1(b) for different crystal thicknesses and angles of incidence. The
solid lines in this figure show the results of the modelled ellipsometry
angles obtained from the simultaneous fitting approach. The ellipsometry data
corresponding to RuCl3 flakes with thicknesses of 2.9, 6.9, and 36.5 nm yield
the same complex refractive index values, confirming the thickness-independent
optical properties of this material.

Further, we observe a very good agreement between the estimated refractive
index of the nanoflakes and the optical properties reported for bulk RuCl3
crystals in previous works [14, 15, 16, 17]. As seen in Fig. 1(c), the
imaginary part of the refractive index ($\kappa$) shows a clear absorption
peak at $\sim$2.05 eV, in good agreement with the $\beta$ resonance reported
in the imaginary part of the dielectric constant [16] and the real part of the
optical conductivity of bulk RuCl3 crystals [17]. The observed increase of
$\kappa$ towards the edges of our measurement range also agrees well with the
reported existence of stronger $\alpha$ and $\gamma$ absorption resonances at
$\sim$1.2 and 3.2 eV, respectively [16, 17].

##  III Optical contrast analysis of RuCl3

Next, we employ the measured complex refractive index of RuCl3 to calculate
its thickness-dependent optical contrast when deposited on top of SiO2/Si
substrates and imaged with an optical microscope in an epi-illumination
configuration. In our simulations, we adopt the following definition for the
optical contrast between the nanoflake and the substrate [18, 19]:
$OC(\lambda)=(R(\lambda)-R_{0}(\lambda))/(R(\lambda)+R_{0}(\lambda))$, where
$R(\lambda)$ and $R_{0}(\lambda)$ represent the wavelength-dependent
intensities of the light reflected by the whole heterostack (RuCl3/SiO2/Si)
and the bare substrate (SiO2/Si), respectively. To calculate $R(\lambda)$ and
$R_{0}(\lambda)$ we employ the transfer matrix method, in which the forward-
and backward-propagating plane waves form the basis into which the optical
electric field in each layer is decomposed [20]. In our simulations, we
include the wavelength-dependent refractive indices of SiO2 and Si [13], and
assume normal incidence of the illumination for simplicity. Although the
angular spread of incident angles given by the numerical aperture (NA) of the
illumination objective lens can play a significant role on the perceived
optical contrast for thick crystals and/or 2D layers with anisotropic optical
constants [21, 22], the relatively thin flakes ($<100$ nm), the low NA of the
objective lens (0.42), and the measured isotropic refractive index of RuCl3
justify our assumption.

###  III.1 Thickness-dependent optical contrast of RuCl3 in the sRGB color
space

In this section, we explore experimentally and numerically the thickness-
dependent optical contrast of RuCl3 on SiO2/Si substrates in the standard RGB
(sRGB) color space under broadband white illumination. Figure 2(a) shows an
optical microscope image in the sRGB color space of RuCl3 flakes with
thicknesses in the range $\sim 2-100$ nm mechanically exfoliated on a SiO2/Si
substrate with a SiO2 thickness of 295 nm, as confirmed by spectroscopic
ellipsometry. The red, green, and blue dots in Fig. 2(b) represent the
experimental optical contrast values in the different channels (R,G, and B,
respectively) extracted from the optical microscope image shown in Fig. 2(a)
and a second spot nearby on the same substrate. The thicknesses of the layers
indicated in Fig. 2 were estimated by a combination of AFM and spectroscopic
ellipsometry measurements, while the optical contrast values and corresponding
error bars represent the optical contrast value and associated uncertainty
calculated from the measured $R(\lambda)$ and $R_{0}(\lambda)$ in each flake
and a nearby spot in the substrate, respectively. The experimental values and
uncertainties of $R(\lambda)$ and $R_{0}(\lambda)$ for all the measured flakes
were obtained from the median and standard deviation, respectively, of 100
pixels.

Figure 2: (a) Optical microscope image, acquired in the standard RGB (sRGB)
color space, of exfoliated RuCl3 flakes with thicknesses ranging from $\sim$ 2
$-$ 100 nm. Scale bar 50${\mu}$m. (b) Comparison of the experimental (dots)
and calculated (dashed lines) optical contrast in the sRGB color space
corresponding to the red (R), green (G), and blue (B) channels. The colorscale
bar shows the calculated color of RuCl3 as function of flake thickness when
deposited on top of a SiO2/Si substrate with a SiO2 thickness of 295.5 nm.

In order to compare the experimental contrast values in the sRGB color space
with the calculated optical contrast, we follow the approach described in
previous works [13, 23, 24, 25, 26]. In this approach, the simulated
wavelength-dependent reflected intensities of the flake and bare substrate
($R(\lambda)$ and $R_{0}(\lambda)$) are used to calculate CIEXYZ color values,
which simulate the color perceived by the human eye. Such transformation
involves an integration over wavelengths including the light source spectrum
(Thorlabs, Inc., product MCWHLP1) and the CIE color-matching functions to
calculate the XYZ tristimulus components [13, 23, 24]. Finally, the calculated
XYZ values are converted to the sRGB color space using a standard
transformation, which takes into account the chromaticity coordinates of the
sRGB color space and the reference white of the light source [13, 27].

The dashed lines in Fig. 2(b) show sRGB optical contrast values for the R, G,
and B channels as function of flake thickness calculated following the
approach described above. We note that in order to get a good agreement with
the experimental values, the simulated optical contrast response of the three
channels must be scaled by a normalization factor $f$ (with $f<1$). We justify
the use of such normalization factor by the need to account for the reduced
spatial and temporal coherence of the illumination light in our experimental
setup (which we assume to be perfect in the simulations), which is known to
result in a reduced visibility of the interference fringes in the reflected
signals while preserving the thickness-dependent positions of the interference
maxima and minima [28]. We find that a normalization factor $f=0.68$
effectively reproduces the observed optical contrast in all three RGB channels
for all measured RuCl3 thicknesses. Notably, the best agreement is found using
the G and B channels, showing goodness-of-fit parameters (GoF) $>$ 0.98, as
compared to a value of $\approx 0.92$ obtained for the R channel, where
$GoF=1-RMSE$, with $RMSE$ being the root-mean-square error.

The good agreement between the experimental and simulated optical contrast
allows us to calculate the apparent color of RuCl3 on a SiO2(295 nm)/Si
substrate as a function of flake thickness. The colorbar in the top side of
Fig. 2(b) shows the calculated apparent colors of the substrate (outer region)
and the RuCl3 (inner region) as a function of thickness. Overall, we find a
good agreement between the measured and calculated apparent color of RuCl3,
which can be used as a fast method for assessing flake thicknesses, as shown
previously for other different 2D materials [29, 24, 26, 30].

Figure 3: (a) Comparison of the experimental (dots) and calculated (dashed
lines) optical contrast of RuCl3 of with thicknesses $\sim$ 2 $-$ 100 nm
measured using bandpass filters at 450 nm (blue) and 600 nm (red) with a 10-nm
bandwidth. The shaded areas represent the calculated optical contrast at the
corresponding central wavelength of the filter $\pm$ 5 nm. (b) Calculated
optical contrast under the monochromatic excitation wavelengths that maximise
the positive and negative optical contrast of RuCl3 with thicknesses in the
range $\sim$ 0 $-$ 10 nm for SiO2 thicknesses of 90 nm (dashed lines) 295 nm
(solid lines).

###  III.2 Thickness-dependent optical contrast of RuCl3 under narrow-band
illumination

Next, we explore the effects that the inclusion of narrow optical bandpass
filters in the illumination path has on the optical contrast characterisation
of RuCl3. This approach has previously shown to reproduce to a good extent the
optical contrast simulated under monochromatic illumination in other 2D
crystals [31, 19, 32, 33]. The blue and red dots in Fig. 3(a) represent the
experimental contrast values measured for the same RuCl3 flakes shown in Fig.
2 when filtering the illumination spectrum with 10-nm-bandwidth filters
centred at 450 and 600 nm, respectively. The blue and red dashed lines
indicate the simulated thickness-dependent evolution of the optical contrast
assuming monochromatic excitation at 450 and 600 nm, respectively. The blue
and red shaded areas show the simulated optical contrast for monochromatic
illumination in the range $450\pm 5$ and $600\pm 5$ nm, respectively. Similar
to the results under broadband illumination, we observe that the simulated
optical contrast agrees well with the experimental values across the whole
thickness range. Again, we find a better agreement between simulated and
experimental values for the shorter illumination wavelength, with GoF values
of 0.99 and 0.98 for the 450 nm and 600 nm filters, respectively.

Once more, the good agreement between the optical contrast measured with
bandpass filters and the simulated optical contrast under monochromatic
illumination allows us to explore the monochromatic wavelengths that enhance
the optical contrast response of RuCl3 with thicknesses below 10 nm on top of
Si substrates with thermally grown SiO2 layers. Note that an enhanced optical
contrast response arises from a combination of large absolute optical contrast
with the underlying substrate and a linear dependence with RuCl3 thickness.
Figure 3(b) summarises the results of our analysis for two standard SiO2
thicknesses of 90 and 295 nm. Our results suggest that monochromatic
illumination wavelengths of $\sim 730$ nm and $\sim 650$ nm yield the largest
optical contrast values for 90-nm and 295-nm-thick SiO2 layers, respectively,
while showing an almost linear dependence with RuCl3 thickness in the explored
range. We note that in both cases the optimised monochromatic illumination
wavelengths result in a negative optical contrast, i.e., the RuCl3 crystal
appears darker than the substrate. We find that monochromatic illumination
wavelengths of $\sim 460$ nm and $\sim 550$ nm give rise to the opposite case
(i.e., RuCl3 appears brighter than the substrate) for 90-nm and 295-nm-thick
SiO2 layers, respectively, at the expense of a slightly lower absolute optical
contrast.

###  III.3 RuCl3 thickness estimation from optical contrast measurements

Finally, in this section we benchmark the performance of the optical contrast
technique to estimate the thickness of RuCl3 nanoflakes using our calibrated
optical microscope and experimental optical contrast dataset shown in Fig. 2
and Fig. 3. In order to do so, we employ an algorithm that estimates the value
of the RuCl3 thickness ($d$) that minimizes the following quantity:

$\displaystyle
RMSE^{total}(d)=\sqrt{\sum_{i}^{N}\frac{[OC_{exp}^{i}-OC_{sim}^{i}(d)]^{2}}{N}},$
(1)

with $N$ representing the total number of experimental and simulated optical
contrast illumination/detection modes $i$ considered for the minimization
algorithm (i.e., sRGB detection for white illumination or bandpass
illumination). In our algorithm, we follow a two-step approach. We start by
considering only the two optical contrast configurations that show the best
overall agreement between experimental and simulated values (G and B channel
detection under broadband illumination) and estimate the crystal thickness
($d$) that minimizes Eq. (1). In the second step, if the value of $d$ returned
by the initial minimization of Eq. (1) for the G and B channels falls outside
the range $3.5\leq d\leq 55$ nm (i.e. where the R channel shows a good
agreement between experiment and simulations), we minimize again Eq. (1) with
the addition of the R channel. This two-step minimization procedure results in
an estimated thickness with a mean deviation $\overline{\Delta d}=-0.2$ nm
from the crystal thickness estimated by AFM/ellipsometry in the range $d<100$
nm with a standard deviation of 1.2 nm.

Figure 4: Bottom panel: RuCl3 thickness estimated from the experimental
optical contrast obtained by a combination of sRGB and monochromatic detection
as a function of the thickness determined by independent AFM and/or
spectroscopic ellipsometry measurements. The horizontal error bars represent
the uncertainty associated with the independent thickness estimation. The
vertical error bars represent the uncertainty resulting from our thickness
estimation approach, which we define as the thickness range for which the
$RMSE^{total}$ defined by Eq. (1) is less than or equal to two times the
$RMSE^{total}$ corresponding to the estimated thickness. The red line
represents the ideal case, where the thickness inferred from the optical
contrast equals the measured thickness. Top panel: difference between the
estimated thickness inferred from the optical contrast and the measured RuCl3
thickness ($\Delta d$). The green and purple shaded areas represent $\Delta d$
intervals of $\pm$1 and $\pm$2 RuCl3 monolayers, respectively. The histogram
in the top-right panel identifies the statistics using monolayer thickness
binning.

Finally, we note that the inclusion of an additional measurement mode (450 nm
band-pass detection) in the second stage of the two-step minimization
algorithm leads to a slight improvement in the thickness estimation
performance. The bottom panel of Fig. 4 shows the RuCl3 thickness estimated
from the experimental optical contrast as a function of the thickness
determined by AFM and spectroscopic ellipsometry measurements. The red line
represents the ideal case, where the thickness inferred from the optical
contrast equals the measured thickness. The top panel shows the difference
between the estimated thickness inferred from the optical contrast and the
measured RuCl3 thickness ($\Delta d$), with the green and purple shaded areas
representing $\Delta d$ intervals of $\pm$1 and $\pm$2 RuCl3 monolayers,
respectively. Finally, the top right panel shows a histogram of the
corresponding $\Delta d$ values, which yields a $\overline{\Delta d}=-0.2$ nm
for the whole thickness range with a standard deviation of 1 nm, i.e., close
to the thickness of a single RuCl3 monolayer (0.8 nm [34, 35]).

##  IV Conclusions

In summary, we report the complex refractive index of $\alpha$-RuCl3
nanoflakes with thicknesses below 40 nm in the visible and near-infrared
wavelength range. Our results show that the optical properties of the
nanoflakes are independent of the crystal thickness and agree well with the
optical constants reported for bulk samples [14, 15, 16, 17]. Compared to
other 2D crystals such as transition-metal dichalcogenide semiconductors [36],
group III–VI metal chalcogenide semiconductors such as InSe [37], or hybrid
perovskites [38], which show a strong modulation of their dielectric response
with the thickness for few-layer crystals, the thickness-independent optical
response of RuCl3 enables a straightforward and reliable comparison between
the experimental and simulated optical reflectance contrast as function of
crystal thickness. In this context, we show that transfer-matrix based
simulations of the thickness-dependent optical contrast of RuCl3 on oxidized
silicon substrates reproduce with good agreement the experimental optical
contrast obtained under different illumination/detection configurations,
including sRGB color-space detection for broadband and narrow-band
illumination. Finally, we show that optical contrast imaging allows us to
retrieve the thickness of the RuCl3 nanoflakes exfoliated on an oxidized
silicon substrate with a mean deviation of $-0.2$ nm for thicknesses below 100
nm with a standard deviation of only 1 nm across the whole thickness range.
These results demonstrate the potential of optical contrast analysis as a non-
invasive, fast, and reliable technique to estimate the thickness of RuCl3
nanoflakes.

###### Acknowledgements.

This work was supported by the EPSRC (grant nos. EP/P029892/1 and
EP/L015110/1), and the ERC (grant no. 725920) (D.A.-P., S. O., and B.D.G). The
work of Y.W. and K.S.B. was supported by the National Science Foundation (NSF)
EPMD program via grant EPMD-2211334. Crystal growth at ORNL was supported by
the U.S. Department of Energy, Office of Science, National Quantum Information
Science Research Centers, Quantum Science Center. M.B.-G. is supported by a
Royal Society University Research Fellowship. B.D.G. is supported by a Chair
in Emerging Technology from the Royal Academy of Engineering.

## Author declarations

### Conflict of Interest

The authors have no conflicts to disclose.

## Data Availability

The dataset generated and analyzed during the current study is available at:
https://researchportal.hw.ac.uk/en/persons/mauro-brotons-i-gisbert/datasets/

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