  We perform a theoretical study of nonlinear optical isolator devices based on
coupled microcavities with gain and loss. Using coupled-mode theory, we derive
a correspondence between the boundary of asymptotic stability in the nonlinear
regime, where gain saturation is present, and the PT-breaking transition in the
underlying linear system. For zero detuning and weak input intensity, the onset
of optical isolation can be rigorously derived, and corresponds precisely to
the PT transition point. When the couplings to the external ports are unequal,
the isolation ratio exhibits an abrupt jump at the transition point, determined
by the ratio of the couplings. This could be exploited to realize an actively
controlled nonlinear optical isolator, in which strong optical isolation can be
switched on or off using tiny variations in the inter-resonator separation.
