Transfer matrices as nonunitary S matrices, multimode unidirectional invisibility, and perturbative inverse scattering

We show that in one dimension the transfer matrix M of any scattering potential v coincides with the S matrix of an associated time-dependent non-Hermitian 2 x 2 matrix Hamiltonian H(tau). If v is real valued, H(tau) is pseudo-Hermitian and its exceptional points correspond to the classical turning points of v. Applying time-dependent perturbation theory to H(tau) we obtain a perturbative series expansion for 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 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.