## 3 Results and Discussion
### Luminescence Properties of YbGAGG: Cr3+  
The absorbance coefficient spectrum of transparent ceramic YbGAGG: Cr3+ is shown in Figure 3a. The shape of the absorbance coefficient (\(\alpha\)) spectrum in the 400-700 nm range aligns with the \({}^{4}\)A\({}_{2}\)-\({}^{4}\)T\({}_{1}\) and \({}^{4}\)A\({}_{2}\)-\({}^{4}\)T\({}_{2}\) absorption bands of Cr3+. On the contrary, at approximately 1000 nm, typical absorption peaks of Yb3+ are observed. Notably, the spectrum only shows absorption bands for Cr3+ and Yb3+, with no Cr4+ signal (at around 1350 nm). These results prove that the oxidation state of chromium in the YbGAGG host is trivalent.  
The orange curve in Figure 3b represents the PL spectrum of the YbGAGG: Cr3+ sample under 660 nm excitation at room temperature. At approximately 1000 nm, the PL band of Yb3+: \({}^{2}\)F\({}_{5/2}\rightarrow{}^{2}\)F\({}_{7/2}\) was observed. The purple curve is the photoluminescence excitation (PLE) spectrum monitoring the PL of Yb3+. Two PLE bands within the 400-700 nm range were attributed to the transition from the \({}^{4}\)A\({}_{2}\) to two excited states of Cr3+: \({}^{4}\)T\({}_{1}\) and \({}^{4}\)T\({}_{2}\)[26], confirming the energy transfer process from Cr3+ to Yb3+. For the PL spectrum (orange curve) in Figure 3b, no PL signal was observed in the 700-800 nm range (where the Cr3+ PL signal is expected). This result is attributed to the excessive concentration of Yb3+, leading to an efficient energy transfer process from Cr3+ to Yb3+.  
Figure 3: (a) Absorption spectrum of YbGAGG: Cr3+ at room temperature; (b) photoluminescence spectrum under 660 nm excitation, and photoluminescence excitation spectrum of Yb3+ from YbGAGG: Cr3+ at room temperature.  
Figure 2: Rietveld refinement of the XRD profiles of YbGAGG: Cr3+, small black circles and the red curve represent the experimental and the calculated values, respectively; vertical bars (green) indicate the position of Bragg peaks. The blue bottom trace depicts the corresponding residuals between the experimental and the calculated intensity values.  
To further investigate the energy transfer process from Cr\({}^{3+}\) to Yb\({}^{3+}\), a luminescence lifetime measurement of Cr\({}^{3+}\) was performed and shown in Figure 4. The PL decay of both YbGAGG: Cr\({}^{3+}\) and GAGG: Cr\({}^{3+}\) (preparation method is the same as for YbGAGG: Cr\({}^{3+}\)) in the range of 600-800 nm were measured under 480 nm excitation. The obtained decay plots were fitted using the following double exponential function:  
\[I=A_{1}\times exp(-t/\tau_{1})+A_{2}\times exp(-t/\tau_{2}) \tag{1}\]  
After fitting the decay curve with a function and obtaining the parameters, the lifetime is calculated using the following formula:  
\[\tau_{ave}=\frac{A_{1}\tau_{1}^{2}+A_{2}\tau_{2}^{2}}{A_{1}\tau_{1}+A_{2}\tau_{ 2}} \tag{2}\]  
Using Formula (2), the lifetimes of the decay curves for YbGAGG:Cr\({}^{3+}\) and GAGG:Cr\({}^{3+}\) were calculated as \(\tau_{\text{CrYb}}\) = 3.86 ms and \(\tau_{\text{Cr}}\) = 0.122 ms, respectively. Substituting these values into the following formula, the energy transfer efficiency was calculated to be \(\eta\) = 96.8%:  
\[\eta=1-\tau_{\text{CrYb}}/\tau_{\text{Cr}} \tag{3}\]