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

TL;DR

This paper introduces a simplified analytical rate-equation model for 1500 nm to 980 nm upconversion in Er$^{3+}$, accurately describing saturation behavior and providing a new metric for assessing material quality.

Contribution

The paper presents a new simplified analytical model for Er$^{3+}$ upconversion that effectively predicts saturation and introduces saturation intensity as a material quality indicator.

Findings

Model accurately reproduces upconversion saturation over wide intensity range.

Saturation intensity correlates with energy transfer and relaxation rates.

Differences in sample performance are mainly due to non-radiative relaxation rates.

Abstract

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.