Analytical solution of the transfer rate equations in LiNbO3:Er3+/Yb3+

The temporal evolution of the energy transfer between Yb3+ and Er3+ ions in lithium niobate (LiNbO3), after selective ytterbium pulsed excitation, is investigated. The numerical integration of the rate equations shows that the temporal behaviour of the energy transfer between the two ions can be split into two stages: a first one dominated by transfer from Yb3+ to Er3+, followed by another characterized by an equilibrium between transfer and back-transfer processes. In this last stage, it is possible to deduce an analytical expression for the relative population and the lifetime of the resonant manifolds, I-4(11/2) (Er3+) and F-2(5/2) (Yb3+). An excellent accordance between calculated lifetimes and those obtained experimentally (after excitation at 920 nm) in samples with different rare-earth concentrations has been observed.