# Topologically protected modes in dispersive materials: the case of undamped
systems

Konstantinos Alexopoulos Department of Mathematics, ETH Zurich, Rämistrasse
101, CH-8092 Zurich, Switzerland. Bryn Davies Department of Mathematics,
Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom.

###### Abstract

This work extends the theory of topological protection to dispersive systems.
This theory has emerged from the field of topological insulators and has been
established for continuum models in both classical and quantum settings. It
predicts the existence of localised interface modes based on associated
topological indices and shows that, when such modes exist, they benefit from
enhanced robustness with respect to imperfections. This makes topologically
protected modes an ideal starting point for building wave guiding devices.
However, in many practical applications such as optics or locally resonant
meta-structures, materials are dispersive in the operating frequency range. In
this case, the associated spectral theory is less straightforward. This work
shows that the existing theory of topological protection can be extended to
dispersive settings. We consider time-harmonic waves in one-dimensional
systems with no damping.

###### Contents

  1. 1 Introduction
  2. 2 Mathematical setting
  3. 3 Interface mode existence
    1. 3.1 Impedance functions
    2. 3.2 Dispersion relation
    3. 3.3 Mirror symmetry
    4. 3.4 Bulk index
    5. 3.5 Frequency existence
      1. 3.5.1 Impedance evaluation
      2. 3.5.2 Impedance property
      3. 3.5.3 Interface mode existence
  4. 4 Zak phase
  5. 5 Asymptotic behaviour
    1. 5.1 Transfer matrix method
      1. 5.1.1 Spectral properties
      2. 5.1.2 Eigenmode decay
  6. 6 Robustness with respect to imperfections
    1. 6.1 Theoretical results
      1. 6.1.1 Permittivity perturbation
      2. 6.1.2 Symmetry perturbation
    2. 6.2 Numerical results
      1. 6.2.1 Permittivity perturbation
      2. 6.2.2 Symmetry perturbation
  7. 7 Conclusion
  8. A Proofs of impedance monotonicity
    1. A.1 Proof of Theorem 3.12
    2. A.2 Proof of Theorem 6.2
  9. B Characterisation of eigenmode symmetries

##  1 Introduction

The existence of strongly localised existence eigenmodes in perturbed periodic
media is one of the foundations of modern wave physics. Such modes allow for
waves of specific frequencies to be strongly localised at desired locations
and are the starting point for the design of many wave guiding and control
devices.

The pre-eiminent theory in the field of periodic waveguides is the notion of
topological protection [12]. This field was inspired by ideas related to the
famous quantum hall effect [13] and was successfully translated to classical
wave systems around 15 years ago [20]. This theory uses topological indices
associated to the underlying periodic structures [5] to predict the existence
of localised edge modes at interfaces and edges of the material. Further, the
invariant nature of topological indices means we can expect that such modes
experience enhanced robustness with respect to imperfections. Given how
desirable a priori robustness properties are for being able to confidently
manufacture functional devices, topological waveguides has emerged as an
important sub-field of wave physics [17].

Alongside the rapid developments in the physical literature, a corresponding
mathematical theory for topologically protected modes has been developed. This
theory first emerged in the setting of the Schrödinger equation [10, 9], but
has since been extended to classical wave systems [15, 6]. Many of the seminal
mathematical studies consider one-dimensional systems, since the fundamental
mechanism requires an interface or edge to be introduced in just one axis of
periodicity. However, multi-dimensional systems can be studied by using _e.g._
integral operators [2, 3] or a reduction to Dirac operators [11, 8, 4].

The existing mathematical theory mostly studies systems with frequency-
independent material parameters. This is not only a natural toy model but
gives a good description of several physical settings (_e.g._ microwaves and
perfect conductors). However, many important materials are dispersive in
desirable frequency ranges. This includes locally resonant systems (_e.g._
coupled Helmholtz resonators [21]) and most metals at optical frequencies.
This work extends the mathematical theory of topologically protected modes to
dispersive settings. The main theoretical challenge for doing so is that the
spectrum of the differential operator can no longer be understood using
standard linear eigenvalue theory. Instead, the spectrum is obtained through
the solution of a non-linear eigenvalue problem. Further, many important
dispersive materials have poles and singularities at certain frequencies [1,
19].

In this work, we will focus on materials whose permittivity is dispersive and
real-valued (meaning there is no damping in the system). We exploit the work
of [1] which characterised the Bloch spectrum as a function of the
singularities of canonical dispersive permittivities. The main idea of this
work is to combine this theory with the work of [6] (and the previous works
upon which it builds, _e.g._ [10, 15]), which uses the surface impedance to
characterise the existence and topological protection of interface modes.

In Section 2, we establish the mathematical setting describing the problem and
the geometry governing the system. In Section 3, we define the concept of the
impedance function and, exploiting the properties of the mirror symmetry of
the periodically repeated cells, we generalise the monotonicity argument used
in [6] and we show that in each band-gap, there exists a unique non-zero
frequency for which an interface mode exists. Then, in Section 4, we define
the Zak phase and we show why it is an invariant property of the system. In
Section 5, using the transfer matrix method, we show the existence of an
asymptotically decaying mode at infinity and we provide the condition that
needs to be satisfied for that mode to unique. Finally, in Section 6, we study
how this interface mode is affected by perturbations in the material
parameters or the symmetry of the system.

##  2 Mathematical setting

Let us first present the mathematical setting in which we will work. We will
consider materials composed of periodically repeating unit cells, within which
the permittivity is allowed to vary as a function of both frequency and
position. A crucial assumption will be that each unit cell is mirror
symmetric. That is, the material’s permittivity is described by a function
$\epsilon(x,\omega)$ which is periodic
($\epsilon(x,\cdot)=\epsilon(x+1,\cdot)$) and symmetric
($\epsilon(x+h,\cdot)=\epsilon(x+1-h,\cdot)$) in its first variable. Further,
our analysis will require an assumption that $\epsilon(\cdot,\omega)$ is a
piecewise non-decreasing function of $\omega$, in the sense that
$\frac{\mathrm{d}\varepsilon}{\mathrm{d}\omega}\geq 0$ whenever the derivative
exists.

In what follows, in order to ease the notation, our setting will be
constituted by frequency dispersive permittivities, piecewise constant with
respect to the space variable. Although, we note that our analysis is the same
and the results still hold for permittivities which respect the mirror
symmetry as mentioned above.

We consider two dispersive materials $A$ and $B$. Each one is of the form of a
semi-infinite array. The arrays are glued together at $x_{0}=0$ and we assume
that the material $A$ expands towards $-\infty$ and the material $B$ expands
towards $+\infty$. In addition, we assume that each material is constructed by
repeating periodically a unit cell. Each unit cell is the product of layering,
i.e. it has particles of two different permittivities $\varepsilon_{1}$ and
$\varepsilon_{2}$. A key assumption is that each permittivity is a dispersive,
continuous function of the frequency $\omega\in\mathbb{R}$ of the system. Let
us denote by $D^{[1]}_{i}$, $i=1,2,\dots,N+1,$ and by $D^{[2]}_{j}$,
$j=1,2,\dots,N,$ the particles of the unit cell with permittivity
$\varepsilon_{1}$ and $\varepsilon_{2}$, respectively. Then, the permittivity
of the system is defined as follows:

$\displaystyle\varepsilon(x,\omega)=\begin{cases}\varepsilon_{1}(\omega),&x\in
D^{[1]}:=\bigcup_{i=1}^{N+1}D^{[1]}_{i},\\\ \varepsilon_{2}(\omega),&x\in
D^{[2]}:=\bigcup_{i=1}^{N}D^{[2]}_{i}.\end{cases}$ (2.1)

We define the sequence $\\{x_{n}\\}_{n\in\mathbb{Z}}$ to be the set of
endpoints of each one of the periodically repeated cells. We take $x_{0}$ to
be the glue point of the two materials and so for $n>0$ we are in material $B$
and for $n<0$, we are in material $A$.

In Figure 1 we provide a schematic depiction of such an example of periodic
cells for the materials $A$ and $B$, respectively. Concerning the construction
characteristics of the periodic cell, for materials $A$ and $B$, we have $3$
particles of permittivity $\varepsilon_{1}$, i.e. $D^{[1]}_{1},D^{[1]}_{2}$
and $D^{[1]}_{3}$, and $2$ particles of permittivity $\varepsilon_{2}$, i.e.
$D^{[2]}_{1}$ and $D^{[2]}_{2}$. We note that our result can be generalised in
settings where the number of particles in the periodic cells is greater. We
also notice the geometry of each one of the periodic cells, i.e. the symmetric
way in which the particles are placed in each periodic cell.

Figure 1: An example of the setting studied. In the unit cell of each material
we have 3 particles of permittivity $\varepsilon_{1}$ and 2 particles of
permittivity $\varepsilon_{2}$. We notice the mirror symmetric way in which
the particles are placed inside the periodic cells. Each material is
constituted by a semi-infinite array created by periodically repeating the
unit cell. Our structure is the result of gluing materials $A$ and $B$ at the
point $x_{0}$.

We are interested in finding eigenvalues

$\displaystyle\mathcal{L}u=\omega^{2}u$ (2.2)

of the ordinary differential operator

$\mathcal{L}u:=-\frac{1}{\mu_{0}}\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{\varepsilon(x,\omega)}\frac{\mathrm{d}u}{\mathrm{d}x}\right).$
(2.3)

In particular, the eigenmodes of interest are those which decay as
$|x|\rightarrow\infty$, such that they are localised in a neighbourhood of the
interface at $x_{0}=0$.

To find candidate eigenvalues $\omega^{2}$ for which localised eigenmodes can
occur, it is valuable to consider the Floquet-Bloch spectrum of the operators
associated to materials A and B. This is the set of eigenmodes which satisfy
the Floquet-Bloch boundary conditions

$\displaystyle
u(x+1)=e^{i\kappa}u(x)\quad\text{and}\quad\frac{\mathrm{d}u}{\mathrm{d}x}(x+1)=e^{i\kappa}\frac{\mathrm{d}u}{\mathrm{d}x}(x),$
(2.4)

for some $\kappa\in\mathcal{B}:=[-\pi,\pi]$. Hence, these eigenmodes belong to
the space of functions

$\displaystyle L^{2}_{\kappa}:=\Big{\\{}f\in L^{2}_{\mathrm{loc}}:\
f(x+1)=e^{i\kappa}f(x),\ \kappa\in\mathbb{R}\Big{\\}}.$

It follows that the localised eigenmodes that we are looking for, have to be
in band gaps of the two materials. This happens because we are looking for
solutions to (2.3), which decay as $|x|\to+\infty$. Thus, if the modes are in
the bands, then from (2.4), we see that their magnitude is preserved. Let us
also note here that we are interested in materials $A$ and $B$ which have
overlapping band gaps, otherwise such modes cannot exist in this setting.

On its own, studying the spectrum of the operator $\mathcal{L}$ is a
complicated task. This is due to the fact that there is a non-linear
dependence of the operator on the eigenvalues $\omega^{2}$ via
$\varepsilon(x,\omega)$. In addition, the permittivity function $\varepsilon$
may present certain aspect, e.g. poles, which make this analysis even harder.
In [1], we have studied how these properties of the permittivity function
affect the structure of the spectrum of the differential operator
$\mathcal{L}$. In general, its spectrum will consist of countably many
spectral bands. Unusual behaviour can occur near to the poles of the
singularity, however, where there can be a countably infinite number of bands
clustered within a finite region of the pole.

We will argue in the same way as in [6]. We will generalise their results to
our dispersive setting (which differs from their dispersive example as the
dispersion appears in the electric permittivity, rather than the magnetic
permeability, in (2.3)). We will include proofs of most statements for
completeness, although some are straighforward generalisations of [6]. We also
note that in the case where the permittivity $\varepsilon$ no longer depends
on the frequency $\omega$ but only on $x$, then we are in a similar setting as
in [15].

##  3 Interface mode existence

The main theoretical challenge is to show that an interface mode exists for
the structure composed of the two materials.

###  3.1 Impedance functions

Let us see a sufficient condition that needs to be satisfied for the interface
mode to exist. We introduce the following notation.

######  Definition 3.1.

Let $f:\mathbb{R}\to\mathbb{C}$ be a continuous function and let
$\alpha\in\mathbb{R}$. Then, we define

$\displaystyle f(\alpha^{-}):=\lim_{x\uparrow\alpha}f(x)\ \ \ \text{ and }\ \
\ f(\alpha^{+}):=\lim_{x\downarrow\alpha}f(x).$

The condition that needs to be satisfied for the interface mode to exist is
continuity of both the solution and its derivative at the interface. Using the
above notation, the continuity conditions that needs to be satisfied at the
interface $x_{0}$ can be written as

$\displaystyle u(x_{0}^{-})=u(x_{0}^{+})\ \ \ \text{ and }\ \ \
\left(\frac{1}{\varepsilon_{1}(\omega)}\frac{\mathrm{d}u}{\mathrm{d}x}\right)(x_{0}^{-})=\left(\frac{1}{\varepsilon_{1}(\omega)}\frac{\mathrm{d}u}{\mathrm{d}x}\right)(x_{0}^{+}).$
(3.1)

We can now give the definition for the impedance functions.

######  Definition 3.2 (Surface impedances).

We define the surface impedances of the problem (2.2), as follows:

$\displaystyle
Z^{-}(\omega)=-\frac{u(x_{0}^{-})}{\frac{1}{\varepsilon_{1}(\omega)}\frac{\mathrm{d}}{\mathrm{d}x}u(x_{0}^{-})}\
\ \ \text{ and }\ \ \
Z^{+}(\omega)=\frac{u(x_{0}^{+})}{\frac{1}{\varepsilon_{1}(\omega)}\frac{\mathrm{d}}{\mathrm{d}x}u(x_{0}^{+})},\
\ \ \omega\in\mathbb{R}.$ (3.2)

Then, the following result is a direct consequence of the continuity
conditions (3.1).

######  Lemma 3.3.

Let $\mathfrak{A}\subset\mathbb{R}$. Then, a necessary and sufficient
conditions for the existence of an interface localised mode in $\mathfrak{A}$
is

$\displaystyle Z^{+}(\omega)+Z^{-}(\omega)=0,\ \ \ \omega\in\mathfrak{A}.$
(3.3)

The question that rises naturally is whether a frequency $\omega\in\mathbb{R}$
exists such that this condition is satisfied and so, the global mode exists.

###  3.2 Dispersion relation

The quasiperiodic Helmholtz problem (2.3)-(2.4) is characterised by a
dispersion relation. This is an expression which relates the quasiperiodicity
$\kappa$ with the frequency $\omega$, such that it describes the spectral
bands $\omega=\omega_{n}(\kappa)$. It is an equation of the form:

$\displaystyle 2\cos(\kappa)=f(\omega),$ (3.4)

where $f:\mathbb{R}\rightarrow\mathbb{R}$ is a function that depends on the
material parameters and the system’s geometry. This function is variously
known as the discriminant or Lyapunov function of the operator $\Lambda$ [14].
Some examples are provided in _e.g._ [1, 16].

The analysis that follows stems from understanding the symmetries of the Bloch
modes at the edged of the band gaps. Hence, it will be helpful to know that
the maxima and minima of any band must occur at $\kappa=0$ or $\kappa=\pm\pi$.
This follows from the fact that the spectral bands are monotonic functions of
$\kappa$ within the reduced Brillouin zone $[0,\pi]$.

######  Lemma 3.4 (Frequency monotonicity).

It holds that the frequency $\omega\in\mathbb{R}$ satisfying the (3.4), as a
function of the quasiperiodicity $\kappa$, is a monotonic function in the
reduced Brillouin zone $[0,\pi]$.

###### Proof.

Indeed, we differentiate (3.4) with respect to the quasiperiodicity $\kappa$
and we get

$\displaystyle-2\sin(\kappa)=\frac{\mathrm{d}f}{\mathrm{d}\omega}\frac{\mathrm{d}\omega}{\mathrm{d}\kappa}.$

We know that in $[0,\pi]$, it holds $-2\sin(\kappa)\leq 0$ with the zeros
occurring in $\\{0,\pi\\}$. It is a direct consequence of this that
$\frac{\mathrm{d}\omega}{\mathrm{d}\kappa}$ keeps a constant sign in
$(0,\pi)$. Also, if $\frac{\mathrm{d}\omega}{\mathrm{d}\kappa}(\kappa)=0$,
then $\kappa\in\\{0,\pi\\}$. Thus, $\omega(\kappa)$ is monotonic in the
reduced Brillouin zone. This concludes the proof. ∎

######  Remark.

We note that the above property holds for any sufficiently smooth function
$f:\mathbb{R}\to\mathbb{R}$.

Thus, since $\omega$ is a monotonic function of $\kappa$ in the reduced
Brillouin zone, it follows that the edges of the spectrum are at $0$ and
$\pi$.

###  3.3 Mirror symmetry

Let us now focus on the geometry inside the periodic cells. The first thing we
observe is the mirror symmetry which is established in the periodic cells of
both materials. Indeed, we observe that for $x\in[0,1]$, we have

$\displaystyle\varepsilon(x,\omega)=\varepsilon(1-x,\omega).$ (3.5)

We introduce the parity operator $\mathcal{P}$, which acts on a function $f$
as follows:

$\displaystyle(\mathcal{P}f)(x):=f(1-x).$

Then, the mirror symmetry implies

$\displaystyle\mathcal{P}\mathcal{L}(-\kappa,\omega)=\mathcal{L}(\kappa,\omega)\mathcal{P},\
\ \ \forall\kappa\in\mathcal{B}.$ (3.6)

As a direct consequence of (3.6), we have the following lemmas.

######  Lemma 3.5.

Let $\omega\in\mathbb{R}$. Then:

  * •

$\mathcal{L}(-\pi,\omega)=\mathcal{L}(\pi,\omega)$.

  * •

The differential operator $\mathcal{L}$ and the parity operator $\mathcal{P}$
commute at $\kappa\in\\{0,\pm\pi\\}$.

###### Proof.

We divide this proof in two parts, one for each result.

  * •

We know that $\omega_{n}(-\kappa)=\omega_{n}(\kappa)$. Then, from (2.3), it
follows directly that $\mathcal{L}(-\pi,\omega)=\mathcal{L}(\pi,\omega)$.

  * •

From (3.6), we get

$\displaystyle\mathcal{P}\mathcal{L}(0,\omega)=\mathcal{L}(0,\omega)\mathcal{P}.$

We also get

$\displaystyle\mathcal{P}\mathcal{L}(-\pi,\omega)=\mathcal{L}(\pi,\omega)\mathcal{P}.$

We have shown that $\mathcal{L}(-\pi,\omega)=\mathcal{L}(\pi,\omega)$. This
gives

$\displaystyle\mathcal{P}\mathcal{L}(\pi,\omega)=\mathcal{L}(\pi,\omega)\mathcal{P}\
\ \ \text{ and }\ \ \
\mathcal{P}\mathcal{L}(-\pi,\omega)=\mathcal{L}(-\pi,\omega)\mathcal{P},$

which is the desired result.

This concludes the proof. ∎

In order to show the dependence of a solution $u$ on the spatial variable $x$
but also on both the quasiperiodicity $\kappa$ and the frequency $\omega$, we
will use the notation $u^{[\kappa]}(x,\omega)$.

######  Lemma 3.6.

Let $\lambda_{n}(\omega_{n}):=\omega^{2}_{n}(\kappa)$ be an eigenvalue of
$\mathcal{L}(\kappa,\omega_{n})$, with $u_{n}^{[\kappa]}(x,\omega_{n})$ being
the associated eigenvector. Then:

  * •

$\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n})$ is an eigenvector
$\mathcal{L}(-\kappa,\omega_{n})$ with the same eigenvalue
$\lambda_{n}(\kappa)$.

  * •

$u_{n}^{[-\kappa]}(x,\omega_{n})$ is also an eigenvector of
$\mathcal{L}(-\kappa,\omega_{n})$ with eigenvalue $\lambda_{n}(\kappa)$.

###### Proof.

We divide this proof in two parts, one for each result.

  * •

We observe that (3.6) is symmetric in term of $\kappa\in\mathcal{B}$, in the
following sense: for all $\kappa\in\mathcal{B}$, it holds that
$-\kappa\in\mathcal{B}$, since we have taken $\mathcal{B}=[-\pi,\pi]$. Thus,
replacing $\kappa$ by $-\kappa$ in (3.6), we get

$\displaystyle\mathcal{P}\mathcal{L}(\kappa,\omega)=\mathcal{L}(-\kappa,\omega)\mathcal{P}.$
(3.7)

We also know that

$\displaystyle\mathcal{L}(\kappa,\omega_{n})u_{n}^{[\kappa]}(x,\omega_{n})=\lambda_{n}(\omega_{n})u_{n}^{[\kappa]}(x,\omega_{n}).$

Hence,

$\displaystyle\mathcal{P}\mathcal{L}(\kappa,\omega_{n})u_{n}^{[\kappa]}(x,\omega_{n})=\mathcal{P}\lambda_{n}(\omega_{n})u_{n}^{[\kappa]}(x,\omega_{n})=\lambda_{n}(\omega_{n})\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n}).$

So, from (3.7), we get

$\displaystyle\mathcal{L}(-\kappa,\omega_{n})\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n})=\lambda_{n}(\omega_{n})\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n}),$

which gives the desired result.

  * •

Let $\lambda_{n}(-\kappa,\omega_{n})$ be an eigenvalue of
$\mathcal{L}(-\kappa,\omega_{n})$, with $u_{n}^{[-\kappa]}(x,\omega_{n})$
being the associated eigenvector, for $\omega_{n}\in\mathbb{R}$ and
$\kappa\in\mathcal{B}$. This implies that

$\displaystyle\mathcal{L}(-\kappa,\omega_{n})u_{n}^{[-\kappa]}(x,\omega_{n})=\lambda_{n}(-\kappa,\omega_{n})u_{n}^{[-\kappa]}(x,\omega_{n}).$

Although, we know that $\omega_{n}(-\kappa)=\omega_{n}(\kappa)$ and we have
that $\lambda_{n}(\kappa,\omega_{n})=\omega_{n}^{2}(\kappa)$. Hence, we get

$\displaystyle\lambda_{n}(\kappa,\omega_{n})=\lambda_{n}(-\kappa,\omega_{n}),$

which gives the desired result.

This concludes the proof. ∎

######  Lemma 3.7.

Let $\lambda_{n}(\kappa,\omega_{n})$ be a non-degenerate eigenvalue of
$\mathcal{L}(\kappa,\omega_{n})$ with $u_{n}^{[\kappa]}(x,\omega_{n})$ being
the associated eigenvector, for $\kappa$ at the band edges, i.e.
$\kappa\in\\{0,\pm\pi\\}$. Then,

$\displaystyle
u_{n}^{[\kappa]}(x,\omega_{n})=\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n})\ \ \
\text{ or }\ \ \
u_{n}^{[\kappa]}(x,\omega_{n})=-\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n}),\ \
\ \text{ for }\kappa\in\\{0,\pm\pi\\}.$

###### Proof.

Since we take $\lambda_{n}(\kappa,\omega_{n})$ to be non-degenerate, there
exists $\mu\in\mathbb{R}$ such that

$\displaystyle
u_{n}^{[-\kappa]}(x,\omega_{n})=\mu\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n}).$
(3.8)

Let us take $\kappa$ to be at the band edges $\kappa\in\\{0,\pm\pi\\}$. We get
for $\kappa=0$

$\displaystyle
u_{n}^{[0]}(x,\omega_{n})=\mu\mathcal{P}u_{n}^{[0]}(x,\omega_{n}).$

We know that $\mathcal{P}\mathcal{P}=Id$. Thus, applying $\mathcal{P}$ on both
sides, we get

$\displaystyle\mathcal{P}u_{n}^{[0]}(x,\omega_{n})=\mu
u_{n}^{[0]}(x,\omega_{n}),$

which gives

$\displaystyle
u_{n}^{[0]}(x,\omega_{n})=\mu^{2}u_{n}^{[0]}(x,\omega_{n})\Rightarrow\mu^{2}=1.$

Thus, either $\mu=1$, in which case
$\mathcal{P}u_{n}^{[0]}(x,\omega_{n})=u_{n}^{[0]}(x,\omega_{n})$, or $\mu=-1$,
in which case
$\mathcal{P}u_{n}^{[0]}(x,\omega_{n})=-u_{n}^{[0]}(x,\omega_{n})$.
Let us now take $\kappa=-\pi$. We have

$\displaystyle
u_{n}^{[\pi]}(x,\omega_{n})=\mu\mathcal{P}u_{n}^{[-\pi]}(x,\omega_{n}).$

Taking $\kappa=\pi$, we have

$\displaystyle
u_{n}^{[-\pi]}(x,\omega_{n})=\mu\mathcal{P}u_{n}^{[\pi]}(x,\omega_{n}).$

Combining these two expressions, we obtain

$\displaystyle
u_{n}^{[\pi]}(x,\omega_{n})=\mu^{2}\mathcal{P}u_{n}^{[\pi]}(x,\omega_{n})\Rightarrow\mu^{2}=1$

and

$\displaystyle
u_{n}^{[-\pi]}(x,\omega_{n})=\mu^{2}\mathcal{P}u_{n}^{[-\pi]}(x,\omega_{n})\Rightarrow\mu^{2}=1.$

Thus, either $\mu=1$, in which case
$\mathcal{P}u_{n}^{[\pi]}(x,\omega_{n})=u_{n}^{[\pi]}(x,\omega_{n})$ and
$\mathcal{P}u_{n}^{[-\pi]}(x,\omega_{n})=u_{n}^{[-\pi]}(x,\omega_{n})$, or
$\mu=-1$, in which case
$\mathcal{P}u_{n}^{[\pi]}(x,\omega_{n})=-u_{n}^{[\pi]}(x,\omega_{n})$ and
$\mathcal{P}u_{n}^{[-\pi]}(x,\omega_{n})=-u_{n}^{[-\pi]}(x,\omega_{n})$. This
concludes the proof. ∎

We will call a function symmetric if $f=\mathcal{P}f$ and anti-symmetric if
$f=-\mathcal{P}f$. Then, it is well known that due to the symmetry of the unit
cell, combined with the periodicity or anti-periodicity that occurs when
$\kappa\in\\{0,\pi\\}$, the modes at the edges of the bands must be either
symmetric or anti-symmetric (see also [6]).

######  Lemma 3.8.

Let $\kappa\in\\{0,\pi\\}$ and let $u_{n}^{\prime}$ denote the spatial
derivative of $u_{n}$. Then, either

$u_{n}$ is symmetric and $u^{\prime}_{n}$ is anti-symmetric,

or

$u_{n}$ is anti-symmetric and $u^{\prime}_{n}$ is symmetric.

###### Proof.

Let us consider $\kappa$ at the band edges $0$ and $\pi$. Then, we know that
either
$u_{n}^{[\kappa]}(x,\omega_{n})=\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n})$ or
$u_{n}^{[\kappa]}(x,\omega_{n})=-\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n})$.
In the first case, we see that $u_{n}$ is symmetric. In addition,

$\displaystyle
u_{n}^{[\kappa]}(x,\omega_{n})=\mathcal{P}u_{n}^{[\kappa]}(x,\omega_{n})=u_{n}^{[\kappa]}(1-x,\omega_{n})\Rightarrow$
$\displaystyle\Big{(}u_{n}^{[\kappa]}\Big{)}^{\prime}(x,\omega_{n})=-\Big{(}u_{n}^{[\kappa]}\Big{)}^{\prime}(1-x,\omega_{n})=-\mathcal{P}\Big{(}u_{n}^{[\kappa]}\Big{)}^{\prime}(x,\omega_{n}),$

which shows that $u^{\prime}_{n}$ is anti-symmetric. Similarly, in the second
case, we have that $u^{\prime}_{n}$ is anti-symmetric. Then,

$\displaystyle
u_{n}^{[\kappa]}(x,\omega_{n})=-u_{n}^{[\kappa]}(1-x,\omega_{n})\Rightarrow\Big{(}u_{n}^{[\kappa]}\Big{)}^{\prime}(x,\omega_{n})=\Big{(}u_{n}^{[\kappa]}\Big{)}^{\prime}(x,\omega_{n})=\mathcal{P}\Big{(}u_{n}^{[\kappa]}\Big{)}^{\prime}(x,\omega_{n}),$

which shows that $u^{\prime}_{n}$ is symmetric. ∎

Since the modes are labeled by $n$, from now on, in order to ease the
notation, we will suppress the dependence on the frequency $\omega_{n}$, since
it appears indirectly at the subscripts, i.e.
$u_{n}^{[\kappa]}(x,\omega_{n})\equiv u_{n}^{[\kappa]}(x)$. A consequence of
the symmetries characterised in Lemma 3.8 is that the modes at the edges of
the spectral bands may have critical points at $x=0$, which can be
characterised based on the modes’ symmetries. These symmetry arguments are
similar to those in [6]. A specific proof for our setting is given in Appendix
B.

######  Lemma 3.9.

Let $\kappa\in\\{0,\pi\\}$. Then, either

$u_{n}^{[\pi]}(0)=0\quad\text{and}\quad\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(0)=0,$

or

$u_{n}^{[0]}(0)=0\quad\text{and}\quad\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(0)=0.$

###  3.4 Bulk index

Let us now define the notion of the bulk index. This is a topological property
of the material which depends on the symmetry of the mode at the edges of a
band.

######  Definition 3.10 (Bulk index).

Let $\mathfrak{S}:=[a,b]$ denote the $n$-th band gap. Then we define the bulk
topological index $\mathcal{J}_{n}$ of the $n$-th band gap as follows:

$\displaystyle\mathcal{J}_{n}:=\begin{cases}+1,\ \ \ \text{ if }u_{n}\text{ is
symmetric at $a$},\\\ -1,\ \ \ \text{ if }u_{n}\text{ is anti-symmetric at
$a$}.\end{cases}$ (3.9)

###  3.5 Frequency existence

Using the symmetry properties that we have proved, we will show that there
exists $\omega\in\mathbb{R}$, such that (3.3) is satisfied.

####  3.5.1 Impedance evaluation

Let us now see what the impedance functions look like at the band edges
$\kappa\in\\{0,\pi\\}$. We introduce the following notation:

$\displaystyle Z^{\pm}_{\kappa}:=Z^{\pm}\text{ at }\kappa,$

for $\kappa\in\mathfrak{B}$. One of the main insights from the work of [6] is
the dependence of the surface impedance on the symmetries of the eigenmodes
(and, by definition, on the bulk index). This can be replicated in our setting
with the following lemma.

######  Lemma 3.11.

It holds that either

$Z^{\pm}_{0}=\pm\infty$ and $Z^{\pm}_{\pi}=0$,

or

$Z^{\pm}_{0}=0$ and $Z^{\pm}_{\pi}=\pm\infty$.

###### Proof.

As mentioned above, from the uniqueness of solution to (2.2), we know that,
for $\kappa\in\\{0,\pi\\}$, we cannot have
$u_{n}^{[\kappa]}(0)=\Big{(}u_{n}^{[\kappa]}\Big{)}^{\prime}(0)=0$. Also, from
Lemma 3.9, we have two cases for the values of $u_{n}$ and $u_{n}^{\prime}$ at
the band edges. We shall treat them separately.

  * •

In the first case, we know that

$\displaystyle u_{n}^{[\pi]}(0)=0\ \ \ \text{and}\ \ \
\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(0)=0.$

This implies that

$\displaystyle\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(0)\neq 0\ \ \ \text{and}\ \
\ u_{n}^{[0]}(0)\neq 0.$

Hence, combining these results at $\kappa=0$, we get

$\displaystyle\lim_{x\uparrow 0}Z^{-}_{0}(\omega)=\lim_{x\uparrow
0}-\frac{u_{n}^{[0]}(x,\omega)}{\frac{1}{\varepsilon_{1}(\omega)}\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(x,\omega)}=\pm\infty\
\ \text{ and }\ \ \lim_{x\downarrow 0}Z^{+}_{0}(\omega)=\lim_{x\downarrow
0}-\frac{u_{n}^{[0]}(x,\omega)}{\frac{1}{\varepsilon_{1}(\omega)}\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(x,\omega)}=\pm\infty.$

At $\kappa=\pi$, we see that

$\displaystyle\lim_{x\uparrow 0}Z^{-}_{\pi}(\omega)=\lim_{x\uparrow
0}-\frac{u_{n}^{[\pi]}(x,\omega)}{\frac{1}{\varepsilon_{1}(\omega)}\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(x,\omega)}=0\
\ \text{ and }\ \ \lim_{x\uparrow 0}Z^{+}_{\pi}(\omega)=\lim_{x\uparrow
0}-\frac{u_{n}^{[\pi]}(x,\omega)}{\frac{1}{\varepsilon_{1}(\omega)}\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(x,\omega)}=0.$

  * •

For the second case, we know that

$\displaystyle u_{n}^{[0]}(0)=0\ \ \ \text{and}\ \ \
\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(0)=0,$

which implies that

$\displaystyle\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(0)\neq 0\ \ \ \text{and}\ \ \
u_{n}^{[\pi]}(0)\neq 0.$

In the exact same reasoning as in the previous case, we get at $\kappa=0$,

$\displaystyle\lim_{x\uparrow 0}Z^{-}_{0}(\omega)=\lim_{x\uparrow
0}-\frac{u_{n}^{[0]}(x,\omega)}{\frac{1}{\varepsilon_{0}}\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(x,\omega)}=0\
\ \text{ and }\ \ \lim_{x\downarrow 0}Z^{+}_{0}(\omega)=\lim_{x\downarrow
0}-\frac{u_{n}^{[0]}(x,\omega)}{\frac{1}{\varepsilon_{0}}\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(x,\omega)}=0$

and at $\kappa=\pi$,

$\displaystyle\lim_{x\uparrow 0}Z^{-}_{\pi}(\omega)=\lim_{x\uparrow
0}-\frac{u_{n}^{[\pi]}(x,\omega)}{\frac{1}{\varepsilon_{0}}\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(x,\omega)}=\pm\infty\
\ \text{ and }\ \ \lim_{x\uparrow 0}Z^{+}_{\pi}(\omega)=\lim_{x\uparrow
0}-\frac{u_{n}^{[\pi]}(x,\omega)}{\frac{1}{\varepsilon_{0}}\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(x,\omega)}=\pm\infty.$

This concludes the proof. ∎

####  3.5.2 Impedance property

In order to ease the notation, let us define:

$\displaystyle E(x,\omega):=\frac{1}{\varepsilon(x,\omega)}\ \ \ \text{ and }\
\ \ \phi:=\frac{\mathrm{d}u}{\mathrm{d}\omega}.$ (3.10)

The following result is due to the work of [6], which says that the surface
impedance functions are strictly decreasing functions of $\omega$, within each
band gap. For dispersive systems, this rests on the aforementioned assumption
that $\epsilon$ is a non-decreasing function of $\omega$. A proof of this
result, specific to our setting, is given in Appendix A.

######  Theorem 3.12.

Let us assume that $\omega\in\mathfrak{A}$, where
$\mathfrak{A}\subset\mathbb{R}$ is a band gap. Then, the surface impedance
decreases with respect to the frequency, i.e.

$\displaystyle\frac{\mathrm{d}Z^{+}}{\mathrm{d}\omega}<0,\ \
\frac{\mathrm{d}Z^{-}}{\mathrm{d}\omega}<0,\ \ \ \text{ for
}\omega\in\mathfrak{A}.$

####  3.5.3 Interface mode existence

Let $\mathcal{L}_{A}$ and $\mathcal{L}_{B}$ denote the differential operator
$\mathcal{L}$ for the material $A$ and the material $B$, respectively. We
define $\mathfrak{A}_{n}^{A}$ and $\mathfrak{A}_{n}^{B}$ to be the $n$-th band
gaps of $\mathcal{L}_{A}$ and $\mathcal{L}_{B}$, respectively. Let them be
given by

$\displaystyle\mathfrak{A}_{n}^{A}:=[\omega_{n,A}^{+},\omega_{n+1,A}^{-}]\ \ \
\text{ and }\ \ \
\mathfrak{A}_{n}^{B}:=[\omega_{n,B}^{+},\omega_{n+1,B}^{-}].$

Let us denote by $\mathfrak{A}_{n}$ the intersection of $\mathfrak{A}_{n}^{A}$
and $\mathfrak{A}_{n}^{B}$. We assume that it is non-empty, i.e.

$\displaystyle\mathfrak{A}_{n}=\mathfrak{A}_{n}^{A}\cap\mathfrak{A}_{n}^{B}\neq\emptyset.$

We denote it by

$\displaystyle\mathfrak{A}_{n}:=[\omega_{n}^{+},\omega_{n+1}^{-}].$

Finally, let us denote by $\mathcal{J}_{n,A}$ and $\mathcal{J}_{n,B}$ the bulk
topological indices associated to the material $A$ and the material $B$,
respectively, in $\mathfrak{A}_{n}$.

######  Theorem 3.13.

If

$\displaystyle\mathcal{J}_{n,A}+\mathcal{J}_{n,B}\neq 0,$

then no interface mode exists. If

$\displaystyle\mathcal{J}_{n,A}+\mathcal{J}_{n,B}=0,$

then there exists a unique frequency $\omega_{m}\in\mathfrak{A}_{n}$, for
which an interface mode exists.

###### Proof.

We will divide the proof in two steps, one for each case.

  * •

Let us consider:

$\displaystyle\mathcal{J}_{n,A}+\mathcal{J}_{n,B}\neq 0.$

We will treat the case $\kappa=0$. The same argument holds for $\kappa=\pi$.
From (3.9), this implies that either $u^{+}_{n,A},u^{+}_{n,B}$ are both
symmetric or both anti-symmetric. Let us treat the case of both being
symmetric. The anti-symmetric case follows the same reasoning. From Theorem
3.12, we have that $Z^{-}$ is decreasing in $\mathfrak{A}_{n}$. Then, from
Lemma 3.11, we have

$\displaystyle Z^{-}_{0}(\omega_{n,A}^{+})=+\infty\searrow
Z^{-}_{0}(\omega_{n+1,A}^{-})=0\ \text{ in }\mathfrak{A}_{n}^{A}.$

Similarly, we get

$\displaystyle Z^{+}_{0}(\omega_{n,B}^{+})=+\infty\searrow
Z^{+}_{0}(\omega_{n+1,B}^{-})=0\ \text{ in }\mathfrak{A}_{n}^{B}.$

We see directly that Lemma 3.3 does not hold, and hence, no interface mode
exists.

  * •

Let us consider:

$\displaystyle\mathcal{J}_{n,A}+\mathcal{J}_{n,B}=0.$

We will treat the case $\kappa=0$. The same argument holds for $\kappa=\pi$.
From (3.9), this implies that either $u^{+}_{n,A}$ is symmetric and
$u_{n,B}^{+}$ is anti-symmetric, or $u^{+}_{n,A}$ is anti-symmetric and
$u_{n,B}^{+}$ is symmetric. Let us focus on the first possibility. For the
second one, the same argument will hold. So, we take

$u^{+}_{n,A}$ symmetric and $u_{n,B}^{+}$ anti-symmetric at $\kappa=0$.

Then, from Lemma 3.11 and Theorem 3.12, we see that

$\displaystyle Z^{-}_{0}(\omega_{n,A}^{+})=+\infty\searrow
Z^{-}_{0}(\omega_{n+1,A}^{-})=0\ \text{ in }\mathfrak{A}_{n}^{A}.$

Using the fact that $u_{n,B}^{+}$ is anti-symmetric, we get, from Lemma 3.11
and Theorem 3.12, that

$\displaystyle Z^{+}_{0}(\omega_{n,B}^{+})=0\searrow
Z^{+}_{0}(\omega_{n+1,B}^{-})=-\infty\ \text{ in }\mathfrak{A}_{n}^{B}.$

Let us now study how $Z_{0}:=Z^{-}_{0}+Z^{+}_{0}$ behaves in
$\mathfrak{A}_{n}$. Since $\mathfrak{A}_{n}\neq\emptyset$, we observe that

$\displaystyle\min{\mathfrak{A}_{n}}\in\Big{\\{}\omega_{n,A}^{+},\omega_{n,B}^{+}\Big{\\}}.$

If $\min{\mathfrak{A}_{n}}=\omega_{n,A}^{+}$, then we see that

$\displaystyle Z_{0}\Big{(}\omega_{n,A}^{+}\Big{)}=+\infty,$

since $Z^{-}_{0}\Big{(}\omega_{n,A}^{+}\Big{)}=+\infty$ and
$Z^{+}_{0}\Big{(}\omega_{n,A}^{+}\Big{)}\in\mathbb{R}_{<0}$. Also, if
$\min{\mathfrak{A}_{n}}=\omega_{n,B}^{+}$, then we see that

$\displaystyle Z_{0}\Big{(}\omega_{n,B}^{+}\Big{)}>0,$

since $Z^{-}_{0}\Big{(}\omega_{n,B}^{+}\Big{)}\in\mathbb{R}_{>0}$ and
$Z^{+}_{0}\Big{(}\omega_{n,B}^{+}\Big{)}=0$. Thus, in any case, we get that

$\displaystyle Z_{0}\Big{(}\min{\mathfrak{A}_{n}}\Big{)}>0.$

Similarly, we have that

$\displaystyle\max{\mathfrak{A}_{n}}\in\Big{\\{}\omega_{n+1,A}^{-},\omega_{n+1,B}^{-}\Big{\\}}.$

Then, if $\max{\mathfrak{A}_{n}}=\omega_{n+1,A}^{-}$, we see that

$\displaystyle Z_{0}\Big{(}\omega_{n+1,A}^{-}\Big{)}<0,$

since $Z^{-}_{0}\Big{(}\omega_{n+1,A}^{-}\Big{)}=0$ and
$Z^{+}_{0}\Big{(}\omega_{n+1,A}^{-}\Big{)}\in\mathbb{R}_{<0}$. Also, if
$\max{\mathfrak{A}_{n}}=\omega_{n+1,B}^{-}$, we have that

$\displaystyle Z_{0}\Big{(}\omega_{n+1,A}^{-}\Big{)}=-\infty,$

since $Z^{-}_{0}\Big{(}\omega_{n+1,A}^{-}\Big{)}\in\mathbb{R}_{>0}$ and
$Z^{+}_{0}\Big{(}\omega_{n+1,A}^{-}\Big{)}=-\infty$. Hence, in any case, we
observe that

$\displaystyle Z_{0}\Big{(}\max{\mathfrak{A}_{n}}\Big{)}<0.$

To sum up these results, we have that

$\displaystyle Z_{0}\Big{(}\min{\mathfrak{A}_{n}}\Big{)}>0\ \ \ \text{ and }\
\ \ Z_{0}\Big{(}\max{\mathfrak{A}_{n}}\Big{)}<0.$

Then, from Theorem 3.12, we deduce that

$\displaystyle\frac{\mathrm{d}Z_{0}}{\mathrm{d}\omega}<0\ \text{ in
}\mathfrak{A}_{n}.$ (3.11)

So, since $Z_{0}$ is continuous and strictly decreasing in $\mathfrak{A}_{n}$,
there exists a unique point $\omega_{m}\in\mathfrak{A}_{n}$ at which
$Z_{0}(\omega_{m})=0$. Therefore, a unique interface mode exists. This
concludes the proof.

∎

######  Remark.

Here, let us recall that we have considered piecewise constant permittivities
with respect to $x$ but still dispersive with respect to $\omega$. Although,
as mentioned in Section 2, the analysis still applies on permittivity
functions which respect the mirror symmetry in each periodic cell and the
results remain the same.

##  4 Zak phase

Let us now study the Zak phase, an invariant associated to one-dimensional
crystals. This describes the topological properties of our system.

As mentioned at the beginning, we work in the space $L^{2}_{\kappa}$. We equip
$L^{2}_{\kappa}$, with the following inner product:

$\displaystyle\langle
f,g\rangle_{:}=\int_{0}^{1}\mu_{0}f(x)\overline{g}(x)\mathrm{d}x,\ \ \ f,g\in
L^{2}_{\kappa}.$ (4.1)

######  Definition 4.1 (Berry connection).

We define the Berry connection $A_{n}(\kappa)$ of the $n$-th band to be

$\displaystyle A_{n}(\kappa):=-i\left\langle\frac{\partial
u^{[\kappa]}_{n}}{\partial\kappa},u_{n}^{[\kappa]}\right\rangle.$ (4.2)

######  Definition 4.2 (Zak phase).

We define the Zak phase $\Theta_{n}$ of the $n$-th gap to be the integral of
the Berry connection $A_{n}(\kappa)$ across the first Brillouin zone, i.e.,

$\displaystyle\Theta_{n}=\int_{-\pi}^{\pi}A_{n}(\kappa)\mathrm{d}\kappa.$
(4.3)

To see that the Zak phase is well defined, we want to check that it is
invariant $\mathrm{mod}(2\pi)$ with respect to phase changes of the eigenmode
$u_{n}^{[\kappa]}$. This is because a normalised eigenmode (with
$\|u_{n}^{[\kappa]}\|=1$) can always be re-defined by changing the phase by an
integer multiple of $\kappa$,

$\displaystyle\tilde{u}^{[\kappa]}_{n}=e^{i\zeta\kappa}u^{[\kappa]}_{n},\
\zeta\in\mathbb{N}.$

Under this transformation, we can calculate that

$\displaystyle\tilde{A}_{n}(\kappa)$
$\displaystyle=-i\left\langle\frac{\partial\tilde{u}^{[\kappa]}_{n}}{\partial
x},\tilde{u}^{[\kappa]}_{n}\right\rangle=-i\left\langle\frac{\partial}{\partial
x}\Big{(}e^{i\zeta\kappa}u^{[\kappa]}_{n}\Big{)},e^{i\zeta\kappa}u^{[\kappa]}_{n}\right\rangle$
$\displaystyle=-i\int_{0}^{1}\mu_{0}\left(e^{i\zeta\kappa}\frac{\partial
u^{[\kappa]}_{n}}{\partial x}(x)+i\zeta
e^{i\zeta\kappa}u^{[\kappa]}_{n}(x)\right)\overline{e^{i\zeta\kappa}u^{[\kappa]}_{n}(x)}\mathrm{d}x$
$\displaystyle=A_{n}(\kappa)+\int_{0}^{1}\mu_{0}\zeta
u^{[\kappa]}_{n}(x)\overline{u^{[\kappa]}_{n}(x)}\mathrm{d}x$
$\displaystyle=A_{n}(\kappa)+\zeta.$

Thus,

$\displaystyle\int_{-\pi}^{\pi}\tilde{A}_{n}(\kappa)\mathrm{d}\kappa=\int_{-\pi}^{\pi}A_{n}(\kappa)\mathrm{d}\kappa+2\zeta\pi,$

which gives

$\displaystyle\tilde{\Theta}_{n}(\kappa)=\Theta_{n}(\kappa)+2\zeta\pi,$ (4.4)

so $\tilde{\Theta}_{n}$ and ${\Theta}_{n}$ are equal $\mathrm{mod}(2\pi)$.

######  Theorem 4.3.

Let us assume that we have mirror symmetry and that the $n$-th eigenvalue of
the operator $\mathcal{L}$ is non-degenerate. Then, it holds that
$\Theta_{n}\mathrm{mod}(2\pi)\in\\{0,\pi\\}$.

###### Proof.

We recall that, for the assumptions for the theorem, $u_{n}^{[-\kappa]}$ and
$\mathcal{P}u_{n}^{[\kappa]}$ are proportional to one another, i.e., there
exists $\mu\in\mathbb{R}$, depending on $\kappa$, such that

$\displaystyle u_{n}^{[-\kappa]}=\mu(\kappa)\mathcal{P}u_{n}^{[\kappa]}.$

In addition, we have taken the modes to satisfy $\|u_{n}\|=1$. Hence, we get
$\|\mu(\kappa)\|=1$ and so, we can write

$\displaystyle
u_{n}^{[-\kappa]}=e^{i\delta(\kappa)}\mathcal{P}u_{n}^{[\kappa]},$

where $\delta(\kappa)$ is a locally smooth function of $\kappa$. Then, we see
that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\kappa}u_{n}^{[-\kappa]}=\frac{\mathrm{d}}{\mathrm{d}\kappa}\Big{(}e^{i\delta(\kappa)}\mathcal{P}u_{n}^{[\kappa]}\Big{)}\Rightarrow\frac{\mathrm{d}}{\mathrm{d}\kappa}u_{n}^{[-\kappa]}=ie^{i\delta(\kappa)}\frac{\mathrm{d}\delta}{\mathrm{d}\kappa}\mathcal{P}u_{n}^{[\kappa]}+\frac{\mathrm{d}\mathcal{P}u_{n}^{[\kappa]}}{\mathrm{d}\kappa}e^{i\delta(\kappa)}.$

Taking the inner product with $u_{n}^{[-\kappa]}$ on both sides, we see that

$\displaystyle\int_{0}^{1}\mu_{0}e^{i\delta(\kappa)}\frac{\mathrm{d}\mathcal{P}u_{n}^{[\kappa]}}{\mathrm{d}\kappa}(x)\overline{u_{n}^{[-\kappa]}}(x)\mathrm{d}x$
$\displaystyle=\int_{0}^{1}\mu_{0}e^{i\delta(\kappa)}\frac{\mathrm{d}\mathcal{P}u_{n}^{[\kappa]}}{\mathrm{d}\kappa}(x)\overline{e^{i\delta(\kappa)}\mathcal{P}u_{n}^{[\kappa]}}(x)\mathrm{d}x$
$\displaystyle=\int_{0}^{1}\mu_{0}\frac{\mathrm{d}u_{n}^{[\kappa]}}{\mathrm{d}\kappa}(1-x)\overline{u_{n}^{[\kappa]}}(1-x)\mathrm{d}x$
$\displaystyle=-\int_{1}^{0}\mu_{0}\frac{\mathrm{d}u_{n}^{[\kappa]}}{\mathrm{d}\kappa}(x)\overline{u_{n}^{[\kappa]}}(x)\mathrm{d}x=-iA_{n}(\kappa)$

and

$\displaystyle\int_{0}^{1}\mu_{0}e^{i\delta(\kappa)}\frac{\mathrm{d}\delta}{\mathrm{d}\kappa}\mathcal{P}u_{n}^{[\kappa]}(x)\overline{u_{n}^{[-\kappa]}}(x)\mathrm{d}x=\frac{\mathrm{d}\delta}{\mathrm{d}\kappa}\int_{0}^{1}\mu_{0}u_{n}^{[-\kappa]}(x)\overline{u_{n}^{[-\kappa]}}(x)\mathrm{d}x=i\frac{\mathrm{d}\delta}{\mathrm{d}\kappa}.$

Thus, we get

$\displaystyle
iA_{n}(-\kappa)=-iA_{n}(\kappa)+i\frac{\mathrm{d}\delta}{\mathrm{d}\kappa}.$

Multiplying by $i$ and integrating over $[0,\pi]$, we get

$\displaystyle-\int^{\pi}_{0}A_{n}(-\kappa)\mathrm{d}\kappa=\int_{0}^{\pi}A_{n}(\kappa)\mathrm{d}\kappa-\delta(\pi)+\delta(0),$

which is

$\displaystyle-\int_{-\pi}^{0}A_{n}(\kappa)\mathrm{d}\kappa=\int_{0}^{\pi}A_{n}(\kappa)\mathrm{d}\kappa-\delta(\pi)+\delta(0).$

Thus, we get

$\displaystyle\Theta_{n}=\delta(\pi)-\delta(0).$

From Lemma 3.7, we have that $\mu(\kappa)\in\\{\pm,1\\}$, for $\kappa=0,\pi$.
This implies that $\delta(\kappa)\mathrm{mod}(2\pi)\in\\{0,\pi\\}$, for
$\kappa=0,\pi$. If
$\delta(\pi)\mathrm{mod}(2\pi)=\delta(0)\mathrm{mod}(2\pi)$, then
$\Theta_{n}=0$. If
$\delta(\pi)\mathrm{mod}(2\pi)\neq\delta(0)\mathrm{mod}(2\pi)$, then the phase
difference is always $\pi$. Thus, we get

$\displaystyle\Theta_{n}\mathrm{mod}(2\pi)\in\\{0,\pi\\}.$

This concludes the proof. ∎

##  5 Asymptotic behaviour

We wish to study the asymptotic behaviour of the modes as $x\to\pm\infty$. For
this, we will make use of the transfer matrix associated to this problem.

In the sections to follow, we will consider permittivity functions
$\varepsilon(x,\omega)$ which are piecewise constant with respect to $x$ in
each particle. This method still applies for permittivities which are mirror
symmetric with respect to $x$, but by suppressing this dependence, we obtain
explicit expressions for the transfer matrices, and so, we have a more
qualitative result.

###  5.1 Transfer matrix method

The transfer matrix method is a way of describing the mode $u$ and its spatial
derivative $u^{\prime}$ at each point on the structure with respect to the
initial data vector $\Big{(}u(0),u^{\prime}(0)\Big{)}^{\top}$.

We define the segment matrices $\mathscr{T}(x,\omega)$ by

$\displaystyle\mathscr{T}(l_{x},\omega):=\begin{pmatrix}\cos(\sqrt{\mu_{0}\varepsilon(x,\omega)}\omega
l_{x})&\frac{\sin(\sqrt{\mu_{0}\varepsilon(x,\omega)}\omega
l_{x})}{\sqrt{\mu_{0}\varepsilon(x,\omega)}\omega}\\\
-\sqrt{\mu_{0}\varepsilon(x,\omega)}\omega\sin(\sqrt{\mu_{0}\varepsilon(x,\omega)}\omega
l_{x})&\cos(\sqrt{\mu_{0}\varepsilon(x,\omega)}\omega l_{x})\end{pmatrix},$
(5.1)

where $l_{x}$ denotes the length of the part of the segment on which $x$ lies.

Let us consider an interval $I$ which is constituted by $N$ segments of length
$l_{i}$ each, $i=1,\dots,N$. Then, we define the transfer matrix
$T_{I}(\omega)$ over the interval $I$ to be

$\displaystyle T_{I}(\omega):=\prod_{i=1}^{N}\mathscr{T}(l_{i},\omega).$ (5.2)

If we define the vector
$\tilde{u}(x):=\Big{(}u(x),u^{\prime}(x)\Big{)}^{\top}$, then the transfer
matrix method describes the mode $u$ at each point $x$ based on the initial
data at a point $x_{0}$, i.e.

$\displaystyle\tilde{u}(x)=T_{[x_{0},x]}(\omega)\tilde{u}(x_{0}).$ (5.3)

It is clear that the segment matrices $\mathscr{T}$ satisfy
$\det(\mathscr{T}(x,\omega))=1$ for any $x$ and $\omega$. Hence, also, the
transfer matrix $T$ satisfies $\det(T(x,\omega))\equiv 1$.

The mirror symmetry induced to the system gives

$\displaystyle\varepsilon(x_{n}+h,\omega)=\varepsilon(x_{n+1}-h,\omega),\ \
h\in[0,1),\ n\in\mathbb{N}\setminus\\{0\\}.$

Let $T^{[j]}_{p}$ denote the transfer matrix over one periodic cell of the
material $j$, with $j\in\\{A,B\\}$. Then, it holds, as in [7, 18], that

$\displaystyle\begin{pmatrix}u(x_{n+1})\\\
u^{\prime}(x_{n+1})\end{pmatrix}=\mathbbm{1}_{\\{n\geq
0\\}}T_{p}^{[B]}\begin{pmatrix}u(x_{n})\\\
u^{\prime}(x_{n})\end{pmatrix}+\mathbbm{1}_{\\{n<0\\}}T_{p}^{[A]}\begin{pmatrix}u(x_{n})\\\
u^{\prime}(x_{n})\end{pmatrix},$ (5.4)

for $n\in\mathbb{Z}$. Also, the symmetry of the system gives

$\displaystyle\begin{pmatrix}u(x_{n-1})\\\
u^{\prime}(x_{n-1})\end{pmatrix}=\mathbbm{1}_{\\{n\geq
0\\}}ST_{p}^{[B]}S\begin{pmatrix}u(x_{n})\\\
u^{\prime}(x_{n})\end{pmatrix}+\mathbbm{1}_{\\{n<0\\}}ST_{p}^{[A]}S\begin{pmatrix}u(x_{n})\\\
u^{\prime}(x_{n})\end{pmatrix},$ (5.5)

for $n\in\mathbb{Z}$, where the matrix $S$ is given by

$\displaystyle S:=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$

Here, let us note that with a direct calculation, we can see that $S=S^{-1}$.
Applying the quasiperiodic boundary conditions, we get, for each material $i$,

$\displaystyle\tilde{u}(x_{n+1})=e^{i\kappa}\tilde{u}(x_{n}).$

Combining this with (5.4), we get the following problem

$\displaystyle\Big{(}T^{[j]}_{p}(\omega)-e^{i\kappa}I\Big{)}\tilde{u}(x_{n})=0.$
(5.6)

####  5.1.1 Spectral properties

Let us now state certain spectral properties of the transfer matrices.

######  Lemma 5.1.

Let $\omega\in\mathbb{R}\setminus\\{\omega_{p}^{(\pm)}\\}$. Then,
$T^{[j]}_{p}$ has real eigenvalues denoted by $\lambda^{[j]}_{1}$ and
$\lambda^{[j]}_{2}$, satisfying $|\lambda^{[j]}_{1}|<1$ and
$|\lambda^{[j]}_{2}|>1$, for $j\in\\{A,B\\}$.

###### Proof.

Let $i\in\\{A,B\\}$. Since $\det(T^{[j]}_{p}(\omega))=1$, we get either

$\displaystyle\lambda^{[j]}_{1},\lambda^{[j]}_{2}\in\mathbb{R},\ \ \text{with
}\ |\lambda^{[j]}_{1}|<1\ \text{ and }\ |\lambda^{[j]}_{2}|>1,$

or

$\displaystyle\lambda^{[j]}_{1},\lambda^{[j]}_{2}\in\mathbb{C},\ \ \text{with
}\ |\lambda^{[j]}_{1}|=|\lambda^{[j]}_{2}|=1\ \text{ and }\
\lambda^{[j]}_{1}=\overline{\lambda^{[j]}_{2}}.$

If the second case holds, then there exists $\kappa$ such that
$\lambda^{[j]}_{1}=e^{i\kappa}$, which implies that
$\det\Big{(}T^{[j]}_{p}(\omega)-e^{i\kappa}I\Big{)}=0$ which gives a
contradiction. Hence, the first case holds, and so, we obtain the desired
result. ∎

We use this result to obtain information about the asymptotic behaviour of the
modes $u(x_{n})$ as $n\to\pm\infty$.

######  Theorem 5.2.

The eigenfrequency $\omega$ of a localised eigenmode of the Helmholtz problem
(2.2) must satisfy

$\displaystyle\begin{pmatrix}-V_{21}^{[B]}(\omega)&V_{11}^{[B]}(\omega)\end{pmatrix}\begin{pmatrix}V_{11}^{[A]}(\omega)\\\
-V_{21}^{[A]}(\omega)\end{pmatrix}=0,$ (5.7)

where $(V_{11}^{[A]}(\omega),V_{21}^{[A]}(\omega))^{\top}$ is the eigenvector
of the transfer matrix $T_{p}^{[A]}$ associated to the eigenvalue
$|\lambda_{1}^{[A]}|<1$ and
$(V_{11}^{[B]}(\omega),V_{21}^{[B]}(\omega))^{\top}$ is the eigenvector of the
transfer matrix $T_{p}^{[B]}$ associated to the eigenvalue
$|\lambda_{1}^{[B]}|<1$.

###### Proof.

For $n\in\mathbb{N}_{>0}$, we have

$\displaystyle\begin{pmatrix}u(x_{n})\\\
u^{\prime}(x_{n})\end{pmatrix}=(T^{[B]}_{p})^{n}(\omega)\begin{pmatrix}u(x_{0})\\\
u^{\prime}(x_{0})\end{pmatrix}=V^{[B]}\begin{pmatrix}(\lambda^{[B]}_{1})^{n}&0\\\
0&(\lambda^{[B]}_{2})^{n}\end{pmatrix}(V^{[B]})^{-1}\begin{pmatrix}u(x_{0})\\\
u^{\prime}(x_{0})\end{pmatrix},$ (5.8)

and for $n\in\mathbb{N}_{<0}$, from the symmetry condition (5.5), we have

$\displaystyle\begin{pmatrix}u(x_{n})\\\ u^{\prime}(x_{n})\end{pmatrix}$
$\displaystyle=ST^{[A]}_{p}S\begin{pmatrix}u(x_{n+1})\\\
u^{\prime}(x_{n+1})\end{pmatrix}=S(T^{[A]}_{p})^{|n|}S\begin{pmatrix}u(x_{0})\\\
u^{\prime}(x_{0})\end{pmatrix}$ (5.9)
$\displaystyle=SV^{[A]}\begin{pmatrix}(\lambda^{[A]}_{1})^{n}&0\\\
0&(\lambda^{[A]}_{1})^{n}\end{pmatrix}(V^{[A]})^{-1}S\begin{pmatrix}u(x_{0})\\\
u^{\prime}(x_{0})\end{pmatrix},$ (5.10)

where $\lambda^{[A]}_{1},\lambda^{[A]}_{2}$, resp.
$\lambda^{[B]}_{1},\lambda^{[B]}_{2}$, are the eigenvalues of $T^{[A]}_{p}$,
resp. $T^{[B]}_{p}$, with $|\lambda^{[A]}_{1}|<1$ and $|\lambda^{[A]}_{2}|>1$,
resp. $|\lambda^{[B]}_{1}|<1$ and $|\lambda^{[B]}_{2}|>1$, and $V^{[A]}$,
resp. $V^{[B]}$, is the matrix of the associated eigenvectors. We are looking
for localised eigenmodes satisfying

$\displaystyle\lim_{n\to\pm\infty}u(x_{n})=0\ \ \ \text{ and }\ \ \
\lim_{n\to\pm\infty}u^{\prime}(x_{n})=0.$

We know that

$\displaystyle\lim_{n\to\pm\infty}(\lambda^{[A]}_{1})^{|n|}=\lim_{n\to\pm\infty}(\lambda^{[B]}_{1})^{|n|}=0\
\ \ \text{ and }\ \ \
\lim_{n\to\pm\infty}(\lambda^{[A]}_{2})^{|n|}=\lim_{n\to\pm\infty}(\lambda^{[B]}_{2})^{|n|}\neq
0.$

Thus, we wish to find $\omega,u(x_{0})$ and $u^{\prime}(x_{0})$ such that

$\displaystyle\begin{pmatrix}-V^{[B]}_{21}&V^{[B]}_{11}\end{pmatrix}\begin{pmatrix}u(x_{0})\\\
u^{\prime}(x_{0})\end{pmatrix}=0\ \ \ \text{ and }\ \ \
\begin{pmatrix}-V^{[A]}_{21}&V^{[A]}_{11}\end{pmatrix}S\begin{pmatrix}u(x_{0})\\\
u^{\prime}(x_{0})\end{pmatrix}=0,$ (5.11)

since
$(V^{[j]})^{-1}=\frac{1}{\det(V)}\begin{pmatrix}V^{[j]}_{22}&-V^{[j]}_{12}\\\
-V^{[j]}_{21}&V^{[j]}_{11}\end{pmatrix}$, $j\in\\{A,B\\}$ and so we multiply
the eigenvalue $(\lambda^{[j]}_{2})^{|n|}$, $j\in\\{A,B\\}$ by the second row
vector. Now, we observe that the second equation in (5.11) shows that
$(u(x_{0}),u^{\prime}(x_{0}))$ is proportional to
$(V^{[B]}_{11},-V^{[B]}_{21})$. Applying this to the first equation in (5.11),
we get

$\displaystyle\begin{pmatrix}-V^{[B]}_{21}&V^{[B]}_{11}\end{pmatrix}\begin{pmatrix}V^{[A]}_{11}\\\
-V^{[A]}_{21}\end{pmatrix}=0,$

which gives the desired result. ∎

####  5.1.2 Eigenmode decay

From this, we obtain the result concerning the decay of the localised
eigenmodes as $n\to\pm\infty$.

######  Corollary 5.2.1.

A localised eigenmode $u$ of (2.2), posed on a medium constituted by two semi-
infinite arrays of different halide perovskites, and its associated
eigenfrequency $\omega$ must satisfy

$\displaystyle u(x_{n})=O\Big{(}|\lambda^{[A]}_{1}(\omega)|^{|n|}\Big{)}\ \
\text{ and }\ \
u^{\prime}(x_{n})=O\Big{(}|\lambda^{[A]}_{1}(\omega)|^{|n|}\Big{)}\ \ \text{
as }n\to-\infty,$ $\displaystyle
u(x_{n})=O\Big{(}|\lambda^{[B]}_{1}(\omega)|^{|n|}\Big{)}\ \ \text{ and }\ \
u^{\prime}(x_{n})=O\Big{(}|\lambda^{[B]}_{1}(\omega)|^{|n|}\Big{)}\ \ \text{
as }n\to+\infty,$

where $\lambda^{[A]}_{1}$ is the eigenvalue of $T^{[A]}_{p}$ satisfying
$|\lambda^{[A]}_{1}(\omega)|<1$ and $\lambda^{[B]}_{1}$ is the eigenvalue of
$T^{[B]}_{p}$ satisfying $|\lambda^{[B]}_{1}(\omega)|<1$.

###### Proof.

This is a direct result from the previous theorem. Indeed, from Theorem 5.2,
we have that $(u(x_{0}),u^{\prime}(x_{0}))^{\top}$ is proportional to
$(V^{[B]}_{11},-V^{[B]}_{21})^{\top}$. Thus, we get from (5.8), for
$n\in\mathbb{Z}_{>0}$

$\displaystyle\begin{pmatrix}u(x_{n})\\\
u^{\prime}(x_{n})\end{pmatrix}=(\lambda^{[B]}_{1})^{n}(\omega)\begin{pmatrix}V^{[B]}_{11}\\\
V^{[B]}_{21}\end{pmatrix}$

and from (5.9), for $n\in\mathbb{Z}_{<0}$,

$\displaystyle\begin{pmatrix}u(x_{n})\\\
u^{\prime}(x_{n})\end{pmatrix}=(\lambda^{[A]}_{1})^{|n|}S\begin{pmatrix}V^{[A]}_{11}\\\
V^{[A]}_{21}\end{pmatrix}.$

This concludes the proof. ∎

##  6 Robustness with respect to imperfections

Now that we have established the existence of an interface mode in each band
gap for dispersive materials, which decays at infinity, we wish to study the
effect of system perturbations. This happens in two different ways:

  * •

Perturbations on the material parameters which have no effect on the symmetry
inside each periodic cell. For this, we will introduce perturbations to the
permittivity functions of the materials.

  * •

Perturbations on the symmetry inside each periodic cell. To study this effect
we will modify the length of two particles in each periodic cell, so that the
symmetry breaks.

###  6.1 Theoretical results

####  6.1.1 Permittivity perturbation

First, we will study changes in the materials while the symmetry is preserved.
To show this effect, we introduce the perturbation function
$f:\mathbb{R}\to\mathbb{R}$, given by

$\displaystyle f(x,\omega):=\begin{cases}&f_{1}(\omega),\ \ x\in D^{[1]},\\\
&f_{2}(\omega),\ \ x\in D^{[2]}.\end{cases}$ (6.1)

The perturbation function $f$ is a piecewise smooth function of the frequency
$\omega$ and satisfies $\frac{\partial f}{\partial\omega}<+\infty$. Then, we
define $\widetilde{\varepsilon}$ to be the perturbed permittivity of the
system, given by

$\displaystyle\widetilde{\varepsilon}(x,\omega):=\varepsilon(x,\omega)+\delta
f(\omega)=\begin{cases}&\varepsilon_{1}(\omega)+\delta f_{1}(\omega),\ \ x\in
D^{[1]},\\\ &\varepsilon_{2}(\omega)+\delta f_{2}(\omega),\ \ x\in
D^{[2]},\end{cases}$

where $\delta>0$ is the perturbation parameter. Then, we obtain the following
perturbed problem

$\displaystyle\begin{cases}&\widetilde{\mathcal{L}}(\kappa,\omega)u=\omega^{2}u,\\\
&u(x+1)=e^{i\kappa}u(x),\end{cases}$ (6.2)

where

$\displaystyle\widetilde{\mathcal{L}}u:=-\frac{1}{\mu_{0}}\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{\widetilde{\varepsilon}(x,\omega)}\frac{\mathrm{d}u}{\mathrm{d}x}\right).$
(6.3)

The following definition follows naturally from the perturbation of the
system.

######  Definition 6.1 (Perturbed impedance function).

We define the surface impedances associated to the perturbed Helmholtz problem
(6.2) by

$\displaystyle\widetilde{Z}^{-}(\omega)=-\frac{u(x_{0}^{-})}{\frac{1}{\widetilde{\varepsilon}_{1}(\omega)}\frac{\mathrm{d}}{\mathrm{d}x}u(x_{0}^{-})}\
\ \ \text{ and }\ \ \
\widetilde{Z}^{+}(\omega)=\frac{u(x_{0}^{+})}{\frac{1}{\widetilde{\varepsilon}_{1}(\omega)}\frac{\mathrm{d}}{\mathrm{d}x}u(x_{0}^{+})},\
\ \ \omega\in\mathbb{R}.$ (6.4)

We wish to show that, as $\delta\to 0$, an interface mode still exists. For
the existence, Lemma 3.3 holds for the perturbed impedance functions. In
addition, we observe that the symmetry of the system has been preserved by the
perturbation of the permittivity. Hence, it is sufficient to show that the
perturbed impedance functions remain decreasing. Then, because of the symmetry
of the system, the rest of the arguments will hold and so, the result will
follow.

######  Theorem 6.2.

Let us assume that $\omega\in\mathfrak{A}$, where
$\mathfrak{A}\subset\mathbb{R}$ is a band gap. Then, the perturbed surface
impedances, as $\delta\to 0$, are decreasing functions of the frequency
$\omega$, i.e.

$\displaystyle\frac{\mathrm{d}\widetilde{Z}^{+}}{\mathrm{d}\omega}<0,\ \
\frac{\mathrm{d}\widetilde{Z}^{-}}{\mathrm{d}\omega}<0,\ \ \ \text{ for
}\omega\in\mathfrak{A}.$

Thus, we have shown that arbitrarily small perturbations on the permittivity
of the system do not affect the existence of an interface mode. In addition,
using the transfer matrix method (explained in Section 5.1), we can get an
explicit expression of the solution to (6.2) with respect to the initial data
$(u(0),u^{\prime}(0))$. Because of the continuous dependence of the solution
on $\delta$, it is direct that, as $\delta\to 0$, this solution converges to
the one of $\eqref{eq:Helmholtz}$. This translates to the following lemma.

######  Lemma 6.3.

Let $u$ denote the asymptotically decaying interface mode of (2.2) in the band
gap $\mathfrak{A}$ and let $u_{\delta}$ denote an asymptotically decaying
interface mode associated to the permittivity perturbed problem (6.2) in the
band gap $\mathfrak{A}_{\delta}$. Then, if
$\mathfrak{A}\cap\mathfrak{A}_{\delta}\neq$, it holds that

$\displaystyle\lim_{\delta\to 0}u_{\delta}=u.$

####  6.1.2 Symmetry perturbation

Let us now introduce a structural perturbation to the system. This will result
in breaking the symmetry inside each periodic cell. This phenomenon is easy to
produce.

We have considered two semi-infinite materials $A$ and $B$, each one
constituted by a unit cell repeated periodically and glued together at
$x_{0}=0$. We have assumed that each periodic cell is made of two different
kinds of particles, one with permittivity $\varepsilon_{1}(\omega)$ and one
with permittivity $\varepsilon_{2}(\omega)$, denoted by $D^{[1]}_{i}$ and
$D^{[2]}_{j}$, with $i=1,\dots,N+1$ and $j=1,\dots,N$, respectively. The
particles are ordered in the following way inside each periodic cell:

$\displaystyle D^{[1]}_{1}\ -\ D^{[2]}_{1}\ -\ D^{[1]}_{2}\ -\ D^{[2]}_{2}\ -\
\dots\ -\ D^{[2]}_{N}\ -\ D^{[1]}_{N+1}.$

We call the position perturbation parameter $\sigma>0$ and we apply the
following procedure. In the material $A$, we increase the size of the particle
$D^{[1]}_{N+1}$ by $\sigma$ and we decrease the size of the particle
$D^{[1]}_{\frac{\lfloor N+2\rfloor}{2}}$ by $\sigma$. Then, in a similar way,
in the material $B$ we decrease the size of particle $D^{[1]}_{1}$ by $\sigma$
and we increase the size of $D^{[1]}_{\frac{\lfloor N+2\rfloor}{2}}$ by
$\sigma$. Hence, we manage to break the mirror symmetry governing each
material. We will view this more analytically later in Figure 4, where we have
considered a specific example for the model.

Let us note here that the position perturbation $\sigma$ cannot take
arbitrarily big values, as the size of the periodic cell has to remain
unaffected. In fact, for this to happen, we require that

$\displaystyle\sigma\leq\min_{A,B}\Big{\\{}|D^{[1]}_{1}|,\Big{|}D^{[1]}_{\frac{\lfloor
N+2\rfloor}{2}}\Big{|}\Big{\\}},$ (6.5)

where $\min_{A,B}$ denotes the minimum over materials $A$ and $B$, and
$|\cdot|$ denotes the size of each particle. Since we work in the one-
dimensional setting, $|\cdot|$ denoted the length.

Once again, as in Section 6.1.1, using the transfer matrix method, we can see
the continuous dependence that the decay mode has on the length of each
particle. Thus, as a direct result, the following Lemma holds.

######  Lemma 6.4.

Let $u$ denote the asymptotically decaying interface mode of (2.2) in the band
gap $\mathfrak{A}$ and let $u_{\sigma}$ denote an asymptotically decaying
interface mode associated to the position perturbation procedure described
above in the band gap $\mathfrak{A}_{\sigma}$. Then, if
$\mathfrak{A}\cap\mathfrak{A}_{\sigma}\neq\emptyset$, it holds that

$\displaystyle\lim_{\sigma\to 0}u_{\sigma}=u.$

###  6.2 Numerical results

Let us now provide an example, in order to solidify the previous analysis. We
consider dispersive particles with permittivity given by

$\displaystyle\varepsilon(x,\omega)=\begin{cases}\varepsilon_{1}(\omega),&x\in
D^{[1]},\\\ \varepsilon_{2}(\omega),&x\in D^{[2]},\end{cases}\ \ \text{ where
}\ \
\varepsilon_{i}(\omega):=\varepsilon_{0}+\frac{\alpha_{i}}{1-\beta_{i}\omega^{2}},\
\ \text{ for }\ i=1,2,$ (6.6)

with $\varepsilon_{0},\alpha_{i},\beta_{i}\geq 0$, for $i=1,2$. This
permittivity function model is inspired from the examples treated in [1] and
represents the undamped case of a Drude material or a halide perovskite. In
order to obtain analytic results we consider the model for the periodic cells
of materials $A$ and $B$ described in Figure 1. We note that, in this case,
each periodic cell is constituted by five particles, three with permittivity
$\varepsilon_{1}(\omega)$ and two with permittivity $\varepsilon_{2}(\omega)$.
The particles have the following ordering:

$\displaystyle D^{[1]}_{1}\ -\ D^{[2]}_{1}\ -\ D^{[1]}_{2}\ -\ D^{[2]}_{2}\ -\
D^{[1]}_{3}.$

Let us denote by $\omega_{i,p}^{(\pm)}$ the poles of the permittivity (2.1),
i.e.

$\displaystyle\omega_{i,p}^{(\pm)}=\pm\frac{\sqrt{\beta_{i}}}{\beta_{i}}.$
(6.7)

We see that

$\displaystyle\frac{\mathrm{d}\varepsilon}{\mathrm{d}\omega}(x,\omega)=\begin{cases}\frac{\mathrm{d}\varepsilon_{1}}{\mathrm{d}\omega}(\omega),\
\ &x\in D^{[1]},\\\
\frac{\mathrm{d}\varepsilon_{2}}{\mathrm{d}\omega}(\omega),\ \ &x\in
D^{[2]}.\end{cases}$

Thus, it holds

$\displaystyle\frac{\mathrm{d}\varepsilon_{1}}{\mathrm{d}\omega}(\omega)$
$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}\omega}\left(\varepsilon_{0}+\frac{\alpha_{1}}{1-\beta_{1}\omega^{2}}\right)=\frac{2\alpha_{1}\beta_{1}\omega}{(1-\beta_{1}\omega^{2})^{2}}\geq
0$

and similarly

$\displaystyle\frac{\mathrm{d}\varepsilon_{2}}{\mathrm{d}\omega}(\omega)\geq
0,$

which gives

$\displaystyle\frac{\mathrm{d}\varepsilon}{\mathrm{d}\omega}(x,\omega)\geq 0,\
\ x\in\mathbb{R}.$

Hence, Theorem 3.13 holds. This implies that if we consider a one-dimensional
photonic crystal of the form of Figure 1 with permittivity given by (6.6),
then, in each band-gap, a topologically protected interface mode exists.

####  6.2.1 Permittivity perturbation

As mentioned previously, we will first study perturbations on the material
parameters while keeping the symmetry inside each periodic cell intact. This
occurs by introducing a perturbation to the permittivity of the particles and
it gives rise to the perturbed system described by (6.2).

We can see that if the perturbation function $f$ is constant, then the
permittivity is not strongly affected. Hence we will consider perturbation
functions which are dispersive with respect to the frequency $\omega$. In
order to simplify the analysis we will take $f_{1}(\omega)=f_{2}(\omega)$, for
all $\omega\in\mathbb{R}$.

We will study the perturbation effect for two separate cases of perturbation
functions:

  1. 1.

Perturbation functions which satisfy the assumptions of the original
permittivity. In particular, we will consider the following:

$\displaystyle f(\omega)=-\frac{1}{\omega^{2}}.$ (6.8)

  2. 2.

Perturbation functions which does not satisfy the assumptions of the original
permittivity. In particular, we will consider the following:

$\displaystyle f(\omega)=\frac{1}{\omega^{2}}.$ (6.9)

For both cases, we see directly the smoothness of the perturbation function.
We will also work inside the first band gap of the problem (6.2). This region
is away from zero, so we ensure that $\frac{\partial
f}{\partial\omega}<+\infty$.

For the first case, we observe that as the perturbation parameter $\delta$
increases, the size of the band gap increases. Because of the monotonicity of
the perturbation function $\phi$, Theorem (3.13) holds and so a unique
interface mode exists. This fact is shown in Figure 2, where we plot for each
value of $\delta$ the value of the frequency $\omega$ of the interface mode.
We observe that as $\delta$ increases, the interface mode frequency increases
as well, which is natural, since the size of the band gap increases. We also
note that, as $\delta\to 0$, the interface mode frequency converges to the one
of the unperturbed interface mode, as shown in Lemma 6.3.

Figure 2: Plot of the frequencies of the interface mode and the edges of the
band gap as perturbations with magnitude $\delta$ that preserve the
monotonicity condition are added. In particular, $\delta$ times the strictly
increasing function (6.8) is added to the permittivity (6.6). This function is
in full accordance with the initial assumptions on the material
permittivities. We see that as $\delta\to 0$, the perturbed interface mode
frequency converges to the unperturbed one, as stated in Lemma 6.3. As
$\delta$ increases, the size of the band gap increases and the interface mode
frequency increases but still remains inside the band gap.

Let us now move to the second case. It holds that as $\delta$ increases, the
size of the band gap decreases. This happens because the perturbation function
$f$, which is decreasing, becomes the dominating factor of the perturbed
permittivity $\widetilde{\varepsilon}$. In Figure 3, we observe that as
$\delta$ increases, the interface mode frequency $\omega$ decreases. In
particular, for large values of $\delta$, the band gap becomes very small. On
the other hand, as $\delta\to 0$, the original permittivity is still the
dominating factor of the perturbed permittivity and so, from Theorem 3.13, the
interface mode still exists and converges to the unperturbed one.

This closing of the band gap poses problems for the physical implementation of
these systems. In particular, since the edge mode approaches the edge of the
band gap, its eigenfrequency will be close to that of a propagating Bloch
mode. As a result, in any physical device, it is likely than any attempt to
use this localised mode for wave guiding will fail as the nearby propagating
mode will also be excited.

Figure 3: Plot of the frequencies of the interface mode and the edges of the
band gap as perturbations with magnitude $\delta$ that do not preserve the
monotonicity condition are added. In particular, $\delta$ times the strictly
decreasing function (6.9) is added to the permittivity (6.6). As a result,
when $\delta$ is sufficiently large, the permittivity function does not
respect the monotonicity assumption. Of course, we see that as $\delta\to 0$,
the perturbed interface mode frequency converges to the unperturbed one, as
stated in Lemma 6.3. As $\delta$ increases, the width of the band gap
decreases. As a result, the perturbed interface mode frequency gets pushed
outside of the band gap. Hence, a localised interface mode no longer exists
for frequency $\omega$ in this band gap. The shaded areas are the spectral
bands and the red line represents the frequency of the perturbed interface
mode as the perturbation parameter $\delta$ increases. We observe that the
band gap becomes small for large values of $\delta$ and that the perturbed
interface mode frequency is pushed very close to the edge of the gap.

####  6.2.2 Symmetry perturbation

Let us now treat the case where the symmetry of the periodic cells breaks. We
will apply the procedure described in Section 6.1.2 on the model described in
Figure 1. In particular, for material $A$, we reduce the size of the particle
$D^{[1]}_{2}$ by $\sigma$ and we increase the size of particle $D^{[1]}_{3}$
by $\sigma$. Similarly, for material $B$, we decrease the size of
$D^{[1]}_{1}$ by $\sigma$ and we increase the size of $D^{[1]}_{2}$ by the
same amount. We depict this position perturbation graphically for the periodic
cell of material $A$ and $B$ in Figure 4.

As it is already mentioned, the position perturbation parameter $\sigma$
cannot take arbitrarily large values, since the length of the periodic cell
has to remain the same. Thus, from the bound (6.5), we see that

$\displaystyle 0\leq\sigma\leq\theta_{2}.$

Perturbed material APerturbed material B Figure 4: Position perturbation on
the periodic cells of each material. We have considered as values
$\theta_{1}=0.1$, $\theta_{2}=0.15$ and $\mu=0.25$. We notice that the largest
possible value for $\sigma$ is $0.15$. We see that by modifying the sizes of
the particles, the mirror symmetry inside the unit cell no longer holds.

Studying the existence of an interface mode under the position perturbation
regime is not in accordance with the argument we have presented in Theorem
3.13. In fact, our argument is based on the mirror symmetry governing the
periodic cells and proposes a sufficient but not necessary condition for an
interface mode to exist in a band gap. This symmetry breaks once the position
perturbation takes place. What we can say with certainty is that, if the
interface mode exists, then, as the position perturbation parameter $\sigma\to
0$, it converges to the unperturbed interface mode (Lemma 6.4). This result is
shown in Figure 5, where we have plotted the interface mode frequency $\omega$
as a function of the position perturbation parameter $\sigma$ inside a band
gap.

Figure 5: Plot of the interface mode frequency $\omega$ as the position
perturbation parameter $\sigma$ increases. We notice that as $\sigma\to 0$,
the perturbed interface mode frequency converges to the one of the unperturbed
system. Also, we see that as $\sigma$ increases, the break of the symmetry
inside the unit cell is more drastic and so, the perturbed interface mode
frequency diverges from the unperturbed one. Although, this divergence is
quite small and the perturbed interface mode frequency remains inside the band
gap. We note that the shaded areas denote the spectral bands.

##  7 Conclusion

Studying a one-dimensional system for dispersive materials, we used the mirror
symmetry and the underlying periodic structure of our setting to show that
localised interface modes exist. We explained the behaviour of the Zak phase,
a topological invariant of the system, and we have provided the condition that
the interface mode frequency needs to satisfy for these modes to decay at
infinity. Finally, we introduced imperfections to the system in the form of
perturbations on the material permittivity and the symmetry of the periodic
cells and we showed how the decaying modes and their associated frequencies
are affected.

## Conflicts of interest

The authors have no conflicts of interest to disclose.

## Acknowledgements

The work of KA was supported by ETH Zürich under the project ETH-34 20-2. The
work of BD was funded by the Engineering and Physical Sciences Research
Council through a fellowship with grant number EP/X027422/1.

##  Appendix A Proofs of impedance monotonicity

###  A.1 Proof of Theorem 3.12

###### Proof of Theorem 3.12.

We assume that we are working in a band gap $\mathfrak{A}$. We apply
$\frac{\mathrm{d}}{\mathrm{d}\omega}$ on both sides of (2.2) and we get, using
(3.10), that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\omega}\left[\frac{\mathrm{d}}{\mathrm{d}x}\left(E(x,\omega)\frac{\mathrm{d}u}{\mathrm{d}x}\right)\right]=\frac{\mathrm{d}}{\mathrm{d}\omega}\Big{(}-\mu_{0}\omega^{2}u\Big{)},$

which is

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{\mathrm{d}}{\mathrm{d}\omega}\left(E(x,\omega)\frac{\mathrm{d}u}{\mathrm{d}x}\right)\right]=-2\mu_{0}\omega
u-\mu_{0}\omega^{2}\phi$

and so

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{\mathrm{d}E}{\mathrm{d}\omega}(x,\omega)u^{\prime}+E(x,\omega)\phi^{\prime}\right]=-2\mu_{0}\omega
u-\mu_{0}\omega^{2}\phi.$ (A.1)

Also, we have

$\displaystyle\frac{\mathrm{d}Z^{+}}{\mathrm{d}\omega}=\frac{\mathrm{d}}{\mathrm{d}\omega}\left(\frac{u}{Eu}\right)=\frac{1}{(Eu^{\prime})^{2}}\left[Eu^{\prime}\phi-u\left(\frac{\mathrm{d}E}{\mathrm{d}\omega}u^{\prime}+E\phi^{\prime}\right)\right].$

Let us define the Wronskian $\mathcal{W}$ by

$\displaystyle\mathcal{W}:=Eu^{\prime}\phi-u\left(\frac{\mathrm{d}E}{\mathrm{d}\omega}u^{\prime}+E\phi^{\prime}\right).$
(A.2)

Then, we get

$\displaystyle\frac{\mathrm{d}Z_{R}}{\mathrm{d}\omega}=\frac{\mathcal{W}}{(Eu^{\prime})^{2}}.$

We observe that

$\displaystyle\mathcal{W}^{\prime}$
$\displaystyle=\frac{\mathrm{d}W}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}\left[Eu^{\prime}\phi-u\left(\frac{\mathrm{d}E}{\mathrm{d}\omega}u^{\prime}+E\phi^{\prime}\right)\right]$
$\displaystyle=E^{\prime}u^{\prime}\phi+E(u^{\prime\prime}\phi+u^{\prime}\phi^{\prime})-u^{\prime}\left(\frac{\mathrm{d}E}{\mathrm{d}\omega}u^{\prime}+E\phi^{\prime}\right)-u\left(\frac{\mathrm{d}E}{\mathrm{d}\omega}u^{\prime}+E\phi^{\prime}\right)^{\prime}$
$\displaystyle=E^{\prime}u^{\prime}\phi+Eu^{\prime\prime}\phi+Eu^{\prime}\phi^{\prime}-(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}-Eu^{\prime}\phi^{\prime}-u\left(\frac{\mathrm{d}E}{\mathrm{d}\omega}u^{\prime}+E\phi^{\prime}\right)^{\prime}$
$\displaystyle=E^{\prime}u^{\prime}\phi+Eu^{\prime\prime}\phi-(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}-u\left(\frac{\mathrm{d}E}{\mathrm{d}\omega}u^{\prime}+E\phi^{\prime}\right)^{\prime}$
$\displaystyle=\phi(Eu^{\prime})^{\prime}-u\left(\frac{\mathrm{d}E}{\mathrm{d}\omega}u^{\prime}+E\phi^{\prime}\right)^{\prime}-(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}.$

Applying (2.2) and (A.1), we get

$\displaystyle\mathcal{W}^{\prime}$
$\displaystyle=-\phi\mu_{0}\omega^{2}u-u(-2\omega\mu_{0}u-\mu_{0}\omega^{2}\phi)-(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}$
$\displaystyle=2\mu_{0}\omega
u^{2}-(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega},$

which gives

$\displaystyle\mathcal{W}^{\prime}=2\mu_{0}\omega
u^{2}-(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}.$ (A.3)

Integrating with respect to $x$ over the material $B$, i.e. on $[0,+\infty]$,
we have

$\displaystyle\lim_{x\to+\infty}\mathcal{W}(x,w)-\mathcal{W}(0,w)=2\mu_{0}\omega\int_{0}^{+\infty}u^{2}\mathrm{d}x-\int_{0}^{+\infty}(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}\mathrm{d}x.$

We know that when the frequency is inside a band gap, the field vanishes as
$x\to+\infty$. This implies that

$\displaystyle\lim_{x\to+\infty}\mathcal{W}(x,w)=0,$

and so

$\displaystyle\mathcal{W}(0,w)=\int_{0}^{+\infty}(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}\mathrm{d}x-2\mu_{0}\omega\int_{0}^{+\infty}u^{2}\mathrm{d}x.$

We know that $\frac{\mathrm{d}\varepsilon}{\mathrm{d}\omega}\geq 0$. This
gives

$\displaystyle\frac{\mathrm{d}E}{\mathrm{d}\omega}=\frac{\mathrm{d}}{\mathrm{d}\omega}\left(\frac{1}{\varepsilon}\right)=\frac{-1}{\varepsilon^{2}}\frac{\mathrm{d}\varepsilon}{\mathrm{d}\omega}\leq
0.$

Hence,

$\displaystyle\int_{0}^{+\infty}(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}\mathrm{d}x\leq
0.$

Therefore, we get

$\displaystyle\frac{\mathrm{d}Z^{+}}{\mathrm{d}\omega}=\frac{1}{(Eu^{\prime})^{2}}\left(-2\mu_{0}\omega\int_{0}^{+\infty}u^{2}\mathrm{d}x+\int_{0}^{+\infty}(u^{\prime})^{2}\frac{\mathrm{d}E}{\mathrm{d}\omega}\mathrm{d}x\right)<0.$

We apply the same argument for $Z^{-}$ and we obtain similarly that

$\displaystyle\frac{\mathrm{d}Z^{-}}{\mathrm{d}\omega}<0.$

This concludes the proof.

∎

###  A.2 Proof of Theorem 6.2

###### Proof of Theorem 6.2.

The proof follows the same reasoning as the one of Lemma 3.3. We first define
$\widetilde{E}$ by

$\displaystyle\widetilde{E}(x,\omega):=\frac{1}{\widetilde{\varepsilon}(x,\omega)}.$

Applying $\frac{\mathrm{d}}{\mathrm{d}\omega}$ on both sides of
$\eqref{eq:Helmholtz:pert}$, we get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}(x,\omega)u^{\prime}+\widetilde{E}(x,\omega)\phi^{\prime}\right]=-2\mu_{0}\omega
u-\mu_{0}\omega^{2}\phi,$ (A.4)

where $\phi:=\frac{\mathrm{d}u}{\mathrm{d}\omega}$. Also, we have

$\displaystyle\frac{\mathrm{d}\widetilde{Z}^{+}}{\mathrm{d}\omega}=\frac{1}{(\widetilde{E}u^{\prime})^{2}}\left[\widetilde{E}u^{\prime}\phi-u\left(\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}u^{\prime}+\widetilde{E}\phi^{\prime}\right)\right].$

We define the perturbed Wronskian $\widetilde{\mathcal{W}}$ by

$\displaystyle\widetilde{\mathcal{W}}:=\widetilde{E}u^{\prime}\phi-u\left(\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}u^{\prime}+\widetilde{E}\phi^{\prime}\right),$
(A.5)

which gives

$\displaystyle\frac{\mathrm{d}\widetilde{Z}^{+}}{\mathrm{d}\omega}=\frac{\widetilde{\mathcal{W}}}{(\widetilde{E}u^{\prime})^{2}}.$

It follows that

$\displaystyle\widetilde{\mathcal{W}}^{\prime}=\phi(\widetilde{E}u^{\prime})^{\prime}-u\left(\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}u^{\prime}+\widetilde{E}\phi^{\prime}\right)^{\prime}-(u^{\prime})^{2}\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega},$

and so, from (6.2) and (A.4), we get

$\displaystyle\widetilde{\mathcal{W}}^{\prime}=2\mu_{0}\omega
u^{2}-(u^{\prime})^{2}\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}.$

Integrating with respect to $x$ over the perturbed material $B$, i.e. on
$[0,+\infty]$, we have

$\displaystyle\lim_{x\to+\infty}\widetilde{\mathcal{W}}(x,w)-\widetilde{\mathcal{W}}(0,w)=2\mu_{0}\omega\int_{0}^{+\infty}u^{2}\mathrm{d}x-\int_{0}^{+\infty}(u^{\prime})^{2}\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}\mathrm{d}x.$

We know that when in a band gap, the field vanishes as $x\to+\infty$. This
implies that

$\displaystyle\lim_{x\to+\infty}\widetilde{\mathcal{W}}(x,w)=0,$

and so

$\displaystyle\widetilde{\mathcal{W}}(0,w)=\int_{0}^{+\infty}(u^{\prime})^{2}\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}\mathrm{d}x-2\mu_{0}\omega\int_{0}^{+\infty}u^{2}\mathrm{d}x.$

We observe that

$\displaystyle\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}=\frac{\mathrm{d}}{\mathrm{d}\omega}\left(\frac{1}{\widetilde{\varepsilon}(x,\omega)}\right)=\frac{\mathrm{d}}{\mathrm{d}\omega}\left(\frac{1}{\varepsilon(x,\omega)+\delta
f(\omega)}\right)=\frac{-1}{\Big{(}\varepsilon(x,\omega)+\delta
f(\omega)\Big{)}^{2}}\left(\frac{\mathrm{d}\varepsilon}{\mathrm{d}\omega}+\delta\frac{\mathrm{d}f}{\mathrm{d}\omega}\right).$

Thus, since $\frac{\partial f}{\partial\omega}<+\infty$, we get

$\displaystyle\lim_{\delta\to
0}\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}=\frac{-1}{\varepsilon(x,\omega)^{2}}\frac{\mathrm{d}\varepsilon}{\mathrm{d}\omega}\leq
0,$

since we have that $\frac{\mathrm{d}\varepsilon}{\mathrm{d}\omega}\geq 0$.
Hence,

$\displaystyle\lim_{\delta\to
0}\int_{0}^{+\infty}(u^{\prime})^{2}\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}\mathrm{d}x\leq
0,$

Therefore, at $x=0$, we get

$\displaystyle\lim_{\delta\to
0}\frac{\mathrm{d}\widetilde{Z}^{+}}{\mathrm{d}\omega}=\lim_{\delta\to
0}\frac{1}{(\widetilde{E}u^{\prime})^{2}}\left(-2\mu_{0}\omega\int_{0}^{+\infty}u^{2}\mathrm{d}x+\int_{0}^{+\infty}(u^{\prime})^{2}\frac{\mathrm{d}\widetilde{E}}{\mathrm{d}\omega}\mathrm{d}x\right)<0.$

The same argument holds for $\widetilde{Z}^{-}$ and so, we obtain that

$\displaystyle\frac{\mathrm{d}\widetilde{Z}^{-}}{\mathrm{d}\omega}<0.$

This concludes the proof. ∎

##  Appendix B Characterisation of eigenmode symmetries

###### Proof of Lemma 3.9.

The first thing that we notice is that we cannot have
$u_{n}^{[0]}(0)=\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(0)=0$ or
$u_{n}^{[\pi]}(0)=\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(0)=0$, since by the
uniqueness of solution to the problem (2.2), it would mean $u_{n}\equiv 0$.
Then, from the Lemma 3.8, we have two different cases. We will treat each one
separately:

  * •

We have $u_{n}$ being symmetric and $u^{\prime}_{n}$ being anti-symmetric. At
$\kappa=\pi$, from the symmetry of $u_{n}$, we have that

$\displaystyle u_{n}^{[\pi]}(0)=\mathcal{P}u_{n}^{[\pi]}(0)=u_{n}^{[\pi]}(1)$

and from (2.4), it holds that

$\displaystyle u_{n}^{[\pi]}(0)=e^{i\pi}u_{n}^{[\pi]}(1)=-u_{n}^{[\pi]}(1).$

Combining the two, we get

$\displaystyle u_{n}^{[\pi]}(0)=0.$

Then, at $\kappa=0$, since $u^{\prime}_{n}$ is anti-symmetric, we have that

$\displaystyle\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(0)=-\mathcal{P}\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(0)=-\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(1).$

But, from (2.4), we have

$\displaystyle\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(0)=e^{0}\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(1)=\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(1).$

Combining the two, we get

$\displaystyle\Big{(}u_{n}^{[0]}\Big{)}^{\prime}(0)=0.$

This gives the desired result.

  * •

The second case is $u_{n}$ being anti-symmetric and $u^{\prime}_{n}$ being
symmetric. At $\kappa=0$, the anti-symmetry of $u_{n}$ gives

$\displaystyle u_{n}^{[0]}(0)=-\mathcal{P}u_{n}^{[0]}(0)=-u_{n}^{[0]}(1)$

and the quasiperiodic boundary condition (2.4) gives

$\displaystyle u_{n}^{[0]}(0)=e^{0}u_{n}^{[0]}(1)=u_{n}^{[0]}(1).$

Combining the two, we get

$\displaystyle u_{n}^{[0]}(0)=0.$

Then, at $\kappa=\pi$, since $u^{\prime}_{n}$ is symmetric, we have that

$\displaystyle\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(0)=\mathcal{P}\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(0)=\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(1).$

But, from (2.4), we have

$\displaystyle\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(0)=e^{i\pi}\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(1)=-\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(1).$

Combining the two, we get

$\displaystyle\Big{(}u_{n}^{[\pi]}\Big{)}^{\prime}(0)=0.$

This gives the desired result.

This concludes the proof. ∎

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