Cloaking via change of variables in electric impedance tomography

Abstract

A recent paper by Pendry et al (2006 Science 312 1780-2) used the coordinate invariance of Maxwell's equations to show how a region of space can be 'cloaked'-in other words, made inaccessible to electromagnetic sensingby surrounding it with a suitable (anisotropic and heterogenous) dielectric shield. Essentially the same observation was made several years earlier by Math. Res. Lett. 10 685-93, 2003 in the closely related setting of electric impedance tomography. These papers, though brilliant, have two shortcomings: (a) the cloaks they consider are rather singular; and (b) the analysis by Greenleaf, Lassas and Uhlmann does not apply in space dimension n = 2. The present paper provides a fresh treatment that remedies these shortcomings in the context of electric impedance tomography. In particular, we show how a regular near-cloak can be obtained using a nonsingular change of variables, and we prove that the change-of-variable-based scheme achieves perfect cloaking in any dimension n 2.

Cloaking of Small Objects by Anomalous Localized Resonance

“…In order to quantify the term invisibility, a process of measurement is defined. This can be done in the framework of geometrical optics [15] or of partial differential equations, Maxwell equations [8], wave and Helmholtz equations, or electrostatic equations in the context of impedance tomography [12]. In such a setting, the cloaking device and the cloaked object can be described by coefficients in the equation.…”

“…A cloak is then defined by a fixed choice of coefficients. In the change of variables method [12] the construction exploits that, in a reference domain with constant coefficients, a small subset with changed coefficients has only a small effect in measurements. After a change of variables, the small subset is a large subset in the physical domain and the new coefficients have extreme values.…”

Inverse problems: seeing the unseen

“…(See [112] for a similar calculation.) The Euclidian conductivity δ jk in N 2 (130) could be replaced by any smooth conductivity bounded from below and above by positive constants.…”

“…The result above was proven in [64,65] for the case of dimension n ≥ 3. The same basic construction works in the two dimensional case [112]. Figure 4 portrays an analytically obtained solution on a disc with conductivity σ .…”

“…Kohn, Shen, Vogelius and Weinstein [112] in an interesting article have considered the case when instead of blowing up a point one stretches a small ball into the cloaked region. In this case the conductivity is non-singular and one gets "almost" invisibility with a precise estimate in terms of the radius of the small ball.…”