arXiv:1912.06759v2 [eess.SP] 13 Apr 2021 ignaTc,Bakbr,V,201UAemi:ellingson e-mail: USA 24061 VA, Blacksburg, Tech, Virginia aiyprmtrzbe hc skont eareasonable a be to known an by identical is are bypassed which elements is commonly the parameterizable, coupling) of is easily responses mutual issue the that for This assuming calculate. accounting to element difficult the that of is and disadvant pattern principal strategy pattern the The this element explicit. since is of the is RIS problems and the This design within loss [2]. spacing and path discrete e.g. between analysis the see relationship elements; practical as from computed for scattered is ideal strategies fields general scattering of two strategy, are sum first there which the for In RIS, the by tering hne safnto fRSsz,ln emty n element and geometry, link size, RIS of states. function transmitter-RIS-rec i a the consideration in as important loss channel path An the paper. is this engineering of RIS cla focus this the individually; of is element RISs consists each electromag by control the scattered of which field magnitude netic in possibly and envision reflectarray phase often of the are changing form RISs some therein. improve be receiver references to to and and order [1] transmitters in e.g. between see manner channel controlled propagation a the in signals scatters fpt oso h I-nbe hne eaiet hto th is of which matter employin that networks channels, the of to reflection technology. design relative elucidate the specular in channel further consideration and be RIS-enabled important to direct the to predict space used of free shown performance is loss the then path model of exhibit This is that (2). and using scenarios specific (1) identify and to with possible designs it making begins (2), with theor work consistent and electromag laws this continuous scaling a determining limits), for (2) useful or o (most designs) usefu surface of array specific (most an function elemen spacings of (1) the a and analysis set either patterns as to presumes element used loss work Thi parameterizable method previous path RIS. the Whereas characterize and reflectarray-type states. geometry, to link passive used size, a los RIS then path of is the sim consisting model a calculating links channel presents for paper a facilitate method This broadly-applicable loss. to RIS-en path yet of otherw example, is study would the channels in that for propagation consideration stations principal way; A mobile blocked. and be desirable stations base a between scatter in to elements signals individually-controllable of array an ..Elnsni ihteDp.o lcrcladCmue E Computer and Electrical of Dept. the with is Ellingson S.W. nlsso ahls eursamto o acltn scat calculating for method a requires loss path of Analysis that device a is (RIS) surface intelligent reconfigurable A Abstract Arcngrbeitlietsrae(I)employs (RIS) surface intelligent reconfigurable —A ahLs nRcngrbeIntelligent Reconfigurable in Loss Path .I I. embedded NTRODUCTION ufc-nbe Channels Surface-Enabled lmn atr ie,the (i.e., pattern element ..Ellingson, S.W. [email protected]. ngineering, practical incident RIS g abled etical eiver sof ss netic for l of s age ple ise an ed ed eirMme,IEEE Member, Senior s; d n e - s t - f . hc a eueu nbte nesadn hstopic. this understanding summary better independent th in concise useful [6]–[9], a be [4], may yields in which paper While addressed this literature. previously in the been analysis in has the confusion of issue channe of source this f reflection a loss been the specular path has of which and of that transmitter-receiver matter to space the relative elucidate channel transmitter-RIS-receiver further to model reflectarray practical typical a sec RIS. to the applicable using are obtained results strategy the wh that spacing, havi half-wavelength confirms elements with essentially of dBi equal 5 consists condition area about design of boundary particular having directivity This the plate RIS. second the where conducting the to perfectly case by a predicted special that corresponds the the and which to in for equal [2], design is strategy than particular loss a simpler derive path but (i.e., to calculated to used strategy then similar pattern is first is element which which parameterizable the spacings) of following array and an loss with path beginning of ascertain. to calculation difficult be speci may and a explicit comprising not states is the element RIS because eleme and the devices spacing, and RIS fo conditions pattern(s), specific boundary useful the of particularly are between implement design not relationship that to and is on) strategy analysis used so this the technology and However path specific RIS. length, how the path the see (e.g., area, of laws” li fields; RIS independent theoretical “scaling with scattered identifying useful varies the for loss and ideal for performance is solved of This be [4]–[6]. can e.g., value boundary that a yields problem equations, w Maxwell’s condition which, with currents) boundary combined surface or electromagnetic impedance to surface (typically subject surface regularly-spaced ous of arrays planar [3]. large elements for approximation the rnmtatnasystem, antenna transmit nen ytmi h direction the in system antenna where I R II. eodcnrbto fti oki h s fthis of use the is work this of contribution second A for method a is work this of contribution principal The continu- an as treated is RIS the strategy, second the In eern oFgr ,tesailpwrdniyicdn on incident density power spatial the 1, Figure to Referring n th P ECEIVED lmn is: element T stettlpwrapidb h rnmte othe to transmitter the by applied power total the is P S WRI THE IN OWER n i = P G T G T ( T ˆ r ( n i ˆ r ) n i ˆ RIS-E r n i ) stegi ftetransmit the of gain the is / and , 4 πr NABLED i,n 2 r i,n stedistance the is C HANNEL -type mits ond hen ich ree (1) fic ng ls, nt e s s 1 r 2

The voltage-like signal observed by the receiver is:

N jφn y = bn PR,n e (7) n=1 X p where φn = (ri,n + rs,n)2π/λ (8) is the phase accrued by propagation over the path that includes th the n element, and the bn’s are complex-valued coefficients representing the controlled responses of the elements. The total power PR observed by the receiver is therefore:

N 2 jφn PR = bn PR,n e (9)

n=1 X p

III. PATH LOSS AND ELEMENT P ATTERN A. Path Loss In order to separate factors associated with the propagation Fig. 1. Geometry for scattering from element n of an N-element RIS. The transmitter and receiver are indicated as T and R respectively. The unit vectors channel (including the RIS) from the antenna gains of the i s ˆrn and ˆrn are used to indicate directions from the transmitter (“incident”) transmitter and receiver, we make the following approxima- and to the receiver (“scattered”), respectively. The unit vector nˆ indicates the i tions: (1) GT (ˆrn) is constant with respect to n; i.e., the gain outward-facing perpendicular (“broadside”) direction. s of the transmitter is constant over the RIS. (2) GR(−ˆrn) is constant with respect to n; i.e., the gain of the receiver is from the transmitter to the nth element. The power captured constant over the RIS. Then: by the nth element is 4 PR =PT GT GR (λ/4π) i i i P = S A (−ˆr ) (2) N 2 n n e n G (−ˆri )G (ˆrs ) · b e n e n ejφn ǫ (10) where A (−ˆri ) is the effective aperture of the element in the n r2 r2 p e n n=1 s i,n s,n i X direction of the transmitter. This is related to the gain Ge(−ˆrn) i i 2 This expression is exact if the transmit and receive antenna of the element by Ae(−ˆrn) = Ge(−ˆrn)λ /4π where λ is wavelength. Thus: systems exhibit isotropic gain. However, the approximation is broadly applicable. In particular, this approximation is suitable i i i 2 Pn = PT GT (ˆrn)Ge(−ˆrn) (λ/4πri,n) (3) for transmit and receive antenna radiation patterns which are only weakly directional, as is the case for mobile stations. The power density at the receiver due to scattering from the th This approximation is also suitable for transmit and receive n element is: antenna systems exhibiting approximately constant gain over s s s 2 the angular span corresponding to the RIS – this is possible Sn = PnGe(ˆrn)/4πrs,n (4) even if the transmit and receive antenna systems form narrow s s where Pn is the power applied by the RIS to the element, ˆrn is beams, as long as the RIS is sufficiently far away. the direction from the element to the receiver, and rs,n is the Equation 10 is in the form of the Friis transmission equation; distance from the element to the receiver. Note that the same therefore the path loss LRIS is given by: · pattern Ge( ) is used for both incidence and scattering, which 2 4 N i s is valid as long as the maximum dimension of the element − λ G (−ˆr )G (ˆr ) L 1 = b e n e n ejφn ǫ (11) is ≪ λ. The power captured by the receiver from the RIS RIS n 2 2 p 4π s ri,nrs,n element is   n=1 X P = Ss G (−ˆrs )λ2/4π (5) R,n n R n B. Element Pattern Model s where GR(−ˆrn) is the gain of the receiver’s antenna system Next, we seek a expression for the element radiation pattern in the direction of the element. Ge(·) that is simple yet broadly-applicable. Since RIS ele- s i The efficiency ǫp = Pn/Pn accounts for the limited ments are commonly envisioned to be electrically-small low- efficiency of practical antenna elements and insertion losses gain elements above a conducting ground screen, we choose associated with components required to implement the desired the popular model (see e.g., [3], Sec. 9.7.3): change in magnitude and phase. If the RIS is passive, ǫp ≤ 1. 2q Combining expressions, we obtain: Ge(ψ)= γ cos (ψ) 0 ≤ ψ <π/2 (12) =0 π/2 ≤ ψ ≤ π (13) λ 4 G (−ˆri )G (ˆrs ) ri −rs e n e n PR,n = PT GT (ˆn)GR( ˆn) 2 2 ǫp 4π ri,nrs,n where ψ is the angle measured from RIS broadside, q de-   (6) termines the gain of the element, and γ is the coefficient 3 required to satisfy conservation of power. Power is conserved Thus, Equation 11 becomes: by requiring the integral of Ge(ψ) over a surface enclosing the 2 4 N − i · 2q0 s · 2q0 element to be equal to 4π sr. It is shown in [3] (Sec. 9.7.3) −1 λ ( ˆrn nˆ) (+ˆrn nˆ) jφn LRIS = 2 bn 2 2 e ǫp that this constraint is satisfied for 256π s ri,nrs,n n=1 X (19) γ = 2(2q + 1) (14)

An appropriate value of q is determined from the broadside IV. FAR CASE gain Ge(ψ = 0) of the element. From Equations 12–14: In this section we consider the “far” case. For the purposes of this paper, the RIS is said to be far from the transmitter if q = Ge(ψ = 0)/4 − 1/2 (15) i ˆrn and ri,n are approximately independent of n; i.e., approx- For general studies of RIS-based wireless communications, imately equal to the same constants ˆri and ri, respectively. s it is awkward to choose q (hence the element gain) to cor- Similarly, the RIS is said to be far from the receiver if ˆrn and respond to a particular element design. Instead, a physically- rs,n are approximately equal to the same constants ˆrs and motivated “benchmark” value of q is preferred. Such a value rs, respectively. No approximation is made for the phases φn: may be obtained by requiring Ae(ψ = 0) to be equal to These values continue to be exact and are not assumed to be (λ/2)2. Under this condition, the physical area of a RIS is independent of n. equal to the sum of the effective apertures of the elements when the elements are separated by λ/2. This criterion is A. Expressions for Path Loss in the Far Case of particular interest because it essentially corresponds to the Under the far approximation, Equation 19 simplifies to: “electromagnetic surface” paradigm described as the “second 2 strategy” in Section I. Invoking this criterion, we require: 4 − · 2q0 · 2q0 N −1 λ ( ˆri nˆ) (+ˆrs nˆ) jφn L = bne ǫp 2 RIS 2 2 2 2 (16) 256π ri rs Ae(ψ =0)= λ /4π Ge(ψ =0)=(λ/2) n=1 X (20) ∼ ∼ This yields γ = π, q =0.285, and
subsequently Ge(ψ = 0) = Equation 20 correctly indicates that path loss for the RIS in 5 dBi. This value is consistent with the gain of typical patch 2 2 the far case is proportional to ri rs , regardless of the chosen antenna elements, which range between 3 dBi and 9 dBi (see coefficients (bn’s). Equation 20 further indicates that path loss e.g. [3], Sec. 11.2). We therefore define the desired benchmark is minimized when the phase of bn is set equal to −φn. value of q to be q0 =0.285. Assuming phase-only control of the elements, one would select −jφn At this point, it should be emphasized that this pattern model bn = e . In this case Equation 20 reduces to: does not require that the elements be spaced by ; however λ/2 4 2q0 2q0 − λ (−ˆri · nˆ) (+ˆrs · nˆ) if the elements are spaced in this manner, the choice of q = q0 1 2 (21) LRIS = 2 N 2 2 ǫp will result in the broadside effective aperture of the RIS being 256π ri rs equal to its physical aperture. Similarly, the choice of q = q0 Recall that the q = q0 element pattern proposed in Sec- is not required, and in fact this parameter can be “tuned” to tion III-B was derived from the constraint that the sum of model other specific element designs. the broadside effective apertures of the elements is equal to In subsequent work, we shall assume that all element the physical area A of the RIS when the element spacing is patterns are identical; i.e., constant with respect to n. In equal to λ/2. If we now commit to this element spacing, then practice, the patterns of elements close to the center of a A = N (λ/2)2 and Equation 21 can be expressed as: large RIS will be nearly symmetric and uniform, whereas the 2 − A patterns of elements near the edge will exhibit some degree L 1 = (−ˆr · nˆ)2q0 (+ˆr · nˆ)2q0 ǫ (22) RIS 4πr r i s p of asymmetry and gain variation [3]. Since the ratio of “edge  i s  elements” to “interior elements” is small for an electrically- Equation 22 indicates that path loss in the far case depends large RIS, this variation will typically not significantly affect only on the physical area of the RIS, and not at all on fre- path loss calculation. quency. This is actually the expected result, as is demonstrated in the next section. C. Alternative Form of the Path Loss Equation B. Consistency with Plate Scattering Theory A useful alternative form of Equation 11 may be obtained using the element pattern model proposed in the previous Electromagnetic plate scattering theory is noted in [7] as section. First, note that cos ψ = ˆr(ψ) · nˆ where ˆr(ψ) is a a possible starting point for RIS scattering models. In this unit vector pointing outward from the element, indicating the section it is shown that the connection to plate scattering direction in which the pattern is being evaluated; nˆ is the theory actually follows naturally from the model developed unit normal vector indicating RIS broadside, i.e., the direction so far, and does not need to be introduced as a postulate. To corresponding to ψ = 0; and “·” denotes the scalar (“dot”) see this, imagine that an RIS in the far case is replaced by a flat perfectly-conducting plate of area , and let us assume product. Now assuming q = q0, we find: A monostatic geometry; i.e., ˆri = −nˆ and ˆrs = +nˆ). In this i i 2q0 Ge(−ˆrn)= π −ˆrn · nˆ ,and (17) case, the radar range equation (see e.g. [3], Sec. 4.6) is: 2q0 s s 3 Ge(+ˆrn)= π (+ˆrn · nˆ) (18) 2 2 2
PR = PT GT GRλ σ/ (4π) ri rs (23) 4

“minimum” “typical” where σ is the broadside monostatic radar cross section of the Freq. fe = 0.1 km 1 km fe = 0.1 km 1 km plate, which is known to be: 0.8 GHz 6.1 m 19.4 m 8.9 m 28.8 m 1.9 GHz 4.0 m 12.6 m 5.8 m 18.2 m 2 2 σ =4πA /λ (24) 2.4 GHz 3.5 m 11.2 m 5.1 m 16.2 m 5.8 GHz 2.3m 7.2m 3.3 m 10.4 m see e.g. [10], Sec. 3.7. Recasting Equation 23 in the form of 28.0 GHz 1.0m 3.3m 1.5m 4.7m 60.0 GHz 0.7m 2.2m 1.0m 3.2m the Friis transmission equation, the path loss Lplate in this scenario is given by: TABLE I SIDE LENGTH OF A SQUARE RIS FOR PATH LOSS EQUAL TO THAT OF THE −1 2 Lplate = (A/4πrirs) (25) SPECULARREFLECTIONCHANNEL (PHYSICAL DIMENSION).

Now note that Equation 22 gives precisely this result for ˆri = −nˆ, ˆr =+nˆ, and ǫ =1. Therefore Equation 22 is consistent “minimum” “typical” s p = 0 1 1 = 0 1 1 with electromagnetic plate scattering theory. Freq. fe . km km fe . km km 0.8 GHz 16.3 λ 51.6 λ 23.7 λ 74.8 λ Since the plate scattering model is both physically-rigorous 1.9 GHz 25.2 λ 79.6 λ 36.5 λ 115.3 λ and simple to compute, it serves as a useful benchmark for 2.4 GHz 28.3 λ 89.4 λ 41.0 λ 129.6 λ 44 0 139 0 63 7 201 5 path loss in channels employing a RIS in the far case. However 5.8 GHz . λ . λ . λ . λ 28.0 GHz 96.6 λ 305.5 λ 140.0 λ 442.7 λ it should be noted that connection between plate scattering and 60.0 GHz 141.4 λ 447.2 λ 204.9 λ 648.0 λ RIS scattering is not universal. The equivalence demonstrated TABLE II here is attributable to the use of the proposed q = q0 element SIDE LENGTH OF A SQUARE RIS REQUIRED FOR PATH LOSS EQUAL TO pattern model with λ/2 element spacing. While various other THESPECULARREFLECTIONCHANNEL (WAVELENGTHS). combinations of element pattern and spacing can yield the same absolute equivalence, it is not true that any combination of element pattern and spacing will yield this equivalence. i.e., the path loss is equal to that of a free-space path of length ri +rs. Subsequently the path loss for the far case RIS channel C. Comparison to Specular Reflection relative to that of the specular reflection channel is: Specular reflection is the component of scattering from an L r + r A 2 electrically-large smooth surface which is distinct from diffrac- S = i s · (−ˆr · nˆ)2q0 (+ˆr · nˆ)2q0 ǫ (27) L r r λ i s p tion originating from the edges of the surface. If diffraction RIS  i s  in the direction of the receiver is negligible, then, from the Note that the ratio of the path losses is dependent on both the perspective of the receiver, the scattering from the surface is physical area of the RIS and frequency. However, it is also well-described as specular reflection. It was noted from the now apparent that LS can be less than, equal to, or greater previous section that the scattering from a RIS in the far case than LRIS. To better understand the situation, it is convenient cannot be interpreted as specular reflection alone, since path to define an “effective focal length” fe as follows: loss is clearly seen to be dependent on the size of the RIS. 1 1 1 ri + rs However, specular reflection is commonly found to be an = + = (28) f r r r r appropriate model for scattering from terrain, buildings, and e i s i s other electrically-large structures encountered in the analysis This expression is the known in the optics literature as the of terrestrial wireless communications systems. When this is “thin lens equation;” however, the reason for defining fe here the case, it is merely because diffraction is either negligible or is simply brevity and convenience. For example, when ri and not specifically of interest. This begs the question: When, if rs are equal, fe = ri/2= rs/2. Also, when ri ≫ rs, fe ≈ rs; ever, is it is appropriate to interpret far case RIS scattering as similarly when ri ≪ rs, fe ≈ ri. Thus, fe generally ranges specular reflection? The short answer is “never,” as we shall between the lesser of ri and rs down to about half the lesser now demonstrate. A second finding from this analysis will be value. Using this concept, Equation 27 simplifies to: a simple guideline for choosing the size of a RIS in the far 2 LS A 0 0 case. = (−ˆr · nˆ)2q (+ˆr · nˆ)2q ǫ (29) L f λ i s p Consider a RIS which is oriented such that Snell’s law of RIS  e  reflection (i.e., angle of reflection equals angle of incidence) is Now we may determine the RIS size for which the path loss satisfied at the RIS. Next, imagine that the RIS is replaced by of these two channels is equal; i.e, LRIS/LS =1. One finds: an infinitely-large flat conducting plate which lies in the plane −1/2 2q0 2q0 previously occupied by the RIS. Since the plate is infinite, A = feλ (−ˆri · nˆ) (+ˆrs · nˆ) ǫp (30) there are no edges and therefore the scattering from the plate is Table I showsh examples of RISs meetingi this criterion. pure specular reflection. Since the plate is flat, the phasefront Two cases are considered: “minimum,” in which ǫ = 1 and curvature of the reflected wave at the point of reflection is p ˆr ·nˆ =1 (i.e., broadside incidence), yielding minimum A; and equal to the phasefront curvature of the incident wave at the i “typical,” in which ǫ =0.5 and −ˆr · nˆ = ˆr · nˆ =0.5. The point of reflection, and the rate at which spatial power density p i s “typical” case corresponds to incidence and scattering 60◦ off decreases is the same after reflection as it was before reflection. broadside with realistic efficiency, representing a practical RIS Subsequently the path loss LS in this case is simply in a disadvantaged geometry. The effective focal length fe = 2 LS = [4π (ri + rs) /λ] (26) 0.1 km could be a scenario in which either ri or rs is ≈ 0.1 km 5

with the other distance being much greater, ri = rs =0.2 km, or any number of intermediate scenarios. Similarly, fe =1 km 0 could be a scenario in which either ri or rs is ≈ 1 km with the other distance being much greater, ri = rs = 2 km, or any number of intermediate scenarios. Table I indicates that -10 LRIS = LS for RIS side-lengths ranging from meters to 10s -20 of meters, depending on frequency and effective focal length.

Table II shows precisely the same result, except now ex- -30 pressed in electrical length; i.e, units of wavelength. Note that side-lengths ranging from O(10λ) to O(100λ) are required, as -40 one might expect. However, additional increases in side-length do not yield performance comparable to specular reflection; -50 to the contrary, the RIS outperforms specular reflection by path gain [dB] relative to equal-length free space increasing margins as the electrical size is increased. This is -60 simply because the RIS focuses the scattered field in order to 0 20 40 60 80 100 aperture side length [wavelengths] minimize path loss, whereas as the infinite conducting plate cannot. With this in mind, note also that a RIS could be used to accurately reproduce specular reflection, but the RIS would Fig. 2. Path gain relative to the equal-length free space channel for ri = 4 ◦ ◦ ◦ need to be both electrically-large and configured to preserve rs = 10 λ. Curves are shown for ψs = 0 (red), 60 (blue), and 75 (green). the rate of change of phasefront curvature. While this is useful in certain applications such as RIS “broadcasting,” it is not a strategy which minimizes path loss in SISO channels. where p is the vector from a common reference point (e.g., Note that the electrical sizes of A indicated in Table II n the origin) to the location of element . For the purposes of increase with frequency. In particular, note that the electrical n − this paper, this is referred to as “beamforming,” and can be size of A increases in proportion to λ 1/2, resulting in almost alternatively interpreted as collimation, or focusing at infinity. an order of magnitude increase as frequency increases from 0.8 GHz to 60 GHz. Beamforming is of practical interest, despite the non-optimum path loss, because in practice it may be easier to determine Finally, it is noted that [6] reports that the minimum RIS directions than positions. The distinction between focusing and size that achieves path loss equal to the path loss for unob- beamforming was not necessary in the far case, since under the structed direct line-of-sight between transmitter and receiver far case approximations they are equivalent. However, when is A = r r λ/r , where r is the straight-line distance from i s d d the RIS is closer (and as correctly noted in [4] in particular), transmitter to receiver. This is simply Equation 30 for the the difference in performance can be large, as we shall now special case of perfectly-efficient isotropic elements (so that see. the third factor in the expression becomes 1) and with ri + rs −1 replaced by r . Figure 2 shows path gain LRIS from Equation 19, computed d 4 for ri = rs = 10 λ. The result is normalized to the path gain −1 LS (Equation 26) for a free space (i.e., no RIS) channel ENERAL ASE V. G C having path length ri + rs. Thus, 0 dB on the vertical axis When the “far” criteria defined at the beginning of Sec- indicates that the RIS-enabled channel exhibits path gain tion IV are not met, one is forced to backtrack to Equation 19. equal to that of the equal-length free space path. Separate In this case, the conclusions in the far case do not apply and curves are shown for focusing, beamforming, and the far analogous conclusions for the general case are not obvious. For approximation (Equation 22). The curves are too close to this reason, the following numerical study is conducted. This readily distinguish, indicating that this scenario meets the far study assumes a planar square RIS which is again modeled approximation criteria. Also, note that an aperture side-length of at least 70λ is required to meet the 0 dB condition under as N elements with pattern q = q0 which are uniformly distributed with λ/2 spacing. The transmitter is broadside to these conditions. the array while the angle ψs to the receiver is varied from Figure 3 shows the results of the same experiment per- ◦ ◦ 3 0 (RIS broadside) to 60 and then 75 . The distances ri and formed for ri = rs = 10 λ; i.e., the same RIS but one rs are held equal and varied to generate the results shown in order of magnitude closer. Again, separate curves are shown Figures 2, 3, and 4. for focusing, beamforming, and the far approximation. In this In the general case, the path loss minimization criterion case, focusing and the far approximation remain too close to −jφn distinguish. Therefore the far approximation is very good for bn = e is referred to as “focusing”. A different option RIS apertures up to at least × ( 4 elements) is to set the phase of bn so as to minimize path loss with 100λ 100λ N = 10 3 respect to the directions (only) of the transmitter and receiver, for distances & 10 λ. The performance of beamforming, on the other hand, is seen to be limited to within a few dB of which requires only that ˆri and ˆrs (and not the distances ri 0 dB for aperture side-lengths greater than about 20λ. In this or rs) be known. Specifically: sense, a RIS operating in beamforming mode might be said −j(2π/λ)pn·ˆri +j(2π/λ)pn·ˆrs bn = e e (31) to be exhibiting performance comparable to that of specular 6

Fig. 3. Path gain relative to the equal-length free space channel for ri = Fig. 4. Path gain relative to the equal-length free space channel for ri = 3 ◦ ◦ ◦ ◦ ◦ ◦ rs = 10 λ. Curves are shown for ψs = 0 (red), 60 (blue), and 75 rs = 10λ. Curves are shown for ψs = 0 (red), 60 (blue), and 75 (green). (green). Beamforming results are shown as unconnected data points to make clear the rapid variation even when varying the aperture size by the minimum amount; i.e., by one additional row and column of elements per sample. reflection. This is a particularly useful insight: If beamforming is to be used, then there is a maximum useful size (given to the physical area of the RIS when the element spacing is by Equation 30) for the RIS, and further increases in size λ/2. When a RIS with the benchmark element pattern and λ/2 will not significantly reduce path loss. (Essentially the same spacing is analyzed using this model, the results are shown to observation is made in [4] using a somewhat different method; be consistent with both electromagnetic plate scattering theory in fact Equation 55 in [4] is Equation 30 for the special and prior work. Other RISs can be accurately represented by case of a perfectly-efficient RIS and broadside monostatic link varying the element parameter q and the element spacing. The geometry.) If, on the other hand, focusing is used, then further principal shortcoming of the model is reduced accuracy when increases in size do significantly reduce path loss. either transmitter or receiver is very close to the RIS. This is Figure 4 shows the results of the same experiment per- not a limitation of the fully polarimetric models of [4], [6], and formed for ri = rs = 10λ. In this scenario, the RIS side-length [8]; and this could be similarly addressed as a straightforward becomes much larger than the distances to the transmitter and generalization of the model presented here. receiver, and so the far approximation is expected to fail. This is observed to be the case, with the results for focusing and the REFERENCES far approximation diverging for side-lengths greater than about [1] M. Di Renzo et al., “Smart Radio Environments Empowered by Recon- ri (or rs). Focusing approaches 45 dB normalized path gain figurable Intelligent Surfaces: How It Works, State of Research, and the for 100λ side-length, compared to 0 ± 6 dB (independent of Road Ahead,” IEEE J. Sel. Areas Comm., vol. 38, no. 11, pp. 2450–2525, side length) for beamforming. Thus – as expected – focusing Nov. 2020. [2] W. Tang, et al., “Wireless Communications with Reconfigurable Intelli- is capable of enormous gain over beamforming when path gent Surface: Path Loss Modeling and Experimental Measurement,” IEEE distances are less than the aperture side-length. Trans. Wireless Comm., vol. 20, no. 1, pp. 421-439, Jan 2021. The model developed in this paper does not account for vari- [3] W.L. Stutzman & G.A. Thiele, Antenna Theory and Design, 3rd Ed., Wiley, 2012. ation in the per-element incident and scattered polarizations [4] E. Bj¨ornson and L. Sanguinetti, “Power Scaling Laws and Near-Field or variation in transmitter and receiver antenna gains over the Behaviors of Massive MIMO and Intelligent Reflecting Surfaces,” IEEE RIS. It is noted in [4] that this may introduce significant error Open J. Comm. Soc., vol. 1, pp. 1306–1324, Sep 2020. [5] D. Dardari, “Communicating With Large Intelligent Surfaces: Fundamen- when ri or rs are small compared to the size of the RIS, as tal Limits and Models,” IEEE J. Sel. Areas Comm., vol. 38, no. 11, is the case in the scenario addressed in Figure 4. While the pp. 2526–2537, Nov. 2020. results shown in Figure 4 exhibit the expected behavior, they [6] M. Najafi, V. Jamali, R. Schober and H.V. Poor, “Physics-based Modeling and Scalable Optimization of Large Intelligent Reflecting Surfaces,” IEEE are nevertheless less accurate and should be used with caution. Trans. Comm., doi: 10.1109/TCOMM.2020.3047098 (early access). [7] O.¨ Ozdogan,¨ E. Bj¨ornson & E. Larsson, “Intelligent Reflecting Surfaces: Physics, Propagation, and Pathloss Modeling,” IEEE Wireless Comm. Let., VI. CONCLUSIONS vol. 9, no. 5, pp. 581-585, May 2020. [8] J.C.B. Garcia, A. Sibille & M. Kamoun, “Reconfigurable Intelligent This paper presented a simple physical model for RIS Surfaces: Bridging the Gap Between Scattering and Reflection,” IEEE scattering and employed the model to develop expressions for J. Sel. Areas Comm., vol. 38, no. 11, pp. 2538–2547, Nov. 2020. path loss. To avoid the need for implementation-specific RIS [9] E. Bj¨ornson, O.¨ Ozdogan,¨ and E.G. Larsson, “Reconfigurable Intelligent Surfaces: Three Myths and Two Critical Questions,” IEEE Comm. Mag., design parameters, a physically-motivated benchmark element vol. 58, no. 12, pp. 90–96, Dec. 2020. pattern model was proposed. This pattern has the useful feature [10] R.F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill, that the sum of the effective apertures of the elements is equal 1961.