1 Path Loss in Reconfigurable Intelligent Surface-Enabled Channels S.W. Ellingson, Senior Member, IEEE Abstract—A reconfigurable intelligent surface (RIS) employs approximation for large planar arrays of regularly-spaced an array of individually-controllable elements to scatter incident elements [3]. signals in a desirable way; for example, to facilitate links In the second strategy, the RIS is treated as an continu- between base stations and mobile stations that would otherwise be blocked. A principal consideration in the study of RIS-enabled ous surface subject to electromagnetic boundary conditions propagation channels is path loss. This paper presents a simple (typically surface impedance or surface currents) which, when yet broadly-applicable method for calculating the path loss of combined with Maxwell’s equations, yields a boundary value a channel consisting of a passive reflectarray-type RIS. This problem that can be solved for the scattered fields; see model is then used to characterize path loss as a function of e.g., [4]–[6]. This is ideal for identifying theoretical limits RIS size, link geometry, and the method used to set the element states. Whereas previous work presumes either (1) an array of of performance and useful “scaling laws” (e.g., how path parameterizable element patterns and spacings (most useful for loss varies with RIS area, path length, and so on) that are analysis of specific designs) or (2) a continuous electromagnetic independent of the specific technology used to implement surface (most useful for determining scaling laws and theoretical the RIS. However this strategy is not particularly useful for limits), this work begins with (1) and is then shown to be the analysis and design of specific RIS devices because the consistent with (2), making it possible to identify specific practical designs and scenarios that exhibit the performance predicted relationship between the boundary conditions and the element using (2). This model is used to further elucidate the matter pattern(s), spacing, and element states comprising a specific of path loss of the RIS-enabled channel relative to that of the RIS is not explicit and may be difficult to ascertain. free space direct and specular reflection channels, which is an The principal contribution of this work is a method for important consideration in the design of networks employing RIS technology. calculation of path loss following the first strategy (i.e., beginning with an array of parameterizable element patterns and spacings) which is similar to but simpler than [2], and which is then used to derive a particular design for which the I. INTRODUCTION calculated path loss is equal to that predicted by the second A reconfigurable intelligent surface (RIS) is a device that strategy in the special case where the boundary condition scatters signals in a controlled manner in order to improve corresponds a perfectly conducting plate having area equal the propagation channel between transmitters and receivers; to the RIS. This particular design consists of elements having see e.g. [1] and references therein. RISs are often envisioned directivity of about 5 dBi with half-wavelength spacing, which to be some form of reflectarray in which control consists of essentially confirms that the results obtained using the second changing the phase and possibly magnitude of the electromag- strategy are applicable to a typical practical reflectarray-type netic field scattered by each element individually; this class of RIS. RISs is the focus of this paper. An important consideration in A second contribution of this work is the use of this RIS engineering is the path loss in the transmitter-RIS-receiver model to further elucidate the matter of path loss of the arXiv:1912.06759v2 [eess.SP] 13 Apr 2021 channel as a function of RIS size, link geometry, and element transmitter-RIS-receiver channel relative to that of the free states. space transmitter-receiver and specular reflection channels, Analysis of path loss requires a method for calculating scat- which has been a source of confusion in the literature. While tering by the RIS, for which there are two general strategies. this issue has been previously addressed in [4], [6]–[9], the In the first strategy, scattering is computed as the discrete analysis in this paper yields a concise independent summary sum of fields scattered from elements; see e.g. [2]. This is which may be useful in better understanding this topic. ideal for practical analysis and design problems since the relationship between path loss and the element pattern and II. RECEIVED POWER IN THE RIS-ENABLED CHANNEL spacing within the RIS is explicit. The principal disadvantage of this strategy is that embedded element pattern (i.e., the Referring to Figure 1, the spatial power density incident on pattern of the element accounting for mutual coupling) is the nth element is: difficult to calculate. This issue is commonly bypassed by i i 2 assuming that the responses of the elements are identical and Sn = PT GT (ˆrn)/4πri,n (1) easily parameterizable, which is known to be a reasonable where PT is the total power applied by the transmitter to the i S.W. Ellingson is with the Dept. of Electrical and Computer Engineering, transmit antenna system, GT (ˆrn) is the gain of the transmit i Virginia Tech, Blacksburg, VA, 24061 USA e-mail: [email protected]. antenna system in the direction ˆrn, and ri,n is the distance 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.