[Paper Review] Approximate cloaking for electromagnetic waves via transformation optics: cloaking vs infinite energy

Hoài-Minh Nguyên, Loc Tran|arXiv (Cornell University)|Nov 1, 2018

Metamaterials and Metasurfaces Applications28 references3 citations

TL;DR

This paper investigates approximate electromagnetic cloaking via transformation optics without a lossy layer, showing that cloaking is achievable even when energy inside the cloaked region blows up in the resonant case. It establishes that the degree of visibility is of order $\rho^3$, improving upon prior $\rho^2$ estimates, and reveals that field behavior depends critically on source compatibility with the system's resonant structure.

ABSTRACT

We study the approximate cloaking via transformation optics for electromagnetic waves in the time harmonic regime in which the cloaking device {\it only} consists of a layer constructed by the mapping technique. Due to the fact that no-lossy layer is required, resonance might appear and the analysis is delicate. We analyse both non-resonant and resonant cases. In particular, we show that the energy can blow up inside the cloaked region in the resonant case and/whereas cloaking is {\it achieved} in {\it both} cases. Moreover, the degree of visibility {\it depends} on the compatibility of the source inside the cloaked region and the system. These facts are new and distinct from known mathematical results in the literature.

Motivation & Objective

• To analyze approximate cloaking for electromagnetic waves using a regularized transformation without a lossy layer, avoiding singularities and practical fabrication issues.
• To investigate the interplay between resonance, energy blow-up, and cloaking performance in the absence of damping.
• To derive optimal estimates for the degree of visibility in both non-resonant and resonant cases.
• To establish the optimality of the $\rho^3$ convergence rate for visibility, improving on prior $\rho^2$ bounds.
• To clarify how source compatibility with the system's resonant modes affects field behavior and invisibility.

Proposed method

• The authors employ a regularized transformation that maps a small ball of radius $\rho$ to the cloaked region, avoiding singularities.
• They analyze the Maxwell system in the time-harmonic regime using asymptotic analysis and separation of variables in spherical coordinates.
• Key equations involve spherical Bessel and Hankel functions to describe field behavior in the cloaked and surrounding regions.
• The method relies on integral representations and trace estimates in negative Sobolev norms to control field behavior across the cloak interface.
• Resonance is analyzed via the behavior of the polarization tensor and the compatibility of the source with the system's eigenmodes.
• Optimality of the convergence rate is proven using asymptotic expansions and lower bounds on field energy in annular regions.

Experimental results

Research questions

• RQ1 Can approximate cloaking be achieved without a lossy layer in transformation optics, despite potential resonance?
• RQ2 What happens to the energy inside the cloaked region when resonance occurs, and how does it affect cloaking performance?
• RQ3 How does the compatibility of the internal source with the system's resonant modes influence the degree of visibility?
• RQ4 Is the $\rho^3$ convergence rate for visibility optimal, and how does it compare to previous $\rho^2$ estimates?
• RQ5 What is the structure of the electromagnetic fields inside the cloaked region in the resonant case with finite energy?

Key findings

• Cloaking is achieved in both non-resonant and resonant cases, even when the energy inside the cloaked region blows up as $\rho \to 0$.
• In the resonant case with incompatible source, the energy inside the cloaked region can diverge, while cloaking still holds.
• The degree of visibility is of order $\rho^3$ for both non-resonant and resonant cases when no source is present inside the cloaked region.
• For a fixed lossy layer in the non-resonant case, the visibility is $\rho^3$, improving upon the prior $\rho^2$ estimate.
• In the resonant case with finite energy, the internal fields satisfy a non-local structure due to mode coupling.
• The $\rho^3$ convergence rate is optimal, as shown by constructing examples where visibility is bounded below by $C\rho$ in the incompatible resonant case.