# Lasing condition for trapped modes in subwavelength–wired PT–symmetric
resonators

Mauro Cuevas  Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET) and Universidad Austral, Facultad de Ingeniería, Pilar Mariano
Acosta 1611 B1629WWA-Pilar-Buenos Aires Argentina  Mojtaba Karimi  Faculty of
Physics, University of Tabriz, 51664, Tabriz, Iran  Carlos J. Zapata–Rodríguez
Department of Optics and Optometry and Vision Science, University of Valencia,
Dr. Moliner 50, Burjassot 46100, Spain<EMAIL_ADDRESS>
###### Abstract

The ability to control the laser modes within a subwavelength resonator is of
key relevance in modern optoelectronics. This work deals with the theoretical
research on optical properties of a PT–symmetric nano–scaled dimer formed by
two dielectric wires, one is with loss and the other with gain, wrapped with
graphene sheets. We show the existence of two non–radiating trapped modes
which transform into radiating modes by increasing the gain–loss parameter.
Moreover, these modes reach the lasing condition for suitable values of this
parameter, a fact that makes these modes to achieve an ultra high quality
factor that is manifested on the response of the structure when it is excited
by a plane wave. Unlike other mechanism that transform trapped modes into
radiating modes, we show that the variation of gain–loss parameter in the
balanced loss–gain structure here studied leads to a variation in the phase
difference between induced dipole moments on each wires, without appreciable
variation in the modulus of these dipole moments. We provide an approximated
method that reproduces the main results provided by the rigorous calculation.
Our theoretical findings reveal the possibility to develop unconventional
optical devices and structures with enhanced functionality.

###### pacs:

81.05.ue,73.20.Mf,78.68.+m,42.50.Pq

Keywords: PT symmetry, graphene, surface plasmons, plasmonics

##  1 Introduction

It is known that, under certain conditions, a plasmonic system can support
trapped electromagnetics modes, which are electromagnetic non radiating
oscillations that stay localized inside the structure [1]. These plasmon modes
are characterized by an ultra–high quality factor provided that the structural
symmetry remains unbroken. In fact, the narrow resonances characterizing these
mode excitations rely on the breaking degree of the symmetry which allows the
trapped mode to couples with free photons [2]. In this sense, plasmon trapped
modes can be considered as a kind of symmetry protected states. Due to Ohmic
loss in the plasmonic material, the eigenfrequencies of the trapped states are
complex valued. However, by introducing gain material elements into the
structure its possible to compensate this material loss and, as a consequense,
to reach the lasing condition for which the eigenfrequency associated to an
eigenmode is real valued [3].

Because of their fundamental properties as well as their potential
applications, the study of hybrid systems composed of gain material media and
plasmonic materials, which are loss media, is a topic of continuous increasing
interest [4, 5, 6, 7, 8]. In areas such as condensed matter and surface
optics, the amplification of eigenmodes by stimulated emission of radiation
has played a key role in the interpretation of a wide variety of experiments,
the understanding of various fundamental properties of solids and the
engineering of nanolaser devices [9, 10, 11]. In particular, new phenomena
associated with parity–time (PT) symmetry have been observed in optical
systems with balanced loss and gain [12, 13]. These optical systems belong to
a large family of non–Hermitian systems, which can have a real spectrum
provided that the system be invariant under combined operations of parity (P)
and time–reversal (T) symmetry [14, 15].

Possibilities have been widened towards PT symmetric structures that
incorporate graphene as plasmonic material. Doped graphene allows the
propagation of surface plasmons with low Ohmic losses, i.e., with high quality
factor, from terahertz to near–infrared range [16]. Moreover, the plasmon
resonance spectrum, and consequently the optical responses of the system, can
be tuned by varying the doped level on graphene. In this context, several
works have focused on the influence that long living and tunable surface
plasmons on graphene have on the optical responses of a PT–symmetric system
[17, 18, 19, 20].

The present work contains all the ingredients above entered and focusing on
the study of a sub–wavelength graphene plasmonic system with balanced volume
losses. In particular, our system is composed of two parallel dielectric
cylinders, one is with loss and the other with the same level of gain, both
wrapped with graphene shets. Here, the sole effect of graphene is to provide
LSP resonances in sub–wavelength wires. Although other more complex structures
such as an array of many or infinite cylinders can be designed to present
PT–symmetry, our motivation is based in the simplicity of the dimer structure
that we have chosen, since it allows us to understand the underlying physics
behind the PT–symmetry. We use a rigorous formalism based on Mie theory to
calculate the eigenmodes dependence with the gain–loss parameter for both
trapped and radiating eigenmodes. To do this, we solve the boundary value
problem with the corresponding boundary conditions without an incident wave
(homogeneous problem). The procedure requires the analytic continuation of the
eigenvalue, the frequency in our case, in the complex plane. Analytic
continuation is inevitable, even in the case of media with no intrinsic
losses, since the open nature of our resonator generates non–null imaginary
parts due to radiation losses.

In addition, as it is well established nowadays, the modal lasing analysis in
an open plasmonic resonator, as our PT–symmetric structure, can be carried out
by applying the lasing eigenvalue problem (see [24] and references therein),
in which the modal eigenfrequencies are assumed to be real valued at the
lasing condition (or lasing threshold). Following this concept, we solve the
homogeneous problem to find the real valued eigenfrequencies, gain–loss
parameters and chemical potentials for each modal lasing condition. Moreover,
we study the eigenmode influence on the optical response of the system when it
is excited by a plane wave near the lasing condition.

By using the quasistatic approximation valid in the long wavelength limit, we
show that despite the graphene ohmic losses, a fact that gives rise to a
complex spectrum, the structure exhibits a set of properties in common with a
PT–symmetric system. For instance, two eigenmode branches coalesce at an
exceptional point and, for the gain–loss parameter above certain threshold,
these both branches are repelled in the direction of the imaginary part of
frequency, making that one of these branches achieves the lasing condition. In
addition, we demonstrate that branches eigenmode, which are trapped modes in
case of null gain–loss parameter, transform into radiating modes that can be
excited by plane wave incidence when the gain–loss parameter is increased.
Unlike previous works [21, 23], where transitions from trapped to radiating
eigenmodes are achieved by producing a difference between the modulus of
individual dipole moments on each particle forming the dimer, i.e., by
producing a weakly asymmetry with respect to the center of the dimer modifying
the dipole moment amplitudes, here, we show that the increment of the
gain–loss parameter leads to a change in the phase difference between these
individual dipole moments, maintaining their modulus values constant.

This paper is organized as follows. In section 2 we present a brief
description of the rigorous method used in this work to calculate the
scattering of a dimmer composed of two graphene wires. From this method, using
the quasistatic approximation, we deduce analytical expressions for
eigenfrequencies and eigenvectors as a function of geometrical and
constitutive parameters that explain the main features calculated with the
rigorous method. In section 3 we present results of two parallel dielectric
cylinders tightly coated with a graphene layer, one of then with small inner
losses and the other one with the same level of gain. Concluding remarks are
provided in Section 4. The Gaussian system of units is used and an
$\mbox{exp}(-i\,\omega\,t)$ time–dependence is implicit throughout the paper,
where $\omega$ is the angular frequency, $t$ is the time coordinate, and
$i=\sqrt{-1}$. The symbols Re and Im are used for denoting the real and
imaginary parts of a complex quantity, respectively.

##  2 Theory

###  2.1 Rigorous description of the fields and scattering efficiencies

We consider a cluster consisting of two parallel and non overlapping
cylindrical dielectric wires, one with gain,
$\varepsilon_{a}=\varepsilon_{1}-i\varepsilon_{i}$ ($\varepsilon_{i}>0$), and
the other with equal loss, $\varepsilon_{b}=\varepsilon_{1}+i\varepsilon_{i}$,
as shown in Figure 1. Both wires have the same radius $R_{a}=R_{b}=R$ and are
wrapped with a graphene sheet. The system is embedded in a lossless and
non–magnetic dielectric denoted as medium $v$ with permittivity
$\varepsilon_{v}$. In this case, a PT symmetry around the central axis,
denoted by $O$, is fulfilled. We assume that the radius $R$ is sufficiently
large to describe the optical properties of the wires as characterized by the
same surface conductivity as planar graphene (see appendix A). We denote by
$r_{j}(\mathbf{r}),\,\phi_{j}(\mathbf{r})$ ($j=a,\,b$) the polar coordinates
of a point at position $\mathbf{r}$ with respect to the local origin $O_{j}$.
A plane wave radiation impinges on the wires with an angle of incidence
$\phi_{inc}$ with respect to the $y$ axis. Although some enhanced optical
effects related to invisibility modes are observed for $s$ polarization
(electric field along the $z$ axis) [22], this work focus on $p$ polarization
(magnetic fields along the $z$ axis) for which the electric field in the
graphene coating induces electric currents directed along the azimuthal
direction $\phi_{j}(\mathbf{r})$ and LSPs exist in the graphene circular
cylinder. In this way, the incident magnetic field (along the $z$ axis) can be
written in a system linked to the $j$–cylinder as [23],

$H_{inc}(\mathbf{r})=e^{ik_{v}r^{j}\sin(\phi_{inc}-\phi^{j})}\sum_{m=-\infty}^{+\infty}(-1)^{m}J_{m}(k_{v}r_{j}(\mathbf{r}))e^{im\phi_{l}(\mathbf{r})}e^{-im\phi_{inc}}$
(1)

where $r^{j},\,\phi^{j}$ are the polar coordinates of the $j$–cylinder,
$k_{v}=\sqrt{\varepsilon_{v}}\frac{\omega}{c}$, is the modulus of the photon
wave vector in medium $v$, $\omega$ is the angular frequency, $c$ is the
vacuum speed of light and $J_{m}(x)$ is the $n$th Bessel function. The
scattered magnetic field in medium $v$ ($r_{j}(\mathbf{r})>R$) can be written
as a superposition of the field scattered by each of the cylinders,

$H^{(v)}_{s}(\mathbf{r})=\sum_{j=a,b}\sum_{m=-\infty}^{+\infty}b_{j\,m}H_{m}(k_{v}r_{j}(\mathbf{r}))e^{im\phi_{j}(\mathbf{r})}$
(2)

where $H_{m}(x)$ is the $n$th Hankel functions of the first kind. Note that
the $j$th term of the summation corresponds to the field scattered by the
$j$th cylinder linked to the local system with origin $O_{j}$. In the region
inside the cylinders, $r_{j}(\mathbf{r})<R$, the transmitted field is written
as

$H^{(j)}(\mathbf{r})=\sum_{m=-\infty}^{+\infty}a_{j\,m}J_{m}(k_{j}r_{j}(\mathbf{r}))e^{im\phi_{j}(\mathbf{r})}$
(3)

where $j=a,\,b$. To find the unknown complex amplitudes of the reflected
$b_{jm}$ and transmitted $a_{jm}$ fields (2) and (3), we use the usual
boundary conditions and the addition theorem for Bessel and Hankel functions
[25]. This theorem allow us to write one of the terms in (2), associated to
the scattered field of one of the cylinders (for example, the $j=b$ cylinder)
in the other local coordinates (the $j=a$ cylinder). In this way, the
scattered field (2) will be represented in the form of expansions in Hankel
functions written in the local coordinate $j=a$. By replacing this expression
and Eq. (3) with $j=a$ into the boundary conditions along the surface
$r_{a}=R$ of the $j=a$ cylinder, one obtain a set of $2$ equations for the
$2\times 2$ unknown amplitudes. Similarly, we can write the scattered field
(2) in the local coordinate $j=b$ and use the boundary conditions on the
surface of the $j=b$ cylinder to obtain other set of 2 equations for the
$2\times 2$ unknown amplitudes. However, we closely follow a variant of the
method, developed in [26], that allows to reduce to half the dimension of the
system of equations. The detailed of this implementation has been given in
[23], and leads to the following system of equations for the amplitudes
$b_{jm}$

Figure 1:  Schematic illustration of the system composed by $2$ dielectric
cylinders wrapped with graphene sheets.

$\left[\begin{array}[]{llll}\,\,\,\,\,\,\,\,\,\,\,\,\overline{\overline{\mathbf{I}}}&-\overline{\overline{\mathbf{S}}}_{a}\cdot\overline{\overline{\mathbf{T}}}_{a\,b}\\\
-\overline{\overline{\mathbf{S}}}_{b}\cdot\overline{\overline{\mathbf{T}}}_{b\,a}&\,\,\,\,\,\,\,\,\,\,\,\,\overline{\overline{\mathbf{I}}}\\\
\end{array}\right]\left(\begin{array}[]{lll}\overline{\mathbf{b}}_{a}\\\
\overline{\mathbf{b}}_{b}\\\
\end{array}\right)=\left(\begin{array}[]{lll}\overline{\overline{\mathbf{S}}}_{a}\cdot\overline{Q}_{a}\\\
\overline{\overline{\mathbf{S}}}_{b}\cdot\overline{Q}_{b}\\\
\end{array}\right),$ (4)

where $\overline{\mathbf{b}}_{l}$ and $\overline{\mathbf{Q}}_{l}$ are vectors
whose coordinates are the elements $b_{lm}$ and

$\displaystyle
Q_{jm}=e^{ik_{v}r^{j}\sin(\phi_{inc}-\phi^{j})}(-1)^{m}e^{-im\phi_{inc}},$ (5)

respectively, $\overline{\overline{\mathbf{T}}}_{lj}$ is the matrix with
elements

$\displaystyle T_{ljnm}=H_{n-m}(k_{v}r_{l}^{j})e^{i(n-m)\phi_{l}^{j}},$ (6)

and $\mathbf{S_{j}}$ is the matrix with elements
$S_{jmn}=s_{jm}\,\delta_{mn}$, where $s_{jm}$ are the elements of the
scattering matrix associated to the $j$th cylinder [27]

$\displaystyle s_{j\,m}=$
$\displaystyle\frac{[k_{j}\varepsilon_{v}J_{m}(y)J^{{}^{\prime}}_{m}(x)-k_{v}\varepsilon_{j}J_{m}^{{}^{\prime}}(y)J_{m}(x)-\frac{4\pi\sigma}{ck_{0}}ik_{j}k_{v}J^{{}^{\prime}}_{m}(x)J^{{}^{\prime}}_{m}(y)]}{k_{v}\varepsilon_{j}J_{m}(x)H^{{}^{\prime}}_{m}(y)-k_{j}\varepsilon_{v}J^{{}^{\prime}}(x)H_{m}(y)+\frac{4\pi\sigma}{ck_{0}}ik_{j}k_{v}J^{{}^{\prime}}_{m}(x)H^{{}^{\prime}}_{m}(y)}$
(7)

where $x=k_{j}R$, $y=k_{v}R$ and the prime denotes the derivative with respect
to the argument.

Knowing the total electromagnetic field allows us to calculate the scattering
cross sections. The time–averaged scattered power is calculated from the
integral of the radial component of the complex Poynting vector flux through
an imaginary cylinder of length $L$ of radius $\rho_{0}$ which envelops the
graphene wire system (see Fig. 1),

$P_{s}=\rho_{0}L\int_{0}^{2\pi}\left\langle\mathbf{S}\right\rangle\cdot\hat{r}\,d\phi,$
(8)
$\displaystyle\left\langle\mathbf{S}(\rho_{0},\phi)\right\rangle\cdot\hat{r}=\frac{c}{8\pi}\Re\left\\{E_{s,\phi}H_{s}^{*}\right\\}=\frac{c}{8\pi}\Re\left\\{\frac{1}{ik_{0}\varepsilon_{v}}\frac{\partial\,H_{s}}{\partial
r}H_{s}^{*}\right\\}.$ (9)

It is convenient to express each of the terms in Eq. (2) in a same coordinate
$O$ [23]. Substituting the obtained expression into Eq. (9) we obtain

$\displaystyle\left\langle\mathbf{S}(\rho_{0},\phi)\right\rangle\cdot\hat{r}=\frac{c}{8\pi}\Re\left\\{\frac{1}{i\sqrt{\varepsilon_{v}}}\sum_{m,n}B_{m}B_{n}^{*}H^{\prime}_{m}(k_{v}\rho_{0})H^{*}_{m}(k_{v}\rho_{0})e^{i(m-n)\phi}\right\\},$
(10)

where

$\displaystyle
B_{m}=\sum_{j=a,b}\sum_{q=-\infty}^{+\infty}b_{jm}J_{m-q}(k_{v}r^{j})e^{i(q-m)\phi^{j}}.$
(11)

Inserting Eq. (10) into (8) and taking into account the wronskian
$W\left\\{J_{m}(x),Y_{m}(x)\right\\}=2/(\pi x)$ ($x=k_{v}\rho_{0}$), after
some algebraic manipulations, we obtain the power scattered by the cylinders,

$P_{s}=\frac{c^{2}L}{2\pi\omega\varepsilon_{v}}\sum_{m=-\infty}^{+\infty}|B_{m}|^{2}.$
(12)

The scattering cross section is defined as the ratio between the total power
scattered by the cylinders, given by Eq. (12), and the incident power
$P_{inc}$ intersected by the area of all the cylinders.

It is known that the scattering cross section and the near to the cylinders
field are strongly affected by complex singularities in the field amplitudes
$b_{jm}$. Singularities occur at complex locations and they represent the
frequency of the eigenmodes supported by the cylinders system. Complex
frequencies of these modes are obtained by solving the homogeneous problem,
i.e., by imposing the vectors $\overline{\mathbf{Q}}_{a}$ and
$\overline{\mathbf{Q}}_{b}$ in Eq. (4) be zero. Then, a condition to determine
eigenfrequencies is to require the determinant of the matrix in Eq. (4) to be
zero,

$\mbox{det}\left[\begin{array}[]{llll}\,\,\,\,\,\,\,\,\,\,\,\,\overline{\overline{\mathbf{I}}}&-\overline{\overline{\mathbf{S}}}_{a}\cdot\overline{\overline{\mathbf{T}}}_{a\,b}\\\
-\overline{\overline{\mathbf{S}}}_{b}\cdot\overline{\overline{\mathbf{T}}}_{b\,a}&\,\,\,\,\,\,\,\,\,\,\,\,\overline{\overline{\mathbf{I}}}\\\
\end{array}\right]=0.$ (13)

This condition corresponds to the full retarded dispersion equation (FR) for
eigenmodes and it determines the complex frequencies
$\omega=\omega_{R}+i\omega_{I}$ ($\omega_{I}<0$) in terms of all the
geometrical and constitutive parameters of the system.

###  2.2 Quasistatic approximation. A simple model based on two coupled
electric dipoles

Although the rigorous treatment represented by Eq. (13) gives us all the
kinematics and dynamics characteristics of the PT–symmetric eigenmodes, the
method lacks of analytical expressions that explain the main dependencies with
both geometrical and constitutive parameters. For the purpose of showing this
dependencies, by applying the quasistatic method, here, we reduce the full
treatment provided by the homogeneous part of Eq. (4) to a simple 2x2 matrix
description as follows.

Assuming that the radius $R$ of cylinders is much smaller than the wavelength
$\lambda=2\pi/\omega$, the problem can be treated using the dipole
approximation where the dimer eigenfunctions calculated with (4) are, as a
good approximation, a superposition of single plasmons with angular momentum
$m=\pm 1$ linked to each cylinder. In this way, only four coordinates,
$b_{j\pm 1}$ ($j=a,\,b$), define the amplitudes vector.

We first consider the case in which the induced dipole moments are along the
$\pm x$ directions, i.e., the local magnetic field associated to each of the
cylinders has a dependence $\approx\sin\phi_{j}$ ($j=a,\,b$). As a
consequence, the amplitudes $b_{j1}=b_{j-1}$ ($j=a,\,b$). Here, the subscript
$j-1$ stand for cylinder $j$ and angular momentum $m=-1$. In this way, the
matrix equation for the modal amplitudes reduces to a 2$\times$2 matrix system
for amplitudes $b_{a1}$ and $b_{b1}$,

$\left[\begin{array}[]{llll}1&-2\,s_{a}\,H_{1}(z)/z\\\
-2\,s_{b}\,H_{1}(z)/z&1\\\
\end{array}\right]\left(\begin{array}[]{lll}b_{a1}\\\ b_{b1}\\\
\end{array}\right)=\left(\begin{array}[]{lll}0\\\ 0\\\ \end{array}\right),$
(14)

where we have used $2H_{1}(z)/z=H_{0}(z)+H_{2}(z)$, $z=k_{v}R_{ab}$ ($R_{ab}$
is the distance between cylinder centers). Using the small argument asymptotic
expansions for Bessel and Hankel functions [25], it follows that $s_{j}$ can
be written as

$\displaystyle s_{j}=\frac{\pi\,k_{v}^{2}}{4}i\alpha_{j},$ (15)

$j=a,\,b$, where

$\displaystyle\alpha_{j}=R^{2}\frac{\varepsilon_{1}\pm
i\varepsilon_{i}-\varepsilon_{v}+g(\omega)}{\varepsilon_{1}\pm
i\varepsilon_{i}+\varepsilon_{v}+g(\omega)},$ (16)

are the dipolar polarizability of the cylinders $j=a$ or $j=b$, respectively,
$g(\omega)=-\frac{\omega_{g}^{2}}{\omega^{2}+i\omega\gamma_{g}}$,
$\omega_{g}^{2}=\frac{4e^{2}\mu}{\hbar^{2}R}$ is the effective plasma
frequency for the dipolar mode [28]. Note that the value of $\alpha_{a}$
differs from $\alpha_{b}$ in the sign of the gain–loss parameter
$\varepsilon_{i}$, signs $+$ and $-$ correspond to $j=a$ and $j=b$
respectively. Near the resonance frequency, the polarizability $\alpha_{j}$
($j=a,\,b$) can be written as

$\displaystyle\alpha_{j}=R^{2}\frac{\varepsilon_{1}\pm
i\varepsilon_{i}-\varepsilon_{v}+g(\omega_{j})}{g^{\prime}(\omega_{j})(\omega-\omega_{j})}=\frac{A_{j}}{\omega-\omega_{j}},$
(17)

where $\omega_{j}=\omega_{jR}+i\omega_{jI}$ ($\omega_{jI}<0$) is the complex
pole of $\alpha_{j}$, i.e., the eigenfrequency of the single graphene cylinder
$j$, $g^{\prime}(\omega)$ is the derivative of $g(\omega)$ and

$\displaystyle A_{j}=R^{2}\frac{\varepsilon_{1}\pm
i\varepsilon_{i}-\varepsilon_{v}+g(\omega_{a})}{g^{\prime}(\omega_{a})}.$ (18)

The single cylinder eigenfrequencies for cylinders $a$ and $b$ are written
as[28]

$\displaystyle\omega_{j}=\frac{\omega_{g}}{\sqrt{\varepsilon_{j}+\varepsilon_{v}}}-i\frac{\gamma_{g}}{2}\approx\omega_{0}+i\omega_{jI},$
(19)

where

$\displaystyle\omega_{0}=\frac{\omega_{g}}{\sqrt{\varepsilon_{1}+\varepsilon_{v}}},$
(20) $\displaystyle\omega_{jI}=-(\Gamma_{sup}\pm\Gamma_{core}),$ (21)

the signs $+$ and $-$ corresponds to $j=a$ and $j=b$, respectively, and

$\displaystyle\Gamma_{sup}=\frac{\gamma_{g}}{2},$
$\displaystyle\Gamma_{core}=\frac{1}{2}\varepsilon_{i}\frac{\omega_{g}}{(\varepsilon_{1}+\varepsilon_{v})^{3/2}},$
(22)

are the damping rates corresponding to the ohmic loss in graphene covers and
dielectric cores, respectively. We have used the fact that
$\varepsilon_{i}<<\varepsilon_{1}+\varepsilon_{v}$ in the last equality in
Eqs. (19). From Eq. (20) we see that the real part of the eigenfrequencies do
not depends on $\varepsilon_{i}$, a fact that also results by solving the
fully retarded dispersion equation for a single graphene cylinder [29].

By taking into account the small argument $z=k_{v}R_{ab}$ in the Hankel
functions, the off diagonal elements in the matrix in Eq. (14) are written as

$\displaystyle-2\frac{\,H(k_{v}R_{ab})}{k_{v}R_{ab}}s_{j}=-2\frac{-2i}{\pi(k_{v}R_{ab})^{2}}\frac{\pi
k_{v}^{2}i}{4}\alpha_{j}=-\frac{\alpha_{j}}{R_{ab}^{2}}.$ (23)

Therefore, the matrix equation (14) takes the form

$\left[\begin{array}[]{llll}\omega-\omega_{a}&-A_{a}/R_{ab}^{2}\\\
-A_{b}/R_{ab}^{2}&\omega-\omega_{b}\\\
\end{array}\right]\left(\begin{array}[]{lll}b_{a1}\\\ b_{b1}\\\
\end{array}\right)=\left(\begin{array}[]{lll}0\\\ 0\\\ \end{array}\right),$
(24)

where $A_{j}$ $j=a,\,b$ are given by Eq. (18). Eq. (24) gives us a simple
description for the dimer dynamic. For large separations,
$A_{j}/R^{2}_{ab}<<\omega_{a}$ (or $R/R_{ab}<<1$), the matrix (24) is diagonal
and thus the eigenfrequencies correspond to that of each individual graphene
wires composing the dimer, i.e., the plasmonic wires do not interact between
them. Taking into account the PT parameters, the real parts of both
frequencies are the same whereas their imaginary parts differ in
$2\Gamma_{core}$. For small enough values of $R_{ab}$, extra diagonal terms
take appreciable values and a splitting between the real parts of
eigenfrequencies occur. From Eq. (B8) we can see that this splitting is
proportional to $R^{2}/R^{2}_{ab}\omega_{0}$. For system (24) to have a
non–trivial solution, its determinant must be equal to zero, a condition which
can be written as

$\displaystyle(\omega-\omega_{a})(\omega-\omega_{b})=\frac{A_{a}A_{b}}{R_{ab}^{4}}.$
(25)

A detailed developed of the right hand side of this equation can be seen in
appendix B. By replacing expression (B8) into Eq. (25) and solving for
$\omega$ eigenfrequencies,

$\displaystyle\omega_{\pm}=-i\frac{\gamma_{g}}{2}+\omega_{0}\pm\sqrt{-\frac{\gamma^{2}}{4}+\omega_{aI}\omega_{bI}+\frac{A_{a}A_{b}}{R_{ab}^{4}}}=$
$\displaystyle\omega_{0}-i\frac{\gamma_{g}}{2}\pm\frac{\omega_{0}}{(\varepsilon_{1}+\varepsilon_{v})}\sqrt{\frac{-\varepsilon_{i}^{2}}{4}+\frac{R^{4}}{R_{ab}^{4}}\varepsilon_{v}^{2}}$
(26)

where in the last equality we have used Eq. (21).

Figure 2:  Trajectory in the complex plane of the eigenfrequencies (2.2) as a
parametric function of the gain–loss parameter $\varepsilon_{i}$. Two
branches, (a) $+x$$-x$ and $+x$$+x$ and (b) $+y$$-y$ and $+y$$+y$, are
observed. As $\varepsilon_{i}$ increases, the two eigenfrequencies approach
each other until they coalesce in an exceptional point. After passing the
exceptional point, they are repelled in their imaginary parts. The real parts
of the permittivity of the cylinders are
$\Re\,\varepsilon_{1}=\Re\,\varepsilon_{2}=2.13$, $\varepsilon_{v}=1$, the
radius $R_{a}=R_{b}=0.03\mu$m and the gap $\Delta=0.04\mu$m. The graphene
parameters are $T=300$K, $\gamma_{a}=\gamma_{b}=0.1$meV and
$\mu_{a}=\mu_{b}=0.5$eV.

This equation shows the dependence of the eigenfrequencies with the gain–loss
optical parameter $\varepsilon_{i}$. For $\varepsilon_{i}=0$, the real parts
of the eigenfrequencies split by
$\Delta\,\omega=\omega_{+}-\omega_{-}=2\frac{R^{2}}{R_{ab}^{2}}\frac{\omega_{0}}{(\varepsilon_{1}+\varepsilon_{v})}\varepsilon_{v}$
while their imaginary parts remain degenerated at the value $-\gamma_{g}/2$.
From equation (24), we see that the eigenvector associated to the upper branch
verify $b_{a1}=-b_{b1}$ pointing out that the induced dipole moments on each
of the cylinders move out of phase, i.e., the state corresponds to a trapped
mode, while the eigenvector associated to the lower branch verify
$b_{a1}=b_{a1}$ pointing out that this mode corresponds to a radiating mode
that can be excited, for example, by an incident plane wave [23]. We use the
notation $+x$$-x$ and $+x$$+x$ to refer to the upper and lower branches,
respectively. Since the terms inside the root have opposed signs between them,
the splitting between upper ($+x$$-x$) and lower ($+x$$+x$) branches
monotonically decreases as $\varepsilon_{i}$ is increased until reaching the
value

$\displaystyle\varepsilon_{ep}=2\frac{R^{2}}{R_{ab}^{2}}\varepsilon_{v},$ (27)

for which the eigenfrequencies coalesce at the exceptional point
$\omega=-i\frac{\gamma_{g}}{2}+\frac{\omega_{g}}{\sqrt{\varepsilon_{1}+\varepsilon_{v}}}$.
At the same time, the imaginary parts of both eigenfrequencies remain equal.
Beyond the exceptional point and for the gain–loss parameter
$\varepsilon_{i}>\varepsilon_{ep}$, the imaginary parts of the
eigenfrequencies bifurcate, while their real parts remain degenerated. In
Figure 2 we illustrate in the complex plane the above described trajectories
for the upper and lower branches (2.2) as parametric functions of the
gain–loss parameter $\varepsilon_{i}$. We have taken the straight segment
$z=0,\,\mbox{with}\,\Re\,z>0$ as the cut line for the square root function in
Eq. (2.2) so that $\sqrt{-w^{2}}=iw$ for $w$ real and positive.

We now consider the case of polarization along the $y$ axis, in which each of
the induced dipole moments are along $\pm y$ direction, i.e., the local
magnetic field associated to each of the cylinders has a dependence
$\approx\cos\phi_{j}$ ($j=a,\,b$). As a consequence, the amplitudes
$b_{j1}=-b_{j-1}$ ($j=a,\,b$). In this case, the matrix equation for these
amplitudes is

$\left[\begin{array}[]{llll}1&-2\,s_{a}\,H^{\prime}_{1}(z)\\\
-2\,s_{b}\,H^{\prime}_{1}(z)&1\\\
\end{array}\right]\left(\begin{array}[]{lll}b_{a1}\\\ b_{b1}\\\
\end{array}\right)=\left(\begin{array}[]{lll}0\\\ 0\\\ \end{array}\right),$
(28)

where we have used $2H^{\prime}_{1}(z)/z=H_{0}(z)-H_{2}(z)$, and
$H^{\prime}_{1}(z)$ is the derivative of $H_{1}(z)$. Following the same steps
as in the case of $x$ polarization, the matrix equation (28) takes the form

$\left[\begin{array}[]{llll}\omega-\omega_{a}&A_{a}/R_{ab}^{2}\\\
A_{b}/R_{ab}^{2}&\omega-\omega_{b}\\\
\end{array}\right]\left(\begin{array}[]{lll}b_{a1}\\\ b_{b1}\\\
\end{array}\right)=\left(\begin{array}[]{lll}0\\\ 0\\\ \end{array}\right),$
(29)

where $A_{j}$ are given by Eq. (18). Note that the matrix in Eq. (29) differs
from that in Eq. (24) only in the signs of the non diagonal terms. Therefore,
the eigenfrequencies corresponding to induced dipole moments along $y$
direction are formally given by Eq. (2.2). Unlike the $x$ polarization case,
for which the upper frequency branch for $\varepsilon_{i}=0$ corresponds to a
trapped mode, From Eq. (29) and taking $\varepsilon_{i}=0$, we see that the
eigenvector associated to the upper branch verify $b_{a1}=b_{b1}$, i.e., it
corresponds to a radiating mode. Conversely, it is straightforward to verify
that the lower frequency branch corresponds to a trapped mode for which
$b_{a1}=-b_{b1}$. It is worth noting that in this case, $\pm y$ oscillations,
it is convenient to take the cut of the complex square root function
$\sqrt{z}$ as the straight line $z=iw$ ($w$ real and positive) so that
$\sqrt{-w^{2}}=-iw$. In this way, beyond the exceptional point the upper
branch $+y$$+y$ moves away from the real axis whereas the lower branch
$+y$$-y$ reaches the real axis, as shown in Figure 2b.

##  3 Results

We consider a system of two dielectric cylinders with permittivities
$\varepsilon_{a}=2.13+i\varepsilon_{i}$ ($\varepsilon_{i}>0$),
$\varepsilon_{b}=2.13-i\varepsilon_{i}$ for lossy and gain cylinders,
respectively. The radii $R_{a}=R_{b}=R=0.03\mu$m and both cylinders are coated
with a graphene monolayer and immersed in vacuum ($\varepsilon_{v}=1$). The
graphene parameters are: temperature $T=300$K, chemical potentials
$\mu_{1}=\mu_{2}=0.5$eV and the carriers scattering rates
$\gamma_{1}=\gamma_{2}=0.1$meV. The positions of the cylinders are
$\mathbf{r}^{a}=-0.05\mu$m$\hat{x}$, $\mathbf{r}^{b}=0.05\mu$m$\hat{x}$
(center to center distance $R_{ab}=0.1\mu$m) and the gap between them is
$\Delta=0.04\mu$m.

Figure 3:  (a) Trajectory of the four branches as a parametric function of the
gain–loss parameter $\varepsilon_{i}$. The lasing threshold is achieved for
branches $+x$$-x$ and $+y$$-y$ at points A and C, respectively. The gain–loss
parameter values are $\varepsilon_{i}=0.166$, for point A, and
$\varepsilon_{i}=0.1715275$, for point C. Points B and D correspond to states
on the $+x$$+x$ and $+y$$+y$ branches for $\varepsilon_{i}=0.166$ and
$\varepsilon_{i}=0.1715275$, respectively. Dashed lines correspond to QA
branches plotted in Figure 2. (b) Scattering cross section for $p$–polarized
incident waves for illumination direction along ($\phi_{inc}=90^{\circ}$) and
perpendicular ($\phi_{inc}=0$) to the axis joining the cylinder centers. The
permittivity of the cylinders are $\varepsilon_{a}=2.13+i\varepsilon_{i}$,
$\varepsilon_{b}=2.13-i\varepsilon_{i}$, the radius $R_{a}=R_{b}=R=0.03\mu$m
and the gap $\Delta=0.04\mu$m. The graphene parameters are $T=300$K,
$\gamma_{a}=\gamma_{b}=0.1$meV and $\mu_{a}=\mu_{b}=0.5$eV.

Figure 3a shows the trajectory of the eigenfrequencies in the complex plane as
a parametric function of the gain–loss parameter $\varepsilon_{i}$ calculated
by solving the full retarded dispersion equation (FR). To solve this equation,
we use a Newton–Raphson method adapted to treat complex variable. Four
branches are observed, two of them ($+x$$+x$ and $+x$$-x$) corresponds to both
cylinders polarized along the $x$ axis (dipole moments along the $x$ axis)
while the other two branches ($+y$$-y$ and $+y$$+y$) corresponds to the case
in which both cylinders are polarized in the $y$ axis (dipole moments along
$y$ axis). We are using the notation of the asymptotic case when
$\varepsilon_{i}=0$ for naming the dimer surface plasmons, so that $+y$$+y$
($+x$$+x$) configuration corresponds to the two dipole moments moving in phase
on the $y$ axis ($x$ axis), and the $+y$$-y$ ($+x$$-x$) configuration
corresponds to the dipole moments oscillating in opposite phase on the $y$
axis ($x$ axis). For instance, the branch $+x$$+x$ start at frequency
$\omega/c=0.8589226-i0.5789274\,10^{-3}\mu$m-1 for $\varepsilon_{i}=0$, where
both dipole moments are in phase, and it moves to the right side leaving away
from the real axis. Moreover, the real part of this trajectory approaches
asymptotically to the value $\omega/c=0.8868\mu$m-1 corresponding to the real
part of the eigenfrequency of a single graphene cylinder [28]. On the
contrary, the branch $+x$$-x$ start at
$\omega/c=0.9107941-i0.253260710\,10^{-3}\mu$m-1 for $\varepsilon_{i}=0$, with
both dipole moments in opposite phase, and it approaches to the real axis
where the lasing threshold at $\omega_{crit}/c\approx 0.8954615$m-1 is reached
for $\varepsilon_{i}=\varepsilon_{crit}=0.166$ (point A in Figure 3a).

On the other hand, Figure 3a also shows the trajectory of the branch
corresponding to the polarization along the $y$ axis. The branch $+y$$-y$
start at frequency $\omega/c=0.8594583-i0.25349871\,10^{-3}\mu$m-1 for
$\varepsilon_{i}=0$, where both dipole moments are in opposed phase, and it
moves toward the right side, reaching the real axis at point C where the
critical eigenfrequency $\omega_{crit}/c=0.8755407\mu$m-1 for a critical value
of the gain–loss parameter $\varepsilon_{crit}=0.1715275$. On the other hand,
the $+y$$+y$ branch start at frequency
$\omega/c=0.9102954-i0.5560595\,10^{-3}\mu$m-1 for $\varepsilon_{i}=0$ and it
moves to the left side leaving away from the real axis. As in the $x$
polarized case, both branches are repelled in the direction of the imaginary
axis allowing the $+y$$-y$ branch to achieve the lasing threshold at point C.

In order to understand the gain–loss compensation near the critical points at
which the eigenfrequencies are almost real, in Figure 3b we plot the
scattering cross sections for a plane wave impinging at an angle $\phi=0$
(electric field along the $x$ axis) and $\phi=90^{\circ}$ (electric field
along the $y$ axis). The corresponding gain–loss parameter is
$\varepsilon_{i}=0.166$ for $\phi=0$, and $\varepsilon_{i}=0.1715$ for
$\phi=90^{\circ}$, i.e., it values are near to the critical values at which
lasing conditions are achieved. In the $\phi=0$ case, we observe that the
scattering cross section is enhanced at frequency near $\omega/c\approx
0.895$m-1 that agree well with the lasing frequency for the $+x$$-x$ branch
calculated by solving the eigenmode problem (point A in Figure 3a). Moreover,
we observe another peak (less intense) near $\omega/c=0.873\mu$m-1, a value
that falls near the real part of the eigenfrequency of an state with
$\varepsilon_{i}=0.166$ but corresponding to the $+x$$+x$ branch (point B in
Figure 3a). A similar behavior presents the scattering curve for
$\phi=90^{\circ}$ and $\varepsilon_{i}=0.1715$. From Figure 3b we observe a
very sharp peak at frequency that coincides with the lasing frequency
$\omega_{crit}/c\approx 0.8755407\mu$m-1 calculated by solving the eigenmode
problem (point C in Figure 3a). Moreover, a second peak is observed at a
frequency $\approx 0.894\mu$m-1 associated to the excitation of the state on
the $+y$$+y$ branch for $\varepsilon_{i}=0.1715$ (point D in Figure 3a).

The question that arises from the above results is how eigenmodes on the
$+x$$-x$ and $+y$$-y$ branches, which correspond to trapped modes for
$\varepsilon_{i}=0$, can be excited with a plane wave by varying the gain–loss
parameter $\varepsilon_{i}$. Furthermore, these branches reach their lasing
threshold for a critical value of the gain–loss parameter. To find a response,
we have calculated the eigenvectors, which contains all the field amplitudes
of the eigenmodes. In particular, we have verified that coefficients with
$|m|\not=1$ are orders of magnitude less than those corresponding to $m=\pm
1$, suggesting that the dimer plasmons can be considered, as a good
approximation, as a superposition of single plasmon with $m=\pm 1$ linked to
each cylinder. As a consequence, to gain further insight into the underlying
physics of these branches excitations, we applied the QA as follows. Without
loss of generality, we consider the case for which the induced dipole moments
on cylinders are in $\pm x$ direction. By replacing Eq. (2.2) into Eq. (24)
and using Eq. (19), we find the following relation between the dimer
amplitudes

$\displaystyle
b_{a1}=\mp\frac{\varepsilon_{v}(R/R_{ab})^{2}}{\sqrt{-\frac{\varepsilon_{i}^{2}}{4}+\varepsilon_{v}^{2}(R/R_{ab})^{4}}+i\frac{\varepsilon_{i}}{2}}\,b_{b1},$
(30)

where the $-$ and $+$ signs correspond to the $+x$$-x$ and $+x$$+x$ branches,
respectively. From this equation we see that the modulus of the $b_{a1}$ and
$b_{b1}$ amplitudes are equal providing that the gain–loss parameter be less
than $\varepsilon_{ep}$. Taking into account the fact that $b_{a-1}=b_{a1}$
and $b_{b-1}=b_{b1}$ and using Eq. (2), we can write the field scattered by
the cylinders as

$H^{(v)}_{s}(\mathbf{r})=2ib_{a1}H_{1}(k_{v}r_{a})\sin(\phi_{a})+2ib_{b1}H_{1}(k_{v}r_{b})\sin(\phi_{b}).$
(31)

Comparing this expression with that corresponding to a single dipole moment
$p$ along the $x$ axis and centered at the origin ($H\approx
pH_{1}(k_{v}r)\sin(\phi)$), we deduce that Eq. (31) corresponds to a
superposition of two fields, one of them due to a dipole moment of amplitude
$p_{a}\approx b_{a1}$ centered at the cylinder $a$ and other due to a dipole
moment of amplitude $p_{b}\approx b_{b1}$ centered at the cylinder $b$. Since
$b_{a1}=b_{b1}$ regardless of $\varepsilon_{i}$
($\varepsilon_{i}<\varepsilon_{ep}$), the induced dipole moments $p_{a}=p_{b}$
and, as a consequence, both emitted fields for each of the dipoles have the
same intensity. In addition, from Eq. (30) we see that the phase difference
$\Phi$ between $b_{a1}$ and $b_{b1}$ amplitudes, and thus between $p_{a}$ and
$p_{b}$, is shifted from $\pi$ to $\pi/2$ when $\varepsilon_{i}$ increases
from $0$ up to above the value $\varepsilon_{ep}$. This fact implies that the
eigenmodes on the $+x$$-x$ trajectory pass from trapped to bright by
increasing the gain–loss parameter. Our calculation confirm this expectation,
as can be seen in Figure 4 where we show plots of the phase $\Phi$ as a
function of the gain–loss parameter $\varepsilon_{i}$ by applying the FR
dispersion rigorous method (continuous line) and using Eq. (30) (dashed line).

Figure 4:  Phase difference between the induced dipoles on each cylinders as a
function of the gain–loss parameter $\varepsilon_{i}$ for the $+x$$-x$ and
$+y$$-y$ branches. The calculations have been carried out by applying the FR
dispersion method (continuous line) and the QA method (dashed line).
Constitutive and geometrical parameters are the same as in Figure 3.

From this Figure we see that the curve calculated with the QA agree well with
that calculated by using the FR dispersion method. In particular, we see that
the phase $\Phi=\pi$ for $\varepsilon_{i}=0$, which means that the eigenmodes
are dark states, and monotonically decreases for until reach the lasing modes
A (for $+x$$-x$ brach) and B (for $+y$$-y$ branch).

It is worth noting some similarities and differences between the eigenmode
branches calculation by using the FR dispersion equation and those calculated
by using the QA. On the one hand, the $+x$$+x$ and $+x$$-x$ branches in Figure
3a are repelled in the direction of the imaginary axis, as predicted by the QA
for a PT–symmetric system (Figure 2), a feature that allows the $+x$$-x$
branch to achieve the lasing threshold at point A. Furthermore, the critical
value (27) calculated by using the QA results $\varepsilon_{ep}=0.18$, a value
that agree well with the gain–loss parameter for which the lasing threshold is
achieved in Figure 3a. Moreover, the phase difference between the induced
dipole moments on each cylinders calculated by FR and QA methods matches quite
well. On the other hand, $+x$$+x$ and $+x$$-x$ branches in Figure 3a do not
start at points with the same value of their imaginary parts as occur for the
branches in Figure 2 and, as a consequence, these trajectories does not
coalesce at an exceptional point as in Figure 2. This is true because the lack
of a radiation losses term in the QA, i.e., the radiation losses would prevent
the system from having all the properties of a full loss compensated
PT–symmetric structure, shown in Figure 2, such as the existence of an
exceptional point.

In order to study the behavior with the chemical potential on graphene, we set
$\mu_{a}=\mu_{b}=\mu$ and vary the values of $\mu$. In Figure 5 we have
plotted the lasing frequency $\omega_{crit}$ and the corresponding gain–loss
parameter $\varepsilon_{crit}$ as functions of $\mu$, calculated with the
rigorous FR method. From Figure 5a we can see that the lasing frequency for
both $+x$$-x$ and $+y$$-y$ branches are increasing functions of $\mu$. This
fact can be understood by taking into account that the real part of the
eigenfrequency corresponding to a single graphene cylinder, which falls
between the lasing frequencies for $+y$$-y$ and $+x$$-x$ branches, is
proportional to $\sqrt{\mu}$ (see Eq. (20)). On the other hand, from Figure 5b
we see that the critical value of the gain–loss parameter for which the lasing
condition is achieved is a decreasing function of the chemical potential
$\mu$. This behaviour has a similarity with that presented by a single
graphene cylinder for which has been demonstrated that the gain level to
achieve the lasing condition decreases with the chemical potential value [29].

Figure 5:  Lasing frequency (a) and critical gain–loss parameter (b) as
functions of the chemical potential $\mu$ ($\mu_{a}=\mu_{b}=\mu$) for branches
$+x$$-x$ and $+y$$-y$. The calculations have been carried out by applying the
FR dispersion method. Geometrical and other constitutive parameters are the
same as in Figure 3.

##  4 Conclusions

In conclusion, we have analytically studied the scattering and the eigenmode
problems for a dimer composed of two graphene coated dielectric cylinders, one
of them with loss and the other with the same level of gain. We have
demonstrated the existence of two branches, corresponding to trapped modes
when $\varepsilon_{i}=0$, that reach the lasing conditions for suitable values
of the gain–loss parameter. While the phase difference between the induced
dipole moments on individual cylinders changes from $\pi$ to a value near
$\pi/2$ when the gain–loss parameter is incremented, a fact that provides the
mode transformation from trapped to radiating modes, the modulus of each
individual dipole moments maintains equals in between.

Other mechanisms to transform a trapped mode into a resonant observable which
can be excited by a plane wave have been reported in other works. All these
methods are based on the introduction of a small asymmetry with respect to the
center of the dimer by producing dissimilar dipole moment modulus on each of
the cylinders. Interestingly, here, we found that in the transformation from
trapped to radiating eigenmodes on both $+x$$-x$ and $+y$$-y$ branches, the
modulus of the individual dipole moments does not change, while it is changed
the phase between them. A distinction between these two kind of mechanism, the
phase variation and modulus variation mechanisms, to transform a trapped mode
into a resonant observable has not been reported before to our knowledge. We
believe that our results will be usefull for a deeper understanding of the
PT–symmetric LSP characteristics and that the LSP effects we have demonstrated
opens up possibilities for practical applications involving sub–wavelength
laser structures.

## Funding

Consejo Nacional de Investigaciones Cientficas y Técnicas (CONICET)

## Acknowledgments

The authors acknowledge the financial support of Consejo Nacional de
Investigaciones Cientficas y Técnicas (CONICET)

## Disclosures

The authors declare no conflicts of interest.

##  Appendix A Graphene conductivity

We consider the graphene layer as an infinitesimally thin, local and isotropic
two–sided layer with frequency–dependent surface conductivity $\sigma(\omega)$
given by the Kubo formula [30], which can be read as
$\sigma=\sigma^{intra}+\sigma^{inter}$, with the intraband and interband
contributions being

$\sigma^{intra}(\omega)=\frac{2ie^{2}k_{B}T}{\pi\hbar^{2}(\omega+i\gamma_{g})}\mbox{ln}\left[2\mbox{cosh}(\mu_{g}/2k_{B}T)\right],$
(A1)
$\displaystyle\sigma^{inter}(\omega)=\frac{e^{2}}{\hbar}\Bigg{\\{}\frac{1}{2}+\frac{1}{\pi}\mbox{arctan}\left[(\hbar\omega-2\mu_{g})/2k_{B}T\right]-$
$\displaystyle\frac{i}{2\pi}\mbox{ln}\left[\frac{(\hbar\omega+2\mu_{g})^{2}}{(\hbar\omega-2\mu_{g})^{2}+(2k_{B}T)^{2}}\right]\Bigg{\\}},$
(A2)

where $\mu_{g}$ is the chemical potential (controlled with the help of a gate
voltage), $\gamma_{g}$ the carriers scattering rate, $e$ the electron charge,
$k_{B}$ the Boltzmann constant and $\hbar$ the reduced Planck constant.

##  Appendix B Developing of the right hand side in equation (25)

The right side of Eq. (25) can be calculated by using Eq. (18) as follows.

$\displaystyle\frac{A_{a}A_{b}}{R_{ab}^{4}}=\frac{a^{4}}{R_{ab}^{4}}\frac{[\varepsilon_{1}+i\varepsilon_{i}-\varepsilon_{v}+g(\omega_{a})]}{g^{\prime}(\omega_{a})}\frac{[\varepsilon_{1}-i\varepsilon_{i}-\varepsilon_{v}+g(\omega_{b})]}{g^{\prime}(\omega_{b})},$
(B1)

where

$\displaystyle
g(\omega_{j})=-\frac{\omega_{g}^{2}}{\omega_{j}^{2}+i\gamma_{g}\omega_{j}},$
(B2)

and

$\displaystyle
g^{\prime}(\omega_{j})=\frac{\omega_{g}^{2}(2\omega_{j}+i\gamma_{g})}{(\omega_{j}^{2}+i\gamma_{g}\omega_{j})^{2}}.$
(B3)

By taking into account that $\gamma_{g}<<|\omega_{j}|$, we can write this
equations as,

$\displaystyle
g(\omega_{j})=-\frac{\omega_{g}^{2}}{\omega_{j}^{2}}\frac{1}{1+i\gamma_{g}/\omega_{j}}\approx-\frac{\omega_{g}^{2}}{\omega_{j}^{2}}\left(1-i\frac{\gamma_{g}}{\omega_{j}}+\frac{\gamma_{g}^{2}}{\omega_{j}^{2}}+...\right),$
(B4)

and

$\displaystyle
g^{\prime}(\omega_{j})\approx\frac{2\omega_{g}^{2}}{\omega_{j}^{3}}\left(1-i\frac{\gamma_{g}}{\omega_{j}}+\frac{\gamma_{g}^{2}}{\omega_{j}^{2}}+...\right)-\frac{\omega_{g}^{2}}{\omega_{j}^{2}}\left(i\frac{\gamma_{g}}{\omega_{j}^{2}}-2\frac{\gamma_{g}^{2}}{\omega_{j}^{3}}+...\right)=$
$\displaystyle\frac{2\omega_{g}^{2}}{\omega_{j}^{3}}\left(1-\frac{3}{2}i\frac{\gamma_{g}}{\omega_{j}}+2\frac{\gamma_{g}^{2}}{\omega_{j}^{2}}+...\right).$
(B5)

Considering the lowest order in $\Gamma_{sup}/\omega_{0}$ and
$\Gamma_{cover}/\omega_{0}$, functions $g(\omega_{j})$ and
$g^{\prime}(\omega_{j})$ are written as

$\displaystyle
g(\omega_{j})\approx-\frac{\omega_{g}^{2}}{\omega_{0}^{2}}\left[1\pm\frac{i}{\omega_{0}}2\Gamma_{core}\right],$
(B6)

and

$\displaystyle
g^{\prime}(\omega_{j})\approx\frac{2\omega_{g}^{2}}{\omega_{0}^{3}}\left[1\pm
i\frac{3}{\omega_{0}}\Gamma_{core}\right].$ (B7)

Signs $+$ and $-$ correspond to $j=a$ and $j=b$, respectively. In this
approximation, Eq. (B1) reduces to

$\displaystyle\frac{A_{a}A_{b}}{R_{ab}^{4}}=\frac{R^{4}}{R_{ab}^{4}}\frac{\omega_{g}^{2}}{(\varepsilon_{1}+\varepsilon_{v})}\varepsilon_{v}^{2}.$
(B8)

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# Tremain equiangular tight frames

Matthew Fickus  Department of Mathematics and Statistics, Air Force Institute
of Technology, Wright-Patterson Air Force Base, OH 45433, USA
<EMAIL_ADDRESS>,  John Jasper  Department of Mathematical Sciences,
University of Cincinnati, Cincinnati, OH 45221, USA<EMAIL_ADDRESS>,
Dustin G. Mixon  Department of Mathematics and Statistics, Air Force Institute
of Technology, Wright-Patterson Air Force Base, OH 45433, USA
<EMAIL_ADDRESS>and  Jesse Peterson  European Office of Aerospace
Research and Development, Air Force Office of Scientific Research, London, UK
<EMAIL_ADDRESS>
###### Abstract.

Equiangular tight frames provide optimal packings of lines through the origin.
We combine Steiner triple systems with Hadamard matrices to produce a new
infinite family of equiangular tight frames. This in turn leads to new
constructions of strongly regular graphs and distance-regular antipodal covers
of the complete graph.

###### Key words and phrases:

equiangular tight frame, Steiner system, Hadamard matrix, strongly regular
graph, distance-regular antipodal cover of the complete graph

###### 2010 Mathematics Subject Classification:

Primary: 42C15, Secondary: 51E10, 05B20, 05C12

This research was supported by an AFOSR Young Investigator Research Program
award, NSF Grant No. DMS-1321779, and AFOSR Grant No. F4FGA05076J002. The
views expressed in this article are those of the authors and do not reflect
the official policy or position of the United States Air Force, Department of
Defense, or the U.S. Government.

##  1\. Introduction

Consider the following problem: Given a finite-dimensional Hilbert space
$\mathcal{H}$ and some positive integer $N$, find vectors
$\varphi_{1},\ldots,\varphi_{N}\in\mathcal{H}$ of unit length that minimize
coherence:

$\kappa:=\max_{\begin{subarray}{c}i,j\in\\{1,\ldots,N\\}\\\ i\neq
j\end{subarray}}|\langle\varphi_{i},\varphi_{j}\rangle|.$

Geometrically, this problem amounts to packing lines through the origin. Given
their resemblance to error-correcting codes, it comes as no surprise that
ensembles of minimal coherence, called Grassmannian frames, lead to
communication protocols with minimal cross-talk. Since their introduction by
Strohmer and Heath in 2003 [14], Grassmannian frames have received
considerable attention in the finite frame theory community [7].

There were several precursors to the modern study of Grassmannian frames. One
notable example is a 1974 paper of Welch [18], which provides various lower
bounds for coherence in terms of the dimension $M$ of $\mathcal{H}$. This
first of these bounds is

(1.1) $\kappa\geq\sqrt{\frac{N-M}{M(N-1)}},$

cf. Rankin’s 1956 paper [13]. This has since been dubbed the Welch bound, and
one can show that the coherence of an ensemble of unit-length vectors meets
equality in the Welch bound precisely when the ensemble forms an equiangular
tight frame [14], meaning $|\langle\varphi_{i},\varphi_{j}\rangle|$ is some
fixed constant for every choice of $i$ and $j$ with $i\neq j$ (i.e., the
vectors are “equiangular”) and furthermore, the operator
$\mathbf{S}\colon\mathcal{H}\rightarrow\mathcal{H}$ defined by

$\mathbf{S}x:=\sum_{i=1}^{N}\langle
x,\varphi_{i}\rangle\varphi_{i}\qquad\forall x\in\mathcal{H}$

is some multiple of the identity operator (i.e., the ensemble forms a so-
called “tight frame”). By exhibiting equality in the Welch bound, equiangular
tight frames (ETFs) are necessarily Grassmannian.

The significance of equiangularity brings us to another precursor of sorts. In
1966, van Lint and Seidel [16] first introduced a useful identification
between real equiangular ensembles and graphs, in which vertices correspond to
vectors, and edges are drawn according to the sign of the corresponding inner
product. Since the Gram matrix
$\mathbf{G}=[\langle\varphi_{i},\varphi_{j}\rangle]_{i,j=1}^{N}$ can be
expressed in terms of the adjacency matrix $\mathbf{A}$ of this graph, one may
then identify spectral properties of $\mathbf{A}$ with those of $\mathbf{G}$.
This is particularly important in the case of real ETFs, as the tightness
criterion implies that $\mathbf{G}^{2}$ is a multiple of $\mathbf{G}$, which
in turn forces $\mathbf{A}$ to satisfy a related quadratic equation. Indeed,
more recent treatments of this identification [17, 9] have established a one-
to-one correspondence between real ETFs and a family of strongly regular
graphs (SRGs), namely, graphs with the property that every vertex has the same
number $k$ of neighbors, that adjacent vertices have the same number $\lambda$
of common neighbors, and that non-adjacent vertices also have the same number
$\mu$ of common neighbors. Given the maturity of the SRG literature, this
identification has compelled finite frame theorists to direct their attention
toward complex ETFs, where open problems abound.

However, recent developments in ETF research leave one questioning the
insurmountable maturity of the SRG literature. In particular, a decades-old
SRG construction by Goethals and Seidel [11] (which leverages a particular
family of real equiangular ensembles) was recently generalized in [6] to
construct new SRGs from known real ETFs exhibiting certain symmetries. The
notion that ETF research could inform the design of graphs was further
suggested by Coutinho, Godsil, Shirazi and Zhan [5], who established that
complex ETFs with Gram matrices consisting of prime roots of unity can be used
to construct distance-regular antipodal covers of the complete graph (referred
to as DRACKNs).

The present paper produces a new infinite family of complex ETFs that the
authors discovered by generalizing Example 7.10 in Tremain’s notes [15]; we
call these Tremain ETFs. The construction can be written as a real ETF
infinitely often, and in some cases, the corresponding SRGs are new (this is
established with the help of Brouwer’s table of strongly regular graphs [1,
2]). Indeed, the real ETFs in this paper are the first to be discovered in the
last two decades. In addition, some complex cases of the construction lead to
a new infinite family of DRACKNs.

In the next section, we provide some preliminary information about ETFs.
Section 3 then gives our main result, namely, the construction of Tremain
ETFs. We conclude in Section 4 with consequences of this construction to
combinatorial design.

##  2\. Preliminaries

Throughout, we use $\\{\delta_{s}\\}_{s\in\mathcal{S}}$ to denote any fixed
orthonormal basis of $\mathbb{C}^{\mathcal{S}}$. For any
$x\in\mathbb{C}^{\mathcal{S}}$, we may refer to the $s$th entry of $x$ with
respect to this basis $x(s):=\langle x,\delta_{s}\rangle$. Also, we identify a
given vector $x\in\mathcal{H}$ with the corresponding linear operator
$x\colon\mathbb{C}\rightarrow\mathcal{H}$ defined by $x(a):=ax$ for every
$a\in\mathbb{C}$. In particular, this identification allows us to express the
adjoint operator $x^{*}\colon\mathcal{H}\rightarrow\mathbb{C}$ given by
$x^{*}:=\langle\cdot,x\rangle$, along with the outer product operator
$xx^{*}$, etc. Finally, for any positive integer $k$, we denote
$[k]:=\\{1,\ldots,k\\}$.

In this paper, it is convenient to scale the vectors in an ETF so that the
inner products between any two vectors lies in the complex unit circle
$\mathbb{T}$. This scaling then forces the squared norm of each vector to be
the reciprocal of the Welch bound (1.1). For example, consider any $N\times N$
complex Hadamard matrix, that is, an $N\times N$ matrix $\mathbf{H}$ with
entries in $\mathbb{T}$ such that $\mathbf{H}\mathbf{H}^{*}=N\mathbf{I}_{N}$.
Then removing any row from this matrix will produce an $(N-1)\times N$ matrix
$\mathbf{\Phi}$ whose columns form a so-called unimodular simplex. The fact
that such vectors form an ETF is easy to check: In this case, $M=N-1$, and so
the Welch bound (1.1) is $1/(N-1)$, and indeed, each column has squared norm
$N-1$, whereas inner products between distinct columns lie in $\mathbb{T}$. If
we let $a$ denote the row of $\mathbf{H}$ that was removed to produce
$\mathbf{\Phi}$, then we say its entries $\\{a_{i}\\}_{i=1}^{N}$ form the
Naimark complement of the columns $\\{\varphi_{i}\\}_{i=1}^{N}$ of
$\mathbf{\Phi}$. The Naimark complement is useful because sums of inner
products satisfy

(2.1)
$\langle\varphi_{i},\varphi_{j}\rangle+a_{i}\overline{a_{j}}=\left\\{\begin{array}[]{cl}N&\text{if
}i=j\\\ 0&\text{else}\end{array}\right.$

by identifying the sum as an inner product between corresponding columns of
$\mathbf{H}$.

Another useful family of ETFs is the so-called Steiner ETFs, first constructed
in [8] from Steiner systems. A $(2,K,V)$-Steiner system is a collection
$\mathcal{V}$ of $V$ points together with a collection $\mathcal{B}$ of
size-$K$ subsets of $\mathcal{V}$ (called blocks) with the property that each
$2$-element subset of $\mathcal{V}$ is contained in exactly one block. We
write $B=|\mathcal{B}|$. Also, it turns out that each point is contained in
the same number $R$ of blocks. These parameters satisfy the following
identities:

(2.2) $VR=BK,\qquad V-1=R(K-1).$

In this paper, we introduce a certain family of operators that will simplify
the construction of Steiner ETFs, and help in the construction of Tremain ETFs
in the next section. In particular, for each $v\in\mathcal{V}$, consider an
embedding operator
$\mathbf{E}_{v}\colon\mathbb{C}^{R}\rightarrow\mathbb{C}^{\mathcal{B}}$ such
that for every $b\in\mathcal{B}$ with $v\in b$, there exists $r\in[R]$ such
that $\mathbf{E}_{v}\delta_{r}=\delta_{b}$. For example, suppose
$\mathcal{B}=\\{b_{1},\ldots,b_{7}\\}$ and that $v$ is a member of $R=3$
blocks, specifically, $b_{1}$, $b_{4}$ and $b_{5}$. Then we may take
$\mathbf{E}_{v}$ to be any linear operator that sends
$\\{\delta_{1},\delta_{2},\delta_{3}\\}$ to
$\\{\delta_{b_{1}},\delta_{b_{4}},\delta_{b_{5}}\\}$. In general, since
$\mathbf{E}_{v}$ maps the orthonormal basis $\\{\delta_{r}\\}_{r=1}^{R}$ onto
an orthonormal sequence $\\{\delta_{b}\\}_{b\in\mathcal{B},~{}\\!v\in b}$ of
the same cardinality, $\mathbf{E}_{v}$ is an isometry. These embedding
operators satisfy a few useful properties:

######  Lemma 2.1.

Given a $(2,K,V)$-Steiner system $(\mathcal{V},\mathcal{B})$ with embedding
operators $\\{\mathbf{E}_{v}\\}_{v\in\mathcal{V}}$, we have

  * (a)

$\sum_{v\in\mathcal{V}}\mathbf{E}_{v}\mathbf{E}_{v}^{*}=K\mathbf{I}_{\mathcal{B}}$,

  * (b)

$\mathbf{E}_{v}^{*}\mathbf{E}_{v}=\mathbf{I}_{R}$, and

  * (c)

for every $v,v^{\prime}\in\mathcal{V}$ with $v\neq v^{\prime}$, there exist
$i,j\in[R]$ such that
$\mathbf{E}_{v}^{*}\mathbf{E}_{v^{\prime}}=\delta_{i}\delta_{j}^{*}$.

###### Proof.

For (a), we first note that, since $\mathbf{E}_{v}$ is an isometry,
$\mathbf{E}_{v}\mathbf{E}_{v}^{*}$ is an orthogonal projection onto its range.
Since $\\{\delta_{b}\\}_{b\in\mathcal{B},~{}\\!v\in b}$ is an orthonormal
basis for this range, we have

$\mathbf{E}_{v}\mathbf{E}_{v}^{*}=\sum_{\begin{subarray}{c}b\in\mathcal{B}\\\
v\in b\end{subarray}}\delta_{b}\delta_{b}^{*}.$

The desired sum can then be simplified using this identity:

$\sum_{v\in\mathcal{V}}\mathbf{E}_{v}\mathbf{E}_{v}^{*}=\sum_{v\in\mathcal{V}}\sum_{\begin{subarray}{c}b\in\mathcal{B}\\\
v\in
b\end{subarray}}\delta_{b}\delta_{b}^{*}=\sum_{b\in\mathcal{B}}\sum_{\begin{subarray}{c}v\in\mathcal{V}\\\
v\in
b\end{subarray}}\delta_{b}\delta_{b}^{*}=K\sum_{b\in\mathcal{B}}\delta_{b}\delta_{b}^{*}=K\mathbf{I}_{\mathcal{B}}.$

Next, (b) is equivalent to $\mathbf{E}_{v}$ being an isometry. For (c),
suppose $v\neq v^{\prime}$. Then

$\langle\mathbf{E}_{v}^{*}\mathbf{E}_{v^{\prime}}\delta_{r^{\prime}},\delta_{r}\rangle=\langle\mathbf{E}_{v^{\prime}}\delta_{r^{\prime}},\mathbf{E}_{v}\delta_{r}\rangle=\langle\delta_{b^{\prime}(r^{\prime})},\delta_{b(r)}\rangle\in\\{0,1\\}\qquad\forall
r,r^{\prime}\in[R],$

where $\delta_{b(r)}:=\mathbf{E}_{v}\delta_{r}$ and
$\delta_{b^{\prime}(r^{\prime})}:=\mathbf{E}_{v}\delta_{r^{\prime}}$. Summing
these quantities gives

$\displaystyle\sum_{r=1}^{R}\sum_{r^{\prime}=1}^{R}\langle\mathbf{E}_{v}^{*}\mathbf{E}_{v^{\prime}}\delta_{r^{\prime}},\delta_{r}\rangle$
$\displaystyle=\sum_{r=1}^{R}\sum_{r^{\prime}=1}^{R}\langle\delta_{b^{\prime}(r^{\prime})},\delta_{b(r)}\rangle$
$\displaystyle=\sum_{\begin{subarray}{c}b\in\mathcal{B}\\\ v\in
b\end{subarray}}\sum_{\begin{subarray}{c}b^{\prime}\in\mathcal{B}\\\
v^{\prime}\in
b^{\prime}\end{subarray}}\langle\delta_{b^{\prime}},\delta_{b}\rangle$
$\displaystyle=\sum_{\begin{subarray}{c}b\in\mathcal{B}\\\ v\in
b\end{subarray}}\sum_{\begin{subarray}{c}b^{\prime}\in\mathcal{B}\\\
v^{\prime}\in b^{\prime}\end{subarray}}\left\\{\begin{array}[]{cl}1&\text{if
}b=b^{\prime}\\\
0&\text{else}\end{array}\right\\}=\\#\\{b\in\mathcal{B}:v,v^{\prime}\in
b\\}=1.$

We therefore conclude that
$\langle\mathbf{E}_{v}^{*}\mathbf{E}_{v^{\prime}}\delta_{r^{\prime}},\delta_{r}\rangle=1$
for exactly one choice of $r$ and $r^{\prime}$, and equals zero otherwise,
thereby implying the result. ∎

These properties are useful in the proof of the following construction:

######  Proposition 2.2 (Steiner ETFs, see [8]).

Given a $(2,K,V)$-Steiner system $(\mathcal{V},\mathcal{B})$ with embedding
operators $\\{\mathbf{E}_{v}\\}_{v\in\mathcal{V}}$ and a unimodular simplex
$\\{\varphi_{s}\\}_{s=1}^{R+1}$ in $\mathbb{C}^{R}$, the ensemble
$\\{\mathbf{E}_{v}\varphi_{s}\\}_{s=1,~{}\\!v\in\mathcal{V}}^{R+1}$ forms an
equiangular tight frame in $\mathbb{C}^{\mathcal{B}}$.

As an example, we use a $(2,3,7)$-Steiner system known as the Fano plane to
construct a Steiner ETF of $28$ vectors in $\mathbb{C}^{7}$ (actually,
$\mathbb{R}^{7}$). The following matrix is organized into $V=7$ blocks, each
of size $B\times(R+1)=7\times 4$, obtained by applying a corresponding
embedding operator $\mathbf{E}_{v}$ to a fixed unimodular simplex extracted
from a $4\times 4$ Hadamard matrix:

$\left[\setcounter{MaxMatrixCols}{28}\begin{smallmatrix}+&-&+&-&+&-&+&-&+&-&+&-&&&&&&&&&&&&&&&&\\\
+&+&-&-&&&&&&&&&+&-&+&-&+&-&+&-&&&&&&&&\\\
+&-&-&+&&&&&&&&&&&&&&&&&+&-&+&-&+&-&+&-\\\
&&&&+&+&-&-&&&&&+&+&-&-&&&&&+&+&-&-&&&&\\\
&&&&+&-&-&+&&&&&&&&&+&+&-&-&&&&&+&+&-&-\\\
&&&&&&&&+&+&-&-&+&-&-&+&&&&&&&&&+&-&-&+\\\
&&&&&&&&+&-&-&+&&&&&+&-&-&+&+&-&-&+&&&&\end{smallmatrix}\right]$

(For convenience, we chose to scale the vectors so that each coordinate has
size $0$ or $1$; the scaling has no effect on whether the columns form an
ETF.) Here, $+$ denotes $1$, $-$ denotes $-1$, and blank entries are zeros. It
is clear from this example that the columns have squared norm $3$ (the number
of blocks incident to each point), columns from a common block have inner
product $-1$ (since they belong to a common embedded simplex), and the
supports of columns from different blocks intersect at a single entry (since
two points determine a block), leading to an inner product of $\pm 1$.
Additionally, tightness follows from the fact that the rows of this matrix are
orthogonal with equal norms.

###### Proof of Proposition 2.2.

First, the unimodular simplex satisfies
$\sum_{s=1}^{R+1}\varphi_{s}\varphi_{s}^{*}=(R+1)\mathbf{I}_{R}$, which gives
tightness:

$\sum_{v\in\mathcal{V}}\sum_{s=1}^{R+1}(\mathbf{E}_{v}\varphi_{s})(\mathbf{E}_{v}\varphi_{s})^{*}=\sum_{v\in\mathcal{V}}\mathbf{E}_{v}\bigg{(}\sum_{s=1}^{R+1}\varphi_{s}\varphi_{s}^{*}\bigg{)}\mathbf{E}_{v}^{*}=(R+1)\sum_{v\in\mathcal{V}}\mathbf{E}_{v}\mathbf{E}_{v}^{*}=K(R+1)\mathbf{I}_{\mathcal{B}},$

where the last step follows from Lemma 2.1(a). Next, Lemma 2.1(b) gives

$\langle\mathbf{E}_{v}\varphi_{s},\mathbf{E}_{v}\varphi_{s^{\prime}}\rangle=\langle\mathbf{E}_{v}^{*}\mathbf{E}_{v}\varphi_{s},\varphi_{s^{\prime}}\rangle=\langle\varphi_{s},\varphi_{s^{\prime}}\rangle,$

which equals $\|\varphi_{s}\|^{2}=R$ when $s=s^{\prime}$, and otherwise
resides in the complex unit circle $\mathbb{T}$. Finally, for $v\neq
v^{\prime}$, Lemma 2.1(c) gives

$\langle\mathbf{E}_{v}\varphi_{s},\mathbf{E}_{v^{\prime}}\varphi_{s^{\prime}}\rangle=\langle\mathbf{E}_{v^{\prime}}^{*}\mathbf{E}_{v}\varphi_{s},\varphi_{s^{\prime}}\rangle=\langle\delta_{i}\delta_{j}^{*}\varphi_{s},\varphi_{s^{\prime}}\rangle=\langle\delta_{j}^{*}\varphi_{s},\delta_{i}^{*}\varphi_{s^{\prime}}\rangle=\varphi_{s}(j)\overline{\varphi_{s^{\prime}}(i)}\in\mathbb{T},$

thereby implying equiangularity. ∎

##  3\. Tremain ETFs

What follows is our main result, which specifically uses $(2,3,V)$-Steiner
systems, known as Steiner triple systems:

######  Theorem 3.1 (Tremain ETFs).

Given a Steiner triple system $(\mathcal{V},\mathcal{B})$ with embedding
operators $\\{\mathbf{E}_{v}\\}_{v\in\mathcal{V}}$, unimodular simplices
$\\{\varphi_{s}\\}_{s=1}^{R+1}$ in $\mathbb{C}^{R}$ and
$\\{\psi_{t}\\}_{t=1}^{V+1}$ in $\mathbb{C}^{\mathcal{V}}$, and corresponding
Naimark complements $\\{a_{s}\\}_{s=1}^{R+1}$ and $\\{b_{t}\\}_{t=1}^{V+1}$ in
$\mathbb{C}$, the ensemble

$\Big{\\{}\mathbf{E}_{v}\varphi_{s}\oplus\sqrt{2}a_{s}\delta_{v}\oplus
0\Big{\\}}_{s=1,~{}\\!v\in\mathcal{V}}^{R+1}\cup\Big{\\{}0_{\mathcal{B}}\oplus\sqrt{\tfrac{1}{2}}\psi_{t}\oplus\sqrt{\tfrac{3}{2}}b_{t}\Big{\\}}_{t=1}^{V+1}$

forms an equiangular tight frame in
$\mathbb{C}^{\mathcal{B}}\oplus\mathbb{C}^{\mathcal{V}}\oplus\mathbb{C}$.

For the sake of illustration, we turn to Example 7.10 in Tremain’s notes [15].
Consider the columns of the following matrix:

$\left[\setcounter{MaxMatrixCols}{36}\begin{smallmatrix}+&-&+&-&+&-&+&-&+&-&+&-&&&&&&&&&&&&&&&&&&&&&&&&\\\
+&+&-&-&&&&&&&&&+&-&+&-&+&-&+&-&&&&&&&&&&&&&&&&\\\
+&-&-&+&&&&&&&&&&&&&&&&&+&-&+&-&+&-&+&-&&&&&&&&\\\
&&&&+&+&-&-&&&&&+&+&-&-&&&&&+&+&-&-&&&&&&&&&&&&\\\
&&&&+&-&-&+&&&&&&&&&+&+&-&-&&&&&+&+&-&-&&&&&&&&\\\
&&&&&&&&+&+&-&-&+&-&-&+&&&&&&&&&+&-&-&+&&&&&&&&\\\
&&&&&&&&+&-&-&+&&&&&+&-&-&+&+&-&-&+&&&&&&&&&&&&\\\
\bullet&\bullet&\bullet&\bullet&&&&&&&&&&&&&&&&&&&&&&&&&\blacksquare&\square&\blacksquare&\square&\blacksquare&\square&\blacksquare&\square\\\
&&&&\bullet&\bullet&\bullet&\bullet&&&&&&&&&&&&&&&&&&&&&\blacksquare&\blacksquare&\square&\square&\blacksquare&\blacksquare&\square&\square\\\
&&&&&&&&\bullet&\bullet&\bullet&\bullet&&&&&&&&&&&&&&&&&\blacksquare&\square&\square&\blacksquare&\blacksquare&\square&\square&\blacksquare\\\
&&&&&&&&&&&&\bullet&\bullet&\bullet&\bullet&&&&&&&&&&&&&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\square&\square&\square&\square\\\
&&&&&&&&&&&&&&&&\bullet&\bullet&\bullet&\bullet&&&&&&&&&\blacksquare&\square&\blacksquare&\square&\square&\blacksquare&\square&\blacksquare\\\
&&&&&&&&&&&&&&&&&&&&\bullet&\bullet&\bullet&\bullet&&&&&\blacksquare&\blacksquare&\square&\square&\square&\square&\blacksquare&\blacksquare\\\
&&&&&&&&&&&&&&&&&&&&&&&&\bullet&\bullet&\bullet&\bullet&\blacksquare&\square&\square&\blacksquare&\square&\blacksquare&\blacksquare&\square\\\
&&&&&&&&&&&&&&&&&&&&&&&&&&&&\Diamond&\Diamond&\Diamond&\Diamond&\Diamond&\Diamond&\Diamond&\Diamond\end{smallmatrix}\right]$

Here, $+$ and $-$ denote $\pm 1$ and blanks are zeros as before, and
furthermore

$\bullet=\sqrt{2},\qquad\blacksquare=\sqrt{\tfrac{1}{2}},\qquad\square=-\sqrt{\tfrac{1}{2}},\qquad\Diamond=\sqrt{\tfrac{3}{2}}.$

The first $28$ columns come from the Steiner ETF of the previous section.
Steiner ETF vectors in a common embedded simplex have inner product $-1$, and
in the above matrix, these vectors recieve an additional entry in the
$\mathbb{C}^{\mathcal{V}}$ component that changes their inner product to $+1$.
Otherwise, the Steiner-based vectors in the Tremain ETF do not interact in the
$\mathbb{C}^{\mathcal{V}}$ or $\mathbb{C}$ components, thereby inheriting
equiangularity from the Steiner ETF. The other vectors are supported solely in
the $\mathbb{C}^{\mathcal{V}}$ and $\mathbb{C}$ components: the portion in
$\mathbb{C}^{\mathcal{V}}$ forms a simplex extracted from an $8\times 8$
Hadamard matrix whose inner products are $-1/2$, and including the
$\mathbb{C}$ component changes these inner products to $+1$. Also, the
Steiner-based vectors have only one entry of support in common with these
other vectors, allowing the inner products to have modulus
$\sqrt{2}\cdot\sqrt{1/2}=1$. Finally, the vectors all have squared norm $5$,
corresponding to the Welch bound (1.1) for $M=16$ and $N=35$ (namely, $1/5$),
and so this ensemble meets equality in the Welch bound, and equivalently forms
an ETF.

###### Proof of Theorem 3.1.

First, since $M=B+V+1$ and $N=V(R+1)+(V+1)$, a persistent application of the
identities (2.2) with $K=3$ simplifies the Welch bound in this case:

$\sqrt{\frac{N-M}{M(N-1)}}=\frac{1}{R+2}.$

It suffices to demonstrate equality in the Welch bound, e.g., to show that
each member of the purported ETF has squared norm $R+2$, and that inner
products between distinct members lie in $\mathbb{T}$. For convenience, we
denote

(3.1)
$\tau_{v,s}:=\mathbf{E}_{v}\varphi_{s}\oplus\sqrt{2}a_{s}\delta_{v}\oplus
0\qquad\text{and}\qquad\tau_{t}:=0_{\mathcal{B}}\oplus\sqrt{\tfrac{1}{2}}\psi_{t}\oplus\sqrt{\tfrac{3}{2}}b_{t}.$

Checking the norms is straightforward:

$\displaystyle\|\tau_{v,s}\|^{2}$
$\displaystyle=\|\mathbf{E}_{v}\varphi_{s}\|^{2}+\|\sqrt{2}a_{s}\delta_{v}\|^{2}=\|\varphi_{s}\|^{2}+2|a_{s}|^{2}=R+2,$
$\displaystyle\|\tau_{t}\|^{2}$
$\displaystyle=\Big{\|}\sqrt{\tfrac{1}{2}}\psi_{t}\Big{\|}^{2}+\Big{\|}\sqrt{\tfrac{3}{2}}b_{t}\Big{\|}^{2}=\tfrac{1}{2}\|\psi_{t}\|^{2}+\tfrac{3}{2}|b_{t}|^{2}=\tfrac{1}{2}V+\tfrac{3}{2}=R+2.$

To demonstrate equiangularity, we proceed in cases:

Case I: Suppose $v=v^{\prime}$ but $s\neq s^{\prime}$. Then
$\langle\mathbf{E}_{v}\varphi_{s},\mathbf{E}_{v}\varphi_{s^{\prime}}\rangle=\langle\mathbf{E}_{v}^{*}\mathbf{E}_{v}\varphi_{s},\varphi_{s^{\prime}}\rangle=\langle\varphi_{s},\varphi_{s^{\prime}}\rangle$
by Lemma 2.1(b), and so the Naimark property (2.1) gives

(3.2)
$\langle\tau_{v,s},\tau_{v^{\prime},s^{\prime}}\rangle=\langle\mathbf{E}_{v}\varphi_{s},\mathbf{E}_{v}\varphi_{s^{\prime}}\rangle+2a_{s}\overline{a_{s^{\prime}}}=\langle\varphi_{s},\varphi_{s^{\prime}}\rangle+2a_{s}\overline{a_{s^{\prime}}}=a_{s}\overline{a_{s^{\prime}}}.$

Case II: Suppose $v\neq v^{\prime}$. Then for every $s,s^{\prime}\in[R+1]$,
Lemma 2.1(c) gives

(3.3)
$\langle\tau_{v,s},\tau_{v^{\prime},s^{\prime}}\rangle=\langle\mathbf{E}_{v}\varphi_{s},\mathbf{E}_{v^{\prime}}\varphi_{s^{\prime}}\rangle=\langle\mathbf{E}_{v^{\prime}}^{*}\mathbf{E}_{v}\varphi_{s},\varphi_{s^{\prime}}\rangle=\langle\delta_{i}\delta_{j}^{*}\varphi_{s},\varphi_{s^{\prime}}\rangle=\varphi_{s}(j)\overline{\varphi_{s^{\prime}}(i)}.$

Case III: Suppose $t\neq t^{\prime}$. Then the Naimark property (2.1) gives

(3.4)
$\langle\tau_{t},\tau_{t^{\prime}}\rangle=\tfrac{1}{2}\langle\psi_{t},\psi_{t^{\prime}}\rangle+\tfrac{3}{2}b_{t}\overline{b_{t^{\prime}}}=\tfrac{1}{2}\Big{(}\langle\psi_{t},\psi_{t^{\prime}}\rangle+b_{t}\overline{b_{t^{\prime}}}\Big{)}+b_{t}\overline{b_{t^{\prime}}}=b_{t}\overline{b_{t^{\prime}}}.$

Case IV: For every $v\in\mathcal{V}$, $s\in[R+1]$ and $t\in[V+1]$, we have

(3.5)
$\langle\tau_{v,s},\tau_{t}\rangle=\Big{\langle}\sqrt{2}a_{s}\delta_{v},\sqrt{\tfrac{1}{2}}\psi_{t}\Big{\rangle}=a_{s}\psi_{t}(v).$

All of these inner products lie in $\mathbb{T}$, and so we are done. ∎

Since Steiner triple systems exist if and only if $V\equiv 1$ or $3\bmod 6$
with $V\geq 3$, and since complex unimodular simplices exist in every
dimension, we have the following corollary:

######  Corollary 3.2.

For every $V\equiv 1$ or $3\bmod 6$ with $V\geq 3$, there exists an
equiangular tight frame of $N$ vectors in $\mathbb{C}^{M}$ with

$M=\tfrac{1}{6}(V+2)(V+3),\qquad N=\tfrac{1}{2}(V+1)(V+2).$

As illustrated by Tremain’s original example above, this construction is real
whenever the unimodular simplices come from real Hadamard matrices. In
particular, a $(V+1)\times(V+1)$ real Hadamard matrix may be obtained from a
Kronecker product of a $(R+1)\times(R+1)$ real Hadamard matrix with a $2\times
2$ real Hadamard matrix. This gives another corollary:

######  Corollary 3.3.

If there exists an $h\times h$ real Hadamard matrix with $h\equiv 1$ or
$2\bmod 3$, then there exists an equiangular tight frame of $N$ vectors in
$\mathbb{R}^{M}$ with

$M=\tfrac{1}{3}(h+1)(2h+1),\qquad N=h(2h+1).$

The following section demonstrates the novelty of these ETFs by using them to
construct new strongly regular graphs and distance-regular antipodal covers of
the complete graph.

##  4\. New SRGs and DRACKNs

Recall that a $(v,k,\lambda,\mu)$-strongly regular graph (SRG) is a $v$-vertex
graph such that every vertex has the same number $k$ of neighbors, that
adjacent vertices have the same number $\lambda$ of common neighbors, and that
non-adjacent vertices also have the same number $\mu$ of common neighbors.
There are two ways that certain ETFs can be associated with SRGs. First, there
is a one-to-one correspondence between real ETFs (with $N>M$) and a certain
class of SRGs, specifically SRGs with parameters satisfying $2\mu=k$:

######  Proposition 4.1 (see [17, 9]).

Suppose there exists an equiangular tight frame of $N$ vectors in
$\mathbb{R}^{M}$ with $N>M$, and set

$\beta=\sqrt{\frac{N-M}{M(N-1)}}.$

Then there exists a strongly regular graph with parameters

$v=N-1,\quad
k=\frac{N}{2}-1+\frac{1}{2\beta}\left(\frac{N}{M}-2\right),\quad\lambda=\frac{3k-v-1}{2},\quad\mu=\frac{k}{2}.$

Combined with Corollary 3.3, this then implies the following:

######  Corollary 4.2.

If there exists an $h\times h$ real Hadamard matrix with $h\equiv 1$ or
$2\bmod 3$, then there exists a strongly regular graph with parameters

$v=(2h-1)(h+1),\quad
k=(h+2)(h-1),\quad\lambda=\tfrac{1}{2}(h^{2}+2h-6),\quad\mu=\tfrac{1}{2}(h+2)(h-1).$

Examining small values of $h\equiv 1$ or $2\bmod 3$ for which there exists a
real Hadamard matrix, we list the corresponding real Tremain ETFs as well as
the parameters of the associated SRGs given by Corollary 4.2:

$h$ | $M$ | $N$ | $v$ | $k$ | $\lambda$ | $\mu$ | New?
---|---|---|---|---|---|---|---
$2$ | $5$ | $10$ | $9$ | $4$ | $1$ | $2$ | No
$4$ | $15$ | $36$ | $35$ | $18$ | $9$ | $9$ | No
$8$ | $51$ | $136$ | $135$ | $70$ | $37$ | $35$ | No
$16$ | $187$ | $528$ | $527$ | $270$ | $141$ | $135$ | No
$20$ | $287$ | $820$ | $819$ | $418$ | $217$ | $209$ | Yes
$28$ | $551$ | $1596$ | $1595$ | $810$ | $417$ | $405$ | ?

Examining Brouwer’s table of strongly regular graphs [1, 2], we see that the
SRG with parameters $v=819$, $k=418$, $\lambda=217$ and $\mu=209$ appears to
be new to the literature. The SRG with parameters $v=1595$, $k=810$,
$\lambda=417$ and $\mu=405$ is too large to be reported on the table.

Of course, the Hadamard conjecture implies that there is an infinite family of
real Tremain ETFs, but this is unnecessary. Indeed, Kronecker powers of the
$2\times 2$ Hadamard matrix easily produce an infinite family of admissible
Hadamard matrices. In addition, it is not difficult to verify using
Dirichlet’s theorem on primes in arithmetic progressions that Paley’s
construction is admissible infinitely often. Moreover, admissible Hadamard
matrices are closed under Kronecker products, so there is actually a large
family of explicit Hadamard matrices available to the real Tremain ETF
construction.

While all real ETFs give rise to an SRG by way of Proposition 4.1, there is
another method of constructing different SRGs from certain ETFs. A special
case of this construction of SRGs was first observed by Goethals and Seidel
[11].

######  Proposition 4.3 (see [6]).

Suppose there exists an equiangular tight frame $\\{\varphi_{i}\\}_{i=1}^{N}$
in $\mathbb{R}^{M}$ with $N>M$, and set

$\alpha=\frac{N}{M},\qquad\beta=\sqrt{\frac{N-M}{M(N-1)}}.$

If there exists $x\in\mathbb{R}^{M}$ such that $\langle
x,\varphi_{i}\rangle=1$ for every $i\in[N]$, then there exists a strongly
regular graph with parameters

$v=N,\quad
k=\frac{N-1}{2}+\frac{\alpha-1}{2\beta},\quad\lambda=\frac{N}{4}-1+\frac{3\alpha-4}{4\beta},\quad\mu=\frac{N}{4}+\frac{\alpha}{4\beta}.$

Note that the additional hypothesis in Proposition 4.3 asks for a vector
$x\in\mathbb{R}^{M}$ such that $\langle x,\varphi_{i}\rangle=1$ for every
$i\in[N]$. We will show that real Tremain ETFs can be constructed to satisfy
this property if the underlying Steiner triple system contains a parallel
class, that is, a subset of the blocks that form a partition of the set of
points. A parallel class of a $(2,K,V)$-Steiner system must contain exactly
$V/K$ blocks, and so the existence of a parallel class requires $K$ to divide
$V$. Fortunately, it is known that there exists a Steiner triple system with a
parallel class whenever $V\equiv 3\bmod 6$ [4]. This leads to the following:

######  Corollary 4.4.

If there exists an $h\times h$ real Hadamard matrix with $h\equiv 2\bmod 3$,
then there exists a strongly regular graph with parameters

$v=h(2h+1),\quad
k=\frac{(2h-1)(h+2)}{2},\quad\lambda=\frac{(h-1)(h+4)}{2},\quad\mu=\frac{h(h+2)}{2}.$

###### Proof.

Consider real Hadamard matrices $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ of sizes
$h\times h$ and $2h\times 2h$, respectively. Note that we may select
$\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ so that each has all $1$s in the first
row. Put $R=h-1$ and $V=2h-1$, let $\\{\varphi_{s}\\}_{s=1}^{R+1}$ denote the
columns of the matrix $\mathbf{\Phi}$ obtained by removing the last row $a$ of
$\mathbf{H}_{1}$, and let $\\{\psi_{t}\\}_{t=1}^{V+1}$ denote the columns of
the matrix $\mathbf{\Psi}$ obtained by removing the first row $b$ of
$\mathbf{H}_{2}$. Since $h\equiv 2\bmod 3$, we have $V\equiv 3\bmod 6$, and so
we may select a Steiner triple system with a parallel class
$\mathcal{P}\subseteq\mathcal{B}$. For each $v\in\mathcal{V}$, pick an
embedding operator $\mathbf{E}_{v}$ such that
$\mathbf{E}_{v}\delta_{1}=\delta_{b}$ for the unique $b\in\mathcal{P}$ such
that $v\in b$. Combine these ingredients to produce a real Tremain ETF
according to Theorem 3.1.

Now pick
$x\in\mathbb{C}^{\mathcal{B}}\oplus\mathbb{C}^{\mathcal{V}}\oplus\mathbb{C}$
of the form $x=\chi\oplus 0\oplus\sqrt{2/3}$, where $\chi(b)=1$ for every
$b\in\mathcal{P}$, and equals zero otherwise. Then for every
$v\in\mathcal{V}$, our choice of $\mathbf{E}_{v}$ ensures that
$\mathbf{E}_{v}^{*}\chi=\delta_{1}$. Appealing to the notation in (3.1), we
therefore have

$\langle
x,\tau_{v,s}\rangle=\langle\chi,\mathbf{E}_{v}\varphi_{s}\rangle=\langle\mathbf{E}_{v}^{*}\chi,\varphi_{s}\rangle=\langle\delta_{1},\varphi_{s}\rangle=\overline{\varphi_{s}(1)}=1.$

Also, $\langle
x,\tau_{t}\rangle=\sqrt{\tfrac{2}{3}}\cdot\sqrt{\tfrac{3}{2}}\overline{b_{t}}=1$,
and so the result follows from Proposition 4.3. ∎

For small values of $h$, we have the following table of SRG parameters that
arise via Corollary 4.4:

$h$ | $M$ | $N$ | $v$ | $k$ | $\lambda$ | $\mu$ | New?
---|---|---|---|---|---|---|---
$2$ | $5$ | $10$ | $10$ | $6$ | $3$ | $4$ | No
$8$ | $77$ | $136$ | $136$ | $75$ | $42$ | $40$ | No
$20$ | $431$ | $820$ | $820$ | $429$ | $228$ | $220$ | Yes
$32$ | $1073$ | $2080$ | $2080$ | $1071$ | $558$ | $544$ | ?
$44$ | $2003$ | $3916$ | $3916$ | $2001$ | $1032$ | $1012$ | ?
$56$ | $3221$ | $6328$ | $6328$ | $3219$ | $1650$ | $1624$ | ?

Again, we consult Brouwer’s table of strongly regular graphs [1, 2] to
identify which SRGs are known. The SRG with parameters $v=820$, $k=429$,
$\lambda=228$ and $\mu=220$ seems to be new, and the last three graphs from
the table above are too large to be reported on the table.

Having constructed a couple of new families of SRGs, we turn to another
interesting class of graphs. An $(n,r,c)$-distance-regular antipodal cover of
the complete graph (DRACKN) is a simple graph $G$ on $rn$ vertices along with
a partition of the vertices into $n$ sets (called fibers) of $r$ vertices such
that

  * (i)

no two vertices in the same fiber are adjacent,

  * (ii)

the edge set contains a perfect matching between any two fibers, and

  * (iii)

non-adjacent vertices have $c$ common neighbors.

DRACKNs have been investigated in [10, 12, 5]. Recently, Coutinho, Godsil,
Shirazi and Zhan [5] proved the following result that produces DRACKNs from
certain ETFs:

######  Proposition 4.5 (Theorem 5.1 in [5]).

Let $p$ be prime. If there exists an equiangular tight frame
$\\{\varphi_{i}\\}_{i=1}^{N}$ in $\mathbb{C}^{M}$ such that
$\langle\varphi_{i},\varphi_{j}\rangle$ is a $p$th root of unity whenever
$i\neq j$, then there exists an $(N,p,c)$-distance-regular antipodal cover of
the complete graph, where

$c=\frac{1}{p}\left(N-2+\frac{2M-N}{\beta
M}\right),\quad\quad\beta=\sqrt{\frac{N-M}{M(N-1)}}.$

In what follows, we show how Tremain ETFs can be constructed to satisfy the
hypothesis of Proposition 4.5 and hence produce DRACKNs. The crucial
ingredient here is the existence of so-called Butson-type Hadamard matrices,
that is, complex Hadamard matrices whose entries are roots of unity.
Notationally, an $H(p,h)$ matrix is any $h\times h$ Hadamard matrix whose
entries are all $p$th roots of unity.

######  Corollary 4.6.

Let $p$ be prime. If there exist $H(p,h)$ and $H(p,2h)$ matrices for some
$h\equiv 1$ or $2\bmod 3$, then there exists a
$(h(2h+1),p,2h^{2}/p)$-distance-regular antipodal cover of the complete graph.

###### Proof.

Consider $H(p,h)$ and $H(p,2h)$ matrices $\mathbf{H}_{1}$ and
$\mathbf{H}_{2}$, respectively. Put $R=h-1$ and $V=2h-1$, let
$\\{\varphi_{s}\\}_{s=1}^{R+1}$ denote the columns of the matrix
$\mathbf{\Phi}$ obtained by removing the first row $a$ of $\mathbf{H}_{1}$,
and let $\\{\psi_{t}\\}_{t=1}^{V+1}$ denote the columns of the matrix
$\mathbf{\Psi}$ obtained by removing the first row $b$ of $\mathbf{H}_{2}$.
Since $h\equiv 1$ or $2\bmod 3$, we have $V\equiv 1$ or $3\bmod 6$, and so we
may select a Steiner triple system with appropriate embedding operators, and
combine these ingredients to produce a Tremain ETF according to Theorem 3.1.

By Proposition 4.5, it suffices to show that the inner products between all
pairs of vectors in this ETF are $p$th roots of unity. These inner products
are expressed in equations (3.2)–(3.5). Since $a_{s}$, $b_{t}$,
$\varphi_{s}(i)$ and $\psi_{t}(v)$ are $p$th roots of unity for every
$s\in[R+1]$, $t\in[V+1]$, $i\in[R]$ and $v\in\mathcal{V}$, we are done. ∎

It is conjectured that there exists an $H(p,\lambda p)$ matrix for every prime
$p$ and every positive integer $\lambda$. One result in this direction is the
following, due to Butson himself:

######  Proposition 4.7 (see [3]).

If $p$ is prime and $k\geq m\geq 0$ are integers, then there exists an
$H(p,2^{m}p^{k})$ matrix.

As such, the construction in Corollary 4.6 gives an infinite family of
DRACKNs. To demonstrate the novelty of this construction, we first note that
by Theorem 5.1 in [5], our DRACKNs are abelian in a certain (technical) sense.
Existing abelian DRACKNs are enumerated in Table 1 in [5], and all of these
either have $n=rc$ or $(n,r,c)=(45,3,12)$. By contrast, the DRACKNs from
Corollary 4.6 satisfy $n-rc=h\neq 0$.

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