1：Experimental Design and Method Selection:

The study involved measuring thermal conductivity, thermal expansion coefficient, and specific heat at constant pressure from 77 to 298 K. Spectroscopic properties and fluorescence lifetimes were characterized at 77, 150, and 298 K. The stimulated emission cross sections were determined using the Füchtbauer-Ladenburg (FL) formula.

2：Sample Selection and Data Sources:

Samples included

3：38 at.% Yb-doped GAGG, 74 at.% Nd-doped GAGG, and 9 at.% Nd-doped GGG crystals. List of Experimental Equipment and Materials:

A physical property measurement system (PPMS-14 and PPMS-9) for specific heat and thermal conductivity measurements, a thermal dilatometer (Linseis L75) for thermal expansion coefficient measurements, and a fluorescence spectrometer (Edinburgh FLSP980) for spectroscopic characterization.

4：Experimental Procedures and Operational Workflow:

Thermal properties were measured using steady-state longitudinal heat-flow method and thermal expansion was calculated using the average linear thermal expansion coefficients formula. Spectroscopic measurements included absorption and fluorescence spectra upon excitation by a diode laser.

5：Data Analysis Methods:

Absorption cross sections were obtained using Lambert-Beer’s law, and emission cross-sections were calculated using the FL equation.