#  Experiments on Anomaly Detection in Autonomous Driving by Forward-Backward
Style Transfers ††thanks: This work results from the KIGLIS project supported
by the German Federal Ministry of Education and Research (BMBF), grant number
16KIS1231.

Daniel Bogdoll23*, Meng Zhang2*, Maximilian Nitsche3, J. Marius Zöllner23
2FZI Research Center for Information Technology, Germany. Email:
<EMAIL_ADDRESS>3Karlsruhe Institute of Technology, Germany.

###### Abstract

Great progress has been achieved in the community of autonomous driving in the
past few years. As a safety-critical problem, however, anomaly detection is a
huge hurdle towards a large-scale deployment of autonomous vehicles in the
real world. While many approaches, such as uncertainty estimation or
segmentation-based image resynthesis, are extremely promising, there is more
to be explored. Especially inspired by works on anomaly detection based on
image resynthesis, we propose a novel approach for anomaly detection through
style transfer. We leverage generative models to map an image from its
original style domain of road traffic to an arbitrary one and back to generate
pixelwise anomaly scores. However, our experiments have proven our hypothesis
wrong, and we were unable to produce significant results. Nevertheless, we
want to share our findings, so that others can learn from our experiments.

###### Index Terms:

autonomous driving, anomaly detection, style transfer, generative models

**footnotetext: These authors contributed equally

##  I Introduction

Driven by the technology of deep neural networks for computer vision,
tremendous advances have been seen in the field of autonomous driving. For
instance, various deep learning models [1] are nowadays able to provide
outstanding performances for the task of object detection. However, in order
to achieve effective deployment for real-world scenarios, reliability issues
must be overcome. That is to say, under rare or unknown conditions, an
autonomous vehicle is required not only to be able to identify the object
classes from the training dataset, but also to detect atypical objects that
have not been included in the training set. Anomaly detection, therefore, is
an active topic in the research field of autonomous driving. As shown in [2],
there are five main strategies to detect anomalies: Confidence scores,
reconstructions, generative approaches, feature extraction methods and
prediction errors. The approach presented here falls into the category of
reconstructive methods, in which an attempt is made to produce a
reconstruction of an original image via a certain process, with which,
differences from the original image can be classified as anomalies.

Figure 1: Overview of the novel anomaly detection approach. An image with
anomalies is transferred into a stylized version of it with a forward style
transfer, based on a target domain of choice. The resulting image is then
transformed back to its original domain. Finally, a difference map between the
original and the re-created image is computed, which can be interpreted as a
map of pixel-wise anomaly scores.

Inspired by approaches in this category, we came up with the idea to use
generative models for style transfer based reconstructions, as shown in Figure
1. We have formulated our hypothesis as follows: Transferring an image from
the road traffic domain, which includes anomalies, such as unknown objects, to
another domain and back to the original domain, will lead to a resulting image
that is similar to the original in the domain-typical aspects and deviates
from it in the anomalous regions, which allows for the detection of anomalies.
This hypothesis was made under the assumption that anomalies might disappear
or get distorted through two successive style changes, which would allow for
the detection of such. This hypothesis to detect anomalies is based on an
assumption with respect to reconstruction errors with generative models, where
“…normal samples are located on a manifold and all anomalous samples are
located outside.” [3].

For the experiments, we used the KITTI [4] and Cityscapes [5] datasets, and
qualitatively evaluated the detection of anomalies with the Fishyscapes [6]
and Lost and Found [7] datasets. However, our experiments have proven our
hypothesis wrong and have shown that the concept as we have implemented it
does not seem suitable for anomaly detection. To share this finding with the
scientific community, we present our hypothesis, experiments, and negative
results, as is often suggested [8, 9, 10, 11, 12, 13], but rarely practiced.

##  II Related Work

Most of the current research in anomaly detection for autonomous driving
focuses on camera-based approaches [2], which we will present in more detail
in the following section. Subsequently, we will give an overview of style
transfer methods.

###  II-A Camera-based Anomaly Detection

Modern anomaly detection methods that work on camera data can be split into
generative, reconstructive, and confidence-based approaches [2]. The latter
detect anomalies by thresholding a derived uncertainty measure of the model
[14, 15]. Moreover, Du et al. [16] focus on the calibration of the neural
network’s confidence score to differentiate in- and outliers more thoroughly.
For this purpose, they shape the decision boundary via virtual outliers
sampled from a learned feature distribution [16]. Reconstructive approaches on
the other hand try to resynthesize the normality of a driving scene. Here,
normality can be understood as what a model learned, given certain training
data. Anomalous objects or events are then detected by comparing the real and
reconstructed scene, resulting in a reconstruction error. However, modern
generative adversarial networks (GANs) have shown outstanding reconstruction
performance even on unseen data [17]. In order to predict normality in
autonomous driving, various machine learning algorithms are being used [2],
like restricted Boltzmann machines [18], autoencoder architectures [19][20],
or normalizing flows [21]. The NFlowJS model proposed by Grcić et al. [22],
however, uses normalizing flow to synthesize outliers during training and to
sensitize the model for potential anomalies. Other generative approaches use
GANs to resynthesize the input image [23][24] and detect anomalous objects on
the road, e.g., by the perceptual loss [25]. While there also exist methods
that are based on feature extraction [2], these are typically older, which is
why we excluded them from this overview.

###  II-B Image-to-Image Style Transfer

Benefiting from the powerful feature extraction abilities of deep neural
networks, image style transfer has been an active research area in image
translation. Specifically, the aim of style transfer is to render an input
image into a new image, which has a similar style and texture as a style image
from a target domain. As pioneers of style transforms, Gatys et al. [26]
firstly proposed their method Neural Style Transfer (NST) to synthesize an
image to match the style of a painting by an artist. Notably, separate control
of the content and style of the picture is achieved by introducing content and
style loss functions. However, due to its application to single image pairs,
this approach cannot be used for large-scale style conversions. Motivated by
Neural Style Transfer, another style transfer model, introduced by Johnson et
al. [27], performs much more efficiently by optimizing an image transform net.

In addition to ordinary neural network models, the emergence of GANs [28, 29]
opens up more possibilities in the area of style transfer. As the name
suggests, the core idea of GANs is to create desired images by simultaneously
training two opposing neural networks. While a generator is responsible for
synthesizing more realistic images under the guidance of a discriminator, the
discriminator is trained to distinguish fake images from real images more
accurately. Specifically, there are two well-known generative models for the
task of image translation. With the help of adversarial and perceptual losses,
the framework Pix2Pix [30] learns the mapping from a source domain to the
target domain by training a model with paired training data. Differently,
CycleGAN [31] employs a cycle consistency loss [32], as also applied by
DualGAN [33], to conduct image-to-image translation under the unsupervised
setting, making style transfers more flexible. Similar works include DiscoGAN
[34], GANimorph [35], ACL-GAN [36], and AttentionGAN [37]. Besides, Park et
al. [38] firstly applied contrastive learning [39] for unpaired image-to-image
translation. Interestingly, the model is trained to learn the invariance
between the patch of the generated image and its corresponding patch in the
input image by using the InfoNCE loss [40].

##  III Approach

Our hypothesis to detect anomalies is based on an assumption in respect to
reconstruction errors with generative models, where “…normal samples are
located on a manifold and all anomalous samples are located outside. Since the
manifold can be learned only where the training data lie…” [3], we were
curious if generative models for style transfers might perform worse in areas
of data points outside of their learned distribution. Thus, the core idea of
our approach is to perform anomaly detection by leveraging the impact of
successive style transformations, as visualized in Figure 1. The forward style
transfer is responsible for the conversion of an input image into a stylized
version of it. The backward style transfer is then tasked to transform the
image back to its original style domain. Finally, a pixel level comparison
between the input and re-constructed image is conducted to recognize anomalies
in the input image.

Forward Style Transfer By forward style transfer, we refer to the process of
transferring an input image to an arbitrary style domain. Thereby, the input
image will be rendered in the style of the target image, while preserving its
content.

Backward Style Transfer The aim of backward style transfer is to transform the
stylized image, which is the output from the forward style transfer, back to
the original style of the input image.

Pixelwise Anomaly Detection Finally, a pixel-by-pixel comparison is made
between the input image and the newly synthesized image. Based on our
hypothesis, we expected to detect anomalies in the input image through
comparison. While we are aware that such a direct comparison between the re-
synthesized image and the original is not sufficient for anomaly detection due
to noise [41], spikes in such difference maps are a sufficient indicator for
the general suitability of the approach.

##  IV Evaluation

In this section, we present qualitative results of our approach. We performed
three different sets of experiments and show the resulting pixelwise anomaly
scores for multiple input images. As we performed the experiments as a first
step to determine whether the methodology is suitable for anomaly detection,
we omitted a quantitative analysis afterwards.

###  IV-A Data Selection

Since the training data should reflect normality in road traffic, we first
chose to utilize the well established KITTI [4] dataset and also performed
experiments with the much larger Cityscapes dataset [5]. For the following
evaluation, datasets with anomalies were necessary. While there exist many
datasets in the field of autonomous driving [42], only a few of them are
suited for the task of anomaly detection. We utilize data samples from the
Fishyscapes [6] and Lost and Found [7] datasets to include both real and
augmented anomalies. As the Lost and Found dataset is part of the Fishyscapes
suite, we might refer to both as Fishyscapes. Both provide ground-truth labels
for out-of-domain pixels, which we consider anomalies. In autonomous driving,
these are also often called corner cases [43] in the category unknown object
[44]. These unknown objects can typically also be seen clearly in the images
directly. Figure 2 shows a selection of animals and small objects. For the art
styles used for the forward passes, we selected a wide variety of styles,
including art from Van Gogh, Giles and pieces in the Ukiyo-e style.

###  IV-B Model Selection

We have performed experiments with two different image style transfer methods.
First, we used CycleGAN [31], which is an established framework for unpaired
image translations. For model-based style transfers, it is important to choose
a model that can be trained with unpaired images as training data, contrary to
approaches such as Pix2Pix [30], where paired images are necessary. Among
those, CycleGAN [31] is the most prominent representative. While there are
newer models, CycleGAN is optimal for experiments, since high quality open-
source code [45] is available and many works have used it already, making it a
mature method.

Second, we utilized the Neural Style Transfer approach [26], which is regarded
as one of the early milestones in the field of style transformations. While
CycleGAN is able to generate stylized images with a high quality, we have no
dynamic control over how much an image should be transformed. This becomes
possible with NST, so that we were able to examine the influence of conversion
degrees. Based on convolutional neural networks (CNN), the Neural Style
Transfer Algorithm is able to render a random input image in the style of a
target image by leveraging the VGG19 model [46], which has been widely used
for feature extraction in various tasks. Moreover, the highlight of the
algorithm is controlling the transfer degrees through two weight parameters of
the loss functions: content weight and style weight. On the one hand, a
content image would be always compared with the generated image, in order to
make sure the semantic content is retained in the process of stylization.
Specifically, the content structure can be preserved through minimizing the
mean squared error of feature maps between a content image and the input
image. On the other hand, through computation of the Gram matrix, a style
image in the target domain would be always compared with the generated image,
such that the synthesized image will have a similar style as the given style
image. Thus, it is a tradeoff problem for the task of combining the content
image with another style image, where it is impossible to meet the two
constraints perfectly at the same time. For instance, the output image could
have the texture of the style image, but lose some content when a larger style
weight is set. On the contrary, a much larger content weight would result in
that the synthesized image is further away from the target style.

###  IV-C Style Transfer with CycleGAN

In this first experiment, we have utilized CycleGan for both the forward and
backward style transfers. First, we will give an overview of the details of
the forward style transfer.

####  IV-C1 Forward Style Transfer

For this experiment, instead of training a CycleGAN model from scratch, a pre-
trained model was directly used for the forward style transfer. Although the
pre-trained CycleGAN model [45] was trained on paintings by Van Gogh as the
target domain and ordinary photos as the source domain, it has shown to have
good generalization capabilities on various datasets, such as KITTI,
Cityscapes, as well as Fishyscapes and Lost and Found.

####  IV-C2 Backward Style Transfer

For the backward style transfer direction, however, the observed performance
of the pre-trained GAN was not sufficient. Therefore, we chose to train the
model from scratch. Notably, in order to perform the task of style transfer
with a CycleGAN model successfully, images from both the source and target
domain are required to be fed into the model for training. Specifically,
paintings by famous artists are commonly used as source images and ordinary
photos are taken for the target domain. For achieving better performance of
the backward transfer, we trained our model on several data sources, which are
presented in Table I.

Group | Source Domain | Target Domain
---|---|---
A | Paintings by Van Gogh | KITTI
B | Paintings by Van Gogh | KITTI + photos
C | Ukiyo-e style paintings | KITTI
D | Ukiyo-e style paintings | KITTI + photos
E | Van Gogh stylized KITTI | KITTI
F | Van Gogh stylized Cityscapes | Cityscapes

Table I: Data sources used for the training of CycleGAN models

We firstly conducted four sets of trainings, shown as groups A to group D.
Paintings by Van Gogh were adopted as the source domain images in both groups
A and B. In group A, we used images from the KITTI dataset as the target
domain, whereas a combination of KITTI data and ordinary photos served as
target style images in group B. Additionally, we employed the Japanese
painting style Ukiyo-e [47] as the source style in group C and D. For the
target domain, group C used KITTI data as done for group A, and group D
combined KITTI with ordinary photos as in group D. However, this set of
experiments did not perform well, even though we observed slightly better
results when using Van Gogh style images and KITTI data. Here, two problems
occurred. On the one hand, some noise was constantly present, which we
attribute to the special texture of Van Gogh paintings. On the other hand, the
performance of the model was erratic, resulting in the occasional lack of much
difference between transformations and the corresponding input images. After
this analysis, we found two directions to resolve the problem. Firstly, rather
than directly using paintings from artists, we took stylized images, which
were the output from the forward style transfer network, to train our model
with more domain-specific data. Secondly, there were too few images from the
source domain for training. With the newly selected data source, we were able
to increase the number of stylized images significantly and performed another
two trainings, as shown in groups E and F. In group E, Van Gogh stylized KITTI
samples were used for the source domain, and KITTI samples were used for the
target domain. In group F, we followed the same approach, but used Cityscapes
data.

####  IV-C3 Pixelwise Anomaly Detection

For the evaluation, an input image chosen from the Fishyscapes dataset was
used as the input. With the final approach as shown in group F of Table I, we
were able to generate images with the backward style transfer pass that were
consistently close to the original ones, as shown in Figure 2.

Figure 2: Evaluation results of the model trained with Van Gogh stylized
Cityscapes images as the source domain and Cityscapes samples as the target
domain. Four different types of anomalies are included in the Fishyscapes
scenes, which are highlighted in red.

This observed performance improvement on the Fishyscapes dataset is largely
due to the fact, that Fishyscapes is an extension of the original Cityscapes
datasets, so the general domain remains unchanged. After the conversion of the
input into a stylized version of it, based on the forward style transfer, we
reconstructed the input image based on the backward style transfer. The re-
synthesized images are generally very close to the original ones, which was a
wanted effect. However, as visible in the last column, we did not observe any
deformations in the area of the anomaly.

###  IV-D Style Transfer with NST

For the second experiment, we used the Neural Style Transfer method for both
the forward and backward style transformations. In the CycleGAN framework,
weight parameters are updated iteratively, until the model converges.
Contrary, with NST, it is a single image that is optimized in the training
process. Thus, only two images are necessary for the process: a content image
and a style image. Therefore, we directly utilize Fishyscapes data samples for
the style transfers.

####  IV-D1 Forward Style Transfer

For the forward style transfer, a sample from Fishyscapes is used as the
content image, and a painting from Van Gogh is taken as the style image in the
optimization process. Content weight and style weight are two important
parameters, that can be set to control how much the image will be stylized. As
recommended by reference implementations, [48][49], the content weight is kept
unchanged at $1e5$, such that the process of style transfer can be easily
controlled by only changing the style weight.

Figure 3: Style transfers with NST with a scene from the Fishyscapes dataset
with lost cargo as anomalies. Each column represents a different style and
each row a different style weight.

To determine, which degree of style transfer is most likely to lead to
distortions of the content, especially anomalies, during the forward style
transfer, we performed a set of small experiments with the style weight. As
visualized in Figure 3, we performed experiments with four different styles
[50, 51, 52, 53], and four different style weights. The larger the style
weight, the better the generated image matches the style image. Since a style
weight of $1e8$ often led to a nearly complete loss of the content
information, we utilized a value of $1e5$ for the further experiment to
maximize the effect of the style transfer, while retaining the content.

####  IV-D2 Backward Style Transfer

To convert the stylized Fishyscapes images back to their original style of a
regular street scenario, we chose an image from the Cityscapes dataset as the
style image. Importantly, this image from Cityscapes does not contain any
anomalies, such that it only works as a style guidance, avoiding an unexpected
influence on the content part.

####  IV-D3 Pixelwise Anomaly Detection

Illustrative results of the second experiment are shown in Figure 4. Here, we
utilized the same styles as introduced in Figure 3. To highlight the effects
of the different styles on the reconstruction and anomaly detections, we show
only one scene from the Fishyscapes dataset.

Figure 4: Evaluation of Neural Style Transfer with a content weight and a
style weight of $1e5$ each. A sample scene from the Fishyscapes dataset is
used as the input. Each row shows the process with a different style.
Anomalies are highlighted in red.

Here, it becomes visible that the results of the backwards style transfers
differ less than the differences of the stylized images might suggest. Again,
the anomalies remain well visible in the reconstruction. The anomaly map in
the last column shows no spikes in these regions, but shows a high level of
difference values in general.

###  IV-E Hybrid Style Transfer

For the third and final experiment, we combined both approaches. Here, we
performed the forward style transfer with the Neural Style Transfer method,
while we used our CycleGAN model, trained with Cityscapes data, for the
backward style transfer. For this experiment, we followed the intuition that
the NST allows for a better stylization control for the forward pass, while
the CycleGAN enables higher quality results for the backward pass. As we have
explained both passes in the previous Sections IV-C and IV-D, here we describe
only the results, which are visualized in Figure 5. It can be seen, that with
such a variety of styles, the GAN, which was only trained with one style, was
not capable to perform the backward style transfer properly. Here, a leakage
of the applied style to the reconstruction occurs with styles, which were not
used for the training of the GAN. Regarding the anomaly detections, we observe
a similar behavior as in the previous experiments. The amount of noise is,
compared to the other two experiments, in a medium range.

Figure 5: Evaluation of the hybrid approach with a sample scene from the
Fishyscapes dataset. Here, NST is responsible for the forward style transfer.
The backward style transfer was carried out by our CycleGAN model. Each row
represents a different style. Anomalies are highlighted in red.

With this presented set of three comprehensive experiments, we examined the
influence of training data, style data, anomaly data, style weights, and style
transfer models for the purpose of anomaly detection based on successive style
transfers. While the results showed high-quality reconstructions, the style
transfers unfortunately reconstructed the areas with anomalies similarly to
the rest of the images. Thus, the difference maps showed primarily noise,
which makes the detection of anomalies with this method unlikely.
Specifically, our experiments have placed a focus on the comparison of
CycleGAN, Neural Style Transfer, and a hybrid combination of both for the
Forward and Backward Style Transfers parts. The results differed in the value
distribution of the difference maps and the amount of noise, but showed no
sufficient differences for the pixels of the anomalies.

##  V Conclusion

The aim of our work was to investigate whether our proposed approach of
successive style transfers is suitable as a novel method for anomaly detection
in the domain of autonomous driving. Therefore, we performed a set of three
extensive experiments. Within these, we transferred images from the road
traffic domain to multiple art domains with a set of forward style transfer
methods and then transferred the results back to the original domain. We
compared these reconstructions with the original images to calculate a
pixelwise anomaly score. Our experiments revealed, that anomalies were still
consistently visible in the reconstructed image, which did not lead to high
anomaly scores in the difference maps within the areas of the anomalies. Based
on our analysis visualizations, these negative results should serve as
inspiration and knowledge to support following research in the field of
anomaly detection in autonomous driving.

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J. Křepelka Jaromír Křepelka, Palacký University Olomouc, Faculty of Science,
Joint Laboratory of Optics of Palacký University and Institute of Physics of
the Czech Academy of Sciences, 17. listopadu 1192/12, 771 46 Olomouc, Czech
Republic tel: +420 585 631 518, e-mail<EMAIL_ADDRESS>
# Exact solution of maximally flat antireflection coatings for coherent and
incoherent light

Jaromír KŘEPELKA  Palacký University Olomouc, Faculty of Science, Joint
Laboratory of Optics of Palacký University and Institute of Physics of the
Czech Academy of Sciences, Olomouc, Czech Republic

###### Abstract

This paper presents two approaches to the precise design of maximally flat
antireflection coatings reducing the reflectance of the substrate to near zero
in a certain region around the central frequency. The first ideal case
concerns coherent light interference, where it is required that for a chosen
central frequency the maximum number of reflectance derivatives of the
reflectance with respect to the frequency is zero. This approach makes it
possible to determine the refractive indices of a system of homogeneous
quarter-wave thin films by solving set of nonlinear equations that can be
found in explicit form for a maximum of four layers; for higher numbers of
layers, the result must be sought numerically. The second case is the
incoherent superposition of light waves, which refers to the determination of
the refractive indices of the layers independently of their thicknesses. For
these two physically ideal cases, the obtained refractive indices of the
layers and the dependence of their reflectivity on the light frequency are
compared. A simple new method for approximating the refractive indices of a
maximally flat system of antireflective layers is also proposed.

###### keywords:

thin layers, thick layers, maximally flat antireflection coating, exact
solution, coherent light, incoherent light

Přesné řešení maximálně plochých antireflexních vrstev pro koherentní a
nekoherentní světlo Tento článek předkládá dva přístupy k přesnému řešení
maximálně plochých antireflexních vrstev snižujících odraznost substrátu téměř
na nulu v určité oblasti kolem centrální frekvence. První ideální případ se
týká koherentní interference světla, kde je požadováno, aby pro zvolenou
centrální frekvenci byl co nejvyšší počet derivací odraznosti podle frekvence
nulový. Tento přístup umožňuje určit indexy lomu soustavy homogenních
čtvrtvlnných tenkých vrstev řešením nelineárních rovnic, které lze nalézt v
explicitní formě maximálně pro čtyři vrstvy, pro vyšší počet vrstev je nutné
výsledek hledat numericky. Druhým případem je nekoherentní superpozice
světelných vln, což se týká stanovení indexů lomu vrstev nezávisle na jejich
tloušťce. Pro tyto dva fyzikálně ideální případy jsou porovnány získané indexy
lomu vrstev a závislost jejich odraznosti na frekvenci světla. Rovněž je
navržena jednoduchá nová metoda pro aproximaci indexů lomu maximálně plochého
systému antireflexních vrstev. tenké vrstvy, tlusté vrstvy, maximálně plochá
antireflexe, přesné řešení, koherentní světlo, nekoherentní světlo The author
(orcid.org/0000-0003-0684-0775) thanks project OP PIK No.
CZ.01.1.02/0.0/0.0/21_374/0027282 “Relief nano/micro structures for optical
components in the automotive industry” SPP 843102011 and LM2023032 “Pierre
Auger Observatory” of the Ministry of Education, Youth and Sports of the Czech
Republic for their support. researchpapertrue

##  1 Introduction

Antireflection coatings are widely used in optical applications to reduce the
reflectance of optical surfaces, e.g. for lenses applied as components of
various devices, photovoltaic cells, optical sensors and more. Their purpose
is to increase the light transmittance of optical system, with the side effect
of improving image contrast by removing stray light caused by multiple
internal reflections. This is especially important for the perfect function of
photographic cameras, video cameras, binoculars, telescopes, laser devices,
etc. To ensure such function, systems of layers are essential, the
mathematically motivated improvement of which began with Strong’s [1] analysis
of the reflectance of a substrate coated with a single layer of arbitrary
thickness based on Fresnel’s relations, which led to the frequently mentioned
condition that the magnitude of the refractive index of the layer should be
equal to the geometric average of the refractive indices of the substrate and
the superstrate materials. This was followed by further theoretical advances
in the analysis of the reflection properties of more complex systems, in
particular thin interference layers, based on the principle of coherent
summation of partial reflections of light waves from each material boundary.
Among many researchers, we should mention Antonín Vašíček [2], a Czechoslovak
pioneer in the theory, preparation and measurement of optical systems with
thin or thick layers. Mathematical simplification of seemingly somewhat
confusing and complex calculations of macroscopically measurable parameters of
thin film systems, based on the transformation of tangential components of
electric and magnetic field intensities of plane monofrequency electromagnetic
waves in isotropic homogeneous media, was then achieved by a large number of
authors starting with F. Abèles [3]. Let us remember, for example, the
Czechoslovak thin film expert Zdeněk Knittl [4]. The matrix approach to the
calculation of parameters of thin film systems for plane waves, representing
the transformation of tangential components of plane electromagnetic waves,
was generalized by Dwight W. Berremann for anisotropic media [5].

The theoretical research possibilities have been extended, among other things,
finding such layer systems that would reduce the reflectance of the substrate
as much as possible over a very wide range of light spectra. For instance in
[6] an exact solution for the refractive indices of quarter-wave systems of
isotropic homogeneous dielectric layers is given for such an impedance
matching, which, assuming perfect interference of all internally reflected
waves (that is for coherent light), achieve maximally flat reflectance versus
frequency (wavelength) in the sense that the maximum possible number of
derivatives of reflectance versus frequency (or wavelength) for a chosen
central frequency (central wavelength) is equal to zero. Such a mathematically
rigorous problem leads to finding the roots of a system of nonlinear
equations, which has an explicit solution for a maximum of 4 layers, for a
larger number of layers it is necessary to find a solution using numerical
methods. In this paper, this approach valid for ideal thin layers is compared
with the approach applicable to ideally thick layers, where none of the
internally reflected waves interferes with the other. Such an aim
mathematically requires finding the minimum of a function of several
variables. It should be noted that the obtained refractive indices of the
layers in both cases require their gradual change from the refractive index of
the superstrate towards the refractive index of the substrate (or vice versa),
which are usually refractive indices that are not readily available
technologically. However, although theoretical research on the design of
antireflection layers seems to have been completed, with the development of
nanotechnology, new possibilities for the realization of antireflection
surfaces are beginning to emerge (see e.g. [13]).

##  2 Theory

###  2.1 Reflectance and transmittance of thin and thick layers

Let us consider the propagation of a plane monofrequency electromagnetic wave
with angular frequency $\omega$ (angular wave number in vacuum is $\omega/c$,
$c$ is the speed of light in vacuum) in the geometry of parallel plane layers
in a stratified isotropic medium [7]. There are $k$ homogeneous layers
deposited on the substrate (rear half-space), the front half-space is
superstrate. Let the electromagnetic wave incident the system from the
superstrate be denoted as R (right) and from the opposite direction as L
(left). Plane waves carry the same R or L designation even inside thin layers,
the resulting superposed electromagnetic field is then a linear combination of
both counter propagating waves. Let us number the layers in the R direction of
the incident wave, starting with 0 (for the superstrate), 1 (the layer last
deposited on the substrate), 2, …, $k$ (the layer deposited on the substrate
as the first), $k+1$ (the substrate). Each medium is characterized by its
generally complex refractive index $n_{j}$, $j=0,1,\dots,k+1$, the imaginary
part of which represents an absorption due to the conductance of the
materials, for ideal dielectric environments the refractive indices are real
quantities.

For simplicity, assume that the superstrate is a dielectric (e.g. air) in
which the direction of propagation of the incident R wave is determined by the
angle of incidence $\theta_{0}$ measured from the normal of the plane material
interface surfaces. Since the law of refraction
$n_{j}\sin(\theta_{j})=n_{0}\sin(\theta_{0})$ holds, i.e. the product of the
refractive index and the sinus of the propagation angle is invariant for all
envolved media (we do not discuss the physical meaning of the generally
complex propagation angles $\theta_{j}$), the normal component of the
propagation vector in the $j$-th medium is

$\pm\dfrac{\omega}{c}\sqrt{n_{j}^{2}-(n_{0}\sin\theta_{0})^{2}}=\pm\dfrac{\omega}{c}n_{j}\cos(\theta_{j}).$

Here, the plus sign applies to the L wave and the minus sign to the R wave in
order to preserve the sense of the direction of wave propagation, assuming
that the time dependence of the plane wave is a harmonic function of time $t$
as $\exp(\mathrm{i}\omega t)$ and the imaginary parts of the refractive
indices of the absorbing materials are chosen to be nonpositive. Maxwell’s
equations provide a solution for such a configuration, which can be briefly
described in the terms below. Denoting $Z_{0}$ the vacuum impedance, then the
admittance (ratio of tangential components of magnetic and electric field
intensities) of the $j$-th medium is

$Y_{j}=\frac{1}{Z_{0}}\times\left\\{\begin{matrix}\sqrt{n_{j}^{2}-(n_{0}\sin\theta_{0})^{2}}=n_{j}\cos(\theta_{j}),\\\
\dfrac{n_{j}^{2}}{\sqrt{n_{j}^{2}-(n_{0}\sin\theta_{0})^{2}}}=\dfrac{n_{j}}{\cos(\theta_{j})},\\\
\end{matrix}\right.$ (1)

where the upper row defines the admittance of the $j$-th medium for the s (TE)
wave and the lower row for the p (TM) wave. The correct calculation of square
roots in the complex number domain should be taken into account when focusing
on absorbing materials or the case of total internal reflection in dielectric
materials.

Vectors of tangential components of the electric $\mathbf{E}_{\mathrm{t},j}$
and magnetic $\mathbf{H}_{\mathrm{t},j}$ field intensities composed of the
counter propagating R ($\mathbf{E}_{\mathrm{tR},j}$,
$\mathbf{H}_{\mathrm{tR},j}$) and L ($\mathbf{E}_{\mathrm{tL},j}$,
$\mathbf{H}_{\mathrm{tL},j}$) waves in the $j$-th environment, are obtained
from the tangential components of (only) electric field intensity but
decomposed into R and L ($\mathbf{E}_{\mathrm{tR},j}$,
$\mathbf{E}_{\mathrm{tL},j}$) waves using the transformation matrix

$\begin{pmatrix}\mathbf{E}_{\mathrm{t},j}\\\ \mathbf{H}_{\mathrm{t},j}\\\
\end{pmatrix}=\begin{pmatrix}\pm 1&1\\\ Y_{j}&\mp Y_{j}\\\
\end{pmatrix}\begin{pmatrix}\mathbf{E}_{\mathrm{tR},j}\\\
\mathbf{E}_{\mathrm{tL},j}\\\ \end{pmatrix},$ (2)

where the upper sign applies to the s-polarized waves and the lower sign to
the p-polarized waves.

We conclude that the signs following directly from Maxwell’s equations (and
often omitted in calculations) are necessary for the correct determination of
the phases of the amplitude reflectances of s and p waves, as it is known from
Fresnel’s formulas for a single boundary between two materials, otherwise the
p wave argument of amplitude reflectance would differ by $\pi$ from the
correct value, which may affect, for example, the evaluation of ellipsometric
measurements or modal analysis of planar waveguides. It should be noted that
when calculating (power) reflectances as quadratic quantities, the change of
sign for the p wave does not apply.

The tangential components of the electric field intensity, but decomposed into
R and L waves, are obtained from the tangential components of the composed
electric and magnetic field intensities using the inverse of (2)

$\begin{pmatrix}\mathbf{E}_{\mathrm{tR},j}\\\ \mathbf{E}_{\mathrm{tL},j}\\\
\end{pmatrix}=\frac{1}{2}\begin{pmatrix}\pm 1&\dfrac{1}{Y_{j}}\\\
1&\mp\dfrac{1}{Y_{j}}\\\
\end{pmatrix}\begin{pmatrix}\mathbf{E}_{\mathrm{t},j}\\\
\mathbf{H}_{\mathrm{t},j}\\\ \end{pmatrix}.$ (3)

If we introduce for the phase change of the plane wave as its propagates
between the boundaries of the $j$-th layer of thickness $h_{j}$ the term
$\varphi_{j}=(\omega/c)h_{j}\sqrt{n_{j}^{2}-(n_{0}\sin\theta_{0})^{2}}$, then
for the transformation of the tangential components of the electric and
magnetic (composed) field intensities between the boundaries of the $j$-th
layer to the tangential components of the electric field intensity of the
counter propagating waves, we obtain for each of the perpendicular
polarizations the interference matrix $\mathbf{M}_{j}$

$\begin{split}&\mathbf{M}_{j}=\begin{pmatrix}\cos(\varphi_{j})&\pm\dfrac{\mathrm{i}}{Y_{j}}\sin(\varphi_{j})\\\
\pm{\mathrm{i}}Y_{j}\sin(\varphi_{j})&\cos(\varphi_{j})\end{pmatrix},\\\
&\begin{pmatrix}\mathbf{E}_{\mathrm{t},j}\\\ \mathbf{H}_{\mathrm{t},j}\\\
\end{pmatrix}=\mathbf{M}_{j}\begin{pmatrix}\mathbf{E}_{\mathrm{t},j+1}\\\
\mathbf{H}_{\mathrm{t},j+1}\\\ \end{pmatrix}.\end{split}$ (4)

Note that the (algebraic) spectral decomposition of the interference matrix
$\mathbf{M}_{j}$ contains its eigenvectors identical to the columns of the
matrix defined in equation (2)

$\begin{split}&\mathbf{M}_{j}=\begin{pmatrix}\pm 1&1\\\ Y_{j}&\mp Y_{j}\\\
\end{pmatrix}\begin{pmatrix}\exp(\mathrm{i}\varphi_{j})&0\\\
0&\exp(-\mathrm{i}\varphi_{j})\\\ \end{pmatrix}\\\
&\times\frac{1}{2}\begin{pmatrix}\pm 1&\dfrac{1}{Y_{j}}\\\
1&\mp\dfrac{1}{Y_{j}}\\\ \end{pmatrix}.\end{split}$ (5)

In some cases, it is advantageous to use the decomposition of interference
matrices in their eigenvectors and eigenvalues for an alternative calculation
of the amplitude parameters of thin layers, in which explicit expressions for
Fresnel reflectances (transmittances) at the boundaries of adjacent materials
occur. From this decomposition it is clear that the matrix approach to the
calculation of the thin film parameters is equivalent to an infinite sum of
partially reflected and transmitted waves with phases shifted by
$\varphi_{j}=\omega\tau_{j}=2\pi\nu\tau_{j}$ during each of their passage
through the layers. Such an approach, with use of the normalized spectral
power density of the source radiation and the Wiener-Khinchin theorem, also
allows us to show (e.g., [8]) how the smooth transition from coherent to
incoherent light works in stratified media.

Then the resulting matrix transmitting the tangential components of the
electric field intensity decomposed into counter propagating waves between the
outer boundaries of the system of $k$ layers is obtained by multiplying the
interference matrices in the correct order

$\mathbf{S}=\frac{1}{2}\begin{pmatrix}\pm 1&\dfrac{1}{Y_{0}}\\\
1&\mp\dfrac{1}{Y_{0}}\\\
\end{pmatrix}\mathbf{M}_{1}\mathbf{M}_{2}\dots\mathbf{M}_{k}\begin{pmatrix}\pm
1&1\\\ Y_{k+1}&\mp Y_{k+1}\\\ \end{pmatrix},$ (6)

$\begin{pmatrix}\mathbf{E}_{\mathrm{tR},0}\\\ \mathbf{E}_{\mathrm{tL},0}\\\
\end{pmatrix}=\mathbf{S}\begin{pmatrix}\mathbf{E}_{\mathrm{tR},k+1}\\\
\mathbf{E}_{\mathrm{tL},k+1}\\\ \end{pmatrix}.$ (7)

The relation (6) for the transfer matrix $\mathbf{S}$ holds if the tangential
components of the electric and magnetic field intensities are continuous at
the boundaries of the layers, which is not generally fullfiled when surface
currents or surface charges occur at the boundaries between the layers.

Using the standard definition of amplitude reflectances $r$ and transmittances
$t$ of waves incidenting the layer system from the superstrate (subscript R)
and from the substrate (subscript L) directions, we can calculate these
macroscopic complex quantities from the elements of the transfer matrix
$\mathbf{S}$ as follows

$r_{\mathrm{R}}=\frac{S_{21}}{S_{11}},\ t_{\mathrm{R}}=\frac{1}{S_{11}},\
r_{\mathrm{L}}=-\frac{S_{12}}{S_{11}},\
t_{\mathrm{L}}=\frac{Y_{k+1}}{Y_{0}}\frac{1}{S_{11}},$ (8)

from where we immediately have

$\mathbf{S}=\frac{1}{t_{\mathrm{R}}}\begin{pmatrix}1&-r_{\mathrm{L}}\\\
r_{\mathrm{R}}&d\end{pmatrix},\
d=t_{\mathrm{R}}t_{\mathrm{L}}-r_{\mathrm{R}}r_{\mathrm{L}}.$ (9)

From the relation between inverse interference matrices and interference
matrices as functions of $\varphi_{j}$ or $n_{j}$ for the cases of dielectric
or absorbing media and from the physical requirement imposed on the transfer
matrix $\mathbf{S}$ defined by equation (6) for the same layer system in
mirror symmetry, we can derive the correlation between the elements of the
transmission matrix and hence the relation for the amplitude reflectances and
transmittances from left and right (see [9]). Using this so-called theorem of
reversibility, we obtain for non-absorbing systems $S_{22}=S_{11}^{*}$,
$S_{21}=S_{12}^{*}$), i.e. that the amplitude reflectances and transmittances
of the system of thin dielectric layers (4 complex numbers) are definitely
determined by only three real numbers using also the condition
$\mathrm{det}(\mathbf{S})=Y_{k+1}/Y_{0}$. For instance we can choose for this
purpose the absolute value of the amplitude reflectance from the left and its
argument and the argument of the amplitude transmittance from the left and
express with their help all macroscopic parameters. This approach based on the
properties of the transfer matrix with respect to mirror symmetry was
generalized for systems of anisotropic thin layers in [10].

Due to the dependence of the divergence of the energy flux of a plane
electromagnetic wave only on the normal (perpendicular) Cartesian coordinate,
the power (energy) transfer in a system of thin layers in a planar arrangement
is determined only by normal component of the Poynting vector calculated using
tangential vectors of the intensity of the electric and magnetic fields
separately for each of the counter propagating waves. Therefore, we will
calculate the (power) reflectances and transmittances of the thin layer system
following the relations

$\begin{split}&\rho_{\mathrm{R}}=\left|r_{\mathrm{R}}\right|^{2},\
\tau_{\mathrm{R}}=\frac{\mathrm{Re}(Y_{k+1})}{\mathrm{Re}(Y_{0})}\left|t_{\mathrm{R}}\right|^{2},\\\
&\rho_{\mathrm{L}}=\left|r_{\mathrm{L}}\right|^{2},\tau_{\mathrm{L}}=\frac{\mathrm{Re}(Y_{0})}{\mathrm{Re}(Y_{k+1})}\left|t_{\mathrm{L}}\right|^{2},\end{split}$
(10)

where Re stands for the real part of a complex number. From this definition we
obtain the transfer matrix for the normal components of Poynting vector

$\mathbf{N}=\frac{1}{\tau_{\mathrm{R}}}\begin{pmatrix}1&-\rho_{\mathrm{L}}\\\
\rho_{\mathrm{R}}&\delta\end{pmatrix},\
\delta=\tau_{\mathrm{R}}\tau_{\mathrm{L}}-\rho_{\mathrm{R}}\rho_{\mathrm{L}}.$
(11)

It means that for the normal components of Poynting vector at both sides of
the thin film system we have

$\begin{pmatrix}P_{\mathrm{nR},0}\\\
P_{\mathrm{nL},0}\end{pmatrix}=\mathbf{N}\begin{pmatrix}P_{\mathrm{nR},{k+1}}\\\
P_{\mathrm{nL},{k+1}}\end{pmatrix}.$ (12)

If we are interested in the transformation of normal components of the
Poynting vector of the waves propagating in one direction or the other, and
thus in determining the reflectance and transmittance of a thick layer, we
must first determine the attenuation of the field propagating through a layer
of thickness $h$ with complex refractive index $n$, if the propagation angle
$\theta_{0}$ is measured in a medium with refractive index $n_{0}$. This
attenuation of the field is determined by the attenuation factor $0<U\leq 1$

$U=\exp\left(-2h\frac{\omega}{c}\left|\mathrm{Im}\sqrt{n^{2}-(n_{0}\theta_{0})^{2}}\right|\right),$
(13)

where the absolute value in the argument of the exp function is given to
emphasize the energy passivity of the medium. With its help we can determine
the transfer matrix of the normal components of the Poynting vector inside
arbitrary thick layer

$\mathbf{N}=\begin{pmatrix}1/U&0\\\ 0&U\end{pmatrix}.$ (14)

The relations for the transfer matrices of the normal components of the
Poynting vector allow them to be combined for any number of systems
alternating thick layers with systems of thin layers (some of which may be
empty), just multiplying them in the correct order. This is of course true if
the tangential components of the electric and magnetic field intensities, and
hence the normal components of the Poynting vector, are continuous at the
boundaries. The resulting product of these matrices then needs only to be
compared with the matrix elements in the equation (11) to obtain the resulting
(power) reflectances and transmittances. For example, for two thin film
systems deposited on both sides of a thick substrate, the resulting
reflectances and transmittances are determined from the relation

$\begin{split}&\frac{1}{\tau_{\mathrm{R}}}\begin{pmatrix}1&-\rho_{\mathrm{L}}\\\
\rho_{\mathrm{R}}&\delta\end{pmatrix}=\frac{1}{\tau_{{\mathrm{R}}1}}\begin{pmatrix}1&-\rho_{{\mathrm{L}}1}\\\
\rho_{{\mathrm{R}}1}&\delta_{1}\end{pmatrix}\begin{pmatrix}1/U_{2}&0\\\
0&U_{2}\end{pmatrix}\\\
&\times\frac{1}{\tau_{{\mathrm{R}}3}}\begin{pmatrix}1&-\rho_{{\mathrm{L}}3}\\\
\rho_{{\mathrm{R}}3}&\delta_{3}\end{pmatrix}\end{split}$ (15)

where $\delta$ has the same meaning for the whole system as in (11) and the
quantities denoted here by the subscript 1 refer to a system of thin layers
deposited from the left on a thick substrate and calculated from the equation
(8) assuming an infinitely extensive substrate from the right. The attenuation
factor $U_{2}$ (equal to one for a dielectric medium) refers to the thick
layer material, and the quantities denoted by the subscript 3 refer to the
system of thin layers deposited on the substrate from the right, while the
thick layer material serves as an half-infinite medium on the left. Any of the
subsystems may be empty.

In this way, for example, the influence of the reflection from the rear
boundary of the substrate on the measured parameters of the layers can be
taken into account, which is otherwise considered as a half-space from which
light is not reflected back to the system.

###  2.2 Maximally flat antireflection system of layers for coherent light

The assumption of perfectly coherent light is equivalent to the propagation of
a monofrequency field in ideal thin dielectric layers, whose interference
effect is highest when the optical thicknesses of the layers are equal to a
quarter of the chosen (central) wavelength or integer multiples thereof.
Therefore, let us look for the refractive indices of the system of quarter-
wave layers that best minimize the reflectance of the substrate around the
central wavelength at perpendicular incidence.

Let us define maximally flat antireflection coatings as those that exhibit
zero value of the highest possible number of reflectance derivatives with
respect to the angular frequency $\omega$ for the central frequency
$\omega_{c}$. Derivatives can alternatively be made with respect to the common
phase change as the wave pass through each layer, i.e., the common phase layer
thickness $\varphi=\varphi_{j}=(\pi/2)\omega/\omega_{c}$, $j=1,\dots,k$. At
the same time, the zero of the first derivative follows from the requirement
of zero reflectance at the central wavelength, as it is a local minimum of
reflectance.

From the Taylor expansion of the interference matrices around $\varphi=\pi/2$
and the above requirement that the derivatives must be zero, we obtain a
system of nonlinear equations of the form [6]

$\begin{matrix}N_{k_{j}}^{(a)}=n_{0}n_{k+1}N_{k_{j}}^{(b)},&1\leq j\leq k,&j\
\mathrm{odd},\\\ N_{k_{j}}^{(a)}=\dfrac{n_{0}}{n_{k+1}}N_{k_{j}}^{(b)},&2\leq
j\leq k,&j\ \mathrm{even},\end{matrix}$ (16)

where the Pohlack’s coefficients are expressed by the formulas [11]

$\begin{matrix}N^{(a)}_{k_{j}}=\displaystyle\sum_{M_{k_{j}}}\dfrac{n_{i_{1}}n_{i_{3}}\dots
n_{i_{j}}}{n_{i_{2}}n_{i_{4}}\dots n_{i_{j-1}}},&1\leq j\leq k,&j\
\mathrm{odd},\\\
N^{(a)}_{k_{j}}=\displaystyle\sum_{M_{k_{j}}}\dfrac{n_{i_{1}}n_{i_{3}}\dots
n_{i_{j-1}}}{n_{i_{2}}n_{i_{4}}\dots n_{i_{j}}}&2\leq j\leq k,&j\
\mathrm{even},\\\
N^{(b)}_{k_{j}}=\displaystyle\sum_{M_{k_{j}}}\dfrac{n_{i_{2}}n_{i_{4}}\dots
n_{i_{j-1}}}{n_{i_{1}}n_{i_{3}}\dots n_{i_{j}}},&1\leq j\leq k,&j\
\mathrm{odd},\\\
N^{(b)}_{k_{j}}=\displaystyle\sum_{M_{k_{j}}}\dfrac{n_{i_{2}}n_{i_{4}}\dots
n_{i_{j}}}{n_{i_{1}}n_{i_{3}}\dots n_{i_{j-1}}}&2\leq j\leq k,&j\
\mathrm{even}.\end{matrix}$ (17)

The summation set $M_{k_{j}}=\left\\{i_{1},i_{2},\dots i_{j}\right\\}$ is
determined by the inequalities

$\begin{split}&1\leq i_{1}\leq k-(j-1),\ i_{1}+1\leq i_{2}\leq
k-(j-2),\dots,\\\ &i_{j-1}+1\leq i_{j}\leq k-(j-j)\end{split}$ (18)

or equivalently

$\begin{split}&j\leq i_{j}\leq k,\ j-1\leq i_{j-1}\leq i_{j}-1,\dots,\\\
&j-(j-1)\leq i_{1}\leq i_{2}-1.\end{split}$ (19)

If both conditions (16) are satisfied, then the reflectance of the maximally
flat antireflection system defined in this way is depending on the common
phase layer thickness $\varphi$ as follows

$\rho(\varphi)=\dfrac{\rho_{0}\cos^{2k}\varphi}{1-\rho_{0}+\rho_{0}\cos^{2k}\varphi},$
(20)

where $\rho_{0}=\left[({n_{0}-n_{k+1})/(n_{0}+n_{k+1}})\right]^{2}$ is the
reflectance of the bare substrate without layers that a maximally flat
antireflection system periodically achieves at frequencies equal to even
multiples of the central frequency.

From the equivalence of summation sets (18) and (19) and the transformation of
the indices $i_{j+1-m}\to i_{m},\ m=1,2,\dots,j$ we obtain partial result that
the first of the equations (16) is solved if following symmetry condition
between the refractive indices of the layers is satisfied

$n_{j}n_{k+1-j}=n_{0}n_{k+1},\quad j=1,2,\dots,k.$ (21)

The second set of equations (16) together with the conditions (21) represent a
complete system of nonlinear equations for unknown refractive indices. In [6]
it is shown that for one to four layers the solution (16) can be found in
explicit form, for a higher number of layers it is necessary to find the
solution of the system of equations of several variables numerically. The time
required for the numerical calculation time increases dramatically with the
number of layers, and the accuracy of the result is affected by the finite
number of digits implemented in the computer software.

###  2.3 Maximally flat antireflection system of layers for incoherent light

The reflectance of a system of $k$ thick dielectric layers with refractive
indices $n_{j}$, $j=1,\dots,k$ located between media with refractive indices
$n_{0}$ (for example air) and $n_{k+1}=n_{s}$ (for example substrate) is
determined analogously according to the relation (15) by multiplying the
transfer matrices of the normal components of the Poynting vector at each
boundary, where the matrices with attenuation coefficients are unit. In more
detail, for zero angle of incidence, the reflectance of each interface from
the left is given by the Fresnel relation

$\rho_{\mathrm{R},j}=\rho_{j}=\left(\dfrac{n_{j-1}-n_{j}}{n_{j-1}+n_{j}}\right)^{2},\quad
j=1,2,\dots,k+1.$ (22)

The reflectance from the right is the same as from the left
$\rho_{\mathrm{L},j}=\rho_{\mathrm{R},j}$, transmittances are
$\tau_{\mathrm{R},j}=\tau_{\mathrm{L},j}=1-\rho_{j}$ and the parameter
$\delta_{j}=\tau_{\mathrm{R},j}\tau_{\mathrm{L},j}-\rho_{\mathrm{R},j}\rho_{\mathrm{L},j}=1-2\rho_{j}$.
Therefore, the transfer matrix of the normal components of the Poynting vector
for each interface $j=1,2,\dots,k+1$ is

$\mathbf{N}_{j}=\frac{1}{\tau_{{\mathrm{R}},j}}\begin{pmatrix}1&-\rho_{{\mathrm{L}},j}\\\
\rho_{{\mathrm{R}},j}&\delta_{j}\end{pmatrix}=\frac{1}{1-\rho_{j}}\begin{pmatrix}1&-\rho_{j}\\\
\rho_{j}&1-2\rho_{j}\end{pmatrix}$ (23)

and thus the resulting transfer matrix $\mathbf{N}$ of the normal components
of the Poynting vector with the resulting reflectances and transmittances is
given by the product of the sub-matrices in the correct order

$\mathbf{N}=\mathbf{N}_{1}\mathbf{N}_{2}\dots\mathbf{N}_{k+1}=\frac{1}{\tau_{\mathrm{R}}}\begin{pmatrix}1&-\rho_{\mathrm{L}}\\\
\rho_{\mathrm{R}}&\tau_{\mathrm{R}}\tau_{\mathrm{L}}-\rho_{\mathrm{R}}\rho_{\mathrm{L}}\
\end{pmatrix}.$ (24)

Hence, for the resulting reflectance from the left (in this case the same as
from the right), we get the expression from the elements of the matrix
$\mathbf{N}$

$\rho_{\mathrm{R}}(n_{1},\dots,n_{k})=-\dfrac{N_{12}}{N_{11}}=\dfrac{N_{21}}{N_{11}},$
(25)

which need not be explicitly stated.

For our purpose, we understand $\rho_{\mathrm{R}}$ as a function of the
refractive indices of the layers with given refractive indices of the
surrounding media for which we are looking for the minimum. Some numerical
minimization methods can be used to find the minimum reflectance of layers
with incoherent light, but there is also an analytical solution. It is
sufficient to set the derivatives of the function
$\rho_{\mathrm{R}}(n_{1},\dots,n_{k})$ with respect to all variables equal to
zero if $n_{0}$ and $n_{s}$ are fixed. From this requirement we obtain simple
recurrent relations

$n_{j}^{2}=n_{j-1}n_{j+1},\quad j=1,2,\dots,k$ (26)

with obvious boundary conditions for $n_{0}$ and $n_{k+1}=n_{s}$.

The solution of the equations (26) can be found in the form

$n_{j}=n_{0}^{1-\frac{j}{k+1}}n_{s}^{\frac{j}{k+1}},$ (27)

which again indicates the validity of the relation (21) given for the coherent
situation, which can be considered as a consequence of the mirror symmetry of
the planar system of layers. Note that we can easily generalize the result
(27) for oblique incidence and s or p polarization when we replace the
refractive indices of the media with their admittances (1).

We can also find explicit expressions for the (minimum) size of the
reflectance of a layer system and incoherent light

$\begin{split}&\rho_{\mathrm{min}}=\dfrac{(k+1)(q^{1/(k+1)}-1)^{2}}{(k+1)q^{2/(k+1)}-2(k-1)q^{1/(k+1)}+k+1},\\\
&q=\dfrac{n_{0}}{n_{s}}\ \mathrm{or\ \ }q=\dfrac{n_{s}}{n_{0}}\end{split}$
(28)

with the same result and expected limit equal to zero for the number of layers
$k$ going to infinity.

##  3 Results of numerical experiments

The figures 1–6 present the results of numerical calculations of the exact
designed maximally flat antireflection systems for three selected numbers of
layers, as described in the theoretical part of the paper. As expected from
the requirement of impedance matching for coherent and incoherent
electromagnetic fields, the refractive index profiles converge to stepwise
smooth transition of their values from the superstrate to the substrate or
vice versa as the number of layers increases. However, each of the two cases
discussed in detail provides this transition with a different curve, but
always such that the symmetry equation (21) is satisfied. It can also be seen
that the band of low reflectance broadens as the number of layers increases,
and in the case of the design for incoherent light the number of local
alternating minima and maxima of reflectance increases (values below
$10^{-15}$ are not shown in the plots). The attempt to further reduce the
reflectivity over a wider wavelength region of the systems thus designed by
varying their thicknesses does not lead to the goal, since the solutions
already obtained represent local extremes in themselves.

Figure 1:  Refractive index profiles of 10 dielectric layers realizing
maximally flat antireflection for coherent light (red line) compared to layers
designed for incoherent light (blue line), substrate refractive index
$n_{s}=2$, superstrate refractive index $n_{0}=1$.

Figure 2:  Dependence of the decimal logarithm of reflectance $\log(\rho)$ on
the relative frequency $\omega/\omega_{c}$ of the system of 10 quarter-wave
layers for the refractive index profiles shown in fig. 1, the red line refers
to the refractive indices of the layer system designed for coherent light, the
blue line is for the refractive indices of layers designed for incoherent
light.

Figure 3:  Refractive index profiles of 20 dielectric layers realizing
maximally flat antireflection for coherent light (red line) compared to layers
designed for incoherent light (blue line), substrate refractive index
$n_{s}=2$, superstrate refractive index $n_{0}=1$.

Figure 4:  Dependence of the decimal logarithm of reflectance $\log(\rho)$ on
the relative frequency $\omega/\omega_{c}$ of the system of 20 quarter-wave
layers for the refractive index profiles shown in fig. 3, the red line refers
to the refractive indices of the layer system designed for coherent light, the
blue line is for the refractive indices of layers designed for incoherent
light.

Figure 5:  Refractive index profiles of 30 dielectric layers realizing
maximally flat antireflection for coherent light (red line) compared to layers
designed for incoherent light (blue line), substrate refractive index
$n_{s}=2$, superstrate refractive index $n_{0}=1$.

Figure 6:  Dependence of the decimal logarithm of reflectance $\log(\rho)$ on
the relative frequency $\omega/\omega_{c}$ of the system of 30 quarter-wave
layers for the refractive index profiles shown in fig. 5, the red line refers
to the refractive indices of the layer system designed for coherent light, the
blue line is for the refractive indices of layers designed for incoherent
light.

Figure 7:  Dependence of the decimal logarithm of reflectance $\log(\rho)$ on
the relative frequency $\omega/\omega_{c}$ of the system of 1000 quarter-wave
layers designed for maximally flat antireflection, the red line refers to the
refractive indices of the layer system designed for coherent light, the blue
line refers to the layer system designed for incoherent light.

For illustration, fig. 7 shows the reflectance versus relative frequency
$\omega/\omega_{c}$ of a system of 1000 quarter-wave layers designed for
maximally flat antireflection in the coherent and incoherent cases, the
refractive indices of the superstrate and substrate are as in the previous
figures. The low reflectance of the order of $10^{-6}$ for incoherent light
can be determined from the easily calculated refractive indices (see equation
27), but this is not the case for coherent light due to the large demands on
computer time.

##  4 Discussion

The equation (27) for the refractive indices of the layers in the case of
incoherent light satisfies the condition (21), originally derived for the
coherent case. The profile of the refractive indices of the antireflection
layers in figs. 1, 3, 5 suggest that it might be possible to match the desired
refractive indices with a suitable curve that would resemble a Gaussian error
function. Indeed, if we assume the sizes of the refractive indices of the
layers to be of the form

$\begin{split}&n_{j}=n_{0}^{1-\alpha^{(k)}_{j}}n_{s}^{\alpha^{(k)}_{j}},\
j=0,\dots,k+1,\\\ &\alpha^{(k)}_{0}=0,\ \alpha^{(k)}_{k+1}=1.\end{split}$ (29)

For example, from the known refractive indices $n_{j}$ we get the requirement
for the size of $\alpha^{(k)}_{j}$

$\alpha^{(k)}_{j}=\dfrac{\log(n_{j}/n_{0})}{\log(n_{k+1}/n_{0})},\
j=0,\dots,k+1.$ (30)

For the incoherent case we obviously have $\alpha^{(k)}_{j}=j/(k+1)$ from
(27).

The auxiliary coefficients $\alpha^{(k)}_{j}$ entering the expression for the
refractive indices of the layers using the Gaussian error function can be
assumed in the approximate form

$\alpha^{(k)}_{j}=\dfrac{1}{2}\left[1+\mathrm{erf}\left(b_{k}(j-\dfrac{k+1}{2})\right)\right],\
j=1,\dots,k.$ (31)

Since the function $\mathrm{erf}(x)$ is odd, such a dependence satisfies the
condition (21), now written as $\alpha^{(k)}_{j}+\alpha^{(k)}_{k+1-j}=1$.
However, the coefficients $b_{k}$ have to be found numerically so that the
fitted refractive indices are as close as possible to the solutions of
equations (16), which can be done numerically by the least squares method. The
refractive indices of the layers obtained by such a matching procedure differ
from the exact values only at the fourth decimal place, as shown in fig. 8,
which would hardly be observable in a graphical representation of the
refractive index profile.

Figure 8:  The difference $\Delta n$ in the refractive indices of the 30
layers obtained by approximation using the Gaussian error function (31) and
the exact refractive indices calculated by solving the equations (16)
depending on the layer order. The coefficient $b_{30}=0.257959458606224$
entering (31) was obtained numerically, the refractive indices of the
surrounding environments are $n_{0}=1$, $n_{s}=2$.

Figure 9:  The size of the optimized coefficients $b_{k}$ entering the
approximation expression (31) depending on the number of layers, the
refractive indices of the environment surrounding the layers are $n_{0}=1$,
$n_{s}=2$.

However, the coefficients $b_{k}$ calculated in this way depend on the number
of layers, as shown in fig. 9. We can also assume their dependence on the
refractive indices of the surrounding media, but we would welcome this
dependence to be as negligible as possible, which would allow us to easily
make antireflection designs close to the theoretical best.

We can also ask how small differences in approximate refractive indices of the
order of $10^{-4}$ from the theoretically accurate values affect the quality
of the resulting antireflection. Even if the deviations from the ideal state
are small, they are still observable, as shown in fig. 10.

It can be supposed that the stepwise smooth profiles of the refractive indices
of the layers according to the relation (31) will not represent poor
antireflection designs, as it is proven by fig. 11 for 100 layers, compared to
the ideal refractive indices designed for coherent light, which cannot be
easily computed explicitly for 100 layers due to computational complexity.

Figure 10:  Dependence of the decadic logarithm of reflectance $\log(\rho)$ on
the relative frequency $\omega/\omega_{c}$ of a system of 30 quarter-wave
layers designed for maximally flat antireflection in coherent light. The red
line refers to the refractive indices of the layers designed according to the
exact algorithm, the blue line refers to the refractive indices of the layers
designed according to the approximate expression (31), refractive indices
$n_{0}=1$, $n_{s}=2$.

Figure 11:  Dependence of the decadic logarithm of the reflectance
$\log(\rho)$ on the relative frequency $\omega/\omega_{c}$ of a system of 100
quarter-wave layers designed according to the relation (31) with parameter
$b_{100}=0.1$ (blue line). For comparison, the red line refers to the
refractive indices of the layers designed according to the exact algorithm,
$n_{0}=1$, $n_{s}=2$.

However, the relation (31) can be used to design an antireflection with a
different refractive index of the substrate with a suitable choice of the
parameter $b_{k}$. For example, fig. 12 shows the result of the calculation
for 30 layers and $n_{0}=1$, $n_{s}=4$ with the choice of $b_{30}=0.258$.
Thus, the expression (31) allows the designer to play with different
antireflection constructions that are more or less close to the exact
solution.

Perhaps it is interesting to calculate in the coherent case the quantities
$P^{(k)}_{j}=2^{k}\alpha^{(k)}_{j}$ and compare them with the Schallenberg’s
approximation [12]. Not surprisingly, if we round the numbers $P^{(k)}_{j}$
computed numerically according to the exact algorithm to integers, we get the
solutions that follow from Schallenberg’s procedure. For example, for 4
layers, according to [12] we have $P^{(4)}=$ 1, 5, 11, 15, while the exact
algorithm gives the numbers $P^{(4)}=$ 1.003732550742073, 5.003732550742090,
10.996267449257910, 14.996267449257919. However, the refractive indices of the
layers show small differences. For $n_{0}=1$, $n_{s}=2$ we get according to
[12] 1.044273782427414, 1.241857812073484, 1.610490331949254, 1.
91520656139714, while the exact algorithm yields values of 1.044442655609480,
1.242058637263131, 1.610229936009292, 1.914896896692627. The relatively small
deviations between the exact and Schallenberg’s procedures are substantial in
order to be able to declare, on the basis of a comparison of the reflectances
around the central frequency, that maximally flat antireflections for the
coherent case provide only solutions of the equations (20) with the resulting
reflectance according to (16). However, Schallenberg’s solution is undoubtebly
slightly better than the approximation using the error function (31).

Figure 12:  Dependence of the decadic logarithm of the reflectivity
$\log(\rho)$ on the relative frequency $\omega/\omega_{c}$ of a system of 30
quarter-wave layers designed according to the relation (31) with parameter
$b=0.258$ (blue line). For comparison, the red line refers to the refractive
indices of the layers designed according to the exact algorithm, $n_{0}=1$,
$n_{s}=4$.

##  5 Conclusions

In this paper two exact solutions of maximally flat antireflection layer
systems are proposed for two ideal cases of coherent (for thin layers) and
incoherent (for thick layers) light. The first case is based on the
requirement that the maximum number of derivatives of the reflectance of the
system of quarter-wave thin layers with respect to the frequency is equal to
zero for the central frequency, while the second case is based on the
requirement of the minimum reflectance of the system of thick layers. The
first case requires finding the zero point of a system of functions of several
variables, which can be done explicitly for systems of at most four layers,
while the second case requires finding the minimum of a function of several
variables for which an explicit solution has been found. The graphs then
present the obtained profiles of the refractive indices and the corresponding
reflectance curves as a function of the normalized light frequency. There is
no speculation about the realization of such profiles, although it cannot be
ruled out that the current development of nanotechnology will ultimately be
able to achieve flat antireflection to the maximum extent. The possibility of
using approximate expressions for layer refractive indices to easily construct
quite good antireflection systems is also discussed.

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