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On the Path-Loss of Reconfigurable Intelligent Surfaces: An Approach Based on Green's Theorem Applied to Vector Fields

Fadil H. Danufane, Marco Di Renzo, Julien de Rosny, Sergei Tretyakov

arXiv:2007.13158v2cs.ITmath.IT

TL;DR

Accurate path-loss models matter for wireless link-budget predictions. This paper formulates RIS-aided path-loss using the vector generalization of Green’s theorem. Its model describes regime- and transformation-dependent behavior, including bounded received power as RIS size grows.

  • Problem

    Path-loss modeling is important for making appropriate link-budget predictions in wireless networks.

    Source: §I · p. 1

  • Method

    The paper formulates RIS-aided free-space path-loss using the vector generalization of Green’s theorem.

    Source: §I · p. 2

  • Results

    The near-field and far-field asymptotic regimes differ and depend on the wave transformations applied by the RIS; received scattered power is bounded as RIS size increases without bound.

    Source: §VII · p. 25

  • Takeaways & Limitations

    The proposed path-loss model is general enough for application to various operating regimes.

    Source: §VII · p. 25

    The stated formulation assumes operation in the electrically-large regime.

    Source: §VII · p. 27

Abstract

from arXiv · show

In this paper, we introduce a physics-based analytical characterization of the free-space path-loss of a wireless link in the presence of a reconfigurable intelligent surface. The proposed approach is based on the vector generalization of Green's theorem. The obtained path-loss model can be applied to two-dimensional homogenized metasurfaces, which are made of sub-wavelength scattering elements and that operate either in reflection or transmission mode. The path-loss is formulated in terms of a computable integral that depends on the transmission distances, the polarization of the radio waves, the size of the surface, and the desired surface transformation. Closed-form expressions are obtained in two asymptotic regimes that are representative of far-field and near-field deployments. Based on the proposed approach, the impact of several design parameters and operating regimes is unveiled.

I. INTRODUCTION

The paper develops a vector-field path-loss model for homogenized RIS links and finds that path-loss depends strongly on surface size and transmission distances, especially in the near field. Its system setup includes reflection and transmission configurations.

I. INTRODUCTION

  • Prior path-loss models largely target discrete half-wavelength-spaced arrays; [9] studies RIS-to-RIS spatial degrees of freedom, while [10] is limited to one-dimensional RISs.The approach in [10] also does not account for the vectorial nature of electromagnetic waves.Source: §I · pp. 1–2
  • The proposed free-space path-loss model uses the vector generalization of Green’s theorem and is expressed as a computable integral.The integral depends on transmission distances, wave polarization, surface size, and the desired surface transformation.Source: §I · p. 2
  • Closed-form path-loss expressions are derived for two asymptotic regimes representative of far-field and near-field transmission.Numerical results are used to validate the accuracy and applicability of the asymptotic formulations.Source: §I · p. 2
  • Path-loss depends strongly on RIS size and transmission distances, especially in the near-field regime.The paper also discusses similarities and differences between near-field and far-field asymptotic regimes.Source: §I · p. 2

II. SYSTEM MODEL

  • The system model places a transmitter, receiver, and zero-thickness rectangular surface in three-dimensional space, with the terminals on the same or opposite sides.The surface may operate in reflection or transmission mode, according to which side of the surface receives the scattered wave.Source: §II · p. 3

A. Source Modeling

The source model permits general electromagnetic sources but uses a dipole antenna for concrete results. The metasurface model treats a homogenizable surface whose local coefficients encode reflection or transmission transformations.

A. Source Modeling

  • The analytical framework allows general electromagnetic sources, while concrete results model the transmitter as a dipole antenna.The dipole’s charge and current densities are specified in terms of its electric dipole moment.Source: A. Source Modeling · p. 5

B. Metasurface Modeling

  • The RIS is modeled as an electrically large metamaterial surface with sub-wavelength elements and inter-element distances.Under the stated homogenization assumption, it can be represented by continuous surface-averaged functions such as susceptibilities.Source: B. Metasurface Modeling · p. 5
  • A local reflection or transmission coefficient represents an engineered complex function that applies a specified transformation to the incident electromagnetic field.The model allows the RIS to change the polarization of the impinging radio waves.Source: B. Metasurface Modeling · pp. 5–6
  • The surface field on the reflection side combines the incident field with a locally weighted field, whereas the transmission-side field is locally weighted without an incident field there.The weighting accounts for the corresponding reflection or transmission coefficient and polarization-conversion efficiency.Source: B. Metasurface Modeling · p. 6
  • The reflection and transmission field formulations apply in the far field of the RIS microstructure, where possible evanescent fields can be ignored.The paper assumes k is much greater than the inverse Tx-to-surface and surface-to-Rx distances.Source: B. Metasurface Modeling · p. 6

III. PRELIMINARIES

The paper uses physical optics and the Stratton–Chu formula to characterize the received field in the presence of a RIS. It then reformulates the field to expose the surface’s contribution and the effect of receive polarization.

III. PRELIMINARIES

  • The approach follows physical optics and is presented as overcoming geometric optics’ limitations.The section also introduces methods for computing recurrent integrals for fields scattered by reflecting and transmitting surfaces.Source: §III · p. 7

A. Received Field at Rx

  • The received field is represented with the Stratton–Chu formula over a closed boundary containing the RIS, with the transmitter’s location differing for reflection and transmission.The observation point lies inside the volume in both cases.Source: A. Received Field at Rx · p. 7
  • The received field can be projected onto a chosen receive-polarization vector, making its polarization dependence explicit.Choosing the projection vector appropriately retrieves the field along the desired direction.Source: A. Received Field at Rx · pp. 8–9
  • Reformulating the field expression makes the RIS surface appear explicitly and yields expressions useful for wireless applications and scaling-law analysis.The authors note that the initial computable integral does not itself give an explicit expression in terms of the surface and its design parameters.Source: A. Received Field at Rx · pp. 8–9
  • The incident and RIS-scattered field contributions generally combine incoherently, but RIS phase terms can be optimized so they add coherently at the receiver.Source: A. Received Field at Rx · p. 9

B. Approximations and Asymptotic Regimes

The paper develops methods for evaluating recurrent integrals and defines electrically-small and electrically-large regimes. These regimes yield distinct approximations, with stationary points dominating the large-regime integral when present.

1) Type-1 Integral:

  • The electrically-small regime is analogous to the Fraunhofer far-field, while the electrically-large regime can hold at shorter distances than the radiating near-field.The regimes are defined using transmission distances relative to the surface’s electrical size.Source: 1) Type-1 Integral · pp. 10–11
  • In the electrically-small regime, the type-1 integral is approximated by linearizing the transmission-distance phase across the surface.The approximation uses the center distances and directional terms Dx and Dy.Source: 1) Type-1 Integral · p. 10
  • The electrically-large condition is formulated for integrals with at least one stationary point, using distance, surface dimensions, wavelength, and phase curvature.Source: 1) Type-1 Integral · pp. 10–11
  • In the electrically-large regime, stationary-phase analysis approximates the type-1 integral from stationary points when they exist.The approximation assumes a nonempty stationary-point set and a nonsingular phase Hessian at those points.Source: 1) Type-1 Integral · p. 11
  • For large-regime type-1 integrals, magnitude scales inversely with k when a stationary point lies in the surface, versus inversely with k^2 when none does.For k ≫ 1, the stationary-point contributions dominate.Source: 1) Type-1 Integral · p. 12

2) Type-2 Integral:

The type-2 integral separates distance-dependent and distance-independent factors. Its approximation is straightforward when both transmission distances greatly exceed the surface diagonal, but not generally when both are much smaller.

2) Type-2 Integral:

  • The type-2 integral has a real-valued factor depending on both transmission distances and another real-valued factor independent of them.Source: 2) Type-2 Integral · p. 12
  • When both center-to-endpoint distances are much larger than the surface diagonal, the type-2 integral can be approximated by treating distances across the surface as nearly constant.The approximation follows from dTx(x, y) ≈ dTx0 and dRx(x, y) ≈ dRx0 in that regime.Source: 2) Type-2 Integral · p. 12
  • When both center-to-endpoint distances are much smaller than the surface diagonal, computing the type-2 integral is not straightforward in general.Source: 2) Type-2 Integral · p. 12

IV. ELECTRIC FIELD IN THE PRESENCE OF A REFLECTING SURFACE

For a reflecting RIS, the paper formulates the received field under stated high-frequency distance assumptions and examines specular reflection. The resulting asymptotic behavior differs between electrically-small and electrically-large surfaces.

IV. ELECTRIC FIELD IN THE PRESENCE OF A REFLECTING SURFACE

  • The reflecting-surface field expression is approximated under assumptions that k is much greater than the inverse transmitter-to-surface and surface-to-receiver distances.The proof can be generalized to avoid these assumptions.Source: §IV · p. 13

A. S is Configured for Specular Reflection

  • In the electrically-large approximation, the main lobe of the sinc terms narrows as the surface dimensions increase.Source: A. S is Configured for Specular… · pp. 13, 15
  • In the electrically-large regime, specular-reflection field magnitude decays with the sum of the transmitter-to-specular-point and specular-point-to-receiver distances, and is independent of surface size.Thus, received power remains bounded as the surface grows large.Source: A. S is Configured for Specular… · p. 14
  • Specular reflection satisfies the law of reflection: incidence and observation angles match in elevation, while azimuths differ by π modulo 2π.Source: A. S is Configured for Specular… · p. 14
  • In the electrically-small regime, reflected-field magnitude grows linearly with surface area and decays with the product of the center-to-endpoint distances.This area scaling does not imply unbounded growth as the surface tends to infinite size, since that limit leaves the electrically-small regime.Source: A. S is Configured for Specular… · p. 14

B. S is Configured for Anomalous Reflection

For anomalous reflection, the paper approximates its analytical field expression to expose scaling trends and show how phase-gradient parameters control reflection direction. The approximation yields distance- and size-dependent behavior in the electrically-large regime and approaches the specular result when the surface is uniform.

B. S is Configured for Anomalous Reflection

  • For anomalous reflection, (30) approximates (29) using a Taylor approximation; they coincide only for a uniform surface configured for specular reflection.The coincidence requires the stated conditions on K1 and K2 as well as αR = βR = 0.Source: B. S is Configured for Anomalous… · pp. 15–16
  • In the electrically-large regime, the reflected-field magnitude decays with a weighted sum of Tx-to-surface and surface-to-Rx distances and is independent of surface size.The approximation is intended to reveal scaling laws and performance trends.Source: B. S is Configured for Anomalous… · pp. 15–16
  • The incidence and reflection angles at the stationary point generally differ and depend on αR and βR, which can be set for a desired reflection angle.When αR = βR = 0, the expressions reduce to the specular case.Source: B. S is Configured for Anomalous… · p. 16
  • In the electrically-small regime, the paper also gives an approximation for the reflected field, whose magnitude is maximized when αR and βR satisfy the stated angle-dependent equalities.The angles are those of incidence and reflection with respect to the center of S.Source: B. S is Configured for Anomalous… · pp. 16–17

C. S is Configured for Focusing

For focusing, the surface phase is set by the sum of the Tx-to-surface and surface-to-Rx distances. The resulting behavior differs from anomalous reflection in general for electrically-large surfaces, while the two configurations are almost equivalent in the electrically-small regime.

C. S is Configured for Focusing

  • For an electrically-large focusing surface, the analysis provides an upper bound rather than a general simplification of the field integral.The integrand lacks a fast oscillating term, so the stationary phase method cannot be applied.Source: C. S is Configured for Focusing · p. 17
  • For an electrically-large focusing surface, the reflected-field magnitude is upper-bounded as surface size grows, so received power remains bounded even for an infinitely large RIS.The analysis uses an upper bound because the integrand has no fast oscillating term for stationary phase analysis.Source: C. S is Configured for Focusing · p. 17
  • In the electrically-large regime, focusing and anomalous reflection generally behave differently, with different distance scaling because their phase profiles are optimized differently.For focusing, the phase is proportional to the sum of the two transmission distances.Source: C. S is Configured for Focusing · pp. 17–18
  • In the electrically-small regime, focusing and anomalous-reflection surfaces are almost equivalent and have the same scaling laws for field magnitude.The focusing phase becomes equivalent to that of an anomalous reflector directed toward the focusing spot.Source: C. S is Configured for Focusing · p. 18

V. ELECTRIC FIELD IN THE PRESENCE OF A TRANSMITTING SURFACE

The transmitting-surface formulation expresses the received electric field projected onto the receive polarization as a direct-path term plus a surface-dependent integral. For specular transmission, the model gives asymptotic expressions and retrieves the transmission law in the electrically-large regime.

V. ELECTRIC FIELD IN THE PRESENCE OF A TRANSMITTING SURFACE

  • The projected received field is formulated as the direct field in the absence of the surface plus an integral term that depends on the surface design and properties.The transmission integral includes phase and polarization-dependent factors.Source: §V · pp. 18–19
  • With unit transmission magnitude, zero transmission phase, and unchanged polarization, the integral terms cancel and the received field matches the field without the surface.This is the special case of transmission without modification of the impinging wave.Source: §V · p. 19

A. S is Configured for Specular Transmission

  • For specular transmission, a constant surface phase gives the electrically-large approximation, and the stationary-point angles satisfy the law of transmission.The law is ϕinc = (ϕrec + π) mod 2π and θinc = θrec.Source: A. S is Configured for Specular… · pp. 19–20
  • In the electrically-small regime, FT(rRx) can be approximated; its intensity in Corollary 9 is similar to Corollary 2, so similar scaling laws and performance trends are obtained.Source: A. S is Configured for Specular… · pp. 19–20

B. S is Configured for Anomalous Transmission

For anomalous transmission, a phase gradient configured by αT and βT determines the transmission direction. The paper gives large- and small-regime field approximations and reports performance trends similar to those of reflecting surfaces.

B. S is Configured for Anomalous Transmission

  • For electrically-large anomalous transmission, the field has an approximation evaluated at the stationary point and incorporating the phase-gradient parameters αT and βT.The setup uses a surface phase gradient k(αT x + βT y) plus a fixed phase.Source: B. S is Configured for Anomalous… · p. 20
  • In the electrically-small regime, the paper provides an approximation for the transmitted field, with a further simplification when transmission magnitude is constant and positive over the surface.The simplified expression uses DαT = αT + Dx and DβT = βT + Dy.Source: B. S is Configured for Anomalous… · pp. 20–21
  • The transmission angles can be optimized through αT and βT, and the field-intensity behavior is similar to that for anomalous reflection.Reflecting and transmitting surfaces have similar performance trends with surface size and transmission distances.Source: B. S is Configured for Anomalous… · pp. 21–22

VI. NUMERICAL RESULTS

The numerical examples compare anomalous reflection or transmission with focusing and examine discretization and distance effects. They show that path-loss scaling depends on the surface transformation, while the asymptotic frameworks track the integral representation in the regions of interest.

A. Anomalous Reflection and Focusing

  • Anomalous transmission yields correct transmission angles and greater coverage than focusing near the RIS, but not at long distances.Source: A. Anomalous Reflection and Focusing · p. 22, Figure 4 · p. 23
  • When the discretization step is λ/2, no significant differences can be observed at the naked eye; when it is greater than λ/2 (e.g., λ), grating lobes are observed.More closely spaced scattering elements yield more accurate estimates of the electric field, especially in the considered near-field regime.Source: A. Anomalous Reflection and Focusing · pp. 23–24

B. Transmission Distance

  • Analytical frameworks for electrically large and small regimes overlap with the integral representation in the regions of interest.Source: B. Transmission Distance · p. 24
  • Anomalous reflectors have weighted-sum and product path-loss regimes, while focusing lenses follow one product law that is sufficiently accurate at short and long distances.The reflector's weighted-sum model may remain accurate up to a few tens of meters; focusing co-phases contributions scattered from the RIS.Source: B. Transmission Distance · pp. 24–25

C. Surface Size

The numerical results examine field intensity as RIS size increases. The intensity remains bounded, while the electrically small approximation can overestimate it.

C. Surface Size

  • The electric-field intensity remains bounded as the RIS diagonal grows without bound, consistent with power conservation.Source: C. Surface Size · p. 25
  • Electrically large and small frameworks and their bounds predict the scaling law, but the electrically small approximation may overestimate intensity even for relatively small surfaces.The overestimate is reported for long transmission distances, including 100 meters in the figure.Source: C. Surface Size · p. 25
  • The received power is bounded regardless of surface size in the far-field of the RIS microstructure.Source: C. Surface Size · p. 25

VII. CONCLUSION

The section addresses RIS-aided transmission path loss, presenting a physics-compliant model across near-field and far-field regimes and an approximate transmission-side received-field expression with its sign choice and boundary term. It also describes received-power scaling by regime and RIS transformation, including that power remains bounded as RIS size grows without bound.

  • The model is physics-compliant and general enough for multiple operating regimes, including near-field and far-field asymptotic regimes.Source: §VII · p. 25
  • Received-power scaling with transmission distance and RIS size differs between near-field and far-field regimes and depends on the RIS wave transformation.Source: §VII · p. 25
  • Received power scattered by an RIS is bounded as its size increases without bound.Source: §VII · p. 25
  • For network evaluation and optimization, a two-law path-loss model can combine the near-field and far-field closed forms, avoiding two-fold integrals while retaining physics-based constraints.The closed forms should be used only when the system setup satisfies their regime of validity; the integral representation is recommended for physical meaningfulness across design parameters.Source: §VII · pp. 25–26

VII. CONCLUSION

  • Equation (65) gives the approximate received field as the projected incident field multiplied by the Green function G(rRx, rTx).The projection is onto the receive polarization vector.Source: §VII · p. 30
  • The transmission-side boundary term Ftran uses the transmitted and incident fields projected onto the receive polarization and can be expressed using Ωinc and Ωtran.The fields include the transmission coefficient and polarization-dependent terms.Source: §VII · p. 30
  • Both proof signs corresponding to (62) and (63) are negative because incident and transmitted propagation is opposite to the outward normal.The proof takes nout = z.Source: §VII · p. 30
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