Fundamental limitations of the McCumber relation applied to Er-doped silica and other amorphous-host lasers

The approximate McCumber procedure is often used to predict the emission cross-section spectrum of the 1.5-μm transition of Er-doped glass fibers from the transition´s measured absorption spectrum. By applying this procedure to a large number of published Er-doped fiber absorption spectra, we demonstrate that its accuracy is actually statistically quite low: it tends to overestimate the peak cross-section (by up to 75%) and predicts an emission spectrum that is erroneously depressed in the S band (below ∼1530 nm) and inflated in the C and L bands. Error levels are substantial and yield unacceptably large errors when modeling Er-doped fiber devices. We provide analytic evidence that this failure is rooted in part in the approximations inherent to the procedure, and in part in a fundamental limitation of the underlying McCumber relation. Specifically, when applied to broad optical transitions, the McCumber relation yields poor predictions of the emission cross-section spectral shape, the error worsening in the L and S bands, with increasing homogeneous broadening, and with increasing bandwidth. The McCumber relation should be avoided for broad laser transitions, which includes most rare-earth transitions in many amorphous hosts.