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Far- versus Near-Field RIS Modeling and Beam Design

Mohamadreza Delbari, George C. Alexandropoulos, Robert Schober, Vahid Jamali

TL;DR

This work establishes a mathematical foundation for modeling RISs in both far- and near-field regimes, introducing a quadratic near-field subregion and end-to-end channel models that incorporate LOS and non-LOS components under rich and poor scattering. It advances RIS beam design with two complementary approaches: an optimization-based method using a W–W^H relaxation and SCA, and an analytical framework employing quadratic and focusing phase profiles for far- and near-field operation, respectively. Through extensive simulations, the authors show that optimization-based designs achieve higher beam quality but incur cubic computational complexity in the RIS size, while analytical designs offer scalable, tunable-beam solutions that are particularly valuable for very large RISs; near-field designs are especially crucial for capturing wavefront curvature effects and enabling robust coverage. The results highlight a trade-off between performance and CSI overhead, demonstrate gains from leveraging non-LOS components and ground reflections at low Rician factors, and suggest that near-field RIS design will be essential as RIS dimensions grow and operating frequencies rise.

Abstract

In this chapter, we investigate the mathematical foundation of the modeling and design of reconfigurable intelligent surfaces (RIS) in both the far- and near-field regimes. More specifically, we first present RIS-assisted wireless channel models for the far- and near-field regimes, discussing relevant phenomena, such as line-of-sight (LOS) and non-LOS links, rich and poor scattering, channel correlation, and array manifold. Subsequently, we introduce two general approaches for the RIS reflective beam design, namely optimization-based and analytical, which offer different degrees of design flexibility and computational complexity. Furthermore, we provide a comprehensive set of simulation results for the performance evaluation of the studied RIS beam designs and the investigation of the impact of the system parameters.

Far- versus Near-Field RIS Modeling and Beam Design

TL;DR

This work establishes a mathematical foundation for modeling RISs in both far- and near-field regimes, introducing a quadratic near-field subregion and end-to-end channel models that incorporate LOS and non-LOS components under rich and poor scattering. It advances RIS beam design with two complementary approaches: an optimization-based method using a W–W^H relaxation and SCA, and an analytical framework employing quadratic and focusing phase profiles for far- and near-field operation, respectively. Through extensive simulations, the authors show that optimization-based designs achieve higher beam quality but incur cubic computational complexity in the RIS size, while analytical designs offer scalable, tunable-beam solutions that are particularly valuable for very large RISs; near-field designs are especially crucial for capturing wavefront curvature effects and enabling robust coverage. The results highlight a trade-off between performance and CSI overhead, demonstrate gains from leveraging non-LOS components and ground reflections at low Rician factors, and suggest that near-field RIS design will be essential as RIS dimensions grow and operating frequencies rise.

Abstract

In this chapter, we investigate the mathematical foundation of the modeling and design of reconfigurable intelligent surfaces (RIS) in both the far- and near-field regimes. More specifically, we first present RIS-assisted wireless channel models for the far- and near-field regimes, discussing relevant phenomena, such as line-of-sight (LOS) and non-LOS links, rich and poor scattering, channel correlation, and array manifold. Subsequently, we introduce two general approaches for the RIS reflective beam design, namely optimization-based and analytical, which offer different degrees of design flexibility and computational complexity. Furthermore, we provide a comprehensive set of simulation results for the performance evaluation of the studied RIS beam designs and the investigation of the impact of the system parameters.
Paper Structure (21 sections, 1 theorem, 15 equations, 9 figures, 1 algorithm)

This paper contains 21 sections, 1 theorem, 15 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background on Far- versus Near-field Regimes
  3. System and Channel Models
  4. Far-Field Channel Model
  5. Near-Field Channel Model
  6. RIS Reflective Beam Design
  7. Design Considerations
  8. CSI Requirement for RIS Beam Design
  9. Beam Design Objective
  10. Optimization-based RIS Beam Design
  11. Problem Formulation for the Far-Field Regime
  12. Problem Formulation for the Near-Field Regime
  13. Beam Design Solution
  14. Complexity Analysis
  15. Analytical Beam Design
  16. ...and 6 more sections

Key Result

Lemma 1

Assuming $\|\mathbf{p}_0\|\gg \|\mathbf{p}\|$, the change in the phase of the electric field across the Rx plane w.r.t. to the Rx center can be expanded as where $\tilde{p}=\frac{p}{p_0}$, $p_0=\|\mathbf{p}_0\|$, $p=\|\mathbf{p}\|$, and $\psi$ is the angle between vectors $\mathbf{p}_0$ and $\mathbf{p}$.

Figures (9)

  • Figure 1: The change of wavefront across the Rx plane. A point source is located at A, the center of the Rx plane is located at B, and C is an arbitrary point on the Rx plane. Function $\Delta\phi(\mathbf{p}_0,\mathbf{p})$, defined in \ref{['eq:phase_diff']}, quantifies the change in the phase of the electric field at point C with respect to the phase at the Rx center B.
  • Figure 2: Illustration of different propagation regimes as a function of the distance to a point source.
  • Figure 3: Far-field and quadratic near-field distances of a square RIS with horizontal and vertical lengths $L$ (i.e., $D = \sqrt{2}L$) for carrier frequencies of $5$ GHz (upper figure) and $28$ GHz (lower figure). The required number $N$ of unit cells is also shown for half-wavelength element spacing.
  • Figure 4: Schematic illustration of an RIS-assisted downlink wireless communication system. The left-hand side figure shows a 3D model whereas the right-hand side figure presents a 2D model of the same scenario. We adopt both 2D and 3D models for our simulation results in Section \ref{['sec: Simulation result']}, whereby a detailed discussion on the definitions and values of the system parameters are provided.
  • Figure 5: Illustration of perfect reflection (the left-hand side figure) and point-source scattering (the right-hand side figure) in the near-field region.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Remark 1