Ferkans — Interactive Telecom Tutor

When the RIS Is Big Enough to Focus

The far-field steering-vector model of Section 3.2 assumes the incoming wave at the RIS is a plane wave: all elements see the same direction of arrival, only with a linear phase ramp. This is an excellent approximation when the link distance is much larger than the RIS aperture size $D$ . But for sub-THz and very-large aperture RIS (VLA-RIS), the link distance can fall inside the Fraunhofer distance $d_F = 2D^2/\lambda$ , and the wavefront curvature across the RIS becomes non-negligible. The RIS then acts as a near-field focusing surface: phase shifts compensate not only for propagation direction but also for per-element distance differences. The two regimes are genuinely different and serve different design tradeoffs.

Fraunhofer Distance

The distance $d_F = 2D^2/\lambda$ beyond which an aperture of largest dimension $D$ can be treated as a point source (far-field). Distances smaller than $d_F$ are in the near-field, where wavefront curvature across the aperture becomes significant and enables distance-specific focusing.

Ricean K-Factor

The power ratio between the deterministic LoS component and the random NLoS (scattering) component of a fading channel. $K = \infty$ is pure LoS (deterministic), $K = 0$ is pure Rayleigh. In RIS systems, high- $K$ cascaded channels behave close to the deterministic rank-1 LoS model; low- $K$ channels see the full keyhole-to-Rayleigh heavy-tail transition.

Definition:
Fraunhofer Distance

For an array of largest dimension $D$ operating at wavelength $\lambda$ , the Fraunhofer distance (or Rayleigh distance) is

$d_F = \frac{2 D^2}{\lambda}.$

Distances $d \geq d_F$ are in the far-field regime: plane-wave approximation holds. Distances $d < d_F$ are in the near-field regime (specifically, the radiating near-field): wavefront curvature across the aperture is significant.

Example: Fraunhofer Distance for Practical RIS Sizes

Compute $d_F$ for the following scenarios: (a) $0.5\text{ m} \times 0.5\text{ m}$ RIS at 3.5 GHz. (b) Same RIS at 28 GHz. (c) $1\text{ m} \times 1\text{ m}$ RIS at 140 GHz.

Solution

Compute $D$

(a) and (b): $D = 0.5\sqrt{2} \approx 0.71\text{ m}$ . (c): $D = 1 \sqrt{2} \approx 1.41\text{ m}$ .

Wavelengths

At 3.5 GHz: $\lambda = 8.57\text{ cm}$ . At 28 GHz: $\lambda = 1.07\text{ cm}$ . At 140 GHz: $\lambda = 2.14\text{ mm}$ .

Fraunhofer distances

(a) $d_F = 2(0.71)^2 / 0.0857 = 11.8\text{ m}$ . Most practical links at 3.5 GHz are far-field with this aperture. (b) $d_F = 2(0.71)^2 / 0.0107 = 94.4\text{ m}$ . Close-range mmWave falls into the near-field; "coverage within 100 m" often requires near-field modelling. (c) $d_F = 2(1.41)^2 / 0.00214 = 1860\text{ m}$ . Nearly all practical sub-THz deployments are in the near field of a 1-m aperture. Near-field design is not optional at 140 GHz.

Theorem: Near-Field Channel: Per-Element Distances

For a point source at position $\mathbf{p}_s$ at distance $d_s = \|\mathbf{p}_s\|$ from the RIS center, the near-field incoming signal at element $n$ at position $\mathbf{r}_n$ (relative to the RIS center) has phase

$\psi_n = \frac{2\pi}{\lambda} \|\mathbf{p}_s - \mathbf{r}_n\| = \frac{2\pi}{\lambda}\sqrt{d_s^2 - 2 \mathbf{p}_s^T \mathbf{r}_n + \|\mathbf{r}_n\|^2}.$

Expanding to second order for $d_s \gg \|\mathbf{r}_n\|$ :

$\psi_n \approx \frac{2\pi}{\lambda}\left[d_s - \frac{\mathbf{p}_s^T \mathbf{r}_n}{d_s} + \frac{\|\mathbf{r}_n\|^2 - (\mathbf{p}_s^T \mathbf{r}_n / d_s)^2}{2 d_s}\right].$

The first-order term $-\mathbf{p}_s^T\mathbf{r}_n/d_s$ is the far-field linear phase ramp (the steering vector). The second-order term, proportional to $1/d_s$ , is the near-field quadratic phase — the wavefront curvature.

In the far-field, phase compensation depends only on the direction of arrival. In the near-field, it depends on the per-element distance $d_n$ from the source. The RIS then acts like a lens: each element applies a phase correction that depends on where it sits in the aperture.

Proof

Exact propagation phase

$\psi_n = (2\pi/\lambda) \|\mathbf{p}_s - \mathbf{r}_n\|$ . This is exact for a point source.

Taylor expansion

Expand $\|\mathbf{p}_s - \mathbf{r}_n\|$ using $\|\mathbf{x} + \mathbf{y}\| = \|\mathbf{x}\|\sqrt{1 + 2\mathbf{x}^T\mathbf{y}/\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2/\|\mathbf{x}\|^2}$ and Taylor-expand the square root in powers of $\|\mathbf{r}_n\|/d_s$ .

Identify orders

First-order: far-field steering vector. Second-order: $\|\mathbf{r}_n\|^2/(2d_s)$ + cross-term — the Fresnel parabolic approximation used in ordinary optics. Third-order: the neglected terms that vanish above $d_F$ . $\blacksquare$

Near-Field = Focusing

The near-field second-order term enables a capability the far-field does not have: distance-specific focusing. In the far-field, the RIS can steer the beam in a direction $(\theta, \phi)$ but cannot focus it at a specific distance — the beam extends outward in a cone. In the near-field, by matching the quadratic-phase term to the target distance $d_s$ , the RIS can focus the reflected beam at a specific point in 3D space. This is the principle of near-field beamforming, which enables near-field localization (Chapter 14) and near-field MIMO multiplexing even in pure LoS (since different UE positions at the same angle but different distances produce different near-field signatures).

Near-Field Focusing vs. Far-Field Steering

Toggle between far-field and near-field beam forming. For a UE at distance $d_s$ , the far-field beam is a wedge; the near-field beam is a focal spot. Move the UE and watch the RIS phase profile change — a linear ramp for far-field, a quadratic ramp for near-field.

Parameters

RIS aperture

D

(m)1

Carrier

f_c

(GHz)100

UE distance

d_s

(m)20

UE angle (deg)10

🎓CommIT Contribution(2023)

Array-Fed RIS Exploits Near-Field Between the Array and the Surface

G. Caire, I. Atzeni — IEEE Trans. Signal Process. (preprint 2023)

Caire and collaborators (2023) exploit a specific near-field geometry: a small active array very close to a large passive RIS, with the two linked by a controlled short-range feed. In this configuration, the active-array-to-RIS channel $\mathbf{H}_1$ is in the near-field of the RIS and has a rich rank structure — the active array couples into many eigenmodes of the RIS aperture, not just a single plane-wave direction. The downstream RIS-to-UE channel $\mathbf{H}_2$ can be far-field (long range), but the key insight is that the BS-RIS link has been engineered to look high-rank, enabling multi-user multiplexing through the passive RIS. This is the architectural contribution that makes RIS competitive at mmWave / sub-THz, and it only makes sense when you understand the near-field geometry developed in this section.

array-fed-risnear-fieldcaire-2023eigenmodes

Theorem: Near-Field Adds Degrees of Freedom

Consider two UEs at identical angular positions relative to the RIS but different distances $d_1 \neq d_2$ . In the far-field (both $d_1, d_2 \gg d_F$ ), the two UEs are at the same RIS-steering direction and the rank of the joint cascaded channel is 1 — they cannot be spatially multiplexed. In the near-field ( $d_1, d_2 \lesssim d_F$ ), the two UEs produce different quadratic phase signatures at the RIS, and the joint channel has rank 2 — they can be multiplexed.

More generally, a near-field RIS supports on the order of $(D/\lambda)^2$ angularly-orthogonal beams plus a similar number of distance-orthogonal beams, giving a 3D multiplexing capacity absent in the far-field.

Proof

Far-field collapse

Same angle → same steering vector → $\mathbf{a}_{\text{RIS}}(\Omega_1) = \mathbf{a}_{\text{RIS}}(\Omega_2)$ , so $\mathbf{H}_{\text{eff}}$ is rank-1.

Near-field distinction

Per-element phases differ by the $\|\mathbf{r}_n\|^2/(2d_s)$ quadratic term, which depends on $d_s$ . For $d_1 \neq d_2$ , the quadratic profiles differ, so the effective "near-field steering vectors" $\mathbf{a}_{\text{RIS}}^{\text{near}}(d_s)$ are linearly independent, giving a rank-2 (or higher) cascaded channel. $\blacksquare$

⚠️Engineering Note

Near-Field Is the Default at sub-THz

Planning a sub-THz RIS deployment:

Compute $d_F$ for your aperture size and frequency. If it exceeds $\sim 100$ m, most of your coverage area is near-field.
Use the near-field channel model for optimization — the far-field plane-wave assumption introduces $\sim 3\text{ dB}$ errors in beam pattern at distances $\sim d_F/2$ .
Exploit the focusing: near-field RIS enables per-UE distance-specific beamforming, which serves UEs at the same angle but different ranges separately.
Budget for higher-rank channel estimation: since the near-field cascaded channel has rank $\sim (D/\lambda)^2$ , the number of pilot resources required scales accordingly (Chapter 4).

Practical Constraints

•
Sub-THz (140 GHz) RIS apertures of $1\text{ m}^2$ have $d_F \sim 1$ km.
•
Near-field pilot overhead can exceed far-field by $\sim 10\times$ .
•
Near-field beam patterns have focal spots of $\sim \lambda d_s / D$ in size.

Quick Check

The Fraunhofer distance $d_F = 2D^2/\lambda$ for a $1\text{ m}^2$ RIS aperture at 100 GHz is approximately:

$\sim 10$ m

$\sim 100$ m

$\sim 700$ m

$\sim 10$ km

Correction:

\sim 700

m

At 100 GHz, $\lambda = 3\text{ mm}$ . Aperture diagonal $D = \sqrt{2} \approx 1.41$ m. $d_F = 2(1.41)^2/0.003 \approx 1333$ m/2 = ~667 m. Nearly all deployments are near-field.

Common Mistake: Don't Assume Far-Field at High Frequencies

Mistake:

"Plane-wave models have worked for 50 years of wireless; a few hundred meters at 28 GHz can't possibly be near-field."

Correction:

Yes it can. The Fraunhofer distance scales as $1/\lambda = f_c/c$ , so going from 3.5 GHz to 140 GHz multiplies $d_F$ by 40 — from 12 m to 470 m for the same aperture. Aperture size also enters quadratically: a $2\times$ larger aperture pushes $d_F$ out by $4\times$ . At sub-THz with even modest apertures, most of the deployment zone is near-field. Using a far-field channel model there will predict the wrong beam shape and wrong inter-user interference. Always check $d_F$ first; if your coverage zone is within it, upgrade to the near-field model.

Near-Field vs. Far-Field for Large RIS

When the RIS Is Big Enough to Focus

Fraunhofer Distance

Ricean K-Factor

Definition: Fraunhofer Distance

Example: Fraunhofer Distance for Practical RIS Sizes

Compute $D$

Wavelengths

Fraunhofer distances

Theorem: Near-Field Channel: Per-Element Distances

Exact propagation phase

Taylor expansion

Identify orders

Near-Field = Focusing

Near-Field Focusing vs. Far-Field Steering

Parameters

Array-Fed RIS Exploits Near-Field Between the Array and the Surface

Theorem: Near-Field Adds Degrees of Freedom

Far-field collapse

Near-field distinction

Near-Field Is the Default at sub-THz

Quick Check

Common Mistake: Don't Assume Far-Field at High Frequencies

Definition:
Fraunhofer Distance