Unidirectional invisibility with 3D parity-time symmetric structures

Unidirectional invisibility in parity-time-symmetric systems has been so far predicted in one-dimensional (1D) structures, i.e., for planar slabs that are reflectionless when excited in one direction, while have strong reflections from the other direction. Here we generalize this concept to three-dimensional (3D) systems. Contrary to the 1D case, where only two directions are relevant, in the 3D case the full 4π solid angle needs to be considered, making this scenario much richer and with far more exciting possibilities. Interestingly enough, we prove that the unidirectional invisibility condition derived here for parity-time symmetric systems is also the optimal cloaking condition for incidence from a particular direction. Our analytical theory is developed for a sub-wavelength sphere uniformly surrounded by resonant plasmonic nanoparticles with gain and loss. A method for the realization of such nanoparticles using a core-shell geometry is also presented.