Abstract
We present a statistical model for studying excitation upconversion and migration caused by energy transfer between ${\mathrm{Er}}^{3+}$ ions in a glass matrix. Assuming a statistically uniform distribution of ions in the matrix we propose an approach for averaging the solution of the rate equations over possible ensemble configurations. The model provides a transcendental equation for the population inversion and the up-conversion rate in the steady-state case, and an integral equation for \ensuremath{\delta}-pulse excitation. Analytical expressions are given for the asymptotes at small and large population inversions, as well as at the initial time for the dynamic case. The model explains well the experimentally observed nonlinear dependence of the up-conversion rate versus population inversion, and the enhancement of the up-conversion caused by the migration.
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