Analytical solutions to rate equations including losses describing threshold pumped fiber lasers

An exact analytical expression to the threshold pump power has been deduced for rate equations including loss coefficients applicable to fiber lasers doped with quasi two-energy-level ions. Following the derivation of a closed-form expression for the variation of the pump power along the fiber, together with analytical solutions to the position dependent atom population density and laser power in terms of the pump power, we have analytically and fully solved rate equations describing threshold pumped fiber lasers. A comparison has also been made between the exact analytical result with the previous reported results, which indicates that various approximation have underestimated the threshold pump powers.