For modeling the performance of fiber amplifiers made from rare-earth-doped fibers, so-called Giles parameters are often used. These comprise two wavelength-dependent quantities: the absorption coefficient α(λ) of the fiber with all laser-active ions in the ground state, and the gain coefficient g*(λ) for the fiber with all laser-active ions in the upper laser level. (The star may be interpreted as indicating the fully excited fiber.)
These equations are based on the assumption that the doping concentration is constant within some volume and zero outside it. However, they can be easily generalized for smooth doping profiles.
Of course, the equations are based on the assumption that only the laser transition contributes to gain or loss in the considered wavelength range. Parasitic background losses due to absorption and scattering in the fiber can be treated separately. Possible additional effects from excited-state absorption should be kept in mind.
In practice, it is often difficult to precisely determine the dopant concentration ndop, the overlap coefficients Γ(λ) and the transition cross sections of a fiber. However, the Giles parameters can be obtained directly from measurements of absorption and gain. (A difficulty for the gain, however, is that one may not achieve full excitation of the laser-active ions.) Amplifier models may then directly be based on the Giles parameters rather than on the not precisely known values of ndop, Γ(λ), σabs(λ) and σem(λ).
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|||C. R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers”, IEEE J. Lightwave Technol. 9 (2), 271 (1991), doi:10.1109/50.65886|