Stochastic Model of Energy-Transfer Processes Among Rare-Earth Ions. Example of Al2O3:Tm3+

Energy-transfer processes strongly affect the performance of lanthanide-doped photonic devices. In this work, we introduce a simple stochastic model of energy-transfer processes and successfully apply it to the example of cross-relaxation (CR) and energy-transfer upconversion (ETU) in amorphous Al2O3:Tm3+ waveguides on silicon intended for lasers operating at similar to 2 mu m. The stochastic model is based on the rate-equation formalism and considers two spectroscopically distinct ion classes, namely single ions and ions with neighbors (pairs and clusters), with the corresponding ion fractions being dependent on the doping concentration. We prove that a more accurate description of the luminescence properties of amorphous Al2O3:Tm3+ is obtained when accounting for the presence of these distinct ion classes. Based on the developed model, we derive microscopic CR and ETU parameters of C-CR = 5.83 X 10(-38) cm(6) s(-1), C-ETU1 = 0.93 x 10(-40) cm(6) s(-1), and C-Eru2 = 7.81 x 10(-40) cm(6) s(-1), and determine the laser quantum efficiency eta(q) of excitation of Tm3+ ions in the upper laser level. For the maximum Tm3+ concentration of 5.0 X 10(20) cm(-3) studied experimentally in this investigation, eta(q) reaches 1.73. Furthermore, the transition cross sections at the pump and laser wavelengths are determined. For the H-3(6) -> F-3(4) transition, the maximum stimulated-emission cross section is sigma(e) = 0.47 X 10(-20) cm(2) at 1808 nm.