Two dimensional invisibility cloaking for Helmholtz equation and non-local boundary conditions

TL;DR

This paper investigates two-dimensional Helmholtz cloaking with approximate, nonsingular transformations, revealing a non-local boundary condition at the cloak interface caused by the infinite phase velocity near the boundary.

Contribution

It introduces a mathematical analysis of 2D Helmholtz cloaking with truncated transformations, highlighting the emergence of non-local boundary conditions in the limit.

Findings

Non-local boundary condition appears on the inner cloak interface.

Approximate cloaks approach ideal cloaking as truncation parameter varies.

Differences observed between 2D and 3D cloaking behaviors.

Abstract

Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic and quantum waves around a region without penetrating it, so that this region is hidden from external observations. The material pa- rameters used to describe these devices are anisotropic, and singular at the interface between the cloaked and uncloaked regions, making physical realization a challenge. These singular material parameters correspond to singular coefficient functions in the partial differential equations modeling these constructions and the presence of these singularities causes various mathematical problems and physical effects on the interface surface. In this paper, we analyze the two dimensional cloaking for Helmholtz equation when there are sources or sinks present inside the cloaked region. In particular, we consider nonsingular approximate invisibility cloaks based on the truncation of the singular transformations. Using such truncation we analyze the limit when the approximate cloaking approaches the ideal cloaking. We show that, surprisingly, a non-local boundary condition appears on the inner cloak interface. This effect in the two dimensional (or cylindrical) invisibility cloaks, which seems to be caused by the infinite phase velocity near the interface between the cloaked and uncloaked regions, is very different from the earlier studied behavior of the solutions in the three dimensional cloaks.