# Surface Ferron Excitations in Ferroelectrics and Their Directional Routing

Xi-Han Zhou  School of Physics, Huazhong University of Science and Technology,
Wuhan 430074, China  Chengyuan Cai  School of Physics, Huazhong University of
Science and Technology, Wuhan 430074, China  Ping Tang  WPI-AIMR, Tohoku
University, 2-1-1 Katahira, 980-8577 Sendai, Japan  R. L. Rodríguez-Suárez
Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306,
Santiago, Chile  S. M. Rezende  Departamento de Física, Universidade Federal
de Pernambuco, 50670-901, Recife, Pernambuco, Brazil  G. E. W. Bauer  WPI-AIMR
and Institute for Materials Research and CSRN, Tohoku University, Sendai
980-8577, Japan  Kavli Institute for Theoretical Sciences, University of the
Chinese Academy of Sciences, Beijing 10090, China  Tao Yu
<EMAIL_ADDRESS>School of Physics, Huazhong University of Science and
Technology, Wuhan 430074, China

(August 28, 2024)

###### Abstract

The duality between electric and magnetic dipoles inspires recent comparisons
between ferronics and magnonics. Here we predict surface polarization waves or
“ferrons” in ferroelectric insulators, taking the long-range dipolar
interaction into account. We predict properties that are strikingly different
from the magnetic counterpart, i.e. the surface “Damon-Eshbach” magnons in
ferromagnets. The dipolar interaction pushes the ferron branch with locked
circular polarization and momentum to the ionic plasma frequency. The low-
frequency modes are on the other hand in-plane polarized normal to their wave
vectors. The strong anisotropy of the lower branch renders directional
emissions of electric polarization and chiral near fields when activated by a
focused laser beam, allowing optical routing in ferroelectric devices.

Introduction.—Electric and magnetic dipoles are said to be dual since the
associated electric ${\bf E}$ and magnetic ${\bf H}$ fields have identical
forms [1, 2]. The elementary excitations of ordered dipoles in both magnets
and ferroelectrics can be classified as massless Goldstone modes that restore
the broken symmetry and Higgs modes, i.e. scalar excitation of the modulus of
the order parameter. Magnons are transverse spin excitation the classify as
Goldstone modes in the absence of applied or anisotropy fields. The
polarization waves of the ferroelectric order, or “ferrons” [3], can have
either character, depending on the microscopic mechanism that triggers the
ferroelectric phase transition.

Magnons in magnetic insulators favor low-dissipation [4, 5] and novel chiral
[6] transport for the spin, momentum, and heat. Among them, Damon-Eshbach (DE)
modes in finite-size magnets, governed by the long-range magnetic dipolar
interaction, are surface modes holding chirality ruled by the cross product of
magnetization, propagation direction, and surface normal [8, 7]. Being
unidirectional normal to the magnetization, they can propagate against
temperature gradients [9, 10, 11, 12], be backscattering immune to the surface
disorder [13, 14], and interact chirally with light [15, 16]. Such modes are
recently demonstrated to be topological, even though they do not lie in a gap
[17]. Surface modes of ferroelectrics that emerge in an exchange-only
transverse Ising model [18] do not take into account electric dipolar
interactions and have not yet been verified by experiments. Recent theoretical
proposals predict thermal transport of electric polarization by bulk ferrons
in nanostructures [19, 20, 21, 22], with initial possible experimental
evidence [23]. Magnetism and ferroelectricity respectively break time-reversal
and inversion symmetry [24].

Figure 1: Surface ferrons excited by a laser spot (red dot) in a ferroelectric
insulator of thickness $d$ with polarization ${\bf P}_{0}$ poled along the
$\hat{\bf y}$-direction. The green arrows illustrate the predicted emission of
directional coherent ferron beams with four-fold symmetry, and the yellow
arrows illustrate the associated electric stray fields with momentum-locked
polarization.

In this Letter, we predict surface ferrons in ferroelectrics that exist by the
long-range dipolar interaction. They generate strong chiral stray electric
fields in the THz regime, as sketched in Fig. 1. We find two branches of
surface ferrons, among which one branch is pushed to the surface ionic plasma
frequency $\sim\Omega_{p}/\sqrt{2}$ by the crucial dipolar interaction, but
the other branch at much lower frequency $\sim\Omega_{p}/10\sim 10$ THz
remains. The polarization of the lower mode is linear and lies in the surface
plane transverse to the wave vector. In contrast to the DE magnons in
ferromagnets, the surface ferrons are therefore not chiral. Nevertheless,
their stray electric fields are circularly polarized with the precession axis
normal and locked to their momenta. Such chirality has been found as well in
conventional surface polaritons and Rayleigh surface acoustic waves [6, 25,
26, 27, 28, 29, 30]. We predict that the dispersion relations of surface
ferrons are highly anisotropic leading to directional emission from a focused
excitation as illustrated by Fig. 1. These results imply useful
functionalities for broadband optical micro-scale routing in electrically
insulating rather than metallic devices [31, 32, 33, 34, 35, 36].

Model and formalism.—We consider a slab of uniaxial ferroelectric insulator of
thickness $d$ in vacuum that is extended in the $x$-$y$ plane and normal to
the $\hat{\bf z}$-axis (Fig. 1). The spontaneous polarization ${\bf P}_{0}$ at
equilibrium is poled along the $\hat{\bf y}$-direction and emits no stray
fields. In electrical insulators, the bare Coulomb interaction between
fluctuations of the electric polarizations ${\bf p}({\bf r})$ is strong and
long-ranged, which renders the problem nontrivial. Assuming small fluctuations
$|{\bf p}|\ll|{\bf P}_{0}|$ we find the collective modes by self-consistently
solving the Landau-Khalatnikov-Tani (LKT) equation of motion of the electric
polarization [37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 47, 45] coupled to the
Maxwell equation for the electric field ${\bf E}^{(d)}$ emitted by ${\bf p}$.

We start from the free energy $F=\int d{\bf r}{\mathcal{F}}({\bf r})$ below
the Curie temperature $T_{c}$ of a uniaxial ferroelectric insulator biased by
an external (DC or AC) electric field ${\bf E}^{(e)}$, with the energy density
as a function of the electric polarization ${\bf P}={\bf P}_{0}+{\bf p}$ [24,
45, 48, 49]

$\displaystyle{\mathcal{F}}({\bf r})$
$\displaystyle=({\alpha}/{2})P_{y}^{2}+({\beta}/{4})P_{y}^{4}+({\lambda}/{2})(P_{x}^{2}+P_{z}^{2})$
$\displaystyle-({1}/{2}){\bf E}^{(d)}\cdot{\bf P}-{\bf E}^{(e)}\cdot{\bf P}.$
(1)

Our results for $T\ll T_{c}$ do not depend importantly on Landau parameters
$\alpha<0$, $\beta>0$, and $\lambda>0$. We are interested in the surface
ferrons and their difference from conventional phonons in dielectrics [47, 50,
51, 52, 39]. The electric polarization is an independent thermodynamic
variable with independent Goldstone and Higgs excitations different from that
of conventional phonons, noting that in displacive ferroelectrics the ferron
equations of motion are closely related to lattice dynamics, however: The
electric polarization $\sum_{i}{Q}_{i}{\bf r}_{i}$ by atoms positioned at
${\bf r}_{i}$ with effective charges $Q_{i}$ is governed by the ionic motion
driven by effective electric fields [39, 53] and elastic forces [50, 51, 52].
In displacive ferroelectrics, phonon dynamics governs the Landau parameters.
On the other hand, purely electronic ferroelectricity can be described by a
Landau functional as well [54].

Here we focus on the LKT equation of motion [37, 38, 39, 40]

$\displaystyle m_{p}\partial^{2}{\bf P}/\partial t^{2}+\gamma{\partial{\bf
P}}/{\partial t}=-{\delta F}/{\delta{\bf P}},$ (2)

which is interpreted as a damped Newton’s equation for an effective mass
$m_{p}=1/(\varepsilon_{0}\Omega_{p}^{2})$, where $\varepsilon_{0}$ is the
vacuum dielectric constant, ${\Omega_{p}}$ is the ionic plasma frequency, and
$\gamma$ is a phenomenological viscous damping constant [39]. ${\bf P}$ is a
source for the electric field that obeys the wave equation [2] $\nabla^{2}{\bf
E}^{(d)}-({1}/{c^{2}})\partial^{2}_{t}{\bf
E}^{(d)}=\mu_{0}\partial^{2}_{t}{\bf
P}-({1}/{\varepsilon_{0}})\nabla(\nabla\cdot{\bf P})$, contributed by both the
electromagnetic radiation via the shift current density $\dot{\bf P}$ and
Coulomb’s law for the bound charge accumulation $-\nabla\cdot{\bf P}$, where
$\mu_{0}$ is the vacuum permeability. For a monochromatic fluctuation ${\bf
P}({\bf r},t)\propto e^{-i\omega t}$ and assuming that its wavelength
$\lambda$ is much smaller than that of light, $k\gg\omega/c$ and $\lambda\ll
2\pi c/\omega\equiv\lambda_{c}$, they do not couple strongly because of the
wave vector mismatch. At the frequency band of our interest $\omega\sim 10$
THz, $\lambda_{c}\sim 190\leavevmode\nobreak\ \mu$m is a long wavelength. The
retardation due to the finite light velocity $c$ is negligible and we may
treat the electric dipolar field in the static limit [2]

${\bf E}^{(d)}({\bf
r},t)=\dfrac{1}{4\pi\varepsilon_{0}}\nabla\int\dfrac{\nabla^{\prime}\cdot{\bf
P}({\bf r}^{\prime},t)}{|{\bf r}-{\bf r}^{\prime}|}d{\bf r}^{\prime}.$ (3)

At equilibrium, a constant electric polarization ${\bf
P}_{0}=\left\\{0,P_{0},0\right\\}$ minimizes the free energy $\delta
F/\delta{\bf P}=0$ when saturated by the static external field
$E^{(e)}_{y}\hat{\bf y}$, and according to $\alpha P_{0}+\beta
P_{0}^{3}=E^{(e)}_{y}$, $P_{0}^{2}\rightarrow-\alpha/\beta$ when
$E_{y}^{(e)}\rightarrow 0$.

In the linear regime with $|{\bf p}|\ll|{\bf P}_{0}|$ and assuming a small
external electric field ${\bf E}^{(e)}\rightarrow 0$, the equation of motion
for the fluctuations reads

$\displaystyle(1/\Omega_{p})^{2}\partial_{t}^{2}p_{x,z}+K_{\perp}p_{x,z}=\varepsilon_{0}E_{x,z}^{(d)},$
$\displaystyle(1/\Omega_{p})^{2}\partial_{t}^{2}p_{y}+K_{\parallel}p_{y}=\varepsilon_{0}E_{y}^{(d)},$
(4)

where $K_{\perp}=\varepsilon_{0}\lambda>0$ and
$K_{\parallel}=\varepsilon_{0}(\alpha+3\beta
P_{0}^{2})\rightarrow-2\varepsilon_{0}\alpha>0$ are dimensionless “stiffness”
constants. The two branches of the eigenmodes describe longitudinal (Higgs-
like) and transverse (Goldstone-like) fluctuations. Reference [21] addressed
only longitudinal modes, disregarding the dipolar interaction that couples
them with the transverse ones. The Coulomb interaction gaps the Goldstone mode
at zero wave number to the ionic plasma frequency $\Omega_{p}$ as derived
below [see also the Supplemental Material (SM) [55]], analogous to the magnon
gap induced by magnetic anisotropies and the Goldstone mode in superconductors
that is pushed to the electron plasma frequency by charge fluctuations [56,
57].

Equations (3) and (4) describe the interplay of the electric polarization and
its dipolar electric field. We expand the solution into plane waves

$\displaystyle{\bf p}({\bf r},t)=\left({\bf a}e^{ik_{z}z}+{\bf
b}e^{-ik_{z}z}\right)e^{i({\boldsymbol{\kappa}}\cdot{\boldsymbol{\rho}}-\omega
t)},$ (5)

where the in-plane coordinate
${\boldsymbol{\rho}}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}$ and momentum
${\boldsymbol{\kappa}}=k_{x}\hat{\bf x}+k_{y}\hat{\bf y}$, which generates the
stray field [25, 6]

${\bf E}^{(d)}=\frac{e^{i({\boldsymbol{\kappa}}\cdot{\boldsymbol{\rho}}-\omega
t)}}{\varepsilon_{0}}\int_{-d}^{0}dz^{\prime}\mathcal{G}(z-z^{\prime})\left(\begin{array}[]{c}a_{x}e^{ik_{z}z^{\prime}}+b_{x}e^{-ik_{z}z^{\prime}}\\\
a_{y}e^{ik_{z}z^{\prime}}+b_{y}e^{-ik_{z}z^{\prime}}\\\
a_{z}e^{ik_{z}z^{\prime}}+b_{z}e^{-ik_{z}z^{\prime}}\end{array}\right),$ (6)

where $\mathcal{G}(z-z^{\prime})$ is the Green-function tensor. Here
${k_{y}}E_{x}^{(d)}={k_{x}}E_{y}^{(d)}$ implies locking of the circular
polarization and wave vector (see SM [55] for details).

Surface ferrons.—The polarization waves interact with their self-fields caused
by the bound surface and bulk electric charges that must be calculated self-
consistently. Substituting Eq. (6) into (4), we find two coupled
characteristic equations (refer to the SM [55] for details of the derivation)
that can solve $\omega$ and $k_{z}$ in terms of $k_{x}$ and $k_{y}$. The first
characteristic equation

$c_{1}c_{2}k^{2}+c_{1}k_{y}^{2}+c_{2}(k_{x}^{2}+k_{z}^{2})=0$ (7)

establishes the relation between the frequency and momentum, where
$c_{1}\equiv-(\omega/\Omega_{p})^{2}+K_{\perp}$ and
$c_{2}\equiv-(\omega/\Omega_{p})^{2}+K_{\parallel}$ are dimensionless. For
bulk excitation, $k_{z}$ is real and Eq. (7) leads to two ferron branches with
band gaps at $\omega\sim\Omega_{p}$ and $\omega\ll\Omega_{p}$, respectively
[55]. The high-frequency branch is longitudinal with ${\bf p}\parallel{\bf k}$
[55], with its frequencies pushed up to the bulk ionic plasma frequency
$\Omega_{p}$ by the Anderson-Higgs mechanism [56, 57]. The low-frequency
branch is a mix of both longitudinal [21] and transverse modes. A complex
$k_{z}=\eta_{1}+i\eta_{2}$ that also obeys the second characteristic equation
[55]

$\displaystyle\left[(c_{1}+1)k_{z}\sin({k_{z}d}/{2})-c_{1}\kappa\cos(k_{z}d/2)\right]$
$\displaystyle\times\left[(c_{1}+1)k_{z}\cos(k_{z}d/2)+c_{1}\kappa\sin(k_{z}d/2)\right]=0,$
(8)

implying the existence of evanescent modes at the surfaces. We solve Eqs. (7)
and (8) numerically for ferroelectric films of arbitrary thickness.

For thick slabs $\coth{(d\eta_{2}/2})\rightarrow 1$ and
$\tanh({d\eta_{2}/2})\rightarrow 1$, so according to Eq. (8), $\eta_{1}=0$ and
$\eta_{2}=-{c_{1}\kappa}/{(1+c_{1})}$ [55]. Substituting $k_{z}=\pm i\eta_{2}$
into Eq. (7) reduces the coupled characteristic Eqs. (7) and (8) to

$\displaystyle k_{z}=\pm i{c_{1}\kappa}/(1+c_{1}),$ $\displaystyle
k_{x}^{2}c_{2}(1+2c_{1})+k_{y}^{2}\left[c_{1}c_{2}+c_{1}(1+c_{1})\right]=0.$
(9)

In most ferroelectrics, the elastic constants and their difference
$\delta=K_{\parallel}-K_{\perp}$ are much smaller than unity at room
temperature. Solving Eq. (9) to leading order in $\delta$ leads to surface
modes

$\displaystyle\omega_{\boldsymbol{\kappa}}^{+}$
$\displaystyle=\Omega_{p}\sqrt{(1+\delta\cos^{2}{\theta_{\boldsymbol{\kappa}}})/2+K_{\perp}},$
$\displaystyle{\omega_{\boldsymbol{\kappa}}^{-}}$
$\displaystyle={\Omega_{p}}\sqrt{\delta\sin^{2}{\theta_{\boldsymbol{\kappa}}}+K_{\perp}},$
(10)

that are gaped and depend on the angle $\theta_{\boldsymbol{\kappa}}$ between
propagation direction and equilibrium polarization.

Figure 2 illustrates the unique features of the dispersion with the
conventional ferroelectric insulator LiNbO3, with which at room temperature
the parameters $\alpha=-2.012\times 10^{9}\leavevmode\nobreak\
\text{Nm}^{2}/\text{C}^{2}$, $\beta=3.608\times 10^{9}\leavevmode\nobreak\
\text{Nm}^{6}/\text{C}^{4}$, and $\lambda=1.345\times
10^{9}\leavevmode\nobreak\ \text{Nm}^{2}/\text{C}^{2}$ [48] lead to
$P_{0}=0.746\leavevmode\nobreak\ {\rm C/m^{2}}$, $K_{\perp}=0.012$, and
$K_{\parallel}=0.036$. The low-frequency branch is located between
$\sqrt{K_{\perp}}\Omega_{p}$ when ${\boldsymbol{\kappa}}\parallel{\bf P}_{0}$
and $\sqrt{K_{\parallel}}\Omega_{p}$ when ${\boldsymbol{\kappa}}\perp{\bf
P}_{0}$. Decreasing the temperature, the longitudinal stiffness
$K_{\parallel}$ is enhanced since $|\alpha|\propto|T-T_{c}|$, leading to an
enhanced peak frequency of low-frequency surface ferron. The high-frequency
branch is located within a narrow range between
$\Omega_{p}\sqrt{(1+2K_{\perp})/2}$ and
$\Omega_{p}\sqrt{(1+K_{\perp}+K_{\parallel})/2}$, exactly around the surface
ionic plasma frequency $\Omega_{p}/\sqrt{2}$.

The group velocity of surface ferrons does not vanish but is at large
wavelengths transverse to the wave-vector direction as shown for the lower
branch in Fig. 2(c). In the SM [55] we show that the gradient term of the
electric polarization in the free energy functional ${g}(\nabla{\bf P})^{2}/2$
substantially enhances the group velocity in the radial direction at larger
wave numbers. In Fig. 2(d) we plot the dispersion of the lower branch using
$g\sim 5.39\times 10^{-10}\leavevmode\nobreak\ \mathrm{J\cdot m^{3}/C^{2}}$ as
recommended for LiNbO3 [58].

Figure 2: Dispersion relation of the surface ferron branches
$\omega_{\boldsymbol{\kappa}}^{-}$ [(a)] and
$\omega_{\boldsymbol{\kappa}}^{+}$ [(b)] for the room-temperature parameters
of LiNbO3. At low frequencies, the surface ferrons are strongly anisotropic.
They can be excited by focused lasers with propagation directions that can be
tuned by its frequency $\omega_{0}$ (see below). (c) is a plot of the group
velocity vector field of the lower branch, while (d) illustrates the effect of
the gradient term of ${\bf p}$ on the dispersion at higher wave numbers.

We next address the characteristics of the surface ferronic modes and their
associated electric stray fields, including polarization and chirality. We
find ${\bf a}=0$ or ${\bf b}=0$ when $k_{z}=-ic_{1}\kappa/(1+c_{1})$ or
$k_{z}=ic_{1}\kappa/(1+c_{1})$ for the surface modes [Eq. (9)] [55]. Both
solutions conclude that the waves are localized near the upper surface of the
ferroelectric slab since they lead to the same exponential decay as a function
of distance from the surface according to Eq. (5). The in-plane momentum
${\boldsymbol{\kappa}}$ is arbitrary, strikingly different from its magnetic
counterpart: surface magnons [7, 8] in ferromagnets are unidirectional at one
surface. We list the characteristics of both surface ferrons in the SM [55]
and focus on the low-frequency branch that allows the directional routing by
the frequency of THz light excitation [59].

The eigenmodes of the low-frequency branch ${\bf
p}({\boldsymbol{\kappa}},z)={\boldsymbol{\cal
P}}({\boldsymbol{\kappa}},z)\exp(i{\boldsymbol{\kappa}}\cdot{\boldsymbol{\rho}}-i\omega_{\boldsymbol{\kappa}}^{-}t)$
and

$\displaystyle\left(\begin{array}[]{c}{\cal P}_{x}({\boldsymbol{\kappa}},z)\\\
{\cal P}_{y}({\boldsymbol{\kappa}},z)\\\ {\cal
P}_{z}({\boldsymbol{\kappa}},z)\end{array}\right)$
$\displaystyle=\xi\left(\begin{array}[]{c}-{\kappa}/({\delta\sin{\theta_{\boldsymbol{\kappa}}}})\\\
{\kappa}/({\delta\cos{\theta_{\boldsymbol{\kappa}}}})\\\
i{\kappa}/(1-\delta\sin^{2}{\theta_{\boldsymbol{\kappa}}})\end{array}\right)e^{z/\Delta_{\boldsymbol{\kappa}}}$
(17)

strongly depend on the propagation direction. Here $\xi$ is a normalization
constant and
$\Delta_{\boldsymbol{\kappa}}=(1-\delta\sin^{2}{\theta_{\boldsymbol{\kappa}}})/({\kappa\delta\sin^{2}{\theta_{\boldsymbol{\kappa}}}})$
is a strongly anisotropic decay length. The polarization of these waves
depends on the stiffness difference
$\delta=K_{\parallel}-K_{\perp}\rightarrow\varepsilon_{0}(-2\alpha-\lambda)$
that depends on temperature and material. For the conventional materials such
as LiNbO3 [48] and LiTaO3 [49], $\delta\ll 1$ at room temperature, such that
$\\{|p_{x}({\boldsymbol{\kappa}})|,|p_{y}({\boldsymbol{\kappa}})|\\}\gg|p_{z}({\boldsymbol{\kappa}})|$.
The fluctuation lies in the plane and, since ${\bf
p}({\boldsymbol{\kappa}})\cdot{\boldsymbol{\kappa}}=0$, is perpendicular to
the momentum. The surface ferrons are therefore linearly polarized and not
chiral.

Ferrons carry polarization and energy in the presence of various driving
forces [19, 20, 21, 22]. However, surface ferrons also betray their presence
by THz evanescent electric stray fields ${\bf E}_{\rm
out}^{(d)}(\boldsymbol{\kappa},\boldsymbol{\rho},z>0)$ outside the slab.
Inserting the eigenmodes (17) into the Coulomb integral (6)

$\displaystyle{\bf E}_{\rm
out}^{(d)}({\boldsymbol{\kappa}})=(\boldsymbol{\hat{\kappa}}+i\hat{\bf
z})(\xi\kappa/\varepsilon_{0})e^{-\kappa
z}e^{i({\boldsymbol{\kappa}}\cdot{\boldsymbol{\rho}}-\omega_{\boldsymbol{\kappa}}^{-}t)}.$
(18)

Since divergence free $\nabla\cdot{\bf E}^{(d)}_{\rm out}=0$, this field is
circularly polarized with the precession axis or “spin” governed by the cross
product of the momentum propagation direction
$\boldsymbol{\hat{\kappa}}={\boldsymbol{\kappa}}/\kappa$ and the surface
normal $\hat{\bf z}$.

Directional routing by laser.—A resonant excitation of the surface ferrons by
photons can best be formulated in a quantum language, expanding the
fluctuations into surface ferron operators $\hat{a}_{\boldsymbol{\kappa}}$ and
eigenmodes Eq. (17) that are normalized as harmonic oscillators [55]:
$\hat{\bf p}({\bf r},t)=\sum_{\boldsymbol{\kappa}}\left({\boldsymbol{\cal
P}}({\boldsymbol{\kappa}},z)\exp(i{\boldsymbol{\kappa}\cdot\boldsymbol{\rho}})\hat{a}_{\boldsymbol{\kappa}}(t)+\text{H.c.}\right)$.
A laser field ${\bf E}^{(e)}({\bf
r})=\sum_{\boldsymbol{\kappa}}\left({\boldsymbol{\cal
E}}({\boldsymbol{\kappa}},z)e^{i{\boldsymbol{\kappa}\cdot\boldsymbol{\rho}}}e^{-i\omega_{0}t}+\text{H.c.}\right)$
with frequency $\omega_{0}$ and Fourier components ${\boldsymbol{\cal
E}}(\boldsymbol{\kappa},z)$ couples with the surface ferron by the Stark
interaction
$\hat{H}_{c}/\hbar=\sum_{\boldsymbol{\kappa}}\omega_{\boldsymbol{\kappa}}^{-}\hat{a}_{\boldsymbol{\kappa}}^{\dagger}\hat{a}_{\boldsymbol{\kappa}}+\sum_{\boldsymbol{\kappa}}(g_{\boldsymbol{\kappa}}e^{i\omega_{0}t}\hat{a}_{\boldsymbol{\kappa}}+{\rm
H.c.})$, where the coupling constant
$g({\boldsymbol{\kappa}})=(-1/\hbar)\int_{-\infty}^{0}dz{\boldsymbol{\cal
P}}({\boldsymbol{\kappa}},z)\cdot{\boldsymbol{\cal E}}(\boldsymbol{\kappa},z)$
follows the last term in Eq. (1). In terms of the damping parameter $\gamma$
in Eq. (2), the ferron decay rate
$\Gamma=\gamma\varepsilon_{0}\Omega_{p}^{2}/2$. During the illumination

$a_{\boldsymbol{\kappa}}(t)\equiv\langle\hat{a}_{\boldsymbol{\kappa}}(t)\rangle=g^{*}({\boldsymbol{\kappa}})e^{-i\omega_{0}t}/(\omega_{0}-\omega^{-}_{\boldsymbol{\kappa}}+i\Gamma),$
(19)

where $\langle\cdots\rangle$ denotes an ensemble average [55]. The
polarization propagation depends on the angle with the ferroelectric order
according to
$\sin^{2}\theta_{\boldsymbol{\kappa}}=\left((\omega_{0}/\Omega_{p})^{2}-K_{\perp}\right)/\delta$
while the momenta are mainly limited by focus of the exciting electric field.
The spatiotemporal electric polarization in the steady state reads

${\bf p}({\bf r},t)=\sum_{\boldsymbol{\kappa}}\frac{\boldsymbol{\cal
P}(\boldsymbol{\kappa},z)g^{*}(\boldsymbol{\kappa})}{\omega_{0}-\omega_{\boldsymbol{\kappa}}+i\Gamma}e^{i({\boldsymbol{\kappa}}\cdot{\boldsymbol{\rho}}-\omega_{0}t)}+\text{H.c.}.$
(20)

In Fig. 3 we show our predictions of the surface response of a standard
ferroelectric such as the LiNbO3 to a focused THz laser spot with room
temperature parameters. We model the laser spot [60, 61, 62] by a linearly
polarized field of Gaussian shape with size $\sigma$, penetration depth $l$,
and amplitude $E_{0}$, i.e., Fourier components ${\boldsymbol{\cal
E}}(\boldsymbol{\kappa},z)=\pi
E_{0}\sigma^{2}\exp[{-(k_{x}^{2}+k_{y}^{2})\sigma^{2}/4}-z^{2}/l^{2}]\hat{\bf
x}$. We choose parameters $2\sigma=100\leavevmode\nobreak\ \mu$m,
$l=200\leavevmode\nobreak\ \mu$m, $E_{0}=50$ kV/cm, and $\omega_{0}=2\pi\times
7$ THz. A ferron damping rate $\Gamma=10^{-3}\omega_{0}$ corresponds to a
propagation length of up to $\sim 7.7$ mm with an exponential decay that is
much longer than the length scales studied here (refer to SM [55] for the
fitting decay length). The surface polarization wave may also be indirectly
excited via its coupling to the phonons activated by THz and mid-infrared
lasers [63]. Analogous to the spin wave tomography of magnets [64], non-
resonant broadband excitation by focused optical lasers can reveal the
spatiotemporal dynamics of surface ferrons.

Figure 3(a) illustrates that the excited surface ferrons propagate
preferentially into four directions at an angle
$|\theta_{0}|=|\arcsin\sqrt{[(\omega_{0}/\Omega_{p})^{2}-K_{\perp}}]/\delta|\sim
63^{\circ}$ with respect to ${\bf P}_{0}$. The “light house” effect, i.e.
strong collimation of the surface-ferron beams in real space, as plotted in
Fig. 3(b), is the caustics caused by the unique group velocity field as
plotted in Fig. 2(c), rendering the characteristic cross-shaped pattern of
$|{\bf p}_{\parallel}({\bf r})|=\sqrt{p_{x}^{2}({\bf r})+p_{y}^{2}({\bf r})}$
at the sample surface $z=0$. The directional routing [35, 36] can be tuned
[55] by varying $\omega_{0}$. The amplitudes are strongly focused into a tens-
of-micrometer channel along $\pi/2-\theta_{0}$, with respect to ${\bf P}_{0}$.
This direction reflects the Fourier transform of
$a({\boldsymbol{\kappa}})\sim\delta(k_{x}^{2}-\tan^{2}\theta_{0}k_{y}^{2})$ or
$k_{x}/k_{y}\sim\tan\theta_{0}$. The propagation phase in
$\exp(i{\boldsymbol{\kappa}}\cdot{\boldsymbol{\rho}})\sim\exp[ik_{y}(y+x\tan\theta_{0})]$
after integration over wave vectors favors $y+x\tan\theta_{0}\sim 0$ or
$x/y\sim-\cot\theta_{0}$ [55]. The ac electric polarization flows in the
channels both parallel and antiparallel to each other, as indicated by the
black arrows in Fig. 3(b).

Figure 3: Directional optical routing of surface ferrons in LiNbO3 resonantly
excited by a laser spot. (a) addresses the amplitudes of the pumped surface
ferrons in reciprocal space. In real space, simultaneously pumped ferrons
interfere constructively in micro-scale channels as shown in (b). (c) and (d)
plot the in-plane ${\bf E}_{\parallel}$ and out-of-plane ${\bf E}_{z}$
components of the stray electric field at $z=0$, respectively. The arrows in
(b) and (c) are a snapshot of the in-plane electric polarization and the stray
electric field.

The electric polarization produces circularly polarized chiral electric stray
fields Eq. (18) with the same spatial cross-like pattern, as shown in Fig.
3(c) and (d) for the in-plane ${\bf E}_{\parallel}$ and out-of-plane $E_{z}$
components at the surface. Measurement of these fields would allow
reconstruction of the underlying electric polarization fluctuation. The in-
plane component oscillates normal to the channel direction, as illustrated by
the black arrows in Fig. 3(c). The magnitude of the near electric field can
exceed that of the incident laser spot, indicating a substantial plasmonic
enhancement [31, 32, 33, 34, 35, 36, 65].

Equation (4) has the same form when the equilibrium polarizations
$\textbf{P}_{0}\parallel\hat{\textbf{z}}$ are out-of-plane and lead to surface
states with polarization fluctuations normal to the interface with isotropic
dispersion. This is again very different from ferromagnets with perpendicular
magnetization that support only bulk magnons.

The present calculations have been carried out in an electrostatic
approximation, which holds for relatively large wave numbers. For small wave
numbers, the ferrons hybridize with the photons or phonons to form surface
ferron-photon polaritons or surface ferron-phonon polarons, which we shall
address in the future.

Conclusions.—In conclusion, we predict that long-range dipolar interaction
causes evanescent polarization waves or surface ferrons in uniaxial
ferroelectrics. The symmetry breaking by the dipolar order can cause strongly
anisotropic emission patterns from a focused terahertz laser spot. The surface
ferrons may be interpreted as gapped Goldstone and Higgs modes of the
ferroelectric order. Their pronounced anisotropy distinguishes them with the
surface phonon polariton in polar dielectrics [50, 51, 52]. These excitations
can be monitored by their strong and chiral stray fields.

###### Acknowledgements.

This work is financially supported by the National Natural Science Foundation
of China under Grant No. 0214012051, and the startup grant of Huazhong
University of Science and Technology (Grants No. 3004012185 and No.
3004012198). P.T. and G.B. acknowledge the financial support by JSPS KAKENHI
Grants No. 19H00645 and No. 22H04965. R.S. and S.R. were financially supported
in Brazil by Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq), Coordenacão de Aperfeicoamento de Pessoal de Nível Superior (CAPES),
Financiadora de Estudos e Projetos (FINEP), and Fundacão de Amparo á Ciência e
Tecnologia do Estado de Pernambuco (FACEPE), and in Chile by Fondo Nacional de
Desarrollo Científico y Tecnológico (FONDECYT) Grant No. 1210641.

## References

  * [1] Duality (electricity and magnetism), en.wikipedia.org/wiki/Duality_(electricity_and_magnetism).
  * [2] J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1998).
  * [3] G. E. W. Bauer, P. Tang, R. Iguchi, and K. Uchida, Magnonics vs. Ferronics, J. Magn. Magn. Mater. 541, 168468 (2022).
  * [4] A. Barman et al., The 2021 Magnonics Roadmap, J. Phys.: Condens. Matter, 33, 413001 (2021).
  * [5] A. Brataas, B. van Wees, O. Klein, G. de Loubens, and M. Viret, Spin insulatronics, Phys. Rep. 885, 1 (2020).
  * [6] Yu, Luo, and Bauer, Chirality as Generalized Spin-Orbit Interaction in Spintronics, Phys. Rep. 1009,1 (2023).
  * [7] L. R. Walker, Magnetostatic Modes in Ferromagnetic Resonance, Phys. Rev. 105, 390 (1957).
  * [8] R. W. Damon and J. R. Eshbach, Magnetostatic modes of a ferromagnet slab, J. Phys. Chem. Solids 19, 308 (1961).
  * [9] T. An, V. I. Vasyuchka, K. Uchida, A. V. Chumak, K. Yamaguchi, K. Harii, J. Ohe, M. B. Jungfleisch, Y. Kajiwara, H. Adachi, B. Hillebrands, S. Maekawa, and E. Saitoh, Unidirectional spin-wave heat conveyer, Nat. Mater. 12, 549 (2013).
  * [10] O. Wid, J. Bauer, A. Müller, O. Breitenstein, S. S. P. Parkin, and G. Schmidt, Investigation of the unidirectional spin heat conveyer effect in a 200 nm thin Yttrium Iron Garnet film, Sci. Rep. 6, 28233 (2016).
  * [11] E. Shigematsu, Y. Ando, S. Dushenko, T. Shinjo, and M. Shiraishi, Spin-wave-induced lateral temperature gradient in a YIG thin film/GGG system excited in an ESR cavity, Appl. Phys. Lett. 112, 212401 (2018).
  * [12] P. Wang, L. F. Zhou, S. W. Jiang, Z. Z. Luan, D. J. Shu, H. F. Ding, and D. Wu, Unidirectional Spin-Wave-Propagation-Induced Seebeck Voltage in a PEDOT:PSS/YIG Bilayer, Phys. Rev. Lett. 120, 047201 (2018).
  * [13] T. Yu, S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Surface dynamics of rough magnetic films, Phys. Rev. B 99, 174402 (2019).
  * [14] M. Mohseni, R. Verba, T. Brächer, Q. Wang, D. A. Bozhko, B. Hillebrands, and P. Pirro, Backscattering Immunity of Dipole-Exchange Magnetostatic Surface Spin Waves, Phys. Rev. Lett. 122, 197201 (2019).
  * [15] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Cavity Optomagnonics with Spin-Orbit Coupled Photons, Phys. Rev. Lett. 116, 223601 (2016).
  * [16] S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Optical Cooling of Magnons, Phys. Rev. Lett. 121, 087205 (2018).
  * [17] K. Yamamoto, G. C. Thiang, P. Pirro, K.-W. Kim, K. Everschor-Sitte, and E. Saitoh, Topological Characterization of Classical Waves: The Topological Origin of Magnetostatic Surface Spin Waves, Phys. Rev. Lett. 122, 217201 (2019).
  * [18] M. G. Cottam, D. R. Tilley, and B. Zeks, Theory of surface modes in ferroelectrics, J. Phys. C: Solid State Phys. 17, 1793 (1984).
  * [19] G. E. W. Bauer, R. Iguchi, and K.-i. Uchida, Theory of Transport in Ferroelectric Capacitors, Phys. Rev. Lett. 126, 187603 (2021).
  * [20] P. Tang, R. Iguchi, K.-i. Uchida, and G. E. W. Bauer, Thermoelectric Polarization Transport in Ferroelectric Ballistic Point Contacts, Phys. Rev. Lett. 128, 047601 (2022).
  * [21] P. Tang, R. Iguchi, K.-i. Uchida, and G. E. W. Bauer, Excitations of the ferroelectric order, Phys. Rev. B 106, L081105 (2022).
  * [22] P. Tang, K.-i. Uchida, and G. E. W. Bauer, Nonlocal Drag Thermoelectricity Generated by Ferroelectric Heterostructures, arXiv:2207.00240.
  * [23] B. Wooten, R. Iguchi, P. Tang, J. S. Kang, K.-i. Uchida, G. E. W. Bauer, and J. P. Heremans, Sci. Adv. 9, eadd7194 (2023).
  * [24] P. Chandra and P. B. Littlewood, A Landau Primer for Ferroelectrics, in Physics of Ferroelectrics, (Springer-Verlag, Berlin, Heidelberg, 2007), pp. 69–116.
  * [25] L. Novotny and B. Hecht, Principles of Nano-Optics, (Cambridge University Press, Cambridge, England, 2006).
  * [26] P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Chiral quantum optics, Nature (London) 541, 473 (2017).
  * [27] K. Y. Bliokh and F. Nori, Transverse and longitudinal angular momenta of light, Phys. Rep. 592, 1 (2015).
  * [28] M. Matsuo, J. Ieda, and S. Maekawa, Mechanical generation of spin current, Front. Phys. 3, 54 (2015).
  * [29] I. A. Viktorov, Rayleigh and Lamb waves: Physical theory and applications (Plenum Press, New York, 1967).
  * [30] K. Y. Bliokh, D. Smirnova, and F. Nori, Quantum spin Hall effect of light, Science 348, 1448 (2015).
  * [31] T. J. Davis and D. E. Gómez, Colloquium: An algebraic model of localized surface plasmons and their interactions, Rev. Mod. Phys. 89, 011003 (2017).
  * [32] H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).
  * [33] S. A. Maier, Plasmonics: Fundamentals and Applications (Springer US, New York, 2007).
  * [34] J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, Theory of surface plasmons and surface-plasmon polaritons, Rep. Prog. Phys. 70, 1 (2006).
  * [35] F. J. Rodríguez-Fortuño, G. Marino, P. Ginzburg, D. O’Connor, A. Martínez, G. A. Wurtz, and A. V. Zayats, Near-Field Interference for the Unidirectional Excitation of Electromagnetic Guided Modes, Science 340, 328 (2013).
  * [36] J. Petersen, J. Volz, and A. Rauschenbeutel, Chiral nanophotonic waveguide interface based on spin-orbit interaction of light, Science 346, 67 (2014).
  * [37] K. Tani, Dynamics of Displacive-Type Ferroelectrics Soft Modes, J. Phys. Soc. Jpn 26, 93 (1969).
  * [38] Y. Ishibashi, Phenomenological theory of domain walls, Ferroelectrics 98, 193 (1989).
  * [39] S. Sivasubramanian, A. Widom, and Y. N. Srivastava, Physical Kinetics of Ferroelectric Hysteresis, Ferroelectrics 300, 43 (2004).
  * [40] A. Widom, S. Sivasubramanian, C. Vittoria, S. Yoon, and Y. N. Srivastava, Resonance damping in ferromagnets and ferroelectrics, Phys. Rev. B 81, 212402 (2010).
  * [41] V. A. Zhirnov, A contribution to the theory of domain walls in ferroelectrics, Zh. Eksp. Teor. Fiz. 35, 1175 (1958) [Sov. Phys. JETP 35, 822 (1959)].
  * [42] Y. Ishibashi and E. Salje, A Theory of Ferroelectric 90 Degree Domain Wall, J. Phys. Soc. Jpn. 71, 2800 (2002).
  * [43] W. Cao and L. E. Cross, Theory of tetragonal twin structures in ferroelectric perovskites with a first-order phase transition, Phys. Rev. B 44, 5 (1991).
  * [44] W. Cao, G. R. Barsch, and J. A. Krumhansl, Quasi-one-dimensional solutions for domain walls and their constraints in improper ferroelastics, Phys. Rev. B 42, 6396 (1990).
  * [45] J. Hlinka and P. Márton, Phenomenological model of a 90∘ domain wall in BaTiO3-type ferroelectrics, Phys. Rev. B 74, 104104 (2006).
  * [46] M. J. Haun, E. Furman, S. Jang, H. McKinstry, and L. Cross, Thermodynamic theory of PbTiO3, J. Appl. Phys. 62, 3331 (1987).
  * [47] A. N. Morozovska, E. A. Eliseev, C. M. Scherbakov, and Y. M. Vysochanskii, Influence of elastic strain gradient on the upper limit of flexocoupling strength, spatially modulated phases, and soft phonon dispersion in ferroics, Phys. Rev. B 94, 174112 (2016).
  * [48] D. A. Scrymgeour, V. Gopalan, A. Itagi, A. Saxena, and P. J. Swart, Phenomenological theory of a single domain wall in uniaxial trigonal ferroelectrics: Lithium niobate and lithium tantalate, Phys. Rev. B 71, 184110 (2005).
  * [49] I. Tomeno and S. Matsumura, Dielectric properties of LiTaO3, Phys. Rev. B 38, 606 (1988).
  * [50] R. Fuchs and K. L. Kliewer, Optical Modes of Vibration in an Ionic Crystal Slab, Phys. Rev. 140, A2076 (1965).
  * [51] H. J. Bakker, S. Hunsche, and H. Kurz, Coherent phonon polaritons as probes of anharmonic phonons in ferroelectrics, Rev. Mod. Phys. 70, 523 (1998).
  * [52] N. S. Stoyanov, D. W. Ward, T. Feurer, and K. A. Nelson, Terahertz polariton propagation in patterned materials, Nat. Mater. 1, 95 (2002).
  * [53] S. H. Zhuang and J.-M. Hu, Role of polarization-photon coupling in ultrafast terahertz excitation of ferroelectrics, Phys. Rev. B 106, L140302 (2022).
  * [54] H. Adachi, N. Ikeda, and E. Saitoh, Ginzburg-Landau action and polarization current in an excitonic insulator model of electronic ferroelectricity, arXiv:2301.11506.
  * [55] Supplemental Material […] for the derivation of the characteristic equations, the surface ferronic modes, the bulk ferronic modes, as well as the quantum approach for the optical routing of surface ferrons.
  * [56] P. W. Anderson, Random-Phase Approximation in the Theory of Superconductivity, Phys. Rev. 112, 1900 (1958).
  * [57] Y. Nambu, Quasi-Particles and Gauge Invariance in the Theory of Superconductivity, Phys. Rev. 117, 648 (1960).
  * [58] M. S. Richman, X. Li, and A. Caruso, Inadequacy of the extrapolation-length method for modeling the interface of a ferroelectric–graphene heterostructure, J. Appl. Phys. 125, 184103 (2019).
  * [59] R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M. Bonn, Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy, Rev. Mod. Phys. 83, 543 (2011).
  * [60] C. Ruchert, C. Vicario, and C. P. Hauri, Spatiotemporal Focusing Dynamics of Intense Supercontinuum THz Pulses, Phys. Rev. Lett. 110, 123902 (2013).
  * [61] X. F. Zang, C. X. Mao, X. G. Guo, G. J. You, H. Yang, L. Chen, Y. M. Zhu, and S. L. Zhuang, Polarization-controlled terahertz super-focusing, Appl. Phys. Lett. 113, 071102 (2018).
  * [62] H. Chen, Z. X. Wu, Z. Y. Li, Z. F. Luo, X. Jiang, Z. Q. Wen, L. G. Zhu, X. Zhou, H. Li, Z. G. Shang, Z. H. Zhang, K. Zhang, G. F. Liang, S. L. Jiang, L. H. Du, and G. Chen, Sub-wavelength tight-focusing of terahertz waves by polarization-independent high-numerical-aperture dielectric metalens, Optics Express 26, 29817 (2018).
  * [63] R. Mankowsky, A. von Hoegen, M. Fórst, and A. Cavalleri, Ultrafast Reversal of the Ferroelectric Polarization, Phys. Rev. Lett. 118, 197601 (2017).
  * [64] Y. Hashimoto, S. Daimon, R. Iguchi, Y. Oikawa, K. Shen, K. Sato, D. Bossini, Y. Tabuchi, T. Satoh, B. Hillebrands, G. E. W. Bauer, T. H. Johansen, A. Kirilyuk, T. Rasing, and E. Saitoh, All-optical observation and reconstruction of spin wave dispersion, Nat. Comm. 8, 15859 (2017).
  * [65] A. Espinosa-Soria, F. J. Rodríguez-Fortuño, A. Griol, and A. Martínez, On-Chip Optimal Stokes Nanopolarimetry Based on Spin–Orbit Interaction of Light, Nano Lett. 17, 3139 (2017).


# Designing, Building, and Characterizing RF Switch-based Reconfigurable
Intelligent Surfaces

Marco Rossanese  NEC Laboratories Europe ,  Placido Mursia  NEC Laboratories
Europe ,  Andres Garcia-Saavedra  NEC Laboratories Europe ,  Vincenzo
Sciancalepore  NEC Laboratories Europe ,  Arash Asadi  TU Darmstadt, Germany
and  Xavier Costa-Perez  i2cat, NEC Labs Europe and ICREA

###### Abstract.

In this paper, we present our experience designing, prototyping, and
empirically characterizing RF Switch-based Reconfigurable Intelligent Surfaces
(RIS). Our RIS design comprises arrays of patch antennas, delay lines and
programmable radio-frequency (RF) switches that enable passive 3D beamforming,
i.e., without active RF components. We implement this design using PCB
technology and low-cost electronic components, and thoroughly validate our
prototype in a controlled environment with high spatial resolution codebooks.
Finally, we make available a large dataset with a complete characterization of
our RIS and present the costs associated with reproducing our design.

††copyright: none††doi: ††isbn: ††conference: Submitted for review to ACM
WiNTECH; 2022††journalyear: ;††price:

##  1\. Introduction

Reconfigurable Intelligent Surfaces (RISs) are well-perceived as a key
technology for next-generation mobile systems (Pan et al., 2021). Despite the
recent hype on the topic, mostly driven by theoretical models and simulations,
empirical studies are scarce due to lack of accessible and affordable RIS
prototypes.

A RIS is essentially a planar structure with passive reflective cells that can
control the electromagnetic response of impinging radio-frequency (RF)
signals, such as changes in phase, amplitude, or polarization. Indeed, RISs
open a new paradigm (Albanese et al., 2022) where the wireless
channel—traditionally treated simply as an optimization constraint—plays an
active role subject to optimization with the potential of increasing the
energy efficiency of mobile networks by $>\\!\\!50\%$ (Yang et al., 2022).

To this end, a RIS must satisfy the following requirements: ($i$) RISs shall
(re-)steer RF signals with minimal power loss; ($ii$) RISs must not use active
RF components; ($iii$) RISs must minimize the energy required to re-configure
their reflective cells; ($iv$) RISs shall be re-configurable in real-time; and
($v$) RISs must be amenable to low-cost production at scale.

Related Work. Among existing literature on prior experiences, (Yezhen et al.,
2020) discloses a $16$x$16$ RIS operating at $28$GHz, whereas (Trichopoulos et
al., 2021) introduces a $16$x$10$ RIS working at sub-$6$GHz with an Arduino
control unit where only groups of elements can be configured. Both solutions
implement a PIN diode-based RIS with a $1$-bit resolution phase shift.
Additionally, the prototype presented in (Dai et al., 2020) obtains a $2$-bit
phase quantization by using $5$ PIN diodes per RIS element, whereas in (Hu et
al., 2020) the use of $3$ PIN diodes allows for $8$ phase states. (Fara et
al., 2021) discloses a $14$x$14$ RIS based on varactor diodes, which allows
continuous control of the phase shifts at the cost of a wide range in control
voltages, which is generally hard to achieve.

Conversely, RF switch-based implementations unveil a lower cost w.r.t. PIN
diodes used to control the reflector units. In particular in (Tan et al.,
2018), a RIS prototype with $14$x$16$ reflectors at $60$GHz is presented. The
unit elements are placed more than one $\lambda$ away to reduce the mutual
coupling at the expense of a tighter maximum scanning angle ($\lambda$ is the
operating wavelength). In (Arun and Balakrishnan, 2020), $40$ reflectors are
mounted on the boards, they are $\lambda/4$ tall, $\lambda/10$ wide, and
separated $\lambda/10$ on both the x and y-axis. Finally, (Dunna et al., 2020)
allows singular configurations of the elements, thereby reducing the number of
utilized pins at controller side. This RIS is made of $4$x$4$ patch antennas
operating at $5$GHz and controlled with $2$-bit phase shifter made with the
transmission line method.

Our design. Besides meeting the usual requirements for a RIS, our approach
provides additional features compared to previous work. First, in contrast to
diode-based approaches, usually constrained to 1-bit phase shifters, our
design has a resolution of 3 bits, enabling _high spatial resolution
codebooks_.

Second, our design permits coordinating multiple ($M$) boards, and hence it
enables _larger-scale structures_ of $M\times N_{x}\times N_{y}$ surfaces in a
modular and flexible manner.

Finally, in addition to producing phase shifts onto impinging signals in a
programmable manner, we can configure individual cells to fully absorb the
energy of RF signals, which open several opportunities. For instance, we can
effectively switch off reflective components, which let us _virtually optimize
the shape and the size of the RIS_ to meet system constraints. We would like
to remark that _switching off_ a cell in a RIS is not trivial when a cell is
specifically designed to reflect signals passively, without a direct energy
feed that could be cut off.

##  2\. RIS design

Figure 1. RIS board and unit cell.

The main purpose of our RIS is to perform 3D beamforming passively, i.e., re-
focus the energy received from impinging RF signals towards specified
directions without active (energy-consuming) RF components. Fig. 1 illustrates
a _board_ consisting of a grid of $N_{x}\times N_{y}$ _unit cells_ distributed
in a 2D array. Unit cells are elements that can reflect RF signals with
configurable phase shifts.

Phase shifts are configured by a microcontroller unit (MCU), which can be
programmed from an external controller. Conventional RIS designs are
characterized by dedicated $N_{x}\times N_{y}$ connections from the MCU to
each unit cell. However, MCUs only support a limited number of such
connections, which constrains the maximum number of cells and, consequently,
the achievable beamforming gains (Mursia et al., 2021). A more scalable
approach is to connect each cell with a pair of buses, denoted as _column/row
cell selection buses_ , that _select_ the cell to be configured, and a _phase
configuration bus_ , which communicates the desired configuration index out of
a discrete set. In this way, we reduce the complexity of the design from
$N_{x}\times N_{y}$ to $N_{x}+N_{y}$ connections per board.

As shown in the right-hand side of Fig. 1, each unit cell connects both
column/row selection buses with an _AND_ gate. Hence, when the MCU sets a high
voltage state in row $x$ and column $y$, the MCU activates the configuration
bus for unit cell $(x,y)$, whereas all the remaining gates across the board
will output a low voltage state ($0$ V). Each cell also integrates a set of
_flip-flop D_ , which exploit the high-state exiting the AND gate as a rising
edge to update and send out the value stored in memory. We designed our RIS
with 3-bit phase shifters, which enable high spatial resolution codebooks.
Therefore, each cell uses _three_ 1-bit phase configuration buses and _three_
flip-flops.

The latter are connected to _configuration ports_ in an _RF switch_. An RF
switch is a component that can redirect the RF signal received from an _input
port_ towards one _output port_ , as indicated by the configuration ports. The
input port is connected to a _patch antenna_ , the ultimate responsible for
interacting with the medium, through a _feeding line_. Each output port
(except one) uses an open-ended _delay line_ with a suitably-designed length
to reflect impinging signals with a specific time delay, shifting the signal
phase.

We reserve one output port of the RF switch to connect an _absorber_ , an
impedance-matching component that absorbs the energy of incoming signals
instead of reflecting them back. We call this configuration _absorption state_
, and it let us virtually optimize the reflective area of the RIS to meet
system constraints. For instance, we can flexibly adapt to different time
constraints when optimizing the RIS configuration (which takes longer the
larger the number of active cells in the RIS. This is illustrated in Fig. 3.
Alternatively, an energy harvester (Mir et al., 2021) may be employed instead
to re-use the dissipated energy to feed a low-consuming MCU, becoming self-
sustainable boards, which we leave for future work.

Our RIS design is modular: as shown in Fig. 3, multiple boards can be
coordinated through a common bus. The disposition of the unit cells across
different boards has been carefully designed to have a separation of
$\lambda/2$, where $\lambda$ is the operating wavelength. Such modular boards
let us increase/decrease the physical area of our structure without
compromising the inter-cell distance, as depicted in Fig. 3.

Figure 2. Shape-adaptive RIS.

Figure 3. Multi-board RIS.

##  3\. Beamforming codebook

A RIS board can be modelled as a uniform planar array (UPA) comprised of
$N=N_{x}N_{y}$ antenna elements (Basar et al., 2019; Mursia et al., 2021).
Hence, we define the array response at the RIS for the steering angles
$\bar{\theta}$, $\bar{\phi}$ along the azimuth and elevation, respectively, as

(1) $\displaystyle\mathbf{a}(\bar{\theta},\bar{\phi})$
$\displaystyle\triangleq\mathbf{a}_{x}(\bar{\theta})\otimes\mathbf{a}_{y}(\bar{\phi})$
$\displaystyle=[1,e^{j2\pi\delta\cos(\bar{\theta})},\ldots,e^{j2\pi\delta(N_{x}-1)\cos(\bar{\theta})}]$
$\displaystyle\otimes[1,e^{j2\pi\delta\sin(\bar{\phi})},\ldots,e^{j2\pi\delta(N_{y}-1)\sin(\bar{\phi})}]\in\mbox{$\mathbb{C}$}^{N\times
1},$

where $\delta$ is the ratio between the antenna spacing and the signal
wavelength (usually $\delta=0.5$). Assuming line-of-sight signal propagation
and a single-antenna transmitter, the channel between the latter and the RIS
is given by

(2)
$\displaystyle\mathbf{g}\triangleq\sqrt{\gamma_{t}}\,\mathbf{a}(\theta_{t},\phi_{t})\in\mbox{$\mathbb{C}$}^{N\times
1},$

where we define the average channel power gain as
$\gamma_{t}\triangleq{\beta_{0}}/{d_{t}^{2}}$, with $\beta_{0}$ the average
channel power gain at a reference distance. $d_{t}$ represents the distance
between the transmitter and the RIS, whereas $\theta_{t}$, $\phi_{t}$ denote
the angles of arrival at the RIS along the azimuth and elevation,
respectively. With similar reasoning, the channel between the RIS and the
single-antenna receiver is given by

(3)
$\displaystyle\mathbf{h}\triangleq\sqrt{\gamma_{r}}\,\mathbf{a}(\theta_{r},\phi_{r})\in\mbox{$\mathbb{C}$}^{N\times
1}.$

The matrix containing the RIS configuration is defined as

(4)
$\displaystyle\mathbf{\Phi}\triangleq\mathrm{diag}[e^{j\psi_{1}},\ldots,e^{j\psi_{N}}]\in\mbox{$\mathbb{C}$}^{N\times
N},$

with $\psi_{n}\in\mathcal{Q},\,\forall n$ and $\mathcal{Q}$ the quantized RIS
phase shift set. Note that, to preserve a tractable model in (4), we have
neglected any phase-dependent reflection coefficient at the RIS. Lastly, the
received signal at the receiver is given by

(5) $\displaystyle
y\triangleq\mathbf{h}^{\mathrm{H}}\mathbf{\Phi}\mathbf{g}\,s+n\in\mbox{$\mathbb{C}$},$

where $s\in\mbox{$\mathbb{C}$}$ is the transmitted symbol and
$n\in\mbox{$\mathbb{C}$}$ is the noise term distributed as
$\mathcal{CN}(0,\sigma_{n}^{2})$.

Let
$\mathbf{v}\\!\triangleq\\!\mathrm{diag}(\mathbf{\Phi}^{\mathrm{H}})\\!\in\\!\mbox{$\mathbb{C}$}^{N\times
1}$ and
$\bar{\mathbf{h}}\\!\triangleq\\!\mathrm{diag}(\mathbf{h}^{\mathrm{H}})\mathbf{g}\\!\in\\!\mbox{$\mathbb{C}$}^{N\times
1}$, such that the power at the receiver is maximized by letting (Wu et al.,
2018)

(6) $\displaystyle\mathbf{v}=\mathrm{exp}[j\,f_{q}(\angle\bar{\mathbf{h}})],$

where $f_{q}(\cdot)$ projects each element of the vector
$\angle\bar{\mathbf{h}}$ in (6) onto the closest element of set $\mathcal{Q}$
to obtain a feasible solution.

It is important to highlight that vector $\bar{\mathbf{h}}$ has the form of a
scaling term times the UPA response vector in (1) for some steering angles
$({\theta},{\phi})$. Hence, in order to design a codebook of RIS beamforming
vectors, we artificially create $N_{\mathrm{B}}$ pairs of
$\\{({\theta_{n}},{\phi_{n}})\\}_{n=1}^{N_{\mathrm{B}}}$ couples and generate
the corresponding UPA response vectors
$\\{\bar{\mathbf{h}}_{n}\\}_{n=1}^{N_{\mathrm{B}}}$. Given the symmetry of the
array response around the $x$-axis for the azimuth and around the $y$-axis for
the elevation, and in the interest of saving measurement time, we sample a
regular grid of points spaced by $3$ degrees in the search space
$[-\pi/2,\pi/2]\times[-\frac{\pi}{4},\frac{\pi}{4}]$, such that
$N_{\mathrm{B}}=1891$. Finally, we obtain the RIS configurations
$\mathcal{V}:=\\{\mathbf{v}_{n}\\}_{n=1}^{N_{\mathrm{B}}}$ for each steering
angle couple $\\{({\theta_{n}},{\phi_{n}})\\}_{n=1}^{N_{\mathrm{B}}}$ in the
codebook by applying the expression in (6).

##  4\. Prototype Implementation

We prototyped our design using a two-layer PCB (Printed Circuit Board). The
substrate material is FR-4, a composite material made of woven fiberglass with
an epoxy resin binder that is flame resistant. Its relative electrical
permittivity is in the range of $\epsilon_{r}=[4.1,4.8]$, and is coated by two
layers of 1-ounce copper (35$~{}\mu\text{m}$). In general, thick substrates
and high permittivity lead to small bandwidths and low efficiency due to
surface waves (Pandey, 2019). Since the operating frequency of our prototype
is $f=5.3$ GHz ($\lambda=56.56$ mm), we chose a substrate thickness of
$h=0.53$ mm, that is in the range $0.003\lambda\leq h\leq 0.05\lambda$ as
suggested in (Pandey, 2019).

Figure 4. Layout of the patch antenna design.

Figure 5. TDR chart of a 0.95mm-width microstrip.

###  4.1. Patch Antenna

Patch antennas are implemented by cutting out a particular shape from the
copper of the board’s upper layer. In this way, the remaining metallic shape
can radiate at the desired frequency while the back layer operates as ground
for the antenna. Following the conventional literature on antenna design, we
used a rectangular shape, as shown in Fig. 5. In more detail, we used the
transmission-line model from (Balanis, 2016) to calculate its width $W$ and
length $L$ as follows:

(7) $\displaystyle W$
$\displaystyle=\frac{\lambda}{2\sqrt{0.5(\epsilon_{r}+1)}}=16.9~{}\text{mm},$
(8) $\displaystyle L$ $\displaystyle=L_{eff}-2\Delta L=13.15~{}\text{mm},$

where
$\epsilon_{eff}=\frac{\epsilon_{r}+1}{2}+\frac{\epsilon_{r}-1}{2}(\frac{1}{\sqrt{1+12\frac{h}{w}}})$
is the effective dielectric constant that takes into account the fact that the
electric field lines reside in the substrate and partially in the air,
$L_{eff}=\frac{c}{2f\sqrt{\epsilon_{eff}}}$ is the effective length, $c$ is
the speed of light, and $\Delta
L=0.412h\cdot\frac{\epsilon_{eff}+0.3}{\epsilon_{eff}-0.258}\cdot\frac{\frac{h}{w}+0.264}{\frac{h}{w}+0.8}$
is an offset to obtain the antenna physical length from $L_{eff}$ (see
(Balanis, 2016) for details).

As shown in Fig. 5, a microstrip connects each antenna to the RF switch. To
this end, we selected an _inset feeding_ approach, with a notch at the edge of
the antenna. This approach allows us to adapt the antenna to a precise
characteristic impedance, which is crucial to maximizing power transfer. Given
this notch (see details later), we refined the geometry derived before with
the parameters shown in Fig. 5 by exhaustive search using a full-wave
simulator (Dassault Systèmes, 2022), thus setting $W=15.5~{}\text{mm}$ and
$L=12.8~{}\text{mm}$.

In the following we describe in detail two crucial parameters, namely ($i$)
the width of the microstrip that connects the RF switch (see Fig. 1), and
($ii$) the position of the notch. The former is essential to guarantee
impedance-matching, and, since the usual characteristic impedance is 50
$\Omega$, the width of all the microstrips must be selected accordingly.

####  4.1.1. Width of the feeding line

To avoid power loss between the feeding line and the RF switch, their
characteristic impedance should match. Therefore, we used the model described
in (Analog Devices, 2009) to estimate a $0.95$-mm line width, which equalizes
the $50\Omega$-impedance of the switch.

To validate this, we printed a $0.95$mm-width microstrip and applied the Time
Domain Reflectometry (TDR) technique using a Vector Network Analyzer (VNA) to
measure the actual characteristic impedance along the line. TDR generates a
pulse with a short rising time that allows us to calculate the impedance along
the line based on the received reflections.

As depicted in Fig. 5 (green), this experiment shows that the actual impedance
along the line is smaller than the predicted $50~{}\Omega$. Such a mismatch
with the impedance of the switch would incur some power loss at every unit
cell and, consequently, poor beamforming gains overall.

Figure 6. Expected radiation pattern of the patch antenna.

Figure 7. $\boldsymbol{S_{11}}$ parameter of two patch antenna samples.

Consequently, we opted for a simple empirical approach: we printed out several
$135$mm-length microstrips with different widths, and applied the TDR method
to each sample. Fig. 5 shows with an orange line the result of the selected
sample, with a width equal to 0.75 mm, which provided the best performance.
Ignoring the large oscillation at the beginning of the line, which is due to
the soldered SMA connector that we used to connect the line and the VNA, the
experiment shows a perfect match with the expected value of $50~{}\Omega$.

####  4.1.2. Notch

The second relevant parameter is the depth of the notch, which should minimize
the amount of reflected power. This is achieved when the impedance of the
antenna matches that of the feeding line. However, the antenna’s impedance
diminishes as one moves towards its center because the current’s intensity is
higher at that point. Hence, it is important to carefully design the depth of
the notch.

Following the analytical method introduced in (Balanis, 2016), we first
estimated the impedance at the bottom edge of the antenna and centered on the
horizontal plane (see red bullet in Fig. 5). At this point, the impedance is
purely resistive, i.e., its reactance is zero, and should be equal to
$R_{edge}\approx 341~{}\Omega$ (we omit the details of the mathematical model,
which can be found in (Balanis, 2016), to reduce clutter). Then, the optimal
depth of the notch can be computed as:

(9) $\displaystyle
h_{\text{notch}}=\frac{L}{\pi}\cos^{-1}\left(\sqrt{\frac{R}{R_{edge}}}\right)\approx
4.9~{}\text{mm},$

where $R$ is the desired impedance (i.e., $50~{}\Omega$).

We attempted to validate this result with the full-wave simulator (Dassault
Systèmes, 2022) and found that a $3.5$-mm depth, cutting across the patch
antenna as shown in Fig. 5, maximizes performance ($\sim\\!\\!30\%$ difference
with respect to the model). Fig. 7 shows with a blue line the amount of power
that is reflected, estimated by the simulator and referred to as S11 parameter
in antenna design. The result shows good performance at 5.5 GHz, the operating
frequency of choice. Fig. 7 shows the simulated radiation pattern, with
minimal backwards propagation. Note that the antenna is not very directive and
has an expected gain of 1.5dBi, which is common for this type of low-cost
antennas.

To validate the patch design, we printed a sample antenna with the
aforementioned parameters. Using our VNA, we measured the empirical $S_{11}$
and plotted the result with a red line in Fig. 7. Perhaps surprisingly, the
minimal-$S_{11}$ frequency point is 5.3 GHz instead of the intended 5.5 GHz.
After some research, we realized that the offset stems from an error on the
nominal permittivity $\epsilon_{r}=4.3$ used in our model/simulations for the
PCB substrate. After some iterations with our simulator, we estimated the real
permittivity to be $\epsilon_{r}=4.66$. In light of this, we changed the
operating frequency to $f=5.3$ GHz.

From Fig. 7, we can also estimate that the bandwidth of our approach, i.e.,
the range of frequencies where the antenna’s $S_{11}$ is $\leq-10$ dB, is
118MHz (note the black horizontal line). We finally note that, at 5.3 GHz, the
$S_{11}$ is $-41$ dB, which corresponds to a Voltage Standing Wave Ration
(VSWR) of $1.018$. This means that the amount of power from the feeding line
that is reflected back is negligible, which was our goal.

Figure 8. RF switch and microstrips. Output port | 7 | 6 | 5 | 1 | 3 | 2 | 4
---|---|---|---|---|---|---|---
$\varphi$ (deg) | 51.42 | 102.85 | 154.28 | 205.71 | 257.14 | 308.57 | 360
$l$ (mm) | 1.21 | 2.42 | 3.64 | 4.85 | 6.08 | 7.28 | 8.49
Table 1. RF switch’s output ports, the length of the associated delay lines,
and the resulting phase shifts.

###  4.2. Phase shifters

At each cell, a specific phase shift $\varphi$ is applied by _routing_ the RF
signal towards a specific delay line, implemented with a microstrip, that
reflects the signal back to the patch antenna. To this end, we use a 3-bit RF
switch SKY13418-485LF (Skyworks Solutions, Inc, 2019), which has 1 input port
(attached to the feeding line), 3 configuration ports (more later), and 8
output ports connected to delay lines of different lengths. Given one
configuration (input-output port mapping encoded as 3 bits in the
configuration ports), the resulting phase shift follows as:

(10) $\varphi=\frac{360\cdot 2l\cdot f}{c\cdot v_{f}},$

where $l$ denotes the distance travelled from the patch antenna to the end of
the delay line (including the switch and the feeding and delay lines), and
$v_{f}$ is the velocity factor of the microstrip material. Note that $2l$
accounts for the round-trip between antenna and delay line.

To estimate $v_{f}$ empirically, we take advantage of the TDR technique used
earlier, which also measures the time $d_{\mu\text{strip}}$ it takes for a
signal to travel through a microstrip of length $l_{\mu\text{strip}}$. As
shown in Fig. 5, $d_{\mu\text{strip}}=1.51$ ns for a line of
$l_{\mu\text{strip}}=135$ mm, which is sufficiently long to force the signal
to travel at least $2\lambda$ and hence enable highly accurate delay
estimates. We then calculate $v_{f}=\frac{v_{\mu\text{strip}}}{c}=0.298\approx
0.3$, where
$v_{\mu\text{strip}}=\frac{l_{\mu\text{strip}}}{d_{\mu\text{strip}}}$ is the
velocity of the signal through the microstrip.

Given $v_{f}$ and the selected microstrips width derived in §4.1.1 ($0.75$
mm), we can calculate the length of the delay lines corresponding to the phase
shifts that need to be encoded into each output port, as indicated in Table 1.
Note that port 8 has no associated phase shift. Instead, this port connects
with a delay line that ends with a $50\Omega$-resistor, which prevents the
signal to be reflected back. We refer to this configuration as “absorption
state”, which enables us to build _virtual_ RISs of any size and shape.
Alternatively, an energy harvester (Mir et al., 2021) can be used to feed the
MCU and effectively make it self-sustainable. The resulting design is shown in
Fig. 8.

###  4.3. Microcontroller Unit (MCU)

As explained in §2, an MCU is in charge of parametrizing the configuration
ports of the RF switch in each unit cell. We have selected the STM32L071V8T6
MCU from STMicroelectronics (STMicroelectronics, 2021), which is low cost (see
§6), high-speed (configuring 100 cells takes $<35$ ms), and low energy-
consuming ($62$ mW in high-performing mode). We note that we have not
optimized the MCU, which we leave for future work. For instance, we only use
the MCU’s high-performance (high-consuming) mode, although it provides low-
consuming modes too. Exploiting these modes could reduce its energy
consumption to the order of $\mu$W, which is amenable to energy harvesters
(Mir et al., 2021).

(a) $\boldsymbol{\theta=0^{\circ},\phi=0^{\circ}}$.

(b) $\boldsymbol{\theta=80^{\circ},\phi=0}$.

Figure 9. Expected beamforming pattern of a 10$\boldsymbol{\times}$10 RIS.

###  4.4. RIS board

The spacing between unit cells (antennas) has to be carefully designed to
maximize beamforming gains. Roughly speaking, a small spacing increases the
probability of mutual coupling, which decreases the efficiency of each antenna
because of surface waves propagation. Conversely, a large spacing leads to
grating lobes, as we demonstrate in §5. All in all, the inter-cell spacing
depends on the maximum steering angle $\theta_{max}$ of the main lobe, which
is given by $d_{max}=\frac{\lambda}{1+\sin(\theta_{max})}$. Note that, though
$d_{max}=\lambda/2$ maximizes the steering angle range of the array,
$\theta_{max}=90^{\circ}$ is not achievable in practice (Delos et al., 2020).

Prior to developing a prototype, we simulated a $10\times 10$ RIS board with a
regular $10\times 10$ planar antenna array. In our first set of simulations,
each antenna element is fed with equal power from an open-ended transmission
line that induces a phase delay that is optimized offline to maximize power
towards the selected azimuth and elevation angles. Figs. 9(a) and 9(b) show
the expected radiation pattern of two different configurations that maximize
power towards an elevation of $\phi=0^{\circ}$ and, respectively, an azimuth
of $\theta=\\{0^{\circ},80^{\circ}\\}$.

On the one hand, we can observe from Fig. 9(a) that the array can achieve a
narrow beampattern, with a Half Power Beamwidth (HPBW) equal to
$10.1^{\circ}$, with a gain of $25$ dBi in the intended direction, which
halves with a 10-degree offset. No back-radiation is expected. On the other
hand, Fig. 9(b) shows that a large steering angle of $\theta=80^{\circ}$
dissipates half the energy towards the opposite direction and drops the power
of the beam to $19.3$ dBi.

(a) $\boldsymbol{0^{\circ}}$ incidence angle.

(b) $\boldsymbol{30^{\circ}}$ incidence angle.

Figure 10. RCS of a 10$\boldsymbol{\times}$10 RIS illuminated by a plane wave.

(a) RIS board.

(b) Testbed.

Figure 11. (a) 10$\boldsymbol{\times}$10 RIS PCB realization, and (b) testbed
in an anechoic chamber with rotating structure.

Note however that the power radiated by a RIS comes from an external source
that _illuminates_ the surface. In order to simulate this behavior, we
generated a plane wave linearly polarized, and measured the resulting Radar
Cross Section (RCS). The RCS estimates how much of the incident power at every
point of the surface is scattered back to the receiver. Hence, we expect high
RCS values in the direction of the main beam and lower in other directions.
This is indeed confirmed in Figs. 10(a) and 10(b) for an incidence angle of
$0^{\circ}$ and $30^{\circ}$, respectively. In both cases, the RIS is
configured to reflect the received signal perpendicularly to the incidence
angle of the received signal, i.e., the RIS should maximize power in the same
direction of the incidence angle, which is confirmed by both figures with a
RCS approximately equal to $12-13$ dBsm in the intended direction. This
highlights the fact that, in real life scenarios, the angle of arrival must be
known to the controller to maximize beamforming gains.

We manufactured 10 boards of 10x10 unit cells each. All the phase and
selection buses, which connect each unit cell with an MCU, are built with
microstrips. Unit cells are deployed in the PCB layout such that the same
inter-cell distance can be maintained across co-located boards. The remaining
elements described in §2 (flip-flops, resistors, etc.), are standard
components assembled on the PCB. The final printout is shown in Fig. 11(a).

Figure 12. Map of the testbed.

##  5\. Empirical Characterization

We characterized one of our 10x10 RIS boards in an 8m$\times$5m anechoic
chamber. Figs. 11(b) and 12 illustrate our testbed. We mounted the board on a
turntable controlled remotely from a master PC, which is also used to
configure the beamforming parameters of the RIS. We use two software-defined
radio devices attached to horn antennas with gain $G=13.5$ dBi to generate
(“TX”) and receive (“RX”) a continuous stream of OFDM QPSK-modulated symbols
with 5 MHz of bandwidth and numerology that meets 3GPP LTE requirements. The
transmission power of TX is -30 dBm per subcarrier, and we sample the
reference signal received power (RSRP) at RX.

The distances RIS-TX and RIS-RX are $d_{\text{RIS}-\text{TX}}\\!=\\!1.1$ m and
$d_{\text{RIS}-\text{RX}}\\!=\\!6.3$ m, respectively. Considering the size of
the RIS and its operating frequency, it is hard to guarantee that
$d_{\text{RIS}-\text{TX}}$ is larger than the far-field threshold, which is
$2\frac{D^{2}}{\lambda}\\!=\\!6.5$ m (Balanis, 2016), where $D\\!=\\!0.43$ m
is the diagonal of the array. Nevertheless, our choice of
$d_{\text{RIS}-\text{TX}}$ is larger than the _reactive_ near-field threshold,
which is $0.62\sqrt{\frac{D^{3}}{\lambda}}\\!=\\!0.73$ m (Balanis, 2016), and
sufficient for our purposes. As shown in Fig. 12, the rotation of the table
determines the azimuth angle $\theta_{r}$, and the location of TX determines
$\theta_{t}$. Conversely, the elevation angles of RIS-TX and RIS-RX are fixed
to $\phi_{t}\\!=\\!33^{\circ}$ and $\phi_{r}\\!=\\!-3^{\circ}$, respectively.

###  5.1. Codebook characterization

Figure 13. Examples of beampatterns for $\boldsymbol{\theta_{t}=0^{\circ}}$.

We begin our experimental campaign by characterizing the codebook generated in
§3. To this end, we test out all the configurations
$\mathcal{V}=\\{\mathbf{v}_{n}\\}_{n=1}^{N_{\mathrm{B}}=1891}$ for a wide
range of $\theta_{r}=[-90^{\circ},90^{\circ}]$ and for
$\theta_{t}=\\{20^{\circ},90^{\circ}\\}$.

For our empirical results, the first observation is that the direction
$(\theta_{n},\phi_{n})$ of the main lobe points towards
$(\theta_{r}-\theta_{t},\phi_{r}-\phi_{t})$, as intended, for all
configurations $\mathbf{v}_{n}\in\mathcal{V}$. These results, hence, validate
our prototype for practically all configurations in $\mathcal{V}$. Figs. 13
and 14 depict some representative configurations
$\mathbf{v}_{n}\in\mathcal{V}$ for both $\theta_{t}$ settings, respectively.
These figures show that the main beam points towards the intended directions.
We note, however, that the gain of the main lobe is penalized when we use
large steering angles (see, e.g., $\theta=60^{\circ}$ in both figures), which
is expected (Delos et al., 2020). Overall, the power received in the intended
direction ranges between $-74$ dBm (for large steering angles) and $-64$ dBm
(for smaller angles), which give us remarkable beamforming gains between
$\sim\\!17$ dB and $\sim\\!27$ dB over the noise floor.

Using the radar range equation in (Balanis, 2016), the peak RCS can be
calculated as
$64\pi^{3}\frac{P_{RX}}{P_{TX}}\cdot(\frac{d_{\text{RIS}-\text{TX}}\cdot
d_{\text{RIS}-\text{RX}}}{\lambda\cdot G})^{2}=11.2$ dBm2. Moreover, the HPBW
is in average around $10^{\circ}$ for all beampatterns. Both results are in
line with our simulations in §4.4.

Figure 14. Examples of beampatterns for $\boldsymbol{\theta_{t}=20^{\circ}}$.

###  5.2. Scalability analysis

(a) 2x2

(b) 4x4

(c) 8x8

(d) 10x10

(e) off2

(f) off3

Figure 15. Different unit cell activation patterns in a 10x10 RIS board.
Colored squares represent deactivated (in black) and activated (in white)
cells.

To assess scalability, we analyze the beamforming gain of our RIS for a
variable number of unit cells. To this goal, we take advantage of the
absorption state available in our design for each cell, and produce virtual
RISs with different sizes by setting the activation patterns shown in Figs.
15(a) to 15(d). For every virtual RIS, we re-optimized its codebook
$\mathcal{V}_{N}$ to account for its effective size and inter-cell spacing.

To ease the analysis, we now fix $\theta_{r}=\theta_{t}=0^{\circ}$ and measure
the power received for every configuration $\mathbf{v}\in\mathcal{V}_{N}$. The
results are shown in Fig. 16, which represent the measured power with a color
range for every combination of $\theta$ (x-axis) and $\phi$ (y-axis) from
$\mathcal{V}_{N}$, and for $N=\\{4,8,64,100\\}$. From these plots, we can
observe how the main beam becomes sharper and carries more power as we
increase $N$. With $N=4$ (top left plot), no configuration produces a
distinguishable beam, which renders a 2x2 RIS ineffective. For the rest, we
note a growing amount of power in the intended direction, respectively, equal
to $-81.8$ dBm ($N=16$), $-71.5$ dBm ($N=64$), and $-66.5$ dBm (all cells are
activated). This behavior is expected: Fig. 19 depicts in red the power
observed at RX with the optimal configuration $\mathbf{v}$ as a function of
$N$, and compares that with the mathematical model in (Tang et al., 2021) (see
eq. (5) therein) represented in blue. Both results are remarkably close to
each other, which validates the ability of our approach to effectively create
virtual surfaces with different shapes.

Figure 16. Received power over the respective optimal codebook
$\boldsymbol{\mathcal{V}_{N}}$ for virtual RISs with different sizes
(activation patterns of Figs. 15(a)-d, respectively).
$\boldsymbol{\theta_{r}=\theta_{t}=0^{\circ}}$.

###  5.3. Other activation patterns

(a) $\boldsymbol{d_{max}=\lambda}$.

(b) $\boldsymbol{d_{max}=1.5\lambda}$.

Figure 17. Received power over the respective optimal codebook
$\boldsymbol{\mathcal{V}_{d_{max}}}$ for virtual RISs generating grating lobes
(activation patterns of Figs. 15(e)-f, respectively). Red dots indicate the
expected location of the lobes.
$\boldsymbol{\theta_{r}=\theta_{t}=0^{\circ}}$.

To conclude our characterization, we study the performance of our RIS
prototype when the inter-cell distance differs from $d_{max}=\lambda/2$ (shown
in Fig. 16 for $N=100$). Like before, we calculated new optimized codebooks
$\mathcal{V}_{d_{max}}$ for $d_{max}=\\{\lambda,1.5\lambda\\}$, and plot in
Fig. 17 the power received at RX for each configuration
$\mathbf{v}\in\mathcal{V}_{d_{max}}$. We do this for both activation patterns
depicted in Fig. 15(e) (“off2”) and 15(f) (“off3”), for $d_{max}=\lambda$ and
$d_{max}=1.5\lambda$, respectively.

By changing $d_{max}$, we also change the density of active cells per board,
$N=25$ for $d_{max}=\lambda$ (Fig. 17(a)) and $N=16$ for $d_{max}=1.5\lambda$
(Fig. 17(b)). As shown earlier, this has a cost in terms of beamforming gains
that is evidenced also in Fig. 17: the maximum power is $-80.4$ dBm and
$-84.2$ dBm for the two cases, respectively. Both plots reveal the presence of
grating lobes, which are symmetrical beams that are denser for larger
$d_{max}$ values. These effects are well understood in the literature of
antenna design and their distance can be estimated using our model in §3. The
figure depicts with red circles the expected location of these lobes, which
match our measurements remarkably well. This further validates our design to
effectively modify the shape of the RIS to the requirements of any given use
case.

##  6\. Reproducibility

Figure 18. Beamforming gains as a function of the total number of antennas
$N$.

Figure 19. Prototype costs over manufacturing scale.

To conclude our paper, we provide some final remarks that shall help
researchers in the RIS domain build on our results (dataset) and/or reproduce
our RIS prototype.

For starters, we publicly release the dataset111We will publish the dataset
upon the paper acceptance. we have generated during our empirical
characterization in §5, which aggregates a total number of $6.8\cdot 10^{6}$
power samples. This data can help other researchers study RIS-related problems
without the need of building a prototype.

Next, we report the costs associated for building our prototype. For this
purpose, we have used prices publicly available. There are three sources of
cost to manufacture each board: ($i$) PCB production (including the patch
antenna and the microstrips for the delay lines and the buses); ($ii$)
additional electronic components (including RF switches, MCU, resistors, flip-
flops, AND gates, etc.); and ($iii$) assembly all components onto the PCB.

According to PCBWay (PCBWay, [n. d.]), producing 10 PCB boards, as specified
in our design, costs $0.22 per unit cell; but it drops to $0.11 and $0.09 when
the production scales up by 20x and 100x, respectively. Concerning additional
electronic components, each 10x10 board bears 300 flip-flops, 100 AND gates,
100 RF switches, and 1 MCU. According to Digikey.de, the cost boils down to
$1.88 per unit cell when purchasing a batch of 1000 of such boards. The
assembly process, according to PCBWay, scales down from $0.51 to $0.05 per
cell when we scale up the number of boards from 10 to 1000 10x10-RISs. Fig. 19
depicts how these costs (normalized per unit cell) evolve with the
manufacturing scale.

##  7\. Conclusion

Reconfigurable Intelligent Surfaces (RISs) can become a major densification
solution for future mobile systems. We designed, prototyped and characterized
an RF switch-based RIS that met our defined requirements in terms of RF
steering, reconfigurability, energy efficiency, and cost. Throughout the
paper, we reported on our learnings along with the related experiments. These
learnings ranged from our experience bridging theory and empirical findings
(e.g., unexpected sensitivity to certain model parameters) to practical
considerations (e.g., costs and hardware constrains). As a result, our
contributions can be used by the research community to: ($i$) build realistic
RF switch-based RIS models, ($ii$) leverage on the dataset provided to further
study related challenges, ($iii$) reproduce our RIS prototype for research
purposes, and ($iv$) estimate the deployment costs at scale.

## References

  * (1)
  * Albanese et al. (2022) Antonio Albanese et al. 2022\.  MARISA: A Self-configuring Metasurfaces Absorption and Reflection Solution Towards 6G. In _IEEE INFOCOM 2022 - IEEE Conference on Computer Communications_.
  * Analog Devices (2009) Analog Devices. 2009\.  Microstrip and Stripline Design: MT-094 tutorial.  https://www.analog.com/media/en/training-seminars/tutorials/MT-094.pdf. (2009).  Accessed: 2022-03-01.
  * Arun and Balakrishnan (2020) Venkat Arun and Hari Balakrishnan. 2020.  RFocus: Beamforming using thousands of passive antennas. In _USENIX NSDI_.
  * Balanis (2016) Constantine A Balanis. 2016\.  _Antenna theory: analysis and design_.  John wiley & sons.
  * Basar et al. (2019) Ertugrul Basar et al. 2019\.  Wireless communications through reconfigurable intelligent surfaces.  _IEEE access_ 7 (2019), 116753–116773.
  * Dai et al. (2020) Linglong Dai et al. 2020\.  Reconfigurable intelligent surface-based wireless communications: Antenna design, prototyping, and experimental results.  _IEEE Access_ 8 (2020), 45913–45923.
  * Dassault Systèmes (2022) Dassault Systèmes. 2022\.  CST Studio Suite: Electromagnetic field simulation software.  https://www.3ds.com/products-services/simulia/products/cst-studio-suite/. (2022).  Accessed: 2022-03-01.
  * Delos et al. (2020) Peter Delos, Bob Broughton, and Jon Kraft. 2020\.  Phased Array Antenna Patterns—Part 2: Grating Lobes and Beam Squint.  _26 Phased Array Antenna Patterns—Series_ (2020), 39.
  * Dunna et al. (2020) Manideep Dunna et al. 2020\.  ScatterMIMO: Enabling virtual MIMO with smart surfaces. In _ACM MobiCom_.
  * Fara et al. (2021) Romain Fara et al. 2021\.  A Prototype of Reconfigurable Intelligent Surface with Continuous Control of the Reflection Phase.  _arXiv preprint arXiv:2105.11862_ (2021).
  * Hu et al. (2020) Jingzhi Hu et al. 2020\.  Reconfigurable Intelligent Surface Based RF Sensing: Design, Optimization, and Implementation.  _IEEE Journal on Selected Areas in Communications_ 38, 11 (2020), 2700–2716.
  * Mir et al. (2021) Muhammad Sarmad Mir et al. 2021\.  PassiveLiFi: rethinking LiFi for low-power and long range RF backscatter. In _ACM MobiCom_.
  * Mursia et al. (2021) Placido Mursia et al. 2021\.  RISMA: Reconfigurable Intelligent Surfaces Enabling Beamforming for IoT Massive Access.  _IEEE Journal on Selected Areas in Communications_ 39, 4 (2021), 1072–1085.
  * Pan et al. (2021) Cunhua Pan et al. 2021\.  Reconfigurable Intelligent Surfaces for 6G Systems: Principles, Applications, and Research Directions.  _IEEE Communications Magazine_ 59, 6 (2021), 14–20.
  * Pandey (2019) Anil Pandey. 2019\.  _Practical microstrip and printed antenna design_.  Artech House.
  * PCBWay ([n. d.]) PCBWay. [n. d.].  PCB prototype and fabrication.  https://www.pcbway.com. ([n. d.]).
  * Skyworks Solutions, Inc (2019) Skyworks Solutions, Inc. 2019\.  SKY13418-485LF: 0.1 to 6.0 GHz SP8T Antenna Switch.  https://www.skyworksinc.com/-/media/SkyWorks/Documents/Products/701-800/SKY13418_485LF_201712F.pdf. (2019).  Accessed: 2022-03-01.
  * STMicroelectronics (2021) STMicroelectronics. 2021\.  STM32L071V8: Ultra-low-power Arm Cortex-M0+ MCU with 64-Kbytes of Flash memory, 32 MHz CPU .  https://www.st.com/en/microcontrollers-microprocessors/stm32l071v8.html. (2021).  Accessed: 2022-03-01.
  * Tan et al. (2018) Xin Tan, Zhi Sun, Dimitrios Koutsonikolas, and Josep M Jornet. 2018.  Enabling indoor mobile millimeter-wave networks based on smart reflect-arrays. In _IEEE INFOCOM 2018-IEEE Conference on Computer Communications_. IEEE, 270–278.
  * Tang et al. (2021) Wankai Tang et al. 2021\.  Wireless Communications With Reconfigurable Intelligent Surface: Path Loss Modeling and Experimental Measurement.  _IEEE Transactions on Wireless Communications_ 20, 1 (2021), 421–439.  https://doi.org/10.1109/TWC.2020.3024887
  * Trichopoulos et al. (2021) Georgios Trichopoulos et al. 2021\.  Design and Evaluation of Reconfigurable Intelligent Surfaces in Real-World Environment.  _arXiv preprint arXiv:2109.07763_ (2021).
  * Wu et al. (2018) Qingqing Wu et al. 2018\.  Intelligent Reflecting Surface Enhanced Wireless Network: Joint Active and Passive Beamforming Design.  _IEEE Transactions on Wireless Communications_ 18, 11 (2018), 5394–5409.
  * Yang et al. (2022) Zhaohui Yang et al. 2022\.  Energy-Efficient Wireless Communications With Distributed Reconfigurable Intelligent Surfaces.  _IEEE Transactions on Wireless Communications_ 21, 1 (2022), 665–679.
  * Yezhen et al. (2020) Li Yezhen et al. 2020\.  A Novel 28 GHz Phased Array Antenna for 5G Mobile Communications.  _ZTE Communications_ 18, 3 (2020), 20–25.