Abstract
Transformation optics constructions have allowed the design of electromagnetic, acoustic and quantum parameters that steer waves around a region without penetrating it, so that the region is hidden from external observations. The material parameters are anisotropic, and singular at the interface between the cloaked and uncloaked regions, making physical realization a challenge. We address this problem by showing how to construct isotropic and nonsingular parameters that give approximate cloaking to any desired degree of accuracy for electrostatic, acoustic and quantum waves. The techniques used here may be applicable to a wider range of transformation optics designs. For the Helmholtz equation, cloaking is possible outside a discrete set of frequencies or energies, namely the Neumann eigenvalues of the cloaked region. At these frequencies or energies the ideal cloak supports trapped states, vanishing outside of the cloaked region; near these energies, an approximate cloak supports almost trapped states. This is in fact a useful feature, and we conclude by giving several quantum mechanical applications.
Highlights
In [4], we showed that the singular cloaking metrics g for electrostatics constructed in [1, 2], giving the same boundary measurements of electrostatic potentials as the Euclidian metric g0 =, cloak with respect to solutions of the Helmholtz equation at any nonzero frequency ω and with any source p
Our analysis is closely related to the inverse problem for electrostatics, or Calderón’s conductivity problem
One measures, for all voltage distributions u|∂ = f on the boundary the corresponding current fluxes, ν · σ ∇u, where ν is the exterior unit normal to ∂. This amounts to the knowledge of the DN map, σ . corresponding to σ, i.e. the map taking the Dirichlet boundary values of the solution to (4) to the corresponding Neumann boundary values, σ : u|∂ → ν· σ ∇u|∂
Summary
Our analysis is closely related to the inverse problem for electrostatics, or Calderón’s conductivity problem. There is a large (infinite-dimensional) family of conductivities which all give rise to the same electrostatic measurements at the boundary This observation is due to Luc Tartar (see [34] for an account.) Calderón’s inverse problem for anisotropic conductivities is the question of whether two conductivities with the same DN operator must be push-forwards of each other. Let = B3; the conductivity σ is a cloaking conductivity on , as it is indistinguishable from σ0, vis-a-vis electrostatic boundary measurements of electrostatic potentials, treated rigorously as bounded, distributional solutions of the degenerate elliptic boundary value problem corresponding to σ [1, 2]. The same construction of σ | −B1 was proposed in Pendry et al [5] to cloak the region B1 from observation by electromagnetic waves at a positive frequency; see Leonhardt [3] for a related approach for Helmholtz in R2
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