⚡ electrical engineering

Far- versus Near-Field RIS Modeling and Beam Design

This chapter establishes the mathematical foundations for modeling and designing reconfigurable intelligent surfaces (RIS) across far- and near-field regimes by analyzing wireless channel characteristics and comparing optimization-based versus analytical beam design approaches through comprehensive performance simulations.

Original authors: Mohamadreza Delbari, George C. Alexandropoulos, Robert Schober, Vahid Jamali

Published 2026-07-27

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CC BY 4.0

Original authors: Mohamadreza Delbari, George C. Alexandropoulos, Robert Schober, Vahid Jamali

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the air around us is filled with invisible ripples, like the waves spreading out when you toss a stone into a pond. These ripples are radio waves, carrying our texts, videos, and calls. Usually, we think of these waves as traveling in straight lines, but in a busy city full of skyscrapers, they often hit walls, bounce off cars, and get lost, leaving us with a weak signal or a dropped call. For decades, engineers have tried to fix this by building bigger towers or adding more antennas, but there's a limit to how much we can squeeze into a device.

Enter a new idea called a "Reconfigurable Intelligent Surface," or RIS for short. Think of an RIS as a giant, magical mirror made of thousands of tiny, programmable tiles. Unlike a regular mirror that just reflects light at a fixed angle, this smart mirror can be told to bend, focus, or split the radio waves exactly how we want them to go. It's like having a conductor for an orchestra of invisible waves, directing them to avoid obstacles and hit their target perfectly. But here's the tricky part: how do you control this mirror when the person you're trying to reach is standing very close to it versus standing far away? The rules of physics change depending on the distance, and getting the math wrong means the signal gets scattered instead of focused.

This paper dives deep into the math behind that magic mirror, specifically looking at the difference between when the mirror is far away from the user (the "far-field") and when it's close by (the "near-field"). The authors, a team of researchers, explain that when the mirror is huge and the user is close, the waves don't look like flat sheets anymore; they look like ripples spreading out from a point. They show that using the old, simple math designed for far-away users doesn't work well for close-up users. Instead, they propose two new ways to design the mirror's settings. The first way is a super-powerful computer method that finds the absolute best settings but takes a lot of time and computing power, like trying to solve a massive puzzle by checking every single piece. The second way is a clever, faster shortcut that uses a specific mathematical pattern (a "quadratic" shape) to get a very good result without needing a supercomputer.

The researchers ran many computer simulations to see how these methods work in real-world scenarios. They found that while the super-powerful computer method does give the strongest signal, it becomes too slow and expensive to use when the mirror gets extremely large. The clever shortcut method, however, scales up beautifully and still delivers a strong, reliable connection. They also discovered that by using these new designs, the mirror can be tuned to focus on a tiny spot for a clear call or spread out to cover a whole room if the exact location of the user isn't known perfectly. This flexibility is crucial because it means the system can handle mistakes in knowing where the user is standing. Ultimately, the paper suggests that for these giant smart mirrors to work in our future networks, we need to stop treating them like simple far-away reflectors and start using these new, distance-aware designs to keep our connections strong, whether we are across the street or right next to the wall.

Technical Summary: Far- versus Near-Field RIS Modeling and Beam Design

Problem Statement

Reconfigurable Intelligent Surfaces (RISs) are emerging as a key technology for creating smart radio environments by manipulating electromagnetic waves. However, as RISs scale to extremely large electrical dimensions to realize features like 3D beam focusing and virtual line-of-sight (LOS) connections, traditional far-field assumptions become insufficient. The large physical size of RISs increases the far-field distance (Rayleigh distance), necessitating the use of near-field models to accurately characterize the wireless channel.

Furthermore, the sheer number of channel coefficients in extremely large RIS systems makes conventional channel estimation and direct optimization of the RIS phase shifts infeasible due to prohibitive overhead. There is a critical need to:

Mathematically distinguish between far-field, near-field, and specific sub-regions (such as the quadratic near-field) to model wave propagation accurately.
Develop RIS beam design strategies that balance performance with computational complexity and channel state information (CSI) acquisition overhead.
Address the trade-off between high-quality beam design (often requiring full CSI and complex optimization) and scalable, low-complexity analytical solutions suitable for extremely large arrays.

Methodology

1. Mathematical Characterization of Propagation Regimes

The authors establish a rigorous mathematical foundation for distinguishing propagation regimes based on the phase variation of the electric field across the receiver (Rx) plane.

Far-Field: Defined where wavefront curvature is negligible, and phase variation is linear. The boundary is defined by the Rayleigh distance, $d_{FF} = 2D^2/\lambda$ , where $D$ is the largest dimension of the array.
Near-Field: Defined where wavefront curvature is significant.
Quadratic Near-Field Sub-Region: A specific region within the near-field where the cubic term of the phase expansion is negligible, allowing the phase variation to be approximated by a quadratic function. The boundary is defined by $d_{qNF} \approx 0.62 \sqrt{D^3/\lambda}$ . This region is shown to cover most of the near-field area of practical interest.

2. Channel Modeling

The paper presents channel models for both regimes, considering LOS and non-LOS (NLOS) components:

Far-Field Models: LOS components are modeled using array steering vectors dependent on angles of departure/arrival (AoD/AoA). NLOS components are modeled via rich scattering (correlation matrices) or poor scattering (cluster-based models).
Near-Field Models: LOS components depend on the exact Euclidean distances between individual Tx/Rx elements and RIS elements. NLOS components are modeled using "Perfect Reflection" (image theory for large surfaces like walls) and "Point Scattering" (small scatterers) models, accounting for double path loss in the latter.

3. RIS Reflective Beam Design

Two primary approaches are proposed for designing the RIS phase-shift configuration ( $\mathbf{w}$ ) to achieve anomalous reflection or beam focusing with tunable beamwidths:

Optimization-Based Approach:
- Formulates the design as a non-convex optimization problem maximizing the minimum Generalized Radar Cross Section (GRCS) over a set of desired incident and observation angles/locations.
- Utilizes a penalty-based method combined with Successive Convex Approximation (SCA) and Semidefinite Relaxation (SDR) to handle the unit-modulus constraint of passive RIS elements.
- Complexity: The computational complexity scales cubically with the number of RIS elements ( $O(N^3)$ ), making it computationally prohibitive for extremely large RISs.
Analytical Approach:
- Derives closed-form solutions based on the phase profiles required to steer or focus beams.
- Far-Field: Uses a quadratic phase-shift profile to create tunable beamwidths, generalizing the standard linear phase-shift beamforming.
- Near-Field: Extends the quadratic phase concept to the quadratic near-field sub-region, utilizing a mapping function to distribute RIS elements across a target illumination area.
- Complexity: Computationally efficient and scalable to extremely large RISs. These solutions can also serve as effective initializations for the optimization-based algorithm.

Key Results

Convergence and Initialization

Simulation results demonstrate that the optimization-based algorithm (Algorithm 1) converges to a locally optimal solution. Crucially, initializing the algorithm with the proposed analytical solution significantly outperforms random initialization. In tested scenarios, analytical initialization yielded a 5 dB performance gain over random initialization, highlighting the analytical solution's value not just as a standalone design but as a robust starting point for iterative optimization.

Beam Patterns and Tunability

Far-Field: Both optimization and analytical methods successfully generate tunable beamwidths. The optimization method provides more uniform illumination, while the analytical method offers a scalable alternative.
Near-Field: Applying far-field designs to near-field scenarios (neglecting wavefront curvature) results in significant performance degradation and poor coverage. Conversely, the proposed near-field designs (both optimization and analytical) successfully focus energy or illuminate specific regions.
Trade-off: The optimization-based design yields the highest quality (most uniform) illumination but at high computational cost. The analytical design is less flexible but highly scalable and effective for very large RISs.

Impact of Scattering and Multipath

In a 3D multipath environment with self-blockage:

Full CSI: Achieves the best Signal-to-Noise Ratio (SNR) but requires frequent, high-overhead channel estimation.
Proposed Scheme (Partial CSI): By leveraging dominant LOS and a few estimated NLOS paths (e.g., ground reflections), the proposed scheme achieves significantly higher SNR than pure LOS focusing, especially when the Rician $K$ -factor is low or when self-blockage occurs.
Convergence: As the Rician $K$ -factor increases (dominant LOS), the performance of the proposed scheme, the LOS-focusing scheme, and the full-CSI scheme converges.

Significance and Claims

The paper claims to provide a comprehensive mathematical foundation for RIS modeling and beam design across both far- and near-field regimes. Its primary contributions are:

Rigorous Regime Definition: It formally defines the quadratic near-field sub-region, demonstrating that it encompasses most practical near-field scenarios, thereby justifying the use of quadratic approximations for analytical design.
Scalable Design Framework: It addresses the scalability challenge of extremely large RISs by contrasting high-complexity optimization with low-complexity analytical solutions. The paper argues that while optimization offers superior performance, analytical solutions are necessary for practical implementation on massive arrays and serve as excellent initializations for optimization.
Robustness to Estimation Errors: The proposed designs with tunable beamwidths are shown to be robust against parameter estimation errors (e.g., inaccurate BS/MU locations) and reduce the overhead of RIS reconfiguration.
Performance vs. Overhead Trade-off: The results illustrate a favorable trade-off where the proposed partial-CSI schemes significantly outperform simple focusing or random reflection, approaching full-CSI performance in high- $K$ regimes while avoiding the prohibitive overhead of full channel estimation.

The authors conclude that their work bridges the gap between theoretical RIS modeling and practical beam design, offering tools to handle the unique challenges posed by the large electrical dimensions of future RIS deployments.