Simple Link-Budget Estimation Formulas for Channels Including Anomalous Reflectors

Reconfigurable intelligent surfaces (RISs) are a promising tool for the optimization of propagation channels for advanced wireless communication systems. These tools are especially relevant for high-frequency (millimeter-band) links where directive antennas are used. RIS panels act as high-gain passive repeaters, whose reflected waves interfere with the waves reflected from the illuminated spots at supporting walls, creating a complex field pattern in the far zone. In this work, we consider a particular scenario of RISs for anomalous reflection and develop a simple link-budget model for nonline-of-sight (NLOS) channels via reflections from finite-size metasurfaces (MSs) designed as anomalous reflectors or splitters. The developed model takes into account diffraction at the RIS panel edges as well as interference with reflections from supporting structures. We take into account realistic losses and confirm the validity of results by numerical simulations.

Simple Link-Budget Estimation Formulas for Channels Including Anomalous Reflectors Sergei Kosulnikov , Francisco S. Cuesta , Xuchen Wang , and Sergei A. Tretyakov , Fellow, IEEE Abstract-Reconfigurable intelligent surfaces (RISs) are a promising tool for the optimization of propagation channels for advanced wireless communication systems. These tools are especially relevant for high-frequency (millimeter-band) links where directive antennas are used. RIS panels act as high-gain passive repeaters, whose reflected waves interfere with the waves reflected from the illuminated spots at supporting walls, creating a complex field pattern in the far zone. In this work, we consider a particular scenario of RISs for anomalous reflection and develop a simple link-budget model for nonline-of-sight (NLOS) channels via reflections from finite-size metasurfaces (MSs) designed as anomalous reflectors or splitters. The developed model takes into account diffraction at the RIS panel edges as well as interference with reflections from supporting structures. We take into account realistic losses and confirm the validity of results by numerical simulations.

I. INTRODUCTION
M ODERN wireless communication technologies need to utilize high-frequency ranges in order to satisfy the growing demand for high-speed mobile networks. Currently developed wireless communication protocols reach millimeter-wavelength bands that require nontrivial solutions to ensure a stable channel between the receiver (Rx) and the transmitter (Tx). Antennas are becoming more and more directive to compensate for higher free-space propagation decay at higher frequencies, ensuring signal propagation at sufficient distances without losing their intensity and stability [1]. At such high distances and directivities, a direct visual link between elements of a network appears to be a must. However, it is often impossible to provide a straight line-ofsight (LOS) wireless channel between two spots. One potential solution is to enhance the wireless signal by modifying the environment. For example, it is not necessary to establish a direct communication channel, but to resort to nonline-of-sight (NLOS) channeling, reflecting the signal from some specifically designed objects in the environment. This approach allows us to overcome some physical barriers that prevent straight connections between the Rx and the Tx. In addition, if a directive reflector is used, it is possible to enhance the signal strength by focusing reflections toward the Rx position.
Described technique of environment improvement via specifically designed reflectors is currently being implemented using reconfigurable intelligent surface (RIS) technology. Reflectors that are considered for RIS implementation can operate not just like classical mirrors but provide additional degrees of freedom, effectively reflecting in nonspecular directions. Most current works on designs and studies of anomalous reflectors are based on the so-called phase gradient method, where the reflectarray technique is used (see a review in [2]). In this method, the surface is modeled by a local reflection coefficient, and the unit cells are designed using the locally periodical approximation. Another prospective approach is based on various versions of nonlocal designs of the metasurface (MS) structure (e.g., [3], [4], [5], [6], [7], [8], [9]), allowing to overcome the angular and efficiency limitations of the classical reflectarray method and design theoretically perfect anomalous reflectors of required functionalities. During the last decade, MSs have shown a very high potential in performing numerous complex operations on reflected or transmitted waves with potential applications in different technological areas (see [10]). This study focuses only on a particular application of anomalous reflection and beam splitting.
In this article, we discuss analytical models for estimations of the link budget, defined as the relation between the power accepted by the transmitting antenna and the power delivered to the load of the receiving antenna. For a free-space link between two antennas, the link budget can be estimated using the classical Friis formula [11], [12]. There are many works that propose approximate models of link budget in the presence of RISs (e.g., [2], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]). However, there is still no complete understanding of the characteristics of an NLOS channel with MSs, especially under focused beam illuminations. A comparative analysis of known link-budget models can be found in [21, Table I] and the corresponding discussion. Most works on link-budget estimations are based on the summation of contributions from scattering from individual elements of the MS (e.g., [13], [14], [15], [16], [17]). Faqiri et al. [22] proposed 0018-926X © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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a numerical approach, where the entire environment, sources, and RISs are modeled as arrays of subwavelength dipoles. While that approach shines in reflection-rich environments, it could be overwhelming in quasi-free-space scenarios, like the one considered in this work. Özdogan et al. [23] proposed a link-budget formula (19) that is similar to that derived in this work for an MS in free space, but with a different angular factor. As discussed in [24], the model of [23] assumes that phase-gradient reflectors can be viewed as infinitely thin sheets of only electric surface currents with appropriately modulated phases, but this assumption is not applicable to impenetrable reflectors. The aim of this article is to build a simple link-budget model to describe communication channels between the Rx and the Tx connected via an NLOS path due to anomalous reflection in a finite-size MS mounted at a uniform (UNI) wall. Importantly, since usually also a part of the wall is illuminated by the Tx antenna beam, the reflected and scattered field is formed by interference of the beams reflected from the MS and the wall. Here, we account for both reflections and present a comparison of the field amplitude contributions due to both reflections, for a typical NLOS scenario. As an application example, we use the developed model to estimate the minimum size of the anomalous reflector to ensure the required level of the link budget for the Rx positioned in the complete shadow of an impenetrable wall. The developed model takes into account realistic parameters of reflectors and antennas for NLOS channels in the millimeter band. Presented numerical examples show particular estimations for an indoor NLOS channel at D-band. The approach presented in this work can be extended to beam splitters and suboptimally operating anomalous reflectors by accounting for the fields created by all propagating Floquet harmonics using the concepts introduced in [25]. Accounting for the corresponding current distribution, it is possible to calculate the received power from an RIS with an arbitrary configuration [2] or from a passive static EM skin [26], [27].
This article is organized as follows. In Section II, we discuss reflected and scattered fields from an MS at a wall and discuss possible simplifications to be applied for an antenna excitation of different directivities while taking into account sidelobe scattering due to finite dimensions of the MS and the illumination area. Section III provides a detailed elaboration and analysis of a link-budget model for reflections from a finite MS. Furthermore, we confirm the validity and accuracy of the introduced model using full-wave simulations. In Section IV, we give an application example, estimating the MS size needed to ensure the desired level of the signal at the Rx position. Section V is devoted to a discussion of the frequency dependence of the link-budget factors and the fundamental benefits of the use of anomalous reflections for high-frequency links. In the end, we present a concluding discussion of the goals and results of this work.

II. FAR-ZONE REFLECTED FIELDS
Let us consider the scenario illustrated by Fig. 1: An MS is mounted on a flat UNI wall and illuminated by a directive antenna with directivity D. The MS panel has a rectangular shape with the area S 1 = a 1 b 1 , where a 1 and b 1 denote the width and height, respectively. The MS is located in the far-field region of the illuminating antenna, at a distance R 1 . Both MS dimensions are assumed to be small when compared to the curvature radius of the illuminating spherical wave, so that it is possible to approximate the incident wave at the MS location by a plane wave.
In a millimeter-wave scenario, the Tx is usually a steerable MIMO antenna array, which mimics the radiative properties of a horn antenna illuminating the MS. At millimeter-wave frequencies, the typical aperture size of an equivalent horn antenna a ant is of the order of a few millimeters. In particular, in our further examples, we will consider operation at 144.75 GHz, for which case, the aperture size is usually smaller than 5 mm. The distance to the far zone can be estimated as R ff = 2a 2 ant /λ ≈ 24 mm [12], [28]. Thus, at distances larger than about 1 m from the antenna and for typical centimeter-scale dimensions of millimeter-wave MSs, such antenna illumination can be considered a plane wave.
For oblique incidence, the area illuminated by a directive antenna has an elliptical shape. We denote the main axes of the ellipse as a 2 (in the plane of Fig. 1) and b 2 (orthogonal to the plane of the figure). These axes can be expressed in terms of the Tx antenna directivity D and the incidence angle θ i as [25] The surface area of the illuminated ellipse can be calculated simply in terms of the product of the two main axes: Here, we consider anomalously reflecting MSs or meta-gratings that are periodical structures designed to create nonspecular reflected waves. According to the Bloch-Floquet theory, the reflection angle θ n of the nth diffracted harmonics created by infinite periodical structures with the periodicity d ms , illuminated by a plane wave at the incidence angle θ i , can be found from (e.g., [5], [29]) sin θ n = sin θ i + 2π kd ms n.
Here, k = 2π/λ is the wavenumber in the surrounding space (usually, free space). In the majority of situations, the desired anomalous reflection corresponds to the n = −1 Floquet harmonic, and, without loss of generality, we will consider this case. If the illuminated area is much larger than the MS area, the variations of the field amplitude over the reflector area can be negligibly small. However, there can be scenarios where the illuminated spot is smaller than the MS, and the incident field amplitude decays to zero at the edges. Also, the illumination beam amplitude can be approximated either as a nearly UNI in the middle with a sharp drop to zero at the edges of the illuminated spot or by a beam with a smooth decay of the amplitude toward the edges of the spot. Thus, we will consider two general models with the UNI and cosine (COS) field distributions of the incident field amplitude in the illuminated spot.

A. Reflected Field Models for the Illuminated Area Larger Than the MS
We consider the illumination scenario of Fig. 1, where transverse-electric (TE) polarization is assumed with the incidence plane being the plane of the picture. The MS is UNI in the direction normal to the picture plane (z-axis). First, we can approximate the incident field by a UNI plane wave, whose amplitude (at the position of the MS) is denoted as E 0 . Following [25], the reflector is modeled using Huygens' principle in terms of equivalent electric and magnetic surface currents. The z-component (normal to the picture plane) of the total scattered field in far zone reads (see detailed derivation and discussion in [25]) Here, r is the position vector pointing the observation point, wall is the reflection coefficient of the surrounding UNI surface wall, n is the macroscopic reflection coefficient for the nth Floquet harmonic of the MS current, θ is the angle between the normal and the direction to the observation point, a ef = (sin θ − sin θ i )a 2 /2, a ef,n = (sin θ − sin θ n )a 1 /2, and δ n is the Kronecker delta (δ 0 = 1 and δ n̸ =0 = 0). The reflection coefficient from the wall wall is determined by the complex permittivity of the wall material at the frequency range of interest, if reflections from the backside of the wall and the objects behind the wall can be neglected. In the literature, there are experimental data for walls made of different materials (see [30], [31], [32], [33] for the millimeter-wave range). There is also diffuse scattering due to wall roughness, which we neglect in this work. Inside the main reflected beam, the effects of diffuse scattering are seen in a reduction of the effective reflection coefficient from the wall.
The field reflected from the MS forms a beam toward the desired anomalous reflection direction, while, under a directedbeam illumination, the field reflected from the illuminated part of the UNI wall forms another beam into the specular direction, as illustrated in Fig. 1. The sidelobes of the two beams overlap, and the total field pattern is formed by their interference. In order to analyze these effects, we present the total field as a sum of the contributions from the currents induced on the MS E UNI MS , as if that would be positioned in free space and not supported by the wall and from the currents on the wall E UNI wall in case there is a hole in the place of the MS For the desired reflected harmonic, if the MS operates perfectly, we have −1 = √ cos θ i / cos θ −1 (e.g., [3], [5]). For phase-gradient reflectors (linear phase gradient of the local reflection coefficient), several harmonics are excited, producing reflections into several diffraction lobes. Thus, the maximum theoretical efficiency for the reflectors designed with the phase-gradient method is limited to η = 4 cos θ i cos θ r /(cos θ i + cos θ r ) 2 [24]. Recently, methods to realize theoretically perfect MSs for anomalous reflection (compromised only by material losses and manufacture imperfection) even at extreme angles have been developed (e.g., [9], [34]). For these surfaces, the corresponding macroscopic reflection coefficients need to be calculated numerically. In both cases, the models are based on the generalized physical-optics approximation, where the induced currents over a finite-size surface are approximated by the solutions for infinite periodic reflectors.
Next, we consider a more realistic illumination scenario, where the illuminated spot does not have a UNI-amplitude field distribution. To model this situation, we consider a COS distribution of the field amplitude, with the maximum at the center of the illuminated spot. Following the calculation steps presented in [25] for a UNI illumination, we calculate the far-zone scattered field for the case when the illumination amplitude depends on the tangential coordinate x as E 0 (x) = E 0 cos (π x/a 2 ). The result for the fields created by the currents on the MS in free space reads The field of the currents flowing only at the wall (with a hole in place of the MS) is equal to Here, f cos (x) = π 2 cos(x)/(π 2 −4x 2 ). The total reflected field is the sum of the above contribution, and it is found as In this model, the illumination phase is approximated as that of a plane wave coming from the incidence direction, only the amplitude is assumed to decay toward the beam edges. The above formulas allow us to estimate the total scattered field at any reflection angle θ . As an example, we consider the fields reflected toward the desired direction of anomalous reflection, both for the UNI and COS distributions of incident amplitude. We will separately study situations when the illuminated spot is larger or smaller than the MS area.

B. Scattered Sidelobes for UNI Incident Field Distribution When the Illuminated Area is Larger Than the MS
The first term of E UNI MS of (4) corresponds to the specular reflection from a finite-size MS. The second term gives the contributions from nonspecular scattering modes. Most often, the MS is designed or configured to reflect predominantly in the direction of the Rx device. This means that the amplitudes of the Floquet harmonics that radiate in other directions are suppressed, and the observation point is at the reflection angle of the desired harmonic n = −1, whose field in the far zone reads If the MS is tuned to reflect toward an Rx located at the angle θ , we set θ = θ −1 . As a particular example, we consider θ = 0, and assume that the surrounding wall reflects as a perfect mirror ( wall = −1). Furthermore, we assume that the MS is a perfect anomalous reflector with −1 = √ cos θ i / cos θ r . In this case, the total scattering from the uniformly illuminated spot on the wall and the MS E UNI scz and nonspecular (anomalous) scattering from the MS E UNI −1 reduce to In Fig. 2(a), the ratio |E UNI −1 /E UNI scz | oscillates around 0 dB revealing that the total scattering from the illuminated area on the wall and the MS is comparable with the field of the desired harmonic of the anomalous reflector. It is reasonable to say that for a directive illumination case (with the antenna directivities 26 and 30 dB chosen as examples), the desired anomalous scattering from the MS is dominant and the total scattering toward the Rx can be approximated by the value of E UNI −1 for highly directive Tx antennas. However, in the case of low directivities (15-dB example), it is found that other scattered components in the direction of the desired anomalous reflection E = E scz − E −1 are not negligible. Fig. 2(c) presents the ratio of | E/E −1 |. Using these results, we can estimate the impact of the sidelobe scattering to the field at the Rx position θ = θ −1 . In the example case of Tx antenna directivity D = 26 dB, the impact is changing with different distances between the illuminated wall/MS and the illuminating antenna. We see that the impact of sidelobes of the specularly reflected beam is negligibly small compared to the reflection from the MS in the majority of practical sizes of the illuminated area and the MS size.
Strong oscillations of the ratio are due to the sinc function in the expression for E scz . To define the upper boundary of E, we can modify the sinc function to represent the envelope value as follows: Using the expression for the upper boundary for E, we calculate its magnitude and compare it with E −1 for different directivities D and distances R 1 , as both values affect the size of the illuminated area S 2 according to (1). In Fig. 2(c), one can see that the upper boundary well describes the maximum values of fast oscillations given by the exact ratio of E/E −1 for illuminations at θ i = 50 • and directivity of the Tx antenna D = 26 dB for the UNI incident field distribution. Fig. 3(a) shows the upper boundary of these oscillations under the same UNI illumination and different directivities. For the UNI illumination, additional scattering produced by the currents induced at the wall and the MS E is negligible for high-directivity antennas or short distances. A practical example can be extracted from Fig. 3(a)

C. Scattered Sidelobes for COS Amplitude Distribution for Illuminated Areas Larger Than the MS
We can write the scattered field produced by the desired harmonic similar to (9), but for the COS model of the illuminating amplitude Next, in analogy with the above study of UNI-amplitude illumination, leading from (3) to (10), we derive from the expression in (8), the corresponding components for COS-distribution illumination as follows: Numerical results for the amplitude ratio of these components |E −1 /E scz | are presented in Fig. 2(b). The curve is very weakly oscillating and close to the 0-dB value. Next, we again estimate the impact of other scattering components comparing them with the desired anomalous reflection | E/E −1 | for the COS field distribution. These results are presented in Fig. 2(c). The curve is also weekly oscillating, but we can apply an additional procedure to estimate the upper boundary of this impact. In this case, the envelope for the f cos (x) function becomes The upper-bound results are plotted in Fig. 2(c), labeled as "COS, upper bound," and in Fig. 3(b) for different antenna Fig. 4. Numerical analysis of the fields scattered from a partially illuminated MS for UNI and COS illuminations: the absolute value of the amplitude ratio of the scattered field that is produced by other than the desire harmonic E to the desired scattering produced by the anomalously reflected mainlobe E −1 . Analysis is for the MS size a 1 = 0.1 m and the incidence angle θ i = 50 • . directivities D. As expected, the impact of the sidelobe fields on the total field along the main anomalous reflection direction is even weaker than in the case of the UNI illumination. In this case, reflection produced by the desired harmonic E COS −1 is the main contributor for all the considered directivities and distances.

D. Reflected Field Models for the Illuminated Area Smaller Than the MS
In some situations, the illuminated area can be smaller than the MS area, that is, S 2 < S 1 . For example, if the Tx antenna is closer to the MS (but still in the far zone), or for very large MS areas S 1 , or if the Tx antenna has very high directivity. Similar to the procedures that we used to derive far-field expressions in (3) and (8), we can write them for scattered fields from the MS when In the case of a perfect anomalous reflector, the reflection coefficient for the parasitic specular reflection (mode n = 0) 0 = 0, and for the anomalous reflection at n = −1, the reflection coefficient for the perfect anomalous reflector equals −1 = √ cos θ i (for θ = θ −1 = 0). Then the corresponding scattered field and the anomalously reflected component read Here, the expression for E UNI −1 corresponds to the field created by the equivalent currents flowing at the illuminated side of the surface (reflected fields), while the other terms are due to the shadow currents. Because the illuminated spot has a finite size, these currents scatter some fields in all directions, and not only into the shadow region.
For the COS field distribution, the corresponding results read We have analyzed the impact of the sidelobe scattering compared with the desired harmonic scattering, similar to the results presented in Fig. 2(c), but for the case when the illuminated area is smaller than the MS area. Analyzing the plots for the total field amplitude and the upper boundary, presented in Fig. 4 for both UNI and COS field distributions, we note that for moderate distances from the Tx antenna to the MS (for R 1 ≈ 5 m) and highly directive antennas, the sidelobe contribution becomes much smaller than for the case of a large illuminated area.
As a general conclusion of this section, we can state that if the illuminating antenna is a highly directive Tx antenna (more than 26 dB in the considered examples) and the distance from the Tx antenna to the MS is moderate or large (R 1 ≈ 5 m in the above examples), it is possible to approximate the field scattered from the MS toward the desired direction by the field of the desired Floquet harmonic, that is, E scz ≈ E −1 . This simplification will be applied next in the following analysis of the link budget.

III. LINK-BUDGET FORMULA
In Section II, we discussed different models of reflection from the MS into the far field and the possible impact of the sidelobes of the specularly reflected beam. We identified conditions when the reflections from the MS dominate over the reflections from the UNI wall and derived simple formulas for the field at the Rx position in the far zone of the metasurface reflector. Next, we cast these results into the form of a generalized Friis formula, providing a convenient tool for link-budget estimations. We focus on the simpler case of the UNI-amplitude illumination when the MS area is smaller than the illuminated spot (S 2 > S 1 ), but using the same approach, it is possible to derive formulas for the COS-distribution illumination model. We will consider several application scenarios of the MS usage, as illustrated in Fig. 5(a)-(c) and start with the case of oblique incidence and an anomalous reflector designed for perfect anomalous reflection to normal direction [ Fig. 5(a)].

A. Perfect Anomalous Reflector Model
In the far zone of the Tx, the field at the MS position is approximately a plane wave with the Poynting vector whose amplitude is Here, E 0 is the complex amplitude of E, and η 0 is the free-space wave impedance. The power density measured at a point located at distance R 1 from a directive antenna can be written in terms of the transmitted power P t and the Tx antenna gain G t . Equating these two expressions we find the amplitude of the electric field created at the MS plane by an antenna of a known gain as follows: If the anomalously reflected field from the MS is dominant at the Rx position, we can use (9) to find the field at the Rx position, which gives For the MS designed for anomalous reflection at θ −1 = 0 and if the observation angle is at the same position θ = θ −1 = 0, we can substitute the reflection coefficient cos θ i and sinc(ka ef,n ) = 1. We arrive at a simple approximate formula for the field at distance |r| = R 2 reflected by a finite-size MS Substituting the incident field value from (22), we get and the corresponding power density The received power is found in terms of the effective area of the receiving antenna A eff,r as Using the general relation between the effective aperture and gain of any reciprocal receiving antenna The received power can be written as Substituting the incident field amplitude, we finally write The ratio of the received and transmitted powers gives us the square of the transmission coefficient between the two ports S 21 of the link as follows: As a numerical example, we estimate S 21 for an ideal anomalous reflector using some particular values for R 1 and R 2 and a typical antenna gain G t = G r = 26 dB (398.1 on the linear scale). The result is presented in Fig. 6.

B. Channel Reciprocity
At this point, we can check the symmetry of the derived model, requiring excitation from a normal direction, and the observation point to be an oblique direction, as it is shown in Fig. 5(b).
Applying the same procedure that we used to derive (25), but taking into account different excitation field amplitude, distance to the observation point, and the scattered harmonic angle, we can write for the reciprocal illumination, similar to (25) The link budget expression becomes which agrees with the reciprocity theorem, because the reflection angle for this illumination is equal to the incidence angle in (31). We note that this is not the case with the model introduced in [23].

C. Model for Reflectors With a Finite Efficiency
Phase-gradient reflectors designed using the generalized reflection law and the locally periodic approximation have limited efficiency, mainly due to parasitic scattering in the directions of all propagating Floquet harmonics [3], [4], [5], [6], [35]. In addition, material losses and various imperfections lead to some dissipation and parasitic scattering. To account for parasitic scattering into undesired propagating Floquet modes, we can use (3). The macroscopic reflection coefficients n can be calculated using the mode-matching method [9] from the known surface impedance of the reflector. However, if the Rx is at the direction of propagation of the desired anomalously reflected wave, the contributions of other harmonics to the field at the Rx are small, and the efficiency drop can be accounted for by the anomalous reflection efficiency η eff for the term −1 in (3) The efficiency can be approximated by the macroscopic reflection coefficient for the desired reflected mode. The reflected field at the Rx becomes (35) and the final equation for the power received after reflection from the MS reads A numerical example for distances R 1 = R 2 = 5 m and the Tx and Rx antenna gain G r = G t = 26 dB is presented in Fig. 7. The introduction of η eff allows us to consider any nonideality and imperfection of anomalous reflectors caused by approximations in design and imperfections of implementations. For phase-gradient reflectors, η eff can be estimated from the mismatch of the characteristic impedances of the incident and reflected plane-wave modes (see [24, Fig. 2]).

D. Generalized Model for Arbitrary Incidence and Reflection Angles
Finally, we can assume an arbitrary incidence angle and an arbitrary desired reflection direction θ = θ r . For a properly Estimated transmission between the Rx and the Tx for an anomalous reflector as a function of efficiency calculated with (36). Here, G t = G r = 26 dB, θ i = 50 • , and R 1 = R 2 = 5 m.
configured RIS producing the main reflection toward the Rx at θ −1 = θ r , the scattered field reads Equation (30) takes the form cos θ i cos θ r (38) and the generalization of (36) reads This link-budget expression can be cast into a form similar to the classical Friis equation. To do that, we introduce an effective gain of the MS acting as a directive reflectarray antenna G m and rewrite the formula as where

E. Numerical Validation of the Model
In this section, we discuss the model of reflections from finite-size perfect anomalous reflectors (24) and (32). An example anomalous reflector design is based on the method originally presented in [9] and further expanded for the surface impedance subelements as a discrete step function in [34]. Here, the surface was realized as a step-wise UNI impedance sheet over a grounded dielectric substrate, where the values of each UNI strip sheet reactance were tuned to maximize the desired anomalous reflection harmonic. This approach allows the simpler realization of these strips as properly shaped metal patches. For the purposes of this work, we model the impedance sheet by its step-wise UNI impedance, not specifying the actual topology of the layer.
The test anomalous reflector acts at the operational frequency 144.75 GHz and reflects TE-polarized waves incident at θ i = 50 • toward the normal direction. For simplicity and clearance, we assume that the ground plane is the perfect electric conductor (PEC). The substrate material is lossless quartz (ϵ r = 4.2), and the thickness equals h sub = 209.5 µm. On top of the substrate, there is an impedance sheet with an optimized sheet reactance distribution. In accordance with the diffraction grating theory, the period of the structure to ensure the desired functionality at the first diffraction order equals d ms = λ 0 / sin(θ −1 ) ≈ 2705.5 µm. In each period, the impedance sheet is divided into eight subcells with the following optimized sheet impedances with the unit of ohm: The structure was simulated using CST Studio Suite software with the frequency domain solver and the adaptive tetrahedral mesh refinement. Simulations of an infinite array predict 99.99% efficiency of the anomalous reflection. Finitesize structures were simulated for different normalized sizes of the MS N = a 1 /d ms . For the finite-size calculations, we used plane-wave excitation of the structure and could not estimate the accuracy of the mesh grid refinement, which is based on the S-parameters difference and is applied only for the cases where ports are defined in the model, whereas infinite structure calculations enable this functionality and give more reliable results.
A comparative case of a design based on the classical linear phase-gradient solution was also studied. Here, we set the same period of the structure, and for simplicity of realization of the desired local reflection coefficients, the impedance sheet subcells were installed on an "open" boundary of the calculated domain. The impedance boundary is divided into eight subcells with the corresponding values: The far-field region for scattering from the MS can be defined as R ff > 2S 1 /λ [36]. In the simulations, we varied N = a 1 /d ms up to 16, so that the observation point at R 2 = 1000λ, which is 2.07 m for 144.75 GHz, is always in the far zone.
We verified the models presented in (24) and (32) using the far-field probe tool in CST Studio Suite that can be installed outside the calculation domain and gives an estimation of the scattered field into the far-field zone. Therefore, reasonable calculation resources are required for this kind of simulation, whereas only one wavelength of space from the structure is used for the calculation domain, but the far-field observation point position may be distanced significantly. We used two excitation regimes: one for the normal incidence and oblique reflection that corresponds to the theoretical model presented in (24) and the reciprocal case of oblique incidence and the observation point installed along the normal direction, which corresponds to the theoretical model of (32). Both theoretical formulas give naturally the same values, thus they are plotted as a single curve. Comparative results are presented in Fig. 8, confirming the theoretical model's validity. Slight deviations of the simulation are caused mostly by numerical issues, for example, due to mesh and or solver inaccuracy, Fig. 8.
Validation of the analytical models of (24) (perfect anomalous reflector) and (32) (anomalous reflector of a finite efficiency) via full-wave simulation of a finite-size MS. R 1,2 are the distances to the observation point for the corresponding illumination scenario. N is the number of supercells of the MS. The incident field amplitude E 0 = 1 V/m. and they are at an acceptable level for this type of full-wave analysis.
In order to create a full-wave simulation model when the MS is excited by an antenna of a final directivity, we used the so-called hybrid assembly simulation in CST Studio Suite. We use a standard rectangular horn antenna model from the Antenna Magus library initially requesting a directivity of 26 dB, but the full wave analysis with Integral Equation solver showed simulated directivity D = 23.8 dB. We used this project as a platform for an assembly simulation and inserted the MS model designed as a perfect anomalous reflector, similar to the case described in this section above. The simulation domain of the MS is analyzed with the Frequency Domain solver, thus allowing an accurate discretization of the model, taking into account near-field interactions of its subwavelength features. As a result, we can estimate the gain of the assembly system of the horn and MS. Polar plot results for both normal and oblique incidences are presented in Fig. 9(a) for the distance between the horn antenna excitation port and the origin of the MS equal to R 1,2 = 1000λ ≈ 2.07 m, in accordance with the geometry presented in Fig. 5(a) and (b). These results are for the largest considered case of the MS with the square size a 1 = 16d ms ≈ 43.3 mm. Finally, we can compare the theoretical predictions by the simple link-budget formulas (31) and (33) with the simulated results applying the Friis equation with the same Rx gain of 23.8 dB and the separation distances R 1,2 = 1000λ. The effect of scattering sidelobes can create interference with other users. This can be estimated using the sidelobe level (SLL), which is the ratio between the most prominent sidelobe and the mainlobe field amplitudes [37]. In the case of normal-to-oblique reflection, the most prominent sidelobe in the angular region of interest (top of Fig. 9(a), where the MS faces free space) is located at θ ≈ 45 • , with the amplitude of 0 dB. Because of that, the SLL is approximately −12.62 dB. Likewise, the prominent sidelobe for the oblique-to-normal reflection scenario is located at θ = 310 • , and also with 0 dB amplitude. As a result, the SLL is about −12.55 dB. Results for different square sizes of the MS are presented in Fig. 9(b), and they show good coincidence with the theoretical predictions. It is worth noting an expected slight overestimation of the received power by the idealized theoretical model based on the UNI approximation of the excitation and on the assumption that the induced currents are not distorted at the edges of the panel. This overestimation is at the level of about 1.5-2 dB, and it can be adjusted by characterizing the efficiency η eff of the implemented MS architecture.
It is worth mentioning that although all the results in this study are presented for a specific configuration of MSs, the model is applicable for reconfigurable MSs for any particular configuration state. It is only necessary to know the macroscopic reflection coefficient for any required state. For perfectly operating reconfigurable surfaces, this coefficient is known as a simple function of the angles of incidence and reflection, −1 = √ cos θ i / cos θ r . Reconfigurability can be provided by integrating tunable elements, for example, varactors, in metallic structures [19]. The only assumption in this work is that the reconfiguration of RISs is adiabatic, that is, slow when compared with the period of oscillations of the incident wave. That is, we do not account for frequency conversion, parametric amplification, and other effects due to possible fast modulations of MSs. Let us also note that the typical bandwidth of discrete impedance MSs considered here is around 20% [34] which is sufficient for supporting wireless links. Recent studies show that the bandwidth can be further improved to above 50% using other design methods [38].

IV. APPLICATION EXAMPLE: ESTIMATION OF THE REQUIRED SIZE OF THE MS
In this section, we consider an example of the practical application of the developed simple link-budget model. We make an estimation of the MS size that grants the desired level of the EM field at the observation point hidden behind an obstacle. The proposed approach is based on a comparison of the field anomalously reflected by the MS panel and the reference field created by the same Tx in free space, at a given distance R ref from the antenna. These two cases are illustrated in Fig. 5(a) and (d). In both cases, we consider fields in the far zone of the antenna and the MS. The reference Tx antenna can be arbitrary.
In Fig. 5(d), we show an antenna located at the Tx point, and the observation point is at the position of the Rx antenna. In the far-field region, the field of any antenna is a spherical wave that decays inversely proportional to the distance, and the field amplitude at the Rx position for the reference case of a free-space link can be written as Here, A is the amplitude of the wave, which is not relevant to this study. Next, we consider the field, reflected from the MS, excited by the same Tx antenna. The MS is located at distance R 1 from the Tx antenna. The observation point is located at a distance R 2 from the MS, in accordance with the geometry presented in Fig. 5(a). Using the above results of (42), we substitute the amplitude of the field incident to the MS from the antenna distanced at R 1 is E 0 = A/R 1 . We require that the anomalously reflected field from the MS given by (24) at the distance R 2 to be equal to the reference field from the horn antenna in free space at the reference distance R ref , or |E scz | = |E ref |. Then the required MS size a 1 (the area is S 1 = a 2 1 ) of a square MS reflector can be expressed as As a numerical example, we can set and there is an explicit dependence on the frequency (or wavelength λ = 2πc/ω). At high frequencies (small wavelength), path-loss is higher (due to medium attenuation), which is usually considered as a fundamental limitation of communication distances at high frequencies, especially millimeter and THz ranges [12], [39]. In stark contrast, if the communication link is via a reflecting MS, the corresponding formula (38) does not contain this wavelength-dependent factor. It appears that this limitation at high frequencies is removed. Let us discuss this important feature. Here, we stress that this discussion is not about the frequency bandwidth of antennas or MSs, and we discuss the path-loss dependence on the central frequency of the used frequency band.
To understand the general trend of path-loss frequency dependence, we need to consider how the antenna gain changes with the frequency range. In the sub-6-GHz frequency range, used in 4G and most of the 5G communication technologies, resonant antennas are used. Typical base-station antennas are electric dipoles or arrays of a few dipoles (to reduce the beamwidth in the vertical plane). The effective area of a resonant dipole antenna at resonance is determined by the wavelength. For a matched resonant electric dipole antenna, the effective area A eff = (3/8π )λ 2 (e.g., [40]). With the use of the general relation between the antenna effective area and gain (28), we find that for this antenna gain does not depend on the resonance frequency and equals 3/2. We see that if resonant antennas are used, then the path-loss indeed increases with the operational frequency as ω 2 , according to (44). For higher frequencies, the antenna size is reduced to maintain the resonant dimensions, while the antenna gain remains the same.
On the other hand, at high frequencies (millimeter and submillimeter ranges, such as 25-28 GHz already in use in some 5G technologies), directive antennas are used. Typical examples are parabolic reflectors and antenna arrays. The size of these antennas is much larger than the wavelength, but at high frequencies, the wavelength is small and this becomes a rather practical solution. In contrast to resonant antennas, the effective area of these antennas is approximately equal to the geometrical area S of the antenna aperture. For these antennas, (44) also apply, and we find that the antenna gain G = (4π S/λ 2 ) increases with the operational frequency as ω 2 , if the antenna size is kept the same. If both Tx and Rx antennas are directive, the path-loss given by (44) actually decreases as ω 2 . This property makes the use of high-frequency channels attractive even for very high frequencies, with the limitation mainly due to higher absorption in the atmosphere at these frequencies.
The other issue is the ray-type propagation mechanism, with much weaker scattering as compared with the sub-6-GHz range. In this respect, we note that when an MS panel on a UNI wall is illuminated by a highly directive beam, reflections from the illuminated spot on the wall form another directed beam in reflection, as illustrated in Fig. 1. In optimization of the propagation channels, this specularly reflected beam can find its use in illuminating other regions of the environment. Note also that the two beams interfere, which means that tuning the MS reflections affects also the sidelobe scattering from the UNI wall.
The developed analytical models for the reflected field in the far zone take into account the fields reflected from both the MS and the wall. In this article, we used the knowledge of the intensity and angular distribution of reflections from the wall only to estimate how much power is scattered toward the direction of the anomalous reflection by the MS, but the formulas can be used to estimate the total field at any point in the far zone. Worth noting that scattering into several directions (beam splitting) can also be considered using the developed model. If the MS is designed to reflect several waves in several directions, we assume that the induced currents contain two or more corresponding Floquet-type current components, and the derived formulas are applicable for all of them, in the same form. The additional lobes can be used to provide communication links for multiple users.

VI. CONCLUSION
In this article, we considered far-zone fields reflected and scattered by anomalous reflectors mounted at a wall and illuminated by focused beams. This configuration is most relevant for the potential use of RISs in the millimeter-wave or higher-frequency bands. In this situation, the reflected field is formed by two directive beams and their interference. One beam is an anomalously reflected beam due to the currents induced at the MS and the other beam is a specularly reflected beam due to the currents induced on the illuminated part of the supporting wall. If the propagation link between the Tx and the Rx can be formed not only via reflections from the MS (an LOS path is possible or there are other reflected rays coming to the Rx), the presented link-budget formula provides an estimate of the MS reflection contribution. The total field is found by adding the usual estimations for the direct path propagation and/or usual specular reflections and diffuse scattering.
We have analyzed the beam interference in the typical situation of the MS configured to reflect toward the Rx and for this case derived a very simple analytical formula for the linkbudget estimation. We have discussed the impact of sidelobes due to the finite sizes of both the MS panel and the illuminated spot of the supporting wall. We have demonstrated that the simple model of UNI illumination can adequately describe the scattered far field. We have validated this approximation with full-wave electromagnetic simulations. Importantly, the model accounts for the frequency and angular dependence of the MS reflection contribution via an equivalent gain factor (41). It is clearly seen that this factor is of the same form as a gain of a large-aperture directive antenna, whose frequency-dependent factor fully compensates for the free-space path loss increase with increasing frequency. We believe that the results of this work will be useful in studies and designs of MS-enhanced propagation environments, especially at millimeter-range and higher frequencies.