I. Introduction
Rare earth doped fiber lasers (RDFLs) have become an important light sources in many areas [1]–[6], and many researchers have devoted to characterizing this kind of lasers and obtained a rich variety of useful results. In present theoretical studies, the rate equations (REs) are still the most widely used method to predict the laser characteristics, and many attempts have been tried to extract analytical results from these nonlinear equations [7]–[11]. Some useful analytical expressions have been derived till now from these REs after taking certain justified approximations based on experimental considerations. However the analytical expression even for the threshold pump power has not been derived yet without these approximations.