Analytical model for the intensity dependence of 1500 nm to 980 nm upconversion in Er$^{3+}$: a new tool for material characterization

We propose a simplified rate-equation model for the 1500 nm to 980 nm upconversion in Er$^{3+}$. The simplifications, based on typical experimental conditions as well as on conclusions based on previously published more advanced models, enable an analytical solution of the rate equations, which reproduces known properties of upconversion. We have compared the model predictions with intensity-dependent measurements on four samples with different optical properties, such as upconversion-luminescence yield and the characteristic lifetime of the $^4I_{13/2}$ state. The saturation of the upconversion is in all cases well-described by the model over several orders of magnitude in excitation intensities. Finally, the model provides a new measure for the quality of upconverter systems based on Er$^{3+}$ -- the saturation intensity. This parameter provides valuable information on upconversion parameters such as the rates of energy-transfer upconversion and cross-relaxation. In the present investigation, we used the saturation intensity to conclude that the differences in upconversion performance of the investigated samples are mainly due to differences in the non-radiative relaxation rates.