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A Primer on Near-Field Beamforming for Arrays and Reconfigurable Intelligent Surfaces

Emil Björnson, Özlem Tugfe Demir, Luca Sanguinetti

arXiv:2110.06661v2cs.ITeess.SP

TL;DR

Physically large arrays and RIS can operate between the Fraunhofer distances of individual elements and the full aperture, where conventional far-field assumptions do not determine focusing behavior. The paper revisits these assumptions with communication-oriented, polarization-aware channel models and characterizes gain and beam-depth regimes. It finds that near-field focusing and finite-depth beams are governed by distances relative to the Fraunhofer array distance rather than that distance alone.

  • Problem

    Large arrays and RIS may place users between the Fraunhofer distances of individual elements and the full array, making the validity of conventional far-field beamforming and near-field focusing unclear.

  • Method

    The paper develops polarization-aware near-field channel models for passive antennas, active arrays, and RIS, then analyzes array gain, focusing depth, and RIS beamwidth.

  • Results

    For far-field focusing, normalized gain remains between 1 and 0.5 over [dF A/10, ∞), while focusing at dF A/10 yields [dF A/20, ∞).

  • Takeaways & Limitations

    Conventional far-field beamforming incurs at most 3 dB loss for z ≥ dF A/10, whereas z < dF A/10 requires spherical-wave-aware finite-depth beamforming.

Abstract

from arXiv · show

Wireless communication systems have almost exclusively operated in the far-field of antennas and antenna arrays, which is conventionally characterized by having propagation distances beyond the Fraunhofer distance. This is natural since the Fraunhofer distance is normally only a few wavelengths. With the advent of active arrays and passive reconfigurable intelligent surfaces (RIS) that are physically large, it is plausible that the transmitter or receiver is located in between the Fraunhofer distance of the individual array/surface elements and the Fraunhofer distance of the entire array. An RIS then can be configured to reflect the incident waveform towards a point in the radiative near-field of the surface, resulting in a beam with finite depth, or as a conventional angular beam with infinity focus, which only results in amplification in the far-field. To understand when these different options are viable, an accurate characterization of the near-field behaviors is necessary. In this paper, we revisit the motivation and approximations behind the Fraunhofer distance and show that it is not the right metric for determining when near-field focusing is possible. We obtain the distance range where finite-depth beamforming is possible and the distance where the beamforming gain tapers off.

I. INTRODUCTION

Near-field phase variations matter for physically large arrays and RIS, motivating distance-specific beamforming models. The paper provides a communication-oriented primer and extends prior analyses with polarization-aware models.

  • Motivation: Near-field phase variations can sharply reduce gain when far-field configurations communicate in the Fresnel region.Elliptical aperture phase responses tuned to propagation distance can recover the loss, producing beams limited in both angle and depth.
  • Motivation: Finite-depth beamforming is available in both the near-field and Fresnel region.The beam width is constrained transversely and along propagation depth.
  • Contribution: The paper develops a primer on near-field channel modeling for active antenna arrays and RIS using communication terminology.It characterizes single passive antennas, antenna-array beamforming, and RIS operation.
  • Contribution: The analysis uses polarization-aware models and provides publicly available code.The models differ from the cited classical works by accounting for polarization.

II. NEAR-FIELD REGION OF A PASSIVE ANTENNA

The paper models antenna fields using polarized spherical waves and identifies when reactive near-field terms can be neglected. Its remaining analysis therefore focuses on distances beyond the reactive near-field.

  • Field model: A transmit antenna is modeled as multiple point sources emitting polarized spherical waves.The electric field expression includes distance-dependent terms derived from the point-source model.
  • Reactive near-field: The last two field terms decay rapidly with distance and mainly affect the reactive near-field.They are often neglected outside the region very close to the antenna.
  • Reactive near-field: 0.975 is the squared parenthesis magnitude at z = λ, supporting omission of the last two terms for z ≥ λ with an electrically small antenna.For a single point-source approximation, the neglected terms have limited influence at these distances.
  • Reactive near-field: The reactive near-field ends approximately at z = 0.62D^3/λ for an electrically large antenna.The rest of the paper considers propagation distances beyond this reactive near-field.

A. Radiative near-field and Fresnel region

The radiative near-field and Fresnel region are characterized by distance-dependent phase and amplitude behavior across an aperture. The Fraunhofer distance marks a far-field lower limit, while the Fresnel region retains phase variation despite comparatively negligible amplitude variation.

  • Far-field characterization: In the far-field, aperture field strength scales inversely with distance and depends angularly rather than on local wavefront curvature.Reciprocity permits equivalent transmitter- and receiver-side interpretations when one side is isotropic.
  • Fraunhofer distance: The Fraunhofer distance is derived from the center-to-corner phase difference caused by spherical-wave curvature.The largest variation occurs for perpendicular incidence.
  • Fraunhofer distance: 3.5 · 10^-3 is the approximation error when dF ≥ 1.2D, establishing an additional far-field lower limit.This corresponds to an angular difference of at most π/8 and an amplitude ratio of approximately 0.92.
  • Fresnel region: The Fresnel region spans 1.2D to dF, where amplitude variations can be neglected but phase variations cannot.It exists only when dF ≥ 1.2D, implying D ≥ 0.6λ.
  • Antenna gain: For D = 2λ, at least 99% of maximum gain is achieved for z ≥ dF, while gain reaches 93% at z = 0.3dF = 1.2D.The Fresnel approximation is accurate for z ≥ 0.3dF in this example.
  • Summary: The Fraunhofer distance is an accurate lower limit of an antenna’s far-field when aperture phase and amplitude variations can be neglected for gain calculation.Communication channels can measure this condition from either direction when the transmitter or receiver is isotropic.

III. NEAR-FIELD REGION OF AN ANTENNA ARRAY √

The section models a planar square antenna array and defines its normalized array gain, then evaluates when maximum gain is achievable. It finds that the Björnson distance, rather than the Fraunhofer array distance, determines the relevant gain transition.

  • Array model: The array consists of identical antennas deployed edge-to-edge on a square grid, analyzed from a receiver perspective with an isotropic transmitter.The reciprocal transmitting-array setup yields the same result.
  • Array gain: The normalized antenna array gain captures both antenna and array gains because they decouple only in the far-field.It is based on the total received power impinging on reference antennas.
  • Distance criteria: The Fraunhofer array distance requires negligible spherical-curvature phase variation across the array, while the Björnson distance captures negligible amplitude variation.The Björnson distance grows with the square root of the antenna count and can be much shorter than the Fraunhofer array distance.
  • Distance criteria: The Björnson distance exceeds the individual-antenna Fraunhofer distance for N ≥1 when D = λ and for N ≥16 when D = λ/4.More generally, dB is larger than dF whenever N ≥(D/λ)2.
  • Numerical example: 625 antennas with length D = λ/4 give dF A = 625dF and dB = 100dF, but at least 95% of maximum gain is achieved for z ≥dB.The Fraunhofer array distance has no evident impact on the plotted gain curves.
  • Model limitations: The scalar-field approximation is tight only for z ≥dB because it neglects angle-dependent polarization losses and effective antenna areas.These variations must be included when analyzing z < dB.

A. Is the Fraunhofer array distance irrelevant?

The Fraunhofer array distance does not determine when near-field focusing is possible or when array gain reaches its maximum. It instead indicates whether spherical-wave processing is needed, while finite beam depth depends on the focal distance.

  • A. Is the Fraunhofer array distance irrelevant?: Matched filtering compensates for inter-antenna phase curvature, allowing near-maximum array gain even when z ≤ dF A.Each individual element can observe a locally plane wave while the array still experiences measurable spherical curvature.
  • A. Is the Fraunhofer array distance irrelevant?: For z ≥ dF A, plane-wave array responses suffice; for z ≤ dF A, matched-filtering weights must account for spherical curvature and transmitter distance.Thus, dF A characterizes receiver-processing requirements rather than the onset of useful array gain.
  • A. Is the Fraunhofer array distance irrelevant?: Finite 3 dB beam depth occurs only when the focal point satisfies F < dF A/10; for more distant focal points, the depth extends to infinity.As F →∞, the lower limit approaches dF A/10, establishing the boundary between finite-depth near-field and conventional far-field beamforming.
  • A. Is the Fraunhofer array distance irrelevant?: The beam depth depends on the focal location, becoming finite for near-field focusing when the focal point is closer than dF A/10.This behavior is illustrated through the maximum normalized gain at different observation distances.
  • A. Is the Fraunhofer array distance irrelevant?: For an array with N = 1002 = 104 elements, far-field focusing yields a −3 dB depth over [dF A/10, ∞), whereas focusing at dB yields roughly [286dF, 667dF].The example uses dB = 400dF and dF A = 104dF; focusing at dB also incurs noticeable maximum-gain loss.

IV. NEAR-FIELD REGION OF AN RIS

The RIS analysis models a plane-wave incident signal reflected toward a receiver through configurable element phases. Its depth-of-focus follows the same distance behavior as continuous matched filtering in a SIMO channel, with a slight gain reduction from nonzero incidence angle.

  • IV. NEAR-FIELD REGION OF AN RIS: The RIS serves a blocked direct path by beamforming a signal from a single-antenna transmitter to a single-antenna receiver.The transmitter-to-RIS path is assumed unobstructed, and the RIS-to-receiver path is also line-of-sight.
  • IV. NEAR-FIELD REGION OF AN RIS: The incident signal is approximated as a plane wave because the transmitter is assumed to be in the RIS far-field.The transmitter is located at distance ρ from the RIS center and emits y-polarized electric intensity Ei.
  • IV. NEAR-FIELD REGION OF AN RIS: The normalized RIS gain compares received power through the RIS with an ideal N-element reference deployment and lies between 0 and 1.This normalization provides a bounded measure of RIS amplification efficiency.
  • IV. NEAR-FIELD REGION OF AN RIS: RIS phase shifts compensate the incident-wave phase variation and produce the same depth-of-focus as continuous matched filtering in a SIMO channel.The resulting re-radiated field is equivalent to that of the original transmitter apart from end-to-end propagation-loss scaling.
  • IV. NEAR-FIELD REGION OF AN RIS: With θi = π/6 and N = 1002 = 104 elements, RIS gain curves have the same shape as the SIMO case, while maximum gain is slightly lower.The reduction results from the nonzero incident angle causing varying phase across each RIS element; the depth and beamwidth remain computable using Theorem 1.

A. Beam width in the focal plane

The RIS beam width in the focal plane increases with focal distance, while the angular beam width remains independent of focal distance. Near-field focusing therefore produces a finite transverse beam width that varies across configurations.

  • Beam-width derivation: The 3 dB beam width in the focal plane is obtained from the normalized RIS gain integral and the decreasing sinc^2 response.The derivation uses the point where sinc^2(x) is approximately 0.5, namely x ≈ 0.443.
  • Angular beam width: The 3 dB angular beam width is the same for any focal distance, but decreases with RIS size and operating frequency.For small angular widths, arctan(x) ≈ x gives a simplified approximation.
  • Focal-distance dependence: The beam width increases with the distance to the focal point at which the RIS phase-shifts are configured to focus.The figure compares focal distances F = dB, dFA/10, and dFA/5.
  • Modeling assumption: The beam-width formulas apply independently of incident angle when the transmitter is in the far-field, because the incident phase is nearly constant over each RIS element.The same expressions also apply to the SIMO channel under the continuous phase-shift approximation.
  • Numerical comparison: A focus at dB = 400dF produces a roughly 20dF 3 dB beam width, while focusing at dFA/10 = 1000dF produces a roughly 50dF width.The 400dF case reaches half the centered RIS gain at xr = ±10dF.
  • Focusing regimes: Far-field focusing gives infinite depth and transverse width, although the angular beam width remains the same across the compared RIS configurations.The apparent growth of angular widths in the heat maps results from the logarithmic zr-axis scale.

V. CONCLUSION

The paper distinguishes electromagnetic near-field definitions from their communication implications for large antenna arrays and RIS. It identifies when conventional beamforming is adequate, when spherical wavefronts must be modeled, and how near-field beams differ spatially.

  • Near-field and Fresnel regions have unequivocal transmit-antenna definitions, but their implications for communication design and performance differ.
  • For z larger than dFA/10, conventional far-field beamforming can be used with at most 3 dB loss.Here dFA is the Fraunhofer array distance.
  • For z < dFA/10, spherical wavefronts must be included, producing a near-field beam with finite depth.
  • With optimized beamforming, maximum antenna array gain is achieved for z ≥ dB, while dFA ≫ dB for large arrays.dB denotes the Björnson distance.
  • The angular beam width is constant across distances, but its physical width is small in the near-field, enabling spatial multiplexing over line-of-sight channels.
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