We show that in one dimension the transfer matrix $\mathbf{M}$ of any scattering potential $v$ coincides with the $S$ matrix of an associated time-dependent non-Hermitian $2\times 2$ matrix Hamiltonian $\mathbf{H}\left(\tau \right)$. If $v$ is real valued, $\mathbf{H}\left(\tau \right)$ is pseudo-Hermitian and its exceptional points correspond to the classical turning points of $v$. Applying time-dependent perturbation theory to $\mathbf{H}\left(\tau \right)$ we obtain a perturbative series expansion for $\mathbf{M}$ and use it to study the phenomenon of unidirectional invisibility. In particular, we establish the possibility of having multimode unidirectional invisibility with wavelength-dependent direction of invisibility and construct various physically realizable optical potentials possessing this property. We also offer a simple demonstration of the fact that the off-diagonal entries of the first-order Born approximation for $\mathbf{M}$ determine the form of the potential. This gives rise to a perturbative inverse scattering scheme that is particularly suitable for optical design. As a simple application of this scheme, we construct an infinite-range unidirectionally invisible potential.

• Received 6 November 2013

DOI:https://doi.org/10.1103/PhysRevA.89.012709