Rate equation analysis and energy transfer of Er³⁺-doped Ga ₂S₃-GeS₂-La₂S₃ glasses

Decay curves of 1.5 μm (⁴I_13/2) and 1.0 μm (⁴I_11/2) emissions from Er³⁺ ions in Ga ₂S₃-GeS₂-La₂S₃ glasses were measured by the excitation of a 1.5 μm and a 980 nm laser diode pump. The decay curves were simulated using rate equations for a model consisting of six levels of the Er³⁺ from the ground state. The rate equations contained five unknown parameters, i.e., excited state absorption cross sections of ⁴I_13/2 → ⁴I_9/2 (σ_esa24) for 1.5 μm excitation and ⁴I _11/2 → ⁴F_7/2 (σ_esa36) for 980 nm excitation, cooperative upconversion coefficients of ( ⁴I_13/2, ⁴I_13/2) → ( ⁴I_9/2, ⁴I_15/2) (C₂) and (⁴I_11/2, ⁴I_11/2) → ( ⁴F_7/2, ⁴I_15/2) (C₃), and an energy back-transfer coefficient of (⁴I_9/2, ⁴I_15/2) → (⁴I_13/2, ⁴I_13/2) (C_2r). These parameters were estimated by fitting the simulated curves to the measured ones. The obtained values were σ_esa24 = 0.3 × 10^-20 cm², σ_esa36 = 0.5 × 10^-20 cm², C ₂ = 30-40 × 10^-18 cm³/s, C₃ = 20-25 × 10^-18 cm³/s, and C_2r = 5 × 10^-18 cm³/s.