# Design of Explainability Module with Experts in the Loop for Visualization
and Dynamic Adjustment of Continual Learning

Yujiang He, Zhixin Huang, Bernhard Sick Accepted at the AAAI-22 Workshop on
Interactive Machine Learning (IML@AAAI’22)

###### Abstract

Continual learning can enable neural networks to evolve by learning new tasks
sequentially in task-changing scenarios. However, two general and related
challenges should be overcome in further research before we apply this
technique to real-world applications. Firstly, newly collected novelties from
the data stream in applications could contain anomalies that are meaningless
for continual learning. Instead of viewing them as a new task for updating, we
have to filter out such anomalies to reduce the disturbance of extremely high-
entropy data for the progression of convergence. Secondly, fewer efforts have
been put into research regarding the explainability of continual learning,
which leads to a lack of transparency and credibility of the updated neural
networks. Elaborated explanations about the process and result of continual
learning can help experts in judgment and making decisions. Therefore, we
propose the conceptual design of an explainability module with experts in the
loop based on techniques, such as dimension reduction, visualization, and
evaluation strategies. This work aims to overcome the mentioned challenges by
sufficiently explaining and visualizing the identified anomalies and the
updated neural network. With the help of this module, experts can be more
confident in decision-making regarding anomaly filtering, dynamic adjustment
of hyperparameters, data backup, etc.

##  1 Introduction

Continual learning enables neural networks to accumulate novel knowledge for
handling unknown tasks sequentially through learning in task-changing
contexts, where distributions of the source domain and target domain can
change over time. It aims to overcome the so-called catastrophic forgetting
problem (McCloskey and Cohen 1989; Ratcliff 1990), i.e., a trained neural
network could forget the obtained knowledge regarding previous tasks after
learning to handle novel tasks. Numerous continual learning algorithms have
been proposed and evaluated in experiments with public datasets, such as
(Kirkpatrick et al. 2017; Zenke, Poole, and Ganguli 2017; Li and Hoiem 2017;
Chaudhry et al. 2018).

Novelties refer to the samples that a pre-trained neural network has never
seen. The bulk of novel samples that appear in applications could result in
concept drift, which leads to a decrease in the performance of the neural
network. In the standard experimental setup of continual learning for
regression tasks, all novelties can be viewed as new tasks for updating the
model using continual learning (He and Sick 2021; He, Huang, and Sick 2021).
The goal is to maintain the performance of the neural network at a high level
throughout its lifecycle. Generally, neural networks learn multiple tasks
sequentially with fixed learning hyperparameters, usually searched beforehand
by intuition or expert prior knowledge. Subsequently, the updated model will
be evaluated on a validation dataset containing known tasks. Standard
evaluation metrics for continual learning are the same as in the traditional
deep learning experiments, e.g., accuracy and precision, which are
understandable but barely comprehensive.

Compared with the performance of the continual learning algorithms in
experiments, an unexpected decrease of the performance could be caused in
real-world applications because of the following shortcomings in the general
setup:

  * •

Confusing anomalous and novel examples. The samples that the current neural
network can not predict accurately can be defined as novelties. The network
should continually learn to update its knowledge memory as data is
accumulated. However, anomalies, i.e., samples that are meaningless for
continual learning, also belong to the novelty group. Anomalies are usually
caused by unforeseen incidents, such as damaged sensors and stopped energy
generators under specific conditions. Learning meaningful novelties can
instill a neural network with new abilities to handle non-stationary data
streams. Still, learning anomalies without restriction could mislead the
neural network to a wrong converged state and decrease its performance.

  * •

Fixed hyperparameters. Learning all tasks with the fixed hyperparameters
indicates that the algorithm must be robust to the chosen hyperparameters.
Hyperparameters choices are usually tuned manually by prior knowledge and
intuition or searched by optimization methods, such as grid search or random
search, based on the currently available datasets. However, the constant
hyperparameters could be sensitive to unknown tasks and then disturb the
evolution of the neural network in the long run.

  * •

A lack of comprehensive evaluation metrics. Standard evaluation metrics, such
as accuracy and F1-score for classification problems, can reflect only the
performance of neural networks regarding the given dataset at a specific time
point, i.e., at the convergence after the last training epoch. We think a set
of comprehensive metrics should evaluate the updating results from various
perspectives besides accuracy. For example, calculating the extent to which
catastrophic forgetting by quantifying the overlap of the distributed
representation (French 1992), or counting the free neurons to estimate the
remaining free space of the neural network, which determines whether
subsequent tasks can be learned continually and successfully (Mirzadeh,
Farajtabar, and Ghasemzadeh 2020). A set of comprehensive metrics can timely
alarm supervisors to react to an unsatisfied update.

  * •

Assessment with access to old datasets. In some cases, dependency on old
datasets for learning and evaluation might violate some regulations, such as
the General Data Protection Regulation, which the European Union introduced to
remedy potential data privacy problems. The regulation states that the data
subject has the right to request the erasure of personal data, which requires
learning and evaluation without access to the old dataset under some
circumstances.

  * •

A lack of visualization methods. Numeric evaluation can give us only an
abstract score for detected anomalies or updated neural networks. A lack of
elaborated visualization leads to researchers’ uncertainty in the identified
anomalies or the updated models. Furthermore, despite being possible to
analyze the generalization performance of neural networks by observing the
downward trend of the losses over training epochs, the progress regarding the
dynamic evolution of neural networks’ hidden layers is still opaque. Humans
can not understand and interpret complex, high-dimensional data without proper
visualization methods.

In this article, we propose thus a conceptual design of an explainability
module with experts in the loop for addressing the problems that are mentioned
above and generally exist in current applications of continual learning. The
goal of the explainability module is to provide sufficient explanation
regarding the identified anomalies and details of model updating to experts.
Based on the interaction between the explainability module and experts, we
expect to achieve real-time monitoring and dynamic management for neural
networks with continual learning.

The remainder of this article starts with a description of our conceptual
design, which integrates and extends the existing techniques in terms of
evaluation and visualization. Then in Sec. 3, we present a specific use case
about how the proposed design can serve as an extension to the existing work
in (He and Sick 2021), an adaptive continual learning framework for solving
regression problems in non-stationary contexts. The article is concluded with
a brief outlook of our further research.

##  2 Design of Explainability Module

Figure 1: An illustration about how the explainability module with experts in
the loop manages the data stream and the neural network dynamically in a
general continual learning application for regression. Based on the data in
the interaction information flow 1, the evaluation bank can compute assessment
scores for detected anomalies or the neural network updated with the new task
and then send the data as well as the scores to the following bank for
visualization. The scores and graphic outputs can offer experts insight into
the details of the data and the model. Sequentially, experts can easily make
an accurate decision and then feed it back via the information flow 2.

Figure 1 illustrates the interaction between a continual learning application
and the proposed explainability module with an expert in the loop. A new task,
also called as an unknown task, is defined differently in the research
community of continual learning. For example, (Lomonaco and Maltoni 2017)
propose three types of new tasks for object recognition, which are new
instances (NI), new classes (NC), and new instances and classes (NIC),
respectively. NI and NC refer to new training patterns of the same classes and
new training patterns belonging to the different classes, correspondingly. NIC
is a combination of NI and NC, i.e., the new training patterns belonging to
known and new classes. In comparison, a new task in regression is hard to
define due to a lack of labels. (He and Sick 2021) consider a new regression
task as a group of samples that decrease the predictive accuracy of the
current neural network. In Fig. 1, we display an example of a data stream
containing a known regression task, a new regression task, and an anomaly.

The neural network can predict the known task and evolve with the new task by
continual learning. The identified anomaly will be transferred to the
explainability module via the interaction information flow 1 for a precise
assessment. Besides the anomalies, the flow 1 also consists of other
application-related details, including the samples from the data stream, the
corresponding predictions, hyperparameters of learning, the updated neural
network, an elaborated updating log, etc. The interaction can happen once an
anomaly is detected or an update is done.

As shown in Fig 1, the explainability module contains three primary
submodules: evaluation bank, visualization bank, and decision bank, each of
which has two blocks. The anomaly block is designed to analyze the potential
anomalies identified in the data stream, and the model block can evaluate the
updated neural network. Experts can observe and judge whether a positive-true
anomaly exists in the data stream and decide how to process it. Besides,
experts can decide to keep a backup of a successfully updated model or restore
a failed updated model with an adjustment of the hyperparameters after
assessing the updating quality based on the evaluation metrics and
visualization. The experts are the final decision maker with professional
knowledge and experience. The main function of our explainability module is to
provide suggestions and explanations to help the experts to make decisions.
The final decisions regarding the anomalies and the updated models are
transferred via the interaction information flow 2.

The remainder of this section elaborates the three submodules of our design
and introduces the existing techniques relevant to implementing the
explainability module.

###  2.1 Evaluation Bank

#### Anomaly Evaluation

The anomaly evaluation block is responsible for evaluating the potential
anomalies by computing an anomaly score, which assists experts in decision-
making.

In a traditional deep learning setup, training and validation datasets are
preprocessed before experiments by discarding anomalies to eliminate the
negative influence of insufficient data and efficiently improve the training
results. However, anomalies in real-world applications are unavoidable,
bringing high but meaningless entropy to the training process and finally
disturbing the updating results. Therefore, retaining meaningful samples and
filtering out anomalies from novelties plays a crucial role in maintaining
neural networks’ performance in continual learning applications.

Lots of machine learning-based anomaly detection methods have been proposed,
including supervised and unsupervised methods. Deep Support Vector Data
Description (Deep SVDD) maps samples from the input space to the output space
using a neural network. It identifies anomalies due to the distance between
the samples and centers of clusters (Ruff et al. 2018). Samples that are far
away from the cluster center of normal samples are marked as anomalies. (Ding
and Fei 2013) propose an unsupervised method for anomaly detection, called the
Isolation Forest Algorithm for the Stream Data (IFASD). It is a variety of
Isolation Forest focusing on multidimensional time series. Besides, (Meng et
al. 2019) present a transformer-based unsupervised method to detect anomaly by
comparing the error between the original time series and the reconstructed
time series. These methods evaluate the quality of the detected anomalies by
calculating a score and comparing the score with a predefined threshold. If
the score is above the threshold, the corresponding novelties are marked as
anomalies rather than a new task for continual learning.

For supervised methods, samples must be labeled manually, which is time-
consuming and expensive in real-world applications (Siddiqui et al. 2018). By
comparison, using unsupervised methods can avoid additional manual labels and
computation overhead. Still, trained models might have a very high false-
positive rate, requiring further expert verification of the identified
anomalies (Attenberg and Provost 2011).

#### Model Evaluation

The model evaluation can assess the quality of the updated model by providing
an evaluation score based on a comprehensive metric.

We consider that a comprehensive evaluation metric should consider not only
the improved prediction accuracy for new tasks but also the dropping accuracy
for old tasks, also known as forgetting, during continual learning. A trade-
off between obtaining knowledge from new tasks and retaining knowledge from
old tasks, the so-called plasticity-stability dilemma (Mermillod, Bugaiska,
and Bonin 2013), is dependent on the specific application scenario. For
example, in the application of renewable energy forecasting, the aging of
power generators is a slow-changing and irreversible process. In this case,
more attention can be paid to how neural networks can fit new data well and
slightly discard the stored memory. However, in the application of
meteorological time series forecasting, weather conditions change seasonally,
where the probability distribution of data shifts periodically over time.
Memory regarding historical data should be consolidated while neural networks
learn new data.

(Lopez-Paz and Ranzato 2017) define three metrics for evaluating continual
learning experiments: Average Accuracy, Backward Transfer, and Forward
Transfer. The goal of the metrics is to assess the influence of learning a new
task on the performance of the trained network regarding all known tasks,
including previous and future tasks. (Díaz-Rodríguez et al. 2018) extend this
work by proposing additional three metrics, i.e., Model Size Efficiency,
Sample Storage Size Efficiency, and Computational Efficiency, which measure
the increment of model memory size, storage overhead of samples for replay-
based strategies, and computational overhead for learning all tasks. For
ranking purposes, these metrics are summed to a continual learning score with
different weighting schemes. Besides, (He, Henze, and Sick 2020; He and Sick
2021) evaluate continual learning strategies in the context of regression
problems in terms of prediction error, fitting error, forgetting ratio, and
training time. The idea behind is similar to (Díaz-Rodríguez et al. 2018).

###  2.2 Visualization Bank

#### Anomaly Visualization

We use the anomaly visualization to generate the visualization outputs of
these anomalies, which can explain the abstract anomaly score in a trustable
and understandable way.

High-dimensional data usually indicates multiple data sources, e.g., multiple
sensors integrated into a complex system, which brings us two main challenges
for anomaly detection. Firstly, it is difficult to manually analyze and
visualize high-dimensional data without losing local and global architecture
information. Secondly, some data sources are dependent, which means that one
faulty source could affect other normal ones. The dependency makes it too
difficult to identify the exact faulty source and when the fault occurs. Our
proposed anomaly visualization module aims to solve these challenges by
figuring out the following questions:

  * •

How to utilize dimension reduction techniques to make the visualization of the
high-dimensional data understandable?

  * •

How to visualize the dependency among multiple data sources for providing
reasonable explanations for detected anomalies?

  * •

How to rank anomaly detection methods if available methods or metrics are
multiple?

#### Model Visualization

The model visualization focuses on visualizing the dynamic evolving process of
the updated model, including the change of model parameters from the random
initialization to convergence. Compared to the learning curve, where each
point represents the prediction error at each time point, we can be more
convinced by the visualization regarding the dynamic evolution during
continual learning, which can show more details of the progress. For achieving
this goal, we need to consider how to preserve the information of the data as
much as possible in the process of dimension reduction.

Common dimension reduction techniques, such as Principal Component Analysis
(PCA), diffusion maps (Coifman and Lafon 2006), multidimensional scaling (MDS)
(Cox and Cox 2008), t-SNE (Van der Maaten and Hinton 2008), can efficiently
preserve local or global structure information. (Moon et al. 2019) propose
PHATE, a method that can visualize high-dimensional data while revealing both
local and global structures. They compare PHATE and other methods on
biological and artificial datasets. Based on PHATE, (Gigante et al. 2019)
propose M-PHATE, which visualizes dynamic evolution of neural networks by
extracting changes of hidden representations over epochs. Besides, (Mirzadeh,
Farajtabar, and Ghasemzadeh 2020) selectively visualize active neurons in each
task using a heat map, which is intuitive but not applicable to giant neural
networks.

Similar to the anomaly visualization, we also need to rank continual learning
algorithms under various evaluation metrics for the model visualization, for
example, to compare different algorithms or neural network architectures in
ensemble learning. Common visualization methods, such as radar charts and line
charts (Díaz-Rodríguez et al. 2018), are intuitive and easily implemented for
ranking purposes. It makes the updated model more transparent and trustworthy.

The inputs of the visualization bank are the interaction information flow 1
and the output of the evaluation bank. Their graphic outputs provide experts
insights into latent representations of data and models and support experts in
making decisions. Furthermore, we expect that the graphic outputs are
interactive rather than static. They can, for example, zoom in/out, display
the exact values, and easily be exported and saved in a readable format.

###  2.3 Decision Bank

#### Anomaly Decision

The scores and graphs of anomalies from the first two banks are sent to
experts for manual analysis. According to (Gruhl et al. 2021), novelties can
be grouped into four classes: sensor failure, component failure, system
failure, and concept drift. Decision bank could offer experts these
suggestions for the categories of the novelties, as well as the corresponding
decisions:

  * •

In the context of sensor failure, where only one of the sensors behaves
abnormally, we can replace the data from the fault sensor with the data from
the highly relevant sensor, rather than discarding all data of these sensors.

  * •

Component failure denotes that multiple low-correlated sensors in a component
behave abnormally at the same time. Low correlation means that these sensors
may not have a cascade relationship, thus indicating the defect of the
component. The whole system might break down if the component is connected
with others. Such novelties should be stored in an anomaly database for
further analysis and physical system maintenance rather than updating.

  * •

System failure is similar to multi-component failure, where faults are
detected in multiple components. A system failure could be caused by multiple
faulty components or a chain reaction caused by a single faulty component.
Like component failure, such a system failure should also be stored in the
anomaly database instead of being learned. Besides, manual intervention could
also lead to system failure. For example, wind turbines have to be stopped by
a shut-off mechanism under the condition of high wind speed to avoid
significant damage. If the manual intervention is long-term and periodical,
such a system failure could be learned by neural networks.

  * •

Concept drift means that the error between predictions of neural networks and
measurements of sensors starts to increase from some time point due to change
of source or target data distribution. It often occurs when, for example,
sensors or devices are replaced, sensor parameters are adjusted, or
environmental and temporal changes. This kind of novelty should be viewed as a
new task for updating.

According to the analysis and suggestions given by the module, experts can
decide to discard these anomalies or use them for updating models. In the case
of too many misclassified novelties in anomaly evaluation, experts can adjust
the related parameters, such as the threshold, or update the anomaly detector
to reduce the false-positive rate.

#### Model Decision

There are three significant decisions for the updated neural network: backup,
rollback, and hyperparameters adjustment. These decisions are relevant.

The backup means storing the neural networks and the datasets after a
successful update. The rollback denotes that the updated model returns to the
backup state as an initialization state for a new update if the update is
failed.

Here, a successful update is defined as the loss function gradually converges
to a low state in learning unknown tasks without a noticeable decrease of
performance of neural networks on old tasks. Parameters of the successfully
updated model should be stored as a backup. According to the types of replay-
based continual learning algorithms, data backup can denote storing training
datasets regarding known tasks in an original format or training a generative
adversarial network with current tasks to generate fake data.

By contrast, a failed update means that the loss can not be converged within
the given training epochs or the updated model loses most knowledge regarding
old tasks in the evaluation phase. The process of optimization using common
continual learning algorithms is minimizing a target loss function with
stochastic gradient descent. Therefore, stochasticity in the training process
could cause a failed update, which leads to catastrophic forgetting.

In addition, the improper hyperparameters or the undetected anomalies in the
training dataset can also lead to a failed update. Therefore, the decision
bank can display all adjustable hyperparameters to experts and provide
explicit suggestions with corresponding explanations. Experts can adjust the
hyperparameters by their intuitions and professional experience for the future
application or a new round of updating under the case where the updating
result is unacceptable.

##  3 Use Case & Outlook

###  3.1 Use Case: The CLeaR Framework

CLeaR, short for continual learning for regression problems, is an adaptive
framework for overcoming the catastrophic forgetting problem in a non-
stationary regression application scenario (He and Sick 2021). Samples in the
real-time data stream are categorized into novelties and familiarities.
Novelties are the samples that the trained neural network can not predict
accurately, while familiarities are the samples that the network is familiar
with. CLeaR distinguishes the two types by computing the error between the
truth and the prediction and comparing the error with a threshold, which is
dynamically adjustable according to the network’s convergence after training.
Novelties and familiarities are stored in two independent buffers for
continual learning. Updating can be triggered under customizable conditions.
For example, (He and Sick 2021) design a novelty buffer with a fixed length
and trigger updating when the novelty buffer is full.

The five problems mentioned in Sec. 1 exist in the current design of CLeaR.
Firstly, the boundary between the novelty and anomaly is ambiguous according
to the categorization, possibly leading to a failed update because of
misclassification. Secondly, the experimental setups of CLeaR in (He and Sick
2021) and (He, Huang, and Sick 2021) are the same, i.e., learning all detected
unknown tasks with a set of fixed parameters that are optimized beforehand.
Thirdly, the influences of two hyperparameters of the CLeaR framework, i.e.,
the novelty buffer size and the threshold factor, on its learning performance
have been analyzed in (He and Sick 2021). The two parameters control the
frequency of triggered updates and adjust the judgment mechanism for novelty
detection, respectively, thus balancing the performance of the updated model
on all tasks and affecting the computational overhead in applications.
Nevertheless, a comprehensive evaluation of the robustness of these
hyperparameters is still missing in the previous research. Because these
components in the framework, such as the type of the neural network and
continual learning algorithms, the size of the buffers, and the novelty
detection method, are customizable depending on the requirements of the
specific application. These hyperparameters should be adjusted dynamically
during the application in order to learn different tasks efficiently. Besides,
the proposed module can improve the explainability and transparency of the
CLeaR framework by providing visualization and comprehensive evaluations.

###  3.2 Outlook

This paper analyzes the problems and shortcomings that can appear when
continual learning is applied to real-world scenarios that have the same setup
as in experiments. For solving these problems, we propose the conceptual
design of the explainability module integrated with various deep learning-
related techniques, such as anomaly detection, evaluation, and visualization.
In our further research, we plan to implement the design and apply it to the
existing the CLeaR framework as an extension. Based on the visual and
explainable outputs of the module, experts can dynamically manage the
evolution of neural networks driven by continual learning, and perform more
reliable evaluation to its updates in task-changing context.

## Acknowledgments

This work is supported within the Digital-Twin-Solar (03EI6024E) project,
funded by BMWi: Deutsches Bundesministerium für Wirtschaft und Energie/German
Federal Ministry for Economic Affairs and Energy. We also want to thank our
colleagues in the Intelligent Embedded Systems Lab at the University of
Kassel, Kristina Dingel and Lukas Rauch, for their constructive reviews and
suggestions.

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# Revisit of generalized Kerker’s conditions using composite metamaterials

Rfaqat Ali<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio
de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brasil

###### Abstract

Achieving zero backward scattering (ZBS) and zero forward scattering (ZFS),
i.e., the so-called the first and second Kerker’s conditions respectively, by
sphere spherical particles is considered to be impossible due to the
unavailability of naturally occurring magnetic materials in the visible
frequency range. We report theoretical modeling to design composite
metamaterials that present large optical magnetic permeability in the visible
frequency range by employing Mie scattering theory and extended Maxwell
Garnett theory. We numerically show that a careful selection of constituents
of a composite metamaterial one can obtain metamaterials with sufficiently
large artificial permeability that eventually provides the Kerker’s criterion
to achieve the Kereker’s conditions. By taking realistic material parameters
we demonstrate that the metamaterials exhibiting ZBS and ZFS have a small
imaginary part of the refractive index than metallic structures that pave a
path to design high-performance nanophotonic devices.

##  I Introduction

Optimal control on directional scattering by spherical particles in the
visible frequency domain is an ultimate goal for researchers to optimize the
efficiency of optical devices nano ; Gao_2017 ; sensing ; nanoen ; Liu2012 ;
plas ; LED . For referential examples, the desire to achieve zero
backscattering plays a crucial role in optical manipulation of spherical
particles chan2011 ; kall2018 ; r.ali  and light management structures used in
photovoltaic devices Ferry2010 ; Akimov2010 . The Mie scattering theory
provides an exact solution to the problem of light scattering by spheres
bohren  and the seminal work on directional scattering was presented by Kerker
et al. in 1983 ker . They revealed proper combinations of relative
permittivity and relative permeability of an unconventional magneto-dielectric
sphere to achieve ZBS and ZFS that are commonly known as the first and second
Kerker’s conditions, respectively. Fulfillment of the first Kerker’s condition
requires a material with identical relative permittivity $\epsilon$ and
relative permeability $\mu$ (i.e. $\epsilon=\mu)$, while the second Kerker’s
condition needs material with $\epsilon=\frac{4-\mu}{\mu+2}$. Observation of
these conditions for the visible light is considered to be impossible due to
the unavailability of such materials that possess large relative magnetic
permeability in the frequency range, thus, Kerker’s method can not be applied
in optical frequencies.

However, in spite of these limitations ZBS and ZFS are successfully
experimentally demonstrated by a dipolar sphere of large permittivity si ;
person . In this case, the incident field equally excites the electric and
magnetic dipoles that produce anisotropic scattering pattern. Since the
anisotropic distribution of the scattered field is governed by the
interference between the electric and magnetic dipoles. For instance, the in-
phase and out of phase oscillations of the electric and magnetic dipoles can
provide an opportunity to observe ZBS and ZFS, respectively Ge ; Lee ;
neito2015 ; Geffrin2012 . Although these proposals are limited to the sphere
of small radius $(r)$ as compared to the incident wavelength $(\lambda)$, such
that $r\ll\lambda$.

Nowadays, the anomalous but very fascinating scattering behavior is observed
by artificially designed so-called metamaterials with unconventional optical
constants angheta2006 ; angheta2013 . They have the ability to present unique
scattering properties like optical cloaking farhat , negative optical
reflection ap1  and efficient control on scattering directionality r.ali  with
valuable interest in the fields of nanoantenna, nano waveguides and optical
manipulations r.ali ; ap2 ; r.alee ; CTchan . The recent advancement in the
field of metamaterials allows to achieve the invisibility of an object through
scattering cancellation by manipulating the optical permittivity and
permeability of the materials farhat ; soukhoulis . Although, it is
challenging to find appropriate materials to design the metamaterial that
presents a strong artificial magnetism in the visible range. Recently, the ZBS
has been shown by a sphere made of composite metamaterial r.ali  by tuning the
permittivity of the sphere in such a way that electric and magnetic multipoles
interfere destructively in the backward direction, thus ZBS is achieved. This
proposal is neither robust against the radius nor accommodates the
permeability of the sphere, while the Kerker’s method is applicable for all
radii.

Motivated by the recent advancement in the field of metamaterials, in this
article, we put forward a composite material platform to design composite
metamaterials that exhibit strong artificial magnetic response in the optical
frequencies range. Furthermore, we engineer the optical constants of the
composite metamaterials to meet the Kerker’s criterion to achieve the ZBS and
ZFS. In order to implement this proposal, we consider a composite metamaterial
made of a dielectric host medium containing dipolar spherical inclusions of
large optical permittivity farhat ; Goia_2003 ; anghetta . Furthermore, we
apply extended Maxwell-Garnett theory to calculate the effective optical
constants of homogenized composite metamaterials in the optical frequency
range by assuming that the sizes of the inclusions are sufficient to overcome
the nonlocal effects Ruppin1973 ; Gubbin_2020 . In addition, the Mie
scattering theory is used to calculate the relevant directional scattering
cross sections of the composite metamaterials.

The rest of the article is organized as follows. Section II is a
methodological part, where the Mie scattering theory and extended Maxwell-
Garnett (EMG) theory are briefly discussed. The main findings of the work are
presented in sections III, where the Kerker’s conditions for composite
metamaterials are analyzed. Finally, we summarize our findings in section IV.

##  II Methodology

###  II.1 Mie Theory

Consider a plane wave with vacuum wavelength $\lambda_{0}$ scattering by a
spherical particle of radius $r$, refractive index $n=\sqrt{\epsilon\mu}$
embedded in a non-absorbing host medium with refractive index
$n_{1}=\sqrt{\epsilon_{1}\mu_{1}}$, where $\epsilon(\epsilon_{1})$ and
$\mu(\mu_{1})$ are the relative permittivity and relative permeability of the
sphere (host medium). The scattering and absorption of incident light by the
sphere is described by the Mie scattering theory in terms of normalized
scattering efficiency $Q_{sca}$, absorption efficiency $Q_{abs}$, extinction
efficiency $Q_{ext}$ and mathematically defined as bohren ; Ge ; Lee ; van

$Q_{sca}=\frac{2}{x^{2}}{\sum_{\ell=1}^{\infty}}{(2\ell+1)(|a_{\ell}|^{2}+|b_{\ell}|^{2}}),$
(1)

$Q_{abs}=\frac{2}{x^{2}}{\sum_{\ell=1}^{\infty}}{(2\ell+1)({\rm
Re}{[a_{\ell}]}-|a_{\ell}|^{2}+\rm Re[b_{\ell}]-|b_{\ell}|^{2}}),$ (2)

$Q_{ext}=Q_{abs}+Q_{sca},$ (3)

where $a_{\ell}$ and $b_{\ell}$ are the Mie scattering coefficients
corresponding to the transverse magnetic and transverse electric modes,
respectively. The index $\ell$ is used to denote $\ell^{th}$ order spherical
harmonic channel and $x=kr$ is size parameter with $k=2\pi/\lambda$. The Mie
scattering coefficients can be derived by employing the boundary conditions on
the surface of the scatterer and written as bohren

$\displaystyle
a_{\ell}=\frac{m\psi_{\ell}(mx)\psi^{\prime}_{\ell}(x)-\mu\psi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}{m\psi_{\ell}(mx)\xi^{\prime}_{\ell}(x)-\mu\xi_{\ell}(x)\psi^{\prime}_{\ell}(mx)},$
(4) $\displaystyle
b_{\ell}=\frac{\mu\psi_{\ell}(mx)\psi^{\prime}_{\ell}(x)-m\psi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}{\mu\psi_{\ell}(mx)\xi^{\prime}_{\ell}(x)-m\xi_{\ell}(x)\psi^{\prime}_{\ell}(mx)},$
(5)

where, $\psi_{\ell}$, $\xi_{\ell}$ are Riccati-Bessel functions Functions ,
$m=n/n_{1}$ is the relative refractive index. In order to study the
directional scattering pattern, the expressions for differential scattering
efficiencies in forward ($\theta=0$) and backward ($\theta=\pi)$ directions
are given as bohren

$Q_{b}|_{\theta=\pi}=\frac{1}{x^{2}}\bigg{|}{\sum_{\ell=1}^{\infty}}{(2\ell+1)(-1)^{l}(a_{\ell}-b_{\ell})}\bigg{|}^{2},$
(6)

$Q_{f}|_{\theta=0}=\frac{1}{x^{2}}\bigg{|}{\sum_{\ell=1}^{\infty}}{(2\ell+1)(a_{\ell}+b_{\ell})}\bigg{|}^{2}.$
(7)

Figure 1: (a) The real and imaginary parts of the effective permittivity
(blue) and effective permeability (purple) of the composite metamaterial made
of $SiO_{2}$ as a host medium and spherical inclusions of permittivity
$\epsilon_{i}=169$ farhat , where inset shows the geometry of the composite
sphere. (b) The backward scattering efficiency by the composite sphere as a
function of incident wavelength and radius is plotted. (c) The backward
scattering efficiency by composite sphere of radius $500\,\rm nm$ and (d)
forward scattering efficiency by composite sphere of radius $70\,\rm nm$ are
displayed as function of incident wavelength. Throughout the figure, the
radius of the spherical inclusions and volume filling fraction are fixed at
$b=20\,\rm nm$ and $f=0.2$ respectively.

It can be seen in Eq. (6), the first Kerker’s condition (i. e. $Q_{b}=0$) can
be achieved by providing $a{}_{\ell}=b{}_{\ell}$ and this can occur for
identical identical permittivity and permeability of the sphere (see Eqs.
4-5). On the other hand, the second Kerker’s condition can be satisfied by
achieving $Q_{f}\simeq 0$, provided that $a_{1}\simeq-b_{1}$, this occurs when
the condition $\epsilon=\frac{4-\mu}{2\mu+1}$ is satisfied in the quasi-static
limit.

It is worthwhile to mention that the optical theorem imposes a severe
condition to achieve ZFS. This theorem of optics relates the extinction cross
section $\sigma^{ext}$ to the forward scattering amplitude $s_{\theta}(0,0)$
and can be expressed as $\sigma^{ext}=\frac{\lambda^{2}_{0}}{\pi}{\rm
Im}[s_{\theta}(0,0)]$ NZFS ; Lee_2018 . In practice, $Q_{f}=0$ (at $2^{nd}$
Kerker’s condition) appears by achieving $a_{1}=-b_{1}$ without necessarily
making both of them zero ker , which not only vanishes the forward scattering
but also nullify the total extinction cross section (one may refer to Eq. (3)
in quasi-static approximation). Subsequently, the non zero $a_{1}$ and $b_{1}$
are implying that the overall scattering in all other directions may still be
significantly different from zero, which seems to be a contradiction to the
fact that extinction cross section vanishes. However, the usual approximation
for the dipole coefficients fail to satisfy energy conservation requirements
NZFS . A more accurate approximation for those coefficients, taking into
account radiative corrections, shows that the forward scattering is not
exactly zero, relaxing the contradiction. Therefore, one cannot completely
suppress the forward scattering, it may be possible to achieve near zero
forward scattering (NZFS) at second Kerker’s condition. In order to
demonstrate these conditions, we use the effective medium theory that will
provide an environment to satisfy the Kerker’s conditions.

###  II.2  Extended Maxwell Garnett Theory

When electromagnetic field propagates through a heterogeneous medium comprises
of non-absorbing host medium of permittivity $\epsilon_{h}$ and small
spherical inclusions of permittivity $\epsilon_{i}$, radius $b$, such that
$b\ll\lambda$. There is a variety of effective medium theories for
homogenization of the heterogeneous medium Choy2015 ; Doyle1989 . The
collective optical response of such a homogenized medium can be discussed by
defining the effective optical constants provided by one of the effective
medium theory. Due to the best performance we use extended Maxwell-Garnett
theory farhat ; Muhlig_2013 ; ali_2020 ; Muhlig_2012 ; max2 ; ruppin2000 ,
which provides the effective permittivity and permeability as functions of
volume filling fraction ${\it f}$, size of the inclusions, permeability and
permittivity of the host medium as $n_{\rm eff}=\sqrt{\epsilon_{{}_{\rm
eff}}\mu_{{}_{\rm eff}}}$, where

$\epsilon_{{}_{\rm
eff}}=\epsilon_{h}\frac{y^{3}+3ifa{}_{1}}{y^{3}-\frac{3}{2}{i}fa{}_{1}},$ (8)

$\mu_{{}_{\rm
eff}}=\mu_{h}\frac{y^{3}+3ifb{}_{1}}{y^{3}-\frac{3}{2}ifb{}_{1}}.$ (9)

Here $y=\sqrt{\epsilon}_{h}\omega b/c,$ is the size parameter inside the host
medium, $\omega$ is the frequency, c is the speed of light, $a_{1}$ and
$b_{1}$ are dipolar Mie coefficients of the inclusions. The size parameter
must be $y\ll 1$ ruppin2000 , otherwise higher order multipoles will
contribute to effective permittivity and permeability and spoil the validity
of EMG theory.

##  III Results and Discussions

We begin our numerical analysis by considering a composite material made of
$SiO_{2}$ as host medium with relative permittivity $\epsilon_{h}=2.1$
sio21965  and dielectric spherical inclusions of radius $20\,\rm nm$, bulk
relative permittivity $\epsilon_{i}=169$ farhat ; Goia_2003  and volume
filling fraction $f$. The effective optical constants of the composite
material are calculated by EMG theory (Sec. II.2). Since the inclusions have a
large refractive index so that the incident field can generate displacement
currents inside the inclusions and produces tiny magnetic dipoles. The
collective effect of these spherical magnetic dipoles generates a remarkable
magnetic response in the composite metamaterials. The effective optical
response of the composite metamaterials is measured in terms of effective
permittivity and effective permeability by using the Eqs. (8-9) that have
external degrees of freedom, like volume filling fraction, permittivities and
size of inclusions. One can perform fine tuning of these parameters to tailor
the resonance at a chosen wavelength for operation.

Figure 2: (a) The schematic diagram of internal homogenization, (b) the real
and imaginary parts of effective permittivity of the effective sphere. (c) The
real and imaginary parts of effective permittivity and permeability of an
arbitrarily arranged effective spheres in an homogeneous medium with
$\epsilon_{h}=\mu_{h}=1$ for filling fraction $f=0.25$ as a function of
incident wavelength.

In Fig. 1(a), the real and imaginary parts of the effective permittivity
(blue) and permeability (purple) of the composite metamaterial are displayed
as a function of incident wavelength. It is clearly seen that the composite
metamaterial shows remarkable magnetic response with a spectral region that
indeed lies in our domain of interest. For instance, the composite
metamaterial shows an identical effective permeability and permeability at the
wavelength of $560\,{\rm nm}$ with relatively small imaginary parts. According
to Kerker’s method, this is an ideal scenario to satisfy the first Kerker’s
condition for all sized spherical particles. Fig. 1(b) shows a color map of
backward scattering efficiency by the composite sphere as a function of the
incident wavelength and radius of the sphere. It is clearly seen that at
$\lambda=\,560{\rm nm}$ the backward scattering is completely suppressed which
occurs due to the in-phase oscillations of the electric and magnetic
multipoles with same amplitudes such that $a_{\ell}=b_{\ell}$, provided by
identical $\epsilon_{\rm eff}$ and $\mu_{\rm eff}$ as shown in Fig. 1(a).
Therefore, the overall backward scattering efficiency defined in Eq. (6) is
reduced to zero regardless of the sizes of the sphere. The drastic reduction
of backward scattering by the composite sphere of radius $500\,\rm nm$ is
shown in Fig. 1(c) by calculating the backward scattering efficiency versus
wavelength. It is clearly shown that the dramatic reduction appears at
$\lambda=560\,{\rm nm}$ and backward scattering amplitude reduces to zero,
thus satisfying the first Kerker’s condition.

In contrast, the second Kerker’s condition requires the condition
$\epsilon_{\rm eff}=\frac{4-\mu_{\rm eff}}{\mu_{\rm eff}+2}$, which surprising
also occurs at a wavelength of $500\,{\rm nm}$. In Fig. 1(d), we calculate the
forward scattering efficiency by a composite sphere of radius $70\,\rm nm$ as
a function of the incident wavelength. The drastic reduction of forward
scattering appears at $\lambda=500\,{\rm nm}$ and satisfying the second
Kerker’s condition. It is worthwhile to mention that the second Kerker’s
condition can be achieved by providing $a_{\ell}=-b_{\ell}$, which only
implies to the dipolar particles (i.e. for $\ell=1$). Otherwise, this effect
might be spoil for a large sphere due to higher-order multipole contributions
to the forward scattering efficiency. Thus, provides nearly zero forward
scattering $Q_{f}\simeq 0$ at $\lambda=500\,{\rm nm}$ for a  dipolar sphere
(of radius $70\,{\rm nm})$.

Now we extend our proposal to design metamaterials by using naturally
occurring materials that will present a significant magnetic response to
satisfy the Kerker’s conditions. The immediate question that arises, is there
any material that possesses large permittivity in the visible region? The
answer is yes, for instance, a carefully designed core-shell system shows a
very high permittivity due to plasmonic resonances with strong spectral
dependence and in the quasi-static approximation these core-shell
nanoparticles can be used as inclusions into a host medium. For simplicity, we
begin with an internal homogenization approach introduced by Chettiar and
Enghetta anghetta  that provides effective permittivity of core-shell
nanoparticle by homogenizing it to an effective sphere of the same dimension.
Through this approach, the effective permittivity of a core-shell nanoparticle
is calculated by equating the polarizabilities of the core-shell to a
homogenized sphere of the same dimension and finally, the expression turns out
as

$\epsilon_{\rm
MG}=\epsilon_{s}\frac{a^{3}(\epsilon_{c}+2\epsilon_{s})+2b^{3}(\epsilon_{c}-\epsilon_{s})}{a^{3}(\epsilon_{c}+2\epsilon_{s})-b^{3}(\epsilon_{c}-\epsilon_{s})}.$
(10)

Figure 3: (a) Backward scattering efficiency by a sphere made of composite
metamaterial (of permittivity given in Fig. 2c) of radius $800\,{\rm nm}$ as a
function of incident wavelength. (b) Forward scattering efficiency by a
composite sphere of radius $120\,{\rm nm}$, where the volume filling fraction
is fixed at $f=0.25.$

In order to implement this proposal, we consider metallic nanosphere as a core
of radius $b$ embedded in dielectric spherical shell of radius $a,$ forming a
core-shell nanoparticle Haguraa_2010 ; Oldenburg_1992 ; Chaudhuri_2012 ;
Hanske . Assuming that the core is silver ($\rm Ag$) and we may use Drude
formula to define it’s permittivity,
$\epsilon_{c}=\epsilon_{\infty}-\frac{\omega^{2}_{p}}{\omega(\omega+i\Gamma)}$,
with $\omega_{p}=9.2eV,$ $\Gamma=0.0212\,eV,$ $\epsilon_{\infty}=5.0,$ where
$1eV=241.8\,THz$ Johnson_1972  and $\omega$ is the angular frequency of
incident light. Furthermore, the core-shell is homogenized to an effective
sphere of radius $a$ with permittivity $\epsilon_{\rm MG}$ provided by Eq.
(10) and displayed in Fig. 2(b) at fixed radii ratio at $\frac{b}{a}=0.84$. It
has been verified that in the quasi-static limit, the effective sphere has a
good agreement to exact Mie solution of a core-shell nanoparticles of the same
dimension crystal  and indeed it can be used as inclusion into a host medium
to fabricate the structures of desired optical and physical properties
anghetta ; crystal ; JJS ; Rapid .

Let  us design a bulk composite medium comprises of randomly arranged
effective (homogenized) nanospheres with permittivity $\epsilon_{\rm MG}$ and
radius $a$ embedded in a host medium with volume filling fraction $f$. In Fig.
2(c) we calculate the effective permittivity (blue) and permeability (purple)
of newly designed composite metamaterial by means of EMG theory at fixed
$f=0.25$ as a function of $\lambda$. In this configuration, we consider radii
of the effective nanosphere about 14 time less than the incident wavelength
which is a safe zone for the validation of the effective medium theory
ruppin2000 .

It can be seen in Fig. 2(c) that the composite metamaterial not only presents
large effective permeability but also  provides identical permittivity and
permeability with week imaginary parts at a wavelength of $650\,{\rm nm}$.
Therefore, according to Kerker’s criterion, it must present zero backward
scattering at this wavelength. We now calculate the directional scattering by
the composite sphere of radius $r$ with permittivity given in Fig. 2(c). The
backward scattering efficiency and forward scattering efficiency as a function
of incident wavelength are displayed in Fig. 3(a) and 3(b), respectively. It
is clearly seen that the backward scattering efficiency is drastically reduced
to zero at $\lambda=650\,{\rm nm}$ which is due to the fact that equal
electric and magnetic response. On the other hand,  the NZFS appears at a
wavelength of $620\,{\rm nm}$ due to out of phase oscillations of electric and
magnetic dipoles, thus satisfying the second Kerker’s condition.

##  IV Conclusions

In this paper, we have theoretically modeled composite metamaterials using Mie
scattering theory and effective medium theory. Our numerical results have
shown that the composite metamaterials present strong artificial magnetic
permeability in the optical frequency domain that appears due to the
displacement currents inside the inclusions. Our findings show that a fine
tuning of the parameters appear in the EMG theory, one can creates a spectral
region with identical permittivity and permeability not only for bulk
homogeneous composite spherical sphere but also for a regularly arranged
nanospherical particles.  Altogether, this proposal not only paves a path to
design new possible structures with artificial magnetic response but also
allows to achieve Kerker’s conditions for low refractive index materials in
the visible frequency range. These structures with unique optical properties
have very low ${\rm Im(m_{eff})}$ (i. e. low absorption) than metallic
structure that make s our proposal a good candidate to design high-performance
optical antennas, metamaterials and other novel nanophotonic devices.

##  V  Acknowledgment

We thank F. Pinheiro, F. Rosa, PAM Neto and S. Iqbal for inspiring
discussions. This work has been supported by the Brazilian agencies National
Council for Scientific and Technological Development (CNPq), Coordination for
the Improvement of Higher Education Personnel (CAPES).

##  VI References

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