Efficient Computation of Scattered Fields From Reconfigurable Intelligent Surfaces for Propagation Modeling

We propose a method to efficiently compute the scattered electric field of a reconfigurable intelligent surface (RIS) for multiple configurations. In contrast to most existing methods that assume that each unit cell scatters an incident wave individually instead of collectively, our method accounts for the mutual coupling of unit cells. This allows us to estimate the scattered fields in the main scattering direction of an RIS, at an accuracy that is comparable to full-wave analysis. Furthermore, combined with ray tracing, the computed scattered fields can be used to model wave propagation in realistic, multipath radio environments with RISs. Hence, our method efficiently addresses three critical considerations for the analysis of RIS-enabled links: mutual coupling between unit cells of an RIS, multipath effects in the channel due to the RIS acting as a diffuse scatterer, and the variability of the RIS scattering properties that requires extensive computational effort to account for.


I. INTRODUCTION
R ECONFIGURABLE intelligent surfaces (RISs) can redirect electromagnetic (EM) waves in a wireless communication channel in real time, to enhance the signal strength at an intended receiver [1].With this functionality, waves can bypass obstructions to improve coverage at nonline-of-sight (NLoS) areas [2].RISs have given rise to the concept of a smart radio environment, where wave propagation is programmable [3].This challenges the standard view of the environment as a fixed aspect of a communication link [4].More importantly, RISs can operate in a low-cost manner because they are passive reflectors.As a result, they are considered as a promising technology for current, emerging, and future generations of communication systems [5].
To evaluate the received signal strength (RSS) at a receiver, several path loss models of RIS-enabled links have been proposed [6], [7], [8], and [9].The path loss models of [6], [7] assumed that unit cells receive incident signals and reradiate them, acting as secondary transmitters.With the known phase difference of unit cells, the total scattered power was found by field superposition.To calculate the received and radiated power of unit cells, a unit cell gain was employed.Unlike the gain of antennas, which is well defined, the unit cell gain of [6] and [7] is a parametric model inspired by the pattern of patch antennas [10], with one free parameter to be determined.In [7], this parameter was determined by setting the unit cell gain equal to the gain of a microstrip patch antenna of dimensions 0.5λ × 0.5λ (λ being the free-space wavelength).This yielded a gain of π (4.97 dBi), whereas [6] used the gain of 8 dBi for unit cells with dimensions of 0.35λ × 0.35λ.Alternatively, one can choose the parameter to fit the model to simulation or measurement data [11]; however, this makes the model nonpredictive.
There is another type of model that uses the analytical solution for the scattered fields from a rectangular flat conducting plate illuminated by an incident plane wave [8], [9].The expression for the scattered fields was derived using physical optics, assuming that the plate was infinitely large [12].For finite-size plates, the analytical fields were approximate and became less accurate as the observation points were farther from the specular directions.By approximating rectangular unit cells as patches, the total scattered field of an RIS was determined by field superposition.However, the geometry of unit cells can be very different from that of a rectangular patch.In fact, extensive research efforts are dedicated to tailoring unit cell geometries of metasurfaces, to achieve various performance objectives [13], [14], [15].Moreover, this model ignored the substrate and the conducting ground plane that play important roles in the response of a unit cell to an incident plane wave.
All aforementioned models rely on the assumption that the total scattered field of an RIS is the superposition of the scattered fields from each unit cell, considered separately from the rest.In reality, though, unit cells are mutually coupled and scatter collectively [16].The limitations of existing models motivated us to use the complex radar cross section (CRCS) of an RIS obtained by full-wave analysis to determine the scattered fields from an RIS, fully accounting for the geometry of the RIS and the mutual coupling between its unit cells in [17] and [18].Moreover, we used the CRCS to model an RIS as a secondary source in ray tracing.This allowed us to derive site-specific propagation models for RIS-enabled links, accounting for multipath effects in the environments, with comparable accuracy and improved efficiency compared to pure full-wave analysis [19].This method has been experimentally validated in [20], by modeling and measuring an actual communication channel with an anomalous reflection metasurface.
RISs are dynamically reconfigured to redirect waves to different users.This is realized by changing the phase shifts of unit cells.A combination of specific phase shifts of the unit cells of an RIS is referred to as a "configuration."As their scattering properties change, so is their corresponding CRCS, which is generally different for each configuration.The method we presented in [17] and [18] requires one fullwave simulation to compute the CRCS for each configuration of an RIS.Note that while full-wave analysis provides accurate results, it is computationally intensive.The raytracing part in the method of [17] and [18] requires negligible time compared to the full-wave simulation part.Since an RIS has a large number of configurations, the need for repeating computationally expensive full-wave simulations for different RIS configurations is an important limitation for this CRCS-based approach.
The aim of this work is to address these limitations, achieving accurate and efficient modeling of RISs in multipath propagation environments, for various RIS configurations.Hence, we introduce the effective unit cell CRCS, taking into account the mutual coupling of unit cells.Based on this, we efficiently compute the scattered fields of the RIS.This concept can be directly integrated with ray tracing, to enable the accurate analysis of realistic communication channels with RISs.For brevity, we use the term "RIS channels" to represent communication channels with RISs (e.g., the geometry of Fig. 1) throughout the article.The proposed method is the first that jointly addresses three critical factors in RIS channel analysis: mutual coupling between unit cells of an RIS, multipath effects in actual radio environments, and the computational challenge of computing the scattered fields of the RIS for a large number of configurations.
The rest of this article is organized as follows.We introduce the problem addressed by this article and a general methodology to solve this problem in Section II.The effective unit cell CRCS is defined in Section III.Next, our proposed method for computing the scattered fields of an RIS is derived in Section IV.Section V validates this method and compares it with the path loss models proposed in [6] and [7].Section VI gives an example of modeling an RIS communication channel of realistic dimensions and demonstrates the accuracy of the proposed method by comparing the results with those obtained by the experimentally validated method of [18].Section VII summarizes our contributions.

II. PROBLEM STATEMENT AND PROPOSED SOLUTION
This section introduces the propagation modeling problem for RIS channels and our proposed solution.It is shown that the bottleneck of this solution is the efficient computation of scattered fields from the RIS for different configurations.

A. Propagation Modeling Problem for RIS Channels
Fig. 1(a) illustrates a 2-D view of an indoor hallway junction RIS channel, where the purpose of the RIS is to redirect waves Two-dimensional view of an indoor RIS channel, where the configuration of the RIS is adapted to the locations (a) and (b) of the receiver; selected rays that exist in the environment are displayed.from the transmitter (Tx) toward NLoS receiver (Rx) points in the second hallway.In this scenario, the RIS configuration is designed to directly illuminate the receiver, meaning that the main-lobe direction of the RIS aligns with the receiver location.
We focus on the scenario where an RIS is placed in the far-field region of a transmitter, and a receiver is placed in the far-field region of the RIS.This is a practical working scenario of an RIS, due to the large dimensions of realistic radio environments compared to the wavelength.Surfaces such as floors, ceilings, and walls, which are situated in front of the transmitter and the RIS and interact with the waves from them, are typically located at an electrically large distance (see Fig. 1).Hence, the far-field distances of the transmitter and the RIS can be estimated by their values in free space.
A notable characteristic of RIS channels is their ability to adapt to the changing location of the receiver, as depicted in Fig. 1(b): the main-lobe direction of the RIS is adjusted to maximize the RSS at the receiver, which is located at a different position.
In an RIS channel, the RIS works as a secondary source, and scattering waves are radiated by the transmitter.The fields from the transmitter and the RIS interact with the environment, leading to enhanced multipath propagation.The reflected, transmitted, and diffracted fields that originated from both the transmitter and the RIS contribute to the total electric field at the receiver, denoted as E Rx .

B. Ray Tracing an RIS Channel
We use ray tracing to compute the electric field at a receiver point of interest.In this work, we use the shooting and bouncing ray (SBR) method with a ray path correction process based on image theory [21], [22], [23], to determine the ray paths that exist in an environment.This method combines the accuracy of the image method and the efficiency of the SBR method, determining exact ray paths.By separately ray tracing the transmitter and the RIS, all the ray paths from these two sources that reach the receiver are computed.The next task is to compute the electric fields of all determined rays.
For simplicity, Fig. 1 displays four rays that reach the receiver (a single reflected and a single diffracted ray from the transmitter, a direct and a single diffracted ray from the RIS).When P rays from the transmitter and Q rays from the RIS reach the receiver, the total electric field is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where E Tx p and E RIS q denote the electric field of the pth ray from the transmitter and that of the qth ray from the RIS, respectively.The field computation of all rays from a transmitter requires its radiation pattern and transmit power.Specifically, the field of the direct ray at a receiver point is [21] where β and η 0 are the free-space wavenumber and impedance, respectively, P t is the power radiated by the transmitter, G t is the gain of the transmitter, E Tx is the normalized radiation pattern of the transmitter, and r dir is the length of the direct ray path.In (2), V Tx = √ η 0 P t G t /(2π ) E Tx defines the far-zone radiation characteristics of the transmitter.Combined with the reflection, transmission, and uniform geometric theory of diffraction (UTD) coefficients, the fields of reflected (e.g., the red dashed rays in Fig. 1), transmitted, and diffracted rays (e.g., the red dotted rays in Fig. 1) are computed.This is the standard field computation method in ray tracing [21].
For a specific RIS channel, the radiation characteristics of the transmitter are known so that E Tx p in (1) can be directly computed with this field computation algorithm.For example, the horizontal and vertical components of the electric field of a single reflected ray are computed by [21 where h and v are the reflection coefficients at walls for horizontal and vertical polarizations of incident fields, V h and V v are the horizontal and vertical components of V Tx , and r ref is the total unfolded length of the reflected ray path.
To apply the same field computation algorithm as that of a transmitter to E RIS q in (1), the far-zone scattering characteristics of the RIS (i.e., the magnitude and phase of the scattered electric fields) for various RIS configurations are required.Let the far-zone scattered electric field at a distance R from the RIS denoted as E sc RIS .Then This parameter is equivalent to V Tx of a transmitter in (2).With (4), the same field computation algorithm for a primary transmitter can be applied to compute the fields of rays from the RIS.For example, we apply (3) to compute the electric field of a ray scattered from the RIS and then reflected on a wall, with V h and V v being horizontal and vertical components of V RIS .The computation of scattered fields from an RIS is explained in Section IV.

III. EFFECTIVE UNIT CELL CRCS
In this section, we introduce the effective unit cell CRCS, which is used to compute the scattered electric fields of unit cells of an RIS, taking into account the mutual coupling effects due to their surrounding unit cells.Our goal is to express the scattered field of the RIS in a way that allows us to quickly recompute it for new configurations.Hence, we utilize the effective unit cell CRCS to resolve the main bottleneck that we pointed out in Section II.

A. Scattered Electric Field of a Unit Cell in Isolation
The CRCS σ is used to compute the scattered electric field at a far-field distance R of a unit cell, illuminated by a plane wave E inc , considered on its own [24] where ê is the polarization vector of the scattered electric field, |σ | and ̸ σ are the magnitude and phase of the CRCS, and |E inc | and ̸ E inc are the magnitude and phase of E inc at the unit cell.We assume that both the incident and reflected fields are linearly polarized.The CRCS depends on both the angles of incidence (θ inc , ϕ inc ) and the angles of observation (θ sc , ϕ sc ).
For brevity and clarity, we suppress the explicit notation of the angle dependencies.It is worth mentioning that the magnitude of CRCS is the scalar RCS.Fig. 2 shows a full-wave simulation setup used to compute the CRCS of a unit cell, which involves simulating the unit cell in a small domain truncated by perfectly matched layers (PMLs).The unit cell structure shown in the figure, presented in [25], is utilized in this work to validate our proposed method in subsequent sections.For a unit cell that can achieve K phase shifts (i.e., K operation states), K full-wave simulations result in K CRCSs.Then, the scattered field of the unit cell for each state is determined by (5), and the scattered field of the RIS for a specific configuration is computed by summing the scattered fields of all unit cells.Now, an RIS consists of an array of such unit cells whose scattering properties are defined by their CRCS, but also by their coupling to surrounding cells.One can interpret these modified scattering properties of the unit cell within an array, as associated with an "effective CRCS" that is different from the standard CRCS (obtained by simulating a single unit cell in isolation, i.e., using the setup of Fig. 2), due to mutual coupling.In the following, we explain how to compute such an effective unit cell CRCS.

B. Magnitude of Effective Unit Cell CRCS
Let us consider an M × N cell RIS, consisting of unit cells that can be configured to K states.We define the effective CRCS of these unit cells in their ith state (i = 1, 2, . . ., K ) as According to the pattern multiplication principle [26], we can compute the scattered field of the RIS when all its unit cells operate in their ith state, as follows: This E sc cell is the scattered electric field of the unit cell in the ith state.It can be calculated according to (5) using the effective CRCS σ eff i .AF is the array factor, when the phase reference point is at the geometric center of the RIS [26] in which Note that (7) assumes that the unit cells have an identical CRCS, which is the effective CRCS.Edge effects that affect the scattering properties of unit cells in the RIS are disregarded here.
On the other hand, according to (5), the magnitude of the scattered electric field of the RIS is where |σ s i | is the RCS, determined by full-wave analysis, of the RIS when all unit cells operating in the ith state.
By comparing (7) with (10), we define the magnitude of the effective unit cell CRCS (or the effective unit cell RCS) for the ith state as Computing σ s i by full-wave analysis of the RIS, i.e., fully accounting for mutual coupling of unit cells, ensures via (11) that the effective RCS also includes mutual coupling, with the exception of edge effects.
Note that directly applying (11) at all observation angles (θ sc , ϕ sc ) is problematic because of the nulls of |AF| in the denominator of (11).To overcome this, we use an interpolation method.This method is introduced next, with an example that uses the linearly polarized unit cell structure in [25, Fig. 2].This unit cell has two operation states (K = 2), switch on and switch off, implemented by diode switches that cause nearly 180 • variation of the phase of the scattered field at 5.8 GHz.Therefore, there are two effective unit cell CRCSs corresponding to switch-on and -off, denoted as |σ eff 1 | and |σ eff 2 |, respectively.We followed the full-wave modeling approach of [25], which employed short and open terminations in place of the diode switches for simplicity.This approach showed good agreement between simulation and measurement results.For more sophisticated modeling of the RIS, equivalent circuit models for both the switch-on and switch-off states of the diodes can be used, and the following algorithm would proceed in the same way.
We consider an incident TE y -polarized wave (refer to the coordinate system in Fig. 2) at angles of incidence of (θ inc , ϕ inc ) = (45 • , 0 • ).All simulation data presented in this article were obtained for this setup, and the Ansys HFSS finite-element solver was used to perform full-wave analysis.The method of computing |σ eff i | is summarized as follows, along with an example for |σ eff 1 | (we consider a 16 × 16 RIS, to obtain σ s 1 ). 1) We sample the local peaks of |AF| that satisfy (in dB scale) where |AF| max is the maximum of |AF|.The local peaks that do not satisfy (12) are excluded because they can result in unrealistically large calculated values of |σ eff i |.This step returns L discrete values |AF l (θ l , ϕ l )|, l = 1, 2, . . ., L, each of which corresponds to specific coordinates (θ l , ϕ l ), as the yellow points shown in Fig. 3(a to obtain a discrete dataset of |σ eff i,l (θ l , ϕ l )|, l = 1, 2, . . ., L, as the yellow points in Fig. 3(b).
3) We use a surface interpolation algorithm to fit the discrete |σ eff i,l (θ l , ϕ l )|, and the obtained results are considered as |σ eff i |.In this work, we used the thin plate spline interpolation [27].The resulting |σ eff i | is a continuous function of θ sc and ϕ sc , which describes the far-zone scattering characteristics of the unit cells with mutual coupling taken into account, as the red dashed curve in Fig. 3(b).Fig. 3(b) compares the obtained effective RCS for the switch-on state the standard RCS.The effective is smaller than the standard RCS, which indicates that the strong mutual coupling decreases the scattered power density of the RIS.In fact, this is consistent with phased arrays: the mutual coupling of antenna elements increases with decreasing interelement distance, leading to degraded directivity and large discrepancies between the actual and predicted patterns calculated by pattern multiplication [28].On the other hand, edge effects in a finite-size array, which are disregarded here, primarily affect sidelobes [29].This is consistent with our results, as most discrepancies between the normalized array factor and the full-wave simulated RCS in Fig. 3(a) are observed for sidelobes at near-grazing angles.

C. Phase of Effective Unit Cell CRCS
Let the full-wave analysis using the setup of Fig. 2 return the phase of the standard unit cell CRCS for the ith state ̸ σ c i .We define the phase of the effective unit cell CRCS for this ith state as where ̸ σ i is the phase difference between ̸ σ eff i and ̸ σ c i that is caused by mutual coupling.The phases of the scattered electric fields of a unit cell of subwavelength dimensions, with and without mutual coupling accounted for, ̸ σ eff i and ̸ σ c i , can be considered constant with respect to angles (θ sc , ϕ sc ).This is due to the nearly uniform distribution of the induced surface currents.Based on this, we approximate ̸ σ i as a constant.
The task now is to find ̸ σ i .We again consider an M × N RIS when all unit cells operating in the ith state; the phase of the corresponding CRCS of the RIS obtained by full-wave analysis is denoted as ̸ σ s i .Then, the phase of the scattered electric field is calculated, according to (5), as On the other hand, from ( 7), we obtain for the phase term.In the specular reflection direction (θ sc = θ inc , ϕ sc = ϕ inc + π ), ψ x = ψ y = 0, so that ̸ AF(θ inc , ϕ inc + π ) = 0, according to (8).By comparing ( 15) and ( 16), we have Substituting ( 17) into ( 14) yields ) With ̸ σ s i and ̸ σ c i found by full-wave analysis, ̸ σ i can be directly calculated by (18) so that ̸ σ eff i is determined using (14).
For the unit cell structure in Fig. 2, using the same setup as in Section III-B, ̸ σ 1 (switch-on) and ̸ σ 2 (switch-off) due to mutual coupling are −111.3• and −128.9 • , respectively.The phases of CRCSs including the standard ones and the effective ones are shown in Fig. 4.

IV. RIS SCATTERED FIELDS AND LINK BUDGET
BASED ON THE EFFECTIVE CRCS In this section, we derive the expressions of the scattered field and path loss of an RIS, based on the effective unit cell CRCS.Fig. 5 is the diagram of a single-input-single-output (SISO) wireless communication system with an RIS.A plane wave from the transmitter impinges on the RIS and is scattered toward the receiver.According to (5), we obtain the scattered electric field of the (m, n) unit cell where  (m, n) unit cell σ eff m,n , and R r m,n is the distance from the (m, n) unit cell to the observation point.Note that σ eff m,n = σ eff i when the unit cell operates in the ith state.
In an RIS-enabled link (Fig. 5), the magnitude and phase of the incident field E inc m,n are determined by [26] E inc m,n = where G t m,n is the gain of the transmitter in the direction toward the (m, n) unit cell, R t m,n is the distance from the transmitter to the (m, n) unit cell, and ̸ E t is the phase of the radiated field of the transmitter.Let us set ̸ E t to 0 and substitute (20) into (19).Next, applying field superposition, we obtain the following expression: This expression assumes that unit cells operating in the same state share the same effective CRCS, despite their different locations and different numbers and states of neighboring cells.For a specific RIS that is illuminated by a plane wave, we compute |σ eff i | and ̸ σ eff i , i = 1, 2, . . ., K , using the methods in Sections III-B and III-C, respectively.This computation involves K full-wave simulations of the RIS.Then, |σ eff m,n | and ̸ σ eff m,n of all unit cells for an RIS configuration are determined, and ( 21) is used to compute the scattered electric field of the RIS, accounting for mutual coupling.
With the expression of (21), the scattered power received by a polarization-matched receiver with an effective area A r in the far-field of the RIS is where A r = G r (θ rx , ϕ rx )λ 2 /(4π ), in which G r is the gain of the receiver, and (θ rx , ϕ rx ) denote the angles that define incident wave direction at the receiver.With (22), the path Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.loss of an RIS-enabled link is obtained Similar to the models proposed in [6], [7], [8], and [9], this expression can be used to evaluate the link budget of an RIS channel, considering the wave propagation from a transmitter to a receiver via the RIS only, as shown in Fig. 5.
V. VALIDATION OF THE RIS SCATTERED FIELD EXPRESSION In this section, we validate the proposed method for computing the scattered electric fields from RISs and compare it with the models of [6] and [7].Ansys HFSS finite-element analysis (FEA) was used to produce reference data.

A. Setup of the RIS-Enabled Link
We consider the RIS-enabled link shown in Fig. 5 at 5.8 GHz.A transmitter with a maximum gain of 20 dBi and transmit power of 30 dBm centered at (−10 ,0 ,10 m) was used.The unit cell structure in Fig. 2 was used to construct a 16 × 16 RIS.We consider the same incident wave as in Sections III-B and III-C, i.e., an incident TE y -polarized wave (refer to the coordinate system in Fig. 2) with (θ inc , ϕ inc ) = (45 • , 0 • ).Hence, the previously obtained effective CRCSs were directly used to compute E sc RIS using (21).

B. Comparison to Existing Models and Full-Wave Analysis
We compared the predicted scattered electric fields obtained by our method, with those obtained by the models proposed in [6] and [7], and HFSS FEA results.
The gain of unit cells defined and used in [6] and [7] is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where α is a parameter that needs to be determined.Although the same gain model was employed, the expressions of path loss in [6] and [7] were different because the effective areas of unit cells had different expressions.We used α = 3 and π/2 − 1 as given in [6] and [7] and assumed that unit cells had 100% efficiency.
On the other hand, the impact of mutual coupling can be evaluated by replacing the magnitude and phase of σ eff m,n in (21), with those of σ c m,n , which is the standard CRCS obtained by simulating the unit cell in isolation.Fig. 6 compares the scattered electric field magnitude and phase of the RIS obtained by the proposed method, the method of using standard CRCS, the methods presented in [6] and [7], and HFSS FEA, for three different RIS configurations (denoted as A, B, and C, respectively, as indicated in the magnitude plots of Fig. 6).Our method can accurately predict scattered field levels along the main lobes.Note that the methods in [6] and [7] do not provide phase information of the predicted scattered electric field.
Standard CRCS results overestimated the main-lobe levels, because they did not include the significant impact of mutual coupling.We also expect discrepancies at the minor lobes of the scattering pattern, because of edge effects that our model omits.
The primary focus of the method is to accurately predict the field levels in the main-lobe directions for different RIS configurations.To examine the accuracy of our method in the main-lobe directions, we show the electric field magnitude as a function of distance along the main scattering directions of the RIS, for the same three configurations, in Fig. 7.The figure includes our results, along with those of [6] and [7], and reference HFSS data.Notably, our method always matches the HFSS results, while the other two methods overestimate the field levels.
The models of [6] and [7] can be calibrated [11], as previously discussed in Section I. To demonstrate this, we selected a different value of α (6.5), for the model in [6], to match its predictions with HFSS for configuration A, as shown in Fig. 7(a).However, there are still discrepancies for configurations B and C.This shows that the calibration of that model for one configuration does not guarantee its accuracy for other configurations, unlike the method we propose that remains accurate for all configurations.

VI. MODELING OF A REALISTIC RIS CHANNEL
In this section, we use the method described in II, combined with (21), to an indoor RIS channel with realistic dimensions.The experimentally validated method of [18] the CRCS of an RIS for each configuration using full-wave analysis) was used produce reference data [20], to the accuracy of the results obtained by the proposed method.

A. Modeling of a Realistic RIS Channel at 5.8 GHz
We consider a T-shaped hallway, shown in Fig. 8.A transmitter a maximum gain of 18.2 dBi that directly illuminated the RIS was employed.The RIS and the angles of incidence were the same as those in Section V.The simulation frequency was 5.8 GHz, and all facets were set as concrete with a relative permittivity of 2.5 and a loss tangent of 0.15.
Figs. 9-11 show the electric fields across the simulation domain for RIS configurations A, B, and C, respectively.Some discrepancies are observed in Figs.9(c) and 10(c) at certain receiver points.The reason is that the electric fields at these points were dominated by the rays from the minor lobe directions of the RIS, where our method exhibited some errors.Nonetheless, the proposed method and the method of [18] produced similar predictions of field levels.Fig. 12 shows that the electric fields along the main lobes obtained by the predicted CRCSs match with reference values well.

B. Discussion
All simulations were performed on a desktop computer with 12 cores.The HFSS finite-element solver took 29, 29, and 26 min to compute the CRCSs for configurations A, B, and C, respectively.The average ray-tracing time for these simulations was only 1 min and 55 s.Therefore, the total simulation time using the method of [18], which combines Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I COMPARISON OF THE PROPOSED METHOD WITH OTHER METHODS FOR ANALYZING RIS CHANNELS TABLE II AVERAGE AND SD OF THE SAMPLED ELECTRIC FIELD LEVELS
ray tracing and full-wave analysis, primarily depends on the full-wave analysis time.In contrast, the proposed method only required 25 and 28 min to conduct full-wave simulations for all switch-on and switch-off RIS configurations.Next, based on the computed effective unit cell CRCSs σ eff 1 and σ eff 2 , the scattered electric fields of other configurations were approximated using (21), with negligible additional time compared to full-wave simulations.
Table I compares the proposed method with several alternative methods for modeling path loss and/or characterizing wave propagation for RIS-enabled links.As discussed earlier, the methods in [6], [7], [8], and [9] ignore the mutual coupling of unit cells and exhibit limited accuracy.Besides, these methods do not consider the multipath effects of realistic RIS channels.These limitations are addressed by the method in [18].But this method requires one full-wave simulation for RIS configuration, reducing its efficiency for analyzing RIS channels with multiple RIS configurations.Pure full-wave methods such as the solution presented in [19] can achieve high accuracy but demand excessive computational resources, surpassing the computational requirements of other methods.In comparison, the proposed method employs an effective CRCS-based model that takes into account mutual coupling and predicts the scattered electric fields of an RIS for various configurations, in an efficient and accurate manner.Combined with ray tracing, the multipath effects present in RIS channels are accounted for, enabling the modeling of realistic scenarios.Therefore, our method combines the accuracy of the approach in [18], with a computational framework that is comparable in efficiency to established path loss models [6], [7], [8], [9].

C. Channel Characterization
With the proposed method, we can quantitatively estimate the electric field levels in the region of interest to evaluate the influence of an RIS on signal fading in a specific environment.We uniformly sampled 81 × 81 points in the region of 0 m ≤ x ≤ 4 m, 4 m ≤ y ≤ 8 m.Fig. 13 shows histograms of the electric field levels at the sampling points.The electric field is clearly stronger with the assistance of the RIS, and the maximum electric field level in the region is approximately doubled.Table II gives the average and the standard derivation (SD) of the sampled electric field levels.The average electric field levels with the RIS for the three configurations are nearly doubled compared to those without the RIS.Also, in the presence of the RIS, the histograms have a longer "tail," as shown by the increased values of standard deviation.

VII. CONCLUSION
We proposed a method for efficiently computing the scattered electric field of an RIS for multiple configurations, taking into account the mutual coupling between unit cells.We introduced the concept of the effective CRCS to characterize the far-field scattering of a unit cell, including cell-to-cell coupling, and derived an expression for the farzone scattered electric field of an RIS.Our method omits edge effects and assumes that unit cells operating in the same state share the same effective CRCS regardless of their different local environments, but it does account for mutual coupling.Hence, it enables the accurate computation of the scattered field in the main scattering direction of an RIS, where the impact of edge effects is negligible.Our method eliminates the need for one full-wave analysis per RIS configuration.Furthermore, our method can be seamlessly integrated with ray-tracing for the propagation modeling of realistic RIS channels, fully accounting for the multipath effects.Therefore, our work advances the relevant state of the art, by bringing the accuracy of CRCS-based and pure full-wave analysis-based propagation modeling methods for RIS-enabled links [17], [18], [19], in a computational framework of comparable efficiency to that of previously established, yet limited in terms of accuracy and path loss models [6], [7], [8], [9].Future work will focus on investigating how to optimize channel performance by properly implementing the channel, fully considering environmental effects based on the proposed method.

Fig. 1 .
Fig. 1.Two-dimensional view of an indoor RIS channel, where the configuration of the RIS is adapted to the locations (a) and (b) of the receiver; selected rays that exist in the environment are displayed.

Fig. 2 .
Fig.2.Structure of the used RIS unit cell.The simulation domain, which is truncated by PMLs, to compute its standard CRCS is also displayed.

Fig. 3 .
Fig. 3. Different parameters for the switch on.(a) Normalized AF, RCS of the RIS obtained by HFSS (the maximum RCS is 13.69 dB), and sampling points (ϕ = 0 • plane) for the configuration of all switches on (with the operation states of the unit cells as denoted at the top-left corner: the yellow color denotes switch on).(b) Standard and effective unit-cell RCSs (ϕ = 0 • plane), the sampled array factor is also displayed.

). 2 )
With |σ s i | computed by full-wave analysis [e.g., |σ s 1 |, the red dashed curve in Fig. 3(a)] using the setup shown in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
inc m,n | and ̸ E inc m,n are the magnitude and phase of the incident electric field of the (m, n) unit cell, |σ eff m,n | and ̸ σ eff m,n are the magnitude and phase of the effective CRCS of the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 6 .Fig. 7 .
Fig. 6.Scattered electric field magnitudes and phases at R = 10 m and ϕ = 0 • plane (with operation states of the unit cells as denoted at the bottom right corner of each magnitude plot; the yellow color denotes switch on while the blue color denotes switch off).(a) Configuration A, main-lobe direction: θ = 10 • .(b) Configuration B, main-lobe direction: θ = 22 • .(c) Configuration C, main-lobe direction: θ = 29 • .The main lobes are highlighted.

Fig. 10 .Fig. 11 .
Fig. 10.Modeling results of the RIS channel with configuration B obtained by our method.(a) Electric field distribution at the z = 1.5 m plane obtained by the predicted CRCS, with the operation states of the unit cells displayed.(b) Electric field at y = 4 m and z = 1.5 m.(c) Electric field at y = 6 m and z = 1.5 m.