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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

e−ikdr

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|2≤1) 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

e−ikdr

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

y−cos 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 θix−cos 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) = Eie−ikirˆ

y,(7a)

Hi(r) = Ei

Z0

e−ikir(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) = Ere−ikrrˆ

y≡Γ0Eie−ikrrˆ

y,(8a)

Hr(r) = Γ0Ei

Z0

e−ikrr(−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

2/σse =σsm/2≡Zs, where Zsis commonly referred to

as the surface impedance, and is given by:

Zs(x) = Z0

1+Γ0eik(sin θi−sin θr)x

cos θi−cos θrΓ0eik(sin θi−sin θr)x(11)

The surface impedance in Eq.(11) involves the parameter:

Γs(x)≡Γ0eik(sin θi−sin 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

iωµ (∇ × ∇ × Ase )−1

∇ × Asm,(13a)

Hr(r) = 1

iωµ (∇ × ∇ × Asm ) + 1

µ∇ × Ase,(13b)

where

Ase =ZZS

µjseG(r−r0)dr0,(14a)

Asm =ZZS

jsmG(r−r0)dr0,(14b)

and

G(r−r0) = e−ik |r−r0|

4π|r−r0|,(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(r−r0)) ≈

−ikG(r−r0)(ˆ

r−ˆ

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(r−r0)Ve(r−r0)dr0,(17a)

Hr(r) = ik

Z0ZZS

Es(r0)G(r−r0)Vm(r−r0)dr0,(17b)

where

Es(r0)=Γs(r0)Eie−ikir0,(18)

and the vectors Veand Vmare explicitly written as:

Ve=

x−x0

|r−r0|

y−y0

|r−r0|cos θr

−((x−x0)2

|r−r0|2+z2

|r−r0|2) cos θr−z

|r−r0|

y−y0

|r−r0|

z

|r−r0|cos θr+y−y0

|r−r0|

,(19a)

Vm=

(y−y0)2

|r−r0|2+z2

|r−r0|2+z

|r−r0|cos θr

−x−x0

|r−r0|

y−y0

|r−r0|

−x−x0

|r−r0|

z

|r−r0|−x−x0

|r−r0|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)/λ, where

Ve(r−r0)≈Ve(r), Vm(r−r0)≈Vm(r), the Poynting

vector takes the simple form:

Sr(r) = k2

2Z0

Θ(r)ZZS

Es(r0)G(r−r0)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(r−r0)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)Eie−ik sin θix0, with Γs(x0) =

Γ0eik(sin θi−sin 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 φ−sin θr)

kLx

2(sin θcos φ−sin 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(r−r0)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 r−r0=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 r−r0=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)/λ, 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|) = e−ik |rm,n−r0

m,n|

4π|rm,n −r0

m,n|'e−ikrm,n

4πrm,n

e−ikr0

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 θi−sin θr)xm,(27a)

Emn =Eie−ik 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

e−ikrmn

4πrmn

lxly

2

,

(28)

where

C=sin klx

2(sin θcos φ−sin θr)

klx

2(sin θcos φ−sin 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 θdθdφ.

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 p∈R+. 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

cospθi→1, and taking into account that lx, lyλand

Θ(θ, φ)C2/4≤1, 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

cospθi→0, 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

(C2→1). (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 C2→1 as lx/λ, ly/λ →0. Al-

ready for lx=ly=λ/5, C2≈1 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)1−p

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 lxly/λ2in 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

= (p−2) π

p+ 1

lxly

λ2(cos θi)−p

2(cos θr)−p

2sin θr,(36)

normalized with lxly/λ2. The results demonstrate that,

for RIS elements with p∼2, 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 lxly/λ2reduces 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

2Z0≡GtUt

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 E−ﬁeld reaching the receiver is therefore:

Er

m,n =Rm,nqAU C Ut

UC GU C Ur

UC

e−ikdr

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 E−ﬁeld reaching the (m, n) element,

Et

m,n, is written as:

Et

m,n =p2Z0PtGtUt

e−ikdt

m,n

√4πdt

m,n

,(40)

where Ut≡Ut(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 )e−ik(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

|r−r0| λ, where we may approximate:

∇ × (jsG(rs)) ≈ −ikG(rs) (ˆ

rs×js),(42a)

∇ × ∇ × (jsG(rs)) ≈ −k2G(rs) (ˆ

rs×ˆ

rs×js),(42b)

with rs≡r−r0and jsbeing any of jse,jsm .

1. TE polarization

For TE polarization, the incident wave is written as:

Ei(r) = Eie−ikirˆ

y,(43a)

Hi(r) = Ei

Z0

e−ikir(cos θiˆ

x+ sin θiˆ

z),(43b)

and the reﬂected wave as:

Er(r)=ΓT E

0Eie−ikrrˆ

y,(44a)

Hr(r) = ΓT E

0Ei

Z0

e−ikrr(−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×ˆ

rs×y)−(ˆ

rs×x) (46a)

VT E

m=−(ˆ

rs×ˆ

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) = Z0Hie−ikir(−cos θiˆ

x−sin θiˆ

z),(47a)

Hi(r) = Hie−ikirˆ

y,(47b)

and the reﬂected wave as:

Er(r) = Z0ΓT M

0Hie−ikrr(cos θrˆ

x−sin θrˆ

z),(48a)

Hr(r)=ΓT M

0Hie−ikrrˆ

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)

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