We describe a general numerical method for calculating short-pulse laser propagation in rare-earth-doped materials,
which are very important as gain materials for solid-state lasers, fiber lasers and optical amplifiers. The split-step, finite
difference method simultaneously calculates changes in the laser pulse as it propagates through the material and
calculates the dynamic populations of the rare-earth energy levels at any position within the material and for times during
and after the laser pulse has passed through the material. Many traditional theoretical and numerical analyses of laser
pulse propagation involve approximations and assumptions that limit their applicability to a narrow range of problems.
Our numerical method, however, is more comprehensive and includes the processes of single- and multi-photon
absorption, excited state absorption (ESA), energy transfer, upconversion, stimulated emission, cross relaxation,
radiative relaxation and non-radiative relaxation. In the models, the rare-earth dopants can have an arbitrary number of
energy levels. We are able to calculate the electron population density of every electronic level as a function of, for
example, pulse energy, dopant concentration and sample thickness. We compare our theoretical results to published
experimental results for rare-earth ions such as Er3+, Yb3+, Tm3+ and Ho3+.