Analytical Modeling of the Path-Loss for Reconfigurable Intelligent Surfaces -- Anomalous Mirror or Scatterer ?
Marco Di Renzo, Fadil Habibi Danufane, Xiaojun Xi, Julien de Rosny, Sergei Tretyakov
aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n Analytical Modeling of the Path-Loss for ReconfigurableIntelligent Surfaces – Anomalous Mirror or Scatterer ?
Marco Di Renzo (1) , Fadil Habibi Danufane (1) , Xiaojun Xi (1) , Julien de Rosny (2) , and Sergei Tretyakov (3)(1)
Universit´e Paris-Saclay, CNRS, CentraleSup´elec, Laboratoire des Signaux et Syst`emes, Gif-sur-Yvette, France (2)
ESPCI, Institut Langevin, CNRS, Paris, France (3)
Aalto University, Dept. of Electronics and Nanoengineering, Helsinki, Finlandemail: [email protected]
Abstract —Reconfigurable intelligent surfaces (RISs) are anemerging field of research in wireless communications. A funda-mental component for analyzing and optimizing RIS-empoweredwireless networks is the development of simple but sufficientlyaccurate models for the power scattered by an RIS. By leveragingthe general scalar theory of diffraction and the Huygens-Fresnelprinciple, we introduce simple formulas for the electric fieldscattered by an RIS that is modeled as a sheet of electromagneticmaterial of negligible thickness. The proposed approach allowsus to identify the conditions under which an RIS of finite size canor cannot be approximated as an anomalous mirror. Numericalresults are illustrated to confirm the proposed approach.
I. I
NTRODUCTION
In contemporary wireless networks, transmitters and re-ceivers can be programmed and controlled for optimizing thesystem performance. The environmental objects (buildings,walls, ceilings, etc.) that constitute the wireless environmentcannot, on the other hand, be customized based on the networkconditions. This status quo has recently been challenged bythe emerging technology of reconfigurable intelligent surfaces(RISs) – Thin sheets of electromagnetic materials that arecapable of shaping the radio waves in arbitrary ways [1], [2].The overarching vision consists of coating the environmentalobjects with RISs and optimizing their properties, in orderto, e.g., reflect an impinging radio wave towards a desireddirection with the objective of capitalizing from multipathpropagation rather than being negatively affected by it [3]-[8].In simple terms, an RIS is the two-dimensional equivalentof a reconfigurable meta-material, and is made of elementaryelements called scattering particles or meta-atoms [9]. Depend-ing on the arrangement and configuration of the scatteringparticles, an RIS is capable of altering the wavefront ofthe radio waves impinging upon it. For example, RISs canmodify the direction of the reflected or refracted waves andtheir polarization, or can encode data onto the shape ofthe scattered waves [10], [11], [12]. The two-dimensionalnature of RISs make them easier to design, less lossy, lessexpensive, and easier to deploy than their three-dimensionalcounterpart. Broadly speaking, RISs are special surfaces thatare engineered to possess properties that cannot be found insurfaces made of naturally occurring materials [12]. Thanks tothese properties, RISs are receiving major attention from thewireless community, and are considered to be the key enablerof the emerging concept of smart radio environments [1].A major open research issue for analyzing the ultimateperformance limits, optimizing the operation, and assessingthe advantages and limitations of RIS-empowered wirelessnetworks is the development of simple but sufficiently accurate models for the power received at a given location in spacewhen a transmitter emits radio waves that illuminate an RIS.Recently, a few research works have tackled this researchissue. In [13], the authors have performed a measurementcampaign in an anechoic chamber and have shown that thepower reflected from an RIS follows a scaling law that dependson many parameters, including the size of the RIS, the mutualdistances between the transmitter/receiver and the RIS (i.e.,near-field vs. far-field conditions), and whether the RIS is usedfor beamforming or broadcasting applications. In [14], theauthors have employed antenna theory to compute the electricfield in the near-field and far-field of a finite-size RIS, andhave proved that an RIS is capable of acting as an anomalousmirror in the near-field of the array. The results are obtainednumerically and no explicit analytical formulation of thereceived power as a function of the distance is given. Similarresults have been obtained in [15]. In [16], the power measuredfrom passive reflectors in the millimeter-wave frequency bandis compared against ray tracing simulations. By optimizingthe area of the surface that is illuminated, it is shown that afinite-size passive reflector can act as an anomalous mirror.The study in [17] relies on the assumption of plane waves andis valid only in the far-field of the RIS (i.e., long distances).In this paper, we leverage the general scalar theory ofdiffraction and the Huygens-Fresnel principle, and introducesimple closed-form expressions to compute the power re-flected from an RIS as a function of the distance betweenthe transmitter/receiver and the RIS, the size of the RIS,and the phase transformation applied by the RIS. With theaid of the stationary phase method, we identify sufficientconditions under which an RIS acts as an anomalous mirror,and, therefore, the received power decays as a function of thereciprocal of the sum of the distances between the transmitterand the RIS, and the RIS and the receiver. For simplicity,the analytical formulas are reported without proof and for aone-dimensional RIS. The proofs, discussions on the boundaryconditions to solve Maxwell’s equations, the impact of thedirect link, the analysis of refraction, and two-dimensionalRISs can be found in the companion journal paper [18].The rest of this paper is organized as follows. In Section II,the system model is introduced. In Section III, the analyticalformulation of the electric field emitted by a point source andscattered by a finite-size RIS is reported. Explicit expressionsof the electric field in the near-field and far-field are given.In Section IV, numerical results are provided to illustratethe scaling laws of the received power as a function of thetransmission distances. Finally, Section V concludes this paper.
II. S
YSTEM M ODEL
In a two-dimensional space, we consider a system thatconsists of a transmitter (Tx), a receiver (Rx), and a flat surface( S ) of zero-thickness. Without loss of generality, we assumethat S is located such that its center coincides with the origin.Furthermore, S lies in the x -axis and spans along [ − L, L ] , i.e., S = { ( x,
0) : − L ≤ x ≤ L } . In other words, S is a straightline. The locations of Tx and Rx are denoted by ( x T , y T ) and ( x R , y R ) , respectively. We consider only the scenario whereTx and Rx are on the same side of the surface S , i.e., wefocus our attention on modeling reflections from the surface S . Therefore, y T and y R take positive values, while there isno restriction on the values taken by x T and x R .Tx is modeled as a point source that emits cylindrical elec-tromagnetic (EM) waves through the vacuum. The EM wavesemitted by Tx travel at the speed of light c . The frequencyof the EM waves is denoted by f , and the wavelength andwavenumber are λ = c/f and k = 2 π/λ , respectively. Weare interested in computing the intensity of the electric fieldemitted by Tx and observed at an arbitrary point, i.e. Rx, on thepositive y -axis, with the exception of the location of the pointsource. In vacuum, the x and y components of the electric fieldare not coupled, and we assume that S does not change thepolarization of the EM waves. Under these assumptions, wecan analyze any components of the electric field. We considerthe tangential (to the surface S ) component of the electricfield, which is denoted by E x ( x R , y R ) .For every point ( x, ∈ S , the Tx-to- S and S -to-Tx distances are denoted by d T ( x ) = q ( x − x T ) + y T and d R ( x ) = q ( x R − x ) + y R , respectively. In particular, d T ( x ) is the radius of the wavefront of the EM wave thatis emitted by Tx and intersects S at ( x, , and d R ( x ) is theradius of the wavefront of the EM wave that originates from S at ( x, and is observed at Rx. With a similar terminology, theangle of incidence of the EM wave at ( x, ∈ S is denotedby θ T ( x ) . It represents the angle formed by the y -axis andthe wavefront of the EM wave that originates from Tx andintersects S at ( x, . The angle of reflection of the EM waveat ( x, ∈ S is denoted by θ R ( x ) , and it represents the angleformed by the y -axis and the wavefront of the EM wave thatis emitted by S at ( x, and is observed at Rx.For simplicity, we assume d T ( x ) ≫ λ and d R ( x ) ≫ λ ,which usually hold true in practical setups [13]. The completeanalysis is available in [18]. Under these assumptions, theelectric field emitted by the point source (Tx) and observed atRx in the absence of S corresponds to the Green function inthe plane, which is well approximated as follows [19]: E x ( x R , y R ) ≈ E exp ( − jkd T R ( x R , y R )) p kd T R ( x R , y R ) (1)where E = − j p π ) exp ( − jπ /4) , j is the imaginaryunit, and d T R ( x R , y R ) = q ( x R − x T ) + ( y R − y T ) is thedistance between Tx and Rx.The surface S is modeled as a spatially-inhomogeneousreflector that is capable of modifying the phase of the incidentfield. We assume that the electromagnetic properties of the surface S vary slowly, as compared with the wavelength, alongthe surface itself. Under this approximation, the surface S canbe well modeled as a local structure: the reflected field at ( x, ∈ S depends, approximately, only on the incident fieldat ( x, ∈ S [20]. More precisely, the reflection coefficient at ( x, ∈ S can be written as follows: Γ r ( x ) = C ( x ) exp ( j Φ( x )) (2)where C ( x ) ∈ R + and Φ( x ) ∈ [0 , π ) denote the amplitudeand phase of the reflection coefficient, respectively. In thispaper, we are interested in analyzing only reflections. Further-more, we assume that the surface S operates in the regime ofa phase-gradient reflector and, therefore, assume C ( x ) = 1 .III. E LECTRIC F IELD R EFLECTED FROM S Based on the assumptions in Section II, the intensity of theelectric field emitted by Tx, reflected by the surface S , andobserved at ( x R , y R ) for y R > is given as follows. Theorem 1.
Let us assume d T ( x ) ≫ λ and d R ( x ) ≫ λ . Theelectric field E x ( x R , y R ) can be formulated as follows: E x ( x R , y R ) = I Z + L − L I ( x ) exp ( − jk P ( x )) dx (3) where I = 1/(8 π ) and: P ( x ) = d T ( x ) + d R ( x ) − Φ ( x ) (4) I ( x ) = 1 p d T ( x ) d R ( x ) (cid:18) y T d T ( x ) + y R d R ( x ) (cid:19) (5) Proof.
It follows by formulating in mathematical terms theHuygens-Fresnel principle by using the general scalar theoryof diffraction and by applying appropriate boundary conditionsat the surface S . The details can be found in [18].The electric field in (3) is formulated in a simple integralform, which, however, does not explicitly unveil the depen-dency of the electric field as a function of the transmissiondistances. Also, the electric field depends on the specific phaseshift Φ( x ) applied by the surface S . In the following threesub-sections, we consider three case studies for the choice of Φ( x ) . Due to space limitations, only the first two case studiesare analyzed in detail. For both cases, we introduce explicitapproximate closed-form expressions for the electric field in(3) for short and long transmission distances. The definitionof long transmission distance is given as follows. Definition 1.
Let us define d Q ( x ) = q ( x Q − x ) + y Q and sin ( θ Q ) = sin ( θ Q (0)) = − qx Q / d Q for Q = { T, R } ,where q = 1 if Q = T and q = − if Q = R , as wellas d Q = q x Q + y Q for Q = { T, R } . The system is saidto operate in the long distance regime if the approximation d Q ( x ) ≈ d Q + qx sin ( θ Q ) holds true for Q = { T, R } .Otherwise, it is said to operate in the short distance regime. Loosely speaking (with a slightly abuse of terminology), thelong and short distance operating regimes can be identifiedwith the far-field and near-field regimes, respectively. A. S Acting as a Uniform Reflecting Surface
In this section, we analyze the case study in which thesurface S operates as a mirror reflector. This operation isobtained by setting Φ ( x ) = φ for x ∈ S in (3), where φ ∈ [0 , π ) is a fixed phase shift. The following twopropositions report approximate closed-form expressions ofthe intensity of the electric field in (3) under the assumptionof short and long distance regime, respectively. Proposition 1.
In the short distance regime, the intensity ofthe electric field in (3) can be approximated as follows: | E x ( x R , y R ) | ≈ √ πk p d T ( x s ) + d R ( x s ) (6) where x s ∈ [ − L, L ] is the unique solution of the equation: x s − x T d T ( x s ) − x R − x s d R ( x s ) = 0 (7) Proof.
It follows from (3) by applying the stationary phasemethod. The details can be found in [18].
Remark 1.
Proposition 1 holds true only if (7) has at leastone solution x s ∈ [ − L, L ] . The case study when this does nothold true can be found in [18]. Similar comments hold forsimilar case studies analyzed in the following sub-sections. From Proposition 1, the following conclusions follow. • In the short distance regime, the surface S behaves as aspecular mirror. In particular, the (end-to-end) intensityof the electric field reflected from the surface decays asa function of the reciprocal of the square root of the sumof the Tx-to- S and S -to-Rx distances. The presence ofthe square root originates from the assumption of two-dimensional space (see the emitted field in (1)). • Equation (6) can be regarded as an approximation of(3) under the condition of geometric optics propagation.More precisely, (6) unveils that the intensity of theelectric field is approximately the same as that obtainedfrom a single ray (i.e., the direction of propagation of thewavefront of the EM wave) that is obtained from the twoline segments that connect Tx with the point x s ∈ [ − L, L ] that fulfills (7), and the latter point with Rx. Therefore,the point x s can be referred to as reflection point. • From the definition of angles of incidence and reflec-tion, we have ( x s − x T )/ d T ( x s ) = sin ( θ T ( x s )) and ( x R − x s )/ d R ( x s ) = sin ( θ R ( x s )) , respectively. From(7), this implies that the angles of incidence and reflectioncoincide at the reflection point x s ∈ [ − L, L ] . In otherwords, (6) allows us to retrieve the law of reflection. Proposition 2.
In the long distance regime, the intensity ofthe electric field in (3) can be approximated as follows: | E x ( x R , y R ) | ≈ π (cid:12)(cid:12)(cid:12)(cid:12) cos ( θ T ) + cos ( θ R ) √ d T d R (cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12) sin ( kL (sin ( θ T ) − sin ( θ R ))) k (sin ( θ T ) − sin ( θ R )) (cid:12)(cid:12)(cid:12)(cid:12) (8) where cos ( θ Q ) = y Q / d Q for Q = { T, R } . Proof. It follows by using the approximation d Q ( x ) ≈ d Q + qx sin ( θ Q ) for Q = { T, R } . The details are in [18].From Proposition 2, we evince that the surface S does notbehave as a specular mirror for long transmission distances.The scaling law that governs the intensity of the electric fieldas a function of the distances is, in addition, no straightforwardto be identified. To shed light on the impact of the transmissiondistances, i.e., d T and d R in (8), we consider the case studyin which Tx and Rx move along two straight lines such thatthe angles θ T and θ R are kept constant (the two angles donot have to be necessarily the same), but d T and d R aredifferent. In this case, we observe from (8) that the intensityof the electric field decays as a function of the square rootof the product of the distances between Tx and the (center ofthe) surface S , and the (center of the) surface S and Rx. Inthis regime, therefore, the surface S is better modeled as ascatterer, since its size is relatively small in comparison withthe transmission distances involved (i.e., d T and d R ).The findings in Propositions 1 and 2 provide us with evi-dence that justifies the validity of Theorem 1, and, therefore,substantiate the approach embraced in this paper for modelingand analyzing RISs in wireless networks. This constitutes thedeparting point of the following two sub-sections. B. S Acting as a Reconfigurable Intelligent Surface
In this section, we analyze the case study in which thesurface S operates as an RIS whose phase Φ ( x ) can beappropriately optimized. In particular, we assume that S actsas an anomalous reflector that is configured for reflecting theEM waves emitted by Tx towards a given direction. Dueto the assumption C ( x ) = 1 , we implicitly ignore parasiticscattering. To this end, Φ ( x ) in (3) is chosen as follows: Φ ( x ) = (cid:0) ¯ φ T − ¯ φ R (cid:1) x + φ /k (9)where φ ∈ [0 , π ) is a fixed phase shift, ¯ φ T = − ¯ x T .p ¯ x T + ¯ y T , ¯ φ R = ¯ x R .p ¯ x R + ¯ y R , and (¯ x T , ¯ y T ) and (¯ x R , ¯ y R ) are parameters that are optimized for obtaining thedesired reflection capability, as detailed in further text.In contrast with Section III-A, this case study needs moreelaboration to unveil the scaling law of the intensity of theelectric field as a function of the distances. To this end, weconsider the specific setup in which ¯ φ T = sin ( θ T (0)) and ¯ φ R = sin ( θ R (0)) , which corresponds to a surface S that isconfigured by taking into account the angles of incidence andreflection of the EM with respect to (0 , . Other case studiesare analyzed in [18]. It is worth mentioning that this setup doesnot necessarily imply (¯ x T , ¯ y T ) = ( x T , y T ) and (¯ x R , ¯ y R ) =( x R , y R ) , which would imply that the locations of Tx (thepoint source) and Rx (the observation point) need to be exactlyknown. Setups corresponding to different locations of Tx andRx, but yielding the same angles of incidence and reflection,are included in the considered case study. For example, Tx andRx move along two straight lines in which the angles with the y -axis at (0 , are kept constant but the distances are not.The following two propositions report approximate expres-sions of the intensity of the electric field in (3) under theassumption of short and long distance regime, respectively. Proposition 3.
Assume ¯ φ T = sin ( θ T (0)) and ¯ φ R =sin ( θ R (0)) . In the short distance regime, the intensity of theelectric field in (3) can be approximated as follows: | E x ( x R , y R ) | ≈ √ πk q − ¯ φ T + q − ¯ φ R q(cid:0) − ¯ φ R (cid:1) d T + (cid:0) − ¯ φ T (cid:1) d R (10) Proof.
The proof is similar to Proposition 1 [18].
Proposition 4.
Assume ¯ φ T = sin ( θ T (0)) and ¯ φ R =sin ( θ R (0)) . In the long distance regime, the intensity of theelectric field in (3) can be approximated as follows: | E x ( x R , y R ) | ≈ L π q − ¯ φ T + q − ¯ φ R √ d T d R (11) Proof.
The proof is similar to Proposition 2 [18].Since ¯ φ T and ¯ φ R in Propositions 3 and 4 do not depend onthe distances, the following conclusions can be drawn. • From (10), RISs behave as anomalous mirrors in the shortdistance regime: the intensity of the electric field decayswith the square root of a weighted sum of the distances,but the angles of incidence and reflection can be different. • From (11), RISs behave as scatterers in the long distanceregime: the intensity of the electric field decays as afunction of the square root of the product of the distances. • In (10) and (11), ¯ φ T is configured based on the directionof incidence (at (0 , and with the y -axis) of the EMwave emitted by Tx, and ¯ φ R is configured based on thedesired direction of reflection (at (0 , and with the y -axis) of the EM wave reflected by the RIS. By optimizing ¯ φ R , RISs can be configured to reflect EM waves towards,predominantly, any directions. The limitations are dis-cussed in, e.g., [10]. This is different from (6), in whichthe direction of reflection and incidence coincide. This isthe difference between specular and anomalous mirrors. C. S Acting as a Passive Reflecting Beamformer
In this section, we analyze the case study in which the sur-face S operates as an RIS whose phase Φ ( x ) is appropriatelyoptimized in order for S to act as a beamformer. The differencewith the previous sub-section can be summarized as follows. • In Section III-B, the desired functionality of the RISconsists of reflecting (or steering) the incident EM wavetowards a predetermined direction . All the receivers lo-cated in the direction of reflection benefit from the RIS.This setup is, therefore, more suitable for RISs that areemployed for broadcasting applications [13]. • In this sub-section, on the other hand, the desired func-tionality of the RIS is to focus the EM wave towards apredetermined location . In this case, a single or a fewreceivers at specific locations benefit from the RIS.In particular, we consider that S acts as a beamformer (or areflecting lens) that focuses the signal towards a single location (¯ x R , ¯ y R ) . To this end, Φ ( x ) in (3) is chosen as follows: Φ ( x ) = q ( x − x T ) + y T + q ( x − ¯ x R ) + ¯ y R (12) From a mathematical point of view, Φ ( x ) in Section III-B is optimized such that the first-order derivative of P ( x ) isequal to zero at x s ∈ [ − L, L ] (if it exists). The phase Φ ( x ) in(12) is, by contrast, optimized such that P ( x ) is equal to zerowhen evaluated at the location of interest, i.e., (¯ x R , ¯ y R ) . Thedesign criterion for optimizing the surface S is, thus, different.The intensity of the electric field can be computed, in thelong distance regime, by using analytical steps similar tothose reported in Sections III-A and III-B [18]. Due to spacelimitations, the details are omitted. Numerical illustrations are,however, reported in the next section in order to showcase thedifference among the three configurations for the surface S .IV. N UMERICAL R ESULTS
In this section, we provide some illustrative numericalresults in order to showcase the difference among the threeconfigurations for S that are elaborated in Section III, and inorder to numerically evaluate the intensity of the electric fieldas a function of the transmission distances. For ease of writing,the directions of incidence and reflection are identified by theangles θ T and θ R , respectively. In all simulation results, weconsider the following setup: (i) L = 0 . m; (ii) f = 28 GHz; (iii) x T = − d T sin ( θ T ) and y T = d T cos ( θ T ) with θ T = π /4 ; (iv) (¯ x T , ¯ y T ) = ( x T , y T ) ; and (v) φ = 0 . Thissetup corresponds to a scenario in which S is employed in themillimeter-wave frequency band, and its size, L , correspondsto, approximately, the diagonal of a two-dimensional surfaceof size 1 m . This is compatible and in agreement with otherrecent papers and experimental activities [13], [16], [21].More precisely, we consider two case studies. • In the first case study, we are interested in illustratingthe difference among the three different configurationsfor S analyzed in Section III. To this end, we plot theintensity of the electric field emitted by a fixed location(Tx) and observed at different locations ( x R , y R ) . Thefollowing setup is considered: (i) d T = 11 m; (ii) ¯ x R = d R sin ( θ R ) and ¯ y R = d R cos ( θ R ) with d R = 5 mand θ R = π /3 . As for the surface S in Section III-B,this setup corresponds to reflecting an EM wave that isincident at an angle of degrees with the y -axis towardsan angle of degrees with the y -axis. As for the surface S in Section III-C, this setup corresponds to focusingan EM wave that is incident at an angle of degreeswith the y -axis towards the single location (¯ x R , ¯ y R ) =(4 . , . m; and (iii) the observation region is chosenin the range x R ∈ [ − , m and y R ∈ [0 , m. • In the second case study, we are interested in illustratingthe different scaling law of the intensity of the electricfield as a function of the transmission distances, and,in particular, in showcasing the two operating regimesthat correspond to short and long transmission dis-tances. To this end, the following setup is considered:(i) x T = − d sin ( θ T ) and y T = d cos ( θ T ) ; (ii) x R = d sin ( θ R ) and y R = d cos ( θ R ) with θ R = π /4 forthe uniform surface in Section III-A and θ R = π /6 for the RIS in Section III-B; and (iii) the Tx-to- S and S -to-Rx distances are the same and are in the range d ∈ [0 , m. Thus, the end-to-end distance is d . Sec. III-A -2 0 2 4 6 8 10 x y Sec. III-B -2 0 2 4 6 8 10 x y Sec. III-C -2 0 2 4 6 8 10 x y -3 Fig. 1: Intensity of the electric field from Theorem 1. -3 -2 I n t en s i t y E l e c t r i c F i e l d Th. 1Prop. 1Prop. 2 -3 -2 I n t en s i t y E l e c t r i c F i e l d Th. 1Prop. 3Prop. 4
Fig. 2: Comparison of short and long distance approximations.
The results corresponding to the first case study are reportedin Fig. 1, which shows the intensity of the electric field ob-tained from (3). The figure substantiates the findings in SectionIII: (i) the angles of incidence and reflections of a uniformreflecting surface are the same (Section III-A); (ii) an RISconfigured as described in Section III-B is capable of steeringthe reflected signal towards desired (anomalous) directions;and (iii) an RIS configured as described in Section III-C iscapable of focusing the signal towards desired locations.The results corresponding to the second case study arereported in Fig. 2, which compares (3) with the approximatedclosed-form expressions obtained in Sections III-A and III-B. The figure substantiates the findings in Section III. Inparticular: (i) the closed-form approximations for the shortdistance regime are accurate for end-to-end distances ( d )up to 100-150 m; and (ii) the closed-form approximations forthe long distance regime are accurate for end-to-end distances( d ) greater than 200-250 m. For the considered setup, weconclude that RISs are capable of acting as anomalous mirrorsfor distances of the order of tens of meters. The range of distances for which the approximation holds depend, amongother parameters, on the size of the surface and the operatingfrequency. In general, the larger the size of the surface is andthe higher the operating frequency is, the more accurate theapproximation as an anomalous mirror becomes, i.e., it can beused for longer transmission distances.V. C ONCLUSION
In this paper, we have leveraged the general scalar theory ofdiffraction in order to obtain approximate closed-form expres-sions of the intensity for the electric field reflected by RISsin the short and long transmission distance regimes. We haveobserved different scaling laws in the two considered operatingregimes. The proposed approach and results constitute a firstattempt to identify appropriate path-loss models for analyzingthe achievable performance of RISs in wireless networks.A
CKNOWLEDGMENT
This work was supported in part by the European Com-mission through the H2020 5GstepFWD project under grant722429 and the H2020 ARIADNE project under grant 871464.R
EFERENCES[1] M. Di Renzo et al., “Smart radio environments empowered by reconfig-urable AI meta-surfaces: An idea whose time has come”, EURASIP J.Wireless Commun. Networking, Vol. 129, 20 pages, May 2019.[2] A. Zappone et al., “Wireless networks design in the era of deep learning:Model-based, AI-based, or both?”, IEEE Trans. Commun., Oct. 2019.[3] C. Liaskos et al., “A new wireless communication paradigm throughsoftware-controlled metasurfaces”, IEEE Commun. Mag., Sep. 2018.[4] C. Liaskos et al., “Using any surface to realize a new paradigm forwireless communications”, Commun. ACM, Nov. 2018.[5] E. Basar et al., “Wireless communications through reconfigurable intel-ligent surfaces”, IEEE Access, Vol. 7, pp. 116753-116773, 2019.[6] K. Ntotnin et al., “Reconfigurable intelligent surfaces vs. relaying: Dif-ferences, similarities, and performance comparison”, ArXiv:1908.08747,submitted, 2019.[7] R. Karasik et al., “Beyond max-SNR: Joint encoding for reconfigurableintelligent surfaces”, ArXiv:1911.09443, submitted, 2019.[8] C. Huang et al., “Holographic MIMO surfaces for 6G wireless networks:Opportunities, challenges, trends”, ArXiv:1911.12296, submitted, 2019.[9] N. Yu et al. “Light propagation with phase discontinuities: Generalizedlaws of reflection and refraction”, Science, vol. 334, Oct. 2011.[10] V. S. Asadchy et al., “Perfect control of reflection and refraction usingspatially dispersive metasurfaces”, Phys. Rev. B, vol. 94, Aug. 2016.[11] N. M. Estakhri et al., “Wave-front transformation with gradient meta-surfaces”, Phys. Rev. X, vol. 6, Oct. 2016.[12] F. Liu et al., “Intelligent metasurfaces with continuously tunable localsurface impedance for multiple reconfigurable functions”, Phys. Rev.Applied, Apr. 2019.[13] W. Tang et al., “Wireless communications with reconfigurable intel-ligent surface: Path loss modeling and experimental measurement”,ArXiv:1911.05326, submitted, 2019.[14] J. C. Bucheli et al., “Reconfigurable intelligent surfaces: Bridging thegap between scattering/reflection”, ArXiv:1912.05344, submitted, 2019.[15] S. W. Ellingson et al., “Path loss in reconfigurable intelligent surface-enabled channels”, ArXiv:1912.06759, submitted, 2019.[16] W. Khawaja et al., “Coverage enhancement for NLOS mmWave linksusing passive reflectors”, ArXiv:1905.04794, submitted, 2019.[17] O. Ozdogan, “Intelligent reflecting surfaces: Physics, propagation, andpathloss modeling”, IEEE Wireless Commun. Lett., 2020 (to appear).[18] M. Di Renzo et al. “On the path-loss of reconfigurable intelligentsurfaces: An approach based on the theory of diffraction”, in submission.[19] E. N. Economou.