Mapping the free-space propagation of light to inhomogeneous non-Hermitian media

Waves typically behave very differently when propagating through a homogeneous medium like free space or through an inhomogeneous medium like a complex dielectric structure. Here we present the surprising result that any wave solution of two- dimensional free space can be mapped to a solution inside a suitably designed non-Hermitian potential landscape such that both solutions share the same spatial distribution of their wave intensity. The mapping we introduce here is broadly applicable and can be used as a design protocol for a special class of non-Hermitian media through which specific incoming waves can propagate in an undistorted way. We illustrate this concept through the example of a beam that maintains its Gaussian shape while passing through a randomly assembled distribution of scatterers with gain and loss.