  We demonstrate how to tailor the losses of nonlinear cavities in order to
suppress their reflection and enhance their non-reciprocal transmission. We
derive analytical expressions predicting the existence of zero-reflection
channels in single and coupled nonlinear cavities, depending on the driving
frequency and loss rates. While suppressing the reflection from a single cavity
imposes a stringent condition on the input-output leakage rates, we demonstrate
that this condition can be significantly relaxed in systems of coupled
cavities. In particular, zero-reflection and non-reciprocity can be achieved
across a range of driving frequencies in coupled cavities by tuning the output
leakage rate alone. Numerical calculations based on the driven-dissipative
Gross-Pitaevksii equation, usually employed to describe microcavity polaritons,
reveal the spatial phenomenology associated with zero-reflection states and
provide design guidelines for the construction of nonlinear optical isolators.
