We report numerical simulations on a fiber compatible, self-focusing, saturable absorber device. Two fiber ends are separated by a bulk, nonlinear medium. An optical beam transmitted from one tapered fiber end, propagates through the nonlinear medium and couples back into the other tapered fiber end. The fiber mode distributions at the ends of the tapered fibers are calculated using a Finite Difference Method (FDM). We apply the beam propagation method to simulate the diffraction and nonlinearity in the nonlinear medium. As a function of initial beam power and the fiber mode design, the coupling efficiency plots are calculated and compared for different nonlinear mediums. Our simulations identify the optimum contrast between low and high input powers.