Statistical model for energy-transfer-induced up-conversion in Er3+-doped glasses

We present a statistical model for studying excitation upconversion and mig ration caused by energy transfer between Er3+ ions in a glass matrix. Assum ing a statistically uniform distribution of ions in the matrix we propose a n approach for averaging the solution of the rate equations over possible e nsemble configurations. The model provides a transcendental equation for th e population inversion and the up-conversion rate in the steady-state case, and an integral equation for delta -pulse excitation. Analytical expressio ns are given for the asymptotes at small and large population inversions, a s well as at the initial time fur 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 caus ed by the migration.