Closed-form expressions for the gain in three- and four-level laser fibers

Exact and approximate theoretical models of the small-signal single-pass gain in single-mode three-level laser fibers are presented. These models are relevant to the behavior of rare-earth-doped (in particular Er³⁺) silica-based fibers. They account for the effects of both the pump and signal-mode structures and pump excited-state absorption (ESA), the latter being present for some pump wavelengths in Er-doped silica fibers. By relying on the modal overlap integral formalism for four-level lasers, relatively simple closed-form expressions for the pump absorption and the gain are derived. These expressions are also applicable to four level laser fibers by a simple adjustment of some of the model parameters