Cloaked electromagnetic, acoustic, and quantum amplifiers via transformation optics.

The advent of transformation optics and metamaterials has made possible devices producing extreme effects on wave propagation. Here we describe a class of invisible reservoirs and amplifiers for waves, which we refer to as Schrödinger hats. The unifying mathematical principle on which these are based admits such devices for any time harmonic waves modeled by either the Helmholtz or Schrödinger equation, e.g., polarized waves in electromagnetism, acoustical waves and matter waves in quantum mechanics. Schrödinger hats occupy one part of a parameter-space continuum of wave-manipulating structures which also contains standard transformation optics based cloaks, resonant cloaks and cloaked sensors. Possible applications include near-field quantum microscopy.

Cloaking a sensor for three-dimensional Maxwell's equations: transformation optics approach.

The ideal transformation optics cloaking is accompanied by shielding: external observations do not provide any indication of the presence of a cloaked object, nor is any information about the fields outside detectable inside the cloaked region. In this paper, a transformation is proposed to cloak three-dimensional objects for electromagnetic waves in sensor mode, i.e., cloaking accompanied by degraded shielding. The proposed transformation tackles the difficulty caused by the fact that the lowest multipole in three-dimensional electromagnetic radiation is dipole rather than monopole. The loss of the surface impedance of the sensor plays an important role in determining the cloaking modes: ideal cloaking, sensor cloaking and resonance.

Cloaking a sensor via transformation optics.

Ideal transformation optics cloaking at positive frequency, besides rendering the cloaked region invisible to detection by scattering of incident waves, also shields the region from those same waves. In contrast, we demonstrate that approximate cloaking permits a strong coupling between the cloaked and uncloaked regions; careful choice of parameters allows this coupling to be amplified, leading to effective cloaks with degraded shielding. The sensor modes we describe are close to but distinct from interior resonances, which destroy cloaking. As one application, we describe how to use transformation optics to hide sensors in the cloaked region and yet enable the sensors to efficiently measure incident waves on the exterior of the cloak, an effect similar to the plasmon-based approach of Alù and Engheta.

Determination of second-order elliptic operators in two dimensions from partial Cauchy data.

We consider the inverse boundary value problem in two dimensions of determining the coefficients of a general second-order elliptic operator from the Cauchy data measured on a nonempty arbitrary relatively open subset of the boundary. We give a complete characterization of the set of coefficients yielding the same partial Cauchy data. As a corollary we prove several uniqueness results in determining coefficients from partial Cauchy data for the isotropic conductivity equation, the Schrödinger equation, the convection-diffusion equation, the anisotropic conductivity equation modulo a group of diffeomorphisms that are the identity at the boundary, and the magnetic Schrödinger equations modulo gauge transformations. The key step is the construction of novel complex geometrical optics solutions using Carleman estimates.

Approximate quantum cloaking and almost-trapped states.

We describe potentials which act as approximate cloaks for matter waves. These potentials are derived from ideal cloaks for the conductivity and Helmholtz equations. At most energies E, if a potential is surrounded by an approximate cloak, then it becomes almost undetectable and unaltered by matter waves originating externally to the cloak. For certain E, however, the approximate cloaks are resonant, supporting wave functions almost trapped inside the cloaked region and negligible outside. Applications include dc or magnetically tunable ion traps and beam switches.

Electromagnetic wormholes and virtual magnetic monopoles from metamaterials.

We describe new configurations of electromagnetic (EM) material parameters, the electric permittivity epsilon and magnetic permeability micro, which allow one to construct devices that function as invisible tunnels. These allow EM wave propagation between the regions at the two ends of a tunnel, but the tunnels themselves and the regions they enclose are not detectable to lateral EM observations. Such devices act as wormholes with respect to Maxwell's equations and effectively change the topology of space vis-à-vis EM wave propagation. We suggest several applications, including devices behaving as virtual magnetic monopoles, invisible cables, and scopes for MRI-assisted surgery.

Anisotropic conductivities that cannot be detected by EIT.

We construct anisotropic conductivities in dimension 3 that give rise to the same voltage and current measurements at the boundary of a body as a homogeneous isotropic conductivity. These conductivities are non-zero, but degenerate close to a surface inside the body.