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We give explicit criteria for the reflectionlessness, transparency, andinvisibility of a finite-range potential in the presence of an incoherent(intensity-dependent) nonlinearity that is confined to the range of thepotential. This allows us to conduct a systematic study of the effects of sucha nonlinearity on a locally periodic class of finite-range potentials thatdisplay perturbative unidirectional invisibility. We use our general results toexamine the effects of a weak Kerr nonlinearity on the behavior of thesepotentials and show that the presence of nonlinearity destroys theunidirectional invisibility of these potentials. If the strength of the Kerrnonlinearity is so weak that the first-order perturbation theory is reliable,the presence of nonlinearity does not affect the unidirectionalreflectionlessness and transmission reciprocity of the potential. We show thatthe expected violation of the latter is a second order perturbative effect.

Unidirectional Reflection and Invisibility in Nonlinear Media with an Incoherent Nonlinearity