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Communication Models for Reconfigurable Intelligent Surfaces: From Surface Electromagnetics to Wireless Networks Optimization

Marco Di Renzo, Fadil H. Danufane, Sergei Tretyakov

arXiv:2110.00833v2cs.ITeess.SP

TL;DR

RIS research requires communication models that are both tractable for wireless-network optimization and consistent with electromagnetic behavior. The paper reviews common models, develops impedance-boundary models through a step-by-step tutorial, and compares local and global designs using power efficiency and reradiated flux. It concludes that approximate global designs can be achieved with locally passive, purely reactive RISs, including large reflection angles and high power efficiency.

  • Problem

    Wireless-network research needs RIS models that are electromagnetically consistent yet sufficiently tractable for signal-level and system-level analysis and optimization.

  • Method

    The paper reviews RIS communication models and develops step-by-step analytical models based on inhomogeneous surface impedance, comparing local and global designs through power efficiency and Poynting-vector flux.

  • Results

    An appropriately designed RIS can theoretically reflect P0 regardless of incidence angle with 100% efficiency, while approximate global designs use purely reactive boundaries without local power amplification.

  • Takeaways & Limitations

    Locally passive RISs with zero electrical resistance can realize approximate global designs while retaining high power efficiency, including for large reflection angles.

Abstract

from arXiv · show

A reconfigurable intelligent surface (RIS) is a planar structure that is engineered to dynamically control the electromagnetic waves. In wireless communications, RISs have recently emerged as a promising technology for realizing programmable and reconfigurable wireless propagation environments through nearly passive signal transformations. With the aid of RISs, a wireless environment becomes part of the network design parameters that are subject to optimization. In this tutorial paper, we focus our attention on communication models for RISs. First, we review the communication models that are most often employed in wireless communications and networks for analyzing and optimizing RISs, and elaborate on their advantages and limitations. Then, we concentrate on models for RISs that are based on inhomogeneous sheets of surface impedance, and offer a step-by-step tutorial on formulating electromagnetically-consistent analytical models for optimizing the surface impedance. The differences between local and global designs are discussed and analytically formulated in terms of surface power efficiency and reradiated power flux through the Poynting vector. Finally, with the aid of numerical results, we discuss how approximate global designs can be realized by using locally passive RISs with zero electrical resistance (i.e., inhomogeneous reactance boundaries with no local power amplification), even for large angles of reflection and at high power efficiency.

I. INTRODUCTION

The introduction presents RISs as engineered surfaces for controlling electromagnetic propagation and motivates electromagnetically consistent, scalable models for wireless-network optimization.

  • Wireless communications transfer information without electrical conductors, primarily through electromagnetic waves.
  • Metasurfaces offer electrically thin alternatives to bulky metamaterials while retaining capabilities for shaping electromagnetic-wave propagation.
  • RISs can make environmental interactions controllable, supporting joint optimization of transmitted waves, propagation, and receiver decoding.
  • RISs are planar arrays of scattering elements that adjust phase shifts, and possibly amplitudes, to steer reradiated waves toward specified directions.
  • Global designs jointly optimize groups of unit cells and can better account for mutual coupling, but periodic optimization may not support every wave transformation or reconfigurable implementation.

C. Research Opportunities and Challenges

The paper identifies open RIS research challenges and reviews communication models used to analyze and optimize RIS configurations. It emphasizes modeling assumptions, implementation constraints, and the trade-off between model simplicity and physical accuracy.

  • Open RIS research challenges include channel estimation, RIS control, deployment, and integration into wireless systems.
  • The tutorial reviews widely used RIS communication models and explains the assumptions and conditions under which they apply.The third model is developed further for RIS optimization problems.
  • Locally Periodic Discrete Model: The locally periodic discrete model represents each unit cell by a reflection coefficient selected from a finite RIS alphabet.The model computes received power as a function of transmitter and receiver locations, RIS placement, and unit-cell configuration.
  • Locally Periodic Discrete Model: Local reflection coefficients are characterized by surrounding a configured unit cell with an infinite homogeneous repetition, then applying those coefficients independently across an inhomogeneous RIS.This can neglect the changed electromagnetic environment created by neighboring cells with different states.

B. Mutually Coupled Antenna Elements

The mutually coupled antenna model represents RIS-assisted links using coupled thin-wire dipoles and tunable impedances. It resembles a MIMO channel while explicitly incorporating mutual coupling and the RIS control circuit.

  • The mutually coupled antenna model explicitly accounts for mutual coupling among RIS elements and the unit-cell control circuit.It is based on mutually coupled antenna theory and applies to multiple-antenna communication systems.
  • The RIS comprises P independently configurable, nearly passive thin-wire dipoles connected to tunable impedances.Each dipole approximates a unit cell, and the physical structure can be generalized.
  • The end-to-end channel matrix maps transmit-generator voltages to receive-antenna voltages and includes direct and RIS-reradiated links.The RIS-reradiated term combines transmitter-to-RIS transfer, RIS-to-receiver transfer, and the tunable-impedance response.
  • The matrices of self and mutual impedances characterize propagation and coupling, while the diagonal tunable-impedance matrix is optimized for wave steering.The fixed impedance matrices can be obtained analytically or with full-wave simulators.
  • Constraining tunable impedances to have positive real parts prevents incident-wave amplification and avoids power amplifiers.A negative resistance is equivalent to requiring a power amplifier.
  • The model is relatively simple to use because its channel matrix resembles a conventional MIMO channel, but it assumes sinusoidal surface currents on thin-wire dipoles.This current assumption is a principal condition governing the model's applicability.

C. Inhomogeneous Sheets of Surface Impedance

The paper presents macroscopic RIS models based on homogenized inhomogeneous sheets of surface impedance and admittance, while stating their assumptions and electromagnetic uses. These models support RIS analysis, synthesis, and optimization, but require subwavelength unit cells and suitable observation distances.

  • C. Inhomogeneous Sheets of Surface Impedance: Homogenizable RISs can be modeled as continuous sheets characterized by macroscopic surface impedances and admittances.This abstraction applies when unit-cell dimensions and spacings are much smaller than the wavelength.
  • C. Inhomogeneous Sheets of Surface Impedance: The homogenized model uses electric surface impedance and magnetic surface admittance, generally represented as dyadic tensors for general wave transformations.These tensors characterize the RIS’s macroscopic electromagnetic response.
  • C. Inhomogeneous Sheets of Surface Impedance: Generalized sheet transition conditions provide boundary equations that characterize the electric and magnetic fields on both sides of the RIS.The equations use incident, reflected, and transmitted fields evaluated at the surface.
  • C. Inhomogeneous Sheets of Surface Impedance: The model enables direct analysis with known surface functions and inverse source problems for designing them according to specified criteria.The paper uses these formulations to study RIS optimization as a function of surface impedance.
  • C. Inhomogeneous Sheets of Surface Impedance: Homogenized impedance-sheet models assume zero thickness and require observation points sufficiently far from finite-size unit-cell effects.The stated rule of thumb is |z| > t/2 + max {d_x, d_y}.
  • C. Inhomogeneous Sheets of Surface Impedance: For an impenetrable reflecting RIS, the model becomes an inhomogeneous impedance boundary with no transmitted field on the opposite side.The paper focuses on this reflection-mode specialization in its electromagnetically consistent formulation.

A. Electromagnetically Consistent Modeling of RISs

The paper fixes notation for electromagnetic modeling, including the universal time dependence, vector operations, complex-part operators, and Euclidean distance.

  • A. Electromagnetically Consistent Modeling of RISs: The paper adopts the universal time dependency e^jωt and defines j as the imaginary unit and ω as angular frequency.It also specifies notation for gradients, vector products, real and imaginary parts, and Euclidean distance.
  • A. Electromagnetically Consistent Modeling of RISs: The notation includes ∇_ξ for gradients with respect to ξ, · and × for scalar and vector products, and ℜ(·) and ℑ(·) for complex parts.The Euclidean distance is denoted by ∥ξ_1 − ξ_2∥.

1) System Model:

The system model places a transmitter, receiver, and flat reflecting RIS in three-dimensional free space, with the RIS represented as a zero-thickness inhomogeneous impedance boundary.

  • 1) System Model:: The system contains a transmitter, receiver, and flat RIS modeled as an inhomogeneous surface-impedance boundary with negligible thickness.The RIS is represented by a flat surface S.
  • 1) System Model:: The RIS is a rectangle in the xy-plane centered at the origin, with side lengths 2L_x and 2L_y.Its sides are parallel to the x- and y-axes.
  • 1) System Model:: The reflecting RIS has the transmitter and receiver on the same side, while the transmitter emits an electromagnetic wave through vacuum.Vacuum is characterized by permittivity ϵ_0 and permeability µ_0.
  • 1) System Model:: The carrier frequency, wavelength, wavenumber, and angular frequency are denoted by f, λ = c/f, k = 2π/λ, and ω = 2πf.The propagation speed is c = 1/√ϵ_0µ_0.
  • 1) System Model:: The transmitter is described by charge and current densities localized in a small volume, and is assumed to lie in the RIS and receiver far-field region.The far-field assumption simplifies the incident-field model.

2) Electric and Magnetic Fields:

The paper constructs incident and reradiated fields under physical optics, parameterizes the RIS response through a complex reflection coefficient, and imposes Maxwell-consistency constraints.

  • 2) Electric and Magnetic Fields:: The incident wave arrives at specified elevation and azimuth angles, while the RIS reradiates toward the receiver, with both propagation directions restricted to the yz-plane.The simplifying assumption is ϕ_i(r_Tx) = ϕ_r(r_Rx) = π/2.
  • 2) Electric and Magnetic Fields:: The RIS-reradiated field is modeled by secondary currents induced on the surface under the physical optics approximation.This approach guides RIS design and optimization without numerically intensive methods such as the method of moments.
  • 2) Electric and Magnetic Fields:: The reflected field uses a complex reflection coefficient whose magnitude and phase describe amplitude and phase control across the RIS.The coefficient’s surface form characterizes RIS reradiation properties.
  • 2) Electric and Magnetic Fields:: The phase function is chosen to steer the incident wave from its incident wavevector toward the desired reflected wavevector.The amplitude function supports non-uniform amplitude control, and amplitude and phase may be coupled by optimization criteria.
  • 2) Electric and Magnetic Fields:: The analytical reflected field must approximately satisfy Helmholtz’s equation, requiring a corresponding constraint during reflection-coefficient optimization.The constraint is exact when the reflection coefficient is constant along the surface, producing a plane wave.
  • 2) Electric and Magnetic Fields:: A zero-divergence condition and a magnetic field derived from Maxwell’s equations complete the electromagnetically consistent reflected-field formulation.Together, the incident and reflected fields and the Helmholtz constraint form the model used for RIS analysis and optimization.

3) Surface Impedance:

The surface-impedance formulation derives electromagnetically consistent RIS fields from tangential incident and reflected fields, linking local impedance properties to reflection angles, passivity, and reradiated modes.

  • Surface-impedance formulation: The surface impedance is defined from the tangential electric and magnetic fields at the RIS surface for an impenetrable, purely reflecting sheet.The transmitted fields are assumed zero, so an electric surface impedance boundary suffices and magnetic surface admittance is redundant.
  • Passivity and reflection: The condition ℜ(Z) > 0 does not generally ensure |ΓS| ≤ 1 because the result depends on the incidence and desired reflection angles.When θi = 0, however, positive real impedance is sufficient for |ΓS| ≤ 1.
  • Passivity and reflection: Equal incidence and reflection angles yield a purely reactive, lossless RIS; incidence exceeding reflection requires local activity, whereas incidence below reflection yields local passivity.These cases correspond respectively to ℜ(Z) = 0, ℜ(Z) < 0, and ℜ(Z) > 0.
  • Passivity and reflection: Unit-amplitude field reflection may require local power amplification or local power loss, depending on the incidence and desired reflection angles.A negative resistance represents local amplification, while positive resistance represents local losses along the surface.
  • Surface-impedance formulation: The reflected field can be represented as a sum of plane waves and, for periodic RISs, is consistent with Floquet-theorem diffracted modes.A constant surface reflection coefficient produces the harmonic corresponding to n = 1 in the Fourier-series representation.
  • Reradiated modes: The Fourier coefficients of the surface reflection coefficient determine how reradiated power is distributed among propagating modes, including intended and interfering waves.The model remains consistent with Floquet theory for periodic RISs and supports analysis of interference arriving from other directions.

B. Power Efficiency and Reradiated Power Flux

The paper formulates RIS power efficiency and reradiated power flux through the Poynting vector, distinguishing local and global unitary-efficiency designs.

  • Power efficiency and reradiated power flux: Power efficiency and reradiated power flux are defined as functions of surface impedance using the Poynting vector.The analysis distinguishes local unitary efficiency based on local power conservation from global unitary efficiency based on average power conservation.

1) Power Efficiency – Surface Poynting Vector:

The surface Poynting vector provides an electromagnetically consistent way to evaluate RIS power efficiency and distinguish local from global designs. Global designs relax local power-flow constraints, enlarging feasible transformations while introducing implementation tradeoffs.

  • Surface Poynting vector: Power efficiency measures reradiated power toward the desired reflection direction relative to incident power and is evaluated from net power flow near the RIS surface.For lossless RISs, unitary efficiency requires the relevant power-flow condition to hold.
  • Local design: A lossless RIS is locally passive when ℜ(Z(rRx, y, rTx)) ≥ 0 for every y, and locally unitary only when ℜ(Z(rRx, y, rTx)) = 0 everywhere.The latter requires purely reactive components without resistive elements.
  • Local design: For phase-gradient reflectors, unitary efficiency occurs only under specular reflection; normal incidence can steer to arbitrary reflection angles without local amplification, but efficiency typically decreases as the angle increases.Transformations toward smaller reflection angles than incidence require locally positive power flow and therefore local power amplification under unit amplitude.
  • Local design: With R(rRx, y, rTx) = 1, the R0 = 1 case can use a lossless RIS without power amplification, whereas the R0 = √C0 case requires power amplification because NZ(y) oscillates between positive and negative values.Amplification may be virtual through surface waves or realized with power amplifiers.
  • Global design: Global unitary efficiency requires only integrated surface power flow PS(rRx, rTx) = 0, allowing positive and negative impedance values and collective reradiation by all RIS elements.A globally optimal amplitude ratio is obtained under periodicity and an integer number of RIS periods, making global designs more flexible than local designs.
  • Local design vs. global design: Approximate globally optimal solutions can be implemented with zero real surface impedance and reactive components, but may produce slightly lower beam-pattern gains and higher sidelobes.Global designs can require resistive or active elements; virtual losses and gains through evanescent Floquet harmonics can avoid active elements.

2) Reradiated Power Flux – Poynting Vector:

The reradiated power flux is computed from the Poynting vector of fields generated by RIS surface currents and quantifies angular power distribution beyond the surface vicinity. Physical-optics and far-field approximations yield analytically tractable expressions, but neglect edge effects and impose validity conditions.

  • Reradiated power flux is the Poynting-vector magnitude at an arbitrary observation point, including radiative near-field and Fraunhofer far-field regions.Unlike surface power efficiency, it is not restricted to z = 0+ near the RIS.
  • In the far field, reradiated power flux is proportional to the RIS radiation pattern, characterizing both the intended main lobe and undesired side lobes.This reveals how incident power is distributed across observation angles.
  • The reradiated electric and magnetic fields at an observation point are uniquely determined by the surface electromagnetic fields through radiation integrals.The observation point must lie sufficiently far from the RIS microstructure for the stated field formulation.
  • Physical-optics approximations produce equivalent surface currents by assuming an infinitely sized RIS and neglecting edge effects.Despite this approximation, the resulting analytical expressions support performance evaluation and RIS optimization.
  • The analytical power-flux expression depends explicitly on the surface reflection coefficient and can evaluate reradiated power as a function of observation angle and surface impedance.Under the stated electromagnetic consistency constraint, the expression is suitable for RIS performance evaluation.

C. Optimization of the Surface Impedance

RIS surface-impedance optimization evaluates candidate designs through the observation-point power flux while accounting for discretization and spurious reflections.

  • RIS optimization evaluates designs using the Poynting-vector power flux Pobs as a function of the observation point.The formulation also addresses how spurious reflections reduce power reradiated toward the intended direction.
  • The surface impedance is discretized into unit cells of length Δy, the spatial resolution for controlling incident-wave amplitude and phase.The discretized surface impedance is the optimization variable used for numerical implementation.

1) Benchmark Solution – Generalized Geometrical Optics:

The generalized geometrical-optics benchmark uses a linear phase gradient to redirect an incident wave toward a prescribed reflection angle. It is a local design that does not necessarily provide locally unitary power efficiency.

  • The generalized geometrical-optics solution is a canonical linear phase-gradient design associated with the generalized law of reflection.It provides the benchmark surface reflection coefficient and impedance without solving an optimization problem.
  • The generalized solution differs from conventional geometrical optics because its reflected direction can differ from the incidence direction rather than being necessarily specular.In conventional geometrical optics, the reflection angle equals the incidence angle.
  • The RIS phase modulation Φ(y) is selected so that an incident wave from θi is reflected toward θr according to Fermat’s principle.The geometrical-optics model represents the incident and reflected waves as rays and does not model their power.
  • The reflection angle is configured by optimizing the first-order derivative of the RIS phase modulation.A linear phase modulation Φ(y) = mk(sin θdesired − sin θi)y yields θr = θdesired.
  • The discretized benchmark uses ΓGO,n = ΓGO(yn) and ZGO,n = ZGO(yn), with the resulting power flux equal to Pobs(ZGO).No optimization problem is required for this benchmark construction.

2) Global Design – Unit Power Efficiency:

The global unit-power-efficiency design is formulated as a constrained optimization over the surface impedance, enforcing Helmholtz consistency and zero surface power flow.

  • A globally optimal unit-power-efficiency RIS design solves a constrained optimization problem over the surface impedance.The resulting impedance is denoted Zglo0 and is designed to have zero surface power flow.
  • The constraint Hn(Z) ≤ ε for all n = 1, 2, …, N − 2 enforces the Helmholtz condition.Here ε is a small positive constant.
  • The global optimization problem need not have a unique solution, allowing additional implementation constraints to select a desirable surface impedance.This nonuniqueness is described as advantageous for system implementation.

3) Approximated Global Design – Purely Reactive Impedance Boundary:

The paper approximates a globally optimized RIS using a purely reactive surface impedance, enforcing zero real impedance while preserving receiver power close to the unconstrained solution.

  • The globally optimized impedance may have a non-zero real part, implying local power gains and losses that complicate implementation.
  • The constrained optimization problem retains the receiver power objective while enforcing Helmholtz constraints and purely reactive impedance.
  • A purely reactive impedance boundary imposes zero real part on every discretized impedance value.

4) Optimization Constraints on Spurious Reflections:

Receiver-focused RIS optimization can leave substantial power in undesired directions, so the formulations add radiation-pattern constraints over specified angular sectors.

  • Optimizing only receiver power does not explicitly control reradiation toward other directions, allowing substantial undesired power.
  • The modified formulations add constraints that limit observed power within selected angular sectors.
  • The purely reactive formulation retains the constraint ℜ(Z_n) = 0 for every impedance element while adding spurious-radiation limits.
  • These optimization problems can be extended from received power to SINR for interference networks with one intended link and multiple interfering links.
  • The presented formulations assume free-space channels modeled by the free-space Green’s function, although extensions to fading and multiple-antenna systems are possible.

IV. NUMERICAL EXAMPLES

Numerical examples compare globally optimized, purely reactive, and geometrical-optics RIS designs, including their power efficiency and unwanted reradiation. Purely reactive designs approximate the main lobe with small receiver-power loss, while added constraints suppress selected spurious modes.

  • The examples evaluate surface impedances, Helmholtz constraints, and reradiated power flux versus observation angle for reflection angles of 30° and 75°.
  • The purely reactive solution offers a good approximation of the main reradiation lobe, with slightly higher side lobes than the unconstrained optimum.
  • Unwanted reflections occur toward the specular direction and the direction symmetric about the desired reradiation angle, consistent with Floquet theory.
  • Adding constraints can suppress the specular reflection and keep side lobes below the main lobe for 30° and 75° target angles.
  • With δ = 10^-4, a feasible purely reactive solution makes the two spurious reradiation intensities smaller than the predefined threshold.
  • Setting the real part of the unconstrained optimum to zero directly produces strong undesired reflections and violates the Helmholtz constraint.
  • The geometrical-optics design loses 4.8 dB at a 75° reflection angle, whereas the purely reactive design loses about 0.27 dB relative to the globally optimum design.
  • Nullifying specular reflection reduces power sent toward that direction and increases power steered toward the receiver, consistent with total power conservation.

V. CONCLUDING REMARKS

The paper reviews RIS communication models and develops electromagnetically consistent surface-impedance optimization formulations. Its scope is slowly varying impedances under physical optics, while more advanced suppression of parasitic scattering requires additional control or design.

  • The paper overviews three widely used RIS communication models and focuses on inhomogeneous surface-impedance boundaries for electromagnetic consistency.
  • Surface power efficiency defines local and global optimization objectives, while Poynting-vector power flux evaluates RIS steering capabilities.
  • The formulations apply when surface impedance varies slowly relative to wavelength and the physical-optics approximation is valid.
  • Theoretically perfect anomalous reflections with complete parasitic-scattering suppression require control of fast-varying surface modes or careful diffraction-grating design.
  • Developing efficient numerical algorithms for the electromagnetically consistent optimization problems is identified as an important extension.
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