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From Steering to Focusing

In the far field, the single degree of freedom a beamformer exploits is direction: a precoder picks an angle $\theta$ , illuminates everything along that ray, and cannot discriminate between a user at $10$ m and a user at $100$ m if they share that angle. In the near field, the beamformer has one more knob: range. The spherical wavefront carries enough information about the distance to the source that the array can concentrate energy at a specific point in 3-D space. We call this beam focusing, in direct analogy with how a lens focuses light to a spot rather than a direction.

Operationally, beam focusing means that two users at the same angle but different ranges can be spatially multiplexed. It also means that energy leaks out of the focus along the range axis only slowly, which gives rise to a notion of depth of focus — the range window within which a focused beam delivers nearly full gain. This depth is finite in the near field and collapses back to infinity as the range approaches the Fraunhofer distance.

,

Definition:
Near-Field Matched Beamformer

Given an array with element positions $\{\mathbf{p}_n\}$ and a target focal point $\mathbf{p}_k$ , the near-field matched (focusing) beamformer is $\mathbf{v}(\mathbf{p}_k) \;\triangleq\; \frac{1}{\sqrt{N_t}}\begin{bmatrix} e^{+j\kappa r_{k,1}} \\ \vdots \\ e^{+j\kappa r_{k,N_t}} \end{bmatrix}, \qquad r_{k,n} = \|\mathbf{p}_k - \mathbf{p}_n\|.$ It is the unit-norm vector that maximises the coherent gain $|\mathbf{v}^{H} \mathbf{a}_{\text{NF}}(\mathbf{p}_k)|$ ; by the Cauchy–Schwarz inequality, the maximum equals $\|\mathbf{a}_{\text{NF}}(\mathbf{p}_k)\| = 1$ . The corresponding single-user LoS received SNR, ignoring noise at the focus, is $\text{SNR}_{\text{focus}} = \frac{P_t\,N_t\,\beta_{k}}{\sigma^2}.$ Equivalently: the near-field matched beamformer still achieves the massive-MIMO beamforming gain of $N_t$ at its focal point.

In the far field, $\mathbf{v}(\mathbf{p}_k) = \mathbf{a}(\theta_k)^*$ depends only on the direction, and all points along the ray through $\mathbf{p}_k$ enjoy the same gain (limited only by path-loss $\beta_{k}$ ). In the near field, the quadratic phase in $r_{k,n}$ creates constructive interference only near the focal point, and destructive interference everywhere else along the ray.

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Theorem: Depth of Focus Along the Broadside Ray

Fix a ULA of aperture $D$ along the $y$ -axis and let the focal point be at $\mathbf{p}_k = (d_k, 0, 0)$ on the broadside ray with $d_k < d_F = 2 D^2/\lambda$ . Consider a test point at $\mathbf{p} = (r, 0, 0)$ and let $g(r) = |\mathbf{v}(\mathbf{p}_k)^H \mathbf{a}_{\text{NF}}(\mathbf{p})|^2$ be the normalised focusing gain. Using the quadratic-phase model of Section 17.2, the half-power (−3 dB) width of $g$ about $r = d_k$ is $\Delta r \;\approx\; \frac{2\, d_F\, d_k^2}{d_F^2 - d_k^2} \quad\text{(depth of focus)}.$ In particular, $\Delta r \to 0$ as $d_k \to 0$ and $\Delta r \to \infty$ as $d_k \to d_F$ , recovering the far-field limit where the focused beam collapses to an infinite pencil beam.

The focused beamformer aligns phases for a range $d_k$ and leaves a quadratic phase residual when you slide the test point along the ray: this residual behaves like a chirp, and its half-power width is inversely proportional to the chirp rate. As $d_k$ approaches $d_F$ the chirp rate goes to zero — the quadratic term disappears, and the beam opens up along the entire ray. For $d_k \ll d_F$ , the chirp rate is large, and the beam has a tight range window.

Show Hint

Write the gain $g(r)$ using the Fresnel expansion of both the focused beamformer and the test-point array response.

The common linear phase cancels; what remains is a quadratic-phase mismatch in $y_n^2$ with coefficient proportional to $(1/r - 1/d_k)$ .

The Fresnel integral $\int_{-D/2}^{D/2} e^{j\alpha y^2}\,dy$ has a half-width $\alpha D^2/2 \lesssim \pi/2$ .

Proof

Quadratic-phase mismatch

Use the Fresnel expansion $r_{k,n} \approx r_k + y_n^2/(2 r_k)$ on broadside ( $\cos\theta = 1$ ). The gain is $g(r) = \frac{1}{N_t^{2}}\left|\sum_{n=1}^{N_t} e^{-j\kappa y_n^2 (1/(2 r) - 1/(2 d_k))}\right|^2.$ The common range phase cancels; only the quadratic residual $\alpha(r) y_n^2/2$ survives, with $\alpha(r) = \kappa\left(\frac{1}{d_k} - \frac{1}{r}\right).$

Replace the sum by a Fresnel integral

With $\lambda/2$ -spaced elements and large $N_t$ , approximate the sum by an integral over $y \in [-D/2, D/2]$ : $g(r) \approx \frac{1}{D^2}\left|\int_{-D/2}^{D/2} e^{-j\alpha(r) y^2/2}\,dy\right|^2.$ This is a Fresnel integral whose magnitude is close to its stationary-phase value (which gives full coherent gain) as long as the total phase sweep across the aperture stays below $\pi/2$ : $|\alpha(r)|\,(D/2)^2/2 \leq \pi/2$ .

Solve for the range window

The boundary condition $|\alpha(r)|D^2/8 = \pi/2$ gives $\left|\frac{1}{d_k} - \frac{1}{r}\right| = \frac{4\pi}{\kappa D^2} = \frac{2\lambda}{D^2} = \frac{1}{d_F}\cdot 1.$ Writing $1/r = 1/d_k \pm 1/d_F$ and expanding, $r_{\pm} = \frac{d_k d_F}{d_F \mp d_k} \quad\Longrightarrow\quad \Delta r = r_+ - r_- = \frac{2\, d_F\, d_k^2}{d_F^2 - d_k^2}.$ As $d_k \to d_F$ the denominator vanishes and $\Delta r$ diverges. As $d_k \to 0$ , $\Delta r$ goes to $0$ like $2 d_k^2/d_F$ . $\blacksquare$

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Depth of Focus Along a Broadside Ray

The beam is focused at range $d_k$ along the broadside ray. The curve shows the gain as you slide a scatterer along the same ray. The shaded band is the theoretical $\Delta r = 2 d_F d_k^2/(d_F^2 - d_k^2)$ ; the dotted vertical lines mark $d_k$ and $d_F$ . As you move $d_k$ from very short ranges toward $d_F$ , watch the depth-of-focus window open up until, at $d_k \approx d_F$ , the beam has effectively infinite depth — the far-field limit.

Parameters

N_t

128

Carrier

f_0

(GHz)28

Focus range

d_k

(m)5

Min scan range (m)1

Max scan range (m)60

2-D Beam-Focusing Map

The array focuses at $(d_{\text{focus}},\varphi_{\text{focus}})$ in the $(\text{range},\text{angle})$ plane. The heatmap is the normalised gain delivered to a scatterer at every grid point. Inside the Fraunhofer distance, the pattern is a localised spot — that is the operational signature of beam focusing. For large focus ranges, the spot elongates along the range axis and eventually merges into a far-field pencil beam that extends indefinitely.

Parameters

N_t

64

Carrier

f_0

(GHz)28

Focus range (m)5

Focus angle (deg)0

Max range (m)30

Angle span (deg)30

Example: Depth of Focus: A $28$ GHz Example

A $64$ -element ULA at $f_0 = 28$ GHz with half-wavelength spacing focuses at $d_k = 5$ m along broadside. Compute $d_F$ and the resulting depth of focus $\Delta r$ . How much range separation between two users at the same angle is needed to make them nearly orthogonal?

Solution

Aperture and $d_F$

$\lambda \approx 1.07$ cm. Aperture $D = 63\cdot 0.00536 = 0.337$ m. $d_F = 2\cdot 0.337^2/0.0107 \approx 21.2$ m.

Depth of focus

$\Delta r = \frac{2\, d_F\, d_k^2}{d_F^2 - d_k^2} = \frac{2\cdot 21.2\cdot 25}{21.2^2 - 25} \approx \frac{1060}{424.4} \approx 2.5\ \text{m}.$ $Two users at the same angle become nearly orthogonal once their range separation exceeds roughly$ \Delta r $— here, about$ 2.5 $m. A$ 28 $GHz array therefore can separate, for instance, a user at$ 4 $m from a user at$ 7$ m along the same broadside direction. That is a spatial degree of freedom the plane-wave model would not see.

Polar-Domain Codebook Construction (Cui–Dai 2022)

Complexity:

O(N_t^{3/2})

codewords;

O(N_t)

correlation per codeword

Input: Aperture

D

, wavelength

\lambda

, angular resolution

N_\theta = N_t

Output: Polar-domain codebook

\mathcal{C} = \{\mathbf{v}(\theta_i, r_j)\}

1. Compute

d_F \leftarrow 2 D^2 / \lambda

.

2. Set angular grid:

\sin\theta_i = -1 + 2(i-1)/N_\theta

for

i = 1,\ldots,N_\theta

.

3. for each

\theta_i

do

4.

\quad

Compute range step

\Delta(1/r) = \lambda/(D^2\cos^2\theta_i)

.

5.

\quad

Set range grid:

1/r_j = j\cdot\Delta(1/r)

for

j = 1, 2, \ldots

up to

1/r_{\min}

.

6.

\quad

for each

r_j

do

7.

\qquad

Append

\mathbf{v}(\theta_i, r_j)

to

\mathcal{C}

using Def. DNear-Field Matched Beamformer.

8. end for

9. return

\mathcal{C}

— total size

\mathcal{O}(N_t^{3/2})

.

The polar-domain codebook replaces the DFT codebook of the far field. Its angular axis is uniform in $\sin\theta$ (like the DFT); its range axis samples $1/r$ uniformly with step $\lambda/(D^2)$ — i.e., one sample per depth of focus. Cui and Dai show that $\mathcal{O}(N_t^{3/2})$ codewords cover the $(\theta, r)$ plane with incoherence at most $\mathcal{O}(1/\sqrt{N_t})$ , which is all a near-field OMP estimator needs.

Beam Steering vs Beam Focusing

Property	Far-Field Beam Steering	Near-Field Beam Focusing
Parameter	Direction $\theta$	Position $(r, \theta)$ (or $(r,\theta,\varphi)$ in 3-D)
Array response	Linear phase in $y_n$	Linear plus quadratic phase in $y_n$
Gain profile along a ray	Infinite pencil beam	Spot of depth $\Delta r = 2 d_F d_k^2/(d_F^2 - d_k^2)$
Codebook	DFT, $\mathcal{O}(N_t)$ beams	Polar, $\mathcal{O}(N_t^{3/2})$ beams
Can separate users co-directional?	No	Yes, if $\|r_1 - r_2\| \gtrsim \Delta r$
Peak beamforming gain	$N_t$ (at $\theta$ )	$N_t$ (at the focal point)
Far-field limit	Native regime	Recovers steering as $d_k \to d_F$

Common Mistake: Near-Field Does Not Cost Array Gain

Mistake:

A misconception is that near-field operation "loses" array gain because the spherical wavefront is less efficient than a plane wave.

Correction:

The coherent beamforming gain at the focal point is exactly $N_t$ — the same as the plane-wave steering gain at an angle, and for the same reason: the matched filter aligns $N_t$ independent phases and sums them coherently. What near-field operation does cost is the far-field assumption that energy extends uniformly along a ray; the beam energy is localised to a spot, which is the feature that enables range separation. Localisation and gain are not in conflict — the localisation is how the same gain gets concentrated.

Why This Matters: Near-Field ISAC and Localisation

Beam focusing blurs the line between communication and sensing. The near-field array response depends on both angle and range, so a near-field sounding of the channel already contains range information — something a conventional cellular array cannot easily measure. Recent work on joint positioning and communication at FR3/FR4 exploits exactly this: a single cell, without multi-cell TDOA, can produce metre-scale position estimates purely from the quadratic-phase signature of the user. Chapter 16 (joint detection and positioning for cell-free) already hinted at this idea; in Chapter 24 we will return to it in the ISAC context.

Historical Note: Beam Focusing in Optical Systems Predates RF by a Century

1900s optics → 2020s RF

The concept of "depth of focus" is a bread-and-butter topic in optical engineering. Born and Wolf's Principles of Optics (first edition 1959, seventh 1999) treats it as one of the first non-trivial results of diffraction theory: a lens with aperture $D$ and focal length $f$ has axial depth of focus scaling as $\lambda(f/D)^2$ . The same scaling, with $f$ replaced by the focal range $d_k$ and $D$ by the aperture of the array, gives the RF near-field depth-of-focus formula.

It is in a real sense ironic that RF engineers spent forty years treating the Fraunhofer distance as the place beyond which everything becomes easy, while optical engineers spent those same forty years working very happily inside the Fresnel region and calling it a feature. Sub-THz wavelengths are now bringing the two communities onto the same ground.

Depth of Focus

The range extent $\Delta r$ over which a near-field matched beamformer maintains at least half its peak coherent gain. For a broadside focus at range $d_k$ on an aperture with Fraunhofer distance $d_F$ , $\Delta r \approx 2 d_F d_k^2/(d_F^2 - d_k^2)$ . Two users at the same angle but range separation exceeding $\Delta r$ are spatially separable by the array.

Polar-Domain Codebook

A near-field codebook of beamforming vectors gridded uniformly in $\sin\theta$ and in $1/r$ , with $\mathcal{O}(N_t^{3/2})$ entries. Introduced by Cui and Dai (2022) as the minimal extension of the classical DFT (far-field) codebook that captures both angular and range information.

Quick Check

A $256$ -element ULA at $60$ GHz, $\lambda/2$ -spaced, focuses at $d_k = 3$ m. What is the approximate depth of focus $\Delta r$ ?

About $3$ mm — beams at mmWave are extremely tight.

About $11$ cm.

Essentially infinite — the Fraunhofer distance is $163$ m, and $3$ m is well inside, so the beam is range-insensitive.

About $d_F = 163$ m.

Correction:

About

11

cm.

$\lambda = 5$ mm, $D = 255\cdot 0.0025 \approx 0.638$ m, $d_F = 2\cdot 0.407 / 0.005 \approx 162.7$ m. In the short-range regime $d_k \ll d_F$ , $\Delta r \approx 2 d_k^2/d_F = 2\cdot 9 /162.7 \approx 0.111$ m $\approx 11$ cm.

Key Takeaway

Near-field beams have a finite depth of focus, and this is a feature, not a bug. The same $N_t$ -fold beamforming gain that the far field delivers along a ray, the near field delivers to a spot of depth $\Delta r = 2 d_F d_k^2/(d_F^2 - d_k^2)$ . Two users at the same angle but different ranges become spatially separable whenever their range separation exceeds $\Delta r$ — a spatial degree of freedom that simply does not exist in the far field.

Beam Focusing and Depth of Focus

From Steering to Focusing

Definition: Near-Field Matched Beamformer

Theorem: Depth of Focus Along the Broadside Ray

Quadratic-phase mismatch

Replace the sum by a Fresnel integral

Solve for the range window

Depth of Focus Along a Broadside Ray

Parameters

2-D Beam-Focusing Map

Parameters

Example: Depth of Focus: A 282828 GHz Example

Aperture and $d_F$

Depth of focus

Polar-Domain Codebook Construction (Cui–Dai 2022)

Beam Steering vs Beam Focusing

Common Mistake: Near-Field Does Not Cost Array Gain

Why This Matters: Near-Field ISAC and Localisation

Historical Note: Beam Focusing in Optical Systems Predates RF by a Century

Depth of Focus

Polar-Domain Codebook

Quick Check

Key Takeaway

Definition:
Near-Field Matched Beamformer

Example: Depth of Focus: A $28$ GHz Example