# Terahertz binding of nanoparticles based on graphene surface plasmons
excitations

Hernan Ferrari  Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET).  Facultad de Ingeniería, Universidad Austral, Mariano Acosta 1611,
Pilar 1629, Buenos Aires, Argentina.  Carlos J. Zapata–Rodrǵuez  Department of
Optics and Optometry and Vision Sciences, University of Valencia, Dr. Moliner
50, 46100, Burjassot, Spain.  Mauro Cuevas<EMAIL_ADDRESS>
###### Abstract

This work studies the optical binding of a dimer composed by dielectric
particles close to a graphene sheet. Using a rigorous electromagnetic method,
we calculated the optical force acting on each nanoparticle. In addition, we
deduced analytical expressions enabling to evaluate the contribution of
graphene surface plasmons (GSPs) to optical binding. Our results show that
surface plasmon on graphene excitations generate multiple equilibrium
positions for which the distance between particles are tens of times smaller
than the photon wavelength. Moreover, these positions can be dynamically
controlled by adjusting the chemical potential on graphene. Normal and oblique
incidence have been considered.

###### keywords:

surface plasmon , graphene , THz nanoparticles binding

###### PACS:

81.05.ue , 73.20.Mf , 78.68.+m , 42.25.Fx

††journal: Journal of Quantitative Spectroscopy and Radiative Transfer

##  1 Introduction

The scattered field by a neutral particle assembly can result, under
particular conditions, in an effective dipole–dipole interaction useful for
optically binding the particle array. Most of the literature in this field has
been devoted to micro–sized particles, since the significant electromagnetic
scattering at this scale makes the strength of optical binding surpassing the
hydrodynamic interactions and the stochastic thermal activation [1, 2].

The significant progress made in experimental techniques and the extensive
wealth of theoretical research have meet the need of optical binding between
nano sized particles. In this context, the effect of localized surface plasmon
excitations on assembly clusters of metallic nanoparticles has been
theoretically and experimentally demonstrated [3, 4, 5, 6]. In addition,
stable optical binding between dielectric nano particles using an evanescent
field formed by total reflection at a dielectric interface has been
demonstrated [7, 8]. Recently, a planar metallic or metamaterial structure was
proposed as an optical binding tool, in which the propagating eigenmodes
excited on the surface play an essential role on the increment of the
interaction force between dielectric nano particles. The binding distance
between nanoparticles is defined by the eigenmode wavelength which, in case of
bound modes such as surface plasmons, is smaller than the photon wavelength
[9, 10].

It is known, apart from the well known surface plasmons supported by a
metallic surface, long livid propagating plasmons can be supported by graphene
from terahertz (THz) to infrared (IR) frequency range. GSPs present good
tunability through electrical or chemical modification of the carrier
densitiy, relative low loss and tightly confined fields. These properties have
been the subject of theoretical and experimental studies to find application
in a wide range of disciplines. For example, GSPs have been proposed as a new
generation of nano antennas for communications in the THz bands [11],
molecular sensors capable to selectively enhance individual spectral features
[12, 13], low frequency spaser (see [14] and Refs. therein) and for enhancing
the spontaneous emission and energy transfer between molecules [15, 16, 17].
Recently, the use of high-quality GSP resonances on twin graphene–coated
dielectric rods to enhance the diffraction radiation intensity in the IR
region has been demonstrated [18].

Regarding to optical tweezers framework, some works have focused on the
optical trapping using graphene as plasmonic material [19, 20]. In these
structures, GSPs provided the channels for capturing a small dielectric
nanoparticle [21] or trapping and sorting nanoparticles [22]. In [23], authors
proposed a nanopatterned graphene structure for particle trapping/levitation.
In addition, optical force on the nonlinear graphene–wrapped nanoparticle has
been investigated [24].

A natural question that arises from the above outstanding characteristics is
whether propagating GSPs can be useful for binding particles, taking advantage
of their great localization to bind particles greatly exceeding the
diffraction limit. In particular, how GSPs enable dielectric nanoparticles to
be bound at distances of few hundred nanometers between them by using low
frequency radiation instead of visible radiation. In consequence, the present
work is theoretically developed in order to respond to these questions.
Specifically, we analytically study the binding conditions for a dimer formed
by dielectric nanoparticles, providing a complete description about the use of
GSPs to create equilibrium positions solely by plane wave radiation. Since the
kinematic properties of GSPs can be controlled by chemical potential
variations, possibilities to dynamically control the equilibrium positions
without changing the geometrical parameters of the structure are opened.

This paper is organized as follows. In section 2, we present a brief
description of the analytical method, based on the Green approach, to
calculate the optical force between two dielectric nanoparticles placed above
a graphene sheet. By using contour integration in the complex plane, we
obtained simple formulas, which contain the GSP contribution to the optical
force, that reproduce the main results obtained by applying the rigorous
theory. In section 3, we exemplify for both normally and obliquely incident
illumination cases, and for different particle shapes. Concluding remarks are
provided in section 4. The international system of units is used and an
$\exp(-i\omega t)$ 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 Optical binding.

We consider a system formed by two dielectric particles placed above of a
plane graphene sheet (at $z=0$) separating two homogeneous half spaces, vacuum
($\varepsilon_{1}=1$) and a non–magnetic and isotropic dielectric medium
($\varepsilon_{2}=2.13$). Both dielectric particles are identical. Spherical
particles are characterized by radius $R_{1}=R_{2}=R$, and other shapes are
taken in such a way the volumes of these objects are equal to that of the
sphere. The relative permittivity of the particles is $\varepsilon_{p}$ and
they are placed at $\mathbf{r}_{A}=z_{0}\hat{z}$ and
$\mathbf{r}_{B}=x_{0}\hat{x}+z_{0}\hat{z}$ ($z_{0}\geq 0$) in the field of an
incident plane wave (see Fig. 1).

Figure 1: Schematic of the problem. Two identical particles of permittivity
$\varepsilon_{p}$ under illumination of a plane wave. The particles are
embedded in vacuum ($\varepsilon_{1}=1$) at a distance $z_{0}$ from the
graphene plane, at $z=0$, that separates the vacuum from a dielectric medium
characterized by a permittivity $\varepsilon_{2}=2.13$. In case of normal
incidence, the electric field can be parallel (parallel polarization) or
perpendicular (perpendicular polarization) to the axis joining the particles.

Taking into account that the size of the particles are smaller than plasmon
and photon wavelengths ($R<\lambda_{sp}<<\lambda$), we can use the dipolar
approximation. In this framework, the time average of the total force acting
on a single particle is [26],

$\mathbf{F(\mathbf{r})}=\frac{1}{2}\mbox{Re}\sum_{j=x,y,z}p_{j}^{*}\nabla
E_{j}(\mathbf{r}),$ (1)

where $E_{j}$ is the $j$ component of the electric field,
$\mathbf{p}=\alpha_{0}\mathbf{E}$ is the induced electric dipole on particle
at $\mathbf{r}$ position,

$\alpha_{0}=\frac{\alpha_{e}}{1-i\frac{k_{0}^{3}}{6\pi\varepsilon_{0}}\alpha_{e}},$
(2)

is the radiation corrected electric polarizability, $k_{0}=2\pi/\lambda$ is
the modulus of the vacuum photon wavevector, $\varepsilon_{0}$ is the vacuum
permittivity and $\alpha_{e}$ is the electrostatic approximation of particle
polarizability. In case of spherical particle,

$\alpha_{e}=4\pi\varepsilon_{0}R^{3}\frac{\varepsilon_{p}-\varepsilon_{1}}{\varepsilon_{p}+2\varepsilon_{1}}.$
(3)

In case of other shapes such as cubes or cylinders particles, we have
calculated the $\alpha_{e}$ polarizability following the accurate formulas
presented in [27].

The electric field in Eq. (1) is given by

$\mathbf{E(\mathbf{r})}=\mathbf{E}_{0}+\frac{k_{0}^{2}}{\varepsilon_{0}}\mathbf{\hat{G}}(\mathbf{r},\mathbf{r}_{A})\mathbf{p}_{A}+\frac{k_{0}^{2}}{\varepsilon_{0}}\mathbf{\hat{G}}(\mathbf{r},\mathbf{r}_{B})\mathbf{p}_{B},$
(4)

where $\mathbf{E}_{0}$ is the superposition of the incident field and the
reflected field on the flat substrate,
$\mathbf{\hat{G}}(\mathbf{r},\mathbf{r}_{j})=\mathbf{\hat{G}}_{0}(\mathbf{r},\mathbf{r}_{j})+\mathbf{\hat{G}}_{s}(\mathbf{r},\mathbf{r}_{j})$
is the sum of the vacuum and the scattered–by–the–surface Green tensors of a
point dipole. Second and third terms in Eq. (4) correspond to the scattered
electric fields on $\mathbf{r}$ position by the electric dipole in particle A
and B, respectively. By solving the self consistent Eq. (4) for
$\mathbf{E}(\mathbf{r}_{j})$ ($j=\mbox{A, B}$) and using Eq. (1) we can
calculate the optical force on particle A or B. An equivalent form using the
effective particle polarizabilities defined in C that simplify the force
calculation was considered in [3, 9]. For example, by using the expression for
$\mathbf{p}_{A}$ given by Eq. (D4) and that equivalent for $\mathbf{p}_{B}$,
Eqs. (4) and Eq. (1), we obtain an expression for the force on particle B,

$\displaystyle\mathbf{F(\mathbf{r}_{B})}=\frac{1}{2}\mbox{Re}\sum_{j=x,y,z}p_{Bj}^{*}\nabla
E_{j}(\mathbf{r})|_{\mathbf{r}_{B}}=\frac{1}{2}\mbox{Re}\sum_{j=x,y,z}p_{Bj}^{*}\nabla\Bigg{\\{}[\mathbf{E}_{0}(\mathbf{r})]_{j}$
$\displaystyle+\frac{k_{0}^{2}}{\varepsilon_{0}}[\mathbf{\hat{G}}(\mathbf{r},\mathbf{r}_{A})\mathbf{p}_{A}]_{j}+\frac{k_{0}^{2}}{\varepsilon_{0}}[\mathbf{\hat{G}}(\mathbf{r},\mathbf{r}_{B})\mathbf{p}_{B}]_{j}\Bigg{\\}}|_{\mathbf{r}_{B}},$
(5)

where $[\cdot]_{j}$ indicates the $j$ coordinate of $\cdot$. An equivalent
equation can be obtained for particle A.

###  2.2 Graphene surface plasmon contribution.

Graphene surface plasmons, eigenmodes of the structure that propagate along
the graphene surface with their electric and magnetic fields decaying
exponentially away from the graphene sheet, can provide channels for enhancing
the optical force between particles. Since Im $\sigma>0$ ($\sigma$ is the
graphene conductivity) in the frequency range considered in this work, only
$p$ polarized GSPs exist, i.e., surface waves with the total magnetic field
parallel to the surface [28]. Thus, the full characteristics of the GSPs can
be obtained by studying the singularities of the analytic continuation of the
reflection coefficient $r_{p}(k_{||})$. Pole singularities occur at complex
locations and they represent the propagation constant $k_{sp}$ of the GSPs.
Since the integration path in Green’s functions

$\displaystyle\mathbf{\hat{G}}_{s}(\mathbf{r}_{B},\mathbf{r}_{A})=\frac{i}{8\pi^{2}}\int_{0}^{\infty}dk_{||}\,\mathbf{f}(k_{||},\rho)\,e^{i\gamma^{(1)}(z_{A}+z_{B})},$
(6)

is set along the real and positive $k_{||}$ axis, the integral, and
consequently the Green’s functions, will be strongly affected by singularities
$k_{||}=k_{sp}$ that are close to that axis. In Eq. (6), $k_{||}$ is the wave
vector parallel to the surface,
$\rho=\sqrt{|\mathbf{r}_{A}-\mathbf{r}_{B}|^{2}-(z_{A}-z_{B})^{2}}$ and
functions $\mathbf{f}(k_{||},\rho)$ are defined in B. Following the same steps
as in Ref. [25], we can extract the GSP contribution from the Green functions
$\mathbf{G_{s}(\mathbf{r},\mathbf{r}_{j})}$ ($j=\mbox{A, B}$) as follows.
Since the procedure is similar for all components of
$\mathbf{\hat{G}}_{s}(\mathbf{r},\mathbf{r}^{\prime})$, we show it for the
$xx$ component. Applying the symmetry properties of Bessel integrals [25],

$\displaystyle\int_{0}^{+\infty}f_{odd}(x)J_{0}(x)dx=\frac{1}{2}\int_{-\infty}^{+\infty}f_{odd}(x)H_{0}^{(1)}(x)dx,$
$\displaystyle\int_{0}^{+\infty}f_{even}(x)J_{1}(x)dx=\frac{1}{2}\int_{-\infty}^{+\infty}f_{even}(x)H_{1}^{(1)}(x)dx,$
(7)

where $f_{odd}(x)$ and $f_{even}(x)$ are odd and even functions, respectively,
of the argument, we obtain

$\displaystyle\hat{G}_{s,xx}(\mathbf{r}_{B},\mathbf{r}_{A})=\frac{i}{8\pi^{2}}\int_{-\infty}^{\infty}dk_{||}\,2\pi$
$\displaystyle\times\frac{k_{||}\gamma^{(1)}}{k_{0}^{2}}\frac{1}{2}[\frac{H_{1}(k_{||}x_{0})}{k_{||}x_{0}}-H_{0}(k_{||}x_{0})]r_{p}(k_{||})\,e^{i2\gamma^{(1)}z_{0}}=$
$\displaystyle-\frac{i}{8\pi^{2}}\int_{-\infty}^{\infty}dk_{||}\,2\pi\frac{k_{||}\gamma^{(1)}}{k_{0}^{2}}\frac{1}{2}H^{\prime}_{1}(k_{||}x_{0})r_{p}(k_{||})\,e^{i2\gamma^{(1)}z_{0}},$
(8)

where we only have considered the $p$ polarization. In the last equality in
Eq. (2.2) we used the fact that $H_{1}(z)/z-H_{0}(z)=-H^{\prime}_{1}(z)$. Note
that we have set $\theta=0$, $z_{A}=z_{B}=z_{0}$ and $\rho=x_{0}$ to write Eq.
(2.2) because both particles are on $x$ axis and at the same distance from the
graphene plane. We now deform the integration path in (2.2) into a semicircle
of large radius ($|k_{||}|\rightarrow\infty$) in the positive imaginary
half–plane $\mbox{Im}k_{||}>0$, avoiding the branch point and pole
singularities, as in Ref. [25]. Then, the residues theorem gives

$\displaystyle\hat{G}_{s,xx}(\mathbf{r}_{B},\mathbf{r}_{A})=\frac{k_{sp}\gamma^{(1)}_{sp}}{4k_{0}^{2}}H^{\prime}_{1}(k_{sp}x_{0})\,e^{i2\gamma^{(1)}z_{0}}\mbox{Res}\,r_{p},$
(9)

where $k_{sp}$ is the propagation constant of GSPs,
$\gamma^{(1)}=\sqrt{k_{0}^{2}-k_{sp}^{2}}$ and

$\displaystyle\mbox{Res}\,r_{p}=\lim_{k_{||}\to k_{sp}}(k_{||}-k_{sp})r_{p}.$
(10)

As $k_{sp}$ is almost real and higher than the modulus of the photon wave
vector $k_{0}$, $\gamma^{(1)}\approx i\gamma_{sp}$ with
$\gamma_{sp}=\sqrt{k_{sp}^{2}-k_{0}^{2}}$ a real number ($k_{sp}>k_{0}$).
Thus, Eq. (9) can be written as,

$\displaystyle\hat{G}_{s,xx}(\mathbf{r}_{B},\mathbf{r}_{A})=i\frac{k_{sp}\gamma_{sp}}{4k_{0}^{2}}H_{1}^{\prime}(k_{sp}x_{0})\,e^{-2\gamma_{sp}z_{0}}\mbox{Res}\,r_{p}.$
(11)

Using the non retarded approximation, i.e., $k_{0}<<k_{sp}$, it follows
$\gamma_{sp}=k_{sp}$ and
$\mbox{Res}\,r_{p}=2k_{sp}/(\varepsilon_{1}+\varepsilon_{2})$ (see D). As a
consequence, Eq. (11) takes the form,

$\displaystyle\hat{G}_{s,xx}(\mathbf{r}_{B},\mathbf{r}_{A})=i\frac{k_{sp}^{3}}{2k_{0}^{2}(\varepsilon_{1}+\varepsilon_{2})}H^{\prime}_{1}(k_{sp}x_{0})\,e^{-2k_{sp}z_{0}}.$
(12)

Taking the limit case for $\mathbf{r}_{A}$ tending to $\mathbf{r}_{B}$, we
obtain

$\displaystyle\hat{G}_{s,xx}(\mathbf{r}_{B},\mathbf{r}_{B})=i\frac{k_{sp}^{3}}{4k_{0}^{2}(\varepsilon_{1}+\varepsilon_{2})}\,e^{-2k_{sp}z_{0}}.$
(13)

Now, we can use Eq. (2.1) to calculate the force in the $x$ direction.

Firstly, we begin by studying the case of normal incidence. In the limit case
in which the diagonal elements of the effective polarizability tensor (D7) are
dominant [9], from Eq. (2.1), the $x$ component of the optical force on
particle B can be approximated by,

$\displaystyle
F^{(GSP,x)}_{x}(\mathbf{r}_{B})=\frac{1}{2}\frac{k_{0}^{2}}{\varepsilon_{0}}\,\Bigg{\\{}\mbox{Re}\,p_{Bx}^{*}\frac{\partial}{\partial
x}[\hat{G}_{s,xx}(\mathbf{r},\mathbf{r}_{A})]|_{\mathbf{r}_{B}}\,p_{Ax}+$
$\displaystyle\mbox{Re}\,p_{Bx}^{*}\frac{\partial}{\partial
x}[\hat{G}_{s,xx}(\mathbf{r},\mathbf{r}_{B})]|_{\mathbf{r}_{B}}\,p_{Bx}\Bigg{\\}},$
(14)

for the case in which the incident electric field is along the $x$ axis, and

$\displaystyle
F^{(GSP,y)}_{x}(\mathbf{r}_{B})=\frac{1}{2}\frac{k_{0}^{2}}{\varepsilon_{0}}\,\Bigg{\\{}\mbox{Re}\,p_{By}^{*}\frac{\partial}{\partial
x}[\hat{G}_{s,yy}(\mathbf{r},\mathbf{r}_{A})]|_{\mathbf{r}_{B}}\,p_{Ay}+$
$\displaystyle\mbox{Re}\,p_{By}^{*}\frac{\partial}{\partial
x}[\hat{G}_{s,yy}(\mathbf{r},\mathbf{r}_{B})]|_{\mathbf{r}_{B}}\,p_{By}\Bigg{\\}},$
(15)

for the case in which the incident electric field is along the $y$ axis. Note
that we have included the superscript $x$ and $y$ in Eqs. (2.2) and (2.2) to
indicate the direction of the incident electric field. Since,
$\hat{G}_{s,xx}(\mathbf{r}^{\prime},\mathbf{r}^{\prime})$ given by Eq. (13) is
imaginary, it is follows that $\frac{\partial}{\partial
x}\hat{G}_{s,xx}(\mathbf{r}^{\prime},\mathbf{r}^{\prime})$ is imaginary,
consequently the last term in Eqs. (2.2) and (2.2) does not contribute to the
force. Taking derivative respect to $x$ variable, and considering
$p_{Ax}=p_{Bx}=p_{x}$ (the field is normally incident on the surface), Eq.
(2.2) is written as

$\displaystyle
F^{(GSP,x)}_{x}(\mathbf{r}_{B})=\mbox{Re}\Bigg{\\{}i\frac{1}{2}\frac{k_{0}^{2}}{\varepsilon_{0}}\,|p_{x}|^{2}\frac{k_{sp}^{4}}{2k_{0}^{2}(\varepsilon_{1}+\varepsilon_{2})}H^{\prime\prime}_{1}(k_{sp}x_{0})\,e^{-2k_{sp}z_{0}}\Bigg{\\}}\approx$
$\displaystyle\frac{1}{4\varepsilon_{0}}\,|p_{x}|^{2}\frac{k_{sp}^{4}}{(\varepsilon_{1}+\varepsilon_{2})}Y_{1}(k_{sp}x_{0})\,e^{-2k_{sp}z_{0}}.$
(16)

In the last equality we have given the plasmonic force for large range
distances, $x_{0}>>1/k_{sp}$. Following the same steps for obtaining Eq.
(2.2), from Eqs. (2.2) and (B) with $\theta=0$, we obtain

$\displaystyle
F^{(GSP,y)}_{x}(\mathbf{r}_{B})\approx\frac{1}{4\varepsilon_{0}}\,|p_{y}|^{2}\frac{k_{sp}^{4}}{(\varepsilon_{1}+\varepsilon_{2})}\frac{Y_{2}(k_{sp}x_{0})}{k_{sp}x_{0}}\,e^{-2k_{sp}z_{0}},$
(17)

where we have used $\frac{d}{dx}(H_{1}(x)/x)=-H_{2}(x)/x$. Equations (2.2) and
(17) give the GSP optical force in $x$ direction between two particles aligned
along the $x$ axis, when the field is normally incident with its electric
field parallel to the $x$ axis (Eq. (2.2)) and parallel to the $y$ axis (Eq.
(17)).

Next, we obtain the plasmon contribution for oblique incidence in the case for
which the plane of incidence coincides with the $x-z$ plane. To do this, by
following the same procedure as that used to obtain Eq. (12), we extract the
GSP contribution from all components of the Green tensor (6) and replace these
into Eq. (2.1),

$\displaystyle
F_{x}(\mathbf{r}_{B})\approx\frac{1}{2}\mbox{Re}\Bigg{\\{}ik_{0}\sin\theta_{i}[p^{*}_{Bx}E_{0x}(\mathbf{r}_{B})+p^{*}_{Bz}E_{0z}(\mathbf{r}_{B})]$
$\displaystyle+\frac{k_{sp}^{4}}{4\varepsilon_{0}(\varepsilon_{1}+\varepsilon_{2})}\,\Bigg{(}-iH_{1}(k_{sp}x_{0})[p_{Bx}^{*}p_{Ax}+p^{*}_{Bz}p_{Az}]$
$\displaystyle-
iH_{1}^{\prime}(k_{sp}x_{0})[p_{Bz}^{*}p_{Ax}-p_{Bx}^{*}p_{Az}]+\frac{i}{2}[p_{Bz}^{*}P_{Bx}-p_{Bx}^{*}P_{Bz}]\Bigg{)}\Bigg{\\}}$
$\displaystyle\times e^{-2k_{sp}z_{0}},$ (18)

where we have used that
$\hat{G}_{s,xy}(\mathbf{r}_{B},\mathbf{r}_{A})=\hat{G}_{s,yz}(\mathbf{r}_{B},\mathbf{r}_{A})=0$
because $\theta=0$ or $\pi$ in Eqs. (B) and (B9) (see B). The first term in
Eq. (2.2) corresponds to the component force along the $x$ axis of the
incident and reflected by the graphene plane surface fields,
$E_{0x}(\mathbf{r}_{B})=E_{i}\cos\theta_{i}[e^{-ik_{0}\cos\theta_{i}z_{0}}-r_{p}e^{ik_{0}\cos\theta_{i}z_{0}}]e^{ik_{0}\sin\theta_{i}x_{B}}$
and
$E_{0z}(\mathbf{r}_{B})=E_{i}\sin\theta_{i}[e^{-ik_{0}\cos\theta_{i}z_{0}}+r_{p}e^{ik_{0}\cos\theta_{i}z_{0}}]e^{ik_{0}\sin\theta_{i}x_{B}}$.
The second term represents the interaction between two dipole moments, both
oriented in the $x$ and $z$ axis, respectively, via surface plasmon
excitations. The third term represents the interaction between dipole $A$ and
$B$, one of them is along the $x$ axis and the other is along the $z$ axis,
and the last term represents the interaction between both components, $x$ and
$z$, of the same dipole moment. The force on particle A can be obtained from
Eq. (2.2) by permuting the indices A and B and changing the sign of function
$H_{1}(k_{sp}x_{0})$ in the second term.

##  3 Results.

In this section, we apply the sketch developed in above sections to calculate
the optical force between particles under plane wave illumination. Firstly,
our calculations are made for spherical particles and then we take care of
others shapes different to spherical ones, like cylinders and cubes. The
relative permittivity of the particles are $\varepsilon_{p}=3.9$. The radius
of the spheres is $R=90$nm and the size of cubes an cylinders are taken in
such a way the volumes of these three objects are equal. We use the large
doping condition for which the graphene conductivity takes a Drude behaviour
(see A).

Figure 2: Optical force as a function of the interparticle distance $x_{0}$
for (a) parallel and (b) perpendicular polarizations, and (c) potential energy
binding normalized to $k_{B}$T for parallel polarization. The potential energy
values have been calculated for an intensity $I=50mW/\mu$m2. The arrow in (c)
indicates the position of zero potential, which coincides with the equilibrium
position 2. The frequency $\omega/c=0.325\mu$m-1, the particle radius $R=90$nm
and the distance $z_{0}=100$nm. The graphene parameters are $\mu_{g}=0.3$eV
and $\gamma_{g}=0.1$meV.

We have normalized the optical force $F$ with respect to the incident density
of power $I=|E_{0}|^{2}/(2Z_{0})$, $Z_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$ is
the vacuum impedance, i.e., we calculated $f=F/I$, where $F$ is given in
pico–Newton ($p$N) and $I$ is given in $mW/\mu$m2.

In all the cases, the optical force has been rigorously calculated. To do
this, the Green tensor components are calculated following the same procedure
as one applied in Ref. [25] (see E).

Figures 2a and 2b show the optical force for normal incidence as a function of
the interparticle distance $x_{0}$, rigorously calculated using Eq. (2.1), for
illumination parallel (electric field parallel to the incidence plane,
$E_{i}=E_{||}$) and perpendicular (electric field perpendicular to the
incidence plane, $E_{i}=E_{\perp}$), respectively, rigorously calculated using
Eq. (2.1). The chemical potential of graphene and frequency are
$\mu_{g}=0.3$meV and $\omega/c=0.325\mu$m-1 (wavelength $\lambda=19.3\mu$m),
respectively. In both cases, we observe that the force periodically varies
with the $x_{0}$ coordinate with a period $\approx 0.8\mu$m. At the incident
field frequency, the real part of the calculated GSP propagation constant is
$\mbox{Re}\,k_{sp}=7.55\mu$m-1, corresponding to a plasmon wavelength
$\lambda_{sp}=0.83\mu$m which agrees well with the numerically calculated
period of the force.

To gain insight about the role of GSPs in the interparticle force, in Figure 2
we plotted the plasmon contributions given by Eqs. (2.2) and (17). We observe
that for $x_{0}>1\mu$m these curves match with those rigorously calculated
with Eq. (2.1), indicating that for interparticle distance values large
enough, surface plasmon excitations dominate the optical binding calculated in
Figure 2. On the contrary, for small values of $x_{0}$ ($x_{0}<1\mu$m), the
surface plasmon contribution departs from the total optical force which is
very approximately equal to the one in free space, i.e., to the optical force
in absence of graphene. In fact, in this range, the force is attractive when
the particles are illuminated with parallel polarization, whereas it is
repulsive when particles are illuminated with perpendicular polarization.
Moreover, in this range the optical force changes rapidly with $x_{0}$.

All these characteristics can be understood by comparing the contributions of
free space and GSP modes in the short distance range between particles.

$F_{x}^{(FS)}\approx\frac{1}{2}\frac{k_{0}^{2}}{\varepsilon_{0}}\mbox{Re}\Bigg{\\{}|p_{j}|^{2}\frac{\partial}{\partial
x_{j}^{\prime}}G_{0,x_{j}x_{j}}(\mathbf{r}^{\prime},\mathbf{r})\Bigg{\\}},$
(19)

where $j=x$ for parallel polarization and $j=y$ for perpendicular
polarization, $G_{0,xx}$ and $G_{0,yy}$ are the components of free Green
tensor (Eq. (B)), $\mathbf{r}=x_{0}\hat{x}+z_{0}\hat{z}$ and
$\mathbf{r}^{\prime}=z_{0}\hat{z}$. Taking the derivative of $G_{0,xx}$ and
$G_{0,yy}$ components of free Green tensor (Eq. (B)), we obtain,
$F_{x}^{(FS)}\approx-|p_{x}|^{2}/x_{0}^{4}$ for parallel polarization and
$F_{x}^{(FS)}\approx|p_{y}|^{2}/x_{0}^{4}$ for perpendicular polarization. The
negative sign indicates that the force is attractive for parallel
polarization, whereas the positive sign corresponds to a repulsive force for
perpendicular polarization. This is consistent with the intuitive idea that in
this range, where the plasmonic contribution can be neglected, the
polarization charges on the surface of particles are asymmetric for parallel
polarization, resulting in an attractive force. On the contrary, for
perpendicular polarization the surface charge distribution on particles is
symmetric, resulting in a repulsive force [5].

Figure 3: Map of the optical force as a function of the interparticle distance
$x_{0}$ and the height $z_{0}$ for $\omega/c=0.325\mu$m-1 and $\mu_{g}=0.3$eV.
All other parameters are the same as in Figure 2.

While the spatial dependence of the free space contribution is $\approx\pm
1/x_{0}^{4}$, the GSP contribution given by Eqs. (2.2) and (17) is
$F_{x}^{(GSP)}\approx 1/x_{0}^{3}$. From the above analysis, we conclude that
free space contributions dominate the optical force interaction at short
distances.

In Figure 2a we have plotted points at stable positions, these are, points on
the $x_{0}$ axis where the optical force pass from positive to negative values
[9]. In this range, the normalized total force reaches a value $\approx
1.4\mbox{x}10^{-3}p$N, i.e., a value of $0.07p$N for intensities of
50$mW/\mu$m2. Taking into account that the period of the force is
$\lambda_{p}=0.83\mu$m, the trapping potential of the optical binding
approaches $4k_{B}$T at room temperature and moderate intensities involved in
optical trapping [9, 10]. This fact can be seen in Figure 2c, where we have
calculated the potential energy of the optical binding for parallel
polarization by integration of the force for an intensity of $50mW/\mu$m2 (we
take the potential energy as null for the equilibrium position 2). To do this,
we used the fact that the particles are smaller than the wavelength, which
results in low values of the imaginary part of the polarizability
$\hat{\alpha}_{j}$ ($j=A,\,B$) and, consequently, the non conservative part of
the force (radiation force) can be neglected [9, 29]. On the other hand, note
that stable first positions are at $x_{0}\approx 1.5\mu$m, a value near 15
times lower than that corresponding to the free space, which falls at
$x_{0}\approx 22\mu$m [not shown in Figure 2], which demonstrates the high
subwavelength binding that is provided by GSPs.

By comparing Figures 2a and 2b, we observe that for $x>1\mu$m, the force
values for perpendicular polarization are one order of magnitude less than
those for parallel polarization. This is because, in this range, where the
force is dominated by the plasmon contribution, the optical force for
perpendicular polarization decays faster than that corresponding to parallel
polarization ($F_{x}^{(y)}\approx Y_{2}(k_{sp}x_{0})/(k_{sp}x_{0})$ and
$F_{x}^{(x)}\approx Y_{1}(k_{sp}x_{0})$). As a consequence, the stable
trapping for moderate intensity, with the parameters considered here, is then
impossible for perpendicular polarization.

In what follows, we only present calculations for parallel polarization.

In Figure 3 we calculated the optical force for normal incidence as a function
of $x_{0}$ and $z_{0}$ for $\mu_{g}=0.3$eV and $\omega/c=0.325\mu$m-1. We see
that the force presents the same periodic behaviour, with respect $x_{0}$
variable, as in Figure 2, passing from positive to negative values with the
period of the plasmon wavelength $\lambda_{sp}=0.83\mu$m. In addition, the
force values quickly decrease with the height $z_{0}$ and practically
disappear for $z_{0}$ values larger than $0.35\mu$m. This can be understood
from Eqs. (2.2) and (17), where we observe an exponential decay of the force
with $z_{0}$ as $\approx\exp(-2k_{sp}z_{0})$.

Figure 4: Map of the optical force as a function of the interparticle distance
and frequency. All other parameters are the same as in Figure 2.

Next, we study the frequency dependence of the optical binding. Figure 4 shows
a map of the optical force as a function of $\omega/c$ frequency and the
interparticle distance $x_{0}$ for $\mu_{g}=0.3$eV. For a fixed frequency
value, we observe a periodic behaviour together with a decay of the
oscillation amplitude with the $x_{0}$ distance. This period decreases with
frequency, which is in agreement with the fact that the GSP propagation
constant, given by Eq. (E2), is an increasing function of frequency. It is
worth noting that, although the force intensity decreases with the
interparticle distance, the dependence with frequency is not like this. For a
fixed $x_{0}$ value, the force intensity increases until it reaches the
maximum value at $\approx 0.5\mu$m-1 and then, the force intensity
monotonously decreases with frequency. This behaviour can be understood from
Eq. (2.2), where we can see that the force depends on the GSP propagation
constant as $F_{x}^{(x)}(x_{0})\approx
k_{sp}^{4}Y_{1}(k_{sp}x_{0})e^{-2k_{sp}z_{0}}$. Then, for a fixed value of
$x_{0}$, the function $F_{x}^{(x)}$ reaches its maximum value for
$k_{sp}=2/z_{0}=20\mu$m-1. From the dispersion relation plotted in Figure 8
for $\mu_{g}=0.3$eV, we find that the corresponding value for
$k_{sp}(\omega)=20\mu$m-1 is $\omega/c=0.52\mu$m-1, a value that agrees well
with that numerically found. Even though for perpendicular polarization, the
force presents a similar behaviour to that for parallel polarization shown in
Figure 4, it is worth noting that, from Eq. (17), the frequency value where
the force for perpendicular polarization reaches the maximum value verify the
condition $k_{sp}(\omega)=3/(2z_{0})=15\mu$m-1, leading to
$\omega/c=0.46\mu$m-1.

Figure 5: Map of the optical force as a function of the interparticle distance
and chemical potential. All parameters are the same as in Figure 2.

In order to study the optical binding dependence with graphene parameters, we
calculate the interparticle force dependence with the chemical potential for
normal incidence. Figure 5 shows the optical force as a function of the
$x_{0}$ distance and the chemical potential. We observe zones for which the
force is positive (red zones) and others where the force is negative (blue
zones). Note that for a fixed chemical potential value, the force is a
periodic function on the interparticle distance $x_{0}$, and that this period
depends on the chemical potential value, being an increasing function of
$\mu_{g}$. This can be understood from the quasistatic expression for the GSP
propagation constant. From Eq. (E2), we see that $k_{sp}$ is a decreasing
function of $\mu_{g}$, thus the period of the function $Y_{1}(k_{sp}x_{0})$ in
Eq. (2.2) and $Y_{2}(k_{sp}x_{0})$ in Eq. (17) increases with $\mu_{g}$.
Unlike the case of surface plasmons on metallic materials, for which the
plasmon kinematic quantities are fixed by an invariable charge density, GSPs
provide a dynamic control on the optical binding period, and consequently, on
the stable interparticle distance by varying the chemical potential of
graphene.

Now, we investigate the optical binding properties under oblique incidence. We
focus on the case in which the plane of incidence coincides with the $x-z$
plane. Figure 6 shows the optical force, rigorously calculated, as a function
of $x_{0}$ for $\theta_{i}=10^{\circ}\,\mbox{and}\,30^{\circ}$. The optical
force on particle B for normal incidence is plotted as a reference.

Figure 6: (a) Optical force and (b) potential energy of interaction between
the particles A and B normalized to $k_{B}$T, as functions of the
interparticle distance $x_{0}$ for $\omega/c=0.325\mu$m-1 The incidence angles
are $\theta_{i}=10^{\circ},\,30^{\circ}$. The curve for normal incidence is
given as a reference. The potential energy values have been calculated for an
intensity $I=50mW/\mu$m2. All parameters are the same as in Figure 2.

From Figure 6a we observe that the force on particle A is not opposite to the
force on particle B, as in the case of normal incidence. This effect is
essentially caused by two mechanisms. The first mechanism is associated with
the incoming photon momentum parallel to the graphene surface: both particles
are pushed in the $+x$ direction, i.e., in the direction of the incident
photon propagation constant parallel to the graphene surface. The second
mechanism is associated with the break of the mirror symmetry imposed by the
oblique incidence, giving rise to different dipole moments induced on
particles A and B and different forces on each of the particles take place. As
a consequence, the equilibrium positions obtained for particle B (positions
where the particle B is fixed) do not coincide with those obtained for
particle A. In addition, as $\theta_{i}$ increases, the equilibrium positions
are lost, as we can see for $\theta_{i}=30^{\circ}$, where the particle A does
not reaches any equilibrium position for $x_{0}>1.5\mu$m. On the other hand,
the potential energy of interaction between particles, i.e., the potential
referred to the relative positions between particles, reaches minimum values,
as can be seen in Figure 6b where we have computed the potential for
$\theta_{i}=10^{\circ}$, $30^{\circ}$ and an intensity $I=50mW/\mu$m2. We
recall that these positions do not correspond to stable equilibrium in which
both particles are fixed, but to positions where the system attains internal
equilibrium. A similar behaviour has been found for a dielectric interface in
the total reflection configuration [3]. By using Eq. (2.2), we have calculated
(Figure 6a) the GSP contribution to the optical force on particle B for
$\theta_{i}=10^{\circ}$ and $30^{\circ}$. As in case of normal incidence, for
an interparticle distance large enough, the plasmonic contribution follows
well the behaviour of the optical force rigorously calculated by using Eq.
(2.1).

Finally, we explore the influence of the shape of particles on the optical
binding. To do this, cubic (with edge length $a$ and cylindrical (with height
$h$ equal to diameter $d$) shapes have been considered. The values of $a$ and
$h$ have been chosen in such a way the volumes of the cubic, cylinder and
sphere coincide. The quasistatic polarizabilities have been calculated using
the formulas presented in [27].

From Figure 7 we see that the optical force curve for the case of two
cylinders placed with the cross section along $x-z$ plane (cylinders along y
axis) matches with that corresponding to spherical particles. This is to be
expected because the value of the polarizability along an axis contained in
the cylinder cross section almost coincides with that of the sphere of the
same volume for $\varepsilon_{p}=3.9$ [27].

Figure 7: Optical force as a function of the interparticle distance $x_{0}$
for normal incidence and various shapes. The radius of the sphere is $R=90$nm
and the volume of cubes and cylinders are equal to that of the sphere. All
other parameters are the same as in Figure 2.

When the cylinders are placed with their symmetry axis along the incident
electric field direction (cylinders along $x$ axis), the force amplitude
increases as a consequence of the polarizability increment along this
direction. On the other hand, the force curve for a cubic shape falls between
the curves corresponding to spheres and cylinders aligned along the $x$ axis.

##  4 Conclusions

In conclusion, we have analyzed the benefits of a graphene sheet as a
nanoparticle binder. Due to super confinement of GSPs, we found equilibrium
interparticle distances of few micrometers for low frequency plane wave
incidence, THz and IR. Moreover, these equilibrium distances can be
dynamically controlled by chemical potential variations, a fact that
highlights the use of graphene for optical binding applications.

In addition, using the dispersion properties of surface plasmons on a
perfectly flat and infinite graphene sheet, we have obtained an analytical
expression for the binding force that quantitatively explains the main results
obtained by applying the rigorous Green’s method.

The possibility to incorporate another graphene sheet forming a graphene
parallel waveguide [31] or by creating a space layer forming an attenuated
total reflection structure [32], allows interesting degree of freedoms to
increase the GSP field and, as a consequence, the depth of the potential well
of the optical force between particles. Although we are planning to report the
results of such studies in future papers, as a first step, here we have
restricted ourselves to performing an analysis of the optical binding
properties on a single graphene sheet. In this way, we believe that our
results are valuable in the framework of optical binding including graphene
plasmonic structures.

## Acknowledgment

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

##  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 [28] , which can be read as
$\sigma_{loc}=\sigma^{intra}+\sigma^{inter}$, 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. For
chemical potential values substantially larger than the thermal energy, i.e.,
$k_{B}T<<\mu_{g}$, the graphene conductivity can be well approached by the
intraband term which takes the Drude form,

$\sigma(\omega)=\frac{ie^{2}\mu_{g}}{\pi\hbar^{2}(\omega+i\gamma_{g})}.$ (A3)

##  Appendix B Green tensor

The Green tensor $\mathbf{\hat{G}}(\mathbf{r},\mathbf{r}^{\prime})$ is defined
by the electric field at point $\mathbf{r}$ generated by an electric dipole
$\mathbf{p}$ located at the source point $\mathbf{r}^{\prime}$ and satisfies
[29],

$\displaystyle\nabla\times\nabla\times\mathbf{\hat{G}}(\mathbf{r},\mathbf{r}^{\prime})-k_{0}^{2}\mathbf{\hat{G}}(\mathbf{r},\mathbf{r}^{\prime})=\mathbf{I}\delta(\mathbf{r}-\mathbf{r}^{\prime})),$
(B1)

where $\mathbf{I}$ is the unit tensor. By applying the superposition method,
the solution of Eq. (B1) can be expressed as a sum of two parts, one of them,
$\mathbf{\hat{G}}_{0}(\mathbf{r},\mathbf{r}^{\prime})$, is associated to the
primary dipole emission of the source and the other,
$\mathbf{\hat{G}}_{s}(\mathbf{r},\mathbf{r}^{\prime})$, takes into account the
field scattered on $z=0$ surface. For $z,\,z^{\prime}>0$ (upper half space),
the expression for these functions have the form [29]:

$\displaystyle\mathbf{\hat{G}}_{0}(\mathbf{r},\mathbf{r}^{\prime})=\frac{e^{ik_{0}d}}{4\pi
d}$
$\displaystyle\times\Bigg{\\{}[1+\frac{ik_{0}d-1}{k_{0}^{2}d^{2}}]\mathbf{I}+\frac{3-i3k_{0}d-k_{0}^{2}d^{2}}{3k_{0}^{2}d^{2}}\hat{r}\hat{r}\Bigg{\\}},$
(B2)

where $d=|\mathbf{r}-\mathbf{r}^{\prime}|$, and

$\displaystyle\mathbf{\hat{G}}_{s}(\mathbf{r},\mathbf{r}^{\prime})=\frac{i}{8\pi^{2}}\int_{0}^{\infty}dk_{||}\,\mathbf{f}(k_{||},\rho)\,e^{i\gamma^{(1)}(z+z^{\prime})},$
(B3)

where $k_{||}$ is the wave vector parallel to the surface,
$\rho=\sqrt{d^{2}-(z-z^{\prime})^{2}}$,
$\gamma^{(j)}=\sqrt{k_{0}^{2}\varepsilon_{j}-k_{||}^{2}}$ ($j=1,\,2$). The
cartesian components of $\mathbf{f}(k_{||},\rho)$ are:

$\displaystyle\mathbf{f}_{xx}(k_{||},\rho)=2\pi$
$\displaystyle\times\Bigg{\\{}\frac{k_{||}}{\gamma^{(1)}}[-\cos(2\theta)\frac{J_{1}(k_{||}\rho)}{k_{||}\rho}+\cos^{2}(\theta)J_{0}(k_{||}\rho)]r_{s}(k_{||})$
$\displaystyle+\frac{k_{||}\gamma^{(1)}}{k_{0}^{2}}[\cos(2\theta)\frac{J_{1}(k_{||}\rho)}{k_{||}\rho}-\cos^{2}(\theta)J_{0}(k_{||}\rho)]r_{p}(k_{||})\Bigg{\\}},$
(B4) $\displaystyle\mathbf{f}_{yy}(k_{||},\rho)=2\pi$
$\displaystyle\times\Bigg{\\{}\frac{k_{||}}{\gamma^{(1)}}[-\cos(2\theta)\frac{J_{1}(k_{||}\rho)}{k_{||}\rho}+\cos^{2}(\theta)J_{0}(k_{||}\rho)]r_{s}(k_{||})]$
$\displaystyle-\frac{k_{||}\gamma^{(1)}}{k_{0}^{2}}[\cos(2\theta)\frac{J_{1}(k_{||}\rho)}{k_{||}\rho}+\sin^{2}(\theta)J_{0}(k_{||}\rho)]r_{p}(k_{||})\Bigg{\\}},$
(B5)
$\displaystyle\mathbf{f}_{zz}(k_{||},\rho)=2\pi\frac{k_{||}^{3}}{k_{0}^{2}\gamma^{(1)}}J_{0}(k_{||}\rho),$
(B6) $\displaystyle\mathbf{f}_{xy}(k_{||},\rho)=\mathbf{f}_{yx}(k_{||},\rho)=$
$\displaystyle\pi\Bigg{\\{}\frac{k_{||}}{\gamma^{(1)}}r_{s}(k_{||})+\frac{k_{||}\gamma^{(1)}}{k_{0}^{2}}r_{p}(k_{||})\Bigg{\\}}J_{2}(k_{||}\rho)\sin(2\theta),$
(B7)
$\displaystyle\mathbf{f}_{xz}(k_{||},\rho)=-\mathbf{f}_{zx}(k_{||},\rho)=-2\pi
i\frac{k_{||}^{2}}{k_{0}^{2}}J_{1}(k_{||}\rho)r_{p}(k_{||})\cos(\theta),$ (B8)
$\displaystyle\mathbf{f}_{yz}(k_{||},\rho)=-\mathbf{f}_{zy}(k_{||},\rho)=-2\pi
i\frac{k_{||}^{2}}{k_{0}^{2}}J_{1}(k_{||}\rho)r_{p}(k_{||})\sin(\theta),$ (B9)

where $\theta$ is the angle between the axis joining the particles and the $x$
axis, and $J_{n}(x)$ is the Bessel function of $n$th order. The complex
amplitude

$\displaystyle
r_{p}=\frac{\frac{\gamma^{(1)}}{\varepsilon_{1}}-\frac{\gamma^{(2)}}{\varepsilon_{2}}+\frac{Z_{0}\sigma}{k_{0}}\frac{\gamma^{(1)}}{\varepsilon_{i}}\frac{\gamma^{(2)}}{\varepsilon_{2}}}{\frac{\gamma^{(1)}}{\varepsilon_{1}}+\frac{\gamma^{(2)}}{\varepsilon_{2}}+\frac{Z_{0}\sigma}{k_{0}}\frac{\gamma^{(1)}}{\varepsilon_{1}}\frac{\gamma^{(2)}}{\varepsilon_{2}}},$
(B10)

is the Fresnel reflection coefficient for $p$ polarization (magnetic field
parallel to the $z=0$ surface), $Z_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$ is the
vacuum impedance, whereas

$\displaystyle
r_{s}=\frac{\gamma^{(1)}-\gamma^{(2)}-Z_{0}k_{0}\sigma}{\gamma^{(1)}+\gamma^{(2)}+Z_{0}k_{0}\sigma},$
(B11)

is the Fresnel reflection for $s$ polarization (electric field parallel to the
$z=0$ surface).

In the case in which the particles are along the $x$ axis, at heights
$z=z^{\prime}=z_{0}$ and separated one from the other a distance $x_{0}$,
i.e., $\mathbf{r}=x_{0}\hat{x}+z_{0}\hat{z}$ and
$\mathbf{r}^{\prime}=z_{0}\hat{z}$, we must take $\theta=0$ in expressions (B)
to (B9). Moreover, Eq. (B) is written as,

$\displaystyle\mathbf{\hat{G}}_{0}(\mathbf{r},\mathbf{r}^{\prime})=\frac{e^{ik_{0}x_{0}}}{4\pi
x_{0}}$
$\displaystyle\times\Bigg{\\{}[1+\frac{ik_{0}x_{0}-1}{k_{0}^{2}x_{0}^{2}}]\mathbf{I}+\frac{3-i3k_{0}x_{0}-k_{0}^{2}x_{0}^{2}}{3k_{0}^{2}x_{0}^{2}}\hat{x}\hat{x}\Bigg{\\}}.$
(B12)

##  Appendix C Incident field and induced dipole moments

To obtain the induced dipole moments due to the incoming and scattered
electric fields, we follows the same steps as in Ref. [9]. In this manner, the
dipole moments are written as

$\mathbf{p}_{A}=\hat{\alpha}_{s}[\mathbf{E}_{0}(\mathbf{x}_{A})+\frac{k_{0}^{2}}{\varepsilon_{0}}\hat{\mathbf{\hat{G}}}(\mathbf{x}_{A},\mathbf{x}_{B})\mathbf{p}_{B}],$
(D1)

and

$\mathbf{p}_{B}=\hat{\alpha}_{s}[\mathbf{E}_{0}(\mathbf{x}_{B})+\frac{k_{0}^{2}}{\varepsilon_{0}}\hat{\mathbf{G}}(\mathbf{x}_{B},\mathbf{x}_{A})\mathbf{p}_{A}],$
(D2)

for particles A and B respectively. The first term in Eq. (D1) (Eq. (D2))
represents the incident and reflected electric field,
$\mathbf{E}_{0}(\mathbf{r})=\mathbf{E}_{i}(\mathbf{r})+\mathbf{E}_{r}(\mathbf{r})$,
whereas the second term represents the field in particle A (B) which is
scattered by particle B (A). Here, $\hat{\alpha}_{s}$ is the particle
polarizability that account the corrections due to the scattering with the
surface,

$\hat{\alpha}_{s}=\frac{\alpha_{0}}{\hat{\mathbf{I}}-\frac{k_{0}^{2}}{\varepsilon_{0}}\alpha_{0}\hat{\mathbf{G}}_{s}(\mathbf{r}_{j},\mathbf{r}_{j})},$
(D3)

where $j=A,\,B$. In Eqs. (D1) and (D2) we have taking into account that
particles A and B are identical. By solving Eqs. (D1) and (D2), we obtain

$\begin{array}[]{ll}\mathbf{p}_{A}=\hat{\alpha}_{s}\frac{\mathbf{E}_{0}(\mathbf{x}_{A})+\frac{k_{0}^{2}}{\varepsilon_{0}}\hat{\mathbf{G}}(\mathbf{x}_{A},\mathbf{x}_{B})\hat{\alpha}_{s}\mathbf{E}_{0}(\mathbf{x}_{B})}{\hat{\mathbf{I}}-\frac{k_{0}^{4}}{\varepsilon_{0}^{2}}\hat{\alpha}_{s}\hat{\mathbf{G}}(\mathbf{x}_{A},\mathbf{x}_{B})\hat{\alpha}_{s}\hat{\mathbf{G}}(\mathbf{x}_{B},\mathbf{x}_{A})}.\end{array}$
(D4)

In case where the incidence is on the $x-z$ plane and for $p$ polarization
(electric field on the incidence plane), the explicit form of the incident
electric field $\mathbf{E}_{i}(\mathbf{r}_{j})$ ($j=A,\,B$) is given by,

$\begin{array}[]{ll}\mathbf{E}_{i}(\mathbf{r}_{A})=E_{i}[\,\cos\theta_{i}\,\cos\phi_{i}\hat{x}+\\\
\,\cos\theta_{i}\,\sin\phi_{i}\hat{y}+\sin\theta_{i}\hat{z}]\,e^{-ik_{0}\cos\theta_{i}\,z_{0}},\\\
\mathbf{E}_{i}(\mathbf{r}_{B})=E_{i}[\,\cos\theta_{i}\,\cos\phi_{i}\hat{x}+\\\
\,\cos\theta_{i}\,\sin\phi_{i}\hat{y}+\sin\theta_{i}\hat{z}]\,e^{ik_{0}(\sin\theta_{i}\,x_{0}-\cos\theta_{i}\,z_{0})},\end{array}$
(D5)

where $E_{i}$ is the amplitude of the incident plane electric field. The
explicit form of the reflected field $\mathbf{E}_{r}(\mathbf{r}_{j})$
($j=A,\,B$) can be obtained from Eq. (D5) using the reflection coefficients
$r_{p}(k_{||})$ defined in Eqs. (B10).

In many cases, for normal incidence for example, the electric fields at both
particle positions are equals, i.e.,
$\mathbf{E}_{0}(\mathbf{r}_{A})=\mathbf{E}_{0}(\mathbf{r}_{B})$. In such
cases, Eq. (D4) can be written as

$\begin{array}[]{ll}\mathbf{p}_{A}=\hat{\alpha}_{A}\mathbf{E}_{0},\end{array}$
(D6)

where we have defined the effective polarizability for particle A as

$\begin{array}[]{ll}\hat{\alpha}_{A}=\hat{\alpha}_{s}\frac{\hat{\mathbf{I}}+\frac{k_{0}^{2}}{\varepsilon_{0}}\hat{\mathbf{G}}(\mathbf{x}_{A},\mathbf{x}_{B})\hat{\alpha}_{s}}{\hat{\mathbf{I}}-\frac{k_{0}^{4}}{\varepsilon_{0}^{2}}\hat{\alpha}_{s}\hat{\mathbf{G}}(\mathbf{x}_{A},\mathbf{x}_{B})\hat{\alpha}_{s}\hat{\mathbf{G}}(\mathbf{x}_{B},\mathbf{x}_{A})}.\end{array}$
(D7)

Expressions for the dipole moment and polatizability on particle B are
obtained from Eq. (D4) and (D7), respectively, by interchanging A by B.

##  Appendix D Graphene plasmon dispersion Characteristics

To obtain the dispersion characteristics of GSPs, we equate to zero the
denominator in Eq. (B10),

$\frac{\gamma^{(1)}}{\varepsilon_{1}}+\frac{\gamma^{(2)}}{\varepsilon_{2}}+\frac{Z_{0}\sigma}{k_{0}}\frac{\gamma^{(1)}}{\varepsilon_{1}}\frac{\gamma^{(2)}}{\varepsilon_{2}}=0,$
(E1)

and solve for $k_{||}$. In the non retarded regime, $k_{0}<<k_{sp}$, the
wavenumber $k_{sp}$ can be approximated by [30]

$k_{sp}=ik_{0}\frac{\varepsilon_{1}\varepsilon_{2}}{Z_{0}\sigma(\varepsilon_{1}+\varepsilon_{2})}=k_{0}\frac{\varepsilon_{1}\varepsilon_{2}\pi\hbar^{2}}{Z_{0}e^{2}\mu_{g}(\varepsilon_{1}+\varepsilon_{2})}(\omega+i\gamma_{g}),$
(E2)

where in the last equality we have used Eq. (A3). Figure 8 shows the
dispersion curves obtained by solving Eq. (E1) and the non retarded expression
Eq. (E2) for various values of chemical potential, $\mu_{g}=0.3,\,0.7$eV. We
observe that results from Eq. (E2) agree well with the values obtained by
solving the full retarded dispersion equation (E1).

Figure 8: GSP dispersion curves for $\mu_{g}=0.3,\,0.7$eV. Solid (dashed) line
is obtained by using the full retarded (FR) Eq. (E1) (the non retarded (NR)
Eq. (E2)).

##  Appendix E Numerical method

The integration path in Eq. (6) is set along the real and positive $k_{||}$
axis, so that the integral will be strongly affected by singularities that are
close to that axis. Pole singularities, zeroes of the denominator in
$r_{p}(k_{||})$ coefficient, occur at complex location and it represents the
propagation constant of the GSP mode. We transform the original oscillatory
integrand function into one to avoid the complex singularities which lie near
the real $k_{||}$ axis and then we apply a numerical quadrature to calculate
the field integrals. To do this, we surround the pole singularity by deforming
the integration path into the complex plane as shown in Figure 9. The path I
is an elliptical path starting at $k_{||}=0$ with the major semi–axis
$k_{||}=a$ and the minor semi–axis $k_{||}=b$. In the region between path I
and the real axis the integrand function is analytical, thus the Cauchy’s
integral theorem implies that an integration on path I will be equal to the
integral on the real axis from 0 to $2a$. The $a$ value should be chosen large
enough to surround the pole singularities, thus $2a>k_{sp}$ must be fulfilled.
In a first step we divide the integration interval into several subintervals
and then apply the numerical quadrature in each subinterval. We have
implemented a 32 point Gauss Legendre quadrature in each of such sub–interval.

Figure 9:  Singularities and path of integration in the complex plane
$k_{||}=$Re $k_{||}+i$Im${k_{||}}$ for the electromagnetic fields. The
original path, along the positive semi–axis, is deformed along an elliptical
path (I) surrounding the singularities, together with the paths (II) and (III)
parallel to the imaginary $k_{||}$ axis using Hankel functions.

Taking into account that,

$J_{n}(z)=\frac{1}{2}[H_{n}^{(1)}(z)+H_{n}^{(2)}(z)],$ (E3)

and the fact that the Hankel funtions of the first kind $H_{n}^{(1)}(z)$ and
the second kind $H_{n}^{(2)}(z)$ decrease faster as long as $|$Im $z|$
increases in the sector Im $z>0$ and Im $z<0$, respectively, the remaining
integration is carried out by deflecting the integration path from the real
axis to a path parallel to the imaginary $k_{||}$ axis as shown in Figure 9,
with Im $k_{||}>0$ for $H_{n}^{(1)}(k_{||}\rho)$ (path II) and with Im
$k_{||}<0$ for $H_{n}^{(2)}(k_{||}\rho)$ (path III). In the region between
path II and the real axis the integrand has no pole singularities, thus
Cauchy’s integral theorem implies that the integral on a closed path in this
region will be zero.Therefore, the integral over path II in the direction
shown in Figure 9 is equal to that from $2a$ to$+\infty$ over the real axis.
In a similar way, one can demonstrate that the integral over path III in the
direction shown in Figure 9 is equal to that from $2a$ to$+\infty$ over the
real axis. In our implementation, we have used a 32 point Gauss Legendre
quadrature to calculate the field integrals on paths I, II and III.

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# A characterization of centrally symmetric convex bodies in terms of visual
cones

E. Morales-Amaya1, J. Jerónimo-Castro2,
and D. J. Verdusco Hernández3

2Facultad de Ingeniería
Universidad Autónoma de Querétaro, México
1-3Facultad de Matemáticas-Acapulco,
Universidad Autónoma de Guerrero, México
1emoralesamaya<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

In this work we prove the following result: Let $K$ be a strictly convex body
in the Euclidean space $\mathbb{R}^{n},n\geq 3$, and let $L$ be a
hypersurface, which is the image of an embedding of the sphere
$\mathbb{S}^{n-1}$, such that $K$ is contained in the interior of $L$. Suppose
that, for every $x\in L$, there exists $y\in L$ such that the support double-
cones of $K$ with apexes at $x$ and $y$, differ by a translation. Then $K$ and
$L$ are centrally symmetric and concentric.

##  1 Introduction

Let $K\subset\mathbb{R}^{n}$ be a convex body, i.e., a compact and convex set
with non-empty interior, $n\geq 3$, and let $x\in\mathbb{R}^{n}\setminus K$.
We call the set

$\bigcup_{y\in K}\operatorname*{aff}\\{x,y\\}$

the solid double-cone generated by $K$ and $x$, where
$\operatorname*{aff}\\{x,y\\}$ denotes the affine hull of $x$ and $y$. The
boundary of the solid double-cone generated by $K$ and $x$ will be called the
support double-cone of $K$ with apex at $x$ and it will be denoted by $C_{x}$.
In what follows, we shall use the names cone or support cone, by simplicity,
to refer to the double cone and double support cone, respectively.

A classical problem in convexity is to determine properties of a convex body
$K\subset\mathbb{R}^{n}$ from the information of its orthogonal projections.
For instance, in dimension 3 one can prove that if all the orthogonal
projections of a body $K$ are circles, then $K$ is a Euclidean ball. One can
see this problem from the following perspective: consider the family of
cylinders where $K$ is inscribed and impose a condition in a particular
section of each of them, which is obtained with a hyperplane perpendicular to
the lines which generates the cylinder. In our example, this means that we
have a convex body $K\subset\mathbb{R}^{3}$ such that for every cylinder
$\Omega$, where $K$ is inscribed, the section $H\cap\Omega$ is a circle, where
$H$ is a plane orthogonal to the lines which determines $\Omega$.

We formulate the following general problem.

Problem 1. _For a given subgroup $G$ of the general linear group
$\operatorname*{GL}(\mathbb{R},n)$, to determine the convex bodies
$K\subset\mathbb{R}^{n}$, $n\geq 3$, such that for every couple of different
cylinders $\Lambda,\Gamma$, circumscribed to $K$, there exists an element
$\Phi\in G$ such that $\Phi(\Lambda)=\Gamma$_.

Kuzminyh [3] proved, for $n=3$, that the assumption $G=O(\mathbb{R},3)$
implies that $K$ is a ball, where $O(\mathbb{R},3)$ is the real orthogonal
group. On the other hand, if $K\subset\mathbb{R}^{n},n\geq 3$, is centrally
symmetric, in virtue of the Aleksandrov Uniqueness Theorem, it follows that
$K$ is a ball since all the projections have the same volume. Recently, L.
Montejano [5] has considered the case where $G$ is the affine subgroup
$A(\mathbb{R},n)$ and he has obtained that $K$ is an ellipsoid.

In virtue that the cylinders are cones with apexes at the infinity, the
original problem mentioned at the beginning of this introduction can be
formulated in the following manner: To determine properties of convex bodies
imposing conditions on the sections of cones where $K$ is inscribed and whose
apexes are contained in a hyperplane. Naturally, we can replace in the
aforesaid problem the condition that the set of apexes is situated in a
hyperplane by the condition that they are contained in a hypersurface $S$. In
particular, we can assume that $S$ is the boundary of a convex body
$M\subset\mathbb{R}^{n}$ such that $K\subset\operatorname*{int}M$. An
interesting example of this type is the well known Matsuura’s Theorem [4]:

Matsuura’s theorem. Let $K\subset\mathbb{R}^{3}$ be a convex body and let $S$
be a closed convex surface which contains $K$ in its interior. If the support
cone of $K$ from every point in $S$ is a right circular cone then $K$ is a
Euclidean ball.

We are interested in the following problem for cones.

Problem 2. _Given a subgroup $G$ of the general linear group
$\operatorname*{GL}(\mathbb{R},n)$ and a hypersurface $S$ which is the image
of an embedding of $\mathbb{S}^{n-1}$, to determine the convex bodies
$K\subset\mathbb{R}^{n}$, $n\geq 3$, such that for every couple of different
cones $\Lambda,\Gamma$, circumscribed to $K$ and with apexes in $S$, there
exists an element $\Phi\in G$ such that $\Phi(\Lambda)=\Gamma$._

A particularly interesting case of Problem 2 is when $G$ is equal to
$O(\mathbb{R},n)$, i.e., we know that all the cones which circumscribes $K$
and with apexes in $S$ are congruent. Very recently, S. Myroshnychenko [6] has
proved the following related result: let $P$ and $Q$ be polytopes contained in
the interior of the ball $B_{r}(n)$, $n\geq 3$, and assume that from every
point in the sphere $r\mathbb{S}^{n-1}$ the support cones of $P$ and $Q$ are
congruent, then $P=Q$.

We denote by $T(\mathbb{R},n)$ the family of translations in $\mathbb{R}^{n}$.
The main result of this work was inspired by Problem 2, however, we involve
$T(\mathbb{R},n)$ which is not a subgroup of $GL(\mathbb{R},n)$, nevertheless
it is an isometry of $\mathbb{R}^{n}$. Our main theorem claims that if
$K\subset\mathbb{R}^{n}$, $n\geq 3$, is a strictly convex body and $L$ is a
hypersurface, which is the image of an embedding of the sphere
$\mathbb{S}^{n-1}$, $K\subset\operatorname*{int}L$, and for every $x\in L$,
there exists $y\in L$ and $\Phi\in T(\mathbb{R},n)$ such that
$C_{y}=\Phi(C_{y})$, then $K$ and $L$ are centrally symmetric and concentric.

More precisely, we are going to prove the following theorem.

###### Theorem 1

Let $K\subset\mathbb{R}^{n},$ $n\geq 3,$ be a strictly convex body and let $L$
be hypersurface which is an embedding of $\mathbb{S}^{n-1}$ such that
$K\subset\operatorname*{int}L$. Suppose that for every $x\in L$ there exist
two points $y\in L$ and $p\in\mathbb{R}^{n}$ such that

$\displaystyle C_{y}=p+C_{x}.$ (1)

Then $K$ and $L$ are centrally symmetric and concentric.

##  2 Proofs and auxiliary results

Let $\mathbb{R}^{n}$ be the Euclidean space of dimension $n$ endowed with the
usual interior product
$\langle\cdot,\cdot\rangle:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$.
We take a orthogonal coordinate system $(x_{1},...,x_{n})$ for
$\mathbb{R}^{n}$. Let $B_{r}(n)=\\{x\in\mathbb{R}^{n}:||x||\leq r\\}$ be the
$n$-ball of radius $r$ centered at the origin, and let
$r\mathbb{S}^{n-1}=\\{x\in\mathbb{R}^{n}:||x||=r\\}$ be its boundary. For
$u\in\mathbb{S}^{n-1}$ and a non-negative $s\in\mathbb{R}$, we denote by
$\Pi(u,s)$ the hyperplane $\\{x\in\mathbb{R}^{n}|\langle u,x\rangle=s\\}$
whose unit normal vector is $u$ and by $\Pi^{*}(u,s)$ the open half-space
$\\{x\in\mathbb{R}^{n}|\langle u,x\rangle<s\\}$. In particular, $\Pi(x,0)$ is
denoted by $x^{\perp}$. For the points $x,y\in\mathbb{R}^{n}$ we will denote
by $\operatorname*{aff}\\{x,y\\}$ the affine hull of $x$ and $y$ and by
$[x,y]$ the line segment with endpoints $x$ and $y$. A convex hypersurface is
the boundary of a convex body in $\mathbb{R}^{n}$. As usual
$\operatorname*{int}K$, $\operatorname*{bd}K$ will denote the interior and the
boundary of the convex body $K$. An embedding of $\mathbb{S}^{n-1}$ in
$\mathbb{R}^{n}$ is a map $\alpha:\mathbb{S}^{n-1}\rightarrow\mathbb{R}^{n}$
such that $\alpha$ is homeomorphic onto its image.

Let $M\subset\mathbb{R}^{n}$ be a set and let $q\in\mathbb{R}^{n}$ be a point.
The set $M^{\prime}=2q-M$ is said to be centrally symmetric to $M$ with
respect to the center $q$. Let $S:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be
the map such that $x\mapsto-x$. The following is en easy consequence of the
definition of centrally symmetric sets.

###### Lemma 1

Let $\Lambda$ be the image of an embedding
$\beta:\mathbb{S}^{n-2}\rightarrow\mathbb{R}^{n}$. Then $S(\Lambda)$ is a
translate of $\Lambda$ if and only if $\Lambda$ is centrally symmetric.

###### Lemma 2

For every $x\in L$, the set $\Lambda=C_{x}\cap C_{y}$ is an embedding of
$\mathbb{S}^{n-2}$, where $y$ satisfies (1). On the other hand, the set
$\Lambda$ is centrally symmetric and, if we choose the center of $\Lambda$ as
the origin of coordinates, then

$\displaystyle-C_{x}=C_{y}.$ (2)

_Proof._ Is easy to see that for a strictly convex body
$K\subset\mathbb{R}^{n}$ and a point $x\in\mathbb{R}^{n}\setminus K$ the set
$C_{x}\cap\operatorname*{bd}K$, the graze of $K$ with respect to $x$, is the
image of an embedding of $\mathbb{S}^{n-2}$. On the other side, we can define
a map $\phi:C_{x}\cap\operatorname*{bd}K\rightarrow\Lambda$ such that $\phi$
is injective, one-to-one and bi-continuous, i.e., we can find an homeomorphism
between $C_{x}\cap\operatorname*{bd}K$ and $\Lambda$. Thus $\Lambda$ is the
image of an embedding of $\mathbb{S}^{n-2}$.

Now, note that every double-cone is a centrally symmetric set with respect to
its apex. Since there is $p\in\mathbb{R}^{n}$ such that $C_{y}=p+C_{x}$, and
$C_{y}$ is a symmetric image of $C_{x}$ with respect to the midpoint of
$[x,y]$, we have that $\Lambda=C_{x}\cap C_{y}$ is a centrally symmetric set
with center at $\frac{x+y}{2}.$ If we choose the origin of coordinates at
$\frac{x+y}{2}$ then $C_{y}=-C_{x}$. $\Box$

###### Lemma 3

Let $x,y\in L$ such that (1) holds. Let $\Pi_{1},$ $\Pi_{2}$, $\Gamma_{1},$
$\Gamma_{2}$, be supporting hyperplanes of $K$ such that
$x\in\Pi_{1}\cap\Pi_{2}$, $\Gamma_{i}$ is parallel to $\Pi_{i},$ $i=1,2,$ and
$\Pi_{1}\cap\Pi_{2}$ is not a supporting $(n-2)$-dimensional plane of $L$.
Denote by $k_{i}$ the contact point of $\Pi_{i}$ with
$\operatorname*{bd}K,i=1,2$ and by $l_{i}$ the contact point of $\Gamma_{i}$
with $\operatorname*{bd}K,i=1,2$. Then the segments $[k_{1},k_{2}]$ and
$[l_{1},l_{2}]$ are parallel.

Figure 1:

_Proof._ Let $\bar{x}$ be any point in $\Pi_{1}\cap\Pi_{2}\cap L$, with
$\bar{x}\neq x$. In virtue of the hypothesis there exists $\bar{y}\in L$ and
$q\in\mathbb{R}^{n}$ such that

$\displaystyle C_{\bar{y}}=q+C_{\bar{x}}.$ (3)

By Lemma 2 and relation (2), we conclude that $y\in\Gamma_{1}\cap\Gamma_{2}$
and by (1) it follows that

$\operatorname*{aff}\\{y,l_{1}\\}=p+\operatorname*{aff}\\{x,k_{1}\\}\textrm{
}\textrm{ and }\textrm{
}\operatorname*{aff}\\{y,l_{2}\\}=p+\operatorname*{aff}\\{x,k_{2}\\},$

i.e.,

$\displaystyle\operatorname*{aff}\\{y,l_{1},l_{2}\\}=p+\operatorname*{aff}\\{x,k_{1},k_{2}\\}$
(4)

(See Fig. 1). Analogously, since $\bar{y}\in\Gamma_{1}\cap\Gamma_{2}$ by Lemma
2 and by (3) we have that

$\displaystyle\operatorname*{aff}\\{\bar{y},l_{1},l_{2}\\}=q+\operatorname*{aff}\\{\bar{x},k_{1},k_{2}\\}.$
(5)

By (4) and (5) it follows

$\operatorname*{aff}\\{l_{1},l_{2}\\}=\operatorname*{aff}\\{y,l_{1},l_{2}\\}\cap\operatorname*{aff}\\{\bar{y},l_{1},l_{2}\\}=$
$=[q+\operatorname*{aff}\\{x,k_{1},k_{2}\\}]\cap[q+\operatorname*{aff}\\{\bar{x},k_{1},k_{2}\\}]=$
$=q+[\operatorname*{aff}\\{x,k_{1},k_{2}\\}\cap\operatorname*{aff}\\{\bar{x},k_{1},k_{2}\\}]$
$=q+\operatorname*{aff}\\{k_{1},k_{2}\\}.$

$\Box$

###### Lemma 4

Let $C\subset\mathbb{R}^{n}$ be a convex cone with apex at the origin $O$ and
let $\Delta_{1},\Delta_{2}:\mathbb{S}^{n-2}\rightarrow\mathbb{R}^{n}$ be two
embeddings of $\mathbb{S}^{n-2}$ in $\operatorname*{bd}C$. Suppose that for
every $u\in\mathbb{S}^{n-1}$ such that
$\operatorname*{lin}\\{u\\}\subset\operatorname*{bd}C$, the sets
$\operatorname*{lin}\\{u\\}\cap\Delta_{1}$ and
$\operatorname*{lin}\\{u\\}\cap\Delta_{2}$ consist of only one point; we
denote by $\alpha(u)$ and by $\beta(u)$, respectively, such intersections.
Suppose that for every two such $u,v\in\mathbb{S}^{n-1},u\not=v,$ the line
segment $[\alpha(u),\alpha(v)]$ is parallel to the segment
$[\beta(u),\beta(v)]$. Then $\Delta_{1},\Delta_{2}$ satisfies the relation

$\displaystyle\Delta_{2}=H(\Delta_{1}),$ (6)

where $H:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is an homothety with center
of homothety at $O$.

_Proof._ Let $H:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be an homothety with
center at $O$ such that $\Delta_{2}\cap H(\Delta_{1})\not=\emptyset$. Let
$x\in\Delta_{2}\cap H(\Delta_{1})$ and let $y\in\Delta_{2}$ such that
$y\not=x$. In virtue of the first part of hypothesis
$\operatorname*{lin}\\{y\\}\not=\operatorname*{lin}\\{x\\}$. We denote by $z$
the point $H(\Delta_{1})\cap\operatorname*{lin}\\{y\\}$. It follows that
$z\not=x$, otherwise $\operatorname*{lin}\\{y\\}=\operatorname*{lin}\\{x\\}$.
We denote by $u,v$ the unit vectors $\frac{x}{\|x\|},\frac{y}{\|y\|}$,
respectively. Then

$x=H(\alpha(u)),z=H(\alpha(v)),x=\beta(u),y=\beta(v).$

In virtue of the hypothesis, the line segment $[\alpha(u),\alpha(v)]$ is
parallel to the segment $[\beta(u),\beta(v)]$. Hence the line segment
$[H(\alpha(u)),H(\alpha(v))]$ is parallel to the segment $[\beta(u),\beta(v)]$
and then $y=z$. Therefore, the relation (6) holds. $\Box$

###### Lemma 5

Let $x$ and $y$ be points in $L$ such that (1) holds. We denote by
$\Omega_{x}$ and $\Omega_{y}$ the intersections $C_{x}\cap\operatorname*{bd}K$
and $C_{y}\cap\operatorname*{bd}K$, respectively. Then $\Omega_{x}$ and
$\Omega_{y}$ are inversely homothetic.

_Proof._ In virtue of the hypothesis $\Delta_{2}=p+\Omega_{x}\subset C_{y}$.
Denote by $\Delta_{1}$ the reflection of $\Omega_{y}$ with respect to $y$.
Consider any two points $a,b\in\Omega_{x}$ and let $\Pi_{a}$ and $\Pi_{b}$ be
the support hyperplanes of $K$ through $a$ and $b$, respectively, which by
hypothesis pass through the point $x$. In other words, $\Pi_{a}$ and $\Pi_{b}$
are support hyperplanes of the cone $C_{x}$. Suppose the hyperplanes
$\Pi_{a^{\prime}}$ and $\Pi_{b^{\prime}}$, parallel to $\Pi_{a}$ and $\Pi_{b}$
through the point $y$, touch $K$ at the points $a^{\prime}$ and $b^{\prime}$,
respectively. Since the cone $C_{y}$ is a translate of the cone $C_{x}$, we
have that $\Pi_{a^{\prime}}$ and $\Pi_{b^{\prime}}$ are support hyperplanes of
the cone $C_{y}$. Without loss of generality, we suppose the origin is at the
point $y$ and consider the unitary vectors $u$ and $v$ in direction of the
lines $a^{\prime}y$ and $b^{\prime}y$, respectively. Consider the points
$\alpha(u),$ $\alpha(v)$, $\beta(u)$, and $\beta(v)$, as defined in Lemma 4.
By Lemma 3 we have that $[a,b]$ and $[a^{\prime},b^{\prime}]$ are parallel and
so $[\alpha(u),\alpha(v)]$ is parallel to $[\beta(u),\beta(v)]$. We have the
conditions of Lemma 4, hence $\Delta_{1}$ is directly homothetic to
$\Delta_{2}$, therefore, $\Omega_{x}$ is inversely homothetic to $\Omega_{y}$.
$\Box$

Figure 2:

Remark 1. From this lemma we can deduce that every pair of parallel support
hyperplanes of $K$ cut in $L$ sections which are inversely homothetic, and
moreover, the center of homothety is in the segment that joins the points of
contact between $K$ and these hyperplanes. With this condition and condition
(1), we could continue with the proof that $K$ and $L$ are centrally symmetric
and concentric, however, we will continue with the proof in a little different
direction. Nevertheless, this suggest that the following conjecture have
chance to be true.

Conjecture 1. Let $K$ and $L$ be convex bodies in $\mathbb{R}^{n}$, $n\geq 3$,
with $K\subset\text{int}L$. Suppose every pair of parallel support hyperplanes
of $K$ cut in $L$ sections which are inversely homothetic with center of
homothety in the segment that joins the points of contact between $K$ and
these hyperplanes. Then $K$ and $L$ are centrally symmetric and concentric.

###### Lemma 6

Let $x,y\in L$ be two points such that (1) holds. Consider the center of
$\Lambda=C_{x}\cap C_{y}$ is at the origin. If $\Omega_{x}$ and $\Omega_{y}$
have ratio of homothety equal to $-1$, then

$\displaystyle-\Omega_{x}=\Omega_{y}.$ (7)

_Proof._ By (2) we have that $-\Omega_{x}\subset-C_{x}=C_{y}$. By the
hypothesis $-\Omega_{x}$ and $\Omega_{y}$ are either translated or equal.
However, Since $-\Omega_{x},\Omega_{y}\subset C_{y}$ they can not be
translated. Thus $-\Omega_{x}=\Omega_{y}$. $\Box$

###### Lemma 7

Let $x,y,\bar{x},\bar{y}\in L$ be points such that (1) holds for each pair
$x,y$ and $\bar{x},\bar{y}$. Suppose that $\Omega_{x}\cap\Omega_{\bar{x}}$ is
non-empty, it has more than one point and the ratio of homothety between
$\Omega_{x}$ and $\Omega_{y}$ is -1. Then the ratio of homothety between
$\Omega_{\bar{x}}$ and $\Omega_{\bar{y}}$ is -1 and the center $c$ of
$\Lambda$ coincides with the center $\bar{c}$ of $\bar{\Lambda}$.

_Proof._ Following the notation of Lemma 3, let $\Pi_{1},$ $\Pi_{2}$,
$\Gamma_{1},$ and $\Gamma_{2}$ be supporting hyperplanes of $K$ such that
$x,\bar{x}\in\Pi_{1}\cap\Pi_{2}$ and $\Gamma_{i}$ is parallel to $\Pi_{i},$
for $i=1,2$. We denote by $k_{i}$ the contact point of $\Pi_{i}$ with
$\operatorname*{bd}K,$ for $i=1,2$ and by $l_{i}$ the contact point of
$\Gamma_{i}$ with $\operatorname*{bd}K,$ for $i=1,2$. By Lemma 3 we have that
the segments $[k_{1},k_{2}]$ and $[l_{1},l_{2}]$ are parallel. On the other
side, since $k_{1},k_{2}\in\Omega_{x}$ and $l_{1},l_{2}\in\Omega_{y}$ and the
ratio of homothety between $\Omega_{x}$ and $\Omega_{y}$ is -1, which is
indeed the ratio of homothety between the segments $[k_{1},k_{2}]$ and
$[l_{1},l_{2}]$ such that $k_{i}$ corresponds to $l_{i}$ (for $i=1,2$), we
have that $\|k_{1}-k_{2}\|=\|l_{1}-l_{2}\|$. The center of the inverse
homothety between $\Omega_{x}$ and $\Omega_{y}$ is the center of the
parallelogram $k_{2}k_{1}l_{1}l_{2},$ which is precisely $c=y-p/2.$ We also
have that the center of symmetry of $\Lambda$ is $c$. Now, since
$k_{1},k_{2}\in\Omega_{\bar{x}}$ and $l_{1},l_{2}\in\Omega_{\bar{y}}$, it
follows that the ratio of homothety between $\Omega_{\bar{x}}$ and
$\Omega_{\bar{y}}$ is $-1$ with center at $c$ as well. $\Box$

Let $N\subset L$ be the set of points $x\in L$ such that, if the point $y\in
L$ has the property that for $x,y$ (1) holds, then the curves $\Omega_{x}$ and
$\Omega_{y}$ are inversely homothetic with factor of homothety equal to $-1$.
We will prove that indeed $N=L$.

###### Lemma 8

We have that $N=L$.

_Proof._ Consider a pair of points $u,v\in L$ such that (1) holds for $u,v$
and $u,v\in N$. The existence of this pair of points is guaranteed by an
standard argument of continuity, however, we give some details of this
argument for the interested reader in the Appendix.

We denote by $D_{u}$ the set of points $z\in L$ such that $[u,z]\cap
K\neq\emptyset$ and denote by $L_{u}$ the set $L\setminus D_{u}$. Let $a$ be
any point in $L_{u}$, with $a\neq u$. Since
$\Omega_{a}\cap\Omega_{u}\neq\emptyset$, by Lemma 7 we have that $a\in N$.
Associated with $a$ we have the open region $L_{a}$, and as before, every
point $z\in L_{a}$ is also in $N$. Continuing with this procedure we construct
an open cover of $L$ such that every open set in this cover belong to $N$.
Since $L$ is a compact set, we may consider indeed a finite subcover of $L$,
hence all the points in $L$ belong to $N$. $\Box$

_Proof of Theorem 1_. First, by Lemma 7 we have that all the curves $\Lambda$
are concentric at $c$. From here we get that the support function
$h_{k}:\mathbb{S}^{n-1}\rightarrow\mathbb{R}$ of $K$ is symmetric, i.e.,
$h_{k}(u)=h_{k}(-u)$, that is, $K$ is centrally symmetric with center at $c$.

Finally, we prove that $L$ is centrally symmetric and concentric with $K$. We
choose a system of coordinates such that the origin $O$ is the centre of $K$.
We observe that if a convex body $M\subset\mathbb{R}^{n}$ is centrally
symmetric with centre at $O$ and $x\in\mathbb{R}^{n}\setminus M$, there exist
$y\in\mathbb{R}^{n}\setminus K$ and $p\in\mathbb{R}^{n}$, both unique, such
that the relation

$C_{y}=p+C_{x}$

holds, it is clear that $y=-x$. If $L$ were not be centrally symmetric and
concentric with $K$, then would exist $x_{0}\in L$ such that $-x_{0}\not\in
L$. Thus would not exist $y_{0}\in L$ and $p\in\mathbb{R}^{n}$ such that
$C_{y_{0}}=p+C_{x_{0}}$ which would contradict the hypothesis of Theorem 1.
Therefore, $L$ is centrally symmetric with center at $O$. $\Box$

Remark 2. The main theorem proved in this article is not true in the plane,
even if all the support cones are congruent. To see this is sufficient to
consider a convex body $K\subset\mathbb{R}^{2}$ and its _isoptic curve_
$K_{\alpha}$, i.e., the set of points $z\in\mathbb{R}^{2}$ such that $K$ is
seen under the constant angle $\alpha$ from $z$ (see for instance [1], [2]).

## References

  * [1] W. Cieślak, A. Miernowski, and W. Mozgawa. _Isoptics of a closed strictly convex curve,_ Lecture Notes in Mathematics 1481, $28-35$ (1991).
  * [2] J. W. Green. _Sets subtending a constant angle on a circle,_ Duke Math. J. 17, $263-267$ (1950).
  * [3] A. V. Kuzminyh. _The isoprojection property of the sphere_ , Soviet Math. Dokl. 14, No. 3, (1973).
  * [4] S. Matsuura. _A problem in solid geometry._ J. Math.. Osaka City Univ. 12, $89-95$, (1961).
  * [5] L. Montejano. _Convex bodies with affinely equivalent projections and affine bodies of revolution_ , arXiv:2005.02290v1.
  * [6] S. Myroshnychenko. _On recognizing shapes of polytopes from their shadows_ , Discrete Comput. Geom. 62, $856-864$, (2019).
  * [7] F. A. Valentine. _Convex sets._ McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, (1964).

##  3 Appendix

Here we prove that there exists two points $u,v\in L$ such that (1) holds for
$u,v$ and $u,v\in N$.

Let $\alpha:[0,1]\rightarrow L$ be a continuous curve such that $\alpha(0)=x$
and $\alpha(1)=y$. In virtue of the hypothesis there exists a continuous curve
$\beta:[0,1]\rightarrow L$ such that $\beta(0)=y$ and $\beta(1)=x$ and, for
every $t\in[0,1]$, the points $\alpha(t)$ and $\beta(t)$ are such the relation
(1) holds for them, i.e.,

$\displaystyle C_{\beta(t)}=p(t)+C_{\alpha(t)},$ (8)

where $p(t)\in\mathbb{R}^{3}$. We define $d(t)$ as the number
$\operatorname*{diam}\Omega_{\beta(t)}-\operatorname*{diam}\Omega_{\alpha(t)}$.
By Lemma 5,

$\displaystyle\operatorname*{diam}\Omega_{\alpha(t)}=r(t)\operatorname*{diam}\Omega_{\beta(t)}.$
(9)

Thus

$d(t)=(1-r(t))\operatorname*{diam}\Omega_{\beta(t)}.$

Since the surfaces $L$ and $\operatorname*{bd}K$ are continuous and $L\cap
K=\emptyset$, it follows that the functions
$r,d:[0,1]\rightarrow\mathbb{R}^{+}$ are continuous. By our assumption
$\operatorname*{diam}\Omega_{\alpha(0)}\not=\operatorname*{diam}\Omega_{\beta(0)}$,
say

$\displaystyle\operatorname*{diam}\Omega_{\alpha(0)}<\operatorname*{diam}\Omega_{\beta(0)},$
(10)

replacing (9), for $t=0$, in (10) we get $r(0)<1$, that is, $d(0)>0$. On the
other hand, in virtue that $\alpha(0)=x=\beta(1)$ and $\alpha(1)=y=\beta(0)$,
from (10) it follows

$\operatorname*{diam}\Omega_{\beta(1)}<\operatorname*{diam}\Omega_{\alpha(1)},$

i.e., $1<r(1)$ and $d(1)<0$. Consequently, by the mean value theorem, there
exists $t^{*}\in[0,1]$ such that $d(t^{*})=0$, that is, $r(t^{*})=1$. Hence

$\displaystyle\operatorname*{diam}\Omega_{\alpha(t^{*})}=\operatorname*{diam}\Omega_{\beta(t^{*})}.$
(11)

(See Fig. 3). Thus $u=\alpha(t^{*})$ and $v=\beta(t^{*})$ are the point what
we are looking for.

Figure 3: