Isotropic transformation optics: approximate acoustic and quantum cloaking

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. For the frequencies or energies corresponding to the Neumann eigenvalues of the cloaked region, the ideal cloak supports trapped states; 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. ∗Department of Mathematics, University of Rochester, Rochester, NY 14627 †Department of Mathematical Sciences, University College London, Gower Str, London, WC1E 6BT, UK ‡Institute of Mathematics, Helsinki University of Technology, FIN-02015, Finland §Department of Mathematics, University of Washington, Seattle, WA 98195 ¶Authors listed in alphabetical order. AG and GU are supported by US NSF, ML by Academy of Finland and YK by UK EPSRC. 1 ar X iv :0 80 6. 00 85 v2 [ ph ys ic s. op tic s] 1 A ug 2 00 8