# Dispersion properties of plasmonic sub-wavelength elliptical wires wrapped
with graphene

Mauro Cuevas12 1Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET)  2 Universidad Austral, Facultad de Ingeniería, Pilar Mariano Acosta
1611 B1629WWA-Pilar-Buenos Aires Argentina<EMAIL_ADDRESS>Ricardo A.
Depine3 3Grupo de Electromagnetismo Aplicado, Departamento de Física, FCEN,
Universidad de Buenos Aires and IFIBA, Ciudad Universitaria, Pabellón I,
C1428EHA, Buenos Aires, Argentina

###### Abstract

One fundamental motivation to know the dispersive, or frequency dependent
characteristics of localized surface plasmos (LSPs) supported by elliptical
shaped particles wrapped with graphene sheet, as well as their scattering
characteristics when these elliptical LSPs are excited, is related with the
design of plasmonic structures capable to manipulate light at sub-wavelength
scale. The anisotropy imposed by the ellipse eccentricity can be used as a
geometrical tool for controlling plasmonic resonances. Unlike metallic case,
where the multipolar eigenmodes are independent of each others, we find that
the induced current on graphene boundary couples multipolar eigenmodes with
the same parity. In the long wavelength limit, a recursive relation equation
for LSPs in term of the ellipse eccentricity parameter is derived and explicit
solutions at lowest order are presented. In this approximation, we obtain
analytical expressions for both the anisotropic polarizability tensor elements
and the scattered power when LSPs are excited by plane wave incidence.

##  1 Introduction

Light scattering properties on metallic wire particles with non-circular cross
section have been extensively studied [1, 2]. The two-dimensional geometrical
anisotropy leads to a splitting in two or more plasmon branches corresponding
to the lowest energy states, instead of one as occur in the circular case,
which are evidenced on the angular optical response of the wire [3]. This fact
together with other outstanding properties have found applications in some
optical topics requiring field hot spots, such as surface enhanced Raman and
optical nano-antennas development [4, 5, 6, 7].

Sub-wavelength structures coated with a graphene sheet provide a suitable
alternative to metallic elements because they exhibit relative low loss and
highly tunable, via electrostatic gating or chemical doping [8], surface
plasmons in the frequency region from microwaves to infrared [9]. The small
plasmon wavelength, which usually reaches values smaller than one tenth of the
wavelength of the photon of the same frequency, deals with the possibility to
build smaller plasmonic constituent elements, a feature positioning the
graphene as a promising platform to the development of controllable plasmon
devices, in particular of a new generation of sensors and modulators from
microwaves to the mid-infrared regimes [10, 11, 12, 13] constituting an
intersection between optics and electronic.

Graphene layers structures has attracted wide attention in many applications
due to a strong adsorption capacity for contaminants [14, 15], electrical
properties [16, 17, 18] and improved light harvesting [19, 20, 21].

In the framework of electromagnetic scattering by sub-wavelength graphene
particles, an extensive wealth of theoretical analysis have been developed,
allowing possible a wide range of applications based on the interaction
between graphene and electromagnetic radiation via LSP mechanisms, including
sensing [22, 23], superscattering [24, 25], low energy spasers [26, 27, 28,
30], PT-symmetric structures tailoring lasing modes [29, 30], and micro and
nano antennas [31, 32, 33, 34].

This work deals with the plasmon properties of an elliptical dielectric wire
wrapped with graphene. The LSP branches corresponding to lower eigenmodes are
obtained by solving the homogeneous problem, i.e., the scattering problem
without external excitation. Unlike the metallic case, where LSP eigenmodes
are uncoupled between them, the current density on graphene coating couples
all multipolar LSPs with the same parity given rise to an infinite set of
coupled equations for the field amplitude. Moreover, the tangential direction
of charge oscillation on graphene covered leads to a counterintuitive
splitting: the low dipolar frequency mode corresponds to polarizability
oscillations along minor ellipse axis while the high dipolar frequency mode
corresponds to polarizability oscillations along the major ellipse axis. In
this framework, analytical expressions of the anysotropic polarizability is
found. We also provided analytical expressions for the scattered power when
the graphene elliptical wire is excited by a plane wave. All scattering curves
are compared with those obtained by applying a rigorous formalism based in the
Green surface integral method [35, 36] valid for arbitrary shaped cross
sections.

This paper is organized as follows. Firstly, in Section 2 we develop an
analytical method based on the separation of variables in elliptical
coordinates and obtain an approximated solution for the electromagnetic field
scattered by an elliptical wire cover with graphene. This approach allows us
to express each of the amplitudes in the multipolar expansion of the fields as
a series of power of the ellipse eccentricity. In Section 3 we present the
results related with the dispersive characteristics and the scattering
calculations for a dielectric wire wrapped with graphene. Finally, 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,
with $\omega$ as the angular frequency, $t$ as the time, and $i=\sqrt{-1}$.
The symbols Re and Im are respectively used for denoting the real and
imaginary parts of a complex quantity.

##  2 Theory

###  2.1 Scattered field equations

We consider an elliptical dielectric cylinder wrapped with graphene sheet. The
elliptical profile has the major semi-axis $a$ along $x$ axis and the minor
semi-axis $b$ along the $y$ axis. We use elliptical coordinates, which are
related with the Cartesian coordinates by following relations

$\begin{array}[]{ll}x=l\cosh(\rho)\cos(\phi),\\\
y=l\sinh{}(\rho)\sin(\phi),\\\ z=z\end{array}$ (1)

$\rho$ and $\phi$ are the radial and angular elliptical coordinates. The unit
vectors $\hat{\rho}$ and $\hat{\phi}$ are normal and tangent along elliptical
shape, respectively (see Figure 1). In this way, the boundary curve of the
ellipse is given by $\rho=\rho_{0}$, $\tanh(\rho_{0})=b/a$, with the major
semi-axis $a=l\cosh(\rho_{0})$ along the $x$ axis and the minor semi-axis
$b=l\sinh(\rho_{0})$ along the $y$ axis. The scale parameter $l$ is related
with $a$ and $b$ by the following $l=\sqrt{a^{2}-b^{2}}$.

Figure 1:  Schematic illustration of the elliptical wire wrapped with
graphene. The wire ($\varepsilon_{1}$ and surface conductivity $\sigma$) is
embedded in a transparent medium with permittivity $\varepsilon_{2}$. Both
media are non magnetic, $\mu_{1}=\mu_{2}=1$.

The magnetic field inside the elliptic cylinder $\rho<\rho_{0}$ can be
expressed as follows:

$H^{(1)}(\rho,\phi)=\sum_{m=1}^{+\infty}a_{m}\sinh(m\rho)\sin(m\phi)+c_{m}\cosh(m\rho)\cos(m\phi),$
(2)

and the field outside the ellipse $\rho>\rho_{0}$ is expressed as,

$H^{(2)}(\rho,\phi)=\sum_{m=1}^{+\infty}b_{m}e^{-m\rho}\sin(m\phi)+d_{m}e^{-m\rho}\cos(m\phi)+H^{(inc)}.$
(3)

where the last term corresponds to the incident magnetic field. Firstly, we
suppose a plane wave incidence with the electric field along the $x$ axis and
amplitude $E_{0}$. In this case, the incident magnetic field in the long
wavelength limit is written as

$H^{(inc)}(\rho,\phi)=-ik_{0}E_{0}\varepsilon_{2}l\sinh(\rho)\sin(\phi).$ (4)

The symmetry imposed by the incident field direction imposes the amplitudes
$c_{m}$ and $d_{m}$ in the field expressions (2) and (3) to be zero. By using
the Ampere-Maxwell equation, we can relate the components of the electric
field and the $z$-component of the magnetic field as follows,

$\mathbf{E}^{(j)}=-\frac{1}{ik_{0}\varepsilon_{j}}\mathbf{\nabla}_{t}\times\hat{z}H^{(j)}$
(5)

$j=1,\,2$,

$\mathbf{\nabla}_{t}=\frac{1}{f(\rho_{0},\phi)}\left[\hat{\rho}\frac{\partial}{\partial\rho}+\hat{\phi}\frac{\partial}{\partial\phi}\right]$
(6)

is the transverse part of the $\mathbf{\nabla}$ operator and
$f(\rho_{0},\phi)=l\sqrt{\cosh^{2}(\rho)-\cos^{2}(\phi)}$. The boundary
condition along the contour of the ellipse $\rho=\rho_{0}$ is written as [37],

$\begin{array}[]{ll}\frac{1}{\varepsilon_{1}}\frac{\partial
H^{(1)}}{\partial\rho}|_{\rho_{0}}=\frac{1}{\varepsilon_{2}}\frac{\partial
H^{(2)}}{\partial\rho}|_{\rho_{0}}\\\
\left[H^{(2)}+H^{(inc)}-H^{(1)}\right]|_{\rho_{0}}=\frac{4\pi\sigma}{c}[E^{(2)}_{\phi}+E^{(inc)}_{\phi}]|_{\rho_{0}}.\end{array}$
(7)

Using field expressions (2) and (3) with $c_{m}=d_{m}=0$ into the boundary
condition (7) and projecting into the Fourier basis, two system of equations
for amplitudes $a_{m}$ and $b_{m}$ are obtained. Then, we can eliminate
amplitudes $a_{m}$ to obtain one set of equations for amplitudes
$B_{m}=b_{m}e^{-m\rho_{0}}$ (see appendix B),

$\displaystyle B_{k}t_{k}+\sum_{m\not=k}^{+\infty}M_{km}B_{m}=Q_{k},$ (8)

with

$\displaystyle M_{km}=\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}ims_{km},$ (9)
$\displaystyle
t_{k}=1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\tanh(k\rho_{0})+M_{kk},$ (10)
$\displaystyle
Q_{k}=ik_{0}E_{0}(\varepsilon_{2}-\varepsilon_{1})l\sinh(\rho_{0})\delta_{k1}+\frac{4\pi\sigma}{c}E_{0}\coth(\rho_{0})s_{k1}.$
(11)

###  2.2 Lowest order eccentricity

Before studying the scattering problem given by Eq. (8), we first study the
homogeneous problem (or eigenmodes problem), i.e., the scattering problem
without an external source ($E_{0}=0$ in Eq. (8)). To do this, we consider the
homogeneous part in Eq. (8) which can be rewritten as

$\displaystyle B_{k}t_{k}+\sum_{m\not=k}^{+\infty}M_{km}B_{m}=0,$ (12)

Since in the limit of null eccentricity $\tanh(k\rho_{0})=1$ for
$\rho=\infty$, the first term in Eq. (12) reduces to the dispersion relation
of a cylinder with circular cross section for the $k$ multipolar order, while
the second term reduce to zero. For small values of eccentricity,
$e=l/b=\sqrt{(a/b)^{2}-1}<<1$, we can expand Eq. (12) in powers of $e$ as
follows. Taking into account that the matrix element is of order $O(2)$ for
$k\not=m$ (see appendix C), $s_{km}\approx e^{2}$, from Eq. (9) we deduce that
$M_{1m}$ is at last of order $O(2)$ and thus we can expand the homogeneous
equation (12) in powers of $e$. The form that Eq. (12) takes for $k=1$ is,

$\displaystyle B_{1}t_{1}+\sum_{m\not=1}^{+\infty}M_{1m}B_{m}=0,$ (13)

and the form that Eq. (12) takes for $m\not=1$ is,

$\displaystyle
B_{m}=-\frac{M_{m1}}{t_{m}}B_{1}-\sum_{n\not=1,m}^{+\infty}\frac{M_{mn}}{t_{m}}B_{n}.$
(14)

Replacing Eq. (14) into Eq. (13), we obtain

$\displaystyle
B_{1}t_{1}-\sum_{m\not=1}^{+\infty}M_{1m}\frac{M_{m1}}{t_{m}}B_{1}-\sum_{m\not=1}^{+\infty}\sum_{n\not=1,m}^{+\infty}M_{1m}\frac{M_{mn}}{t_{m}}B_{n}=0.$
(15)

Following the same steps as described above, we solve Eq. (12) by iteration,

$\displaystyle
t_{1}-\sum_{m\not=1}^{+\infty}M_{1m}\frac{M_{m1}}{t_{m}}-\sum_{m\not=1}^{+\infty}\sum_{n\not=1,m}^{+\infty}M_{1m}\frac{M_{mn}}{t_{m}}\frac{M_{n1}}{t_{n}}+...=0.$
(16)

This equation is the dispersion relation of elliptical LSPs on graphene. To
find the LSP characteristics, we consider the lowest order, $e^{4}$, in the
dispersion relation (16),

$\displaystyle t_{1}-M_{13}\frac{M_{31}}{t_{3}}=0,$ (17)

which can be made explicit using Eqs. (9) and (10) as follows

$\displaystyle\left(1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\tanh(\rho_{0})+\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}is_{11}\right)$
$\displaystyle\times\left(1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\tanh(3\rho_{0})+3\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}is_{33}\right)+3\left(\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}\right)^{2}(s_{13})^{2}=0,$
(18)

where we have used $s_{13}=s_{31}$ and the fact that $s_{12}=s_{21}=0$. The
expressions for $s_{1m}$ and $s_{33}$ until order $e^{2}$ can be obtained from
Eq. (C) (see appendix C),

$\displaystyle s_{11}=\frac{1}{b}\left[1-\frac{3}{8}e^{2}\right],$
$\displaystyle s_{13}=\frac{1}{b}\frac{1}{8}e^{2},$ $\displaystyle
s_{33}=\frac{1}{b}\left[1-\frac{1}{4}e^{2}\right].$ (19)

We note that for null eccentricity, the matrix element $s_{13}=0$ and
$\tanh(\rho_{0})=\tanh(3\rho_{0})=1$, as a consequence the dispersion relation
(2.2) reduces to a product between two factors, both corresponding to a
circular cylinder. One of these factors corresponds to the dispersion relation
of the dipolar order ($m=1$) and the other corresponds to the hexapolar order
($m=3$). For small values of eccentricity, the matrix element $s_{13}\not=0$
and consequently the last term in Eq. (2.2) is not null, with which Eq. (2.2)
represents the elliptical LSP dispersion relation. Note that the coupling
mechanism between the dipolar and the hexapolar orders is evidenced by the
presence of the last term in the dispersion equation (2.2).

It is worth nothing that Eq. (2.2) is the lowest order of the elliptical LSP
dispersion relation. This is true because the truncation of Eq. (15) at order
$O(4)$ in eccentricity admit only modes $m\leq 4$. In addition, for higher
orders, $O(2N)$ for example, Eq. (15) couples $m=1,\,3,\,5,...,2N-1$
multipolar orders.

At the same order in eccentricity as we have written Eq. (17), from Eq. (8) we
can obtain the dipolar coefficient $b_{1}$ for the scattering of a plane wave
(non homogeneous problem) polarized along $x$ axis,

$b_{1}=E_{0}e^{\rho_{0}}\widetilde{b_{1}},$ (20)

where

$\displaystyle\widetilde{b_{1}}=\frac{A_{x}}{t_{1}t_{3}-M_{13}M_{31}},$ (21)

and

$\displaystyle
A_{x}=\left(ik_{0}b[\varepsilon_{2}-\varepsilon_{1}]+\frac{4\pi\sigma
s_{11}}{c\tanh(\rho_{0})}\right)$
$\displaystyle\times\left(1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\tanh(3\rho_{0})+\frac{4\pi\sigma
i3s_{33}}{ck_{0}\varepsilon_{2}l\sinh(\rho_{0})}\right)$
$\displaystyle-\left(\frac{4\pi\sigma}{c}\right)^{2}\frac{\cosh(\rho_{0})is_{13}s_{31}}{k_{0}\varepsilon_{2}l\sinh^{2}(\rho_{0})}$
(22)

Once the scattering amplitude $b_{1}$ is found, the polarizability
$\alpha_{x}$ is given by (see appendix D)

$\alpha_{x}=\frac{i(a+b)}{4k_{0}}\widetilde{b_{1}}.$ (23)

Following the same steps that allow us to obtain Eq. (17) but taking the
electric field along the $y$ axis, we obtain the lowest order dispersion
equation for $y$ polarization,

$\displaystyle\left(1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\coth(\rho_{0})+\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}iv_{11}\right)\left(1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\coth(3\rho_{0})+3\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}iv_{33}\right)$
$\displaystyle+3\left(\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}\right)^{2}(v_{13})^{2}=0,$
(24)

$\displaystyle v_{11}=\frac{1}{a}\left[1+\frac{3}{8}e^{2}\right],$
$\displaystyle v_{13}=v_{31}=\frac{1}{a}\frac{1}{8}e^{2},$ $\displaystyle
v_{33}=\frac{1}{a}\left[1+\frac{1}{4}e^{2}\right].$ (25)

At the same order in eccentricity as we have written Eq. (2.2), the dipolar
coefficient $d_{1}$ for the scattering of a plane wave polarized along $y$
axis,

$d_{1}=E_{0}e^{\rho_{0}}\widetilde{d_{1}},$ (26)

where $d_{1}$ is the dipolar amplitude of the scattered field in medium 2 for
$y$ polarization and

$\displaystyle\widetilde{d_{1}}=\frac{A_{y}}{q_{1}q_{3}-N_{13}N_{31}},$ (27)

where

$\displaystyle
A_{y}=\left(ik_{0}b[\varepsilon_{2}-\varepsilon_{1}]+\frac{4\pi\sigma
v_{11}}{c\coth(\rho_{0})}\right)$
$\displaystyle\times\left(1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\coth(3\rho_{0})+\frac{4\pi\sigma
i3v_{33}}{ck_{0}\varepsilon_{2}l\cosh(\rho_{0})}\right)$ (28)
$\displaystyle-\left(\frac{4\pi\sigma}{c}\right)^{2}\frac{\sinh(\rho_{0})iv_{13}v_{31}}{k_{0}\varepsilon_{2}l\cosh^{2}(\rho_{0})},$
(29)

and

$\displaystyle
q_{k}=1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\coth(k\rho_{0})+N_{kk},$ (30)
$\displaystyle N_{mk}=\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}imv_{km}.$ (31)

The corresponding polarizability $\alpha_{y}$ is given by (see appendix D)

$\alpha_{y}=\frac{i(a+b)}{4k_{0}}\widetilde{d_{1}}.$ (32)

It is interesting to note that in the limit of $\rho_{0}\rightarrow\infty$,
the polarizabilities (23) and (32) tend to the value corresponding to the
circular case,

$\alpha_{c}=\varepsilon_{2}\frac{b^{2}}{2}\left[\frac{\varepsilon_{1}-\varepsilon_{2}+\frac{4\pi\sigma
i}{ck_{0}b}}{\varepsilon_{1}+\varepsilon_{2}+\frac{4\pi\sigma
i}{ck_{0}b}}\right].$ (33)

Polarizabilities (23) and (32) constitutes the central object for optical
interactions where the quasistatic regime is applicable [38], such as enhanced
and confined optical near–fields [39], metasurface and metagrating
applications [40] and nanoparticles bonding [41, 42, 43]. In the present work,
we use these expressions to calculate the power scattered by the particle in
the quasistatic limit as follows. Taking into account that the radiated power
by a point dipole $\mathbf{p}$ is [44],

$P=\frac{\pi\omega^{3}}{4c^{2}}|\mathbf{p}|^{2},$ (34)

and neglecting third order contributions, we can replace the induced dipole
moment components $p_{j}=\alpha_{j}\,E_{0}$ ($j=x,\,y$) to calculate the power
scattered by the elliptical particle in the dipolar limit. Considering that
the incident power is $P_{0}=\frac{c}{8\pi}E_{0}^{2}2L_{i}$, $L_{i}=a,\,b$ for
$x$ and $y$ polarization, respectively, the normalized power scattered by the
particles is written as

$\frac{P_{i}}{P_{0}}=\frac{\pi^{2}\omega^{3}}{L_{i}c^{3}}|\alpha_{i}|^{2}.$
(35)

##  3 Results

In this section we use the formalism developed in the above section to
calculate the eigenfrequencies and the scattering cross section curves. In all
the examples the wire is immersed in vacuum ($\varepsilon_{2}=\mu_{2}=1$), the
dielectric core has a permittivity $\varepsilon_{1}=3.9$ and permeability
$\mu_{1}=1$. The graphene parameters are $T=300$K and $\gamma_{g}=0.1$ meV.

In order to explore the effects that the departure from the circular geometry
has on the dispersive characteristics of LSPs, we compare the results obtained
for all elliptical shapes with those obtained in the circular case with the
same perimeter. We assume that the perimeter is sufficiently large to describe
the optical properties of the wires as characterized by the same local surface
conductivity as planar graphene (see appendix A). Moreover, since the aroused
interest on graphene due to plasmonic properties in THz and IR frequency
regions, graphene–coated wires with a micro-sized cross section have become an
attractive platform for optical applications (see [45, 46, 47] and Refs.
therein). In this way, we have chosen a perimeter equal to $\pi\,\mu$m for all
examples, excepting those presented in Fig. 5a where we have selected
$\pi/2\,\mbox{and}\,\pi/4\,\mu$m. Even though nonlocal effects could appear in
our system for frequencies lower than characteristic resonance frequencies,
this is not interesting for our purposes (see appendix E).

Firstly, we evaluate the coupling mechanism between multipolar orders provided
by the graphene current. To do this, by solving Eq. (8) we calculate the
amplitude modulus $|b_{m}|$ for $m=1$ and $3$ (both non-null lowest order) as
a function of $\omega/c$ frequency and for three values of eccentricity,
$a/b=1.05,\,1.2$ and $1.4$. Without loss of generality we made the calculation
for horizontal (or $x$) polarization (electric field parallel to $x$ axis).
For the lowest eccentricity value of $a/b=1.05$ we observe that the amplitude
$b_{1}$ reaches it maximum value ($\approx 10$) at $\hbar\omega=31.52$meV
corresponding to the dipolar excitation. At the same frequency, the amplitude
$b_{3}$ reaches a local maximum ($\approx 0.1$), evidencing the coupling
between third and first order modes. This means that the amplitude $b_{3}$ not
only contributes to the scattered field at the resonance frequency of the
$m=1$ order (where the $|b_{1}|$ amplitude reaches the absolute maximum
value), a fact that also occur on metallic nanoparticles [1, 48, 49, 50] for
which the field amplitudes are decoupled, but also reaches a local maximum at
this frequency.

Figure 2:  Amplitudes $b_{m}$ modulus ($m=1$ and $3$) as a function of
$\omega/c$ frequency for $a/b=1.05,\,1.2,\,1.4$ and for $x$ polarization. The
perimeter of the ellipse is $\pi\,\mu$m and chemical potential
$\mu_{g}=0.5eV$.

On the other hand, the amplitude $b_{3}$ reaches the absolute maximum value at
$\hbar\omega\approx 59$meV corresponding to the hexapolar order excitation,
while the first order amplitude $b_{1}$ shows a local maximum and minimum at
this frequency value. As eccentricity values are increasing, two effects are
observed: on the one hand, dipolar and hexapolar order frequencies moves to
higher values and, on the other hand, the coupling mechanism between different
orders appear more visible, as can be seen in Figure 2 where the local maximum
value of $b_{3}$ curve at the resonant dipolar frequency (where the $b_{1}$
amplitude reaches the absolute maximum value) increases with the value of
$a/b$. The same behavior is observed near $\hbar\omega\approx 59$meV, where
the local variation of the $b_{1}$ amplitude at the hexapolar resonant
frequency is more visible with the $a/b$ increment. These results show how a
non–zero surface conductivity on graphene couples different orders of the same
parity (see second term in Eq. (8)) causing all these modes to reach a local
maximum at the resonant frequency of one of them.

In the vertical (or $y$) polarization (electric field parallel to $y$ axis)
case (not shown in Figure 2), we obtain similar results with the only
exception that the resonant frequency is a decreasing function of the
eccentricity $a/b$, instead of being increasing as occur in the $x$
polarization case.

In order to gain insight about the resonant frequency dependence with the
eccentricity of the wire, we solve the homogeneous problem at lowest order in
eccentricity to obtain the dispersion equation for lower modes. Figure 3 shows
the dispersion relation as a function of $a/b$ ($a/b\leq 1.4$) and the
scattering curves for both horizontal and vertical polarization for $a/b=1.2$
($e=0.66$). From Figure 3a we see two branches, the upper branch, calculated
with Eq. (17), corresponds to $x$ polarization and the lower branch,
calculated with Eq. (2.2), corresponds to $y$ polarization. For null
eccentricity, $a=b$, these two branches converge to the value
$\hbar\omega=34.28$meV corresponding to the case of circular cross section. As
the value of eccentricity increases, the gap between two branches increases
leading to an increment in the splitting between resonance frequencies when
the structure is illuminated with a plane wave. In Figure 3b we plotted the
scattering curves for plane wave incidence by using Eq. (35) with $i=x$ and
$i=y$ for $x$ and $y$ polarization, respectively, (continuous line). The
calculation also was made by using a rigorous method based on the Green
surface integral method (GSIM) (dashed line) [35]. Two peaks are observed. The
lower frequency peak, angle of incidence $\theta=90^{\circ}$, corresponds to
excitation of $y$-polarized LSPs, i.e., surface plasmons whose polarizability
$\alpha_{y}$ is along the $y$ axis. The upper frequency peak, angle of
incidence $\theta=0$, correspond to excitation of $x$-polarized LSPs, i.e.,
surface plasmons whose polarizability $\alpha_{x}$ is along the $x$ axis. Both
resonance frequencies agree well with the values calculated from the
dispersion equation. We also observe a small red shifting of the resonance
peaks calculated using GSIM with respect to those calculated using Eq. (35).

Figure 3:  (a) Real part of the eigenfrequencies as a function of
$\frac{a}{b}$ ($\frac{a}{b}=\sqrt{e^{2}+1}$). (b) Scattering cross section for
incidence direction parallel to $y$ axis (electric field along $x$ axis) and
parallel to y axis (electric field along the $y$ axis). Continuous lines
correspond to the calculus using Eq. (35) and dashed line curves correspond to
calculation using GSIM. The arrows indicate the incident direction. The
perimeter of the ellipse is $\pi\,\mu$m and chemical potential
$\mu_{g}=0.5$eV.

It is worth noting that unlike metallic structures, where the lower modal
frequency is associated to oscillations along the major axis (the $x$ axis in
the present case) and the upper modal frequency is along the minor axis (the
$y$ axis in the present case), from Figure 3 we see that the situation is
completely other way around for graphene wrapped wires. This fact is due to
the geometrical difference between the charge oscillations on each of the
systems. While in the metallic case the displacement of charge is along the
induced electric field, in the graphene particles case, the movement of charge
is along the boundary elliptical curvature.

In order to study the behavior with graphene parameters, we calculate the
eigenfrequency dependence on chemical potential. Figure 4 shows the
eigenfrequency branches, one for $x$ polarization and the other for $y$
polarization, for various values of the chemical potential,
$\mu_{g}=0.6,\,0.7,\,0.8,\,0.9,\,1$eV. We observe that eigenfrequencies of the
upper and lower branches increase with the chemical potential increment. This
is consistent with the fact that frequency plasmonic resonances for circular
cross section cylinders are proportional to $\sqrt{\mu_{g}}$ [52].

Figure 4:  Real part of the eigenfrequencies as a function of $\frac{a}{b}$
($\frac{a}{b}=\sqrt{e^{2}+1}$) for $\mu_{g}=0.6,\,0.7,\,0.8,\,0.9,\,1$eV.
Solid lines correspond to $x$ polarization and dashed lines corresponds to $y$
polarization. Other parameters are the same as in Figure 3

Finally, we calculate the dependence of the eigenfrequencies with the size and
the contrast between external and internal constitutive parameters. In Figure
5a we observe that both branches, corresponding to $x$ and $y$ polarizations,
increase their frequencies as the ellipse perimeter is decreased from $\pi$ to
$\pi/4\,\mu$m. This fact can be understood from the circular case studied in
[52], where we have demonstrated that the eigenfrequency depends on the radius
as $\omega\approx R^{-1/2}$. In Figure 5b we have plotted the eigenfrequency
branches for three values of the permittivity of the internal medium,
$\varepsilon_{1}=3.9,\,3,\,2.13$. We observe that Re$\,\hbar\omega$ increases
with the decreasing of the permittivity value. This behavior is consistent
with the fact that the eigenfrequency depends on permittivities as
$\omega\approx(1+\varepsilon_{1})^{-1/2}$ for circular graphene wires [52].

Figure 5:  Real part of the eigenfrequencies as a function of $\frac{a}{b}$
($\frac{a}{b}=\sqrt{e^{2}+1}$). (a) $\varepsilon_{1}=3.9$ and various values
of perimeters: $\pi,\,\pi/2,\,\pi/4\,\mu$m, (b) for perimeter equal to
$\pi\mu$m and various values of permittivities,
$\varepsilon_{1}=3.9,\,3,\,2.13$. The chemical potential is $\mu_{g}=0.5$eV.
Solid lines correspond to $x$ polarization and dashed lines corresponds to $y$
polarization.

##  4 Conclusion

In conclusion, we have analytically studied the dispersive characteristics for
an ellipical dielectric wire wrapped with graphene. We have found two branches
corresponding to the lowest frequency band, one of them corresponds to
eigenmodes with a dipole moment along the major axis and the other with a
dipole moment along the minor axis of the wire elliptical cross section.
Interestingly, we found that contrary to what happens in the metallic
plasmonic case, the low dipolar frequency branch corresponds to polarizability
oscillations along minor ellipse axis and the high dipolar frequency eigenmode
corresponds to polarizability oscillations along the major ellipse axis. We
have found analytical expressions for the polarizability elements along the
ellipse axis which reduces to that of the circular shaped wire as the
eccentricity parameter tends to zero.

We think that the results provided in this work can be employed for a deeper
interpretation of the dispersive characteristics of elliptical graphene
plasmons, which opens up possibilities for practical applications using
structures capable to manipulate light at sub-wavelength scale.

##  Appendix A Graphene conductivity

The graphene layer is considered as an infinitesimally thin layer with a
frequency-dependent surface conductivity $\sigma(\omega)$ given by the Kubo
formula [51], $\sigma=\sigma^{intra}+\sigma^{inter}$, with the intraband and
interband contributions given by

$\sigma^{intra}(\omega)=\frac{2ie^{2}k_{B}T}{\pi\hbar^{2}(\omega+i\gamma_{g})}\mbox{ln}\left[2\mbox{cosh}(\mu_{g}/2k_{B}T)\right],$
(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 Boundary conditions

By replacing Eqs. (2) and (3) into Eq. (7) we obtain a set of two coupled
equations for amplitudes $a_{m}$ and $b_{m}$,

$\sum_{m=1}^{+\infty}\left(\frac{a_{m}}{\varepsilon_{1}}\cosh(m\rho_{0})m+m\frac{b_{m}}{\varepsilon_{2}}e^{-m\rho_{0}}\right)\sin(\phi)=\frac{1}{\varepsilon_{2}}\frac{\partial
H^{(inc)}}{\partial\rho}$ (B1)
$\displaystyle\sum_{m=1}^{+\infty}\left(b_{m}e^{-m\rho_{0}}-a_{m}\sinh(m\rho_{0})+\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}imb_{m}e^{-m\rho_{0}}\frac{1}{f(\rho_{0},\phi)}\right)$
$\displaystyle\times\sin(m\phi)=$
$\displaystyle-H^{(inc)}|_{\rho_{0}}+\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}i\frac{1}{f(\rho_{0},\phi)}\frac{\partial
H^{(inc)}}{\partial\rho}|_{\rho_{0}}.$ (B2)

Note that the left hand side of Eq. (B1) is written as a sine Fourier series,
then we can expand the right hand side as a sine series and find, performing
Fourier integral, a lineal equation between amplitudes $a_{m}$ and $b_{m}$,

$\frac{a_{k}}{\varepsilon_{1}}\cosh(k\rho_{0})k+\frac{b_{k}}{\varepsilon_{2}}ke^{-k\rho_{0}}=-ik_{0}E_{0}l\cosh(\rho_{0})\delta_{k1}.$
(B3)

Contrary, Eq. (B) presents a difficulty related with the graphene current
factor $f^{-1}(\rho_{0},\phi)$. Multiplying Eq. (3) by $\sin(k\phi)$ ($k\geq
1$) and integrating in $[-\pi,\,\pi]$, we obtain,

$\displaystyle
b_{k}e^{-k\rho_{0}}-a_{k}\sinh(k\rho_{0})+\sum_{m=1}^{+\infty}\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}imb_{m}e^{-m\rho_{0}}s_{km}=$
$\displaystyle-h_{k}+\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}idh_{k},$ (B4)

where

$\displaystyle
s_{km}=\frac{1}{\pi}\int_{-\pi}^{\pi}\frac{1}{f(\rho_{0},\phi)}\sin(m\phi)\sin(k\phi)d\phi$
(B5) $\displaystyle
h_{k}=\frac{1}{\pi}\int_{-\pi}^{\pi}H^{(inc)}|_{\rho}\sin(k\phi)d\phi$ (B6)
$\displaystyle
dh_{k}=\frac{1}{\pi}\int_{-\pi}^{\pi}\frac{1}{f(\rho_{0},\phi)}\frac{\partial
H^{(inc)}}{\partial\rho}|_{\rho_{0}}\sin(k\phi)d\phi.$ (B7)

Note that both functions $H^{(inc)}|_{\rho_{0}}$ and $\frac{\partial
H^{(inc)}}{\partial\rho}|_{\rho_{0}}$ have an angular dependence
$\approx\sin(\phi)$ and thus $h_{k}\approx\delta_{k1}$ and $dh_{k}\approx
s_{km}\delta_{m1}$. Therefore, Eq. (B) can be written as,

$\displaystyle
b_{k}e^{-k\rho_{0}}-a_{k}\sinh(k\rho_{0})+\sum_{m=1}^{+\infty}\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}imb_{m}e^{-m\rho_{0}}s_{km}=$
$\displaystyle
ik_{0}E_{0}\varepsilon_{2}l\sinh(\rho_{0})\delta_{k1}+\frac{4\pi\sigma}{c}E_{0}\coth(\rho_{0})s_{k1}.$
(B8)

Note that in absence of graphene, i.e. $\sigma=0$, the system of Eqs. (B3) and
(B) are uncoupled for field amplitudes. Contrary, in presence of graphene, the
graphene current term in the left hand side of Eq. (B) couples orders with the
same parity. This is true because the matrix elements $s_{km}\not=0$ provided
that $m+k$ be even (Eq. (B5)).

By eliminatig the amplitudes $a_{m}$ from Eqs. (B3) and (B), we obtain an
equation for amplitudes $b_{m}$,

$\displaystyle
b_{k}e^{-k\rho_{0}}\left(1+\frac{\varepsilon_{1}}{\varepsilon_{2}}\tanh(k\rho_{0})+\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}iks_{kk}b_{k}e^{-k\rho_{0}}\right)+$
$\displaystyle\sum_{m\not=k}^{+\infty}\frac{4\pi\sigma}{ck_{0}\varepsilon_{2}}imb_{m}e^{-m\rho_{0}}s_{km}=$
$\displaystyle
ik_{0}E_{0}(\varepsilon_{2}-\varepsilon_{1})l\sinh(\rho_{0})\delta_{k1}+\frac{4\pi\sigma}{c}E_{0}\coth(\rho_{0})s_{k1}.$
(B9)

##  Appendix C Explicit form of $s_{km}$ matrix elements

We expand the current factor as power of $\sin(\phi)$,

$\displaystyle\frac{1}{h(\rho_{0},\phi)}=\frac{1}{\sinh(\rho_{0})\sqrt{1+\sin^{2}(\phi)e^{2}}}=$
$\displaystyle\frac{1}{\sinh(\rho_{0})}[1-\sin^{2}(\phi)\frac{1}{2}e^{2}+\frac{3}{8}\sin^{4}(\phi)e^{4}-\frac{5}{16}\sin^{6}(\phi)e^{6}$
$\displaystyle+\frac{35}{128}\sin^{8}(\phi)e^{8}-\frac{63}{226}\sin^{10}(\phi)e^{10}$
$\displaystyle+\frac{231}{1024}\sin^{12}(\phi)e^{12}-\frac{429}{2048}\sin^{14}(\phi)e^{14}+...].$
(C1)

Note that the expansion (C) is valid for $a/b<\sqrt{2}$. By replacing Eq. (C)
into Eq. (B5), we obtain

$\displaystyle
s_{km}=\frac{1}{b\pi}\int_{-\pi}^{\pi}[1-\sin^{2}(\phi)\frac{1}{2}e^{2}+\frac{3}{8}\sin^{4}(\phi)e^{4}$
$\displaystyle-\frac{5}{16}\sin^{6}(\phi)e^{6}+\frac{35}{128}\sin^{8}(\phi)e^{8}-\frac{63}{226}\sin^{10}(\phi)e^{10}$
$\displaystyle+\frac{231}{1024}\sin^{12}(\phi)e^{12}-\frac{429}{2048}\sin^{14}(\phi)e^{14}+...]$
$\displaystyle\times\sin(m\phi)\sin(k\phi)d\phi.$ (C2)

From this equation, we can see that $s_{km}\approx e^{2}$ ($s_{km}$ is of
order $O(2)$) for $k\not=m$. For example, if we consider $k=1$, it is
straightforward verify,

$\displaystyle
s_{1m}=\frac{1}{b}[\delta_{m1}-\frac{e^{2}}{2}\left(\frac{3}{4}\delta_{m1}-\frac{1}{4}\delta_{m3}\right)$
$\displaystyle+\frac{3}{8}e^{4}\left(\frac{5}{8}\delta_{m1}-\frac{5}{16}\delta_{m3}+\frac{11}{16}\delta_{m5}\right)+...].$
(C3)

We can see that $s_{11}$ is of order $O(0)$, $s_{13}$ is of order $O(2)$,
$s_{15}$ is order $O(4)$, …, $s_{1m}$ is order $O(m-1)$.

##  Appendix D Field of a dipole moment placed at the origin

We consider a line dipole source (whose axis lies along the $z$ axis) with a
dipole moment $\mathbf{p}$ is placed at the origin. The magnetic field is
along the $z$ axis ($\mathbf{H}(\mathbf{r})=\phi(\mathbf{r})\hat{r}$). The
wave equation for $\phi(\mathbf{r})$ when the retardation is negligible reads

$\displaystyle\nabla^{2}\phi(\mathbf{r})=-4\pi
ik_{0}\mathbf{p}\times\nabla\delta(\mathbf{r}).$ (D1)

The solution $\phi(\mathbf{r})$ of Eq. (D1) can be written as

$\displaystyle\phi(\mathbf{r})=ik_{0}\mathbf{p}\times\nabla f(\mathbf{r}),$
(D2)

where $f(\mathbf{r})$ satisfies

$\displaystyle\nabla^{2}\phi(\mathbf{r})=-4\pi\delta(\mathbf{r}).$ (D3)

By solving Eq. (D3) we obtain $\phi(\mathbf{r})=-2\log(\mathbf{r})$. Then, the
magnetic field $\phi(\mathbf{r})$ of a point dipole calculated using Eq. (D2)
in elliptical coordinates (1) is given by

$\displaystyle\phi(\mathbf{r})=-2ik_{0}\frac{p_{x}\sinh{\rho}\sin\phi-
p_{y}\cosh{\rho}\cos\phi}{l(\cosh^{2}{\rho}-\sin^{2}{\phi})},$ (D4)

where $p_{x}$ and $p_{y}$ are the Cartesian components of the dipole moment
$\mathbf{p}$. For $\rho$ values large enough, this expression can be rewritten
as

$\displaystyle\phi(\mathbf{r})=-4ik_{0}\frac{p_{x}\sin\phi-
p_{y}\cos\phi}{l}e^{-\rho}.$ (D5)

By comparing this equation (with $p_{y}=0$) with the first term in the
scattered field given in Eq. (3) we obtain the induced electrical dipole as a
function of the $b_{1}$ scattering amplitude

$p_{x}=\frac{il}{4k_{0}}b_{1}=\frac{il}{4k_{0}}E_{0}e^{\rho_{0}}\widetilde{b_{1}}=\frac{i(a+b)}{4k_{0}}E_{0}\widetilde{b_{1}},$
(D6)

where the sub-index stand for the direction of the dipole moment. In the last
equality in Eq. (D6) we have used $le^{\rho_{0}}=a+b$. Finally, dividing Eq.
(D6) by $E_{0}$, the corresponding polarizability $\alpha_{x}$ is obtained.

In the similar way, by comparing Eq. (D5) (with $p_{x}=0$) with the second
term in Eq. (3) we obtain the induced electrical dipole

$p_{y}=\frac{il}{4k_{0}}d_{1}=\frac{il}{4k_{0}}E_{0}e^{\rho_{0}}\widetilde{d_{1}}=\frac{i(a+b)}{4k_{0}}E_{0}\widetilde{d_{1}}.$
(D7)

By dividing Eq. (D7) by the incident field amplitude $E_{0}$, we obtain the
$\alpha_{y}$ polarizability (32).

##  Appendix E Nonlocal response in graphene conductivity

The objective of this section is to provide information about the relation
between the local graphene conductivity response considered in this work and
the dimensions of the proposed structure. The local approximation begins to
break down as a reduction of the structure takes place. The nonlocality or
spatial dispersion in graphene conductivity arises when the plasmon phase
velocity $v_{SP}$ is slow and comparable with the electron Fermi velocity
$v_{F}\approx 10^{12}\,\mu m/s$. Since, we focused on cylindrical structures
whose cross section is a slight deviation from the circular cross section, an
estimation of their size can be made by considering a cylinder with circular
cross section and radius $R$ ($a<R<b$). Taking into account the small size of
the cylinder, $R/\lambda<<1$ ( $\lambda=2\pi/\omega$ the wavelength), the
phase velocity of plasmons $v_{SP}=k_{SP}/\omega$ can be estimated by using
the quasistatic approximation. In this way, the surface plasmon effective
momentum $k_{SP}$, which is along the azimutal angle ($\phi$ axis), can be
written as [52],

$\displaystyle k_{SP}=\frac{m}{R},$ (E1)

where $m$ is the eigenmode order. If we consider $m=1$ (dipolar order), the
factor

$\displaystyle\frac{v_{F}}{v_{SP}}=\frac{v_{F}}{R\,\omega}\approx\frac{10^{12}\mu\mbox{m}/s}{R\,\frac{\omega}{c}\,3\times
10^{14}\mu m/s}=\frac{1}{R\,\frac{\omega}{c}\,300}.$ (E2)

The smaller $\frac{v_{F}}{v_{SP}}$ in Eq. (E2), the better the local
conductivity approximation. For example, if we consider a value
$\frac{v_{F}}{v_{SP}}<1/10$, graphene local conductivity differs less than
1$\%$ from that considering non local effects [38]. As a consequence, our
method is acceptable provided that

$\displaystyle R\,\frac{\omega}{c}>\frac{1}{30}.$ (E3)

Equation (E3) establishes a reasonable limit for the applicability of our
model. Figure 6 shows a map with two regions. In one of them the nonlocal
effects can be neglected (upper region), whereas in the other (lower region)
these effects become important. The curve separating these regions corresponds
to $R$ values, named $R_{c}$, for which the equality in Eq. (E3) is fulfilled.

Figure 6:  Regions of the plane $\mbox{R}-\mu_{g}$ (radius and chemical
potential) where, according to Eq. (E3), local and non local effects take
place. The curve separating these two regions corresponds to critical values
of the radius, $R_{c}$, that verify the equality in Eq. (E3) for varying
$\mu_{g}$ values.

## Acknowledgment

The authors acknowledge the financial supports of Universidad Austral
O04-INV00020 and Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET).

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