On the Complexity of Electromagnetic Far-Field Modeling

TL;DR

The paper addresses the challenge of modeling electromagnetic far-field interactions for general finite-size antenna architectures, including reflective and transformable surfaces. It develops a Maxwell-based framework and proves that the far-field interaction operator $oldsymbol{ ext{T}}$ is arbitrarily well approximable by finite-rank operators, enabling finite-parameter representations. A vector-spherical-harmonics based construction yields a sequence of finite-rank operators $oldsymbol{ ext{T}}_L$ with super-exponential decay of the approximation error once the rank exceeds the effective bandwidth $L_{ extnormal{B}} = igl\uparrow k a igr rbracket$, with explicit constants $oldsymbol{ ext{alpha}}$ and $oldsymbol{ extbeta}(L)$ such that $igl\|oldsymbol{ ext{T}}-oldsymbol{ ext{T}}_Ligr ext{op}igr ext{ ext{}} \,\le \,oldsymbol{ ext{alpha}} e^{-oldsymbol{ extbeta}(L) L}$. This result supports efficient, accurate digital modeling for modern wireless systems (including RIS-like architectures) by reducing the problem to a finite-dimensional representation whose accuracy improves rapidly with $L$. The paper thus provides a principled basis for low-complexity, high-fidelity EM modeling across a broad class of antenna systems.

Abstract

Modern wireless systems are envisioned to employ antenna architectures that not only transmit and receive electromagnetic (EM) waves, but also intentionally reflect and possibly transform incident EM waves. In this paper, we propose a mathematically rigorous framework grounded in Maxwell's equations for analyzing the complexity of EM far-field modeling of general antenna architectures. We show that-under physically meaningful assumptions-such antenna architectures exhibit limited complexity, i.e., can be modeled by finite-rank operators using finitely many parameters. Furthermore, we construct a sequence of finite-rank operators whose approximation error decays super-exponentially once the operator rank exceeds an effective bandwidth associated with the antenna architecture and the analysis frequency. These results constitute a fundamental prerequisite for the efficient and accurate modeling of general antenna architectures on digital computing platforms.

Figures (2)

• Figure 1: Analyzed problem setup: We consider a radiating structure that occupies the volume $\mathcal{V}$, has $M$ ports, and is embedded in free space. Outside the structure’s volume, in the region $\overline{\mathcal{V}}$, the current density ${\color{pink}\bm{\mathsf{J}}}_1$ is impressed. This excitation gives rise to an induced current density ${\color{pink}\bm{\mathsf{J}}}_2$ within the radiating structure, which in turn induces the electromagnetic field $({\color{pink}\bm{\mathsf{E}}}_2,{\color{pink}\bm{\mathsf{H}}}_2)$.
• Figure 2: We consider the interaction of a radiating structure with (i) the circuit-theoretic power waves at its $M$ ports and (ii) the spherical power waves sufficiently far away in the radiating structure's far-field region. The former are characterized by the phasor vectors ${\color{pink}\bm{\mathsf{a}}}$ and ${\color{pink}\bm{\mathsf{b}}}$; and the latter by the angular spectrum ${\color{pink}\bm{\mathsf{f}}}^\swarrow$ of the incoming converging wave and the angular spectrum ${\color{pink}\bm{\mathsf{f}}}_2^\nearrow$ of the component of the outgoing diverging wave that is directly induced by the current density ${\color{pink}\bm{\mathsf{J}}}_2$.

Theorems & Definitions (17)

• Definition 1: Radiating Structure
• Remark 1
• Remark 2
• Definition 2: Circuit-Theoretic Power Waves
• Remark 3
• Definition 3: Spherical Power Waves
• Remark 4
• Remark 5
• Remark 6
• Theorem 1: Finite-Rank Representability of Far-Field Interactions