Exactness of the Born approximation and broadband unidirectional invisibility in two dimensions

Abstract

Achieving exact unidirectional invisibility in a finite frequency band has been an outstanding problem for many years. We offer a simple solution to this problem in two dimensions that is based on our solution to another more basic open problem of scattering theory, namely, finding potentials v(x, y) whose scattering problem is exactly solvable via the first Born approximation. Specifically, we find a simple condition under which the first Born approximation gives the exact expression for the scattering amplitude whenever the wave number for the incident wave is not greater than a given critical value alpha. Because this condition only restricts the y dependence of v(x, y), we can use it to determine classes of such potentials that have certain desirable scattering features. This leads to a partial inverse scattering scheme that we employ to achieve perfect (nonapproximate) broadband unidirectional invisibility in two dimensions. We discuss an optical realization of the latter by identifying a class of two-dimensional isotropic active media that do not scatter incident transverse waves with wave numbers in the range (alpha/root 2, alpha] and source located at x = infinity while scattering the same waves if their source is relocated to x = -infinity.