Near-Field Engineering in RIS-Aided Links: Beamfocusing Analytical Performance Assessment

Reconfigurable intelligent surfaces (RISs) are typically utilized in the far-field as an effective means of creating virtual line-of-sight (LOS) links to mediate non-LOS propagation in wireless communications via beamforming. Owing to their large surface and the multitude of scatterers, the use of RISs can be extended in the near-field, to transform the incident beam into a focused beam that is able to address the challenges of high frequencies more efficiently than conventional beamforming. In this paper we explain from a physics’ standpoint how the RIS can engineer wavefronts to transform the incident beam into a focused beam targeted at the user, and we employ the angular spectrum representation approach to describe analytically the dynamics of beamfocusing. We derive analytical expressions that provide the necessary insight into the dependencies and trade-offs between crucial parameters, such as the incident beam’s footprint on the RIS, the intended focal distance of the reflected beam, and the link topology. To assess the beamfocusing efficiency we provide metrics that are crucial for future applications, including energy efficient communications, wireless power transfer, tracking and localization.


I. INTRODUCTION
One of the most common functionalities of reconfigurable intelligent surfaces (RISs) regards redirecting an incident beam towards any desired direction, beyond specular reflection, essentially creating virtual line-of-sight (LOS) links to mediate non-LOS propagation.Owing to this unique feature, the RIS has been proposed as a means to bypass blockage, especially in high frequency communications, such as the terahertz (THz) band [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].As wireless communications are nowadays shifting to higher frequencies with the aim to meet the perpetual demand for increased bandwidth, it gradually becomes clear that, to achieve and maintain high quality of service, future networks are envisioned to be equipped with functionalities beyond conventional beamforming.
The associate editor coordinating the review of this manuscript and approving it for publication was Zaharias D. Zaharis .
A direct consequence of frequency upscale is that, for a certain aperture, a radiating element becomes electrically large, and the transition from the near-to the far-field moves to larger distances.Objects and users that at GHz frequencies were located in the far-field of the antennas, are now found within the near-field of large-scale antennas and RISs operating in the THz band, for example.The availability of electrically large surfaces opens up new opportunities for manipulating the wavefront of the radiated wave, to acquire curvature beyond the typical far-field planar form [11], [12], [13].For example, by shaping the curvature of a beam into spherical wavefronts, beams that would typically diffract can now counteract spreading, and focus.While the RIS has been so far extensively studied within the context of beamforming, the possibility for beamfocusing has only recently been addressed [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].With beamfocusing, the incident power can be concentrated at controllable distances from the RIS, towards any desired direction.Therefore, beamfocusing is ideal for dramatically increasing the received power in small areas, a key element for energy efficient communications [17], [24], [25], [26], [27] and localization applications [28], [29], [30], [31].
To exploit the full potential of beamfocusing, it is fundamental to understand the requirements for efficient and controllable formation of focal areas.The beamfocusing efficiency will depend on crucial parameters, such as the positioning of the RIS with respect to the transmitter and receiver, the properties of the beam footprint on the RIS, the reflection angle and the desired focal distance [32].Therefore, there is a need for analytical models that can clarify the bounds imposed by the system design parameters on the link performance.
In this work, the RIS beamfocusing capabilities are studied in terms of the received power.The main contributions are summarized as follows.
• An exact analytical model for beamfocusing at oblique angles is derived.The model provides the spatial distribution of the focused beam and the power delivered to the user at any location.
• The derivation of the analytical model is based on explicit electromagnetic modeling, by means of the angular spectrum representation approach, which captures the full-wave propagation characteristics of the RIS-reflected beam, as predicted by Maxwell's equations.
• The analytical model provides insight into the interplay between crucial parameters, such as the positioning of the RIS with respect to the transmitter and receiver, and the properties of the beam footprint on the RIS.
• Metrics to assess the beamfocusing efficiency are introduced.
• The approach followed in this work provides simple guidelines for algorithm design and performance optimization.

II. SYSTEM MODEL
Let us consider a RIS located at the origin of the coordinate system shown in Fig. 1.A beam illuminates the RIS, and is subsequently reflected by the RIS towards a desired direction.
The RIS reshapes the incident wavefront so that the reflected beam is focused at a desired focal distance.The directions of incidence and reflection are defined by the wavevectors k i and k r , respectively, which are expressed with respect to the elevation (θ) and azimuth (ϕ) angles of incidence (subscript i) and reflection (subscript r), as shown in Fig. 1  power delivery to the UE, it is necessary to bring the focal point at the UE location.This requires that the reflected beam is directed towards the UE, i.e. θ r = θ UE and that the focal distance is equal to the RIS-UE distance, i.e. f 0 = d UE .
Given a certain AP-RIS-UE topology, we would like to know how the incident beam is redistributed by the RIS upon reflection, and what is the power delivered to the UE from the reflected beam, which is focused at a desired location.To this end, we need to calculate the field reflected from the RIS at any desired observation point (x, y, z).Here, we will follow the angular spectrum representation approach adopted in [18], according to which the reflected field can be calculated anywhere in the semi-infinite space z > 0, using only (a) the footprint of the incident field on the RIS and (b) the phase introduced by the RIS.

A. FOOTPRINT OF INCIDENT BEAM ON THE RIS
The footprint of the incident beam on the RIS depends on the properties of the AP antenna.For most practical cases the main lobe of the incident beam can be modeled by a Gaussian distribution [9], [18], [33], [34].Without loss of generality we consider a y−polarized beam propagating on the xz−plane, the E-field of which at z = 0 is written as

RIS
) exp (−jk sin θ i x)ŷ, (1) where E 0 is a complex constant, w RIS the radius of the footprint at normal incidence (θ i = 0), k = 2π/λ is the free-space wavenumber and λ the wavelength.Note that this model takes into account the fact that under oblique incidence (θ i ̸ = 0) the footprint acquires elliptical shape, with the major axis of the ellipse residing on the x−axis [33].

B. RIS REFLECTION COEFFICIENT
To perform beamfocusing it is necessary to transform the incident beam into spherical wavefronts that converge at the desired location, characterized by the focal distance f 0 and the angle θ r , as schematically shown in Fig. 1(c).The role of the RIS is to introduce the necessary phase φ(x, y), in order to reshape the incident wavefront into the desired form, i.e. to add φ(x, y) to the phase of the incident wavefront.For polarization-preserving RIS, the footprint of the reflected beam on the RIS can be written as where (x, y) = |R| exp(−jφ(x, y)) is the RIS reflection coefficient and E i (x, y) the incident field at the RIS plane, i.e. the incident beam footprint.R is a complex constant that accounts for possible loss upon reflection (|R| ≤ 1) and φ(x, y) is the phase introduced by the RIS.
For conventional beam steering, the RIS corrects the phase tilt of the incident wave, which has the form exp (−jk sin θ i x) (see (1)), by adding a phase with the opposite slope and introduces a linear phase exp (−jk sin θ r x), to form a wavefront that directs the wave towards the desired direction.Hence, for beam steering on the xz-plane, the RIS introduces the phase (for steering out of plane, see e.g.[9]).For beamfocusing, after correcting the incident phase, the RIS must provide a phase of the form exp −jk (f 0 cos θ r ) 2 + (x − f 0 sin θ r ) 2 + y 2 , and the phase introduced by the RIS is expressed as Note that, in the limit f 0 → ∞, (4) approaches (3) (we may omit the global phase), i.e. corresponds to the phase for conventional beamsteering.

III. ANGULAR SPECTRUM REPRESENTATION APPROACH
Let us consider the field reflected from the RIS at any observation point (x, y, z), E r (r) = E r (x, y, z).In this notation, the reflected field right before departing from the RIS, i.e. on the xy-plane where the RIS surface resides, is E r (x, y, 0) ≡ E RIS (x, y).Its two-dimensional Fourier transform is y)e −j(k x x+k y y) dxdy, (5) where x, y are the Cartesian transverse coordinates and k x , k y the corresponding spatial frequencies.Similarly, the inverse Fourier transform reads Note that the field E RIS and its Fourier transform ÊRIS represent vectors and, hence, the Fourier integrals hold separately for each vector component.The field has to satisfy Maxwell's equations, which for free-space propagation reduce to the vector Helmholtz equation (∇ 2  + k 2 )E(r) = 0. Expressing similarly the reflected field E r (x, y, z) via its Fourier transform and inserting it into the Helmholtz equation, we find that the Fourier spectrum Êr of the reflected field evolves as where After performing the inverse Fourier transform of ( 7) we find for arbitrary z which is known as the angular spectrum representation [35], [36].The result of ( 9) states that the field at any z > 0 is determined entirely by its Fourier spectrum at z = 0. Hence, to calculate the reflected field, only knowledge of E RIS (x, y) is required, which is given by (2), using (1) and ( 4).Then, the reflected field at any observation point (x, y, z) is calculated using (5), the Fourier transform of E RIS (x, y), and inserting the result into (9).In (9), while the integration is performed in the entire R 2 domain, components with imaginary k z correspond to evanescent waves, which do not propagate.Therefore, we may reduce the domain of integration within the range k 2 x + k 2 y < k 2 .This technique provides the reflected power density distribution for any illumination conditions, from full to partial RIS illumination [33].Full illumination refers to the case where the incident beam footprint extends beyond the size of the RIS.In this case the integration domain in ( 5) is limited within the area of the RIS.Partial illumination refers to the case where the incident beam footprint is smaller than the RIS [9], [18], [34].Hence, we may neglect the RIS boundary and perform the integration of (5) in the entire R 2 domain; the RIS size becomes effectively infinite.Importantly, in this case, we may simplify the integration of (9), to express the reflected power density in compact analytical form.For example, we can take advantage of the fact that the integrand in (9) contributes to the integration essentially at a narrow region around k r = k sin θ r , where the k−content of the reflected beam is distributed.Therefore, for plane waves or beams of finite extent that typically have a narrow k−content, we may expand k z at the direction of propagation of the reflected wave, i.e. around k x = k r , k y = 0.This leads to the approximation of (8) (see Appendix A for details) For θ r = 0 • , this approximation reduces to the well-known parabolic form [18].We may further simplify the analytical calculation of ( 9) by taking advantage of the relatively large curvature of φ in (4), required to focus the beam at distances larger than the RIS size.In this case we may expand (4) around the center of the RIS to obtain (see Appendix A for details) For θ r = 0 • , i.e. beam focusing along the z-axis, (11) reduces to the simple parabolic form φ(x, y) = −k sin θ i x + kf 0 + k(x 2 + y 2 )/2f 0 [18].Using (1), ( 2) and ( 11) in ( 5) we calculate the k-spectrum of the reflected beam at the RIS plane, ÊRIS (k x , k y ).The reflected field at any observation point on the xz−plane is calculated by inserting this result and ( 10) into ( 9), and the resulting power density S r = |E r | 2 /2Z 0 is given by where Z 0 is the free-space wave impedance and z R = πw 2 RIS /λ is the Rayleigh length.Note that, for f 0 → ∞, (12) yields the received power for conventional beam forming [9], [18].The choice , where P t is the total power (Watts) of the incident beam.
Along the propagation direction of the focused beam, i.e. for x = r sin θ r , z = r cos θ r , the argument of the exponential term in (12) becomes zero, leading to a maximum power density expressed by the remaining prefactor By differentiating (13) with respect to r, the slope for r → 0 acquires the positive value 4P t |R| 2 cos θ i /π f 0 w 2 RIS , while vanishes for r → ∞.Therefore, there is always a maximum, which is expected at r = f 0 .To understand how beamfocusing depends on the involved parameters, next we examine the derived expressions with examples of realistic scenarios.

IV. IMPACT OF RIS FOOTPRINT AND FOCAL DISTANCE ON RECEIVED POWER
Let us consider a D-band indoor scenario with operation frequency of 150 GHz.The AP is equipped with a directional antenna of tunable gain that transmits a beam of constant power P t = 1 W towards the RIS.Depending on the position and orientation of the AP relative to the RIS, footprints of different size and ellipticity are captured by the RIS, all with the same total power.The RIS is lossless (R = 1) and focuses the incident beam towards θ r = 0 • .In Fig. 2 we use (12) to examine beamfocusing as a function of the beam footprint, for intended focusing at f 0 = 4 m focal distance.For relatively small footprint (w RIS = 0.1 m) we observe inefficient focusing, i.e. the power density is maximized sooner than f 0 , with maximum that is relatively weak.However, with increasing footprint (w RIS = 0.2 m, 0.4 m), the maximum power density improves dramatically and the focal point f 0 is quickly approached.
In Fig. 3 we examine the same scenario as a function of the focal distance, for constant footprint w RIS = 0.2 m.The relatively large footprint enables efficient beamfocusing at the desired distance and the maximum power density reduces with increasing f 0 .
To understand the observations in these two examples, let us examine the power density at the focal distance.By setting r = f 0 in (13) we obtain The form of ( 14) implies that, for constant incident power P t , the power density at a certain observation distance r is expected to increase with larger footprints (and/or angles of incidence), while it is expected to decrease with larger focal distances (and/or angles of reflection), in accord with our observations.The two general examples of Fig. 2 and Fig. 3 aim to provide insight into the underlying mechanisms of beamfocusing, beyond a specific AP-RIS-UE topology.
Next, we examine explicitly the impact of the topology on beamfocusing.

A. IMPACT OF AP POSITION ON RECEIVED POWER
To understand how beamfocusing depends on the distance and orientation of the AP relative to the RIS, let us now consider a mobile AP with antenna of 30 dB constant gain.In Fig. 4(a) the AP moves along the direction θ AP = 30 • and the RIS focuses the incident beam at f 0 = 4 m focal distance towards the direction θ r = 60 • .Due to oblique incidence (θ AP ̸ = 0), the incident footprint has shape that is elliptical (see ( 1)), and size that increases with increasing distance d AP .For d AP = 1, 2, 3 m, the radius of the footprint becomes w RIS = 9, 18, 27 cm, respectively (see Appendix B for details).Therefore, although the incident power is constant (P t = 1 W), the maximum power density at the focal point increases with increasing AP-RIS distance. In

B. IMPACT OF FOCAL DISTANCE ON RECEIVED POWER
The power density at the UE can be calculated using (12) at the UE location, i.e. for x = d UE sin θ UE , z = d UE cos θ UE .To maximize the power at the UE we need to ensure that the focal point is located at the UE, i.e. that θ r = θ UE and f 0 = d UE .In this case, the focal point essentially follows the UE as it moves, guaranteeing that the maximum power is delivered.Solving in terms of d UE we can retrieve the RIS-UE distance at which the power density acquires a desired threshold, S th , that is The form of (15) implies that the trajectory of constant power forms a circle with radius d UE (0)/2, centered at (x = 0, z = d UE (0)/2).As an example, in Fig. 5(a) we seek the UE positions at which the power density is constant and equal to S th = 0.4 W/cm 2 , when the beam is focused at focal distance f 0 = 4 m.The UE positions that satisfy the imposed criterion are denoted by the white circle, which is expressed analytically by (15).The circle essentially marks the locus of constant power density delivered by several different focused beams.The examples marked with the letters A, B and C depict three such beams, and their power density evolution along each individual direction is shown in 29540 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

V. BEAMFOCUSING EFFICIENCY ASSESSMENT A. POWER ENHANCEMENT OF BEAMFOCUSING VS. BEAMFORMING
From the examples demonstrated so far, it is evident that beamfocusing offers the advantage of high power concentration at confined regions in space.This is particularly practical for increasing the power at the receiver, without the need for higher gain at the transmitter, e.g. for energy-efficient communications and power transfer applications [11].To quantify the achievable enhancement in the received power we calculate the ratio of (13) over the received power in the absence of focusing (f 0 → ∞), accounting for conventional steering.The enhancement factor, e f ≡ P r /P r (f 0 → ∞), at r = f 0 yields In Fig. 6 we present ( 16), as a function of z R /f 0 and cos θ i / cos θ r , which serves as a universal plot, capturing any combination of all the involved parameters.In our examples, where z R /f 0 > 12 and cos θ i / cos θ r > 0.5, the minimum achievable enhancement is in the order of e f ∼ 10 2 .

B. FWHM OF FOCAL AREA
The ability to control the size of the focal spot is crucial for partitioning the radial distance along the focusing direction, e.g. for localization applications [11], [20], [21].Under the constraint of constant incident beam power, the size of the focal spot changes inversely with the beam maximum, which in turn depends on both the focusing direction and the focal distance, as captured by (14) and the examples studied so far.The extent of the focal spot can be expressed by its Full Width Half Maximum (FWHM) along the propagation direction, w FWHM , which for θ i = θ r = 0 • is given by (see Appendix C for derivation) The functional form of (17) dictates that w FWHM increases with f 0 and reduces with w RIS .As an example, in Fig. 7(a) we plot the trace of beam maximum (dashed black lines) and FWHM of focal spot (shaded areas) as a function of the steering angle θ r , for w RIS = 0.25 m and variable f 0 .Indeed, as f 0 increases, the FWHM increases as well.Note, however, that according to (17) we can achieve a focal spot with constant FWHM, regardless of the focal distance, if we increase w RIS with increasing f 0 .For example, as demonstrated in Fig. 7(b), using (17) with f 0 = 4 m and w RIS = 0.25 m, we find that upon changing f 0 to 8 m and 2 m, we can achieve a constant FWHM if we choose w RIS to be 0.5 m and 0.125 m, respectively.The constant ratio w RIS /f 0 in this example is not accidental; note that, in the limit z R ≫ f 0 , (17) takes the simple form indicating that a proportional change in f 0 and w RIS leads to the same FWHM.In our examples z R /f 0 > 12, i.e. we are well within the asymptotic limit of ( 17), as can be verified in Fig. 7(c).The simple form of ( 18) provides a straightforward way to partition the radial distance in segments of equal size, essentially forming zones of constant width.The formation of such zones and, in particular, the ability to control their extent, is crucial for tracking and localization applications, where a sensing-based communication scheme is able to detect the UE distance within controllable resolution, and transmit data with high quality of service.

VI. MULTI-BEAM OPERATION
For multiple users that reside in the RIS near field, the RIS can split the incident beam into a multitude of focused beams, to simultaneously serve all users.In this case, each reflected beam is associated with an individual reflection coefficient n (x, y) = |R| exp(−jφ n (x, y)), accounting for the n th user, with φ n (x, y) given by (4) (or its approximation, (11)).The angle θ r and the focal distance f 0 have distinct values for each n, θ r,n and f 0,n , respectively, thus accommodating reflected beams, directed and focused towards each user.The footprint at the RIS is then written as the linear superposition which is a generalization of (2) to N beams.The 1/ √ N prefactor is chosen to ensure that the incident power is preserved and equally distributed between the reflected beams, which are assumed to be spatially separated so that they interfere weakly (see Appendix D for details).Using (19) to integrate (9) leads to the analytical expression for multiple focused beams.Note that, because the total footprint is a linear superposition of N individual contributions, the total reflected field can be calculated in a straightforward manner, by solving for the n th E-field, E r,n , and expressing the total field as E r,total = N n=1 E r,n .The total power density, which is ∼ |E r,total | 2 , will eventually be cast in a rather long and unpractical analytical form; yet, we can take advantage of the fact that, for spatially separated beams that do not interfere significantly with each other, |E r,total In this case, we can apply the expressions derived in our work, which describe the dynamics of a single beam, to each of the simultaneously generated focused beams, individually.
As an example, in Fig. 8 we examine N = 3 simultaneously generated focused beams, and we compare the full numerical propagation of the beams using (9), with the analytical expression (12), applied to each of the three beams, individually.The incident beam has total power P t = 1 W and footprint radius w RIS = 20 cm, and is split into three beams of equal power upon reflection.The beams are reflected towards the directions characterized by the angles θ r,1 = −3 • , θ r,2 = 0 • and θ r,3 = 6 • , and the focal lengths f 0,1 = 4.5 m, f 0,2 = 3.5 m and f 0,3 = 5.5 m, respectively.In Fig. 8(a) we present the numerical solution of ( 9), as a function of the propagation distance z, using (19).The dashed lines denote cross-sections at distance z = 3 m, 4 m and 5 m, which are shown in panels (b),(c) and (d), respectively.Together with the numerical crosssections (green solid lines) is also shown the analytically calculated power density (black dashed lines), using (12) with = n , separately applied for n = 1, 2, 3. Note how, despite the beam interference that dominates close to the RIS, the analytical expression successfully reproduces the numerically calculated beam profile, especially at distances where the beams evolve into distinct, spatially separated beams.These observations enable us to use our analytical results to any number of simultaneously generated focused 29542 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
beams that are spatially separated, thus keeping the simplicity of our analytical expressions and the associated metrics, and generalizing the analysis and conclusions to multiple focused beams.Importantly, we can go beyond conventional beamfocusing, where an incident wave is focused into a single focal area, to design beams directed towards multiple directions with multiple focal points, for broadcast to selected multiple users.
Our framework can be applied to general multi-user scenarios, in conjuction with a multiple access scheme.For example, in a Time Division Multiple Access (TDMA) scheme, the RIS focuses the incident beam towards different directions and distances at different time slots, to successively serve each user.Similarly, in a Space Division Multiple Access (SDMA) scheme, different areas of the RIS are devoted to simultaneously focus multiple beams towards multiple users.In environments with severe multipath, where the instantaneous focused power may change due to fading, our framework can be directly extended to model beams with statistical properties (see e.g.[34]).

VII. CONCLUSION
As wireless communications are nowadays shifting to higher operation frequencies, taking advantage of the near-field offered by electrically large RISs opens up new opportunities for manipulating the wavefront of beams, to enrich communications with functionalities beyond conventional beamforming.In this work, we studied near-field engineering in RIS-aided links, in which the RIS transforms the incident beam into a focused beam, and we assessed the RIS beamfocusing capabilities analytically, using explicit electromagnetic modeling.To describe beamfocusing with a physically consistent model, the derivations were based on the angular spectrum representation approach, which captures the dynamics of free-space wave propagation in compliance with Maxwell's equations.With our model, we demonstrated the dependencies and trade-offs between crucial parameters, such as the incident beam's footprint on the RIS, the intended focal distance of the reflected beam, and the link topology.To assess the beamfocusing efficiency we provided metrics that are crucial for future applications, such as energy efficient communications, tracking and localization, and we demonstrated the theoretically expected performance with examples of typical D-band indoor scenarios.

APPENDIX A VALIDITY OF APPROXIMATIONS
To calculate (10) we expand (8) around the direction of propagation, which is characterized by k x = k r , k y = 0, where k r = k sin θ r .First, expansion around k y = 0 leads to where we have retained all terms up to 2 nd order.Next, expansion of (20) around k x = k r leads to where we have retained all terms up to 2 nd order.Substitution of k r = k sin θ r in (21) leads to the result of (10).
To calculate (11) we expand (4) around the center of the RIS, i.e. at x = 0, y = 0. Keeping all terms up to 2 nd order leads to the result of (11).
To verify the validity of the approximations ( 10) and ( 11) we have compared the analytically derived (12) with exhaustive full-wave numerical propagation examples under extreme conditions, e.g. at large angles of incidence and reflection.In this section we demonstrate the validity of the approximations ( 10) and ( 11) through comparison with their respective full form.In the following examples the AP is located in front of the RIS (θ AP = 0 • ), creating a footprint of w RIS = 0.15 m on the RIS.The RIS focuses the incident beam along three different directions characterized by θ r = 0 • , 30 • , 60 • .
In Fig. 9(a) we plot (10) for θ r = 0 • , which approximates the parabolic form of (8) in a relatively large region around k x = k y = 0, where the k-content of the reflected beam is located.This is illustrated in Fig. 9(b), where we show a cross-section of (10) along k y /k 0 = 0 (red dashed line) as well as the full form of k z for comparison (solid black line).The gray region marks the cross-section of the beam's k-content, verifying that within its extent, where integration takes place, ( 8) and (10) coincide.In the remaining panels we present the respective plots for θ r = 30 • (Fig. 9(c),(d)) and θ r = 60 • (Fig. 9(e),(f)).Note that the footprint used in these examples is relatively narrow and leads to a relatively wide kcontent.For larger footprint, the beam k-content is narrower, and the error in ( 10) is further suppressed.
In Fig. 10(a) we plot (11), the approximation of (4), which for θ r = 0 • has a parabolic form centered at x = y = 0. Using the full form of φ, in Fig. 10(b) we plot the relative phase error, which we define as (φ − φ approx.)/φ, where φ accounts for (4) and φ approx.for (11).The solid circle marks the FWHM of the incident beam.In the remaining panels we present the respective plots for θ r = 30 • (Fig. 10(c),(d)) and θ r = 60 • (Fig. 10(e),(f)).Note how the center of the parabolic phase moves along the x-axis with increasing θ r .In all cases the relative error is practically below 0.1%.

APPENDIX B DERIVATION OF FOOTPRINT RADIUS IN TERMS OF THE ANTENNA GAIN
The footprint of the AP beam on the RIS can be expressed using ( 1) with E 0 = 4Z 0 P t cos θ i /π w 2 RIS , as with expressing the AP radiation pattern at distance d AP , where the RIS is located.The power density of ( 22) can be alternatively expressed in terms of G t , the AP gain, as: where d is the radius of the sphere centered at the AP.For pencil beams d ∼ d AP and, upon simple inspection of ( 22) and ( 24), we reach the final result

APPENDIX C DERIVATION OF FWHM OF FOCAL AREA
To determine the FWHM of the focal spot we need to find the distance at which the power density is maximized.For θ i = θ r = 0 • , differentiation of (13) with respect to the propagation distance r yields the focal distance at which the power density is maximized and becomes equal to Using (27) to solve (13) for S r = S r (r = r focal )/2 in terms of w RIS , yields two solutions w RIS± corresponding to the two edges of the focal spot along r, where the threshold criterion is fulfilled.The FWHM of the focal spot along r is then w FWHM = w RIS+ − w RIS− .

APPENDIX D RIS REFLECTION COEFFICIENT FOR MULTI-BEAM OPERATION
The total reflected field can be expressed as the linear superposition of the field of the individual beams, E r,total = N n=1 E r,n .At z = 0 the total field E r,total becomes simply the footprint E RIS , and the individual fields are expressed in terms of the incident field as E r,n = w n n E i , where n is the reflection coefficient for the n th beam with w n ∈ R accounting for the weight with which the n th beam is reflected.Hence, the footprint takes the form For a lossless RIS, the power of the reflected beam must be equal to that of the incident, i.e. = N n=1 w 2 n (|R| = 1 for lossless RIS) and, hence, (30) leads to N n=1 w 2 n = 1.For equally distributed power among the reflected beams w 1 = w 2 = . . .w N , which leads to w n = 1/ √ N .
(a).In this work we consider beam steering on the xz−plane and, therefore, we may neglect the angle ϕ.The incident beam is generated by an access point (AP), the location of which is characterized by the distance d AP from the RIS center and the angle θ AP ≡ θ i , as shown in Fig.1(b).The location of the user equipment (UE) is characterized by the distance d UE from the RIS center and the angle θ UE .As shown in Fig.1(c), the reflected beam may point towards any direction and, hence, to maximize the

FIGURE 1 .
FIGURE 1. System model of a RIS-aided beamfocusing link.(a) The incident field, impinging along the direction characterized by the angles θ i , ϕ i , is focused along the direction characterized by the angles θ r , ϕ r (b) Cross-section of the focused beam on the xz−plane, illustrating the AP and UE locations relative to the RIS.(c) Schematic illustration of the spherical wavefronts of the focused beam that are directed at angle θ r and converge at distance f 0 .The power at the UE is maximized when θ r = θ UE and f 0 = d UE .

FIGURE 2 .
FIGURE 2. Impact of incident beam's footprint on the RIS, w RIS , on beamfocusing.Cross-section of power density distribution on the xz−plane of a beam focused along the direction θ r = 0 • with f 0 = 4 m, and (a) w RIS = 0.1 m, (b) w RIS = 0.2 m, and (c) w RIS = 0.4 m.(d) Power density along the beam center for the cases shown in panels (a)-(c).

FIGURE 3 .
FIGURE 3. Impact of focal distance f 0 on beamfocusing.Cross-section of power density distribution on the xz−plane of a beam focused along the direction θ r = 0 • with w RIS = 0.2 m, and (a) f 0 = 2 m, (b) f 0 = 4 m, and (c) f 0 = 8 m.(d) Power density along the beam center for the cases shown in panels (a)-(c).

FIGURE 4 .
FIGURE 4. Impact of AP distance and orientation with respect to the RIS, on beamfocusing.Power density along the beam center, for AP locations characterized by (a) θ AP = 30 • and d AP = 1, 2, 3 m (b) d AP = 2 m and θ AP = 0 • , 30 • , 60 • .In all examples the AP beam is focused by the RIS towards the focal point characterized by θ r = 60 • and f 0 = 4 m.

Fig. 4 (
b) the AP moves along a circular trajectory at constant distance d AP = 2 m from the RIS, creating a footprint with w RIS = 18 cm.As the AP departs from the RIS normal, the incident footprint becomes more elliptical, in turn increasing in size, and improving the maximum power density at the focal area, as shown in the examples for θ AP = 0 • , 30 • , 60 • .

FIGURE 5 .
FIGURE 5. Beamfocusing at the UE position.(a) Spatial distribution of power density for the three example beams A, B, C, focusing along θ r = 0 • , 30 • , 60 • , respectively.The white circle marks the locus of UE positions, where the received power is constant.(b) Cross-section along the propagation direction of beams A, B, C shown in (a).(c) Power density of beamfocusing at the UE position, as a function of the power density threshold S th .The contour lines show examples of S th , and represent the locus of UE positions, where the same received power can be achieved for each individual case.

Fig. 5 (
Fig. 5(b).Requiring different threshold leads to a circle of different radius and the collection of all such possibilities is shown in Fig. 5(c) as a function of S th .

FIGURE 6 .
FIGURE 6. Enhancement factor, e f , expressing the increase in the achievable power delivery to the UE due to beamfocusing, over the respective with beamforming.

FIGURE 7 .
FIGURE 7. Properties of focal spot.(a) Trace of beam maximum (dashed black lines) and FWHM of focal spot (shaded areas) as a function of the steering θ r for w RIS = 0.25 m and variable f 0 .(b) Same as in (a) with tunable w RIS , in order to achieve the same FWHM for the different f 0 's.(c) Plot of (17) and asymptotic limit for z R ≫ f 0 .

FIGURE 8 .
FIGURE 8. Multiple simultaneously focused beams, for broadcast to selected multiple users.(a) Power density of numerically propagated reflected field.The incident beam has footprint w RIS = 20 cm and carries total power P t = 1 W, which is split equally by the RIS into three beams, characterized by θ r = −3 • , 0 • , 6 • , and f 0 = 4.5 m, 3.5 m, 5.5 m, respectively.The white dashed lines mark cross-sections of the beams at (b) z = 3 m, (c) z = 4 m, and (d) z = 5 m, where comparison with the analytical model of (12) for single beam is also shown.
|E i | 2 .For beams that are spatially separated we may simplify |E r,total | 2 ≈ | 2 , which leads to |E RIS | 2 ≈ N n=1 |w n n | 2 |E i | 2 .Note that N n=1 |w n n | 2 = N n=1 w 2 n |R| 2 29544VOLUME 12, 2024Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.N n=1 |E r,n