Active exterior cloaking for the 2D Helmholtz equation with complex wavenumbers and application to thermal cloaking

TL;DR

This paper develops a method for active exterior cloaking in 2D Helmholtz equations with complex wavenumbers, enabling cloaking in media with dispersion, loss, and gain, and applies it to thermal cloaking.

Contribution

It extends existing cloaking techniques to complex wavenumbers and provides new bounds for cloaking quality, applicable to various physical media.

Findings

Method works for media with dispersion, loss, and gain

Provides bounds on cloaking effectiveness

Demonstrates cloaking for the heat equation

Abstract

We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same approach works for complex wavenumbers which makes it applicable to a variety of media, including media with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's addition formulas with complex arguments, we obtain new estimates that allow to quantify the quality of the cloaking effect. We illustrate our results by applying them to achieve active exterior cloaking for the heat equation.