Broadband quasistatic passive cloaking: bounds and limitations in the near-field regime

TL;DR

This paper establishes fundamental limits on passive cloaking of dielectric objects in the quasistatic regime, showing that perfect cloaking over a finite frequency range is impossible and providing bounds based on physical parameters.

Contribution

It introduces a new representation theorem for the Dirichlet-to-Neumann map and derives quantitative bounds on cloaking capabilities using variational principles and sum rules for passive systems.

Findings

Passive cloaking cannot be perfect over a finite frequency interval.

Derived bounds depend on size, location, and material properties of the cloak and object.

Limits apply to both lossy and lossless cloaking configurations.

Abstract

We consider here several aspects of the following challenging question: is it possible to use a passive cloak to make invisible a dielectric inclusion on a finite frequency interval in the quasistatic regime of Maxwell's equations for an observer close to the object? In this work, by considering the Dirichlet-to-Neumann (DtN) map, we not only answer negatively this question, but we go further and provide some quantitative bounds on this map that provide fundamental limits to both cloaking as well as approximate cloaking. These bounds involve the following physical parameters: the length and center of the frequency interval, the volume of the cloaking device, the volume of the obstacle, and the relative permittivity of the object. Our approach is based on two key tools: i) variational principles from the abstract theory of composites and ii) the analytic approach to deriving bounds from sum rules for passive systems. To use i), we prove a new representation theorem for the DtN map which allows us to interpret this map as an effective operator in the abstract theory of composites. One important consequence of this representation is that it allows one to incorporate the broad and deep results from the theory of composites, such as variational principles, and to apply the bounds derived from them to the DtN map. These results could be useful in other contexts other than cloaking. Next, to use ii), we show that the passivity assumption allows us to connect the DtN map (as function of the frequency) with two important classes of analytic functions, namely, Herglotz and Stieltjes functions. The sum rules for these functions, combined with the variational approach, allows us to derive new inequalities on the DtN map which impose fundamental limitations on passive cloaking, both exact and approximate, over a frequency interval. We consider both cases of lossy and lossless cloaks.