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Abstract and Figures

A Reconfigurable Intelligent Surface (RIS) redirects and possibly modifies the properties of incident waves, with the aim to restore non-line-of-sight communication links. Composed of elementary scatterers, the RIS has been so far treated as a collection of point scatterers with properties similar to antennas in an equivalent massive MIMO communication link. Despite the discrete nature of the RIS, current design approaches often treat the RIS as a continuous radiating surface, which is subsequently discretized. Here we investigate the connection between the two approaches in an attempt to bridge the two seemingly opposite perspectives. We analytically find the factor that renders the two approaches equivalent and we demonstrate our findings with examples of RIS elements modeled as antennas with commonly used radiation patterns and properties consistent with antenna theory. The equivalence between the two theoretical approaches is analyzed with respect to design aspects of the RIS elements, such as gain and directivity, with the aim to provide insight into the observed discrepancies, the understanding of which is crucial for assessing the RIS efficiency.
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Reconfigurable Intelligent Surface: MIMO or radiating sheet?
Sotiris Droulias1, and Angeliki Alexiou1
1Department of Digital Systems, University of Piraeus, Piraeus 18534, Greece.
A Reconfigurable Intelligent Surface (RIS) redirects and possibly modifies the properties of inci-
dent waves, with the aim to restore non-line-of-sight communication links. Composed of elementary
scatterers, the RIS has been so far treated as a collection of point scatterers with properties similar
to antennas in an equivalent massive MIMO communication link. Despite the discrete nature of the
RIS, current design approaches often treat the RIS as a continuous radiating surface, which is sub-
sequently discretized. Here we investigate the connection between the two approaches in an attempt
to bridge the two seemingly opposite perspectives. We analytically find the factor that renders the
two approaches equivalent and we demonstrate our findings with examples of RIS elements modeled
as antennas with commonly used radiation patterns and properties consistent with antenna theory.
The equivalence between the two theoretical approaches is analyzed with respect to design aspects
of the RIS elements, such as gain and directivity, with the aim to provide insight into the observed
discrepancies, the understanding of which is crucial for assessing the RIS efficiency.
I. INTRODUCTION
The primary role of a Reconfigurable Intelligent Surface
(RIS) is to mediate a non-line-of-sight link by redirecting
the incident beam from the transmitter to the receiver
and possibly modifying its characteristics, in order to op-
timize the beamforming efficiency and to maximize the
signal power at the receiver. Its operation is similar to
that of a mirror, however the reflection is not limited to
specular; depending on the properties of the RIS elements
(periodically distributed scatterers forming the RIS sur-
face), the incident beam can be redirected in a control-
lable manner [1, 2]. In recent years, there has been a con-
siderable effort to incorporate the functionalities offered
by RISs in millimeter-wave (mmWave) (30-100 GHz) and
terahertz (THz) band (0.1-10 THz) communications [3–
5].
So far, several techniques have been theoretically pro-
posed for the design of the desired RIS properties [6–20]
and relevant experiments have been performed in order
to verify the predicted RIS performance [12–15, 21–27].
To achieve the desired wave manipulation, the design in-
volves the determination of the appropriate surface prop-
erties, such as surface impedance (or effective electric
and magnetic surface conductivities). Usually the RIS
is treated as a continuous surface, i.e. as a radiating
sheet that locally satisfies the boundary conditions, en-
suring that an incident plane wave is reflected towards
the desired direction. The solution, which is exact for
surfaces of infinite extent, leads to the prescription of a
continuous local wave impedance at the RIS surface. Ide-
ally, a continuous surface of finite extent characterized by
the prescribed impedance will steer the incident wave to-
wards the desired direction, however for the sake of prac-
tical implementation the surface must be discretized, in
essence rendering the continuous surface a collection of
discrete scatterers.
sdroulias@unipi.gr
Alternatively, due to the inherently discrete nature
of the RIS elements, it is more natural to treat the RIS
as a planar distribution of scatterers. The analysis can
be simplified if the RIS elements are considered as point
scatterers that bear the properties of conventional an-
tennas, such as gain and directivity [28–33]. As with an-
tennas, the local currents that induce radiation depend
on the particular design of the RIS elements. However,
while actual antennas are fed directly by external cur-
rents, here the currents are excited by external waves; the
local phase and amplitude of the incident waves drive the
phase and amplitude, respectively, of the locally excited
currents in the RIS elements. By controlling the proper-
ties of the RIS elements (e.g., by incorporating tunable
resistive and reactive elements [21–27]), the amplitude
and phase of the current oscillation at each individual
scatterer can be tuned, similarly to how phased arrays
are controlled. As a result, the entire distribution of
scatterers emits radiation with prescribed amplitude and
phase, essentially re-radiating the incident wave towards
the desired direction, which is macroscopically perceived
as the RIS steering the incident beam.
FIG. 1. Schematic representation of a RIS steering an inci-
dent wave to a prescribed direction. The RIS can be analyzed
as a homogeneous radiating sheet (top) or, equivalently, as a
distribution of point scatterers (bottom) that bear the prop-
erties of conventional antennas.
arXiv:submit/4312376 [physics.app-ph] 18 May 2022
2
On the one hand, because the former of the two ap-
proaches involves a continuous radiating sheet of infinite
extent that is subsequently discretized, the question is
how well the local RIS elements retain the properties pre-
scribed by the continuous infinite radiating sheet upon
discretization and what is the effect of the finite RIS
size. On the other hand, because the latter of the two
approaches involves the collective response of individual
scatterers, therefore by definition implementing a finite-
sized RIS, the question is how well the properties of the
individual antenna-like scatterers can reproduce the ac-
tual field radiated from the RIS.
In this work we demonstrate the equivalence between
the two approaches in an attempt to bridge the two seem-
ingly opposite perspectives. We analyze each approach
separately and we find the connection between the two,
on the basis that they must both lead to the same scat-
tered wave. We investigate how the properties of the
continuous sheet are related to the properties of point
scatterers that bear characteristics consistent with the
antenna theory, and we discuss commonly used models in
recent theoretical works. We find that, overall, the treat-
ment of the RIS as point scatterers may overestimate the
scattered field and, therefore, a correction factor must be
taken into account. By analyzing the correction factor
by means of realistic radiation patterns, we demonstrate
how the RIS performance is affected by design properties
of the RIS elements, such as gain and directivity, and we
discuss the implications on the predicted power at the
receiver.
II. DISCRETE VS CONTINUOUS APPROACH
In this section we briefly summarize the treatment of the
RIS under both approaches and we derive the received
power, which is the relevant quantity in RIS-aided links.
A. RIS as MIMO
Let us start with the configuration shown in Fig. 2. The
RIS is replaced by a distribution of M×Npoint scat-
terers that bear the properties of conventional antennas
and are periodically distributed on the xy-plane with
periodicity lxand lyalong the xand ydirections, re-
spectively. The RIS can be perceived as having M×N
inputs (M×Nelements being externally excited) and
M×Noutputs (the same M×Nelements re-radiating),
i.e. the RIS operation can be described in terms of a
matrix A= (ap,q)C(M×N)×(M×N), similarly to how
antennas are described in massive MIMO links:
Er
p=
M×N
X
q=1
ap,qEt
q,(1)
where Et
qis the field amplitude at the position of the qth
element from the transmitter (t) or access point (AP),
FIG. 2. RIS as MIMO. The RIS is replaced by a distribution
of M×Npoint scatterers (here shown 4 ×5 = 20 elements)
periodically arranged on the xy-plane with periodicity lxand
lyalong the xand yaxis, respectively. Each scatterer is char-
acterized by the same radiation pattern, U0, and the angles
θt,r
m,n and φt,r
m,n denote the elevation and azimuth angle, re-
spectively, from the (m, n) element to the transmitter (t) or
receiver (r) antenna, which is located at distance dt,r
m,n from
the (m, n) element.
and Er
pis the field amplitude at the receiver (r) or user
equipment (UE) from the pth element. The off-diagonal
elements of matrix Aaccount for possible coupling be-
tween the RIS elements, i.e. when the excited fields in
one element affect the local currents of neighboring ele-
ments, particularly if the RIS elements are closely spaced.
In the absence of such coupling the matrix Abecomes
diagonal, and all elements ap,q with p=qcan be or-
ganized in a new matrix B= (bm,n)CM×N, so that
their position on the matrix can be essentially associated
with their geometric location on the RIS, as illustrated
in Fig. 2. Using matrix Bwe may simply write:
Er
m,n =bm,nEt
m,n,(2)
where bm,n is a complex parameter relating the field in-
cident to element (m, n) to the field scattered from the
same element, and depends on the design of the RIS el-
ement. Assuming that all elements are identical, having
the same normalized radiation pattern U0and, hence,
common aperture AU C and gain GU C , and that the re-
ceiver antenna is at distance dr
m,n from the (m, n) RIS
element, the general form of the parameter bm,n can be
expressed as (see Appendix for derivation):
bm,n =Rm,nqAU C Ut
UC GU C Ur
UC
eikdr
m,n
4πdr
m,n
,(3)
where k= 2π/λ (λis the free-space wavelength) and
Ut,r
UC U0(θt,r
m,n, φt,r
m,n), with θm,n , φm,n denoting the el-
evation and azimuth angle from the (m, n) element to
3
the transmitter (t) and to the receiver (r), as illustrated
in Fig. 2. Rm,n is a complex coefficient, with |Rm,n|ac-
counting for power loss (|Rm,n|21) and arg(Rm,n) for
the excitation phase of the (m, n) element.
The total field at the receiver, Er, is simply given by
the sum of the field contributions from all elements and,
hence, the received power Pris expressed via the receiver
aperture, Ar, and the power density, Sr=|Er|2/2Z0, as:
Pr=ArSr=Ar
2Z0X
mX
n
Er
m,n
2
=
Ar
2Z0X
mX
n
Rm,nqAU C Ut
UC GU C Ur
UC Et
m,n
eikdr
m,n
4πdr
m,n
2
,
(4)
where Z0is the characteristic impedance of air, Ar=
Grλ2/4π, and Gris the antenna gain of the receiver.
By properly assigning the magnitude and phase of
Et
m,n, different RIS illumination conditions can be stud-
ied, e.g. illumination by plane waves, spherical waves
[30–33] (see Appendix for details) or beams [34, 35].
B. RIS as radiating sheet
When the RIS is considered as a radiating sheet, the
starting point is to replace the actual RIS with an electri-
cally thin surface of infinite extent, which carries electric
and magnetic currents that circulate in such a manner
so that the sheet redirects incident plane waves similarly
to how the RIS does. The directions of incidence and
reflection are defined by the wavevectors kiand kr, re-
spectively, which are expressed with respect to the eleva-
tion (θ) and azimuth (φ) angles of incidence (subscript
0i0) and reflection (subscript 0r0) as:
ki=k(sin θicos φiˆ
x+ sin θisin φiˆ
ycos θiˆ
z),(5a)
kr=k(sin θrcos φrˆ
x+ sin θrsin φrˆ
y+ cos θrˆ
z),(5b)
For simplicity we will consider plane waves traveling on
the xz-plane, i.e. with φi=π,φr= 0, and hence the
above wavevectors become:
ki=k(sin θixcos θiz),(6a)
kr=k(sin θrx+ cos θrz).(6b)
The incident polarization can be always expressed as a
superposition of TE (components Hx, Ey, Hz) and TM
(components Ex, Hy, Ez) polarizations, hence here we
will examine TE waves and the analysis for TM waves
follows similar steps (see Appendix for details).
FIG. 3. RIS as radiating sheet. The RIS is a continuous
surface of size Lx×Ly, which steers incident plane waves.
The wavevectors kiand krcorrespond to the incident (i) and
reflected (r) wave and are characterized by the elevation and
azimuth angles θi, φiand θr, φrrespectively.
1. Formulation for infinite sheet
For TE-polarization the incident wave is written as:
Ei(r) = Eieikirˆ
y,(7a)
Hi(r) = Ei
Z0
eikir(cos θiˆ
x+ sin θiˆ
z),(7b)
where r=xˆ
x+yˆ
y+zˆ
zis the observation vector and the
triplet (Ei,Hi,ki) forms a right-handed orthogonal sys-
tem, in compliance with Maxwell’s equations. Assuming
that the sheet preserves the incident polarization, the
reflected wave is also TE-polarized and is similarly ex-
pressed as:
Er(r) = Ereikrrˆ
yΓ0Eieikrrˆ
y,(8a)
Hr(r) = Γ0Ei
Z0
eikrr(cos θrˆ
x+ sin θrˆ
z),(8b)
where Γ0is a complex constant associating the incident
with the reflected wave amplitude as Er= Γ0Ei. The
presence of the reflecting sheet does not allow for waves
transmitted in the z < 0 region and, hence, the boundary
conditions that the fields must satisfy at z= 0 are written
as:
ˆ
z×(Ei+Er)|z=0 = +jsm,(9a)
ˆ
z×(Hi+Hr)|z=0 =jse.(9b)
where jse,jsm are the surface (sheet) electric and mag-
netic current, respectively. The latter are given by:
jse =σseEloc ,(10a)
jsm =σsmHloc ,(10b)
4
where σse,σsm , are the electric and magnetic sheet con-
ductivity, respectively, accounting for the effective mate-
rial properties of the RIS, and Eloc = (Ei+Er)/2 and
Hloc = (Hi+Hr)/2, are the local fields, i.e. the fields at
z= 0. The solution of the boundary conditions imposes
2se =σsm/2Zs, where Zsis commonly referred to
as the surface impedance, and is given by:
Zs(x) = Z0
1+Γ0eik(sin θisin θr)x
cos θicos θrΓ0eik(sin θisin θr)x(11)
The surface impedance in Eq.(11) involves the parameter:
Γs(x)Γ0eik(sin θisin θr)x(12)
which is usually termed as surface reflection coefficient.
While the result of Eq.(11) guarantees perfect transfor-
mation of a single incident plane wave to a single reflected
plane wave, deviations in experimentally realized Zsand
possibly the demand for incorporation of active elements
may complicate practical implementations [36]. To facil-
itate fabrication and to optimize the RIS performance,
alternative techniques have been proposed, generalizing
the form of Zsand, therefore, of Γs[10–17].
2. Formulation for finite-size sheet
The surface impedance found for the infinite sheet can
be also used for finite-sized sheets, which are more rel-
evant to actual RISs. However, due to the finite size of
the surface, an incident plane wave is not reflected into
just a single plane wave and, to determine the scattered
field, the contribution of the surface currents must be in-
tegrated across the RIS surface.
Let us consider the radiating sheet of size Lx×Ly
shown in Fig. 3, the observation vector r=xˆ
x+yˆ
y+zˆ
z,
and a vector r0=x0ˆ
x+y0ˆ
y, which is confined on the sheet
surface S. If the surface currents jse,jsm are known, then
the fields reflected from the sheet at rcan be written as
[36, 37]:
Er(r) = 1
iωµ (∇ × ∇ × Ase )1
∇ × Asm,(13a)
Hr(r) = 1
iωµ (∇ × ∇ × Asm ) + 1
µ∇ × Ase,(13b)
where
Ase =ZZS
µjseG(rr0)dr0,(14a)
Asm =ZZS
jsmG(rr0)dr0,(14b)
and
G(rr0) = eik |rr0|
4π|rr0|,(15)
is the Green’s function for the Helmholtz equation. For
surfaces that are significantly larger than λ, so that the
edge effects can be neglected, as is usually the case in
practical situations, we may use the field equivalence
principle [36, 37] and replace the surface currents with:
jse =ˆ
z×Hs(r0),(16a)
jsm =ˆ
z×Es(r0),(16b)
where Es(r0), Hs(r0) are the reflected fields at z= 0.
For |r|  λ, we may approximate ∇ × (jsG(rr0))
ikG(rr0)(ˆ
rˆ
r0)×js, and, for TE-polarization, the
reflected fields from the finite-sized sheet can be explicitly
expressed everywhere in space, in terms of the local fields
at the RIS surface, as:
Er(r) = ik ZZS
Es(r0)G(rr0)Ve(rr0)dr0,(17a)
Hr(r) = ik
Z0ZZS
Es(r0)G(rr0)Vm(rr0)dr0,(17b)
where
Es(r0)=Γs(r0)Eieikir0,(18)
and the vectors Veand Vmare explicitly written as:
Ve=
xx0
|rr0|
yy0
|rr0|cos θr
((xx0)2
|rr0|2+z2
|rr0|2) cos θrz
|rr0|
yy0
|rr0|
z
|rr0|cos θr+yy0
|rr0|
,(19a)
Vm=
(yy0)2
|rr0|2+z2
|rr0|2+z
|rr0|cos θr
xx0
|rr0|
yy0
|rr0|
xx0
|rr0|
z
|rr0|xx0
|rr0|cos θr
.(19b)
The power density at ris given by the Poynting vec-
tor, i.e. Sr(r) = 1
2Re(Er×H
r), and involves the
calculation of the integrals in Eq.(17). In the far-field,
i.e. approximately after distance 2 max(L2
x, L2
y), where
Ve(rr0)Ve(r), Vm(rr0)Vm(r), the Poynting
vector takes the simple form:
Sr(r) = k2
2Z0
Θ(r)ZZS
Es(r0)G(rr0)dr0
2
ˆ
r,(20)
where
Θ(r)≡ |Ve×Vm|
=y2
r2+z2
r2+ 2z
rcos θr+x2
r2+z2
r2cos θr2(21)
and ˆ
r=r/r is the unit vector (r=|r|=px2+y2+z2).
In spherical coordinates the observation vector is written
as r=rsin θcos φˆ
x+rsin θsin φˆ
y+rcos θˆ
z(θ: elevation
angle, φ: azimuth angle), and Θ becomes:
Θ(θ, φ) = sin φ2(1 + cos θcos θr)2
+ cos φ2(cos θ+ cos θr)2.(22)
5
The power at the observation point that is captured by
a receiver with antenna aperture Ar, is then given by
Pr=ArSr(r)·ˆ
r, or:
Pr(r) = Ar
k2
2Z0
Θ(θ, φ)ZZS
Es(r0)G(rr0)dr0
2
.(23)
To derive Eq.(23) we made no assumptions on the
amplitude and phase of Esand, therefore, the received
power can be calculated for any illumination conditions.
For plane wave illumination, in particular, we may use
Eq.(18), i.e. Es(x0)=Γs(x0)Eieik sin θix0, with Γs(x0) =
Γ0eik(sin θisin θr)x0. Then, the received power yields the
following analytical expression:
Pr=Ar
k2
2Z0
Θ(θ, φ)|Γ0Ei|2LxLy
4πr 2
×
sin kLx
2(sin θcos φsin θr)
kLx
2(sin θcos φsin θr)
sin kLy
2sin θsin φ
kLy
2sin θsin φ
2
.
(24)
III. RELATION BETWEEN MODELS
Let us consider a RIS consisting of M×Nelements,
periodically arranged with periodicity lxand lyalong
the xand ydirections, respectively. Under the MIMO
approach, the RIS is a planar distribution of M×N
point scatterers, while under the sheet approach, the
RIS is a continuous rectangular surface with dimensions
Lx=M×lxand Ly=N×ly.
To find a correspondence between the MIMO and sheet
parameters, we need to spatially quantize the continuous
properties of the sheet, and we therefore start with di-
viding the continuous sheet into Mrows and Ncolumns.
As a result, the surface of total area Lx×Lyis replaced
by M×Nsmall rectangles of area lx×lyeach, and the
integral involved in Eq.(23) is discretized. Hence, the
received power is expressed by a sum over all rectangles:
Pr(r) = Ar
k2
2Z0
Θ(θ, φ)
×X
mX
nZZSmn
Es(r0)G(rr0)dr0
2
,(25)
where m= 1 . . . M ,n= 1 . . . N , and Smn denotes the
area of the (m, n) rectangle.
Equation (25) reproduces the result of Eq.(23) exactly,
regardless of the size of the rectangles, because the inte-
gration is still performed across the entire sheet area.
The question is whether we can replace the integrals on
each rectangle with a quantity that accurately reproduces
the same result; this will render the sheet a discrete grid,
which can be directly associated with the (inherently dis-
crete) MIMO model.
FIG. 4. Connection between MIMO and sheet model. The
continuous radiating sheet of size Lx×Lyis divided in M×
Nsmall rectangles of size lx×ly, and the total field at the
UE is expressed as a discrete sum of all contributions from
each continuous rectangular element. Vectors rand r0are
the position vectors of the UE and of point P0on the surface,
respectively, in the global coordinate system xyz. Point P0
belongs to the (m, n) element with local coordinate system
x0y0z0on which its position is expressed via r0
mn; hence, the
vector defined by point P0and the UE is rr0=rmn r0
mn.
Let us define a Cartesian grid formed by the centers
of all small rectangles. The continuous coordinates that
enter the integration are quantized, with xm=m×lxand
yn=n×lydenoting the discrete position of the (m, n)
RIS element on the Cartesian grid. The position of point
P0on the sheet defined by vector r0, can be also expressed
on the local coordinate system of the (m, n) rectangle, in
which P0is located, by the vector r0
m,n, as shown in Fig. 4.
The observation point defined by rcan be represented
on the local coordinate system of the (m, n) element by
rm,n and, hence, noting that rr0=rm,n r0
m,n, we
may now transfer the integration at the local coordinate
system of each rectangle. Additionally, we may simplify
the integration in Eq.(25) if we assume sufficiently small
rectangles; because the far-field of each rectangle starts
approximately after distance 2max(l2
x, l2
y), for rectan-
gle side in the order of λ/2, the far-field is in the order
of λand we may approximate:
G(|rm,n r0
m,n|) = eik |rm,nr0
m,n|
4π|rm,n r0
m,n|'eikrm,n
4πrm,n
eikr0
m,n ,
(26)
where k=kˆ
r. We may now perform the integration
locally on each rectangle Smn. Using the discrete values
of Γ and Eiat the grid points,
Γmn =|Γ(xm, yn)|eik(sin θisin θr)xm,(27a)
Emn =Eieik sin θixm,(27b)
6
respectively, we find that the received power at the ob-
servation point is:
Pr=Ar
k2
2Z0
Θ(θ, φ)C2X
mX
n
ΓmnEmn
eikrmn
4πrmn
lxly
2
,
(28)
where
C=sin klx
2(sin θcos φsin θr)
klx
2(sin θcos φsin θr)
sin kly
2sin θsin φ
kly
2sin θsin φ.(29)
For plane wave illumination, Eq.(28) reproduces exactly
the power predicted by Eq.(24) for point scatterers with
|Γmn|= Γ0, regardless of the discretization density. This
is guaranteed by the analytically calculated factor C
given by Eq.(29), which results from the condition that
Eshas constant magnitude, Ei, across each rectangle.
1. Correspondence between MIMO and sheet parameters
To find an equivalence between the MIMO and sheet
model, the parameters in Eq.(4) and Eq.(28) must be
such that, under the same illumination conditions, the
received power is the same for all observation points. Of-
ten, it is assumed that Rmn Γmn and AU C lxly
[31, 32]. As a result, the properties of the small radiat-
ing rectangles of the sheet model are associated with the
properties of the MIMO elements via:
Rm,n Γmn (30a)
AUC lxly,(30b)
GUC =4π
λ2AUC 4π
λ2lxly,(30c)
Ut
UC = 1,(30d)
Ur
UC Θ(θ , φ)
4C2.(30e)
With Eqs.(30), Eqs.(4) and (28) become identical, en-
suring the equivalence between the two models.
To verify the equivalence, as an example, let us con-
sider a RIS of size 10 cm ×10 cm, operating at 150
GHz (λ= 2 mm) and consisting of 250 ×250 elements
with periodicity λ/5 along both xand ydirections. A
y-polarized plane wave propagating on the xz-plane with
Ei= 1 V/m illuminates the RIS at incidence angle θi=0
and the RIS redirects the wave towards angle θrwith
|Γ0|= 1. The receiver can move freely on the same plane
at constant distance r= 20 m from the RIS and has an-
tenna gain Gr= 20 dB. The received power as a function
of the observation angle θis shown in Fig. 5 for two cases,
namely θr= 30o[Fig. 5(a)] and θr= 60o[Fig. 5(b)]. The
analytical result of Eq.(24) (solid red line) is overlapped
with the numerically calculated Eq.(28) for the MIMO
model (dashed black line), using the parameters given
by Eq.(30).
Evidently, while both approaches lead to the calcula-
tion of the same received power, the derived properties
FIG. 5. Equivalence between the sheet and MIMO model. A
TE-polarized wave with Ei= 1 V/m, normally incident on
the RIS (θi=0), is steered by the RIS with |Γ0|= 1 towards
(a) θr= 30o, (b) θr= 60o. The RIS is operating at 150 GHz
and consists of 250 ×250 elements with periodicity λ/5 along
both xand ydirections, forming a reflecting surface of total
size 10 cm ×10 cm. The received power predicted by the
sheet model (red solid line) and the MIMO model (dashed
black line) is calculated at distance 20 m from the RIS.
in Eqs.(30) can hardly represent realistic scattering ele-
ments, because of the form of the radiation pattern Ur
UC .
This is illustrated in Fig. 6, where the radiation pattern
of a single RIS element is shown as a function of the
RIS element size (lx×ly) for θr= 30oin Fig. 6(a) and
θr= 60oin Fig. 6(b). For lx=ly=λ/20 the radia-
tion pattern practically converges to the limit of a point
scatterer; however, the shape of the radiation pattern de-
pends strongly on the chosen reflection angle and changes
qualitatively with decreasing element size. Note how, for
large element size (lx=ly=λ/2), the radiation pattern
is directed towards the reflection angle. All these ob-
servations reveal that, although mathematically exact,
the equivalence in Eqs.(30) refers to complex and non-
practical scatterers. Ideally, what would be of relevance
to the equivalent MIMO model is scatterers with radia-
tion pattern that (a) has the same qualitative form for
both reception and emission operation, and (b) does not
depend on the chosen reflection angle; the wave steering
should result from the different excitation phase among
the RIS elements, all having identical radiation patterns.
7
FIG. 6. Radiation pattern Ur
UC of a single RIS element un-
der the sheet-MIMO equivalence given by Eq.(30). The ra-
diation pattern is shown as a function of the RIS element
size (lx×ly), for a RIS reflecting towards (a) θr= 30o, and
(b) θr= 60o. The shape of the radiation pattern depends
strongly on the chosen steering angle and changes qualita-
tively with decreasing element size. Note how, for large el-
ements (lx=ly=λ/2), the radiation pattern is directed
towards the steering angle.
2. Correspondence consistent with antenna theory
In antenna theory [38] it is convenient to express all rel-
evant quantities with respect to the radiation intensity
U(θ, φ) = r2Srad (θ, φ) (W/unit solid angle), where Srad
is the power density emitted by the antenna at distance
r; the total power is obtained by integrating over the en-
tire solid angle Prad =R2π
0Rπ
0U(θ, φ) sin θdθ.
Under the MIMO approach, we can attribute to each
RIS element a radiation pattern U(θ, φ) = UmaxU0(θ, φ),
where Umax is the maximum value of U(θ , φ) and U0(θ, φ)
is the normalized radiation pattern. Then, the antenna
parameters of the RIS elements, including the directiv-
ity DUC , gain GU C and receiving aperture AU C , are ex-
pressed in terms of U(θ, φ) as:
Ut
UC U0(θt
m,n, φt
m,n),(31a)
Ur
UC U0(θr
m,n, φr
m,n),(31b)
DUC = 4πUmax
R2π
0Rπ
0U(θ, φ)sinθdθdφ(31c)
GUC =e0DU C (31d)
AUC =λ2
4πGUC ,(31e)
where e0accounts for the antenna efficiency, which mod-
els possible absorption and scattering losses (0 < e0<1),
and θm,n, φm,n denote the elevation and azimuth angle
from the (m, n) element to the transmitter (t) and to
the receiver (r), as previoulsy illustrated in Fig. 2. Note
that, under plane wave illumination, all elements are ex-
cited under the same angle of incidence, i.e. θt
m,n θi,
φt
m,n φi. Hence, Ut
UC is a common constant for all
elements and we can simply write Ut
UC U0(θi, φi). Ad-
ditionally, in the far-field of the RIS where θr
m,n θ,
φr
m,n φwe may simplify Ur
UC U0(θ , φ), i.e. write
Ur
UC as a function of the observation angle.
In the above analysis we have defined all relevant an-
tenna parameters except for Rmn, which is associated
with the scattering strength of the RIS and is, there-
fore, closely related to e0. At this point we are free
to set Rmn = Γmn and use e0as a separate parameter
or we may incoroporate e0into a new reflection coeffi-
cient Rmn =e0Γmn and use the directivity instead of
the gain in Eq.(31). Alternatively, we may define a new
antenna efficiency or correction factor e0
0=e0Γmn and
omit Rmn from the analysis; all approaches are mathe-
matically equivalent. From simple inspection of Eq.(4)
and Eq.(28) we find that the unknown e0must satisfy:
e0=4π
DUC
lxly
λ2sΘ(θ, φ)C2
4Ut
UC Ur
UC
(32)
A RIS described by antenna elements with properties
given by Eqs.(31),(32) is consistent with the antenna the-
ory and reproduces successfully the scattered field pre-
dicted by the sheet model. Note that, due to Θ, C2and
Ur
UC , the correction factor e0that adjusts the maximum
intensity is a function of the observation angle; this im-
plies that e0, besides adjusting the gain, also modifies
qualitatively the radiation pattern.
To gain some insight into the correction factor e0, next
we will consider some typical antenna radiation patterns
that are commonly used in related works to describe the
response of subwavelength scatterers.
3. Examples of RIS element properties consistent with
antenna theory
The radiation intensity of the major lobe of many anten-
nas is frequently represented by U0(θ , φ) = cospθ, where
the positive real pmodifies the antenna directivity [38].
Using this form for U0, the properties of the RIS elements
are analytically expressed as:
Ut
UC = cospθi,(33a)
Ur
UC = cospθ, (33b)
DUC = 2(p+ 1),(33c)
GUC = 2(p+ 1)e0(θ , φ),(33d)
AUC =λ2
4π2(p+ 1)e0(θ, φ),(33e)
e0(θ, φ) = 4π
2(p+ 1)
lxly
λ2sΘ(θ, φ)C2
4 cospθicospθ.(33f)
As an example, the problem previously examined in
Fig. 5 is revisited in Fig. 7 using Eqs.(33) with p= 2, i.e.
the radiation pattern of the RIS elements is U0(θ, φ) =
cos2θ. The numerically calculated received power is
8
shown both with and without using the correction factor
(dashed black and solid gray lines, respectively). Clearly,
without the correction factor, i.e. by setting e0= 1, the
MIMO model overestimates the power density practically
across the entire observation range and, importantly, at
the steering angle θr, where the UE is located and the
redirected power density is maximized.
4. Commonly used RIS element properties
Last, we would like to note that the properties of the RIS
elements are considered in some works as [31, 32]:
Ut
UC = cospθi,(34a)
Ur
UC = cospθ, (34b)
AUC =lxly,(34c)
GUC = 2(p+ 1) (34d)
Rm,n = Γmn.(34e)
where pR+. This model borrows elements from the
antenna theory (form of UUC and corresponding GUC ),
however considers a 100% aperture efficiency (AU C /lxly)
and does not account for the necessary correction fac-
tor. Therefore, the scattering predicted by this model is
overestimated.
IV. ANALYSIS OF CORRECTION FACTOR
The correction factor captures the discrepancy in the
power predicted by the two approaches; note that the
ratio of the received power calculated with the sheet
model over the received power calculated with the MIMO
model is simply 1/e2
0. For small incident angles where
cospθi1, and taking into account that lx, lyλand
Θ(θ, φ)C2/41, with a simple inspection of Eq.(33f),
we see that 1/e2
0>1, i.e. the MIMO model overesti-
mates the received power with respect to the sheet model.
The opposite may occur for large incident angles where
cospθi0, however requires very directive scatterers,
which is hard to achieve with subwavelength elements.
The presence of the factor ΘC2in e0implies that,
to achieve equivalence between the discrete and contin-
uous models, the radiation pattern of the RIS elements
must depend explicitly on the angle θrat which the inci-
dent wave is steered. Although this could be a possibility,
what is here sought after is an equivalent that is closer to
the phased array picture; the incident wave is redirected
due to linear phase lag between the identical radiation
patterns of the individual elements, rather than due to
changes in the radiation patterns that promote radiation
towards a prescribed direction. This means that, in re-
alistic implementations where the scatterers have a fixed
radiation pattern, deviations should be expected and, for
the same incident wave, the magnitude of the scattered
wave will depend on the qualitative form of the scatterer
FIG. 7. Equivalence between the sheet and MIMO model
for RIS elements modeled as antennas with radiation pat-
tern U(θ, φ) = cos2θand properties consistent with antenna
theory. A TE-polarized wave with Ei= 1 V/m, normally in-
cident on the RIS (θi=0), is steered by the RIS with |Γ0|= 1
towards (a) θr= 30o, (b) θr= 60o. Without the correction
factor (solid gray line), i.e. by setting e0= 1, the antenna
model overestimates the analytically predicted power (solid
red line); the deviation is lifted upon incorporating e0(dashed
black line), thus restoring the sheet-MIMO equivalence. The
RIS is operating at 150 GHz and consists of 250×250 elements
with periodicity λ/5 along both xand ydirections, forming
a reflecting surface of total size 10 cm ×10 cm. The received
power is calculated at distance 20 m from the RIS.
radiation pattern. Importantly, due to the dependency
of e0on θr, the discrepancy in the received power be-
tween the discrete and continuous model is expected, in
general, to change with θr.
To gain more insight into the correction factor e0,
in Fig. 8 we consider a RIS consisting of elements with
tunable radiation pattern of the form U0(θ, φ) = cospθ,
i.e. complying with Eqs.(33). The radiation pattern for
elements with p= 1,2,4 is shown in Fig.8(a). In the re-
maining panels, e0is calculated as a function of the RIS
element size (lx=ly), for the case of normal incidence
(θi= 0). In Fig. 8(b) e0is shown for a RIS that redi-
rects the wave towards θr= 60o. The results show that,
with decreasing element size, the density of the scatter-
ers increases and e0becomes smaller in order to ensure
the same scattered field. Additionally, e0depends on the
observation angle θand has an asymmetry with respect
to θ= 0o, which is gradually lifted as the element size
9
FIG. 8. Correction factor e0for RIS elements modeled as
antennas with radiation pattern U(θ, φ) = cospθand prop-
erties consistent with antenna theory. (a) Polar plot of RIS
element radiation pattern for p= 1, 2 and 4. (b) Correction
factor as a function of the RIS element size (lx=ly) for a
RIS that steers a normally incident wave (θi= 0o) towards
θr= 60o. The slight asymmetry with respect to θ= 0ois due
to the factor C2, and is lifted with decreasing element size
(C21). (c) Correction factor at observation angle along
the scattering direction, i.e. θ=θr, as a function of the RIS
element size (lx=ly); along this direction, C2= 1 regardless
of the element size. Note that, with decreasing element size,
e0balances the increasing density of scatterers, ensuring the
same scattered field. For the examples shown here e0<1,
i.e. without taking into account the correction factor, the
scattering is overestimated.
becomes more subwavelength. This is due to the factor
C2, which converges to C21 as lx/λ, ly0. Al-
ready for lx=ly=λ/5, C21 and the asymmetry in
the correction factor is practically lifted. In Fig. 8(c) e0
is calculated along the steering angle (θ=θr,φ= 0),
where the RIS performance is most relevant. Along this
direction Θ = 4 cos2θrand C2= 1, the latter holding
independently from the RIS element size. In this case,
the antenna efficiency takes the simple form:
e0=4π
2(p+ 1)
lxly
λ2(cos θi)p
2(cos θr)1p
2,(35)
where the observation angle spans the entire range of θr,
i.e. from 0o(along the z-axis) to almost 90o(normally to
the RIS). The results of Fig. 8(c) demonstrate the impact
of the RIS element directivity on the dependence of e0on
FIG. 9. Sensitivity analysis of correction factor e0for ele-
ments with radiation pattern of the form U0(θ, φ) = cospθ.
Normalized plots of (a) e0, and (b) e0
∂θr, as a function of the
exponent p, which expresses the RIS element directivity.
θr. In particular, for highly directive elements that pro-
mote forward emission strongly, e0increases with increas-
ing angle to compensate the preferential emission (e.g.
example with p= 4). On the contrary, for weakly di-
rective elements that promote forward emission slightly,
e0decreases with increasing angle (e.g. example with
p= 1). The turning point between the two extremes
occurs for elements with p= 2, for which Θ = 4 cos2θr
and, therefore, Ur
UC = cos2θrexactly compensates Θ for
all angles [see Eq.(32)]. In this case the incident wave
can be steered to any direction with constant e0, i.e. the
discrepancy between the sheet and MIMO model does
not depend on θr; this can be seen in Eq.(35) by setting
p= 2, and also in the examples presented in Fig. 7.
To further quantify the analysis with respect to the
RIS element directivity, in Fig. 9(a) the correction factor
along the steering direction [Eq.(35)] is calculated as a
function of the exponent p. The results are normalized
with lxly2in order to provide a universal diagram for
RIS elements with radiation pattern of the form cospθ,
under normal incidence (θi= 0o); for different incident
angles the universal plot is just scaled by a factor of
cospθi. To emphasize the qualitative change of e0across
the critical value p= 2 (marked with the dashed line),
the colormap is saturated within the range (1 . . . 3).
To estimate the sensitivity of e0on the steering angle
θrin Fig. 9(b) we plot the calculated derivative:
∂e0
∂θr
= (p2) π
p+ 1
lxly
λ2(cos θi)p
2(cos θr)p
2sin θr,(36)
normalized with lxly2. The results demonstrate that,
for RIS elements with p2, the sensitivity of e0on
θrpractically remains weak for any desired reflection
angle. The same can be achieved for RIS elements with
arbitrary directivity, as long as the reflection angles are
small. In essence, while the MIMO and sheet models are
equivalent in the entire parameter space, there is a range
[green region in Fig. 9(b)] within which the sensitivity
of e0on θrcan be minimized for RIS elements with
fixed radiation pattern. Note that, with decreasing RIS
10
element size, the factor lxly2reduces and the sen-
sitivity decreases, in turn expanding the respective range.
V. DISCUSSION AND CONCLUSION
Designs based on continuous sheet models provide the
necessary surface impedance for designing RIS elements
to steer the wave to a prescribed direction. However, in
view of the various ways of implementing realistic RIS
elements, the question that is naturally raised is how a
specific design performs for different steering angles and
whether certain designs promote the scattered waves to-
wards certain directions. Here, the study of e0essentially
attempts to provide a qualitative answer to this question,
which is important for assessing the RIS efficiency and
utilization in realistic indoor or outdoor scenarios. In
general, the design and study of RIS elements requires
full wave simulations that take into account all details
that cannot be captured by the simplified antenna mod-
els. For example, in the analysis presented herein it has
been implied that only the phase of the scatterers is exter-
nally controlled, i.e. that it is possible to tune the phases
of the scatterers without affecting the amplitude of their
response. In practice, because the scatterer design usu-
ally involves resonant modes, by tuning the phases of the
RIS elements it is also possible that their amplitudes are
affected as well [21, 24–27]. This dependence may have
implications on the RIS performance, which may become
further complicated if coupling between the RIS elements
is also present. Last, we would like to note that the find-
ings in this work, although based on TE-polarized waves,
are directly applicable to TM-polarization. In fact, the
parameter Θ, which was found to play a crucial role in
the equivalence between the MIMO and sheet models, is
identical for TM waves (see Appendix for details) and
the observations are therefore general and applicable to
any linear polarization.
In this work we analyzed the RIS under two frequently
used approaches, where scattering from the RIS is usually
perceived as resulting from either the collective excitation
of local (discrete) scatterers or from the global response
of a continuous radiating surface. We demonstrated the
equivalence between the two models on the basis that
they must both lead to the same scattered wave, and
we discussed models commonly used in recent theoreti-
cal works. Overall, we found that the treatment of the
RIS as point scatterers may overestimate the scattered
field and, therefore, a correction factor must be taken
into account. By using point scatterers with properties
consistent with the antenna theory we showed how the
qualitative form of the RIS element radiation pattern has
implications on the discrepancy in the received power be-
tween the two approaches, which was found to depend on
the steering angle. With our analysis we aim to bridge
the gap between RIS approaches that have different ori-
gin, i.e. antenna theory vs. effective medium approach,
and to provide insight into the possible observed discrep-
ancies between the theoretical models, the understanding
of which is crucial for assessing the RIS efficiency.
ACKNOWLEDGMENTS
This work has received funding from the European
Commission’s Horizon 2020 research and innovation pro-
gramme ARIADNE under grant agreement No. 871464.
APPENDIX
A. RIS as MIMO
In this section we derive the result of Eq.(3) and we also
derive the E-field scattered by the RIS under spherical
wave illumination.
Let us consider an AP at distance dt
m,n from the (m, n)
RIS element, equipped with a directive antenna with gain
Gtthat emits total power Pttowards the RIS. The power
density at the (m, n) RIS element is:
St=|Et
m,n|2
2Z0GtUt
Pt
4π(dt
m,n)2,(37)
where Et
m,n is the field amplitude at the position of
the (m, n) element from the transmitter (t) or access
point (AP). The power received by the RIS element is
Pinc
UC =StAU C Ut
UC and the power subsequently emitted
is Prf l
UC =|Rm,n |2Pinc
UC . The power density at the receiver
due to the (m, n) RIS element is:
Sr=GUC Ur
UC
Prf l
UC
4π(dr
m,n)2=GU C Ur
UC |Rm,n |2StAU C Ut
UC
4π(dr
m,n)2
(38)
and the Efield reaching the receiver is therefore:
Er
m,n =Rm,nqAU C Ut
UC GU C Ur
UC
eikdr
m,n
4πdr
m,n
Et
m,n,(39)
By properly assigning the magnitude and phase of
Et
m,n, different RIS illumination conditions can be stud-
ied. For example, assuming that the transmitter antenna
is at distance dt
m,n from the (m, n) RIS element, has gain
Gt, and emits total power Ptin spherical wavefronts to-
wards the RIS, the Efield reaching the (m, n) element,
Et
m,n, is written as:
Et
m,n =p2Z0PtGtUt
eikdt
m,n
4πdt
m,n
,(40)
where UtUt(θt, φt) is the normalized radiation pattern
of the transmitter, with θt, φtdenoting the elevation and
azimuth angle from the transmitter to the RIS element.
11
Using Eqs.(39),(40), the field scattered from the (m, n)
element at the position of the receiver takes the com-
monly used form [30–33]:
Er
m,n =q2Z0PtGtUtAUC Ut
UC GU C Ur
UC
|Rm,n|eiarg(Rm,n )eik(dt
m,n+dr
m,n)
4πdt
m,ndr
m,n
.(41)
B. RIS as radiating sheet
In this section we derive Θ for both TE and TM polar-
izations. For the derivation we will work at distances
|rr0|  λ, where we may approximate:
∇ × (jsG(rs)) ≈ −ikG(rs) (ˆ
rs×js),(42a)
∇ × ∇ × (jsG(rs)) ≈ −k2G(rs) (ˆ
rs׈
rs×js),(42b)
with rsrr0and jsbeing any of jse,jsm .
1. TE polarization
For TE polarization, the incident wave is written as:
Ei(r) = Eieikirˆ
y,(43a)
Hi(r) = Ei
Z0
eikir(cos θiˆ
x+ sin θiˆ
z),(43b)
and the reflected wave as:
Er(r)=ΓT E
0Eieikrrˆ
y,(44a)
Hr(r) = ΓT E
0Ei
Z0
eikrr(cos θrˆ
x+ sin θrˆ
z),(44b)
where ΓT E
0has been simply denoted as Γ0in the main
text [see Eq.(8)]. Using Eq.(16) with Eq.(44) we find:
jse =cos θr
Z0
ΓT E
0Eiˆ
y,(45a)
jsm =ΓT E
0Eiˆ
x.(45b)
Inserting Eq.(45) into Eqs.(13), (14) and using Eq.(42),
we find:
VT E
e= cos θr(ˆ
rs׈
rs×y)(ˆ
rs×x) (46a)
VT E
m=(ˆ
rs׈
rs×x)cos θr(ˆ
rs×y),(46b)
the components of which are written explicitly in
Eq.(19), and ΘT E ≡ |VT E
e×VT E
m|, which is simply de-
noted as Θ in the main text.
2. TM polarization
For TM polarization, the incident wave is written as:
Ei(r) = Z0Hieikir(cos θiˆ
xsin θiˆ
z),(47a)
Hi(r) = Hieikirˆ
y,(47b)
and the reflected wave as:
Er(r) = Z0ΓT M
0Hieikrr(cos θrˆ
xsin θrˆ
z),(48a)
Hr(r)=ΓT M
0Hieikrrˆ
y,(48b)
Using Eq.(16) with Eq.(48) we find:
jse = ΓT M
0Hiˆ
x,(49a)
jsm = cos θrZ0ΓTM
0Hiˆ
y.(49b)
Inserting Eq.(49) into Eqs.(13), (14) and using Eq.(42),
we find:
VT M
e=VT E
m(50a)
VT M
m= +VT E
e.(50b)
Hence:
VT M
e×VT M
m=VT E
m×VT E
e=VT E
e×VT E
m.(51)
and:
ΘT M ≡ |VT M
e×VT M
m|=|VT E
e×VT E
m|= ΘT E .(52)
[1] A. Epstein and G. V. Eleftheriades, Huygens’ metasur-
faces via the equivalence principle: design and applica-
tions, J. Opt. Soc. Am. B 33, A31 (2016).
[2] M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini,
C. Yuen, J. de Rosny, and S. Tretyakov, Smart radio en-
vironments empowered by reconfigurable intelligent sur-
faces: How it works, state of research, and the road
ahead, IEEE Journal on Selected Areas in Communica-
tions 38, 2450 (2020).
[3] D. Headland, Y. Monnai, D. Abbott, C. Fumeaux, and
W. Withayachumnankul, Tutorial: Terahertz beamform-
ing, from concepts to realizations, APL Photonics 3,
12
051101 (2018), https://doi.org/10.1063/1.5011063.
[4] O. Tsilipakos, A. C. Tasolamprou, A. Pitilakis, F. Liu,
X. Wang, M. S. Mirmoosa, D. C. Tzarouchis, S. Abadal,
H. Taghvaee, C. Liaskos, A. Tsioliaridou, J. Georgiou,
A. Cabellos-Aparicio, E. Alarc´on, S. Ioannidis, A. Pitsil-
lides, I. F. Akyildiz, N. V. Kantartzis, E. N. Economou,
C. M. Soukoulis, M. Kafesaki, and S. Tretyakov, Toward
intelligent metasurfaces: The progress from globally tun-
able metasurfaces to software-defined metasurfaces with
an embedded network of controllers, Advanced Optical
Materials 8, 2000783 (2020).
[5] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, Intel-
ligent reflecting surface-aided wireless communications:
A tutorial, IEEE Transactions on Communications 69,
3313 (2021).
[6] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P.
Tetienne, F. Capasso, and Z. Gaburro, Light prop-
agation with phase discontinuities: Generalized laws
of reflection and refraction, Science 334, 333 (2011),
https://www.science.org/doi/pdf/10.1126/science.1210713.
[7] A. Epstein and G. V. Eleftheriades, Passive lossless huy-
gens metasurfaces for conversion of arbitrary source field
to directive radiation, IEEE Transactions on Antennas
and Propagation 62, 5680 (2014).
[8] V. S. Asadchy, M. Albooyeh, S. N. Tcvetkova, A. D´ıaz-
Rubio, Y. Ra’di, and S. A. Tretyakov, Perfect control of
reflection and refraction using spatially dispersive meta-
surfaces, Phys. Rev. B 94, 075142 (2016).
[9] N. Mohammadi Estakhri and A. Al`u, Wave-front trans-
formation with gradient metasurfaces, Phys. Rev. X 6,
041008 (2016).
[10] A. Epstein and G. V. Eleftheriades, Synthesis of pas-
sive lossless metasurfaces using auxiliary fields for reflec-
tionless beam splitting and perfect reflection, Phys. Rev.
Lett. 117, 256103 (2016).
[11] Y. Ra’di, D. L. Sounas, and A. Al`u, Metagratings: Be-
yond the limits of graded metasurfaces for wave front
control, Phys. Rev. Lett. 119, 067404 (2017).
[12] V. S. Asadchy, A. D´ıaz-Rubio, S. N. Tcvetkova, D.-H.
Kwon, A. Elsakka, M. Albooyeh, and S. A. Tretyakov,
Flat engineered multichannel reflectors, Phys. Rev. X 7,
031046 (2017).
[13] A. D´ıaz-Rubio, V. S. Asadchy, A. Elsakka, and
S. A. Tretyakov, From the generalized reflec-
tion law to the realization of perfect anomalous
reflectors, Science Advances 3, e1602714 (2017),
https://www.science.org/doi/pdf/10.1126/sciadv.1602714.
[14] A. M. H. Wong and G. V. Eleftheriades, Perfect anoma-
lous reflection with a bipartite huygens’ metasurface,
Phys. Rev. X 8, 011036 (2018).
[15] A. D´ıaz-Rubio, J. Li, C. Shen, S. A. Cum-
mer, and S. A. Tretyakov, Power flow-conformal
metamirrors for engineering wave reflec-
tions, Science Advances 5, eaau7288 (2019),
https://www.science.org/doi/pdf/10.1126/sciadv.aau7288.
[16] J. Budhu and A. Grbic, Perfectly reflecting metasurface
reflectarrays: Mutual coupling modeling between unique
elements through homogenization, IEEE Transactions on
Antennas and Propagation 69, 122 (2021).
[17] A. D´ıaz-Rubio and S. A. Tretyakov, Macroscopic mod-
eling of anomalously reflecting metasurfaces: Angular
response and far-field scattering, IEEE Transactions on
Antennas and Propagation 69, 6560 (2021).
[18] M. Di Renzo, F. Habibi Danufane, X. Xi, J. de Rosny,
and S. Tretyakov, Analytical modeling of the path-loss for
reconfigurable intelligent surfaces – anomalous mirror or
scatterer ?, in 2020 IEEE 21st International Workshop
on Signal Processing Advances in Wireless Communica-
tions (SPAWC) (2020) pp. 1–5.
[19] Z. Wan, Z. Gao, F. Gao, M. D. Renzo, and M.-S. Alouini,
Terahertz massive mimo with holographic reconfigurable
intelligent surfaces, IEEE Transactions on Communica-
tions 69, 4732 (2021).
[20] K. Dovelos, S. D. Assimonis, H. Q. Ngo, B. Bellalta, and
M. Matthaiou, Electromagnetic modeling of holographic
intelligent reflecting surfaces at terahertz bands (2021),
arXiv:2108.08104 [cs.IT].
[21] H. Yang, X. Cao, F. Yang, J. Gao, S. Xu, M. Li, X. Chen,
Y. Zhao, Y. Zheng, and S. Li, A programmable metasur-
face with dynamic polarization, scattering and focusing
control, Scientific Reports 6, 35692 (2016).
[22] L. Zhang, X. Q. Chen, S. Liu, Q. Zhang, J. Zhao, J. Y.
Dai, G. D. Bai, X. Wan, Q. Cheng, G. Castaldi, V. Galdi,
and T. J. Cui, Space-time-coding digital metasurfaces,
Nature Communications 9, 4334 (2018).
[23] F. Liu, O. Tsilipakos, A. Pitilakis, A. C. Tasolamprou,
M. S. Mirmoosa, N. V. Kantartzis, D.-H. Kwon, J. Geor-
giou, K. Kossifos, M. A. Antoniades, M. Kafesaki, C. M.
Soukoulis, and S. A. Tretyakov, Intelligent metasurfaces
with continuously tunable local surface impedance for
multiple reconfigurable functions, Phys. Rev. Applied 11,
044024 (2019).
[24] L. Dai, B. Wang, M. Wang, X. Yang, J. Tan, S. Bi,
S. Xu, F. Yang, Z. Chen, M. D. Renzo, C.-B. Chae, and
L. Hanzo, Reconfigurable intelligent surface-based wire-
less communications: Antenna design, prototyping, and
experimental results, IEEE Access 8, 45913 (2020).
[25] X. Pei, H. Yin, L. Tan, L. Cao, Z. Li, K. Wang, K. Zhang,
and E. Bj¨ornson, Ris-aided wireless communications:
Prototyping, adaptive beamforming, and indoor/outdoor
field trials (2021), arXiv:2103.00534 [cs.IT].
[26] R. Fara, P. Ratajczak, D.-T. P. Huy, A. Ourir, M. D.
Renzo, and J. D. Rosny, A prototype of reconfigurable
intelligent surface with continuous control of the reflec-
tion phase (2021), arXiv:2105.11862 [eess.SP].
[27] A. Pitilakis, O. Tsilipakos, F. Liu, K. M. Kossifos, A. C.
Tasolamprou, D.-H. Kwon, M. S. Mirmoosa, D. Manessis,
N. V. Kantartzis, C. Liaskos, M. A. Antoniades, J. Geor-
giou, C. M. Soukoulis, M. Kafesaki, and S. A. Tretyakov,
A multi-functional reconfigurable metasurface: Electro-
magnetic design accounting for fabrication aspects, IEEE
Transactions on Antennas and Propagation 69, 1440
(2021).
[28] ¨
O. ¨
Ozdogan, E. Bj¨ornson, and E. G. Larsson, Intelligent
reflecting surfaces: Physics, propagation, and pathloss
modeling, IEEE Wireless Communications Letters 9, 581
(2020).
[29] R. J. Williams, E. de Carvalho, and T. L. Marzetta,
A communication model for large intelligent surfaces,
in 2020 IEEE International Conference on Communica-
tions Workshops (ICC Workshops) (2020) pp. 1–6.
[30] S. W. Ellingson, Path loss in reconfigurable in-
telligent surface-enabled channels, 2021 IEEE 32nd
Annual International Symposium on Personal, In-
door and Mobile Radio Communications (PIMRC)
10.1109/pimrc50174.2021.9569465 (2021).
13
[31] W. Tang, M. Z. Chen, X. Chen, J. Y. Dai, Y. Han,
M. Di Renzo, Y. Zeng, S. Jin, Q. Cheng, and T. J. Cui,
Wireless communications with reconfigurable intelligent
surface: Path loss modeling and experimental measure-
ment, IEEE Transactions on Wireless Communications
20, 421 (2021).
[32] A. A. A. Boulogeorgos and A. Alexiou, Coverage analysis
of reconfigurable intelligent surface assisted thz wireless
systems, IEEE Open Journal of Vehicular Technology ,
1 (2021).
[33] K. Ntontin, A.-A. A. Boulogeorgos, D. G. Selimis,
F. I. Lazarakis, A. Alexiou, and S. Chatzinotas, Re-
configurable intelligent surface optimal placement in
millimeter-wave networks, IEEE Open Journal of the
Communications Society 2, 704 (2021).
[34] G. Stratidakis, S. Droulias, and A. Alexiou, Analytical
performance assessment of beamforming efficiency in re-
configurable intelligent surface-aided links, IEEE Access
9, 115922 (2021).
[35] G. Stratidakis, S. Droulias, and A. Alexiou, Impact of
reconfigurable intelligent surface size on beamforming ef-
ficiency, in 2021 IEEE 32nd Annual International Sym-
posium on Personal, Indoor and Mobile Radio Commu-
nications (PIMRC) (2021) pp. 1–5.
[36] M. D. Renzo, F. H. Danufane, and S. Tretyakov, Com-
munication models for reconfigurable intelligent surfaces:
From surface electromagnetics to wireless networks opti-
mization (2021), arXiv:2110.00833.
[37] S. J. Orfanidis, Electromagnetic Waves and Antennas
(Orfanidis, S. J., 2016) https://www.ece.rutgers.edu/ or-
fanidi/ewa/.
[38] C. A. Balanis, Antenna theory: analysis and design
(Wiley-Interscience, 2005).
ResearchGate has not been able to resolve any citations for this publication.
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