⚡ electrical engineering

From Focusing to Con-Focusing: Optimal Power Transfer in Line-of-Sight Near-Field MIMO

This paper demonstrates that in line-of-sight near-field MIMO links with extended apertures, traditional beamfocusing becomes suboptimal beyond a specific Fresnel number threshold, leading to the proposal of a universal "con-focusing" strategy where both apertures aim at a common point to achieve order-optimal power transfer without requiring channel knowledge or geometry exchange.

Original authors: Marouan Mizmizi, Sanzhar Yergaliev

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Marouan Mizmizi, Sanzhar Yergaliev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A New Way to Shine a Light

Imagine you have two large, flat flashlights facing each other across a room. One is the Transmitter (sending the signal), and the other is the Receiver (catching the signal).

For a long time, engineers believed that to get the best signal between these two flashlights, the sender should "focus" its beam exactly onto the center of the receiver, like a laser pointer hitting a bullseye. This is called Focusing.

This paper argues that this old rule is wrong in many modern situations. The authors discovered that if both flashlights are large and close together, trying to hit the center of the receiver actually wastes energy. Instead, the best strategy is often to "steer" the beam straight, or to find a special "sweet spot" between the two devices where both flashlights aim at the same point in space. They call this new strategy Con-focusing.

The Problem: The "Spotlight" Mistake

To understand why the old rule fails, let's use an analogy of a projector and a screen.

The Old Rule (Focusing): Imagine you are a projector. You zoom your lens so that the image is perfectly sharp on the center of a large movie screen.
- The Result: The image is incredibly bright right in the center. But because you zoomed in so tight, the light spreads out very quickly. By the time the light reaches the top and bottom edges of the screen, it's gone. You are wasting a huge amount of energy because the "screen" (the receiver) is too big to fit inside your tiny, sharp spot.
- The Paper's Finding: If the receiver is large, focusing on its center is like trying to fill a swimming pool with a water pistol aimed at a single drop of water. Most of the pool stays dry.
The "Far-Field" Rule (Steering): Imagine you point the projector straight ahead without zooming. The beam is wide and flat.
- The Result: The light covers the whole screen, but it's not as intense in the middle as the focused beam was.
- The Paper's Finding: When the two flashlights are close and large, this "wide beam" approach actually delivers more total power to the receiver than the "sharp focus" approach.

The Discovery: The "Sweet Spot" (Con-focusing)

The authors asked: "Is there a way to get the best of both worlds?"

They found a mathematical "sweet spot" called Con-focusing.

The Analogy: Imagine two people standing on opposite sides of a hallway, holding flashlights. Instead of one person aiming at the other, they both aim their flashlights at a specific point on the floor between them.
Why it works: At this specific point, the beam coming from the first flashlight expands just enough to perfectly fill the second flashlight. It's like a handshake in the middle of the room where both hands fit perfectly together.
The Magic: This strategy works for any distance and size. It doesn't require knowing the exact channel conditions (like a GPS lock); it just requires knowing the geometry (how far apart they are).

The "Fresnel Number": The Ruler of the Room

The paper introduces a concept called the Fresnel Number. Think of this as a "Room Size Score."

Low Score (Small Room / Small Flashlights): If the score is low, the old "Focusing" rule (aiming at the center) still works best.
High Score (Large Room / Huge Flashlights): Once the score gets above a specific magic number (1.947), the old rule breaks.
- The Crossover: At this magic number, the "Focusing" strategy suddenly becomes worse than "Steering."
- The Penalty: If you keep focusing when you should be steering, you lose signal strength very fast. The paper says you lose 10 decibels of power for every step you take deeper into the "large room" zone. That's like turning your volume down by 90% for every small step you take.

The Solution: A Simple, Blind Strategy

The authors propose a strategy called Con-focusing that is "order-optimal." In plain English, this means it is nearly perfect and impossible to beat significantly, even with complex math.

How it works: You don't need to know the complex details of the air or the signal path. You just need to know the distance between the two devices.
The Trick: You aim both devices at a point where they "see" each other at the same angle.
- If the devices are the same size, this point is exactly halfway between them.
- If one is bigger, the point shifts closer to the smaller one.
The Benefit: This strategy is so good that it performs almost as well as the theoretical maximum possible power transfer. It is robust, simple, and doesn't require expensive equipment to calculate complex channel maps.

Summary of the "Rules of the Road"

Small Receivers: If the receiver is tiny (like a single antenna), Focusing (aiming at the center) is still the best.
Large Receivers: If the receiver is big (like a large wall of antennas), Focusing is a trap. It wastes energy.
The Switch: Once the "Room Size Score" (Fresnel Number) passes 1.947, you must stop focusing on the center.
The New Best: Use Con-focusing. Aim both devices at a shared point in the middle so their beams match up perfectly.
The Result: This simple geometric trick recovers the lost power and is the most efficient way to send signals between large, close-range devices.

In short: Stop trying to hit the bullseye if the target is a whole wall. Instead, aim both flashlights at the middle of the wall so they meet perfectly.

Technical Summary: From Focusing to Con-Focusing in Line-of-Sight Near-Field MIMO

Problem Statement
This paper addresses the optimal power transfer strategy for a single-user Line-of-Sight (LoS) MIMO link where both the transmitter (TX) and receiver (RX) possess extended apertures (large arrays). While the prevailing consensus in near-field communications (specifically XL-MIMO) dictates that "beamfocusing" (matching phase to a spherical wavefront converging on the user) is superior to "beam steering" (planar wavefronts) within the Rayleigh distance, this conclusion is derived almost exclusively for Multiple-Input Single-Output (MISO) scenarios where the receiver is a single antenna.

The authors investigate the generalization of this problem to a MIMO link where the user also carries an extended aperture. They posit that the standard "focus on the user" prescription may be suboptimal or even detrimental when the receiver aperture is large relative to the diffraction-limited focal spot generated by the transmitter. The core question is: under what conditions does focusing on the user's center outperform far-field steering, and what is the optimal strategy when neither extreme is ideal?

Methodology
The study employs a theoretical framework based on continuous-aperture approximations of Uniform Linear Arrays (ULAs) under pure LoS conditions (no multipath).

System Model: The link is defined by transmit length $L_t$ , receive length $L_r$ , distance $D$ , and wavelength $\lambda$ . The analysis relies on two dimensionless parameters: the link Fresnel number $N_F = L_t L_r / (\lambda D)$ and the aperture ratio $\rho = L_r / L_t$ (assuming $\rho \le 1$ ).
Beamforming Family: The authors analyze a family of unit-modulus (phase-only) beamformers defined by focal depths $r_t$ and $r_r$ . The residual phase after beamforming is characterized by a quadratic form involving curvatures $A$ and $B$ .
Analytical Derivation: The array gain is reduced to a field integral over the joint aperture. The authors derive closed-form expressions for the power gain of two specific strategies:
- Focusing: $r_t = r_r = D$ (focusing on the opposite array's center).
- Steering: $r_t = r_r = \infty$ (planar phase ramp).
Optimization: The authors identify the "confocal locus" in the curvature space where the residual phase becomes separable, allowing for coherent addition of signals. They derive the specific geometric point that maximizes gain along this locus.

Key Contributions and Results

Inversion of the Focusing Advantage:
The paper proves that the superiority of focusing is not universal. For equal apertures ( $\rho = 1$ ), there exists a universal crossover point at $N_F^* \approx 1.947$ .
- Below $N_F^*$ , focusing slightly outperforms steering.
- Above $N_F^*$ , steering significantly outperforms focusing. In the deep near-field ( $N_F \gg 1$ ), the power gain of focusing decays as $O(N_F^{-2})$ , while steering decays only as $O(N_F^{-1})$ . This results in a signal-to-noise ratio (SNR) gap that widens by 10 dB per decade of the Fresnel number.
- The "MISO intuition" (that focusing is always better) is shown to be the limiting case where the receive aperture vanishes ( $\rho \to 0$ ).
The Con-Focusing Law:
The authors propose a new strategy called con-focusing. Instead of focusing on the opposite array, both apertures aim at a common axial point $P$ from which they subtend equal angles.
- Geometry: The optimal focal depths are $r_t^* = D/(1+\rho)$ and $r_r^* = \rho D/(1+\rho)$ .
- Optimality: This strategy is order-optimal among all rank-one schemes. It attains the rank-one eigenbound (the theoretical maximum for a single beam) in the leading constant.
- Degeneracy: For equal apertures ( $\rho=1$ ), the con-focusing point moves to infinity, degenerating into plain steering. As $\rho \to 0$ , it collapses onto the user, recovering the standard MISO focusing prescription.
Closed-Form Gains and Crossover:
The paper provides exact closed-form expressions for the gains of focusing and steering in the continuous-aperture limit. It proves that for equal apertures, the ratio of gains crosses unity exactly once at $N_F = 1.947$ , independent of the number of elements, wavelength, or distance.
Blind Acquisition Protocol:
The authors demonstrate that the optimal con-focusing strategy requires no channel state information (CSI) or geometry exchange. It depends only on the range $D$ . They propose an adaptive maximum-likelihood protocol to estimate $D$ using received power measurements alone, allowing the system to reach the optimum within a single beam-refinement round.
Robustness and Discretization:
The analysis accounts for discrete array sampling, showing that the continuous results hold provided the element spacing satisfies Nyquist conditions relative to the Fresnel number. The paper also proves that the $(N_F, \rho)$ trade-off map is covariant under array rotations (tilting and rolling), where the effective apertures are simply their projections onto the broadside plane.

Significance and Claims
The paper claims to resolve a fundamental misconception in near-field MIMO literature: that focusing on the user is the optimal strategy for all near-field links. The authors assert that for links between two extended apertures, "focusing on the user" is often the worst possible choice in the deep near-field, leading to a 10 dB/decade performance loss compared to steering.

The significance of the work lies in:

Unification: It reconciles the classical MISO result, holographic surface literature, and symmetric aperture inversions into a single operating region map governed by $(N_F, \rho)$ .
Constructive Solution: It provides a simple, geometry-only "con-focusing" law that is asymptotically optimal, requires no channel knowledge, and is easily acquirable.
Theoretical Rigor: It establishes the exact location, uniqueness, and universality of the crossover point and characterizes the asymptotic laws of power transfer in the near-field MIMO regime.

The authors emphasize that the trade-off is governed by aperture extent (physical size), not the number of discrete elements, and that the "focusing advantage" is strictly an asymmetry phenomenon that vanishes as the apertures become equal.