Electromagnetic Model of Reflective Intelligent Surfaces
11 Electromagnetic Model of Reflective Intelligent Surfaces
Filippo Costa,
Senior Member, IEEE and Michele Borgese,
Member, IEEE
We present an accurate and simple analytical model for thecomputation of the reflection amplitude and phase of ReflectingIntelligent Surfaces. The model is based on a transmission-line circuit representation of the RIS which takes into accountthe physics behind the structure including the effect of allrelevant geometrical and electrical parameters. The proposedrepresentation of the RIS allows to take into account the effectof incidence angle, the mutual coupling among elements, theeffect of the interaction of the periodic surface with the RISground plane. It is shown that, by using the proposed approach,it is possible to maximize the power received by a user in aRIS-assisted link without recurring to onerous electromagneticsimulations. The proposed model aims at filling the gap inthe design of RIS assisted communications algorithms that areusually disconnected from physical implementation issues andrealistic performance of these surfaces.
Index Terms —Equivalent Circuit model (ECM), ReflectiveIntelligent Surfaces (RIS), Intelligent Reflective Surfaces (IRS),Metasurfaces, Transmission Line (TL).
I. I
NTRODUCTION
Reflecting Intelligent surface (RIS) assisted wireless com-munication has recently emerged as a promising solution toenhance the spectrum and energy efficiency of future wirelesssystems [1]–[5]. Specifically, a RIS allows to control thewireless propagation environment via an array of reconfig-urable passive reflecting elements [6]–[8]. RIS technology isattractive from an energy consumption point of view sinceit is possible forwarding the incoming signal without em-ploying any power amplifier like in MIMO arrays [9]–[11],but rather by suitably designing the phase shifts applied byeach reflecting element in order to constructively combinethe reflected signal. RIS can be also employed for physicalinformation encoding in the so called massive backscatterwireless communication (MBWC) scheme [12]. RIS struc-tures have the advantage of being easily integrable in thecommunication environment because of the easy deploymentinto buildings, ceilings of factories or into human clothing[13]. A RIS is essentially a periodic metasurface [14] withcapacitive elements located at sub-wavelength distance loadedwith active components in order to control the phase and theamplitude of the reflection coefficient. For instance, they canbe tuned such that signals bouncing off a RIS are combined Filippo Costa is with University of Pisa, Dipartimento di Ingeg-neria dell’Informazione, Pisa, Italy. (corresponding author, e-mail: fi[email protected])Michele Borgese is with the Research and Development Department,SIAE MICROELETTRONICA, 20093 Cologno Monzese, Italy. (e-mail:[email protected]) Intelligent Reflecting Surface (IRS) is alternatively used in other works.The acronyms IRS and RIS can be used interchangeably as they refer to thesame physical device. In this paper we used the term RIS.
Fig. 1: Three dimensional layout of the analyzed problem. constructively to increase signal in the position of the intendedreceiver or destructively to avoid leaking signals to undesiredreceivers [15]. The modeling, analysis and design of RISs forapplication to wireless networks is a multidisciplinary researchtopic covering communication theory, computer science andelectromagnetism. A major limitation of current research onRISs in wireless networks is the lack of accurate models thatdescribe the reconfigurable meta-surfaces as a function of theirEM properties.The vast majority of research works available to date rely onthe assumption that metasurfaces act as perfect reflectors witha controlled phase reflection. Some papers employ a fittingempirical model which is reasonable at normal incidence [6].Actually, the response of RIS to the radio waves dependson the geometrical and electrical properties of the designedsurface, on the choice of substrates (low-loss materials leadto lower losses but the cost of the surface scales rapidly withthe size), the characteristics of the tunable components, butthe also depends on the properties of the impinging EM wavesuch as angle of incidence, the angle of reflection [16], po-larization [17]. Physics based models which take into accountthe aforementioned design parameters are therefore of cru-cial importance to order to avoid too optimistic performancepredictions [18]–[20]. This approach is limiting since sometheoretical results may be invalidated by the impossibility of apractical implementation in the real world and also because theproposed theoretical design do not have a corresponding peaceof hardware behind. Moreover, designing RIS by neglectingsome important aspects such as the change of the response atoblique incidence (spatial dispersion) does not allow to achieve a r X i v : . [ ee ss . SP ] F e b optimal performance. The hard separation between communi-cation and EM aspects is not beneficial. It would be importantthat communication algorithms may be supported by somerealistic, accurate but fast EM models. It is important that theEM counterpart of the problem is treated in real time with anaccurate but still analytical approach since, given the alreadyhigh computational burden associated with the introduction ofRIS in the communication scenario, it cannot be acceptable aslow EM analysis including all the geometrical details of theperiodic pattern. We present an accurate and simple analyticalmodel for the computation of the phases of the RIS whichtakes into account incidence angle and the coupling amongelements. Moreover, our approach also considers the effectsof the interaction of the periodic surface with the RIS groundplane. The models also consider accurately the behavior ofthe RIS as a function of the incidence angle. By using theproposed approach, it is possible to derive the scattered fieldsin the space without recurring to onerous electromagneticsimulations. We will show the accuracy of the model byconsidering an example of RIS which connects a receiver toa transmitter.The paper is organized as follows. In Section II theaddressed problem is discussed and motivated. Section IIIpresents the analytical model of the RIS and its accuracy isverified by comparing the reflection coefficients with electro-magnetic simulations. The model is employed in Section IVwhere a practical example of RIS-assisted communication isanalysed. Finally conclusions are drawn in Section V.II. A DDRESSED P ROBLEM
The aim of the work is to present a physically based simpleand accurate model for Reflecting Intelligent Surfaces. Themodel should take into account the properties of RIS which arethe reconfigurability, the oblique incidence properties (spatialdispersion), the mutual coupling among closely spaced cellsand reflection losses. Moreover, the model can guarantee adirect link between the reflection phases and the physicallayout of the RIS. RIS are synthesized through thin resonantcavities that fully reflect incident waves with an arbitraryreflection phase. The reconfigurability can be achieved invarious way but the simplest approach to achieve the full re-configurability of the RIS is based on the adoption of varactordiodes within each unit cell. Fig. 1 shows the physical aspectof the structure with an periodic surface composed by squaremetal patches loaded with varactor diodes. In order to focusa significant amount of the power towards the user, the RISdimensions might be considerably large because the collectedpower is related to the physical area of the reflecting surface.Consequently, it is not uncommon that both the transmitter andthe received can be located in the near field of the surface [21],[22]. Therefore it is of vital importance that the model of thesurface is able to consider the specific incident and reflectedangle for each unit cell.
A. Reconfigurability
When a AIS is loaded with active components to obtain aRIS, the reflection properties of each unit cell of the surface can be controlled as a function of a direct current (DC)voltage. By adequately changing the polarization state ofthe varactors, it is possible to perform the beamforming sothat each user can be reached with the maximum availablepower. It has to be pointed out that the beamforming canbe performed both in the near field of the surface (nearfield focusing) and in the far field synthesizing the classicalfar field beam directed toward the user. To this aim, in themicrowave regime, varactor [23] or PIN diodes [24], whoseEM characteristics (say, capacitances) can be dramaticallytuned through varying applied voltages, can be incorporatedwithin the surface. By independently controlling the voltagesof each unit cell, phase/amplitude profiles can be accuratelysynthesized in order to shaping and controlling the reflectedfields. Examples of varactor-controlled electrically steerablereflector in the microwave regime can be found since theearly 2000s [25]–[27]. Alternative approaches to tune EMproperties of metadevices rely on the free carrier doping inconductive materials with electrical gating and photoexcitationmethods. Conventional semiconductors (e.g., GaAs, Silicon,and germanium), atomically thin 2D materials (e.g., graphene;MoS2), and transparent conducting oxides or nitrides suchas Indium Thin Oxide (ITO), Aluminum-doped Zinc Oxide(AZO) are widely used in the spectral range covering THzand up to the visible regime [28], [29]. Mechanical tuningoffers another effective way to switch the EM propertiesof metadevices by reconfiguring the shape and surroundingenvironment of meta-atoms by the means of MEMS/NEMS,elastic subtract, or microfluidics [28]–[30].
B. Oblique Incidence
One important aspect of these structures is their workingmechanism which is based on the interference between theperiodic metasurface and the ground plane. Consequently, thereflection properties of an electromagnetic wave striking upona RIS, strongly depend on the incidence angle. In electromag-netic language, it is said that the device is spatially dispersive.This aspect is completely overlooked in most of the paperspublished on this subject. For this reason, a physics based butstill simple model is extremely important for the correct designof the RIS. Two meaningful situations can be distinguishedwhen an electromagnetic wave impinges on the surface: paral-lel polarization (TM - Transverse Magnetic) and perpendicularpolarizations (TE - Transverse Electric). At normal incidence,for a symmetric surface topology, the behaviour of the surfacefor parallel and perpendicular polarization coincides. However,for off-normal angles the behaviour of the RIS for TE andTM polarization should be analyzed separately. In Fig. 2 theparallel and perpendicular polarization incidence are shown.In the parallel polarization, the electric field lays in the planeof incidence whereas the magnetic field is orthogonal to thesame plane. On the contrary, for the parallel polarization case,the magnetic field lays in the plane of incidence whereas theelectric field is orthogonal to the same plane.
C. Mutual Coupling
Another key factor in the analysis of RIS is the effectof the mutual coupling. The electromagnetic analysis of RIS E iz E ix zE i E s θ i H i H s xθ s (a) H iz H ix zH i E s θ i E i H s xθ s (b) Fig. 2: Oblique incidence on a RIS. (a) Parallel polarization (or Tran-verse Magnetic - TM) (b) Perpendicular polarization (or TransverseElectric - TE). is based on the local periodicity approach [31]–[33]. It isindeed assumed that the size of the characteristics of theelements varies gradually along the surface and this justify theapproximation of studying a single unit cell response with aninfinite set of identical elements. Once a parametric simulationof the elements with different sizes is performed, a databaseof phases is created and thus a correspondence betweengeometrical parameters and phase response of a particularpixel of the periodic surface can be created. The analysis ofthe phase response of the periodic surface for a certain elementconfiguration has a twofold advantage. The first advantageis that the periodic surface can be analysed by using theFloquet theorem which consider the single unit cell placed inan infinite array of elements all identical to each other. Thisapproach, is much more efficient (in terms of computationaltime required for the EM simulation) than studying an isolatedunit cell. The second advantage is that the mutual couplingbetween neighboring unit cells is automatically included inthe calculations because the unit cell is analysed in an infinitearray and not in a standalone configuration. For this reason, wepresent in the following section an approach for the modellingof a periodic Impedance surface configuration. More detailsregarding the Floquet theorem are provided in Appendix-C
D. Reflection Losses
A physical surface with a phase-controlled reflection coef-ficient is strongly affected by losses. The losses in the surfacedepend on multiple factors: dielectric losses due to the useof real materials (glass-reinforced epoxy laminate materialsuch as FR4 are cost-effective but they also exhibit highlylosses), ohmic losses due to the current flowing on imperfectconductors and losses in the active components. It is importantto underline that an efficient strategy to limit the effect oflosses consists in keeping small the periodicity of the lattice[34]–[36]. Moreover, the adoption of a pattern with smallperiod is also the best strategy for achieving wideband RIS[36], [37]. The usual practice of designing reflectarrays with λ/ spaced elements is not the best configuration for thebandwidth maximization or for loss minimization. However,reducing the period determines the increase of the numberof active components on the periodic surface required toachieve reconfigurability. Consequently, a suitable compromisebetween these two requirements should be found.III. EM M ODEL OF
RISThe not reprogrammable version of the surface is alsoknown as Artificial Impedance Surface (AIS) [17], [37], [38].The AIS comprises a periodic surface printed on the top ofa grounded dielectric slab. The periodic surface can vary inshape but is essentially a two–dimensional capacitive sheetformed by disconnected metal obstacles. An effective approachto model the behavior of periodic surfaces is to establish ananalogy with lumped filters. A Transmission-Line (TL) circuitanalysis, unlike full-wave simulations, provides immediateresults and a good physical insights into the design propertiesof the structure. In low frequency bands, when the periodicityof the periodic surface is much larger than the operatingwavelength λ , the periodic surface can be efficiently analyzedby using homogenized theory [17]. In the intermediate fre-quency range, where the periodicity is smaller but comparablewith the operating wavelength, the periodic surface elementcan be resonant. In this region the periodic surface can stillbe modelled by using the circuit theory but the values ofthe lumped parameters need to be retrieved by using a full-wave simulations followed by an inversion procedure [39].When the operating wavelength becomes shorter than guidedwavelength, grating lobes emerge (i.e., higher Floquet spaceharmonics become propagating) and a single impedance wouldnot suffice for the description of the properties of the sur-face [40]. The conventional model of an artificial impedancesurface consists in a parallel connection between the surfaceimpedance of the metasurface and inductive impedance of atransmission line closed on a short circuit [17], [41]. As shownin Fig. 3, the input impedance of the AIS structure ( Z v ) isequal to the parallel connection between impedance of theperiodic surface Z surf and the impedance of the groundeddielectric slab Z T E/T Md : Z v = Z surf Z T E/T Md Z surf + Z T E/T Md (1)Once computed the input impedance of the structure, thecomplex reflection coefficient can be calculated as follow:
Γ = Z v − ζ Z v + ζ (2)Two meaningful cases should be distinguished when the re-flection coefficient is computed for oblique incidence: parallelpolarization (TM - Transverse Magnetic) in which the electricfield lies on the plane of incidence and the perpendicularpolarization (TE - Transverse Electric) in which the electricfield is perpendicular with respect to the plane of incidence.The two oblique incidence cases are represented in Fig. 2. Fig. 3: Equivalent circuit model of a RIS.
The thin grounded dielectric slab behaves as an inductor.Consequently, its impedance can be computed analytically asfollows: Z T E/T Md = jZ T E/T M tan ( k z d ); (3)where Z T E/T M represents the characteristic impedance ofthe substrate along the equivalent transmission line towards z direction: Z T E = ωµk z ; Z T M = k z ωε ε r (4) k z = k (cid:112) ε r − sin θ i is the normal propagation constantwithin the substrate, k is the propagation constant of theincident plane wave and θ i is the incidence angle. At normalincidence the TE and TM case equals to each other and theimpedance of the medium is simply ζ / √ ε r , where ζ is thefree space impedance ( ζ = (cid:112) µ /ε ), where ε and µ are thefree space dielectric permittivity and magnetic permeability,respectively. However, at oblique incidence two cases must beanalysed since the behaviour of the surface change drasticallydepending on the polarization of the field [17], [41]. Theimpedance of a periodic surface can be represented througha lumped equivalent circuit if the periodicity of the latticeis lower than one wavelength. The specific geometry of theunit cell of the periodic surface determines the impedanceproperties of the surface and its circuit representation. Usually,the capacitance and the inductance are determined by the shapeof the periodic element while the resistance is influenced bythe losses of the metal of the printed structure. For instance, theimpedance of an array of patches ( Z patch ) can be representedby a capacitance only up to a certain frequency where thegranting lobe onset: Z T E/T Mpatch = R patch + 1 jωC T E/T Mpatch (5)On the contrary, more complicated geometries have also aconsiderable inductive part in the surface impedance [40].It is important to highlight that the impedance of a periodicstructure strongly depends on the incidence angle ( θ i ) of theEM wave impinging on the AIS. For the case of periodic patches arrays, the capacitance for the TE and the TM po-larization can be expressed as [17], [42]: C T Epatch = D y ε ε eff π ln (cid:18) (cid:16) πwy Dy (cid:17) (cid:19) (cid:16) − k k eff sin θ i (cid:17) (6) C T Mpatch = 2 D x ε ε eff π ln (cid:16) πw x D x (cid:17) (7)where ε eff = (1 + ε r ) / is the effective permittivity ofthe uniform medium surrounding the periodic surface, k eff isthe wave number of the incident wave vector in the effectivehost medium. D x , D y are the periodicity of the lattice and w x and w y are the gaps between the squares patches along x and y planar directions. The averaged formulas can beemployed for computing the reflection coefficient both fornormal and oblique incidence. The presented patches arraysmodel which is analyzed for the case of azimuthal incidenceangle equal to zero, can be extended to a generic angle due tothe geometrical symmetry of the unit cell. The aforementionedaveraged expressions of the capacitance refer to cases in whichthe dielectric thickness is sufficiently thick so that high-orderFloquet harmonics can be neglected [41]. In order to takeinto account the effect of higher-order Floquet harmonics thecapacitance of the periodic surface can be corrected as [41]: C T E/T Mpatch = C T E/T Mpatch − C patch − ground (8)where C patch − ground is a correction terms which takes intoaccount the capacitance between the patch and the groundplane: C patch − ground = 2 Dε π ln (1 − e − πd/D ) (9)when D x = D y = D . If the substrate thickness d be-comes much smaller than the cell periodicity D , the influenceof higher-order (evanescent) Floquet modes reflected by theground plane starts to play an important role and, when theratio d/D approaches to zero, the correction term tends toinfinite as well as the capacitance of the patch array. If d is small compared to the wavelength but large compared to D , the interaction between the grid and the metal plane isnegligible.The analysis of the RIS surface is performed by adding anadditional lumped capacitor, in parallel with the capacitancecreated by the patch array [42]. The effect of the diode istaken into account by placing the lumped element impedancein parallel to the impedance of the periodic surface. As is wellknown, a varactor diode can be represented as a series of aresistor and a capacitor: Z var = R var + jωL var + 1 jωC var (10)where C var represents the variable capacitance of the diode.The series inductance L var depends on the size of the lumpedcomponent and must be included in the varactor model [43]to take into account the self resonance of the component. Thelosses of the varactor are taken into account by the series resistor R var . The bandwidth of the structure is proportional toequivalent inductance of the grounded substrate. Consequently,the thicker the substrate, the larger the operating bandwidthand therefore the smother the phase slope as a function offrequency. Concerning the element shape, the patch with a varysmall edge gap represents the optimal solution to maximize theoperating bandwidth of the AIS [37].In order to assess the accuracy of the proposed circuitmodel, an example of reconfigurable surface is analysed.The selected parameters are the following: D = 5 mm , d = 1 . mm, w = 0 . mm and C var = [0 . − . pF.The reflection phase of the AIS without the varactor diodeis initially analysed at normal and oblique incidence bothby using the proposed model and by using a commercialelectromagnetic simulator, that is CST Microwave Studio. Thelayout of the unit cell analysed with CST is shown in Fig. 4.The amplitude and phase reflection coefficient of the surfaceare reported in Fig. 5 for normal and oblique incidence. Asevident, the agreement between the full-wave results and theproposed model is satisfactory for both normal and obliqueincidence with only a small frequency shift between CSTsimulations and the model. However, it has to be pointed outthat the shift diminishes as the discretization mesh is refinedin the electromagnetic solver. The main hurdle in computingthe Radar Cross Section (RCS) of these finite structureshosting several resonators placed close to a ground plane isthat the resonators and the metallic surface act as a Fabry-Perot interference device with multiple reflection contributionsinvolved. In order to accurately capture these phenomena,a very fine local mesh is required to achieve good results.The next step is to introduce the varactors which allowsthe active control of the properties of the AIS. In this case,the surface can be classified as Intelligent Reflective Surfacesince the reflection phase properties can be controlled an eachunit cell by though a DC voltage. Each unit cell should becontrolled independently in order to guarantee a completereconfiguration. An architecture for achieving the full controlof the elements in the two planar direction was presented in[25].The reflection phase of the surface controlled by varying thecapacitance of the varactor diode is shown in Fig. 6. The theproposed transmission line (TL) model is extremely accuratefor both amplitude and phase reflection of the RIS. In Fig. 7,the surface impedance of the periodic surface loaded withvaractor diode is also shown. The behaviour of the impedanceis shown as a function of the frequency for several values ofthe capacitance of the varactor. The impedance obtained byusing the proposed model is compared with the impedanceextracted from CST according to [44]. It is evident that theproposed model predicts very well the behaviour of the surfaceimpedance till well above the resonance frequency of the RIS.It is also interesting to point out that the equivalent surfaceimpedance turns from capacitive to inductive behaviour forrelevant varactor capacitance values (i.e. C var = 0 . pF).This is due to the fact that the varactor inductance plays animportant role in the impedance behaviour. It has to be pointedout that both series inductance is considered in the CST model.The inductance of the lumped component is a distributed (a) (b) Fig. 4: 3D layout of the Varactor loaded RIS simulated with CSTMicrowave Studio. The simulation relies on the Floquet theoremwhich considers the unit cell placed in an infinite periodic array:(a) analysed unit cell, (b) equivalent infinite array.
Model - = 0°CST - = 0°Model - = 60°CST - = 60° (a)
TE-amplitude
Model - = 0°CST - = 0°Model - = 60°CST - = 60° (b)
TM-amplitude
Model - = 0°CST - = 0°Model - = 60°CST - = 60° (c)
TE-phase
Model - = 0°CST - = 0°Model - = 60°CST - = 60° (d)
TM-phase
Fig. 5: (a) TE and (b) TM reflection phase for normal ( θ i = θ s = θ = 0 ◦ ) and oblique incidence ( θ i = θ s = θ = 60 ◦ ). The equivalentTL model is compared with the results obtained with CST MicrowaveStudio. (a) amplitude (b) phase Fig. 6: (a) Amplitude and (b) phase of the reflection coefficient atnormal incidence for a RIS composed by a varactor loaded AIS. Theresults of the model are compared with CST Microwave Studio.) (a)
Real Part (b)
Imaginary part
Fig. 7: (a) Real and (b) imaginary part of the surface impedance Z surf as a function of frequency for different values of the varactorcapacitance. -6-5-4-3-2-10 (a) TE amplitude -6-5-4-3-2-10 (b)
TM amplitude -150-100-50050100150 (c)
TE phase -150-100-50050100150 (d)
TM phase
Fig. 8: (a) TE amplitude, (b) TM amplitude, (c) TE phase (d) TMphase. The geometrical and electrical parameters used for the RISare the following: D x = D y = D = 5 mm, w x = 0 . mm, ε r =4 . − j . , d = 1 . mm, σ c = 58 . × S/m , R var = 0 . . inductance created by the presence of the lumped capacitorbetween two neighbouring patches. Once that the accuracy ofthe model has been verified, the proposed simple model can beadopted to synthesize the phase properties of the RIS. To thisaim, it is necessary to select some geometrical parameters forthe RIS and then create a database of reflection coefficients asa function of diode varactor and incidence angle. Therefore,four matrix are needed: two for the reflection amplitude forTE and TM polarization and two for the reflection phase forTE and TM polarization. By using the proposed model, it istrivial to create a lookup table which links the capacitance statewith the precise phase and amplitude value of the reflectioncoefficient for the required incidence angle and polarization.An example of bidimensional maps are reported in Fig. 8where the reflection coefficient as a function of the varactorcapacitance and incident angle is shown through a color map.It is immediately clear how the phase value of the surfaceis characterized by a consistent phase variation for a certaincapacitance value of the varactor. Also the reflection amplitudeof the cell changes as a function of the angle both for TE and TM polarization. In the reported color maps, the ohmic lossesin the periodic surface, the losses in the dielectric substrate( ε r = 4 . − j . ) and the losses introduced by the seriesresistance of the varactor are considered ( R var = 0 . ).For the case of patches array with D x = D y = D and w x = w y = w , ohmic losses in the periodic surface areconsidered as [45]: R patch = (cid:18) DD − w (cid:19) R s (11)where R s = σ c δ , ω is the angular frequency, δ is the skindepth of metal ( δ = √ σ c ωµ ), µ is the magnetic permeabilityin a classical vacuum and the dielectric losses in the substrateare considered while the resistance of the varactor is set to ω .The conductivity of copper for the metallic surface is σ c =58 . × S/m . A. Reproducible Research
The simulation results for can be reproduced using the codeavailable at: https://github.com/MicheleBorgese/Intelligent-SurfacesIV. L
INK BUDGET OF
RIS
ASSISTED COMMUNICATION
The link between the transmitter and receiver can be mod-elled as a classical backscattering communication system. Thegeometry of the communication path from the transmitted tothe receiver passing though the RIS is shown in Fig. 9. The RISis placed in the xy -plane of a Cartesian coordinate system, andthe geometric center of the RIS is aligned with the origin of thecoordinate system. Let M and N denote the number of rowsand columns of the regularly arranged unit cells of the RIS,respectively. The size of each unit cell is D x and D y along x and y axis respectively. The received signal power in RIS-assisted wireless communications can be computed accordingto eq.(12) which is shown in the next page. In this equation, λ is the wavelength of the impinging plane wave, Γ m,n is thereflection coefficient of the ( m, n ) th cell, which is evaluatedby the proposed analytical approach described in section III.The pedexes m, n denote the m th row and n th column of theprogrammable surface. The symbols r tm,n , and r rm,n identifythe distance of the ( m, n ) th unit cell from the transmitter(TX) and the receiver (RX), respectively. The elevation angleand azimuth angles of transmitter and receiver with respect to ( m, n ) th unit cell are identified by θ tm,n , ϕ tm,n , θ rm,n , ϕ rm,n .The polarization of the reflection coefficient can parallel orperpendicular depending on the polarization of the transmittingantenna.As shown in Fig. 9, the transmitter emits a signal towardsthe RIS with power P t through an antenna with power gain G t ( θ txn,m ) . The signal is reflected by the RIS and then receivedby the receiver with a gain G r ( θ rxn,m ) . It is assumed that thepolarization of the transmitter and receiver are always properlymatched, even after the transmitted signal is reflected by theRIS. The radiation pattern of the transmitting and receivingantenna can be approximated through a cos q function and thus T X ( x t , y t , z t ) RX ( x r , y r , z r ) r t n , m r r n , m xyz ϕ r θ txn,m θ rxn,m θ r θ t r t r r ϕ t x (cid:48) z (cid:48) ϕ rn,m θ rn,m θ tn,m ϕ tn,m Blocking Object D x D y z (cid:48) x (cid:48) Fig. 9: Layout of the RIS-assisted non-line of sight communication scenario. P r = P t (4 π ) λ M (cid:88) m =1 N (cid:88) n =1 G t ( θ txm,n ) G r ( θ rxm,n ) σ ( θ tm,n , θ rm,n ) | Γ m,n | e − j ( (cid:54) Γ m,n ) e − j k ( r rm,n + r tm,n ) | r tm,n | | r rm,n | (12)the gain of the transmitting/receiving antenna to the specificunit cell of the RIS can be computed according to: G ( θ ) = 4 π cos q ( θ ) (cid:90) π (cid:90) π/ cos q ( θ ) dθdφ (13)The gain is only a function of the θ angle. The recevidepower calculation in eq.(12) is similar to the one followedin [46]. Differently from [46], we used the concept of RadarCross Section (RCS) [47], which is more common for scat-tering problems. Anyway, the results are consistent with [46]since the maximum RCS of the unit cell can be approximatedas the gain of the unit cell multiplied by its area. The bistaticRCS of a unit cell can be computed according to bistatic RCSof a metallic plate. Assuming a parallel polarized incidentplane wave (Fig. 2(a)) on the yz -plane (this is commonlycalled T M x in scattering problems), the RCS in eq.(12) canbe written as follows: σ ( θ tm,n , θ rm,n ) = 4 π (cid:18) D x D y λ (cid:19) cos ( θ tm,n ) ×× sin (cid:16) kD y (cid:0) sin θ rm,n + sin θ tm,n (cid:1)(cid:17) kD y (cid:0) sin θ rm,n + sin θ tm,n (cid:1) (14)where θ tm,n ∈ [ − π , π ] and θ sm,n ∈ [ − π , π ] . The RCStowards a generic direction is directly proportional to thephysical area of the scattering object.In order to verify the importance of the TL model of the RISalso for oblique incidence, we present an example of a RIS-assisted communication between two antennas. Let us assumeto have a transmitting antenna in the position ( − , , cmand a receiving antenna in the position (20 , , cm. The RISis composed by × cells with a size of mm. It is assumed that both the transmitting and receiving antennas are pointedtowards the centrer of the surface and are linearly polarizedwith a field along y -direction (this implies that the obliqueincidence on the surface is TE polarized). The distance of thetransmitter and receiver from the RIS are compact in orderto maintain low the time needed to draw the two dimensionalfield maps. However, the proposed model can be applied towhatever distance from the RIS taking into account that thelarger the RIS size, the larger the amount of energy deflectedtowards the receiver position. Indeed, the same principles usedin the design of reflectarray or reflector antennas [32], [33]hold for addressing which would be the most suitable panelsize given a certain location of the transmitter . In the caseof reflectarray antennas, the parameters known as illuminationefficiency and spillover efficiency are used to determine thecorrect size of the reflector but in a communication scenariothe location of the transmitter with respect to the RIS isusually unknown and the optimization of the RIS size is notneeded. However, if the location of the transmitter with respectto the RIS is known, the RIS size may be optimized basedon spillover efficiency and illumination efficiency. The idealphases of the surface to maximize the power at the receiverlocation can be found analytically according to: (cid:54) (Γ m,n ) = k ( r rm,n + r tm,n ) (15)However, in order to design an actual RIS which physicallyimplements the behaviour of the ideal surface, the idealphases found according to (15) should be translated into thegeometrical parameters of the RIS (shape of the unit cell, peri-odicity, dielectric thickness and dielectric permittivity) and thecapacitance values provided by the varactors connected in eachunit cell. This process can be advantageously accomplishedthrough the model presented in section II. Two cases arepossible: the incidence angle with respect to each unit cell isneglected and all the phases are computed at normal incidence; alternatively, the incident angle for each unit cell is consideredand the exact capacitance value needed for achieving thedesired phase at the incidence angle specific for the unit cell iscomputed. It is evident that, while the optimal phases can besynthesized by properly selecting the geometrical parametersof the unit cell and the capacitance state, the ideal amplitudescannot be synthesized since a realistic surface realizationis accompanied by some intrinsic absorption losses of thesurface. A 2D plot of the power radiated in the plane parallelto both transmitting and receiving antenna is reported inFig. 10 for three cases: ideal, normal incidence phases, obliqueincidence phases. As is evident, the optimal situation is the onewith the ideal phase values and the ideal amplitude values(perfect reflection). If the phases are optimized consideringnormal incidence for all elements, a considerable drop of thereceived power is obtained both because of the non idealitiesdue to the absorption phenomena and more importantly for thephase detuning. The third approach, the proposed one, whichrelies on the oblique incidence model instead allows to limitthe power drop with respect to the ideal situation since thephase detuning due to the spatial dispersion (oblique incidenceeffect) of the surface is eliminated. In this example a powergain of 3.9 dB is achieved by using the proposed approachcompared to the normal incidence approach. In general, theachieved power gain considering the oblique incidence can belower or even much higher if the impinging EM waves comefrom grazing angles with respect to the normal to the surface.The only case in which the normal incidence approach canguarantee the same performance of the proposed approach isthe particular situation where the EM wave impinges at normalincidence. The optimized values of the varactor capacitancesare reported in Fig. 11. Both the cases of the normal incidencemodel and the proposed oblique incidence model are shown.The same figure reports also the percentage error of thecapacitances synthesized by neglecting the dependence onincidence angle. V. C ONCLUSION
An accurate analytical model for the synthesis of reflectingintelligent surfaces has been presented. The model is basedon an analytical representation of the surface with a trans-mission line circuit approach. The RIS reflection coefficient iscontrolled through varactors connecting periodic patches. Thevaractors can be programmed through a biasing network toadjust locally the reflection coefficient phase of the RIS in or-der to achieve the desired beam forming and the correspondingmaximization of the power on the receiver position. It has beenshown that the use of the proposed model, which is capableof catching the reflection coefficient behaviour both at normaland oblique incidence for both for TE and TM polarizations,provide a consistent power gain in the design of RIS-assistedcommunications. A
PPENDIX
ARCS
OF A C ONDUCTING F LAT P LATE
The RCS of a plane wave on a flat plate can be computedboth for TE and TM polarized waves. The derivation are pretty -60 -40 -20 0 20 4010152025 P R =36.9 dBRIS -10010203040 (a) ideal -60 -40 -20 0 20 4010152025 P R =31.1 dBRIS -100102030 (b) normal incidence model -60 -40 -20 0 20 4010152025 P R =34.8 dBRIS -100102030 (c) proposed model Fig. 10: Received power ( P r ) on the xz -plane: (a) ideal, (b) normalincidence model, (c) proposed model. Both the normal incidencemodel and the proposed (oblique incidence) model include the effectof reflection losses due to non-idealities of the RIS (ohmic loss,substrate loss, varactor loss) but the oblique incidence model alsoconsider the specific angle of incidence for each unit cell for derivingthe optimal varactor state. similar for the two polarizations but what is changing are thetangential components of the surface current excited on themetal plate. We present the TM case but similar considerationshold for TE waves. Let us assume to have a parallel polarized( T M x ) plane wave on the yz -plane impinging with an angle θ i upon a rectangular electric perfectly conducting flat plate ofdimensions a and b . The incident electric and magnetic fieldscan be written as follow: E i = E (cid:0) ˆ a y cos θ i + ˆ a z sin θ i (cid:1) e − jβ ( y (cid:48) sin θ i − z (cid:48) cos θ i ) (16) H i = E ζ ˆ a x e − jβ ( y (cid:48) sin θ i − z (cid:48) cos θ i ) (17)The surface current induced in the metallic place reads: J s = ˆ a z × (cid:0) H i + H s (cid:1) | z =0 = 2ˆ a y H i | z =0 == ˆ a y E ζ e − jβy (cid:48) sin θ i (18) (a) normal incidence model (b) proposed model (c) Percentage error
Fig. 11: Capacitances optimized with (a) normal incidence ( C normalvar )model and with (b) oblique incidence ( C obliquevar ) model, (c) percent-age error of the capacitance computed with normal incidence model: Error (%) = | ( C obliquevar − C normalvar ) / ( C obliquevar ) |× . (a) -90 -60 -30 0 30 60 90-50-40-30-20-100 (b) Fig. 12: (a) Uniform
T M x plane wave incident on a rectangularconducting plate; (b) far-field scattering of square a metallic plate( a = b = 15 cm) of with an incident angle of θ i = θ s = 38 . ◦ computed with the double summation approach (according to (12))considering the metallic surface composed by × perfectlyconductive unit cells and with the RCS relation for a flat plate(according to (22)). The scattered field of each curve is normalizedto the maximum value which is reached when θ i = θ s (bistaticscattering). The surface current for the reported case is only in the y -direction and it is different from zero only in the regionoccupied by the metallic surface. The scattered field can becomputed by evaluating the integral of the flat current on thefinite sized surface. The finite integral has the form of a sinc function. Once computed the E θ and the E φ component of thescattered field [47], the radar cross section of the metallic platecan be computed in three dimensional space can be written asfollow: σ ( θ i , θ s , ϕ s ) = 4 π (cid:18) abλ (cid:19) ×× (cid:0) cos θ s sin ϕ s + cos ϕ s (cid:1) (cid:18) sin XX (cid:19) (cid:18) sin YY (cid:19) (19)where X and Y are defined as: X = ka θ s cos ϕ s (20) Y = kb (cid:0) sin θ s sin ϕ s − sin ϕ i (cid:1) (21)If we restrict the analysis to the plane of incidence, yz plane,the expression of the RCS simplifies as [48]: σ ( θ i , θ s ) = 4 π (cid:0) abλ (cid:1) cos ( θ i ) (cid:18) sin ( kb ( sin θ s − sin θ i )) kb (sin θ s − sin θ i ) (cid:19) (22)A PPENDIX BP OWER RECEIVED WITH A METALLIC PLATE IN FAR FIELD
The expression for the calculation of the power reflected bythe RIS can be simplified to an analytical form in case that thetransmitter and the received are far from the surface and thereflection coefficient of all unit cells are identical. Indeed, inthis case, the surface just introduce a phase delay to the wholeimpinging rays. The Perfect Electric Conductor (PEC) case isa particular case of uniform surface where all the elementsare metallic and thus they are characterized by a reflectioncoefficient equal to -1. In this case the power scattered patternshould agree with the canonical obtained by using the classicalexpression for radar cross section of a metallic plate [47].Indeed, in case of N × M identical metallic elements, thedouble summation in eq. (12) leads to: P r = P t (4 π ) λ ( M N ) G t ( θ tx ) G r ( θ rx ) σ ( θ t , θ r ) | r t | | r r | (23)By substituting the closed form expression for the unit cellsRCS, we obtain: P r = P t (4 π ) ( M N D x D y ) G t ( θ tx ) G r ( θ rx ) | r t | | r r | cos ( θ t ) ×× sin (cid:16) kD y (sin θ r − sin θ t ) (cid:17) kD y (sin θ r − sin θ t ) (24)The power is maximized on if the incident and scatteredangle are the same but with a 180° rotation of the azimuthangle (bistatic case). Moreover, the relation highlight that,once fixed the gain of the transmitter and receiver, the poweravailable at the receiver only depends on the size of theRIS and on the transmitter-surface distance distance, r t andsurface-receiver distance, r r and not on the frequency. In thebistatic case, the received power can be simplified as: P r = P t (4 π ) ( D RISx D RISy ) G t ( θ tx ) G r ( θ rx ) | r t | | r r | cos ( θ t ) (25)where D RISx = M D x and D RISy = N D y .A PPENDIX CF LOQUET THEOREM FOR PERIODIC STRUCTURES
The reflecting intelligent surface is basically a periodicstructure perturbed by the local change of active components.The analysis of the surface can be efficiently performed byanalysing a single unit cell under the local periodicity approachand thus by exploiting the Floquet theorem. This approachallows to consider the effect of neighbouring cells in thereflection coefficient and consequently the effect of the mutualcoupling. For simplicity, we describe a monodimensionalperiodic surface lying on the xz -plane and with a periodicityalong x -direction as shown in Fig. 13. A similar approach isvalid for two-dimensional periodic surfaces. z xD x Fig. 13: Bidimensional representation of a periodic ArtificialImpedance Surface.
In presence of a periodic structure as the one reported inFig. 13, any electromagnetic field component can be describedas: φ ( x + D x , y, z ) = e − jk x D x φ ( x, y, z ) (26)The exponential factor indicates the complex phase shiftbetween neighbouring cells ( k x = β − jα ) , implying that thefield components differ only for a phase term. This relation iscalled Floquet Theorem [49]. The field component satisfyingeq.(26) can be written described as: φ ( x, y, z ) = e − jk x D x P ( x, y, z ) (27)where P is a periodic function ( P ( x, y, z ) = P ( x + D x , y, z )) which can be thereforeexpanded in a Fourier series: P ( x, y, z ) = ∞ (cid:88) n = −∞ a n ( z, y ) e − j πnDx x (28)where a n is a functions that describes the field variation in yz -plane. By using eq.(27) and eq.(28), the field can be finallywritten as: φ ( x, y, z ) = ∞ (cid:88) n = −∞ a n ( x, y ) e − jk xn x (29)where k xn = k x + 2 πD x n, with n = 0 , ± , ± ... (30) The terms of the summation in eq.(29) are usually referredas Floquet spatial harmonics . The Floquet harmonics consistof a set of plane waves which can be in propagation (thepropagation constant β is purely real) or in cut-off (thepropagation constant β is purely imaginary) but all of themcontribute in the determination of the local field distribution.R EFERENCES[1] T. S. Rappaport, Y. Xing, O. Kanhere, S. Ju, A. Madanayake, S. Mandal,A. Alkhateeb, and G. C. Trichopoulos, “Wireless communications andapplications above 100 ghz: Opportunities and challenges for 6g andbeyond,”
Ieee Access , vol. 7, pp. 78 729–78 757, 2019.[2] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment:Intelligent reflecting surface aided wireless network,”
IEEE Communi-cations Magazine , vol. 58, no. 1, pp. 106–112, 2019.[3] ——, “Beamforming optimization for wireless network aided by intel-ligent reflecting surface with discrete phase shifts,”
IEEE Transactionson Communications , vol. 68, no. 3, pp. 1838–1851, 2019.[4] S. Hu, F. Rusek, and O. Edfors, “Beyond massive mimo: The potentialof data transmission with large intelligent surfaces,”
IEEE Transactionson Signal Processing , vol. 66, no. 10, pp. 2746–2758, 2018.[5] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, andL. Hanzo, “Intelligent reflecting surface aided mimo broadcasting forsimultaneous wireless information and power transfer,”
IEEE Journalon Selected Areas in Communications , vol. 38, no. 8, pp. 1719–1734,2020.[6] S. Abeywickrama, R. Zhang, Q. Wu, and C. Yuen, “Intelligent reflectingsurface: Practical phase shift model and beamforming optimization,” arXiv preprint arXiv:2002.10112 , 2020.[7] D. Dardari, “Communicating with large intelligent surfaces: Fundamen-tal limits and models,”
IEEE Journal on Selected Areas in Communica-tions , vol. 38, no. 11, pp. 2526–2537, 2020.[8] M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen,J. de Rosny, and S. Tretyakov, “Smart radio environments empowered byreconfigurable intelligent surfaces: How it works, state of research, andthe road ahead,”
IEEE Journal on Selected Areas in Communications ,vol. 38, no. 11, pp. 2450–2525, 2020.[9] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectralefficiency of very large multiuser mimo systems,”
IEEE Transactionson Communications , vol. 61, no. 4, pp. 1436–1449, 2013.[10] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massivemimo for next generation wireless systems,”
IEEE communicationsmagazine , vol. 52, no. 2, pp. 186–195, 2014.[11] P. Rocca, G. Oliveri, R. J. Mailloux, and A. Massa, “Unconventionalphased array architectures and design methodologies—a review,”
Pro-ceedings of the IEEE , vol. 104, no. 3, pp. 544–560, 2016.[12] H. Zhao, Y. Shuang, M. Wei, T. J. Cui, P. Del Hougne, and L. Li,“Metasurface-assisted massive backscatter wireless communication withcommodity wi-fi signals,”
Nature communications , vol. 11, no. 1, pp.1–10, 2020.[13] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, andC. Yuen, “Reconfigurable intelligent surfaces for energy efficiency inwireless communication,”
IEEE Transactions on Wireless Communica-tions , vol. 18, no. 8, pp. 4157–4170, 2019.[14] M. Borgese and F. Costa, “A simple equivalent circuit approach foranisotropic frequency-selective surfaces and metasurfaces,”
IEEE Trans-actions on Antennas and Propagation , vol. 68, no. 10, pp. 7088–7098,2020.[15] P. Mursia, V. Sciancalepore, A. Garcia-Saavedra, L. Cottatellucci,X. Costa-P´erez, and D. Gesbert, “Risma: Reconfigurable intelligentsurfaces enabling beamforming for iot massive access,”
IEEE Journalon Selected Areas in Communications , 2020.[16] W. Chen, L. Bai, W. Tang, S. Jin, W. X. Jiang, and T. J. Cui,“Angle-dependent phase shifter model for reconfigurable intelligent sur-faces: Does the angle-reciprocity hold?”
IEEE Communications Letters ,vol. 24, no. 9, pp. 2060–2064, 2020.[17] O. Luukkonen, C. Simovski, G. Granet, G. Goussetis, D. Lioubtchenko,A. V. Raisanen, and S. A. Tretyakov, “Simple and accurate analyticalmodel of planar grids and high-impedance surfaces comprising metalstrips or patches,”
IEEE Transactions on Antennas and Propagation ,vol. 56, no. 6, pp. 1624–1632, June 2008.[18] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M.-S. Alouini, andR. Zhang, “Wireless communications through reconfigurable intelligentsurfaces,”
IEEE Access , vol. 7, pp. 116 753–116 773, 2019. [19] G. C. Alexandropoulos, G. Lerosey, M. Debbah, and M. Fink, “Recon-figurable intelligent surfaces and metamaterials: The potential of wavepropagation control for 6g wireless communications,” arXiv preprintarXiv:2006.11136 , 2020.[20] M. Najafi, V. Jamali, R. Schober, and H. V. Poor, “Physics-based mod-eling and scalable optimization of large intelligent reflecting surfaces,” IEEE Transactions on Communications , 2020.[21] S. W. Ellingson, “Path loss in reconfigurable intelligent surface-enabledchannels,” arXiv preprint arXiv:1912.06759 , 2019.[22] E. Bj¨ornson and L. Sanguinetti, “Power scaling laws and near-fieldbehaviors of massive mimo and intelligent reflecting surfaces,”
IEEEOpen Journal of the Communications Society , vol. 1, pp. 1306–1324,2020.[23] F. Costa, A. Monorchio, S. Talarico, and F. M. Valeri, “An active high-impedance surface for low-profile tunable and steerable antennas,”
IEEEAntennas and Wireless Propagation Letters , vol. 7, pp. 676–680, 2008.[24] S. Genovesi, A. Monorchio, M. B. Borgese, S. Pisu, and F. M. Va-leri, “Frequency-reconfigurable microstrip antenna with biasing networkdriven by a pic microcontroller,”
IEEE Antennas and Wireless Propa-gation Letters , vol. 11, pp. 156–159, 2012.[25] D. F. Sievenpiper, J. H. Schaffner, H. J. Song, R. Y. Loo, and G. Tang-onan, “Two-dimensional beam steering using an electrically tunableimpedance surface,”
IEEE Transactions on antennas and propagation ,vol. 51, no. 10, pp. 2713–2722, 2003.[26] D. F. Sievenpiper, “Forward and backward leaky wave radiation withlarge effective aperture from an electronically tunable textured surface,”
IEEE transactions on antennas and propagation , vol. 53, no. 1, pp.236–247, 2005.[27] H. Li, F. Costa, Y. Wang, Q. Cao, and A. Monorchio, “A switchable andtunable multifunctional absorber/reflector with polarization-insensitivefeatures,”
International Journal of RF and Microwave Computer-AidedEngineering , p. e22573.[28] Q. He, S. Sun, L. Zhou et al. , “Tunable/reconfigurable metasurfaces:physics and applications,”
Research , vol. 2019, p. 1849272, 2019.[29] O. Tsilipakos, A. C. Tasolamprou, A. Pitilakis, F. Liu, X. Wang, M. S.Mirmoosa, D. C. Tzarouchis, S. Abadal, H. Taghvaee, C. Liaskos et al. ,“Toward intelligent metasurfaces: The progress from globally tunablemetasurfaces to software-defined metasurfaces with an embedded net-work of controllers,”
Advanced Optical Materials , vol. 8, no. 17, p.2000783, 2020.[30] X. Wang, A. D´ıaz-Rubio, H. Li, S. A. Tretyakov, and A. Al`u, “Theoryand design of multifunctional space-time metasurfaces,”
Physical ReviewApplied , vol. 13, no. 4, p. 044040, 2020.[31] D. M. Pozar, S. D. Targonski, and H. Syrigos, “Design of millimeterwave microstrip reflectarrays,”
IEEE transactions on antennas andpropagation , vol. 45, no. 2, pp. 287–296, 1997.[32] P. Nayeri, A. Z. Elsherbeni, and F. Yang, “Radiation analysis approachesfor reflectarray antennas [antenna designer’s notebook],”
IEEE Antennasand Propagation Magazine , vol. 55, no. 1, pp. 127–134, 2013.[33] M. Borgese, F. Costa, S. Genovesi, and A. Monorchio, “An iterativedesign procedure for multiband single-layer reflectarrays: Design andexperimental validation,”
IEEE Transactions on Antennas and Propaga-tion , vol. 65, no. 9, pp. 4595–4606, Sept 2017.[34] F. Costa and A. Monorchio, “Closed-form analysis of reflection lossesin microstrip reflectarray antennas,”
IEEE transactions on antennas andpropagation , vol. 60, no. 10, pp. 4650–4660, 2012.[35] J. Ethier, M. Chaharmir, and J. Shaker, “Loss reduction in reflectarraydesigns using sub-wavelength coupled-resonant elements,”
IEEE trans-actions on antennas and propagation , vol. 60, no. 11, pp. 5456–5459,2012.[36] D. Pozar, “Wideband reflectarrays using artificial impedance surfaces,”
Electronics letters , vol. 43, no. 3, pp. 148–149, 2007.[37] F. Costa, S. Genovesi, and A. Monorchio, “On the bandwidth of high-impedance frequency selective surfaces,”
IEEE Antennas and WirelessPropagation Letters , vol. 8, pp. 1341–1344, 2009.[38] S. Maci, M. Caiazzo, A. Cucini, and M. Casaletti, “A pole-zero matchingmethod for ebg surfaces composed of a dipole fss printed on a groundeddielectric slab,”
IEEE Transactions on Antennas and Propagation ,vol. 53, no. 1, pp. 70–81, 2005.[39] F. Costa, A. Monorchio, and G. Manara, “Efficient analysis of frequency-selective surfaces by a simple equivalent-circuit model,”
IEEE Antennasand Propagation Magazine , vol. 54, no. 4, pp. 35–48, Aug 2012.[40] ——, “An overview of equivalent circuit modeling techniques of fre-quency selective surfaces and metasurfaces,”
Appl. Comput. Electro-magn. Soc. J. , vol. 29, no. 12, pp. 960–976, 2014.[41] F. Costa, S. Genovesi, A. Monorchio, and G. Manara, “A circuit-basedmodel for the interpretation of perfect metamaterial absorbers,”
IEEE Transactions on Antennas and Propagation , vol. 61, no. 3, pp. 1201–1209, 2012.[42] F. Costa and A. Monorchio, “Design of subwavelength tunable andsteerable fabry-perot/leaky wave antennas,”
Progress In Electromagnet-ics Research , vol. 111, pp. 467–481, 2011.[43] S. A. Note, “Varactor spice models for rf vco applications,”
SkyworksProprietary information, February , 2010.[44] F. Costa, A. Monorchio, and G. Manara, “An equivalent-circuit modelingof high impedance surfaces employing arbitrarily shaped fss,” in .IEEE, 2009, pp. 852–855.[45] ——, “Analysis and design of ultra thin electromagnetic absorbers com-prising resistively loaded high impedance surfaces,”
IEEE Transactionson Antennas and Propagation , vol. 58, no. 5, pp. 1551–1558, 2010.[46] 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 withreconfigurable intelligent surface: Path loss modeling and experimentalmeasurement,”
IEEE Transactions on Wireless Communications , 2020.[47] C. A. Balanis,
Advanced engineering electromagnetics . John Wiley &Sons, 2012.[48] ¨O. ¨Ozdogan, E. Bj¨ornson, and E. G. Larsson, “Intelligent reflectingsurfaces: Physics, propagation, and pathloss modeling,”
IEEE WirelessCommunications Letters , vol. 9, no. 5, pp. 581–585, 2019.[49] T. Itoh, “Periodic structures for microwave engineering,” in