406/473 nm Pump-Band Absorption Cross Sections and Derivative-Based Line-Shape Descriptors in Er³⁺/Ho³⁺:Y₃Ga₅O₁₂

We establish a general, device-oriented procedure to extract absolute pump-band metrics from room-temperature UV–Vis (ultraviolet–visible) absorbance—including the absorption coefficient α(λ), per-active-ion cross-section σ_eff(λ), the effective per-active-ion absorption cross section σ_eff(λ) and derivative-based line-shape descriptors . As a representative case study, the procedure is applied to nanocrystalline Er³⁺/Ho³⁺:Y₃Ga₅O₁₂ over the 350–700 nm spectral range. After baseline correction and line-shape inspection assisted by the numerical second derivative of the absorbance, we extract conservative peak positions and the full width at half maximum across the visible 4f–4f manifolds. At the technologically relevant pump wavelengths near 406 nm (Er-addressing) and 473 nm (Ho-addressing) bands, resulting absorption coefficients are α = 0.313 ± 0.047 cm⁻¹ and α = 0.472 ± 0.071 cm⁻¹, respectively. The corresponding per-active-ion σ_eff of (3.62 ± 0.54) × 10⁻²² cm² and (5.46 ± 0.82) × 10⁻²² cm², referenced to the measured optical path length L = 0.22 ± 0.03 mm (approximately 15% propagated relative uncertainty; explicit 1/L rescaling). Cross sections are reported per total active-ion density (Er³⁺ + Ho³⁺). The spectra exhibit Stark-type substructure only partially resolved at room temperature; the second derivative highlights hidden components, and we report quantitative descriptors (component count, mean spacing, curvature-weighted prominence, and pump detuning) that link line-shape structure to absolute pump response. These device-grade metrics enable rate-equation modelling (pump thresholds, detuning tolerance), optical design choices (path length, single/multi-pass or cavity coupling), and host-to-host benchmarking at 295 K. The procedure is general and applies to any rare-earth-doped material given an absorbance spectrum and path length.