Near-Field Beamforming Gain Degradation

• Near-field beamforming gain degradation is the loss of optimal array gain when far-field beamforming techniques are applied in environments with spherical-wavefront curvature and dynamic user conditions.
• It arises from mismatches in range and angle, wideband beam-split, and hardware inefficiencies that distort coherent signal addition.
• Compensation strategies such as true-time delay, adaptive hybrid beamforming, and dynamic reconfiguration are essential for restoring near-field focusing performance.

Near-field beamforming gain degradation quantifies the loss in achievable array gain and focusing performance when classical far-field (planar-wave) beamforming architectures are deployed in regimes where spherical-wavefront curvature, user movement, wideband operation, or hardware losses undermine optimal coherent addition. The phenomenon is fundamental to advanced wireless systems employing extremely large antenna arrays (ELAAs), dynamic metasurface antennas (DMAs), reconfigurable intelligent/refractive surfaces (RIS/RRS), and terahertz (THz) bands, where both the array aperture and carrier frequency drive users into the radiative near-field. Rigorous characterizations leverage analytic models, closed-form correlation expressions, and design guidelines that link gain degradation to geometric, electromagnetic, and hardware variables (Gavriilidis et al., 29 Nov 2025).

1. Fundamental Analytical Framework

Near-field beamforming assumes a discrete or continuous aperture of N elements (e.g., planar array of dimension D, carrier wavelength λ), focusing energy on user coordinates p = (r, θ) with steering vector

$$[\mathbf a(r,θ)]_{i,n} = \exp(-j\,2\pi\,r_{i,n}(r,θ)/\lambda)$$

where $r_{i,n}(r, θ)$ is the true spherical distance from element (i,n) to the user.

Gain Degradation Due to Focus Mismatch

The normalized correlation (array gain) between the ideal steering vector and a mismatched focus (r + Δr, θ + Δθ) is

$$\rho(Δr,Δθ) = \frac{\bigl|\mathbf a^H(r,θ)\,\mathbf a(r+Δr,θ+Δθ)\bigr|^2}{N^2}$$

This decomposes for large N into range and angle factors: $\rho(Δr,Δθ)\approx I^2(a(Δr))\,\Lambda^2(Δθ)$ with Fresnel integral for range mismatch (I(·)) and Dirichlet sinc for angle (Λ(·)). The direction of fastest degradation follows from the gradient and Hessian of ρ in the (Δr, Δθ) plane.

Rayleigh Distance and Near-Field Region

The Rayleigh threshold for far-field applicability is

$d_{\rm Rayleigh} = \frac{2 D^2}{\lambda}$

Users within $r < d_{\rm Rayleigh}$ experience non-planar phase fronts, and focusing must account for spherical geometry.

2. Generalizations: Wideband, Geometry, and Hardware Effects

Wideband and Double Beam-Split Effects

Frequency-flat beamformers exhibit "beam-squint" and "beam split," where each frequency focuses at a distinct spatial point: $$G(f) \approx \left| \sum_{n=0}^{N-1} e^{j\Delta\phi_n(f)} \right|/N$$ with per-element phase errors accumulating across the bandwidth. For RIS/RRS, the phenomenon is compounded by frequency-selective hardware responses, e.g.,

$$\Delta\phi_i(m) = -2\pi \frac{(f_m-f_c)}{c} d_i + [\Phi_i(f_m)-\theta_i^c]$$

yielding severe gain loss (up to 90%) at band edges unless frequency-dependent compensation is implemented (Lin et al., 2 Jan 2025, Qiu et al., 2024, Yu et al., 2024).

Array Geometry and Finite Beam Depth

Large apertures induce finite range of tight focusing ("beam depth," BD). Analytical expressions, e.g., for rectangular array,

$$G_{\rm rect}(r) \approx \frac{\bigl[C^2(\eta\sqrt{a})+S^2(\eta\sqrt{a})\bigr]\;\bigl[C^2(\sqrt{a})+S^2(\sqrt{a})\bigr]}{(\eta a)^2}$$

with Fresnel parameter $a = d_{FA} / (4 z_{\rm eff}(1+\eta^2))$. The 3 dB beam depth is the interval where $G(r) \ge 1/2$.

Shape ordering: ULA < circular < square in terms of BD for fixed aperture area (Kosasih et al., 2023).

3. Architectures and Compensation Strategies

True-Time Delay (TTD) and Hybrid Precoding

Sub-connected TTD chains at the array or RIS allow phase compensation across frequencies. For instance, the analog part on subcarrier f is

$$T(f) = \operatorname{blkdiag}( e^{-j2\pi f t_1}, \dots, e^{-j2\pi f t_{M_{\rm RF}}} )$$

enabling realignment of beams over ultra-wide bandwidths. DLDD and PDF architectures for IRS/RIS partition the aperture and apply time delays at the sub-surface or subarray levels, restoring near-flat gain (Qiu et al., 2024, Cui et al., 2021, Guo et al., 2024).

Dynamic/Adaptive Hybrid Beamforming

Dynamic subarray activation (e.g., in ELAA systems) mitigates near-field loss by sparsely connecting elements with strong instantaneous gain and adaptively updating phase shifts: $$\mathbf F_{\rm RF}(n,l) \in \{0\} \cup \mathcal F,\quad |\mathbf F_{\rm RF}(n,l)| =1/\sqrt{N_t}$$ achieving full-array gain recovery for near-field users (Liu et al., 2024).

4. Impact of Loss and Estimation Uncertainty

Microstrip and Metasurface Losses

Loss mechanisms (modeled as exponential decay with attenuation α) reduce both peak gain and the beam depth: $$\Gamma(r_0+\Delta r;\,\alpha) \approx n^{-2} K^2(t_z(\Delta r),\,w)$$ where n is the effective element factor, and K(·) involves erfi functions. Design rules cap α(N_e d_e) ≲ 3 to avoid excessive narrowing (Gavriilidis et al., 15 Mar 2025, Gavriilidis et al., 29 Nov 2025).

Velocity and Position Estimation

Beamforming gain is exquisitely sensitive to user movement, especially in the transverse direction, and the minimum estimation accuracy must ensure that predicted beam gain lies above a prescribed threshold (quantified by the beam coherence time): $T_c^{(K)} = C_K/u$ where $C_K$ is the minimum displacement for K% gain retention.

Velocity mismatch analysis in MLAs reveals that transverse velocity error is the dominant cause of gain loss, whereas radial errors induce only phase rotation without amplitude attenuation (Alshumayri et al., 9 Nov 2025).

5. Beamforming Gain Degradation in RIS/RRS and THz Systems

RIS/RRS Wideband and Fresnel Zone-Based Designs

For large RIS in mmWave/THz, spherical-wavefront phase mismatch across subcarriers (narrowband design at f_c only) yields frequency-dependent focal shift: $\Delta r(f) \approx r_c (1 - f_c/f)$ Fresnel-zone approaches partition the surface and set equal phase shifts per zone, turning the gain degradation problem into a Fourier spectrum shaping over zones, which can be solved via stationary phase or GS algorithms for nearly flat band gain (Yu et al., 2024).

Compensation in Reconfigurable Refractive Surfaces

Per-element tunable time-delays (Delayed-RRS structure) jointly mitigate geometric and frequency-selective phase errors, with joint digital/analog/delay optimization algorithms iteratively restoring ~95% of the lost wideband focusing (Lin et al., 2 Jan 2025).

6. System Design Implications and Practical Guidelines

• Positioning/CSI Resolution: The required position accuracy follows directly from the closed-form beam correlation (solve $I^2(a(\Delta r_K)) = K/100$, $\Lambda^2(\Delta \theta_K )= K/100$).
• CSI Update Frequency: The beam coherence time is dictated by user speed and the minimum d for K% gain loss.
• Aperture–Bandwidth Tradeoff: There is a fundamental ceiling: $$B L \leq 2c [\gamma_1\gamma_2]_{\max(\tau)}/|\sin\theta_{\rm worst}|$$, limiting simultaneous scaling of array size and operational bandwidth (Deshpande et al., 2022).
• Subarray/TTD Partitioning: Subarray size and number of TTDs must be chosen to keep each sub-section in its own far-field, determined by the effective Rayleigh distance (tight gain-based criterion).
• Active/Passive Array Choices: Near-field gain loss in passive RIS can be compensated by increasing active feeder power or number of elements, with SVD-based eigenmode alignment at optimal near-field separation distances (Tiwari et al., 2022).
• Material Loss Management: Low-loss substrates and short feed lengths are essential for robust DMA beam depth; if low α is infeasible, element segmentation is recommended (Gavriilidis et al., 15 Mar 2025).

7. Summary Table: Key Quantities for Near-Field Gain Degradation

Quantity	Formula/Definition	Physical Meaning
Rayleigh distance	$r_{\rm Rayleigh}=2D^2/\lambda$	Far-/near-field boundary
Beam correlation under mismatch	$\rho(Δr,Δθ)\approx I^2(a(Δr))\Lambda^2(Δθ)$	Relative array gain at offset
Beam coherence time	$T_c^{(K)}=C_K/u$	Update interval for K% gain retention
Wideband gain deviation	$G(f)\approx\sinc^2(\pi(f-f_c)\tau_{\max})$	Sinc attenuation from beam-squint
Loss-induced gain reduction	$\Gamma(r_0+\Delta r;\alpha )$	Array gain with microstrip attenuation
Effective Rayleigh distance	$d_{ER}(\theta)=C_\Delta\cos^2\theta 2D^2/\lambda$	Practical gain-based boundary

Near-field beamforming gain degradation thus emerges as a core limiting factor for advanced wireless architectures, dictating required granularity in user tracking, array partitioning, and hardware innovation for robust high-frequency, ultra-large array MIMO and ISAC deployments (Gavriilidis et al., 29 Nov 2025, Qu et al., 2023, Gavriilidis et al., 15 Mar 2025, Deshpande et al., 2022, Qiu et al., 2024, Wang et al., 2024, Tiwari et al., 2022, Cui et al., 2021, Kosasih et al., 2023, Yu et al., 2024, Lin et al., 2 Jan 2025, Liu et al., 2024, Alshumayri et al., 9 Nov 2025, Wang et al., 25 Jun 2025).