Perturbative unidirectional invisibility

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) = n(0) + f (x), where n(0) is real, f (x) is complex valued, and vertical bar f (x)vertical bar << 1 <= n(0). 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 n(0) = 1 may be used to generate a hierarchy of unidirectionally invisible PT-symmetric index profiles with n(0) > 1. The results pertaining to unidirectional invisibility for n(0) > 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) = n(0) + zeta e(iKx), zeta and K real, and vertical bar zeta vertical bar << 1 can support unidirectional invisibility only for n(0) = 1. We then construct unidirectionally invisible index profiles of the form n(x) = n(0) + Sigma(l) z(l)e(iKlx) with z(l) complex, K-l real, vertical bar z(l)vertical bar << 1, and n(0) > 1.