Perturbative unidirectional invisibility

Abstract

We outline a general perturbative method of evaluating scattering features of finite-range complex potentials and use it to examine complex perturbations of a rectangular barrier potential. In optics, these correspond to modulated refractive index profiles of the form n(x)=n0+f(x), where n0 is real, f(x) is complex valued, and |f(x)| 1≤n0. We give a comprehensive description of the phenomenon of unidirectional invisibility for such media, proving five general theorems on its realization in PT-symmetric and non-PT-symmetric material. In particular, we establish the impossibility of unidirectional invisibility for PT-symmetric samples whose refractive index has a constant real part and show how a simple scaling transformation of a unidirectionally invisible PT-symmetric index profile with n0=1 may be used to generate a hierarchy of unidirectionally invisible PT-symmetric index profiles with n0>1. The results pertaining to unidirectional invisibility for n0>1 open the way for the experimental studies of this phenomenon in a variety of active materials. As an application of our general results, we show that a medium with n(x)=n0+ζeiKx, ζ and K real, and |ζ| 1 can support unidirectional invisibility only for n0=1. We then construct unidirectionally invisible index profiles of the form n(x)=n0+eiKx with z complex, K real, z 1, and n0>1.