# Geometrical and topological properties of transmission resonance and
artificial mirage

Youjun Deng  School of Mathematics and Statistics, HNP-LAMA, Central South
University, Changsha, Hunan, China<EMAIL_ADDRESS>,  Hongyu Liu
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong,
China<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Xianchao Wang
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong,
China<EMAIL_ADDRESS>and  Wei Wu  Department of Mathematics, Hong Kong
Baptist University, Hong Kong, China<EMAIL_ADDRESS>
###### Abstract.

Transmission eigenfunctions are certain interior resonant modes that are of
central importance to the wave scattering theory. In this paper, we present
the discovery of novel global rigidity properties of the transmission
eigenfunctions associated with the Maxwell system. It is shown that the
transmission eigenfunctions carry the geometrical and topological information
of the underlying domain. We present both analytical and numerical results of
these intriguing rigidity properties. As an interesting application, we
propose an illusion scheme of artificially generating a mirage image of any
given optical object.

Keywords: electromagnetic scattering; Maxwell system; transmission
eigenfunctions; surface localization; topological structure; artificial mirage

2010 Mathematics Subject Classification: 35P25, 58J50, 35R30, 78A40

##  1\. Introduction

We are concerned with the spectral geometry of the interior transmission
eigenvalue problem arising in the time-harmonic electromagnetic (EM)
scattering described by the Maxwell system. Let $\Omega$ be a bounded
Lipschitz domain in $\mathbb{R}^{3}$ such that
$\mathbb{R}^{3}\backslash\overline{\Omega}$ is connected. Let $\varepsilon$
and $\mu$ be bounded positive functions such that
$\mathrm{supp}(1-\varepsilon)\cup\mathrm{supp}(1-\mu)\subset\Omega$. In the
physical context, $\varepsilon$ and $\mu$ are the optical parameters of the
space medium, and respectively signify the electric permittivity and magnetic
permeability. Throughout, we assume that $\mu\equiv 1$. Consider a pair of
incident EM waves $(\mathbf{E}^{i},\mathbf{H}^{i})$ that are entire solutions
to the following Maxwell system:

$\nabla\wedge\mathbf{E}^{i}-\mathrm{i}k\mathbf{H}^{i}=\mathbf{0},\quad\nabla\wedge\mathbf{H}^{i}+\mathrm{i}k\mathbf{E}^{i}=\mathbf{0}\quad\mbox{in}\
\ \mathbb{R}^{3},$ (1.1)

where $k\in\mathbb{R}_{+}$ and $\mathrm{i}:=\sqrt{-1}$. Here, $k$ signifies
the wavenumber of the EM wave propagation. The EM inhomogeneity
$(\Omega,\varepsilon)$ interrupts the incident fields and leads to the
scattered fields $\mathbf{E}^{s}$ and $\mathbf{H}^{s}$. Let
$\mathbf{E}(\mathbf{x})=\mathbf{E}^{i}(\mathbf{x})+\mathbf{E}^{s}(\mathbf{x})$
and
$\mathbf{H}(\mathbf{x})=\mathbf{H}^{i}(\mathbf{x})+\mathbf{H}^{s}(\mathbf{x})$
denote the total electric and magnetic fields, respectively. The EM scattering
is governed by the following Maxwell system:

$\left\\{\begin{aligned}
&~{}~{}\nabla\wedge\mathbf{E}-\mathrm{i}k\mathbf{H}=\mathbf{0},&&\mbox{in}\
\mathbb{R}^{3},\\\
&~{}~{}\nabla\wedge\mathbf{H}+\mathrm{i}k\varepsilon\mathbf{E}=\mathbf{0}&&\mbox{in}\
\mathbb{R}^{3},\\\
&\lim_{|{\mathbf{x}}|\rightarrow\infty}\left(\mathbf{H}^{s}\wedge{\mathbf{x}}-|{\mathbf{x}}|\mathbf{E}^{s}\right)={\mathbf{0}}.\end{aligned}\right.$
(1.2)

The last limit in (1.2) is known as the Silver-Müller radiation condition,
which characterizes the outgoing nature of the scattered fields and holds
uniformly in the angular variable
$\hat{\mathbf{x}}:=\mathbf{x}/|\mathbf{x}|\in\mathbb{S}^{2}:=\left\\{\mathbf{x}\in\mathbb{R}^{3};|\mathbf{x}|=1\right\\}$.
The well-posedness of the scattering problem (1.2) can be conveniently found
in [25, 28]. There exists a unique pair of solutions
$({\mathbf{E}},{\mathbf{H}})\in$
$H_{\mathrm{loc}}(\mathrm{curl},\mathbb{R}^{3})\times
H_{\mathrm{loc}}(\mathrm{curl},\mathbb{R}^{3})$ which admits the following
asymptotic expansions as $|\mathbf{x}|\rightarrow\infty$:

$\mathbf{Q}(\mathbf{x})=\mathbf{Q}^{i}(\mathbf{x})+\frac{{e}^{\mathrm{i}k|\mathbf{x}|}}{|\mathbf{x}|}\mathbf{Q}_{\infty}(\hat{\mathbf{x}})+\mathcal{O}\left(\frac{1}{|\mathbf{x}|^{2}}\right),\quad\mathbf{Q}:=\mathbf{E},\mathbf{H}.$
(1.3)

The functions $\mathbf{E}_{\infty}(\hat{\mathbf{x}})$ and
$\mathbf{H}_{\infty}(\hat{\mathbf{x}})$ in (1.3) are respectively referred to
as the electric and magnetic far field patterns, and satisfy the following
one-to-one correspondence,

$\mathbf{E}_{\infty}(\hat{\mathbf{x}})=-\hat{\mathbf{x}}\wedge\mathbf{H}_{\infty}(\hat{\mathbf{x}})\quad\mbox{and}\quad\mathbf{H}_{\infty}(\mathbf{x})=\hat{\mathbf{x}}\wedge\mathbf{E}_{\infty}(\hat{\mathbf{x}}),\quad\forall\hat{\mathbf{x}}\in\mathbb{S}^{2}.$

We next consider a particular case that non-scattering, a.k.a invisibility,
occurs, namely $\mathbf{Q}_{\infty}\equiv\mathbf{0}$. In such a case, by
Rellich’s Theorem [15], one has $\mathbf{Q}^{s}=\mathbf{0}$ in
$\mathbf{R}^{3}\backslash\overline{\Omega}$. With such an observation, one can
directly verify that $\mathbf{Q}_{1}=\mathbf{Q}|_{\Omega}$ and
$\mathbf{Q}_{2}=\mathbf{Q}^{i}|_{\Omega}$ fulfil the following PDE system:

$\left\\{\begin{aligned}
&\nabla\wedge\mathbf{E}_{1}-\mathrm{i}k\mathbf{H}_{1}=\mathbf{0},&&\nabla\wedge\mathbf{H}_{1}+\mathrm{i}k\varepsilon\mathbf{E}_{1}=\mathbf{0}\quad&&\mbox{in}\
\ \Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&\nabla\wedge\mathbf{E}_{2}-\mathrm{i}k\mathbf{H}_{2}=\mathbf{0},&&\nabla\wedge\mathbf{H}_{2}+\mathrm{i}k\mathbf{E}_{2}=\mathbf{0}\quad&&\mbox{in}\
\ \Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&\nu\wedge\mathbf{E}_{1}=\nu\wedge\mathbf{E}_{2},&&\nu\wedge\mathbf{H}_{1}=\nu\wedge\mathbf{H}_{2}\quad&&\mbox{on}\
\ \partial\Omega,\end{aligned}\right.$ (1.4)

where and also in what follows, $\nu$ signifies the exterior unit normal
vector to $\partial\Omega$. The system (1.4) is referred to as the Maxwell
transmission eigenvalue problem. It is clear that $\mathbf{E}_{j}=\mathbf{0}$
and $\mathbf{H}_{j}=\mathbf{0}$, $j=1,2$, are trivial solutions. If there
exist nontrivial solutions, $(\mathbf{E}_{j},\mathbf{H}_{j})_{j=1,2}$ are
called the transmission eigenfunctions and $k$ is the associated transmission
eigenvalue. In this paper, we shall be mainly concerned with real transmission
eigenvalues. Hence, if no scattering/invisibility occurs, then the
restrictions of the incident and total EM fields are transmission
eigenfunctions and the wavenumber is an eigenvalue. After eliminating
$\mathbf{H}_{j}$, $j=1,2$, in (1.4), we have the following reduced formulation
of the transmission eigenvalue problem for $\mathbf{E}_{j}\in
H(\mathrm{curl},\Omega)$, $j=1,2$:

$\left\\{\begin{aligned}
&\nabla\wedge(\nabla\wedge\mathbf{E}_{1})-k^{2}\varepsilon\mathbf{E}_{1}=\mathbf{0},&&\nabla\cdot(\varepsilon\mathbf{E}_{1})=\mathbf{0}&&\mbox{in}\
\ \Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&\nabla\wedge(\nabla\wedge\mathbf{E}_{2})-k^{2}\mathbf{E}_{2}=\mathbf{0},&&\nabla\cdot\mathbf{E}_{2}=\mathbf{0}&&\mbox{in}\
\ \Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&\nu\wedge\mathbf{E}_{1}=\nu\wedge\mathbf{E}_{2},&&\nu\wedge(\nabla\wedge\mathbf{E}_{1})=\nu\wedge(\nabla\wedge\mathbf{E}_{2})&&\mbox{on}\
\ \partial\Omega.\end{aligned}\right.$ (1.5)

For a convenient reference, if one considers the transverse EM scattering (cf.
[17]), one can deduce the following transmission eigenvalue problem associated
with the Helmholtz equation in $\mathbb{R}^{2}$:

$\Delta w+k^{2}\mathbf{n}^{2}w=0,\ \ \Delta v+k^{2}v=0\ \ \text{in}\ \
\Omega;\ \ \ w=v,\ \ \partial_{\nu}w=\partial_{\nu}v\ \text{on}\ \
\partial\Omega,$ (1.6)

where $\mathbf{n}:=\sqrt{\epsilon}$ is known as the refractive index of the
inhomogeneous medium, and $w,v$ correspond to the transverse parts of
$\mathbf{E}$ and $\mathbf{H}$.

The spectral study of the transmission eigenvalue problems has a long and
colourful history in the literature and has become a central topic in the
inverse scattering theory. The spectral properties of transmission eigenvalues
have been intensively and extensively investigated in the literature,
including the existence, infiniteness and Weyl’s law, and we refer to [9, 10,
14] and the references cited in for the state-of-the-art development on this
aspect. Recently, the spectral geometry of transmission eigenfunctions has
received considerable attentions in the literature in different physical
contexts [2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 16, 18, 20], and we refer [24] for
survey and review on this aspect. It is pointed out that most of the
aforementioned works are concerned with local geometric properties of the
transmission eigenfunctions except [13, 19], where global rigidity properties
of the transmission eigenfunctions were discovered. Our study in the current
article is strongly motivated by the related study in [13], where it is
established that the transmission eigenfunctions associated with the Helmholtz
system (1.6) carry the geometric information of the underlying domain
$\Omega$. In fact, it is shown that there exists an infinite sequence of
transmission eigenfunctions whose $L^{2}$-energies concentrate on
$\partial\Omega$. This geometrical property has been further used to develop a
super-resolution wave imaging scheme associated with the acoustic scattering.
In this paper, we shall show that the Maxwell transmission eigenfunctions in
(1.5) carry both the geometrical and topological properties of the underlying
domain $\Omega$. First, we construct sequences of eigenfunctions
$(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)})_{n\in\mathbb{N}_{+}}$ associated
with eigenvalues $k_{n}\rightarrow\infty$ whose $L^{2}$-energies localize
around $\partial\Omega$. In fact, we construct both mono-localized and bi-
localized eigenfunctions. Here, by a mono-localized eigenfunction
$(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)})$, we mean that only one of the
pair of eigenfunctions is surface-localized while the other one is not, and by
bi-localization, we mean both $\mathbf{E}_{1}^{(n)}$ and
$\mathbf{E}_{2}^{(n)}$ are surface-localized. In the radial case with constant
refractive index, we give the analytical construction, whereas for the general
case, we provide numerical verifications. Second, we show that the
transmission eigenfunctions carry the topological structure of the underlying
domain $\Omega$ when it has multiple connected components. Using the obtained
geometrical and topological results, we propose a novel and interesting
application of generating the mirage imaging of a given optical object at a
customized location. The main idea is to design suitable illuminating EM
fields that are related to the underlying object as well as the associated EM
transmission eigenfunctions.

The rest of the paper is organized as follows. Sections 2 and 3 are
respectively devoted to the geometrical and topological properties of the EM
transmission eigenfunctions. In Section 4, we consider the application to
artificial mirage.

##  2\. Surface-localized transmission eigenmodes

To begin with, we introduce a qualitative definition of surface-localization
for a function $\mathbf{Q}\in L^{2}(\Omega)^{3}$. In what follows, for a
sufficiently small $\delta\in\mathbb{R}_{+}$, we set

$\mathcal{N}_{\delta}(\partial\Omega):=\\{\mathbf{x}\in\Omega;\
\mathrm{dist}(\mathbf{x},\partial\Omega)<\delta\\}.$

######  Definition 2.1.

Given a function $\mathbf{Q}\in L^{2}(\Omega)^{3}$. It is said to be surface-
localized if there exists a sufficient small constant
$\delta\in\mathbb{R}_{+}$, such that

$\frac{\|\mathbf{Q}\|_{L^{2}(\Omega\backslash\mathcal{N}_{\delta}(\partial\Omega))^{3}}}{\|\mathbf{Q}\|_{L^{2}(\Omega)^{3}}}\ll
1.$ (2.1)

By (2.1), if a function is surface-localized, then its $L^{2}$-energy
concentrates on $\partial\Omega$. In what follows, we shall make the
qualitative relation in (2.1) more quantitative.

Our main finding in this section can be summarized in the following theorem.

######  Theorem 2.1.

Let $\Omega$ be a simply connected Lipschitz domain and $\varepsilon$ be a
positive constant with $\varepsilon\neq 1$. Consider the Maxwell transmission
eigenvalue problem (1.4). Then there exists a sequence of mono-localized
eigenfunctions
$(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)})_{n\in\mathbb{N}_{+}}$, and there
also exists a sequence of bi-localized eigenfunctions
$(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)})_{n\in\mathbb{N}_{+}}$, associated
with the transmission eigenvalues $k_{n}\rightarrow\infty$.

In what follows, we shall rigorously prove Theorem 2.1 in the case that
$\Omega$ is radially symmetric, whereas in the general case, we only provide
numerical verifications. In the numerics, we can actually consider variable
$\varepsilon$ and the conclusion in Theorem 2.1 still holds true.

###  2.1. Transmission eigenvalues for a ball

In this subsection, we briefly discuss the transmission eigenvalues for the
Maxwell system (1.5) in the radially symmetric case associated with a constant
refractive index. This has been derived in [29]. Nevertheless, some of the
technical ingredients shall be needed in our subsequent analysis of the
transmission eigenfunctions and for self-containedness and easy reference, we
present them in what follows. Without loss of generality, we assume that
$\Omega$ centres at the origin, namely,
$\Omega=\\{\mathbf{x}\in\mathbb{R}^{3}:\,|\mathbf{x}|<R\in\mathbb{R}_{+}\\}$.
We also let $\epsilon_{0}=\sqrt{\varepsilon}\in\mathbb{R}_{+}$. Then (1.5) can
be rewritten as

$\begin{cases}\begin{aligned}
&\nabla\wedge(\nabla\wedge\mathbf{E}_{1})-k^{2}\epsilon_{0}^{2}\mathbf{E}_{1}=0,&&\nabla\cdot(\epsilon_{0}^{2}\mathbf{E}_{1})=0&&\mbox{in}\
\ \Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&\nabla\wedge(\nabla\wedge\mathbf{E}_{2})-k^{2}\mathbf{E}_{2}=0,&&\nabla\cdot\mathbf{E}_{2}=0&&\mbox{in}\
\ \Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
&\nu\wedge\mathbf{E}_{1}=\nu\wedge\mathbf{E}_{2},&&\nu\wedge(\nabla\wedge\mathbf{E}_{1})=\nu\wedge(\nabla\wedge\mathbf{E}_{2})&&\mbox{on}\
\ \partial\Omega.\end{aligned}\end{cases}$ (2.2)

Since $\epsilon_{0}$ is a positive constant, we can expand the solutions
$\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ of the system (2.2) into the Fourier
series[15],

$\displaystyle{\bf
E}_{1}(\mathbf{x})=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}a_{n}^{m}\mathcal{M}_{n}^{m}(\mathbf{x})+\sum_{n=1}^{\infty}\sum_{m=-n}^{n}b_{n}^{m}\mathcal{N}_{n}^{m}(\mathbf{x}),\vskip
6.0pt plus 2.0pt minus 2.0pt$ $\displaystyle{\bf
E}_{2}(\mathbf{x})=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}\widetilde{a}_{n}^{m}\widetilde{\mathcal{M}}_{n}^{m}(\mathbf{x})+\sum_{n=1}^{\infty}\sum_{m=-n}^{n}\widetilde{b}_{n}^{m}\widetilde{\mathcal{N}}_{n}^{m}(\mathbf{x}),\vskip
6.0pt plus 2.0pt minus 2.0pt$

where

$\displaystyle\mathcal{M}_{n}^{m}(\mathbf{x})=\nabla\wedge\\{\mathbf{x}j_{n}(k\epsilon_{0}|\mathbf{x}|)Y_{n}^{m}(\hat{\mathbf{x}})\\},\quad\mathcal{N}_{n}^{m}(\mathbf{x})=\frac{1}{\mathrm{i}k}\nabla\wedge\mathcal{M}_{n}^{m}(\mathbf{x}),$
(2.3)
$\displaystyle\widetilde{\mathcal{M}}_{n}^{m}(\mathbf{x})=\nabla\wedge\\{\mathbf{x}j_{n}(k|\mathbf{x}|)Y_{n}^{m}(\hat{\mathbf{x}})\\},\quad\
\
\,\widetilde{\mathcal{N}}_{n}^{m}(\mathbf{x})=\frac{1}{\mathrm{i}k}\nabla\wedge\widetilde{\mathcal{M}}_{n}^{m}(\mathbf{x}).$

Here, $j_{n}(\mathbf{x})$ denotes the spherical Bessel functions of the first
kind and $Y_{n}^{m}$ denotes a spherical harmonic function of degree $n$ and
order $m$. Moreover, one can verify that $\mathcal{M}_{n}^{m}$ and
$\mathcal{N}_{n}^{m}$ are the solutions to

$\nabla\wedge(\nabla\wedge\mathbf{E}_{1})-k^{2}\epsilon_{0}^{2}\mathbf{E}_{1}=0,\quad\nabla\cdot(\epsilon_{0}^{2}\mathbf{E}_{1})=0,$

and $\widetilde{\mathcal{M}}_{n}^{m}$ and $\widetilde{\mathcal{N}}_{n}^{m}$
are the solutions to

$\nabla\wedge(\nabla\wedge\mathbf{E}_{2})-k^{2}\mathbf{E}_{2}=0,\ \ \ \
\nabla\cdot\mathbf{E}_{2}=0.$

In particular, $\mathcal{M}_{n}^{m}$ and $\widetilde{\mathcal{M}}_{n}^{m}$
denote the TE (transverse electric) waves; $\mathcal{N}_{n}^{m}$ and
$\widetilde{\mathcal{N}}_{n}^{m}$ denote the TM (transverse magnetic) waves.

For the TE modes, using the spherical coordinates $(\rho,\theta,\phi)$,
$\mathcal{M}_{n}^{m}$ can be rewritten as

$\displaystyle\mathcal{M}_{n}^{m}$ $\displaystyle=\nabla\wedge\\{\rho
j_{n}(k\epsilon_{0}\rho)Y_{n}^{m}(\theta,\phi)\,\hat{\bm{\rho}}\\}$ (2.4)
$\displaystyle=\frac{1}{\rho^{2}\sin\theta}\left|\begin{array}[]{ccc}\hat{\bm{\rho}}&\rho\hat{\bm{\theta}}&\rho\sin\theta\hat{\bm{\phi}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\frac{\partial}{\partial\rho}&\frac{\partial}{\partial\theta}&\frac{\partial}{\partial\phi}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \rho
j_{n}(k\epsilon_{0}\rho)Y_{n}^{m}(\theta,\phi)&0&0\end{array}\right|$
$\displaystyle=\frac{1}{\sin\theta}j_{n}(k\epsilon_{0}\rho)\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\phi}\hat{\bm{\theta}}-j_{n}(k\epsilon_{0}\rho)\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\theta}\hat{\bm{\phi}},$

where $\rho=|\mathbf{x}|$ and

$\displaystyle\hat{\bm{\rho}}=\sin\theta\cos\phi\,\bm{e}_{1}+\sin\theta\sin\phi\,\bm{e}_{2}+\cos\theta\,\bm{e}_{3},$
$\displaystyle\hat{\bm{\theta}}=\cos\theta\cos\phi\,\bm{e}_{1}+\cos\theta\sin\phi\,\bm{e}_{2}-\sin\theta\,\bm{e}_{3},$
$\displaystyle\hat{\bm{\phi}}=-\sin\phi\,\bm{e}_{1}+\cos\phi\,\bm{e}_{2}.$

Due to

$\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}Y_{n}^{m}(\theta,\phi)}{\partial\phi^{2}}+n(n+1)Y_{n}^{m}(\theta,\phi)=0,$

we can derive that

$\displaystyle\nabla\wedge\mathcal{M}_{n}^{m}=\frac{1}{\rho^{2}\sin\theta}\left|\begin{array}[]{ccc}\hat{\bm{\rho}}&\rho\hat{\bm{\theta}}&\rho\sin\theta\hat{\bm{\phi}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\frac{\partial}{\partial\rho}&\frac{\partial}{\partial\theta}&\frac{\partial}{\partial\phi}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
0&\rho\frac{1}{\sin\theta}j_{n}(k\epsilon_{0}\rho)\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\phi}&-\rho\sin\theta
j_{n}(k\epsilon_{0}\rho)\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\theta}\end{array}\right|$ (2.5)
$\displaystyle=-\frac{j_{n}(k\epsilon_{0}\rho)}{\rho}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}Y_{n}^{m}(\theta,\phi)}{\partial\phi^{2}}\right)\hat{\bm{\rho}}$
$\displaystyle\quad\quad+\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\theta}\hat{\bm{\theta}}+\frac{1}{\rho\sin\theta}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\phi}\hat{\bm{\phi}}$
$\displaystyle=\frac{1}{\rho}j_{n}(k\epsilon_{0}\rho)n(n+1)Y_{n}^{m}(\theta,\phi)\hat{\bm{\rho}}+\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))\frac{1}{\rho}\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\theta}\hat{\bm{\theta}}$
$\displaystyle\quad\quad+\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))\frac{1}{\rho\sin\theta}\frac{\partial
Y_{n}^{m}(\theta,\phi)}{\partial\phi}\hat{\bm{\phi}}.$

Since the surface gradient in the spherical coordinates is given by

$\nabla_{s}f=\frac{\partial
f}{\partial\theta}\hat{\bm{\theta}}+\frac{1}{\sin\theta}\frac{\partial
f}{\partial\phi}\hat{\bm{\phi}},$

equations (2.4) and (2.5) can be rewritten as

$\displaystyle\mathcal{M}_{n}^{m}$
$\displaystyle=j_{n}(k\epsilon_{0}\rho)\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}},$
(2.6) $\displaystyle\nabla\wedge\mathcal{M}_{n}^{m}$
$\displaystyle=\frac{1}{\rho}j_{n}(k\epsilon_{0}\rho)n(n+1)Y_{n}^{m}(\theta,\phi)\hat{\bm{\rho}}+\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))\nabla_{s}Y_{n}^{m}(\theta,\phi).$

Similarly, one can deduce that

$\displaystyle\widetilde{\mathcal{M}}_{n}^{m}$
$\displaystyle=j_{n}(k\rho)\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}},$
(2.7) $\displaystyle\nabla\wedge\widetilde{\mathcal{M}}_{n}^{m}$
$\displaystyle=\frac{1}{\rho}j_{n}(k\rho)n(n+1)Y_{n}^{m}(\theta,\phi)\hat{\bm{\rho}}+\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\rho))\nabla_{s}Y_{n}^{m}(\theta,\phi).$

According to the boundary condition to (2.2), it yields

$\displaystyle
a_{n}^{m}\hat{\bm{\rho}}\wedge\mathcal{M}_{n}^{m}=\widetilde{a}_{n}^{m}\hat{\bm{\rho}}\wedge\widetilde{\mathcal{M}}_{n}^{m},$
$\displaystyle
a_{n}^{m}\hat{\bm{\rho}}\wedge(\nabla\wedge\mathcal{M}_{n}^{m})=\widetilde{a}_{n}^{m}\hat{\bm{\rho}}\wedge(\nabla\wedge\widetilde{\mathcal{M}}_{n}^{m}).$

Applying (2.6) and (2.7) with a straightforward calculation, one can obtain

$\displaystyle\Big{(}a_{n}^{m}j_{n}(k\epsilon_{0}\rho)-\widetilde{a}_{n}^{m}j_{n}(k\rho)\Big{)}\,\nabla_{s}Y_{n}^{m}(\theta,\phi)=0,$
$\displaystyle\left(a_{n}^{m}\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))-\widetilde{a}_{n}^{m}\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\rho))\right)\hat{\rho}\wedge\nabla_{s}Y_{n}^{m}(\theta,\phi)=0,\quad\rho=R.$

Noting that $a_{n}^{m}\neq 0$ and $\widetilde{a}_{n}^{m}\neq 0$, the
eigenvalues $k$’s of the TE modes are positive zeros of the following function

$f_{n}^{\mathrm{TE}}(k)=\left(j_{n}(k\epsilon_{0}\rho)\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\rho))-j_{n}(k\rho)\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))\right)_{\rho=R}.$ (2.8)

Using the recurrence relation of the derivatives of the spherical Bessel
functions,

$\displaystyle\frac{\partial
j_{n}(k\rho)}{\partial\rho}=k\left(\frac{n}{k\rho}j_{n}(k\rho)-j_{n+1}(k\rho)\right),$
(2.9) $\displaystyle\frac{\partial
j_{n}(k\epsilon_{0}\rho)}{\partial\rho}=k\epsilon_{0}\left(\frac{n}{k\epsilon_{0}\rho}j_{n}(k\epsilon_{0}\rho)-j_{n+1}(k\epsilon_{0}\rho)\right),$

equation (2.8) can be rewritten as

$f_{n}^{\mathrm{TE}}(k)=k\Big{(}\epsilon_{0}j_{n}(kR)j_{n+1}(k\epsilon_{0}R)-j_{n+1}(kR)j_{n}(k\epsilon_{0}R)\Big{)},\quad
n\geq 1.$ (2.10)

On the other hand, for the TM modes, one can derive that

$\displaystyle\mathcal{N}_{n}^{m}(\mathbf{x})$
$\displaystyle=\frac{1}{\mathrm{i}k}\nabla\wedge\mathcal{M}_{n}^{m}(\mathbf{x})$
$\displaystyle=\frac{1}{\mathrm{i}k\rho}j_{n}(k\epsilon_{0}\rho)n(n+1)Y_{n}^{m}(\theta,\phi)\hat{\bm{\rho}}+\frac{1}{\mathrm{i}k\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))\nabla_{s}Y_{n}^{m}(\theta,\phi).$

Noting that
$\nabla\wedge\left(\nabla\wedge\mathcal{M}_{n}^{m}\right)-k^{2}\epsilon_{0}^{2}\mathcal{M}_{n}^{m}=0$,
together with (2.3) and (2.6), one can derive that

$\nabla\wedge\mathcal{N}_{n}^{m}=\nabla\wedge\left(\frac{1}{\mathrm{i}k}\nabla\wedge\mathcal{M}_{n}^{m}\right)=-\mathrm{i}k\epsilon_{0}^{2}\mathcal{M}_{n}^{m}=-\mathrm{i}k\epsilon_{0}^{2}j_{n}(k\epsilon_{0}\rho)\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}}.$

Similar results also hold for $\widetilde{\mathcal{N}}_{n}^{m}$. According to
last equation of (2.2), it yields

$\displaystyle
b_{n}^{m}\hat{\bm{\rho}}\wedge\mathcal{N}_{n}^{m}=\widetilde{b}_{n}^{m}\hat{\bm{\rho}}\wedge\widetilde{\mathcal{N}}_{n}^{m},$
$\displaystyle
b_{n}^{m}\hat{\bm{\rho}}\wedge(\nabla\wedge\mathcal{N}_{n}^{m})=\widetilde{b}_{n}^{m}\hat{\bm{\rho}}\wedge(\nabla\wedge\widetilde{\mathcal{N}}_{n}^{m}).$

By following a similar argument as the TE model case, one has

$\displaystyle\frac{1}{\mathrm{i}k}\left(b_{n}^{m}\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))-\widetilde{b}_{n}^{m}\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\rho))\right)\hat{\bm{\rho}}\wedge\nabla_{s}Y_{n}^{m}(\theta,\phi)=0,$
$\displaystyle-\mathrm{i}k\Big{(}b_{n}^{m}\epsilon_{0}^{2}j_{n}(k\epsilon_{0}\rho)-\widetilde{b}_{n}^{m}\,j_{n}(k\rho)\Big{)}\nabla_{s}Y_{n}^{m}(\theta,\phi)=0,\quad\rho=R.$

Noting that $b_{n}^{m}\neq 0$ and $\widetilde{b}_{n}^{m}\neq 0$, the
eigenvalues $k$’s for TM modes are positive zeros of the following function

$f_{n}^{\mathrm{TM}}(k)=\left(j_{n}(k\rho)\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\epsilon_{0}\rho))-\epsilon_{0}^{2}j_{n}(k\epsilon_{0}\rho)\frac{1}{\rho}\frac{\partial}{\partial\rho}(\rho
j_{n}(k\rho))\right)_{\rho=R}.$

Using the recurrence relation (2.9), the last equation can be rewritten as

$\displaystyle f_{n}^{\mathrm{TM}}(k)=$
$\displaystyle\frac{(1-\epsilon_{0}^{2})(1+n)}{\rho}j_{n}(kR)j_{n}(k\epsilon_{0}R)$
(2.11)
$\displaystyle+k\epsilon_{0}\Big{(}\epsilon_{0}j_{n}(k\epsilon_{0}R)j_{n+1}(kR)-j_{n}(kR)j_{n+1}(k\epsilon_{0}R)\Big{)},\quad
n\geq 1.$

Based on the above discussion, the transmission eigenvalues are those $k$’s
(nontrivial solutions) for $f_{n}^{\mathrm{TE}}(k)=0$ or
$f_{n}^{\mathrm{TM}}(k)=0$ for $n\in\mathbb{N}_{+}$. In particular, from
(2.10) and (2.11), one can find that transmission eigenvalues depend on the
parameter $n$, namely the order of the spherical Bessel functions $j_{n}$.

###  2.2. Mono-localized transmission eigenmodes

In this section, we construct a sequence of transmission eigenvalues
$\\{k_{n}\\}_{n\in\mathbb{N}_{+}}$ and prove that one of the corresponding
pair of eigenfunctions
$\left(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)}\right)$ is surface-localized
while the other is not. In what follows, for simplification, we only consider
the case with $\epsilon_{0}>1$ and the case of $0<\epsilon_{0}<1$ can be
deduced by a similar argument.

######  Lemma 2.1.

Let $\Omega=\\{\mathbf{x}\in\mathbb{R}^{3}:|\mathbf{x}|<R\in\mathbb{R}_{+}\\}$
and $\epsilon_{0}>1$ be a constant. Let $E_{\Omega}$ denote the set of
transmission eigenvalues of (2.2). Then there exists a sequence
$\\{k_{n}\\}_{n\in\mathbb{Z}_{+}}\subset E_{\Omega}$, such that for
sufficiently large $n$ , there holds

$k_{n}\in\left(\frac{r_{n,s_{1}(n)}}{R},\frac{r_{n,s_{2}(n)}}{R}\right),$
(2.12)

where $r_{n,s}$ denote the $s$-th positive root of the spherical Bessel
function $j_{n}(|\mathbf{x}|)$ for a fixed order $n$,

$s_{1}(n):=\left[\left(n+\frac{1}{2}\right)^{\gamma_{1}}\right],\quad
s_{2}(n)=\left[\left(n+\frac{1}{2}\right)^{\gamma_{2}}\right],\quad
0<\gamma_{1}<\gamma_{2}<1,$ (2.13)

and $[\,t\,]$ signifies the integer part of a real number $t$. Moreover, one
has

$k_{n}>\frac{n+\frac{1}{2}}{R},\quad n=1,2,3,\cdots.$ (2.14)

###### Proof.

For the TE models, from equation (2.10), we have

$f_{n}^{\mathrm{TE}}\left(\frac{r_{n,s_{1}}}{R}\right)f_{n}^{\mathrm{TE}}\left(\frac{r_{n,s_{2}}}{R}\right)=\frac{r_{n,s_{1}}r_{n,s_{2}}}{R^{2}}j_{n+1}(r_{n,s_{1}})j_{n}(r_{n,s_{1}}\epsilon_{0})j_{n+1}(r_{n,s_{2}})j_{n}(r_{n,s_{2}}\epsilon_{0}).$

Similarly, for the TM models, from equation (2.11), one has

$f_{n}^{\mathrm{TM}}\left(\frac{r_{n,s_{1}}}{R}\right)f_{n}^{\mathrm{TM}}\left(\frac{r_{n,s_{2}}}{R}\right)=\frac{r_{n,s_{1}}r_{n,s_{2}}\epsilon_{0}^{4}}{R^{2}}j_{n+1}(r_{n,s_{1}})j_{n}(r_{n,s_{1}}\epsilon_{0})j_{n+1}(r_{n,s_{2}})j_{n}(r_{n,s_{2}}\epsilon_{0}).$

Next, we will show that there exists $s_{1}$ and $s_{2}$ such that

$j_{n+1}(r_{n,s_{1}})j_{n}(r_{n,s_{1}}\epsilon_{0})j_{n+1}(r_{n,s_{2}})j_{n}(r_{n,s_{2}}\epsilon_{0})<0.$

Based on the relationship between the Bessel and spherical Bessel function

$j_{n}(z)=\sqrt{\frac{\pi}{2z}}J_{n+1/2}(z),\quad z>0,$ (2.15)

one can derive that the positive root $r_{n,s}$ of spherical Bessel function
$j_{n}(z)$ has the following sharp upper and lower bounds[30]

$n+\frac{1}{2}-\frac{a_{s}}{2^{1/3}}\left(n+\frac{1}{2}\right)^{1/3}<r_{n,s}<n+\frac{1}{2}-\frac{a_{s}}{2^{1/3}}\left(n+\frac{1}{2}\right)^{1/3}+\frac{3}{20}a_{s}^{2}\frac{2^{1/3}}{\left(n+\frac{1}{2}\right)^{1/3}},$
(2.16)

where $a_{s}$ is the $s$-th negative zero of the Airy function and has the
representation

$a_{s}=-\left(\frac{3\pi}{8}(4s-1)\right)^{2/3}(1+\sigma_{s}),\quad
0\leq\sigma_{s}\leq 0.130\left(\frac{3\pi}{8}(4s-1.051)\right)^{-2}.$

By noting the choice of $s_{1}$ and $s_{2}$ in (2.13) for sufficiently large
$n$, it holds that

$\displaystyle
r_{n,s_{1}}=\left(n+\frac{1}{2}\right)\left(1+C_{0}(n+1/2)^{2(\gamma_{1}-1)/3}+o((n+1/2)^{2(\gamma_{1}-1)/3})\right),$
(2.17) $\displaystyle
r_{n,s_{2}}=\left(n+\frac{1}{2}\right)\left(1+C_{0}(n+1/2)^{2(\gamma_{2}-1)/3}+o((n+1/2)^{2(\gamma_{2}-1)/3})\right),$

where $C_{0}$ is a positive constant. Note that the Bessel function admits the
following asymptotic formula [23, P. 129]

$J_{n}(z)=\sqrt{\frac{2}{\pi\sqrt{z^{2}-n^{2}}}}\cos\left(\sqrt{z^{2}-n^{2}}-\frac{n\pi}{2}+n\arcsin(n/z)-\frac{\pi}{4}\right)\Big{(}1+o(1)\Big{)},$
(2.18)

for $z>n$ and $n\rightarrow\infty$. Combing (2.15), (2.17), and (2.18),
through a straightforward calculation, one obtains

$j_{n}\left(r_{n,s_{i}}\epsilon_{0}\right)=\frac{1}{C_{n,\epsilon_{0}}^{(i)}}\cos\left((n+1/2)\left(\sqrt{\epsilon_{0}^{2}-1}-\frac{\pi}{2}+\arcsin\frac{1}{\epsilon_{0}}+\mathcal{O}\Big{(}(n+1/2)^{\varsigma_{i}}\Big{)}\right)-\frac{\pi}{4}\right)\Big{(}1+o(1)\Big{)},$

where $\varsigma_{i}:={2(\gamma_{i}-1)/3},\ i=1,2,$ and

$C_{n,\epsilon_{0}}^{(i)}=\big{(}n+{1}/{2}\big{)}\left(\epsilon_{0}^{2}(\epsilon_{0}^{2}-1)\right)^{-1/4}\left(1+\mathcal{O}\Big{(}\left(n+{1}/{2}\right)^{\varsigma_{i}}\Big{)}\right).$

Correspondingly, one can derive that

$j_{n+1}\left(r_{n,s_{i}}\right)=C_{n,1}^{(i)}\cos\left(\,(n+3/2)\mathcal{O}\Big{(}(n+1/2)^{\varsigma_{i}}\Big{)}-\frac{\pi}{4}\right)\Big{(}1+o(1)\Big{)},$

where

$C_{m,1}^{(i)}=\mathcal{O}\left(\Big{(}n+\frac{1}{2}\Big{)}^{1+\epsilon_{i}/4}\right).$

Without loss of generality, we suppose that

$j_{n-1}\left(r_{n,s_{1}}\right)j_{n}\left(r_{n,s_{1}}\,\epsilon_{0}\right)>0.$

We now show that there exists at least one choice of
$s_{2}=(n+1/2)^{\gamma_{2}}$ such that

$j_{n-1}\left(r_{n,s_{2}}\right)j_{n}\left(r_{n,s_{2}}\,\epsilon_{0}\right)<0,$

that is,

$\displaystyle\cos\left((n+3/2)\mathcal{O}\Big{(}(n+1/2)^{\varsigma_{2}}\Big{)}-\frac{\pi}{4}\right)$
(2.19)
$\displaystyle\quad\cdot\cos\left((n+1/2)\left(\sqrt{\epsilon_{0}^{2}-1}-\frac{\pi}{2}+\arcsin\frac{1}{\epsilon_{0}}+\mathcal{O}\Big{(}(n+1/2)^{\varsigma_{2}}\Big{)}\right)-\frac{\pi}{4}\right)<0.$

Noting that the above two cosine functions never have the same frequency, so
it is easy to realize (2.19) by modifying $\gamma_{2}$ w.r.t
$\varsigma_{2}(\gamma_{2})$. Thus one has
$k_{n}\in(r_{n,s_{1}(n)}/R,r_{n,s_{2}(n)}/R)$. (2.14) is then followed
trivially from (2.17) and the proof is complete. ∎

From Lemma 2.1, we prove that there exists a sequences of transmission
eigenvalues. Next, we show the geometrical properties of the the corresponding
transmission eigenfunctions
$\left(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)}\right)$.

######  Theorem 2.2.

Let $\Omega=\\{\mathbf{x}\in\mathbb{R}^{3}:|\mathbf{x}|<R\in\mathbb{R}_{+}\\}$
and $\epsilon_{0}>1$ be a constant. Let
$(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)})$ be the pair of transmission
eigenfunctions associated with eigenvalue $k_{n}$ in (2.12) for Maxwell’s
transmission eigenvalue problem (2.2). Then it holds that the eigenfunction
$\mathbf{E}_{2}^{(n)}$ is surface-localized but the other eigenfunction
$\mathbf{E}_{1}^{(n)}$ is not surface-localized.

###### Proof.

Let $\Omega_{\tau}:=\\{\mathbf{x}:|\mathbf{x}|<\tau,\ \tau<R\\}$ and $k_{n}$
be the eigenvalues defined in (2.12). For the TE models, the eigenfunctions
are given by

$\displaystyle\mathbf{E}_{1}^{(n)}={a}_{n}^{m}\rho\,j_{n}(k_{n}\epsilon_{0}\rho)\,\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}},$
(2.20)
$\displaystyle\mathbf{E}_{2}^{(n)}=\widetilde{a}_{n}^{m}\rho\,j_{n}(k_{n}\rho)\,\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}}.$
(2.21)

Correspondingly, for the TM models, the eigenfunctions are given by

$\displaystyle\mathbf{E}_{1}^{(n)}=\frac{{b}_{n}^{m}}{\mathrm{i}k_{n}\rho}\left(j_{n}(k_{n}\epsilon_{0}\rho)n(n+1)Y_{n}^{m}\hat{\bm{\rho}}+\frac{\partial}{\partial\rho}\Big{(}\rho
j_{n}(k_{n}\epsilon_{0}\rho)\Big{)}\nabla_{s}Y_{n}^{m}(\theta,\phi)\right),$
(2.22)
$\displaystyle\mathbf{E}_{2}^{(n)}=\frac{\widetilde{b}_{n}^{m}}{\mathrm{i}k_{n}\rho}\left(j_{n}(k_{n}\rho)n(n+1)Y_{n}^{m}\hat{\bm{\rho}}+\frac{\partial}{\partial\rho}\Big{(}\rho
j_{n}(k_{n}\rho)\Big{)}\nabla_{s}Y_{n}^{m}(\theta,\phi)\right).$ (2.23)

To begin with, for a fixed $\tau$, we prove that there exists sufficient large
$n$ such that $k_{n}\tau<n+1/2$. Combining (2.12) and (2.17), we can derive
that

$k_{n}=\frac{1}{R}\left(n+\frac{1}{2}\right)\left(1+C_{0}(n+1/2)^{\varsigma}\right),\quad\varsigma\in[\varsigma_{1},\varsigma_{2}],$

where $C_{0}$ is a positive constant and $\varsigma_{i}={2(\gamma_{i}-1)/3}$,
$i=1,2$. Thus, there exists a sufficiently large $n$, such that

$k_{n}\tau=\frac{\tau}{R}\left(n+\frac{1}{2}\right)\left(1+C_{0}(n+1/2)^{\varsigma}\right)<n+\frac{1}{2},$
(2.24)

for any fixed $\tau<R$.

Next, we prove that the transmission eigenfunctions $\mathbf{E}_{2}^{(n)}$ are
surface-localized around the boundary $\partial\Omega$. From [23, P.129], one
has the following asymptotic formula:

$J_{n}(nz)=\frac{z^{n}\mathrm{e}^{n\sqrt{1-z^{2}}}}{(2\pi
n)^{1/2}(1-z^{2})^{1/4}(1+\sqrt{1-z^{2}})^{n}}\Big{(}1+o(1)\Big{)},\quad
0<z<1.$ (2.25)

By a simple calculation, one can find that

$\frac{z\mathrm{e}^{\sqrt{1-z^{2}}}}{1+\sqrt{1-z^{2}}}<1,\quad 0<z<1.$

Therefore, from (2.25), we can derive the following asymptotic expansion:

$\begin{split}j_{n}(k_{n}\tau)=&\sqrt{\frac{\pi}{2k_{n}\tau}}J_{n+1/2}(k_{n}\tau)\\\
=&\frac{1}{2\sqrt{k_{n}\tau}\left(1-\frac{k_{n}\tau}{n+1/2}\right)^{1/4}}\left(n+\frac{1}{2}\right)^{-1/2}\left(\frac{k_{n}\tau}{n+1/2}\frac{\mathrm{e}^{\sqrt{1-\Big{(}\frac{k_{n}\tau}{n+1/2}\Big{)}^{2}}}}{1+\sqrt{1-\Big{(}\frac{k_{n}\tau}{n+1/2}\Big{)}^{2}}}\right)^{n+1/2}\Big{(}1+o(1)\Big{)}\\\
<&C_{1}\left(n+\frac{1}{2}\right)^{-1}\Big{(}1+o(1)\Big{)},\end{split}$

where $C_{1}$ is a positive constant. According to [1, p.370], one has

$\displaystyle n+\frac{1}{2}\leq
r_{n,1}^{{}^{\prime}}<r_{n,1}<r_{n,2}^{{}^{\prime}}<r_{n,2}<r_{n,3}^{{}^{\prime}}<\cdots,$
(2.26) $\displaystyle
j^{\prime}_{n}(z)=\frac{\sqrt{\pi}(z/2)^{n-1}}{4\Gamma(n+1/2)}\mathop{\Pi}\limits_{s=1}^{\infty}\left(1-\frac{z^{2}}{r_{n,s}^{{}^{\prime}2}}\right),$
(2.27)

where $r_{n,s}^{{}^{\prime}}$ denotes the $s$-th positive zero of
$j_{n}^{{}^{\prime}}(z)$ that is derivative of $j_{n}(z)$. Using (2.24) and
(2.26), one can deduce that $k_{n}\tau<r^{\prime}_{n,1}$ for sufficiently
large $n$. Thus, by (2.27), we find that $j_{n}(k_{n}\rho)$ is a monotonically
increasing with respect to $\rho\in(0,\tau)$. Hence, we obtain

$j_{n}(k_{n}\rho)<j_{n}(k_{n}\tau)<C_{1}\left(n+\frac{1}{2}\right)^{-1}\Big{(}1+o(1)\Big{)},\quad
0<\rho<\tau.$ (2.28)

One the other hand, for sufficiently large $n$ , one can choose $\tau_{1}$

$\tau_{1}=\frac{r_{n,1}^{\prime}}{k_{n}},\quad\tau<\tau_{1}<R.$ (2.29)

Thus, we have

$n+\frac{1}{2}<r_{n,1}^{\prime}=k_{n}\tau_{1}.$

Using the asymptotic expansion (2.18), one can deduce that

$\begin{split}&\int_{\tau}^{R}j_{n}^{2}(k_{n}\rho)\,\rho\,\mathrm{d}\rho\geq\int_{\tau_{1}}^{R}j_{n}^{2}(k_{n}\rho)\,\rho\,\mathrm{d}\rho\\\
\geq&\frac{1}{k_{n}\sqrt{k_{n}^{2}R^{2}-(n+\frac{1}{2})^{2}}}\\\ &\
\cdot\int_{\tau_{1}}^{R}\cos^{2}\left(\sqrt{k_{n}^{2}\rho^{2}-\left(n+\frac{1}{2}\right)^{2}}-\frac{\left(n+\frac{1}{2}\right)\pi}{2}+\left(n+\frac{1}{2}\right)\arcsin\left(\frac{n+\frac{1}{2}}{k_{n}\rho}\right)-\frac{\pi}{4}\right)\mathrm{d}\rho\\\
=&C_{2}\frac{R(R-\tau_{1})}{2}\left(n+\frac{1}{2}\right)^{-1-\varsigma/2}\Big{(}1+\mathcal{R}(n)\Big{)},\end{split}$
(2.30)

where $C_{2}$ is a positive constant and the remaining term $\mathcal{R}(n)$
has the approximation

$\begin{split}&\lim_{n\rightarrow\infty}\mathcal{R}(n)\\\
=&\lim_{n\rightarrow\infty}\int_{\tau_{1}}^{R}\sin
2\left(\sqrt{k_{n}^{2}\rho^{2}-\left(n+\frac{1}{2}\right)^{2}}-\frac{\left(n+\frac{1}{2}\right)\pi}{2}+\left(n+\frac{1}{2}\right)\arcsin\left(\frac{n+\frac{1}{2}}{k_{n}\rho}\right)\right)\,\mathrm{d}\rho\\\
=&0.\end{split}$

Thus, for the TE models, using (2.21), (2.28) and (2.30), it holds that

$\displaystyle\frac{\|\mathbf{E}_{2}^{(n)}\|_{L^{2}(\Omega_{\tau})}^{2}}{\|\mathbf{E}_{2}^{(n)}\|^{2}_{L^{2}(\Omega)}}=$
$\displaystyle\frac{\int_{0}^{\tau}\int_{0}^{\pi}\int_{0}^{2\pi}\Big{|}j_{n}(k_{n}\rho)\,\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\bm{\hat{\rho}}\Big{|}^{2}\rho^{2}\sin\theta\,\mathrm{d}\phi\mathrm{d}\theta\mathrm{d}\rho}{\int_{0}^{R}\int_{0}^{\pi}\int_{0}^{2\pi}\Big{|}j_{n}(k_{n}\rho)\,\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\bm{\hat{\rho}}\Big{|}^{2}\rho^{2}\sin\theta\,\mathrm{d}\phi\mathrm{d}\theta\mathrm{d}\rho}$
$\displaystyle\leq$
$\displaystyle\frac{j_{n}^{2}(k_{n}\tau)\int_{0}^{\tau}\,\rho^{2}\,\mathrm{d}\rho}{\tau_{1}\int_{\tau_{1}}^{R}j_{n}^{2}(k_{n}\rho)\rho\,\mathrm{d}\rho}$
$\displaystyle<$
$\displaystyle\frac{C_{1}^{2}\left(n+\frac{1}{2}\right)^{-2}\tau^{3}/3}{C_{2}\left(n+\frac{1}{2}\right)^{-1-\varsigma/2}\tau_{1}R(R-\tau_{1})/2}$
$\displaystyle=$
$\displaystyle\frac{2C_{1}^{2}\tau^{3}}{3C_{2}\tau_{1}R(R-\tau_{1})}\left(n+\frac{1}{2}\right)^{-1+\varsigma/2}\rightarrow
0,\quad\mathrm{as}\quad n\rightarrow\infty.$

Moreover, for the TM models, according to the recurrence relation of spherical
bessel function

$\frac{\partial}{\partial\rho}j_{n}(\rho)=\frac{j_{n-1}(\rho)-j_{n+1}(\rho)}{2},$

and formula (2.14), then the eigenfunction (2.23) can be rewritten as

$\displaystyle\mathbf{E}_{2}^{(n)}=$
$\displaystyle\frac{\widetilde{b}_{n}^{m}}{\mathrm{i}}\Big{(}\frac{n(n+1)}{k_{n}\rho}j_{n}(k_{n}\rho)Y_{n}^{m}(\theta,\phi)\bm{\hat{\rho}}+\frac{j_{n}(k_{n}\rho)}{k_{n}\rho}\nabla_{s}Y_{n}^{m}(\theta,\phi)$
(2.31)
$\displaystyle\quad+\frac{j_{n-1}(k_{n}\rho)-j_{n+1}(k_{n}\rho)}{2}\nabla_{s}Y_{n}^{m}(\theta,\phi)\Big{)}$
$\displaystyle=$
$\displaystyle\frac{\widetilde{b}_{n}^{m}}{\mathrm{i}}\Big{(}\frac{n(n+1)}{k_{n}\rho}j_{n}(k_{n}\rho)Y_{n}^{m}(\theta,\phi)\bm{\hat{\rho}}$
$\displaystyle\quad+\frac{j_{n-1}(k_{n}\rho)-j_{n+1}(k_{n}\rho)}{2}\nabla_{s}Y_{n}^{m}(\theta,\phi)\left(1+\mathcal{O}\left(n^{-1}\right)\right)\Big{)}.$

Due to the Orthogonality property of the surface gradient to spherical
harmonic function

$\int_{\mathbb{S}^{2}}\nabla_{s}Y_{n}^{m}\cdot\nabla_{s}Y_{n^{\prime}}^{m^{\prime}}\mathrm{d}s=n(n+1)\delta_{nn^{\prime}}\delta_{mm^{\prime}},$

together with $\nabla_{s}Y_{n}^{m}(\theta,\phi)=\rho\nabla
Y_{n}^{m}(\theta,\phi)$ on $\partial\Omega_{\rho}$, and equation (2.30), it
holds that

$\displaystyle\frac{\|\mathbf{E}_{2}^{(n)}\|_{L^{2}(\Omega_{\tau})}^{2}}{\|\mathbf{E}_{2}^{(n)}\|^{2}_{L^{2}(\Omega)}}$
$\displaystyle=\frac{\int_{0}^{\tau}\left(\frac{n(n+1)}{k_{n}\rho}\,j_{n}(k_{n}\rho)\right)^{2}\rho^{2}+n(n+1)\left(\frac{j_{n-1}(k_{n}\rho)-j_{n+1}(k_{n}\rho)}{2}\right)^{2}\rho^{2}\,\mathrm{d}\rho}{\int_{0}^{R}\left(\frac{n(n+1)}{k_{n}\rho}\,j_{n}(k_{n}\rho)\right)^{2}\rho^{2}+n(n+1)\left(\frac{j_{n-1}(k_{n}\rho)-j_{n+1}(k_{n}\rho)}{2}\right)^{2}\rho^{2}\,\mathrm{d}\rho}$
$\displaystyle\leq\frac{\int_{0}^{\tau}\frac{n(n+1)}{k_{n}^{2}}j_{n}^{2}(k_{n}\rho)+(j_{n-1}^{2}(k_{n}\rho)+j_{n+1}^{2}(k_{n}\rho))/2\,\rho^{2}\,\mathrm{d}\rho}{\int_{\tau_{1}}^{R}\frac{n(n+1)}{k_{n}^{2}}j_{n}^{2}(k_{n}\rho)\rho\,\mathrm{d}\rho/R}$
$\displaystyle<\frac{C_{3}\left(n+\frac{1}{2}\right)^{-2}\tau}{C_{2}\left(n+\frac{1}{2}\right)^{-1-\varsigma/2}(R-\tau_{1})/2}$
$\displaystyle=\frac{2C_{3}\tau}{C_{2}(R-\tau_{1})}\left(n+\frac{1}{2}\right)^{-1+\varsigma/2}\rightarrow
0,\quad\mathrm{as}\quad n\rightarrow\infty,$

where the constant $C_{3}$ is defined by

$C_{3}=1+\frac{k_{n}^{2}}{n(n+1)}\tau^{2}.$

Hence, the transmission eigenfunctions of $\mathbf{E}_{2}^{(n)}$ for both TE
and TM models are surface-localized on the boundary $\partial\Omega$.

Finally, it remains to prove that the eigenfunctions $\mathbf{E}_{1}^{(n)}$
are not surface-localized around the boundary $\partial\Omega$. Without loss
of generality, we only consider the TE models case and the TM models case can
be proved in a similar manner.

From (2.29), one can deduce that

$\displaystyle\int_{0}^{R}j_{n}^{2}(k_{n}\epsilon_{0}\rho)\rho^{2}\,\mathrm{d}\rho$
$\displaystyle=\frac{1}{\epsilon_{0}^{3}}\int_{0}^{\epsilon_{0}R}j_{n}^{2}(k_{n}\rho^{\prime})\rho^{\prime
2}\,\mathrm{d}\rho^{\prime}$
$\displaystyle=\frac{1}{\epsilon_{0}^{3}}\int_{0}^{\frac{r^{\prime}_{n,1}}{k_{n}}}j_{n}^{2}(k_{n}\rho^{\prime})\rho^{\prime
2}\,\mathrm{d}\rho^{\prime}+\frac{1}{\epsilon_{0}^{3}}\int_{\frac{r^{\prime}_{n,1}}{k_{n}}}^{\epsilon_{0}R}j_{n}^{2}(k_{n}\rho^{\prime})\rho^{\prime
2}\,\mathrm{d}\rho^{\prime}$
$\displaystyle<\frac{2}{\epsilon_{0}^{3}}\int_{\frac{r^{\prime}_{n,1}}{k_{n}}}^{\epsilon_{0}R}j_{n}^{2}(k_{n}\rho^{\prime})\rho^{\prime
2}\,\mathrm{d}\rho^{\prime}$
$\displaystyle=2\int_{\frac{r^{\prime}_{n,1}}{\epsilon_{0}k_{n}}}^{R}j_{n}^{2}(k_{n}\epsilon_{0}\rho)\rho^{2}\,\mathrm{d}\rho.$

By following similar arguments, for any $\tau\in(\frac{R}{\epsilon_{0}},R)$,
then

$\frac{r^{\prime}_{n,1}}{k_{n}\epsilon_{0}}<\frac{R}{\epsilon_{0}}<\tau<R,$

and so for $\tau\leq t<R$, one has

$\begin{split}&\int_{\frac{r^{\prime}_{n,1}}{k_{n}\epsilon_{0}}}^{t}j_{n}^{2}(k_{n}\epsilon_{0}\rho)\rho^{2}\,\mathrm{d}\rho\\\
&=\int_{\frac{r^{\prime}_{n,1}}{k_{n}\epsilon_{0}}}^{t}\frac{\rho\cos^{2}\left(\sqrt{k_{n}^{2}\epsilon_{0}^{2}\rho^{2}-\left(n+\frac{1}{2}\right)^{2}}-\frac{\left(n+\frac{1}{2}\right)\pi}{2}+\left(n+\frac{1}{2}\right)\arcsin\left(\frac{n+\frac{1}{2}}{k_{n}\rho}\right)-\frac{\pi}{4}\right)}{k_{n}\epsilon_{0}\sqrt{k_{n}^{2}\epsilon_{0}^{2}\rho^{2}-(n+\frac{1}{2})^{2}}}\mathrm{d}\rho\\\
&\approx\frac{1}{k_{n}^{2}\epsilon_{0}^{2}}\int_{\frac{r^{\prime}_{n,1}}{k_{n}\epsilon_{0}}}^{t}\frac{\rho}{\sqrt{\rho^{2}-\left(\frac{n+1/2}{k_{n}\epsilon_{0}}\right)^{2}}}\,\mathrm{d}\rho\\\
&=\frac{1}{k_{n}^{3}\epsilon_{0}^{3}}\left(\sqrt{k_{n}^{2}\,\epsilon_{0}^{2}\,t^{2}-\left(n+\frac{1}{2}\right)^{2}}-\sqrt{r^{{}^{\prime}2}_{n,1}-\left(n+\frac{1}{2}\right)^{2}}\right).\end{split}$

Combining the last two estimations, we obtain

$\displaystyle\frac{\|\mathbf{E}_{1}^{(n)}\|_{L^{2}(\Omega_{\tau})}^{2}}{\|\mathbf{E}_{1}^{(n)}\|^{2}_{L^{2}(\Omega)}}=$
$\displaystyle\frac{\int_{0}^{\tau}\int_{0}^{\pi}\int_{0}^{2\pi}\Big{|}j_{n}(k_{n}\epsilon_{0}\rho)\,\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}}\Big{|}^{2}\rho^{2}\sin\theta\,\mathrm{d}\phi\mathrm{d}\theta\mathrm{d}\rho}{\int_{0}^{R}\int_{0}^{\pi}\int_{0}^{2\pi}\Big{|}j_{n}(k_{n}\epsilon_{0}\rho)\,\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}}\Big{|}^{2}\rho^{2}\sin\theta\,\mathrm{d}\phi\mathrm{d}\theta\mathrm{d}\rho}$
$\displaystyle\geq$
$\displaystyle\frac{\int_{\frac{r^{\prime}_{n,1}}{k_{n}\epsilon_{0}}}^{\tau}j_{n}^{2}(k_{n}\epsilon_{0}\rho)\rho^{2}\,\mathrm{d}\rho}{2\int_{\frac{r^{\prime}_{n,1}}{k_{n}\epsilon_{0}}}^{R}j_{n}^{2}(k_{n}\epsilon_{0}\rho)\rho^{2}\,\mathrm{d}\rho}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\sqrt{k_{n}^{2}\,\epsilon_{0}^{2}\,\tau^{2}-\left(n+\frac{1}{2}\right)^{2}}-\sqrt{r^{{}^{\prime}2}_{n,1}-\left(n+\frac{1}{2}\right)^{2}}}{\sqrt{k_{n}^{2}\,\epsilon_{0}^{2}\,R^{2}-\left(n+\frac{1}{2}\right)^{2}}-\sqrt{r^{{}^{\prime}2}_{n,1}-\left(n+\frac{1}{2}\right)^{2}}}$
$\displaystyle\approx$
$\displaystyle\frac{1}{2}\frac{\sqrt{\epsilon_{0}^{2}(\tau/R)^{2}-1}}{\sqrt{\epsilon_{0}^{2}-1}}>0\quad
n\rightarrow\infty.$

The proof is complete. ∎

###  2.3. Bi-localized transmission eigenmodes

In this section, we also construct a sequence of transmission eigenvalues
$\\{k_{n}\\}_{n\in\mathbb{N}_{+}}$ and prove that both corresponding pair of
eigenfunctions $\left(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)}\right)$ are
surface-localized.

######  Lemma 2.2.

Let $\Omega=\\{\mathbf{x}\in\mathbb{R}^{3}:|\mathbf{x}|<R\in\mathbb{R}_{+}\\}$
and $\epsilon_{0}>1$ be a constant. Let $E_{\Omega}^{\prime}$ denote the set
of transmission eigenvalues of (2.2). For a fixed value
$s_{0}\in\mathbb{N}_{+}$, there exists a subsequence
$\\{k_{n}\\}_{n\in\mathbb{N}_{+}}\subset E_{\Omega}^{\prime}$, such that for
sufficient large $n$, there holds

$k_{n}\in\left(\frac{r_{n,s_{0}}}{\epsilon_{0}R},\frac{r_{n,s_{0}+1}}{\epsilon_{0}R}\right).$
(2.32)

Moreover, one has

$\frac{n+\frac{1}{2}}{\epsilon_{0}R}<k_{n}\leq\frac{n+\frac{1}{2}}{R},\quad
n=1,2,3,\cdots.$ (2.33)

###### Proof.

For any fixed $s_{0}\in\mathbb{N}_{+}$, from (2.16) and (2.26), there exists a
sufficiently large $n$ such that

$\frac{r_{n,s_{0}+1}}{\epsilon_{0}}\leq n+\frac{1}{2}\leq r_{n,1}^{\prime}.$
(2.34)

By the monotonicity of the Bessel function before the first local maximal, we
have

$j_{n}(k)\geq 0,\quad k\in(0,r_{n,1}^{\prime}].$

Noting that the positive root of $j_{n}(|x|)$ are interlaced with those of
$j_{n+1}(|x|)$ [27], hence we have

$j_{n+1}(r_{n,s_{0}})\cdot j_{n+1}(r_{n,s_{0}+1})<0.$

Therefore, from the last two equations and (2.10), we can derive that

$\displaystyle
f_{n}^{\mathrm{TE}}\left(\frac{r_{n,s_{0}}}{\epsilon_{0}R}\right)f_{n}^{\mathrm{TE}}\left(\frac{r_{n,s_{0}+1}}{\epsilon_{0}R}\right)$
$\displaystyle=\frac{r_{n,s_{0}}r_{n,s_{0}+1}}{R^{2}}j_{n}\left(\frac{r_{n,s_{0}}}{\epsilon_{0}}\right)j_{n+1}(r_{n,s_{0}})\,j_{n}\left(\frac{r_{n,s_{0}+1}}{\epsilon_{0}}\right)j_{n+1}(r_{n,s_{0}+1})$
$\displaystyle\leq\frac{r_{n,s_{0}}r_{n,s_{0}+1}}{R^{2}}j_{n}^{2}\left(r_{n,1}^{\prime}\right)j_{n+1}(r_{n,s_{0}})j_{n+1}(r_{n,s_{0}+1})<0.$

Hence, there exists
$k_{n}\in\left({r_{n,s_{0}}}/{(\epsilon_{0}R)},{r_{n,s_{0}+1}}/{(\epsilon_{0}R)}\right)$
such that $f_{n}^{\mathrm{TE}}\left(k_{n}\right)=0$. By a similar argument,
one can also verify that $f_{n}^{\mathrm{TM}}\left(k_{n}\right)=0$ and it
proves (2.32). Furthermore, from (2.26) and (2.34), one can deduce (2.33) and
it completes the proof. ∎

######  Theorem 2.3.

Let $\Omega=\\{\mathbf{x}\in\mathbb{R}^{3}:|\mathbf{x}|<R\in\mathbb{R}_{+}\\}$
and $\epsilon_{0}>1$ be a constant. Let
$(\mathbf{E}_{1}^{(n)},\mathbf{E}_{2}^{(n)})$ be the pair of transmission
eigenfunctions associated with eigenvalue $k_{n}$ in (2.12) for Maxwell’s
transmission eigenvalue problem (2.2). Then it holds that both eigenfunctions
$\mathbf{E}_{1}^{(n)}$ and $\mathbf{E}_{2}^{(n)}$ are surface-localized around
the boundary $\partial\Omega$.

###### Proof.

Without loss of generality, we only consider the TE modes and the TM modes can
be proved in a similar manner. Let
$\Omega_{\tau}:=\\{\mathbf{x}:|\mathbf{x}|<\tau,\ \tau<R\\}$ and $k_{n}$ be
the eigenvalues defined in (2.12). Using (2.21) and (2.15), one has

$\displaystyle\frac{\|\mathbf{E}_{2}^{(n)}\|_{L^{2}(\Omega_{\tau})}^{2}}{\|\mathbf{E}_{2}^{(n)}\|^{2}_{L^{2}(\Omega)}}$
$\displaystyle=\frac{\int_{0}^{\tau}\int_{0}^{\pi}\int_{0}^{2\pi}\Big{|}j_{n}(k_{n}\rho)\,\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}}\Big{|}^{2}\rho^{2}\sin\theta\,\mathrm{d}\phi\mathrm{d}\theta\mathrm{d}\rho}{\int_{0}^{R}\int_{0}^{\pi}\int_{0}^{2\pi}\Big{|}j_{n}(k_{n}\rho)\,\nabla_{s}Y_{n}^{m}(\theta,\phi)\wedge\hat{\bm{\rho}}\Big{|}^{2}\rho^{2}\sin\theta\,\mathrm{d}\phi\mathrm{d}\theta\mathrm{d}\rho}$
$\displaystyle=\frac{\int_{0}^{\tau}J_{n+\frac{1}{2}}^{2}(k_{n}\rho)\rho\,\mathrm{d}\rho}{\int_{0}^{R}J_{n+\frac{1}{2}}^{2}(k_{n}\rho)\rho\,\mathrm{d}\rho}.$

Set $f(\rho)=J_{n+\frac{1}{2}}^{2}(k_{n}\rho)\rho$, through a straightforward
calculation, one can verify that $f(\rho)$ is a convex function on $[0,R]$ for
$k_{n}\leq(n+1/2)/R$. Therefore, the area of the triangle under tangent of
$f(R)$ is smaller than $\int_{0}^{R}f(\rho)\mathrm{d}\rho$, that is,

$\frac{R^{2}J_{n+\frac{1}{2}}^{3}(k_{n}R)}{2J_{n+\frac{1}{2}}(k_{n}R)+4k_{n}RJ_{n+\frac{1}{2}}^{{}^{\prime}}(k_{n}R)}\leq\int_{0}^{R}J_{n+\frac{1}{2}}^{2}(k_{n}\rho)\rho\,\mathrm{d}\rho.$

Thus it holds that

$\displaystyle\frac{\|\mathbf{E}_{2}^{(n)}\|_{L^{2}(\Omega_{\tau})}^{2}}{\|\mathbf{E}_{2}^{(n)}\|^{2}_{L^{2}(\Omega)}}$
$\displaystyle\leq\frac{\tau^{2}J_{n+\frac{1}{2}}^{2}(k_{n}\tau)\Big{(}2J_{n+\frac{1}{2}}(k_{n}R)+4k_{n}RJ_{n+\frac{1}{2}}^{{}^{\prime}}(k_{n}R)\Big{)}}{R^{2}J_{n+\frac{1}{2}}^{3}(k_{n}R)}$
$\displaystyle\leq\tau^{2}\left(\frac{J_{n+\frac{1}{2}}^{2}(k_{n}\tau)}{J_{n+\frac{1}{2}}^{2}(k_{n}R)}\right)^{2}\left(\frac{2}{R^{2}}+\frac{4k_{n}}{R}\cdot\frac{J_{n+\frac{1}{2}}^{{}^{\prime}}(k_{n}R)}{J_{n+\frac{1}{2}}(k_{n}R)}\right).$

By Lemma 2.3 in [19], one can show that there exists positive constants
$C_{3}$ and $\gamma$ such that

$\frac{J_{n+\frac{1}{2}}^{{}^{\prime}}(k_{n}R)}{J_{n+\frac{1}{2}}(k_{n}R)}<C_{3}\left(n+\frac{1}{2}\right)^{\gamma}.$

In addition, using the Calini formula, one can find that there exists positive
constants $C_{4}$ and $\delta(\tau,\epsilon_{0})\in(0,1)$, such that

$\frac{J_{n+\frac{1}{2}}^{2}(k_{n}\tau)}{J_{n+\frac{1}{2}}^{2}(k_{n}R)}<C_{4}(1-\delta)^{n+\frac{1}{2}}.$

Hence, combining the last three inequalities, one can obtain that

$\frac{\|\mathbf{E}_{2}^{(n)}\|_{L^{2}(\Omega_{\tau})}^{2}}{\|\mathbf{E}_{2}^{(n)}\|^{2}_{L^{2}(\Omega)}}\rightarrow
0,\quad\mathrm{as}\ n\rightarrow\infty.$

The proof is complete. ∎

######  Remark 2.1.

We would like to point out that there is no contradiction between the mono-
localized eigenmodes and bi-localized modes, since the transmission
eigenvalues for mono-localized eigenmodes satisfy $k_{n}>\frac{n+1/2}{R}$
while the transmission eigenvalues for bi-localized eigenmodes satisfy
$\frac{n+1/2}{\epsilon_{0}R}<k_{n}<\frac{n+1/2}{R}$. In fact, for a fixed
value $n\in\mathbb{N}_{+}$ and $\epsilon_{0}>1$, the mono-localized eigenmodes
would have a better chance for occurring.

###  2.4. Numerics

In this section, we present several numerical examples to show geometrical
properties of transmission eigenfunctions. Here, we shall use the curl
conforming finite element method approximation as proposed in [29] and briefly
describe it in the sequel. Multiplying the first two equation in (1.5) by a
test functions $\varphi\in H_{0}^{1}(\mathrm{curl},\Omega)$ and integrating by
parts, one obtains

$\displaystyle(\nabla\wedge\mathbf{E}_{1},\nabla\wedge\varphi)-k^{2}\varepsilon(\mathbf{E}_{1},\varphi)=0,\quad$
$\displaystyle\forall\varphi\in H_{0}^{1}(\mathrm{curl},\Omega),$ (2.35)
$\displaystyle(\nabla\wedge\mathbf{E}_{2},\nabla\wedge\varphi)-k^{2}(\mathbf{E}_{2},\varphi)=0,\quad$
$\displaystyle\forall\varphi\in H_{0}^{1}(\mathrm{curl},\Omega).$

To enforce the boundary conditions
$\nu\wedge(\nabla\wedge\mathbf{E}_{1})=\nu\wedge(\nabla\wedge\mathbf{E}_{2})$
weakly, we multiply it by a test function $\psi\in
H^{1}(\mathrm{curl},\Omega)$ and integrate by parts

$(\nabla\wedge\mathbf{E}_{1},\nabla\wedge\psi)-k^{2}\varepsilon(\mathbf{E}_{1},\psi)=(\nabla\wedge\mathbf{E}_{2},\nabla\wedge\psi)-k^{2}(\mathbf{E}_{2},\psi).$
(2.36)

Hence, the variational formulation for (2.2) is to find
$(\mathbf{E}_{1},\mathbf{E}_{2})\in H(\mathrm{curl},\Omega)\times
H(\mathrm{curl},\Omega)$ satisfying (2.35) and (2.36), together with the
essential boundary condition $\nu\wedge\mathbf{E}_{1}=\nu\wedge\mathbf{E}_{2}$
on $\partial\Omega$.

Let $\mathcal{T}_{h}$ be a regular tetrahedral mesh for $\Omega$ and $V_{h}$
be the curl conforming element space of Nédélec

$V_{h}=\\{\mathbf{E}\in
H(\mathrm{curl},\Omega):\mathbf{E}|_{K}(\bm{x})=\bm{\alpha}+\bm{\beta}\wedge\bm{x},\
K\in\mathcal{T}_{h},\ \bm{\alpha},\bm{\beta}\in(P_{n})^{3}\\},$

where $P_{n}$ denotes the space of homogeneous polynomials of order $n$. We
also define a subspace $V_{h}^{0}=V\cap H_{0}(\mathrm{curl},\Omega)$. Let
$\\{\bm{\eta}_{i}:\,i=1,2,\cdots m\\}$ denote a basis for $V_{h}^{0}$ and
$\\{\bm{\eta}_{i}:\,i=1,2,\cdots m\\}\cup\\{\bm{\zeta}_{j}:\,j=1,2,\cdots
n\\}$ denote a basis for $V_{h}$, respectively. To enforce the boundary
condition $\nu\wedge\mathbf{E}_{1}=\nu\wedge\mathbf{E}_{2}$, we set

$\mathbf{E}_{1}^{h}=\sum_{i=1}^{m}u_{1}^{(i)}\bm{\eta}_{i}+\sum_{j=1}^{n}w^{(j)}\bm{\zeta}_{j},\quad\mathbf{E}_{2}^{h}=\sum_{i=1}^{m}u_{2}^{(i)}\bm{\eta}_{i}+\sum_{j=1}^{n}w^{(j)}\bm{\zeta}_{j},$

to be the finite element approximations of $\mathbf{E}_{1}$ and
$\mathbf{E}_{2}$, respectively. Thus, the discrete form of (2.35) and (2.36)
can be written as

$AX=k^{2}BX,$ (2.37)

where

$X=[u_{1}^{(1)},\cdots,u_{1}^{(m)},u_{2}^{(1)},\cdots,u_{2}^{(m)},w^{(1)},\cdots,w^{(n)}]\in\mathbb{C}^{2m+n},$

and

$A=\left[\begin{array}[]{ccc}S_{II}&0&S_{IB}\\\ 0&S_{II}&S_{IB}\\\
S_{IB}^{\top}&-S_{IB}^{\top}&0\\\ \end{array}\right],\quad
B=\left[\begin{array}[]{ccc}M_{2,II}&0&M_{2,IB}\\\ 0&M_{1,II}&M_{1,IB}\\\
M_{2,IB}^{\top}&-M_{1,IB}^{\top}&M_{2,BB}-M_{1,BB}\\\ \end{array}\right].$

Here, the stiffness and mass matrices are given by

$\displaystyle S_{II}=(\nabla\wedge\bm{\eta}_{i},\nabla\wedge\bm{\eta}_{j}),$
$\displaystyle S_{IB}=(\nabla\wedge\bm{\eta}_{i},\nabla\wedge\bm{\zeta}_{j}),$
$\displaystyle M_{2,II}=(\varepsilon\bm{\eta}_{i},\bm{\eta}_{j}),$
$\displaystyle M_{1,II}=(\bm{\eta}_{i},\bm{\eta}_{j}),$ $\displaystyle
M_{2,IB}=(\varepsilon\bm{\eta}_{i},\bm{\zeta}_{j}),$ $\displaystyle
M_{1,IB}=(\bm{\eta}_{i},\bm{\zeta}_{j}),$ $\displaystyle
M_{2,BB}=(\varepsilon\bm{\zeta}_{i},\bm{\zeta}_{j}),$ $\displaystyle
M_{1,BB}=(\bm{\zeta}_{i},\bm{\zeta}_{j}).$

Here, the assembly of the above matrices is implemented by an open-source PDE
solver FreeFEM [22] with the first order curl conforming method. The
eigenvalues and eigenvectors for the non-symmetric eigenvalue problem (2.37)
are computed by sptarn function in MATLAB, which is based on the Arnoldi
algorithm with spectral transformation.

(a) $E_{1}^{(1)}$

(b) $E_{1}^{(2)}$

(c) $E_{1}^{(3)}$

(d) $E_{2}^{(1)}$

(e) $E_{2}^{(2)}$

(f) $E_{2}^{(3)}$

Figure 1.  Mono-localized transmission eigenmodes. Slice plots of
eigenfunctions $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ for a unit ball, where
$k=3.3318$. The top row and bottom row denote the three components of
eigenfunctions of $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ respectively.

(a) $E_{1}^{(1)}$

(b) $E_{1}^{(2)}$

(c) $E_{1}^{(3)}$

(d) $E_{2}^{(1)}$

(e) $E_{2}^{(2)}$

(f) $E_{2}^{(3)}$

Figure 2.  Bi-localized transmission eigenmodes. Slice plots of eigenfunctions
$\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ for a unit ball, where $k=3.2296$. The
top row and bottom row denote the three components of eigenfunctions of
$\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ respectively.

(a) $E_{1}^{(1)}$

(b) $E_{1}^{(2)}$

(c) $E_{1}^{(3)}$

(d) $E_{2}^{(1)}$

(e) $E_{2}^{(2)}$

(f) $E_{2}^{(3)}$

Figure 3.  Mono-localized transmission eigenmodes. Slice plots of
eigenfunctions $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ for a torus domain, where
$k=3.1782$. The top row and bottom row denote the three components of
eigenfunctions of $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ respectively.

######  Example 1.

In this example, we consider a constant refractive index case and set
$\varepsilon=64$. We first choose a unit ball as the domain $\Omega$. The mesh
is generated by the open-source soft GMSH [21] and the number of the degree of
freedom is $\mathrm{DoF}=23089$. We use an interval $[10,12]$ to search for
transmission eigenvalues and the corresponding eigenfunctions. Figure 1 shows
the slice plots of transmission eigenfunctions associated with transmission
eigenvalue $k=3.3318$. It is clear that the eigenfunctions are surface-
localized around the boundary for $\mathbf{E}_{2}$ but not surface-localized
around the boundary for $\mathbf{E}_{1}$, which fulfil the mono-localized
transmission eigenmodes. Moreover, Fig. 2 presents a pair of transmission
eigenfunctions that satisfy the bi-localized transmission eigenmodes with
$k=3.2296$. One can find that the localized behavior is obvious for
$\mathbf{E}_{2}$ while not obvious for $\mathbf{E}_{1}$, which is because the
eigenvalue $k$ is not sufficiently large. In fact, the transmission
eigenfunctions of $\mathbf{E}_{1}$ almost vanish in the centre area of the
unit ball and are localized near the boundary. In addition, we must emphasize
that bi-localized cases are much fewer than mono-localized cases in numerics.

We now consider a more general domain, i.e., torus domain. The major radius is
$1.5$ and the minor radius is $0.5$. In order to resolve the model, we
introduce finer meshes and the number of the degree of freedom for torus is
$\mathrm{DoF}=52108$. Interval $[11,12]$ is used for searching transmission
eigenvalues and eigenfunctions. All computational results only fulfil the
mono-localized transmission eigenmodes as shown in Fig. 3.

######  Example 2.

(a) $E_{2}^{(1)}$

(b) $E_{2}^{(2)}$

(c) $E_{2}^{(3)}$

(d) plane $x=0$

(e) plane $y=0$

(f) plane $z=0$

Figure 4.  Mono-localized transmission eigenmodes. Slice plots of
eigenfunctions $\mathbf{E}_{2}$ for a cuboid domain, where $k=3.3448$. The top
row denotes the three components of eigenfunctions of $\mathbf{E}_{2}$ and the
bottom row denotes slice plots of $\mathbf{E}_{2}^{(1)}$ at three planes.

In this example, we consider a case with a piecewise constant refractive
index. We construct a two-layer cuboid domain centered at the origin, where
the edge lengths of the inner cuboid are $(0.8,0.6,0.4)$ and $(1,0.8,0.6)$ for
the outer cuboid. Here, we set $\varepsilon=4$ inside the inner cuboid and
$\varepsilon=400$ between the inner and outer cuboids. To provide a more
accurate solution with less computational cost, we establish a relatively
rough mesh in the inner cuboid but a fine mesh outside the inner cuboid and
the number of the degree of freedom is DoF $=65394$. Fig. 4 presents the
transmission eigenfunctions $\mathbf{E}_{2}$ for the mono-localized
transmission eigenmodes with $k=3.3448$. In the figures, the dotted black
lines denote the contour of the inner cuboid, which is the interface of
different refractive indices. One can observe that the transmission
eigenfunctions are actually surface-localized around the boundary even if the
refractive index is piecewise-constant, see Fig. 4 (d)(e)(f).

##  3\. Topological properties of Maxwell transmission eigenfunctions

In what follows, with a bit abuse of notations, we let $\varepsilon,\mu$ be
symmetric-positive-definite-matrix valued functions in $\Omega$, and consider
the Maxwell system

$\nabla\wedge\mathbf{E}=\mathrm{i}k\mu\mathbf{H},\quad\nabla\wedge\mathbf{H}=-\mathrm{i}k\varepsilon\mathbf{E}\quad\mbox{in}\
\ \Omega.$ (3.1)

We recall the following transformation-invariant property of the Maxwell
system.

######  Lemma 3.1 (Lemma 2.2 in [26]).

Consider a bi-Lipschitz and orientation-preserving coordinate transformation
$\widetilde{\mathbf{x}}=\mathbf{F}(\mathbf{x}):\Omega\rightarrow\widetilde{\Omega}$.
Let $D\mathbf{F}$ denote the Jacobian matrix of $\mathbf{F}$. Assume that
$(\mathbf{E},\mathbf{H})\in H(\mathrm{curl},\Omega)\times
H(\mathrm{curl},\Omega)$ are EM fields to (3.1); then for the pull-back fields
given by

$\widetilde{\mathbf{Q}}=(\mathbf{F}^{-1})^{*}\mathbf{Q}:=(D\mathbf{F}^{T})\mathbf{Q}\circ\mathbf{F}^{-1},\quad\mathbf{Q}=\mathbf{E},\mathbf{H},$

we have $(\mathbf{E},\mathbf{H})\in H(\mathrm{curl},\Omega)\times
H(\mathrm{curl},\Omega)$, which satisfy the Maxwell system

$\widetilde{\nabla}\wedge\widetilde{\mathbf{E}}=\mathrm{i}k\tilde{\mu}\widetilde{\mathbf{H}},\quad\widetilde{\nabla}\wedge\widetilde{\mathbf{H}}=-\mathrm{i}k\widetilde{\varepsilon}\widetilde{\mathbf{E}},$

where $\widetilde{\nabla}$ signifies the differentiation in the
$\widetilde{\mathbf{x}}$-coordinate, and $\tilde{\varepsilon}$ and
$\tilde{\mu}$ are the push-forwards of $\varepsilon$ and $\mu$ via
$\mathbf{F}$, defined by

$\tilde{\zeta}=\mathbf{F}_{*}\zeta:=|D\mathbf{F}|^{-1}D\mathbf{F}\cdot\zeta\cdot
D\mathbf{F}^{T}\circ\mathbf{F}^{-1},\quad\zeta:=\varepsilon,\mu.$

Using Lemma 3.1, one can readily obtain the following transformation property
of the transmission eigenvalue problem.

######  Theorem 3.1.

Let $\Omega,\widetilde{\Omega}$ and $\mathbf{F}$ be the same as those in Lemma
3.1. Assume that $\mathbf{Q}_{j}\in H(\mathrm{curl},\Omega)$, $j=1,2$ and
$\mathbf{Q}=\mathbf{E},\mathbf{H}$, satisfy the following transmission
eigenvalue problem:

$\left\\{\begin{aligned}
&\nabla\wedge\mathbf{E}_{j}-\mathrm{i}k\mu_{j}\mathbf{H}_{j}=\mathbf{0}&&\mbox{in}\
\Omega,\\\
&\nabla\wedge\mathbf{H}_{j}+\mathrm{i}k\varepsilon_{j}\mathbf{E}_{j}=\mathbf{0}&&\mbox{in}\
\Omega,\\\ &\nu\wedge\mathbf{E}_{j}=\nu\wedge\mathbf{H}_{j}&&\mbox{on}\
\partial\Omega,\ \ j=1,2,\end{aligned}\right.$ (3.2)

the the pull-back fields
$\widetilde{\mathbf{Q}}_{j}=(\mathbf{F}^{-1})^{*}\mathbf{Q}_{j}$ are solutions
to the following transmission eigenvalue problem:

$\left\\{\begin{aligned}
&\nabla\wedge\widetilde{\mathbf{E}}_{j}-\mathrm{i}k\tilde{\mu}_{j}\widetilde{\mathbf{H}}_{j}=\mathbf{0},&&\mbox{in}\
\widetilde{\Omega},\\\
&\nabla\wedge\widetilde{\mathbf{H}}_{j}+\mathrm{i}k\tilde{\varepsilon}_{j}\widetilde{\mathbf{E}}_{j}=\mathbf{0}&&\mbox{in}\
\widetilde{\Omega},\\\
&\nu\wedge\widetilde{\mathbf{E}}_{j}=\nu\wedge\widetilde{\mathbf{H}}_{j}&&\mbox{on}\
\partial\widetilde{\Omega},\ \ j=1,2,\end{aligned}\right.$ (3.3)

where
$(\tilde{\varepsilon}_{j},\tilde{\mu}_{j})=\mathbf{F}_{*}(\varepsilon_{j},\mu_{j})$,
$j=1,2$.

###### Proof.

Using Lemma 3.1, one can readily have that the EM fields in (3.2) and (3.3)
are related by the pull-back while the material parameters are related by the
push-forward through the transformation $\mathbf{F}$. Hence, it is sufficient
to show that the transmission conditions on $\partial\Omega$ and
$\partial\widetilde{\Omega}$ are invariant under the transformation, which can
be straightforwardly shown by following the argument in [26] in proving Lemma
3.1. ∎

Let us consider a specific case by letting $\widetilde{\Omega}$ be obtained
through a translation of $\Omega$, namely
$\widetilde{\Omega}=\mathbf{z}+\Omega$ with $\mathbf{z}\in\mathbb{R}^{3}$. In
such a case, we call $\widetilde{\Omega}$ a copy of $\Omega$. Suppose
$\mathbf{E}_{j}(\mathbf{x})$, $\mathbf{x}\in\Omega$ and $j=1,2$, are
transmission eigenfunctions to (1.5) associated with $(\Omega,\varepsilon)$.
Using Theorem 3.1, it is straightforward to verify that
$\widetilde{\mathbf{E}}_{j}(\mathbf{x}-\mathbf{z})$,
$\mathbf{x}\in\widetilde{\Omega}$ and $j=1,2$, are transmission eigenfunctions
associated with $(\widetilde{\Omega},\tilde{\varepsilon})$, where
$\tilde{\varepsilon}(\mathbf{x})=\varepsilon(\mathbf{x}-\mathbf{z})$ for
$\mathbf{x}\in\widetilde{\Omega}$. This fact shall be used in our subsequent
study of the artificial electromagnetic mirage.

Next, we consider the case that the underlying domain $\Omega$ has multiply
connected components. To that end, we first recall that for a fixed
transmission eigenvalue $k\in\mathbb{R}_{+}$ in (1.5), the corresponding
eigenspace is finite-dimensional [10]. In what follows, we let
$\mathcal{S}_{k,(\Omega,\varepsilon)}$ signify the eigenspace associated with
the transmission eigenvalue $k$ and the medium $(\Omega,\varepsilon)$ in
(1.5).

######  Theorem 3.2.

Consider the transmission eigenvalue problem (1.5) and assume that
$(\Omega,\varepsilon)=\cup_{j=1}^{l}(\Omega_{j},\varepsilon_{j})$, where each
$\Omega_{j}$ is a simply-connected component of $\Omega$. Then it holds that

$\mathrm{dim}(\mathcal{S}_{k,(\Omega_{j},\varepsilon_{j})})=p_{j}\quad\mbox{and}\quad\mathcal{S}_{k,(\Omega_{j},\varepsilon_{j})}=\mathrm{Span}\\{(\mathbf{E}_{1}^{\alpha_{j}},\mathbf{E}_{2}^{\alpha_{j}})_{\alpha_{j}=1}^{p_{j}}\\}$

if and only if

$\mathrm{dim}(\mathcal{S}_{k,(\Omega,\varepsilon)})=\sum_{j=1}^{l}p_{j}:=p\quad\mbox{and}\quad\mathcal{S}_{k,(\Omega,\varepsilon)}=\mathrm{Span}\\{(\mathbf{E}_{1}^{\alpha},\mathbf{E}_{2}^{\alpha})_{\alpha=1}^{p}\\},$

where $(\mathbf{E}_{1}^{\alpha},\mathbf{E}_{2}^{\alpha})$ is of the form

$(\mathbf{E}_{1}^{\alpha},\mathbf{E}_{2}^{\alpha})=(\mathbf{E}_{1}^{\alpha_{j}},\mathbf{E}_{2}^{\alpha_{j}})\chi_{\Omega_{j}}+(\mathbf{0},\mathbf{0})\chi_{\cup_{\beta=1,\beta\neq
j}^{l}\Omega_{\beta}},$

for a certain $1\leq j\leq l$.

###### Proof.

First, it is straightforward to verify that $k$ is a transmission eigenvalue
associated with $(\Omega,\varepsilon)$ if and only if $k$ is a transmission
eigenvalue associated with each $(\Omega_{j},\varepsilon_{j})$ for
$j=1,2,\ldots,l$.

Next, by using (1.5), it is directly verified that the non-trivial functions
defined in (3.2) are a pair of transmission eigenfunctions associated with
$(\Omega,\varepsilon)$. On the other hand, if
$(\mathbf{E}_{1},\mathbf{E}_{2})$ are transmission eigenfunctions to (1.5), it
can be shown that
$(\mathbf{E}_{1}|_{\Omega_{j}},\mathbf{E}_{2}|_{\Omega_{j}})$ are transmission
eigenfunctions associated with $(\Omega_{j},\varepsilon_{j})$. Hence, one has

$(\mathbf{E}_{1}|_{\Omega_{j}},\mathbf{E}_{2}|_{\Omega_{j}})\in\mathcal{S}_{k,(\Omega_{j},\varepsilon_{j})}=\mathrm{Span}\\{(\mathbf{E}_{1}^{\alpha_{j}},\mathbf{E}_{2}^{\alpha_{j}})_{\alpha_{j}=1}^{p_{j}}\\},$

which readily implies that $\mathcal{S}_{k,(\Omega,\varepsilon)}$ is spanned
by the $p$ functions defined in (3.2).

The proof is complete.

∎

Consider the special example again as above by letting $\Omega$ be simply-
connected and $\widetilde{\Omega}$ be its copy. In order to further simplify
the exposition, we let $\varepsilon$ be constant. According to Theorem 3.2,
the multiplicity of the eigenvalue $k$ associated with
$(\Omega\cup\widetilde{\Omega},\varepsilon)$ doubles that of
$(\Omega,\varepsilon)$. Moreover, the corresponding eigenfunctions are given
by

$(\mathbf{E}_{1},\mathbf{E}_{2})\chi_{\Omega}+(\mathbf{0},\mathbf{0})\chi_{\widetilde{\Omega}},\quad(\mathbf{0},\mathbf{0})\chi_{\Omega}+(\widetilde{\mathbf{E}}_{1},\widetilde{\mathbf{E}}_{2})\chi_{\widetilde{\Omega}},$
(3.4)

as well as the corresponding linear combination of the above two types of
resonant modes. Here, $(\mathbf{E}_{1},\mathbf{E}_{2})$ are the resonant modes
associated with $(\Omega,\varepsilon)$ and
$(\widetilde{\mathbf{E}}_{1},\widetilde{\mathbf{E}}_{2})$ are the resonant
modes associated with $(\widetilde{\Omega},\varepsilon)$, and by Theorem 3.1,
$\widetilde{\mathbf{E}}_{j}(\mathbf{x})=\mathbf{E}_{j}(\mathbf{x}-\mathbf{z})$,
$\mathbf{x}\in\widetilde{\Omega}$. This fact shall also be used in our
subsequent study of the artificial mirage.

Finally, we would like to point out that Theorem 3.2 can easily extended to
the case with more general transmission eigenvalue problems as considered in
Theorem 3.1. However, we shall not explore this point further.

##  4\. Artificial electromagnetic mirage

In this section, we propose a novel and interesting application of the
geometrical and topological properties establish in the previous sections for
the transmission resonances. The practical scenario can be described as
follows.

Let $(\Omega,\varepsilon)$ be an optical object, where $\Omega$ is simply-
connected and $\varepsilon$ is constant. Our goal is to generate certain
incident electromagnetic field $(\mathbf{E}^{i},\mathbf{H}^{i})$ impinging
upon $(\Omega,\varepsilon)$ with the following properties:

  * •

$(\mathbf{E}^{i},\mathbf{H}^{i})$ is a solution to (1.1);

  * •

the propagation of the EM fields is nearly non-interrupted by the presence of
the physical object $(\Omega,\varepsilon)$;

  * •

the total wave field is localized on the boundary of a predefined set
$\widetilde{\Omega}\subset\mathbb{R}^{3}\backslash\bar{\Omega}$, where
$\widetilde{\Omega}=\Omega+\mathbf{z}$ is simply a translation of $\Omega$.
This copy is referred to as an artificial mirage image of the optical object
$(\Omega,\varepsilon)$.

Now we introduce the scheme to get the desired
$(\mathbf{E}^{i},\mathbf{H}^{i})$ given $\Omega,\varepsilon$ and $\mathbf{z}$.
First, we recall the Fourier extension of the EM fields, which is also known
as the Herglotz approximation. For a tangential vector field $\mathbf{g}\in
L^{2}(\mathbb{S}^{2})^{3}$ on the unit sphere, we define

$\mathbf{E}_{\mathbf{g}}(\mathbf{x}):=\int_{\mathbb{S}^{2}}e^{\mathrm{i}k\mathbf{x}\cdot\mathbf{d}}\mathbf{g}(\mathbf{d})\,ds(\mathbf{d}),\quad\mathbf{H}_{\mathbf{g}}(\mathbf{x}):=\frac{1}{\mathrm{i}k}\nabla\wedge\mathbf{E}_{\mathbf{g}}(\mathbf{x})$
(4.1)

to be the Herglotz fields associated with $\mathbf{g}$. For any regular domain
$\Sigma$ with a connected complement in $\mathbb{R}^{3}$, it is known that the
solution $(\mathbf{E},\mathbf{H})\in H(\mathrm{curl},\Sigma)\times
H(\mathrm{curl},\Sigma)$ to the following system

$\nabla\wedge\mathbf{E}-\mathrm{i}k\mathbf{H}=\mathbf{0},\quad\nabla\wedge\mathbf{H}+\mathrm{i}k\mathbf{E}=\mathbf{0}\quad\mbox{in}\
\Sigma,$ (4.2)

can be approximated by the Herglotz fields to an arbitrary accuracy; see [31].
It is noted that the Herglotz approximation result in [31] requires a bit
stronger regularities on the EM fields in (4.2), which we shall always assume
in our subsequent discussion.

Recall that $(\widetilde{\Omega},\varepsilon)$ is an exact copy of
$(\Omega,\varepsilon)$, where $\widetilde{\Omega}:=\mathbf{z}+\Omega$ and
$\mathbf{z}\in\mathbb{R}^{3}$. Consider the transmission eigenvalue problem
(1.5) associated with
$(\Omega,\varepsilon)\cup(\widetilde{\Omega},\varepsilon)$. By Theorem 3.2,
and in particular the example discussed after the theorem, we know that the
corresponding transmission eigenfunctions are given by (3.4). By the
geometrical results in Theorem 2.3 in Section 2, we know that there exist
infinitely many eigenfunctions associated with $(\Omega,\varepsilon)$ which
are surface-localized around $\partial\Omega$. Hence, we can choose one of
such resonant modes such that $\mathbf{E}_{2}$ is surface-localized around
$\partial\Omega$ and so $\widetilde{\mathbf{E}}_{2}$ is surface-localized
around $\partial\widetilde{\Omega}$, (see Fig. 5 for a schematic
illustration).

Figure 5.  Contour plot of the transmission eigenfunction that is vanished on
the left domain but localized on the right domain.

Next, we consider the second resonant mode in (3.4) and let
${\mathbf{E}}_{\mathbf{g}}^{i}$ be the Herglotz field associated with
$\mathbf{0}\cdot\chi_{\Omega}+\widetilde{\mathbf{E}}_{2}\cdot\chi_{\widetilde{\Omega}}$,
i.e.

$\mathbf{E}^{i}_{\mathbf{g}}\approx\mathbf{0}\cdot\chi_{\Omega}+\widetilde{\mathbf{E}}_{2}\cdot\chi_{\widetilde{\Omega}}\quad\mbox{in}\
\ \Omega\cup\widetilde{\Omega}.$ (4.3)

Let $\mathbf{H}_{\mathbf{g}}^{i}$ be given according to the second formula in
(4.1) associated with $\mathbf{E}_{\mathbf{g}}^{i}$. Clearly,
$(\mathbf{E}_{\mathbf{g}}^{i},\mathbf{H}_{\mathbf{g}}^{i})$ are entire
solutions to (1.1), and can be used as incident fields, namely

$(\mathbf{E}^{i},\mathbf{H}^{i})=(\mathbf{E}_{\mathbf{g}}^{i},\mathbf{H}_{\mathbf{g}}^{i}).$
(4.4)

Next, we consider the electromagnetic scattering problem (1.2) associated with
the optical object $(\Omega,\varepsilon)$ and incident field
$(\mathbf{E}^{i},\mathbf{H}^{i})$. It is straightforward to deduce from (1.1)
and (1.2) that the corresponding scattering field
$(\mathbf{E}^{s},\mathbf{H}^{s})$ satisfies

$\left\\{\begin{aligned}
&~{}~{}\nabla\wedge\mathbf{E}^{s}-\mathrm{i}k\mathbf{H}^{s}=\mathbf{0}&&\mbox{in}\
\mathbb{R}^{3},\\\
&~{}~{}\nabla\wedge\mathbf{H}^{s}+\mathrm{i}k\varepsilon\mathbf{E}^{s}=\mathrm{i}k(1-\varepsilon)\mathbf{E}^{i}&&\mbox{in}\
\mathbb{R}^{3},\\\
&\lim_{|{\mathbf{x}}|\rightarrow\infty}(\mathbf{H}^{s}\wedge{\mathbf{x}}-|{\mathbf{x}}|\mathbf{E}^{s})={\mathbf{0}}.\end{aligned}\right.$
(4.5)

By (4.3), we see that

$\mathrm{i}k(1-\varepsilon)\mathbf{E}^{i}\approx\mathbf{0}\ \ \mbox{in}\
\mathbb{R}^{3}.$

Therefore, by the well-posedness of the forward scattering system (4.5), we
readily conclude that

$\mathbf{E}^{s},\mathbf{H}^{s}\approx\mathbf{0}\quad\mbox{in}\ \
\mathbb{R}^{3}\backslash\overline{\Omega}.$

That is, for the total fields $\mathbf{E}$ and $\mathbf{H}$, one has

$(\mathbf{E},\mathbf{H})\approx(\mathbf{E}_{\mathbf{g}}^{i},\mathbf{H}_{\mathbf{g}}^{i})\quad\mbox{in}\
\ \mathbb{R}^{3}\backslash\overline{\Omega}.$ (4.6)

Finally, by (4.3) and (4.6), we clearly have that the total electric field
$\mathbf{E}$ is localized around $\partial\widetilde{\Omega}$. That is, a
“shining” area, i.e. $\partial\widetilde{\Omega}$, is produced and the
electromagnetic fields inside $\widetilde{\Omega}$ are nearly zero.

To summarize, given an optical object $(\Omega,\varepsilon)$, if one uses the
incident fields designed in (4.3) and (4.4) to illuminate the object, an
artificial mirage copy $(\widetilde{\Omega},\varepsilon)$ will be produced. In
what follows, we shall provide a few examples to illustrate the artificial
mirage effect. However, in order to obtain eminent surface-localizing
behaviour of the transmission eigenfunctions, one needs to compute the
transmission eigenfunctions associated to (1.5) with either very large
$\varepsilon$ or very large $k$. This requires extensive and heavy costs in
numerically computing the three-dimensional transmission eigenvalue problem
(1.5) as well as the corresponding scattering system (1.2). Due to limited
computational resources, we could only afford to conduct the two-dimensional
simulations, which correspond to the TM (transverse magnetic) scattering.

Figure 6.  Contour plots of the wave field for a disk with different
frequencies. (a) $k=7.38$, (b) $k=8.66$, (c) $k=10.08$, (d) $k=14.09$.

######  Example 3.

In this example, we present a mirage phenomenon for a unit disk centering at
$(-2,0)$. The electric permittivity is given by $\epsilon_{0}=4$ in the disk
and $\epsilon_{0}=1$ outside the disk. Figure 6 shows total field with
different frequencies, i.e., transmission eigenvalues, where the solid black
line denotes the exact disk and the dotted red denotes the virtual disk. One
can easily observe that the wave field is surface-localized around the
boundary virtual disk, and the virtual object could be easily identified. By
comparing the results in Figure 6, we can observe that the total field has a
better localized result when the wavenumber is larger.

######  Example 4.

In this example, we are concerned with the mirage phenomenon for a more
general case. Let $\Omega$ be the triangle shaped domain which is described by
the parametric representation

$\mathbf{x}(t)=0.5(2+0.2\cos 3t)(\cos t,\ \sin t)\quad t\in[0,\,2\pi],$

where the center is given by $(-2,0)$ and permittivity $\epsilon_{0}=4$ for
$\mathbf{x}\in\Omega$. Figure 7 presents the wave field around the exact
domain (solid black line) and virtual domain (dotted red line). It
demonstrates very good imaging performance of the mirage for general convex
domain.

Figure 7.  Contour plots of the wave field for a triangle-shaped domain with
different frequencies. (a) $k=13.25$, (b) $k=16.84$.

## Acknowledgment

The work of Y. Deng was supported by NSF grant of China No. 11971487 and NSF
grant of Hunan No. 2020JJ2038. The work of H. Liu was supported by a startup
grant from City University of Hong Kong and Hong Kong RGC General Research
Funds (projects 12301218, 12302919 and 12301420). The work of X. Wang was
supported by the Hong Kong Scholars Program grant under No. XJ2019005 and NSF
grant of China(No. 11971133 and No. 12001140). The work of W. Wu was supported
by the tier-2 startup grant from Hong Kong Baptist University and Hong Kong
RGC General Research Funds (projects 12302219 and 12300520).

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