Effective Beamfocusing Rayleigh Distance (EBRD)

• EBRD is a metric that defines the maximum focal range within which a focused beam maintains a finite 3 dB beamdepth, enabling spatial resolution in both range and angle.
• It generalizes the classical Rayleigh distance by linking beamforming gain loss to focal depth while incorporating array geometry and angle dependencies for precise near-field characterization.
• EBRD guides the design of large arrays such as massive MIMO, RIS, and ELAA by balancing aperture size, wavelength, and gain-loss thresholds to optimize spatial multiplexing and performance.

The Effective Beamfocusing Rayleigh Distance (EBRD) is a metric that delineates the spatial region around a large array or intelligent surface where near-field (NF) beamfocusing remains practically feasible before the array’s ability to resolve fine structure in the range direction fundamentally vanishes. EBRD generalizes the classical Rayleigh or Fraunhofer distance—classically a phase error benchmark—by tying the near-field/far-field boundary to beamforming gain and focal depth, and is central to performance characterization and design of extremely large antenna arrays (ELAAs), reconfigurable intelligent surfaces (RISs), modular linear arrays (MLAs), and array architectures with complex topologies. EBRD provides a rigorous, geometry- and angle-dependent criterion for the validity of depth-focusing or spatial multiplexing in modern array-based communication and sensing systems (Cui et al., 2021, Kosasih et al., 12 May 2025, Hussain et al., 18 Jun 2025, Hussain et al., 16 Nov 2025, Kosasih et al., 4 Dec 2024, Björnson et al., 10 Nov 2025, Mestre et al., 11 Feb 2025).

1. Definition and Physical Principle

EBRD is the maximum focal distance $r_0$ (from the array or surface aperture) within which a finite beamdepth—in the sense of 3 dB width around the focal range—exists for a focused beam. For points beyond EBRD, the 3 dB beamdepth diverges and the array transitions to the far-field regime, where focusing in range is no longer possible and only directional (angle-only) selectivity remains. Formally, for a focusing configuration at range $r_0$, the EBRD is: $$\mathrm{EBRD} = \max\{ r_0: \Delta(r_0) < \infty \}$$ where $\Delta(r_0)$ is the axial span in which the normalized array gain remains above a chosen threshold (typically $-3$ dB) (Hussain et al., 18 Jun 2025, Hussain et al., 16 Nov 2025, Cui et al., 2021). For planar, linear, rectangular, modular, or circular array architectures, closed-form expressions (sometimes angle-dependent) are available.

The physical meaning is that, inside EBRD, spherically focused beamforming enables spatial resolution in both range and angle (enabling, e.g., range-domain multiple access), while outside EBRD only classical angular selectivity remains.

2. Classical Rayleigh Distance vs. EBRD

The classical Rayleigh distance $R$ for an aperture of length $D$ and wavelength $\lambda$,

$R = \frac{2 D^2}{\lambda},$

marks the onset of the radiating near field based on a phase error threshold ($\pi/8$ at the array edge). However, this phase-error-based definition does not adequately capture the practical range over which focusing in depth remains effective for communication rate, power, or spatial multiplexing.

EBRD, in contrast, is defined with respect to beamforming gain (or SNR) loss, or (equivalently) the existence of a finite 3 dB beamdepth. For arrays with $N$ elements, EBRD universally scales sublinearly with $R$, and often substantially contracts the spatial region of true near-field focusing. For example, with a planar or linear aperture (Cui et al., 2021, Hussain et al., 16 Nov 2025): $\text{EBRD} = C_\Delta \cdot R$ where $C_\Delta$ is a gain-loss–dependent factor (e.g., $C_\Delta \approx 0.3$–$0.5$ for typical thresholds), and additional cosine/elevation-angle terms in non-isotropic array topologies.

Comparative Table: Classical vs. EBRD

Criterion	Classical Rayleigh Distance ($R$)	EBRD
Basis	Edge phase error ($\pi/8$)	Beamforming gain loss or finite 3 dB beamdepth
Geometry dependence	Omnidirectional	Explicitly angle-dependent for general arrays
Value	$2 D^2 / \lambda$	$C_\Delta$–scaled, typically much smaller than $R$
Physical meaning	Onset of spherical phase effects	Last range where beam can be focused (in depth)

3. Analytical Expressions for EBRD Across Array Architectures

Uniform Linear Arrays (ULA)

• For a ULA of aperture $D$ and boresight focus:

$$\mathrm{EBRD}_{\text{ULA}}(\varphi) = \frac{R}{4\,\alpha_{3\mathrm{dB}}\,\cos^2\varphi}$$

where $R=2D^2/\lambda$, $\alpha_{3\mathrm{dB}}$ is a root of the Fresnel integral beamgain equation, and $\varphi$ the azimuth angle (Hussain et al., 16 Nov 2025, Mestre et al., 11 Feb 2025, Hussain et al., 18 Jun 2025).

• The gain-based formulation uses the normalized Fresnel kernel, with the boundary set by a tolerable beamforming gain loss $\Delta$ (Cui et al., 2021).

Uniform Rectangular Arrays (URA)

• Given $N_1 \times N_2$ layout, aperture $D = d \sqrt{N_1^2 + N_2^2}$, and aspect ratio $\eta = N_1/N_2$:

$$R_{\mathrm{EB}} = \frac{\eta}{4\alpha_{3\mathrm{dB}}(1+\eta^2)\sin\theta \sqrt{1-\sin^2\theta\sin^2\varphi}}$$

For a square URA ($\eta = 1$), $R_{\mathrm{EB}}$ is minimized; wider/taller aspect ratios extend EBRD at expense of larger beamdepth (Hussain et al., 18 Jun 2025).

RIS/Planar Surfaces

• For a RIS (or equally, a planar UPA) focusing at $z=F$, with aperture $D$:

$$z_R = 2x_{3\mathrm{dB}}\,\frac{\lambda F^2}{D^2}, \quad x_{3\mathrm{dB}} \approx 0.1197$$

Angular and quantization (bit-depth) effects further affect the gain, but the EBRD remains set chiefly by $D$ and $\lambda$ (Björnson et al., 10 Nov 2025).

Modular Linear Arrays (MLA)

• For $L$ subarrays (each $N$ elements), total ML array aperture $D_{\mathrm{array}} = L N \delta + (L-1)\Delta$:

$${\rm EBRD}_{\rm MLA} = \frac{2 D_{\rm array}^2}{\lambda}$$

The 3 dB beamdepth scales as $F^2 / {\rm EBRD}$ (Kosasih et al., 12 May 2025, Kosasih et al., 4 Dec 2024).

Uniform Circular Arrays (UCA)

• For a UCA of radius $R = D/2$, the EBRD is:

$$r^{\mathrm{EBRD}}_{\mathrm{UCA}}(\theta) = \frac{\pi r_R}{16\,\alpha_{3\mathrm{dB}}\,\sin^2\theta}$$

with $r_R = 2D^2/\lambda$, $\alpha_{3\mathrm{dB}}$ the first positive root of the Bessel-threshold equation, and $\theta$ the elevation angle (Hussain et al., 16 Nov 2025).

Holographic and Two-Sided Feasibility

In the holographic regime, beamfocusing is possible only if array size exceeds approximately $4.4\lambda$; EBRD becomes a two-sided interval, with both minimum and maximum permissible ranges for focusing, determined by positivity of the local quadratic-form SNR expansion (Mestre et al., 11 Feb 2025).

4. Dependence on Geometry, Angle, and System Parameters

EBRD is not a fixed "radius" but an angle- and geometry-sensitive boundary.

• Array aperture ($D$): EBRD $\propto D^2$; larger aperture increases both near-field region and total achievable EBRD.
• Wavelength ($\lambda$): EBRD $\propto 1/\lambda$; higher carrier frequency (smaller $\lambda$) pushes EBRD further out.
• Angular factors: For ULAs, EBRD decreases as $1/\cos^2\varphi$; for UCAs, $1/\sin^2\theta$; for URAs, more elaborate directional dependencies emerge (Hussain et al., 18 Jun 2025, Hussain et al., 16 Nov 2025).
• Gain-loss tolerance ($\Delta$): Stricter gain-loss thresholds shrink EBRD; larger $\Delta$ correspond to permissive performance loss, expanding EBRD (Cui et al., 2021).
• Bit quantization and element count: For RIS and quantized arrays, EBRD is robust to phase quantization, as confirmed by experimental validation (Björnson et al., 10 Nov 2025).

5. Implications for Array Design and System Performance

EBRD is essential for practical system design:

• Spatial multiplexing: Within EBRD, arrays support range-division multiple access (RDMA), with sharply focused beams enabling simultaneous users in both angle and range (Hussain et al., 18 Jun 2025, Kosasih et al., 12 May 2025).
• Beamforming architecture selection: EBRD guides when planar phase-shifter-based (far-field) beamformers suffice and when spherical (true-time-delay–capable) near-field focusing is required (Cui et al., 2021).
• Aperture–performance trade-off: Compact square (URA/UCA) configurations yield narrow beamdepth but limit near-field coverage; wide/tall arrangements extend EBRD but increase beamdepth, affecting spatial isolation and multiplexing granularity (Hussain et al., 18 Jun 2025, Hussain et al., 16 Nov 2025).
• RIS and element reduction: For MLAs or separated ULAs, large apertures with antenna "holes" can achieve a desired EBRD with fewer elements, efficiently balancing physical resources and near-field coverage (Kosasih et al., 4 Dec 2024, Kosasih et al., 12 May 2025).
• Experimental confirmation: Practical RISs (including 1-bit, 1024-element at 28 GHz) confirm accurate EBRD predictions, highlighting robustness of the focusing effect under real-world nonidealities (Björnson et al., 10 Nov 2025).

6. Limitations and Extensions

EBRD is fundamentally tied to the narrowband or center-frequency case. While it delivers reliable bounds in single-carrier or narrowband contexts, the presence of severe frequency-dependent "beam split" effects or very wideband signals may necessitate a subcarrier-wise EBRD analysis, as the effective beamforming range further contracts with bandwidth (Cui et al., 2021). In the modular and holographic regimes, anomalous lobing or sharp spatial nulls can appear, imposing practical restrictions on achievable focusing even at ranges within theoretical EBRD (Kosasih et al., 4 Dec 2024, Mestre et al., 11 Feb 2025).

A rigorous EBRD analysis requires careful attention to the underlying assumptions used in the Taylor/Fresnel expansions, as well as array manifold and spatial sampling constraints. The accuracy of the closed-form formulas is highest for large $N$, regular spacings, and when users are located close to the central focal axis; nonidealities such as mutual coupling, element gain variations, and multipath typically reduce, rather than increase, the effective focal region.

7. Summary Table: EBRD in Representative Array Topologies

Array Type	EBRD Closed Form	Principal Angle Dependency	Reference
ULA	$\frac{R}{4\alpha_{3\mathrm{dB}}\cos^2\varphi}$	$1/\cos^2\varphi$	(Hussain et al., 16 Nov 2025)
URA	See Eqn. (2) in (Hussain et al., 18 Jun 2025)	Geometry via $\eta$, angle, aspect	(Hussain et al., 18 Jun 2025)
RIS/planar UPA	$2x_{3\mathrm{dB}}\frac{\lambda F^2}{D^2}$	None for on-axis; angle if off-broadside	(Björnson et al., 10 Nov 2025)
MLA (modular)	$\frac{2 D_{\mathrm{array}}^2}{\lambda}$	None for on-axis	(Kosasih et al., 12 May 2025)
UCA	$\frac{\pi R}{16\alpha_{3\mathrm{dB}}\sin^2\theta}$	$1/\sin^2\theta$	(Hussain et al., 16 Nov 2025)
Holographic	$[D_{\min}, D_{\max}]$ via positivity region, see (Mestre et al., 11 Feb 2025)	Both lower and upper distance, angle	(Mestre et al., 11 Feb 2025)

EBRD is now an indispensable analytic and design metric in the characterization of next-generation multi-antenna platforms, especially for massive-MIMO, ELAA, RIS, and modular array deployments seeking to exploit spatial focusing not just in angle but in range, and hence unlock the full spatial degrees of freedom available in the near field.