# Symmetry-induced quasicrystalline waveguides

Bryn Davies Department of Mathematics, Imperial College London, South
Kensington Campus, London, SW7 2AZ, UK (email: bryn.davies@imperial.ac.uk).
Richard V. Craster11footnotemark: 1

###### Abstract

Introducing an axis of reflectional symmetry in a quasicrystal leads to the
creation of localised edge modes that can be used to build waveguides. We
develop theory that characterises reflection-induced localised modes in
materials that are formed by recursive tiling rules. This general theory
treats a one-dimensional continuous differential model and describes a broad
class of both quasicrystalline and periodic materials. We present an analysis
of a material based on the Fibonacci sequence, which has previously been shown
to have exotic, Cantor-like spectra with very wide spectral gaps. Our approach
provides a way to create localised edge modes at frequencies within these
spectral gaps, giving strong and stable wave localisation. We also use our
general framework to make a comparison with reflection-induced modes in
periodic materials. These comparisons show that while quasicrystalline
waveguides enjoy enhanced robustness over periodic materials in certain
settings, the benefits are less clear if the decay rates are matched. This
shows the need to carefully consider equivalent structures when making
robustness comparisons and to draw conclusions on a case-by-case basis,
depending on the specific application.

Keywords: fractal spectrum, transfer matrix, aperiodic tiling, non-linear
recursion relation, chaos, robust edge modes

##  1 Introduction

A widespread goal of wave physicists and engineers is to design and build
devices that efficiently direct wave energy along specific paths. A common
approach is to start with an inhomogeneous material that has a spectral gap (a
range of frequencies that do not propagate through the material) and perturb
the material by introducing a defect or interface; if done correctly, this
perturbation will create _edge modes_ that are strongly localised along the
path of the defect. The eigenfrequency of these edge modes typically falls
within the spectral gaps of the original material [1, 2, 3]. This strategy is
typically employed for periodic materials, where the spectral gap structure is
easy to understand using Floquet-Bloch theory [4] and the technique has led to
a rich variety of waveguides, capable of guiding waves robustly along
intricate paths [5, 6, 7]. In this work, we will show how to generalise this
principle to non-periodic settings, aiming to realise the under utilised
potential of such structures. In particular, the potential advantages of
quasicrystalline waveguides include very large spectral gaps, even at high
frequencies, [8], very broadband effects, high quality factors [9] and
robustness [10, 9, 11, 12].

The media studied here are generated recursively by tiling different materials
according to tiling rules. The set of such patterns is very broad and has been
studied extensively by mathematicians. They gained popular attention following
the discovery of an aperiodic set of tiles by Berger in 1966 [13], which
disproved previous conjectures that if a set of tiles can tile the plane, then
they can always be arranged to do so periodically [14]. Many such aperiodic
tilings have since been discovered, most notably those composed of just two
tiles that were discovered by Penrose in the 1970s [15]. Following the
discovery of a metallic alloy with quasicrystalline atomic structure (by Dan
Shechtman, who would win a Nobel prize for this work [16]), the excitement
spread beyond mathematics and the field of quasicrystals was born.

The main example we will use to demonstrate our results will be based on a
Fibonacci tiling rule, where each new term in the sequence is formed by
combining the two previous terms. The wave transmission properties of such
materials have been studied extensively, such that their spectral gap
structure is well understood [17, 18, 19, 20, 21, 22, 23]. It has been shown
that this structure has self similarity [22] and demonstrates a Cantor-like
emergence of nested spectral gaps, converging to a fractal pattern [22]. These
properties have even been demonstrated experimentally for a system of
structured rods [23]. Typically, these previous studies have used periodic
approximants to understand the structure of the spectrum for increasing
iterations in the Fibonacci sequence. These _supercell_ methods have the
advantage that Floquet-Bloch theory can be used as the structure is periodic.
As the size of the repeating supercell increases, the Floquet-Bloch spectrum
becomes increasingly intricate, yielding the spectrum of the quasicrystal in
the limit (this is shown nicely in Figure 7 of [24]).

A common feature of the previous studies of waves in Fibonacci quasicrystals,
and also of the present work, is that the materials considered are one-
dimensional. Some general principles for understanding the diffraction of one-
dimensional two-material substitutional systems were established by [25]. A
convenient property of one-dimensional materials is that their properties can
be described concisely using transfer matrices. A review of the use of
transfer matrices for modelling wave propagation in one-dimensional lossless
media, including quasicrystals, can be found in [26]. To study
quasicrystalline unbounded materials we will need to consider the limit of the
product of infinitely many transfer matrices, ordered according to the
relevant tiling rule. This is a subtle problem in general [27, 28], however in
the specific case of the Fibonacci problem the traces of successive products
of transfer matrices can be related by a second-order non-linear recursion
relation, discovered by Kohmoto in 1983 [17], which will simplify the
analysis.

It has long been known that quasicrystalline patterns can be obtained by
projecting a non-commensurate slice of a higher-dimensional periodic pattern
(known as a _superspace_). If we are able to relate the properties of the two
materials, then this offers a very promising approach for understanding the
wave transmission properties of quasicrystals, since the properties of the
periodic superspace material are straightforward to understand using Floquet-
Bloch analysis. This was utilised by [29] to predict the density of states in
quasicrystalline materials, obtained via projection. Similarly, projection of
periodic materials has been used to derive homogenised regimes for
quasicrystalline materials [30, 31]. There are outstanding questions related
to understanding the significance of Bloch wavenumbers in the superspace
material and it is not clear how to relate edge modes in the superspace and to
those in the projected quasicrystal. We will not explore these important
questions in this work, but highlight this as an appealing avenue for future
study.

In this work, we will create localised modes by introducing an axis of
reflectional symmetry to the material. This strategy has been used previously
to create reflection-induced edge modes, for example in quasicrystalline
arrays of point scatterers on elastic plates by [10] and, experimentally, in a
quasicrystalline acoustic waveguide by [32]. We will derive novel theory for
characterising these modes, which emerge as eigenmodes of a continuous
differential model. The model we shall consider will be one dimensional, as we
will focus on the axis perpendicular to the axis of reflection in order to
capture the decaying modes (see Figure 1 for an example). The other type of
localised edge modes that have been studied extensively are those that occur
at the edges of a quasicrystalline material with finite dimensions [33, 34,
35, 12, 36, 37, 38].

Figure 1: The waveguides studied in this work are formed by taking a pattern
of lamina, arranged according to an iterative tiling rule, and introducing an
axis of reflectional symmetry. Waves of specific frequencies are strongly
guided along the interface formed by the reflection. This sketch shows an
example of such a material, along with a representation of a typical decaying
wave form. The decaying mode sketched along the axis perpendicular to the axis
of reflection; we study one-dimensional problems in this work, considering
only this axis.

In the setting of localised modes in periodic media, one of the main avenues
of investigation is the study of modes that are _topologically protected_.
These are localised modes that, due to the topological properties of the
underlying periodic medium, experience enhanced robustness with respect to
imperfections [5, 6]. For non-periodic structures, it is less clear how to
define the topological indices which these theories rely on. One quantity of
interest is the integrated density of states (IDS), which captures the number
of eigenfrequencies below a given frequency and has been shown to be
topologically invariant in certain settings. For quasicrystalline systems
formed by parametrically modulating a periodic structure (with rational and
irrational values of the parameter yielding periodic and quasiperiodic
systems, respectively), it has been shown that the IDS can be used to
attribute a unique label to each spectral gap [38]. It was, further, shown
that this label can be used to predict which gaps will support edge modes in
finite crystals [38]. This argument can be understood using the fact that
certain parameter values yield commensurate structures and changes in the IDS
can be used to predict the number of edge modes that cross the spectral gap in
the interval of parameter space between two subsequent commensurate values
[33]. However, it remains an open question to connect, in general, these
topological properties to the robustness of edge modes created in any of the
spectral gaps.

Even if the link between topological properties and robustness of edge modes
remains unclear for quasicrystalline settings, several studies have
highlighted interesting robustness properties. In particular, the robustness
of reflection-induced edge modes was explored by [10], who randomly perturbed
the positions of a quasicrystalline array of point scatterers on elastic
plates. They observed that the reflection-induced edge modes are robust in the
sense that they persist in spite of the random imperfections and their
eigenfrequencies were “only slightly shifted”. Some robustness conclusions
have also been observed for the edge states in finite-dimensional
quasicrystals [11, 12]. We explore robustness of the reflection-induced edge
modes that occur in our setting by randomly perturbing the material
parameters. Our results suggest that their robustness is comparable to a
periodic mode with the same decay rate.

The general results we develop in this work, which are presented in Section 2,
apply to any one-dimensional medium that can be constructed by a recursive
tiling rule, provided the constituent materials satisfy a symmetry condition
(which could be relaxed, in principle, but would lead to less concise
expressions). In particular, the material does not necessarily have to be
either quasicrystalline or periodic. In Sections 3 and 4, we will present an
example of each, for comparison. The quasicrystalline example will be based on
a Fibonacci material, building on the previous studies of such structures. We
will then present a simple periodic example, in Section 4, before studying the
relative robustness of the quasicrystalline and periodic materials in Section
5.

##  2 General theory of reflection-induced edge modes

###  2.1 Problem setting

We consider a class of media that are constructed using iterative tiling
algorithms that repeatedly join together two different bounded media. We label
the two bounded media as $A$ and $B$ and assume that they both have unit
width, such that they can be described by the functions

$c_{A}:[0,1)\to(0,\infty)\quad\text{and}\quad c_{B}:[0,1)\to(0,\infty).$ (1)

Subsequently, we can introduce some notation to describe media constructed by
repeatedly joining $A$ and $B$ together. We note at this early stage that we
use the convention $0\notin\mathbb{N}$.

######  Definition 2.1 (Tiling labels).

Given $N\in\mathbb{N}$ and a set of labels
$X_{1},X_{2},\dots,X_{N}\in\\{A,B\\}$, we will use the label $X_{1}X_{2}\cdots
X_{N}$ for the tiled material described by the function $c_{X_{1}X_{2}\cdots
X_{N}}:[0,N)\to(0,\infty)$ which satisfies

$c_{X_{1}X_{2}\cdots X_{N}}(x)=c_{X_{n}}(x-n+1),\qquad\text{for
}x\in[n-1,n),\,n=1,\dots,N.$

For example, we will write AB to mean the medium formed by joining $A$ and
$B$, so that $c_{AB}(x)$ is equal to $c_{A}(x)$ for $0\leq x<1$ and equal to
$c_{B}(x-1)$ for $1\leq x<2$. We can now introduce the notion of a reflective
recursive medium, which will be the main object of study in this work:

######  Definition 2.2 (Recursive media).

Given two starting labels $A$ and $B$, we define a _sequence of recursive
media_ to be a sequence of tiling labels $\\{M_{N}:N\in\mathbb{N}\\}$ such
that $M_{1}$ is a tiling of finitely many labels from $\\{A,B\\}$ and, for
$N\geq 2$, $M_{N}$ is a tiling of $M_{N-1}$ with a combination of finitely
many labels from $\\{M_{1},\dots,M_{N-1}\\}$. That is, $M_{N}$ is given by

$M_{N}=M_{N-1}F(M_{1},\dots,M_{N-1}),$

where $F(M_{1},\dots,M_{N-1})$ is a tiling of finitely many elements of
$\\{M_{1},\dots,M_{N-1}\\}$.

######  Definition 2.3 (Reflected recursive media).

Given a sequence of recursive media $\\{M_{N}:N\in\mathbb{N}\\}$ we define a
_reflected recursive medium_ to be the medium obtained by letting $N\to\infty$
and then reflecting in $x=0$. That is, the medium described by the function
$\overline{c}_{M_{\infty}}:\mathbb{R}\to(0,\infty)$ which is such that

$\overline{c}_{M_{\infty}}(x)=\begin{cases}c_{M_{\infty}}(x)&x\geq 0,\\\
c_{M_{\infty}}(-x)&x<0.\end{cases}$

The class of reflective recursive media is broad and contains many different
types of media. For example both the periodic media $M_{\infty}=AAAA\dots$ and
$M_{\infty}=ABABABAB\dots$ fall into this framework (see Section 4 for more on
this). In Section 3, we will study a version of the famous Fibonacci
quasicrystal, which is not periodic. The spectral gaps of similar materials
were previously studied by [17, 18, 20, 21, 22, 23, 39]. This example will be
referred to as the Fibonacci medium and is given by the sequence of recursive
media $\\{F_{N}:N\in\mathbb{N}\\}$ defined by

$F_{1}=A,\quad F_{2}=AB\quad\text{and}\quad F_{N}=F_{N-1}F_{N-2}\quad\text{for
}N\geq 3.$ (2)

This sequence reads $F_{3}=ABA$, $F_{4}=ABAAB$, $F_{5}=ABAABABA$, and so on.
It is closely related to the Fibonacci sequence. In particular, the number of
$A$s in $F_{N}$ is equal to the $N$th Fibonacci number, while the number of
$B$s in $F_{N}$ is give by the $(N-1)$th Fibonacci number. The reflected
material formed by this quasicrystalline tiling rule is sketched in Figure 2.

Figure 2: The central quasicrystalline example considered in this work is
formed by a Fibonacci-type tiling rule, which is reflected to create an
interface. Localised edge modes are created, which decay exponentially away
from this interface.

###  2.2 Transfer matrices

We recall some basic properties of transfer matrices, which will be important
in what follows. See [26] for a review. Given a one-dimensional Helmholtz
equation, posed on a bounded domain of length $L>0$,

$\frac{\mathrm{d}^{2}u}{\mathrm{d}x^{2}}+\frac{\omega^{2}}{c^{2}(x)}u=0,\quad
x\in[0,L],$ (3)

where $c\not\equiv 0$, we can define a transfer matrix $T(\omega)$ to be such
that

$\begin{pmatrix}u(L)\\\
u^{\prime}(L)\end{pmatrix}=T(\omega)\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}.$ (4)

We can see that $T$ is given in terms of the functions $\psi_{1,\omega}$ and
$\psi_{2,\omega}$:

$\displaystyle\begin{cases}\left(\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+\frac{\omega^{2}}{c^{2}(x)}\right)\psi_{1,\lambda}=0,\\\
\psi_{1,\omega}(0)=1,\\\
\psi_{1,\omega}^{\prime}(0)=0,\end{cases}\quad\text{and}\qquad\begin{cases}\left(\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+\frac{\omega^{2}}{c^{2}(x)}\right)\psi_{2,\lambda}=0,\\\
\psi_{2,\omega}(0)=0,\\\ \psi_{2,\omega}^{\prime}(0)=1,\end{cases}$ (5)

as

$\displaystyle
T(\omega)=\begin{pmatrix}\psi_{1,\omega}(L)&\psi_{2,\omega}(L)\\\
\psi_{1,\omega}^{\prime}(L)&\psi_{2,\omega}^{\prime}(L)\end{pmatrix}.$ (6)

######  Lemma 2.4.

Any transfer matrix $T$ must be such that, for all frequencies
$\omega\in\mathbb{R}$, it holds that $T(\omega)\in\mathrm{SL}(2,\mathbb{R})$,
where $\mathrm{SL}(2,\mathbb{R})$ is the group of $2\times 2$ real matrices
with determinant 1.

###### Proof.

Using the definitions of $\psi_{1,\omega}$ and $\psi_{2,\omega}$, we have that

$\displaystyle 0$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:psi12}}}{{=}}\int_{0}^{L}\psi_{1,\omega}(x)\psi_{2,\omega}^{\prime\prime}(x)-\psi_{2,\omega}(x)\psi_{1,\omega}^{\prime\prime}(x)\>\mathrm{d}x$
(7)
$\displaystyle=\int_{0}^{L}\left(\psi_{1,\omega}(x)\psi_{2,\omega}^{\prime}(x)-\psi_{2,\omega}(x)\psi_{1,\omega}^{\prime}(x)\right)^{\prime}\>\mathrm{d}x$
(8)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:psi12}}}{{=}}\psi_{1,\omega}(L)\psi_{2,\omega}^{\prime}(L)-\psi_{2,\omega}(L)\psi_{1,\omega}^{\prime}(L)-1\stackrel{{\scriptstyle\eqref{eq:Texplicit}}}{{=}}\det(T(\omega))-1,$
(9)

so it follows that $\det(T(\omega))=1$ for all $\omega\in\mathbb{R}$. ∎

Since the determinant of a matrix is the product of its eigenvalues, elements
of $\mathrm{SL}(2,\mathbb{R})$ fall into one of three categories:

######  Lemma 2.5 (Partition of $\mathrm{SL}(2,\mathbb{R})$).

For a given a matrix $T\in\mathrm{SL}(2,\mathbb{R})$, one of the following
must hold:

  1. 1.

$T$ has two real eigenvalues $\lambda_{1}$ and $\lambda_{2}$ which satisfy
$|\lambda_{1}|<1$ and $|\lambda_{2}|>1$. In this case, $|\mathrm{tr}(T)|>2$
and $T$ is said to be hyperbolic.

  2. 2.

$T$ has one real eigenvalue $\lambda_{1}$ which satisfies $\lambda_{1}=\pm 1$.
In this case, $|\mathrm{tr}(T)|=2$ and $T$ is said to be parabolic.

  3. 3.

$T$ has two complex eigenvalues $\lambda_{1}$ and $\lambda_{2}$ which satisfy
$\lambda_{1}=\lambda_{2}^{*}$ (with ∗ denoting the complex conjugate). In this
case, $|\mathrm{tr}(T)|<2$ and $T$ is said to be elliptic.

Several other simple properties will also be useful. Firstly, you can multiply
transfer matrices to solve (3) when two domains have been joined together.
That is, if

$\begin{pmatrix}u(L/2)\\\
u^{\prime}(L/2)\end{pmatrix}=T_{1}(\omega)\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}\quad\text{and}\quad\begin{pmatrix}u(L)\\\
u^{\prime}(L)\end{pmatrix}=T_{2}(\omega)\begin{pmatrix}u(L/2)\\\
u^{\prime}(L/2)\end{pmatrix},$

then, clearly,

$\begin{pmatrix}u(L)\\\
u^{\prime}(L)\end{pmatrix}=T_{2}(\omega)T_{1}(\omega)\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}.$ (10)

In addition to this, since $T(\omega)$ is always invertible, we can write that

$\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}=T^{-1}(\omega)\begin{pmatrix}u(L)\\\
u^{\prime}(L)\end{pmatrix}.$ (11)

In order to be able to easily relate $T$ and $T^{-1}$, we will assume that $A$
and $B$ are each symmetric. In this case, the straightforward change of
variables $x\mapsto-x$ in (4) yields the following lemma, which was previously
used in [3]:

######  Lemma 2.6 (Symmetric units).

Given a function $c:[0,1]\to(0,\infty)$ which is symmetric in the sense that

$c(x)=c(1-x),\quad\text{for all }x\in[0,1],$ (12)

then the associated transfer matrix, as defined in (4), satisfies

$T(\omega)^{-1}=ST(\omega)S,\quad\text{where }S=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$

These properties mean that for reflected recursive media, as defined above, if
we know the transfer matrices $T_{A}$ and $T_{B}$ corresponding to the
starting media $A$ and $B$, then we are able to describe fully how waves
propagate through the medium in either direction. The assumption that $A$ and
$B$ are symmetric could be relaxed, however the resulting formulas would be
less concise. In the examples presented in Sections 3 and 4, we will take $A$
and $B$ to be homogeneous materials, which is already sufficient to reveal
spectra with many rich and exotic properties.

###  2.3 Main spectral results

In general, we will use the notation $T_{X}$ to denote the transfer matrix
associated to the medium with label $X$. In particular, given a sequence of
recursive media $\\{M_{N}:N\in\mathbb{N}\\}$ we can define a corresponding
sequence of transfer matrices $\\{T_{M_{N}}:N\in\mathbb{N}\\}$ to be such that

$\begin{pmatrix}u(N)\\\
u^{\prime}(N)\end{pmatrix}=T_{M_{N}}(\omega)\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix},$ (13)

and we have that if $M_{N}=M_{N-1}X_{1}^{N}X_{2}^{N}\cdots X_{m(N)}^{N}$ then
$T_{M_{N}}=T_{X_{m(N)}^{N}}\cdots T_{X_{2}^{N}}T_{X_{1}^{N}}T_{M_{N-1}}$.

######  Lemma 2.7 (Symmetric propagation).

Given a sequence of recursive media $\\{M_{N}:N\in\mathbb{N}\\}$ which is
based on intial media that are symmetric in the sense of (12), it holds that

$\begin{pmatrix}u(-N)\\\
u^{\prime}(-N)\end{pmatrix}=ST_{M_{N}}(\omega)S\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix},\quad\text{for all }N\in\mathbb{N}.$

###### Proof.

Let $N\in\mathbb{N}$ and suppose that we have some sequence of labels
$X_{1}^{N},X_{2}^{N},\dots,X_{M}^{N}\in\\{A,B\\}$, with
$M=M(N)=\sum_{n=1}^{N}m(n)$, such that $M_{N}=X_{1}^{N}X_{2}^{N}\dots
X_{M}^{N}$. Then, it holds that $T_{M_{N}}=T_{X_{M}^{N}}\dots
T_{X_{2}^{N}}T_{X_{1}^{N}}$. Conversely, we have that

$\displaystyle\begin{pmatrix}u(-N)\\\
u^{\prime}(-N)\end{pmatrix}=T_{X_{M}^{N}}^{-1}\dots
T_{X_{2}^{N}}^{-1}T_{X_{1}^{N}}^{-1}\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}=ST_{X_{M}^{N}}\dots
T_{X_{2}^{N}}T_{X_{1}^{N}}S\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}=ST_{M_{N}}S\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix},$

where we have used Lemma 2.6 to infer that
$T_{X_{n}^{N}}^{-1}=ST_{X_{n}^{N}}S$. ∎

######  Theorem 2.8.

Suppose that the function $\overline{c}_{M_{\infty}}$ describes a reflected
recursive medium which is based on initial media that are symmetric in the
sense of (12) and let $(\omega,u)\in\mathbb{R}\times L^{2}(\mathbb{R})$ be a
solution to the problem

$\frac{\mathrm{d}^{2}u}{\mathrm{d}x^{2}}+\frac{\omega^{2}}{\overline{c}_{M_{\infty}}^{2}(x)}u=0.$

Assuming that $\omega\in\mathbb{R}$ is such that

$\min\sigma(T_{M_{N}}(\omega))\to 0,\quad\text{as }N\to\infty,$ (I)

then $u(x)\to 0$ and $u^{\prime}(x)\to 0$ as $x\to\pm\infty$ with $u\not\equiv
0$ if and only if the normalised eigenvector associated to the smallest
eigenvalue of $T_{M_{N}}(\omega)$, denoted by $(v_{1,N}\ v_{2,N})^{\top}$,
satisfies

$\text{either}\quad
v_{1,N}=o\left(\frac{1}{\max\sigma(T_{M_{N}}(\omega))}\right)\quad\text{or}\quad
v_{2,N}=o\left(\frac{1}{\max\sigma(T_{M_{N}}(\omega))}\right)\quad\text{as
}N\to\infty.$ (II)

###### Proof.

In light of (13) and Lemma 2.7, the property that $u(x)\to 0$ and
$u^{\prime}(x)\to 0$ as $x\to\pm\infty$ is equivalent to $u(N)\to 0$ and
$u^{\prime}(N)\to 0$ as $N\to\pm\infty$. Notice also that the values of $u(0)$
and $u^{\prime}(0)$ fully determine $u(x)$, for a given $\omega$. It is also
important to realise that (I) implies that $T_{M_{N}}$ is hyperbolic for large
$N$, as it has an eigenvalue which decays to zero. This means, firstly, that
$\max\sigma(T_{M_{N}}(\omega))\to\infty$ as $N\to\infty$, such that (II) is a
decay condition on either $v_{1,N}$ or $v_{2,N}$. Secondly, it means that,
without loss of generality, we can assume that $T_{M_{N}}(\omega)$ is
hyperbolic for all $N$.

Suppose first that (I) and (II) hold. Thus, using hyperbolicity, we can write
that

$\begin{pmatrix}u(N)\\\
u^{\prime}(N)\end{pmatrix}=\begin{pmatrix}v_{1,N}&w_{1,N}\\\
v_{2,N}&w_{2,N}\end{pmatrix}\begin{pmatrix}\lambda_{1,N}&0\\\
0&\lambda_{2,N}\end{pmatrix}\begin{pmatrix}w_{2,N}&-w_{1,N}\\\
-v_{2,N}&v_{1,N}\end{pmatrix}\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix},$ (14)

where $|\lambda_{1,N}|<1$ and $\lambda_{2,N}=\lambda_{1,N}^{-1}$ are the
eigenvalues of $T_{M_{N}}(\omega)$ with associated normalised eigenvectors
$(v_{1,N},v_{2,N})^{\top}$ and $(w_{1,N},w_{2,N})^{\top}$. Some algebra gives

$\begin{pmatrix}u(N)\\\
u^{\prime}(N)\end{pmatrix}=\lambda_{1,N}\begin{pmatrix}v_{1,N}\\\
v_{2,N}\end{pmatrix}\begin{pmatrix}w_{2,N}&-w_{1,N}\end{pmatrix}\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}+\lambda_{2,N}\begin{pmatrix}w_{1,N}\\\
w_{2,N}\end{pmatrix}\begin{pmatrix}-v_{2,N}&v_{1,N}\end{pmatrix}\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}.$ (15)

We can use Lemma 2.7 to derive a similar formula for $-N$:

$\begin{pmatrix}u(-N)\\\
-u^{\prime}(-N)\end{pmatrix}=\lambda_{1,N}\begin{pmatrix}v_{1,N}\\\
v_{2,N}\end{pmatrix}\begin{pmatrix}w_{2,N}&-w_{1,N}\end{pmatrix}\begin{pmatrix}u(0)\\\
-u^{\prime}(0)\end{pmatrix}+\lambda_{2,N}\begin{pmatrix}w_{1,N}\\\
w_{2,N}\end{pmatrix}\begin{pmatrix}-v_{2,N}&v_{1,N}\end{pmatrix}\begin{pmatrix}u(0)\\\
-u^{\prime}(0)\end{pmatrix}.$ (16)

Assumption (I) means that $\lambda_{1,N}\to 0$ as $N\to\infty$ so first term
in both (15) and (16) vanishes for large $N$. We can take care of the second
terms thanks to (II), since choosing either $u(0)=0$ (if $v_{1,N}\to 0$) or
$u^{\prime}(0)=0$ (if $v_{2,N}\to 0$) means these terms will also vanish as
$N\to\infty$. Therefore, $u(x)\to 0$ and $u^{\prime}(x)\to 0$ as
$x\to\pm\infty$.

Conversely, suppose that (I) holds and that $u(x)\to 0$ and $u^{\prime}(x)\to
0$ as $x\to\pm\infty$. Once again, we can use hyperbolicity and use (15) and
(16) to see that, thanks to (I),

$\lim_{N\to\infty}\lambda_{2,N}\begin{pmatrix}-v_{2,N}&v_{1,N}\end{pmatrix}\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix}=0\quad\text{and}\quad\lim_{N\to\infty}\lambda_{2,N}\begin{pmatrix}-v_{2,N}&v_{1,N}\end{pmatrix}\begin{pmatrix}u(0)\\\
-u^{\prime}(0)\end{pmatrix}=0.$ (17)

Adding and subtracting these gives that

$\left(\lim_{N\to\infty}\lambda_{2,N}\,v_{2,N}\right)u(0)=0\quad\text{and}\quad\left(\lim_{N\to\infty}\lambda_{2,N}\,v_{1,N}\right)u^{\prime}(0)=0.$
(18)

Since $(v_{1,N}\ v_{2,N})^{\top}$ is a unit vector (for all finite $N$) and
(I) implies that $\lambda_{2,N}\to\infty$, we must have that (II) holds in
order for $u(0)$ and $u^{\prime}(0)$ to not be both zero. ∎

Theorem 2.8 is the main result of this work, the implications of which we will
explore in the following sections. We will refer to the condition (I) as the
spectral gap condition as it describes which frequencies are able to propagate
arbitrarily far through the medium: those that are not are said to be in a
spectral gap of the structure. The condition (II) will be referred to as the
edge mode condition. While (I) describes the required behaviour of the
eigenvalues of the sequence of transfer matrices, (II) is the required
condition for the solution to be in the eigenspace corresponding to the
decaying eigenvalue. The analogue of this result for periodic structures was
proved in Theorem 2.4 of [3].

The main challenge with understanding the conditions (I) and (II) is dealing
with the fact that, typically, $\|T_{M_{N}}\|\to\infty$ as $N\to\infty$. This
property is well known in the setting of random media [40]. We can employ some
tricks to make this easier to handle, both numerically and for making analytic
statements. For the edge mode condition (II) we can normalise the transfer
matrices $T_{M_{N}}$ to have unit norm, since the normalised eigenvectors are
unchanged. For the spectral gap condition (I) it will be more convenient to
study the trace of the transfer matrices, instead of their minimum eigenvalue.
Using Lemma 2.5, and the fact that the trace of a matrix is the sum of its
eigenvalues, we have the following result:

######  Lemma 2.9 (Transfer matrix traces).

Given a sequence of transfer matrices $\\{T_{N}\in\mathbb{R}^{2\times
2}:N\in\mathbb{N}\\}$, it holds that $\min\sigma(T_{N})\to 0$ as $N\to\infty$
if and only if $|\mathrm{tr}(T_{N})|\to\infty$ as $N\to\infty$.

This is useful as it means the spectral gap condition (I) can be understood by
dealing with a scalar quantity which, in the case of the Fibonacci medium
studied in the next section, will satisfy a simple recursion relation.

######  Remark 2.10 (Defects).

The convenient form of the edge mode condition (II) is due to the choice of
reflectional symmetry for the interface. If other interfaces or defects were
chosen, this would need to be modified accordingly. For example, the approach
developed here could be combined with the methods of [3] to model localised
modes in recursive media with a defect (with finite size) introduced. This
defect could, for example, be a patch of local periodicity within the
otherwise quasicrystalline structure; a material of this type was proposed by
[41] (shown in Fig. 3). Introducing dislocations to periodic structures is a
well-known approach for creating edge modes that are topologically protected
and which have tunable properties [42, 43].

######  Remark 2.11 (Bound states in the continuum).

This work does not say anything about the existence of bound states in the
continuum. By this we mean states which are localised (so $u(x)\to 0$ and
$u^{\prime}(x)\to 0$ as $x\to\pm\infty$) but for which neither of the
eigenvalues of $T_{M_{N}}(\omega)$ converges to zero as $N\to\infty$ (so there
exists $c>0$ such that $\min\sigma(T_{M_{N}})\geq c$ for all $N$). These have
been studied in a range of different periodic and finite-dimensional systems,
where they typically arise due to interactions between two nearby resonances
with different quality factors [44, 45, 46].

##  3 Application to a Fibonacci quasicrystal

###  3.1 Definition and basic properties

The main example that we will consider in this work is the Fibonacci medium
that was defined in (2) and is given by the sequence of recursive media
$\\{F_{N}:N\in\mathbb{N}\\}$ defined by $F_{1}=A$, $F_{2}=AB$ and
$F_{N}=F_{N-1}F_{N-2}$ for $N\geq 3$. This sequence is closely related to the
Fibonacci sequence. In particular, the number of $A$s in $F_{N}$ is equal to
the $N$th Fibonacci number, the number of $B$s in $F_{N}$ is given by the
$(N-1)$th Fibonacci number and the total number of labels in $F_{N}$ is given
by the $(N+1)$th Fibonacci number.

For simplicity, we will consider the specific case where $A$ and $B$ are both
homogeneous materials, such that $c(x)=1$ in $A$ and $c(x)=r$ in $B$, with $r$
assumed to be positive and not equal to 1. These materials satisfy the
required assumptions of symmetry and their associated transfer matrices have
convenient expressions, given by

$T_{A}=\begin{pmatrix}\cos(\omega)&\frac{1}{\omega}\sin(\omega)\\\
-\omega\sin(\omega)&\cos(\omega)\end{pmatrix}\qquad\text{and}\qquad
T_{B}=\begin{pmatrix}\cos(r\omega)&\frac{1}{r\omega}\sin(r\omega)\\\
-r\omega\sin(r\omega)&\cos(r\omega)\end{pmatrix}.$ (19)

###  3.2 Spectral gap condition

The spectral properties of the Fibonacci material $F_{N}$, for
$N\in\mathbb{N}$, have been studied _e.g._ by [17, 18, 19, 20, 21, 22, 23].
These studies typically concern the case where some finite $N$ is fixed and
then $F_{N}$ is used as the repeating unit cell in a periodic material. They
show several important properties, such as its self similarity [22], the
Cantor-like emergence of nested spectral gaps [22] and the positions of the
largest spectral gaps [21, 19]. A similar result was obtained by [39], where
the positions of the main spectral gaps was predicted using a simple
homogenisation approach that counts the number of $A$s and $B$s in each
$F_{N}$.

Given Lemma 2.9, we can recast the spectral gap condition (I) from Theorem 2.8
in terms of the trace of the transfer matrices. This is helpful since we can
derive a recursion relation for the trace of $T_{F_{N}}$, a result first shown
by [17] and repeated here for completeness:

######  Lemma 3.1 (Recursion relation).

Let $x_{N}=\mathrm{tr}(T_{F_{N}})$ where $F_{N}$ is the Fibonacci medium
defined in (2). Then, it holds that

$x_{N+1}=x_{N}x_{N-1}-x_{N-2},\quad\text{for }N\geq 3.$

###### Proof.

We have that $T_{F_{N}}=T_{F_{N-2}}T_{F_{N-1}}$ for $N\geq 3$, which can be
rearranged to give $T_{F_{N-2}}^{-1}=T_{F_{N-1}}T_{F_{N}}^{-1}$. Similarly,
$T_{F_{N+1}}=T_{F_{N-1}}T_{F_{N}}$. Adding these two equations gives

$T_{F_{N+1}}+T_{F_{N-2}}^{-1}=T_{F_{N-1}}T_{F_{N}}+T_{F_{N-1}}T_{F_{N}}^{-1}.$
(20)

Taking the trace and using the fact that
$\mathrm{tr}(T_{F_{N-1}}T_{F_{N}})+\mathrm{tr}(T_{F_{N-1}}T_{F_{N}}^{-1})=\mathrm{tr}(T_{F_{N-1}})\mathrm{tr}(T_{F_{N}})$
gives the result. ∎

Lemma 2.9 tells us that a frequency satisfies the spectral gap condition (I)
if the sequence of traces $x_{N}$ diverges. Two examples of these sequences
$x_{N}$ are shown in Figure 3, one of which remains bounded (its absolute
value is always less than 2, at least for the first 100 terms computed) and
one which diverges quickly. Understanding the difference between these two
cases is challenging. The system appears to exhibit chaotic behaviour, in the
sense that a small change in the starting values of the sequence can cause the
system to switch between the two cases. This behaviour is often observed in
non-linear recursion relations.

Studying Figure 3 might lead to the hypothesis that if $|x_{n}|>2$ for some
$n$, then $x_{N}$ will diverge as $N\to\infty$. This is compounded by noticing
that many sequences behave like the one shown in Figure 4(a), which is bounded
in absolute value by 2 but then diverges quickly as soon as it records a value
with magnitude greater than 2. Furthermore, similar results hold for other
second-order non-linear recursion relations. For example, it is known that if
the recursion relation that defines the Mandelbrot set ever exceeds $\pm 2$,
then it will diverge. In this case, however, no such result is possible and it
turns out that there are periodic orbits which take arbitrarily large values.
An example of such an orbit is shown in Figure 4(b), which is periodic with
period 6 and can be chosen such that the multiple of 3 terms take arbitrarily
large values. This periodic orbit is used to prove the following lemma, which
makes this notion precise.

(a) $r=0.92$, $\omega=1.65$.

(b) $r=0.92$, $\omega=1.5$.

Figure 3: The boundedness or divergence of the sequence of transfer matrix
traces determines if a frequency is in a spectral gap. For the
quasicrystalline Fibonacci medium, (a) shows a frequency for which the
sequence is bounded, at least for the first 100 iterations, whereas (b) shows
a frequency for which the sequence diverges quickly.

######  Lemma 3.2 (Arbitrarily large bounded sequences).

There exist parameter values $(\omega,r)\in(0,\infty)^{2}$ such that the
solutions to the recursion relation from Lemma 3.1 can be arbitrarily large
but the spectral gap condition (I) is not satisfied. That is, given any
$K\in(0,\infty)$, we can find $(\omega,r)\in(0,\infty)^{2}$ such that the
corresponding sequence of traces, as defined in Lemma 3.1, satisfies
$K<\max_{N\in\mathbb{N}}|x_{N}|<\infty$.

###### Proof.

First, notice that if $x_{1}=0$, $x_{2}\neq 0$ and $x_{3}=0$, then the
recursion relation will give a sequence that is periodic with period 6, given
by $0,x_{2},0,0,-x_{2},0,0,x_{2},0,0,-x_{2},0,\dots$. If we choose $\omega$
and $r$ such that

$\omega=(2n+1)\frac{\pi}{2}\quad\text{and}\quad
r=\frac{2m+1}{2n+1},\quad\text{for }n,m\in\mathbb{N},$ (21)

then we have that $x_{1}=0$, $|x_{2}|=r+\tfrac{1}{r}$ and $x_{3}=0$. Let
$K\in(0,\infty)$. For any given $n$, we can choose $m$ sufficiently large such
that $|x_{2}|>K$. ∎

(a) $r=0.92$, $\omega=1.55$.

(b) $r=\tfrac{7}{3}$, $\omega=\tfrac{3\pi}{2}$.

Figure 4: Sequences of transfer matrix traces, corresponding to the
quasicrystalline Fibonacci medium, can take arbitrarily large values but be
bounded. (a) Many orbits are observed to be bounded by $\pm 2$ (shown in
dashed lines) before diverging quickly once they escape this region. However,
it is not generally true that once an orbit leaves $[-2,2]$ it will diverge.
For example, the orbit in (b) is periodic with period 6 but can be chosen so
that it periodically takes arbitrarily large values. In this case, every third
term alternates between $x_{3}=-\tfrac{58}{21}\approx-2.76$ and
$x_{6}=\tfrac{58}{21}\approx 2.76$. However, increasing $r$ can cause these
terms to take a value that is arbitrarily large (see Lemma 3.2).

In spite of the fact that it is not possible to prove a simple bound for the
maximum permitted value of $|x_{N}|$ in a bounded sequence, we are able to
prove a similarly useful result. In particular, the following lemma says that
if $x_{N}$ has left the interval $[-2,2]$ and starts doubling, in the sense
that three successive terms are each more than double the previous term, then
it will continue doing so indefinitely.

######  Lemma 3.3 (Doubling condition).

The recursion relation from Lemma 3.1 has the property that if there is some
$n\in\mathbb{N}$ such that $2<|x_{n-2}|$, $2|x_{n-2}|<|x_{n-1}|$ and
$2|x_{n-1}|<|x_{n}|$, then $2|x_{N}|<|x_{N+1}|$ for all $N\geq n$.

###### Proof.

Let $\gamma=|x_{n-2}|$ and let $\alpha=|x_{n}|$. Then, by hypothesis, we have
that $\gamma>2$ and $\alpha>4\gamma$. Subsequently, we see that

$|x_{n+1}|\geq|x_{n}||x_{n-1}|-|x_{n-2}|>2\alpha\gamma-\gamma=2\alpha(\gamma-1)+2\alpha-\gamma>2\alpha,$
(22)

where the last inequality follows since $\gamma-1>1$ and $2\alpha-\gamma>0$.
This shows that $|x_{n+1}|>2|x_{n}|$. For $N>n$, the result follows by
induction. ∎

Lemma 3.3 is the key to our numerical search for parameter values satisfying
(I) as we know that once this condition is satisfied we can stop iterating as
the sequence is guaranteed to diverge. For example, the sequence in Figure
3(b) satisfies the doubling condition from Lemma 3.3 after 44 iterations. The
result of performing this search for different values of $\omega$ and $r$ is
shown in Figure 5. Figure 5(a) shows spectral gaps, where (I) holds, in white.
This is designed to be analogous to a Bloch band diagram for a periodic
structure. Figure 5(b) is the same plot but with the spectral gaps coloured
according to the number of iterations that was required for the doubling
condition from Lemma 3.3 to be attained.

(a) The points at which the spectral gap condition (I) does not hold are shown
in black, such that the spectral gaps appear in white. Two zoomed regions are
shown in the plots on the right.

(b) The points at which the spectral gap condition (I) does not hold are shown
in black and points in the spectral gaps are coloured according to the number
of iterations that was required for the doubling condition from Lemma 3.3 to
be satisfied. Two zoomed regions are shown in the plots on the right.

Figure 5: The Fibonacci medium exhibits a complex structure of spectral gaps,
that has self-similar and fractal properties. For each pair of material
contrast $r$ and frequency $\omega$ in this plot, we check if the spectral gap
condition (I) from Theorem 2.8 holds by iterating the sequence of traces
(using Lemma 3.1) until the doubling condition from Lemma 3.3 is attained.

Figure 5 shows an exotic pattern of spectral gaps. Particularly from the
zoomed in plots (on the right), we can see a fractal-like pattern of gaps
emerging. Some of the main features of Figure 5 can be explained easily.
First, the lack of spectral gaps when $r=1$ is due to the two materials, $A$
and $B$, being the same in this case. Additionally, there are horizontal and
hyperbolic stripes where there are no spectral gaps. In particular, these are
when either $\omega=(2n+1)\tfrac{\pi}{2}$ or $r\omega=(2n+1)\tfrac{\pi}{2}$,
for some $n\in\mathbb{N}$, in which case either $T_{A}$ or $T_{B}$ is equal to
the $2\times 2$ identity matrix (up to a factor of $\pm 1$).

######  Remark 3.4 (Rainbow devices).

Given the rich hierarchy of spectral gaps exhibited by the Fibonacci
structure, some of which are very large (_cf._ [8]), there are many other
potential applications of this material, beyond the symmetry-induced
waveguides considered in this work. For example, Figure 5 shows how the
spectral gaps are shifted by modulating the material contrast parameter. This
property could be used to create a _rainbow device_ which has spatially graded
material parameters and, as a result of the local spectral gap being gradually
shifted, localises different frequencies at different locations. Rainbow
devices based on similar graded materials have been used successfully in many
different settings: see [47, 48] and references therein.

###  3.3 Edge mode condition

The edge mode condition (II) is slightly easier to handle than the spectral
gap condition (I). In particular, we can normalise the transfer matrices,
since we only need the directions of the eigenvectors, which deals with the
issues posed by the fact that $\|T_{M_{N}}\|\to\infty$ as $N\to\infty$. Thus,
given a set of values $\omega$ and $r$ which falls in a spectral gap, we can
test if an edge mode is supported by iterating the sequence of transfer
matrices and checking if (II) is checking for large iteration numbers. The
results of doing this are shown in Figure 6, where we examine two zoomed in
regions of the spectrum and check values in the main spectral gaps for the
edge mode condition (II).

Two examples of the localised edge modes are shown in Figure 7. These modes
are computed for a finite-sized piece of the material, using finite
differences. We take material contrast parameter $r=2$ and study a material
with 55 pieces of material on either side of the interface (that is, $N=10$ in
the Fibonacci tiling algorithm). Second-order finite difference approximations
are used, with a step size $h=0.01$, and Dirichlet boundary conditions are
imposed at either end of the material. It is noticeable that while the modes
fluctuate a little as they decay, particularly at higher frequencies, they
nonetheless have a strongly decaying envelope.

Edge modeEdge modes Figure 6: The reflected Fibonacci medium supports
localised edge modes within its spectral gaps. We check if the edge mode
condition (II) from Theorem 2.8 holds for different material contrasts $r$ and
frequencies $\omega$ lying within spectral gaps, as determined by (I).
Spectral gaps are shown in white, as in Figure 5, and the red lines denote the
locations of predicted localised edge modes. Two different zoomed in regions
of the spectrum are shown.

(a) $\omega=1.416$.

(b) $\omega=4.869$.

Figure 7: The localised edge modes supported by the reflected Fibonacci medium
decay exponentially quickly away from the interface. We take material contrast
parameter $r=2$ and study a finite material with 55 pieces of material on
either side of the interface (the corresponds to $N=10$). The modes are shown
alongside an approximate decay rate (the dotted line), which is approximated
using transfer matrix eigenvalues and the formula (24).

In addition to predicting the frequencies at which localised edge modes will
exist, it is useful to estimate the rate at which they decay away from the
interface. In one-dimensional waveguides based on periodic media, this can be
estimated using the eigenvalues of the transfer matrices on either side of the
interface, see [3]. In our quasicrystalline setting, an approximate bound on
the rate of decay can be similarly achieved by studying the decaying sequence
of transfer matrix eigenvalues. In particular, the edge mode condition (II)
means the mode lies in the limiting eigenspace of the decaying eigenvalues.
Thus, if $N$ is large, we can make the approximation

$\begin{pmatrix}u(\\#_{F_{N}})\\\
u^{\prime}(\\#_{F_{N}})\end{pmatrix}\approx\min\sigma(T_{F_{N}}(\omega))\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix},$ (23)

where $\\#_{F_{N}}$ is the number of labels in $F_{N}$ (and, since $A$ and $B$
are assumed to have unit length, is equal to the length of $F_{N}$). Assuming
that the decay rate is approximately constant across the length of the
material, we reach the expression

$\begin{pmatrix}u(x)\\\
u^{\prime}(x)\end{pmatrix}\approx\exp\left(\frac{\log(\min\sigma(T_{F_{N}}))}{\\#_{F_{N}}}|x|\right)\begin{pmatrix}u(0)\\\
u^{\prime}(0)\end{pmatrix},$ (24)

where we want $N$ to be large, in order to get a reasonable approximation.
Note that since $\min\sigma(T_{F_{N}}))\to 0$ as $N\to\infty$,
$\log(\min\sigma(T_{F_{N}})))<0$ for large $N$ so we have an exponentially
decaying mode. Figure 7 shows two localised edge modes along with the
approximate envelope from (24). In this case, we take $N=10$ (so
$\\#_{F_{N}}=55$) and get a reasonable prediction of the overall shapes of the
modes (in spite of the crude approximations that were used to derive (24)).

######  Remark 3.5 (Lyapunov exponents).

In the setting of random media, localisation lengths are typically quantified
using Lyapunov exponents [49, 50, 51, 52]. Using our notation, if
$\\#_{M_{N}}\in\mathbb{N}$ is the number of labels in $M_{N}$, then the
(maximal) Lyapunov exponent is defined as
$\gamma=\lim_{N\to\infty}{\log\|T_{M_{N}}\|}/{\\#_{M_{N}}}$. The fact that
$\gamma\geq 0$ in general for random media follows from Furstenberg [40]. The
quantity $1/\gamma$ can then be interpreted as a measure of the localisation
length [49, 50, 40]. For the Fibonacci medium studied here, which is
deterministic, the Lyapunov exponent can be computed explicitly. It is also
easy to see that
$0\leq\gamma\leq\varphi^{-1}\log\|T_{A}\|+\varphi^{-2}\log\|T_{B}\|$, where
$\varphi=(1+\sqrt{5})/2$ is the golden ratio. However, in spite of being easy
to work with, the Lyapunov exponent provides a much less tight bound on the
decay of the eigenmode than the eigenvalue sequence used in (24). This is to
be expected, since the Lyapunov exponent $\gamma$ doesn’t contain any
information about the reflection that induced the defect. Conversely,
information about the specific choice of defect was crucial to the derivation
of (24) (and we could modify the result accordingly for different types of
defects and interfaces).

##  4 Application to a periodic material

The framework developed in Section 2 applies to any material that can be
defined by a recursive tiling rule. While quasicrystalline media are the
interesting case for which the theory was developed, these methods can also be
used to describe periodic media. For example, given an initial material
$P_{1}$ (which we would prefer to be inhomogeneous, to give interesting
phenomena), we define $\\{P_{N}:N\in\mathbb{N}\\}$ as

$P_{N+1}=P_{N}P_{1}.$ (25)

The reflected material based on this rule is depicted in Figure 8, where
$P_{1}=AB$.

Figure 8: The theory developed in this work can also be used to model
reflected periodic media, which support localised edge modes that decay away
from the reflection-induced interface.

In the case of this periodic medium, the conditions (I) and (II) from Theorem
2.8 are particularly easy to understand. In fact, they reduce to the formulas
that were derived in [3]. The periodicity means that successive transfer
matrices $T_{P_{N}}$ are easy to characterise since
$T_{P_{N}}=(T_{P_{1}})^{N}$. This means that the spectral gap condition (I),
which says that $\min\sigma(T_{P_{N}})\to 0$, is equivalent to the condition
that the minimum eigenvalue of $T_{P_{1}}$ has magnitude less than 1. This
simple condition, leads directly to the pattern of spectral gaps plotted in
Figure 9.

Figure 9 also shows, in the two right-hand plots, the localised edge modes
that exist for certain parameter values within the spectral gaps. This is
based on (II) from Theorem 2.8, which is also straightforward in the case of a
periodic material since the eigenvectors of $T_{P_{N}}=(T_{P_{1}})^{N}$ are
the same as those of $T_{P_{1}}$. This means we simply need to check if the
eigenvector associated to the smaller eigenvalue of $T_{P_{1}}$ is parallel to
either $(1,0)^{\top}$ or $(0,1)^{\top}$. Some examples of the periodic modes
are shown in Figure 10, computed using a finite-difference approximation of a
finite-sized piece of the material (analogous to Figure 7). Here, upper bounds
for the decay rates are shown, according to the formula

$|u(x)|\leq\exp\left[\log(\min\sigma(T_{P_{1}}))|x|\right],$ (26)

where $\log(\min\sigma(T_{P_{1}}))<0$ since $\min\sigma(T_{P_{1}})<1$. This
bound was proved in Corollary 2.5 of [3] (where tighter estimates for the
decay rate were computed using high-frequency homogenisation).

Figure 9: The theory developed in this work can be used to characterise the
spectral gaps of periodic media, as well as the localised edge modes created
by introducing an axis of reflectional symmetry. On the left, the pattern of
spectral gaps is shown, by checking if the condition (I) is satisfied.
Spectral gaps are shown in white (this is analogous to Figure 5(a) for the
Fibonacci material). On the right, values of the material contrast $r$ and the
frequency $\omega$, that lie within the spectral gaps, are checked for the
edge mode condition (II). Point supporting edge modes are shown in red (this
is the analogue of Figure 6).

(a) $\omega=1.916$.

(b) $\omega=4.383$.

Figure 10: The reflected periodic structure supports localised modes supported
that decay away from the interface. These modes are simulated for a finite
material with 55 pieces of material on either side of the interface, with
material contrast parameter $r=2$. The modes decay exponentially quickly, at a
rate that can be bound from above using the eigenvalues of the transfer
matrices (shown with a dotted line).

######  Remark 4.1 (The Mandelbrot set).

A natural question to ask, when faced with materials that yield second-order
recursion relations like the one in Lemma 3.1, is whether any particular
choices of reflected recursive materials lead to relations that have been
studied previously. The most notable, and widely studied, of which is the
quadratic recursion relation that defines the Mandelbrot set: the Mandelbrot
set is given by $c\in\mathbb{C}$ such that the recursion relation
$z_{N+1}=z_{N}^{2}+c$ with $z_{1}=c$ remains bounded for all $N\in\mathbb{N}$.
If we have a periodic medium which is given by $P_{N+1}=P_{N}P_{N}$, then it
holds that $T_{P_{N+1}}=T_{P_{N}}^{2}$ and taking the trace of this gives that
$y_{N}=\mathrm{tr}(T_{P_{N}})$ satisfies

$y_{N}=y_{N-1}^{2}-2\det(T_{P_{N}})=y_{N-1}^{2}-2.$ (27)

Since $c=-2$ is in the Mandelbrot set, if $\omega$ is such that the starting
material satisfies $\mathrm{tr}(T_{P_{1}}(\omega))=-2$, then $\omega$ is in a
spectral band of the material (_i.e._ (I) from Theorem 2.8 is not satisfied).
For example, if $P_{1}$ is a homogeneous material of unit length, then we have
that $\mathrm{tr}(T_{P_{1}}(\omega))=2\cos(\omega)$, from which we can deduce
that $\omega=(2n+1)\pi$ is in a spectral band, for all $n\in\mathbb{N}$. While
invoking properties of the Mandelbrot set to prove that a homogeneous material
is transparent at certain frequencies is clearly a wasted effort, this
suggests a valuable path of future investigation: using the known properties
of widely studied fractals (see _e.g._ [53] for a summary) to reveal
properties of novel quasicrystalline media without the need for extensive new
analysis.

##  5 Robustness of quasicrystalline waveguides

An important consideration when designing waveguides that are based on making
perturbations to induce edge modes is their stability with respect to
imperfections. When devices are manufactured, errors and defects will
inevitably be introduced and it is important to understand how these
undesirable perturbations interact with the one that was made to create the
interface. Thus, for the waveguide to be of practical use, it is important
that it retains its waveguiding properties when random errors are introduced
to the structure.

Most metamaterial devices inherently have some robustness, in the sense that
their properties depend continuously on small imperfections [48]. However, a
significant goal in the field is to optimise this robustness. The main
breakthrough in this direction has been the discovery of so-called
_topologically protected_ edge modes, which leverage topological properties of
the underlying periodic media to create modes which experience enhanced
robustness to imperfections [5, 6]. Previous studies have successfully defined
topological indices associated to quasicrystalline media, such as the
integrated density of states (IDS) [38, 33], and [54] showed how these ideas
could be extended to discrete Fibonacci systems. However, it is not generally
clear how to link these properties to the robustness of edge modes induced by
perturbations such as reflections.

Some studies have pointed to quasicrystalline media having appealing
robustness properties [10, 9, 11, 12]. A possible (and very approximate)
explanation of this is that since quasicrystals already appear to be
‘disordered’ on short scales, and the underlying order is only clear when you
examine the long-range patterns in the structure, making local perturbations
is less disruptive to its properties than in a periodic medium. This
robustness was shown qualitatively for the specific case of reflection-induced
edge modes (in arrays of point scatterers on elastic plates) by [10]. We can
likewise examine this hypothesis in the setting of the reflection-induced
waveguides studied here.

Our robustness study is based on perturbing the values of the material
parameters $c$. Both the quasicrystalline Fibonacci example presented in
Section 3 and the periodic example presented in Section 4 are based on
piecewise constant building blocks of unit length. We introduce random
imperfections to the structure by generating independent random variables
$X_{n}\sim N(0,\sigma^{2})$, for $n\in\mathbb{Z}$, and making perturbations on
each constant building block according to

$c(x)\mapsto(1+X_{n})c(x),\quad x\in[n,n+1),n\in\mathbb{Z}.$ (28)

We additionally enforce that $c>0$ by taking $\max\\{c,\epsilon\\}$ for some
small $\epsilon$ (for the simulations presented here, $\epsilon=10^{-3}$ is
used). Parts of the spectra, perturbed according to (28), are shown in Figure
11 for several different geometries. In each case, increasing perturbation
size $\sigma$ is shown on the horizontal axis with the eigenfrequencies of the
localised modes shown with red diamonds. In each case, the profile of the
unperturbed ($\sigma=0$) localised mode is shown inset. The same finite-
difference approximations that were used for Figures 7 and 10 were used for
these simulations, with step size $h=0.01$.

(a) Fibonacci.

(b) Periodic.

(c) Stretched periodic.

Figure 11: A comparison of the robustness of the reflection-induced edge modes
in Fibonacci and periodic media. We repeatedly compute the spectrum for
randomly perturbed media, for 100 different perturbations sizes $\sigma$ that
are evenly spaced between 0 and 0.1, with $\sigma$ being the standard
deviation of the normally distributed perturbations $N(0,\sigma^{2})$. Each
point corresponds to an element of the spectrum, with the red diamonds
indicating modes that are localised (the unperturbed localised mode is shown
inset). In each case, we simulate materials composed of 110 units (55 either
side of the interface) with material contrast $r=2$.

The initial conclusion to draw from Figure 11(a) is that the Fibonacci
material exhibits strong robustness with respect to imperfections. Even for
large perturbation sizes, the localised mode persists and its eigenfrequency
appears to remain stable. This is particularly striking when compared with
Figure 11(b), which is the equivalent plot for the periodic material, as
sketched in Figure 8. The eigenfrequency is greatly affected by the random
perturbations and is mixed up with other localised modes that are created by
the perturbations. However, another important difference between these two
materials is the rate at which the unperturbed eigenmodes decay (the modes are
shown inset in Figure 11). The reflection induced edge mode in the periodic
material decays significantly faster than the mode in the Fibonacci
quasicrystal. If we construct a ‘stretched’ version of the periodic material,
with $P_{1}=AAAABBBB$ instead of $P_{1}=AB$, then the localised mode is
similarly stretched by a factor of (approximately) four, such that its decay
rate is similar to the mode in the Fibonacci quasicrystal. A robustness
analysis for this configuration is shown in Figure 11(c), where we see that
the eigenfrequency of the localised mode is much more robust, with a degree of
stability that is comparable to that of the Fibonacci quasicrystal.

The conclusion to draw from this brief robustness study is that while
quasicrystalline waveguides may have some beneficial robustness properties,
care is needed to make comparisons with _e.g._ periodic waveguides. For
example, the robustness benefits may not be clear if decay rates are matched.
Whether the comparison should be made between materials with the same sized
building blocks or between those that exhibit similar decay rates may depend
on the specific application that one has in mind.

##  6 Concluding remarks

This work has established general results for characterising reflection-
induced localised modes in materials formed by recursive tiling rules. We have
demonstrated the power of this framework by using it to study these modes in
the famous Fibonacci quasicrystal. This revealed the rich fractal structure of
spectral gaps and demonstrated that adding a reflection creates exponentially
localised modes with eigenfrequencies within these spectral gaps. We also
showed that these symmetry-induced quasicrystalline waveguides may enjoy
enhanced robustness properties, as observed by previous studies [10, 9, 11,
12], but that the comparison with the equivalent periodic waveguide requires
care and depends on the specific application in mind.

One of the main challenges with studying quasicrystalline media is that it is
difficult to formulate general statements and methods. Many previous studies
have used methods that rely on the specific properties of the tiling rule, so
yield results that are specifically tailored to the quasicrystal in question.
This is also true of the results in Section 3 of this work, for example, many
of which rely on the recursion relation from Lemma 3.1. However, the general
framework established in Section 2 provides a starting point for more
systematic studies, which could reveal the properties of the vast collection
of unexplored quasicrystals.

A notable feature of the approach developed in this work is that the
theoretical results are computationally expensive to implement numerically.
The results are iterative in nature, as they rely on the recursive tiling
rules that define the materials, and require many iterations to be computed in
order to understand the materials’ properties. This is common to many studies
[21, 22, 23, 20, 19]. However, as demonstrated in Section 4 of this work, the
application of this theory to periodic materials is significantly more
straightforward. It might be possible to take advantage of the fact that
quasicrystals are projections of higher-dimensional periodic structures to
perform this analysis more concisely. Similar _superspace lifting_ approaches
have been used previously to reveal new insight into the properties of
quasicrystals [29, 30, 31], however relating the eigenmodes of the
quasicrystal and the lifted periodic system is non-trivial in general.

The results of this paper, and related works, are reducing the barriers to
using quasicrystals in waveguiding devices. Thanks to the convenience of
Floquet-Bloch analysis, periodic waveguides are ubiquitous in wave physics.
Conversely, the wave transmission properties of quasicrystals can be
fiendishly difficult to understand, so their use in applications has been
limited. Given their exciting properties (such as large spectral gaps and
robustness), the rapidly growing sophistication of techniques for studying
quasicrystalline wave systems has the potential to unlock a new class of
powerful wave devices based on quasicrystals.

## Acknowledgements

BD and RVC were funded by the EC-H2020 FETOpen project BOHEME under grant
agreement 863179. The authors would like to thank Jordan Docking for pointing
out the periodic orbit that was used in Lemma 3.2 as well as Ian Hooper and
Henry Putley for their insightful comments that were valuable in refining the
robustness analysis presented in Section 5. The code used for the numerical
examples presented in this article is available at
https://doi.org/10.5281/zenodo.7006818.

## Competing interests

The authors declare no competing interests.

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