A simple regenerative fiber amplifier with a nonlinear refractive index is described and analyzed. Using standard equations of operation, we model the circulating intensity within the resonant cavity and determine the nonlinear phase response to changes in this intensity over the entire range of initial cavity detuning. Simple stability criteria are established and described. It is shown that the presence of sufficient nonlinearity leads to stable regions of operation with consistent phase-intensity relationships. Based on these results, the phase of the amplifier output is predicted relative to the injected field for the entire range of initial detuning