## 3 Results and Discussions
### Time-Integrated Acquisitions at Four Conditions of the Irradiation-Sample Complex
The medium trace marked by the green diamonds of the medium size represents the integration of the afterglow irradiance of the illumination-delivery fiber. As the time of integration increases, this trace starts to ascend noticeably at 0.1 s and gradually merges with the bottom trace. The merging of the ascending medium trace with the ascending bottom trace below indicates that the slope of the medium trace was smaller than that of the bottom trace. The top trace marked by the red circles of the largest size represents the integration of the delayed luminescence of the yeast in the presence of the afterglow of the illumination-delivery fiber. This top trace ascends earlier than the medium trace and remains above the medium trace over the entire range of time. The time differentiation of the time-integrated illuminance of a trace shown in Figure 8A would then correspond to the delayed luminescence at the time of the end of the duration of the integration. The time differentiation results of the three traces of Figure 8A are presented in Figure 8B, with the addition of model-fitting of the respective traces over the time range of 10 us to 100 s. The three traces were found to be fitted to the following model:  
\[R(t)=A\cdot t^{-b}\cdot exp\left(-\frac{t}{\tau}\right)+\Omega \tag{13}\]  
The three traces at the descending order, when counted from the top, were fitted by the following models in the form of Equation (13) that gave a goodness-of-fit of 0.92, 0.93, and 0.95, respectively:  
\[R_{DL}(t)=27.48\cdot t^{-0.75}\cdot exp\left(-\frac{t}{407.89}\right)+0.48 \tag{14}\]  
\[R_{AG}(t)=6.32\cdot t^{-0.90}\cdot exp\left(-\frac{t}{977.90}\right)+0.48 \tag{15}\]  
\[R_{UPE}(t)=5.14\cdot t^{-0.98}\cdot exp\left(-\frac{t}{15.32}\right)+0.50 \tag{16}\]  
We note that none of the three traces received a good fit using the combination of simple exponential functions indicated by Equation (11). The model form of Equation (13)  
Figure 8: (**A**) The difference between each of the three non-baseline traces of Figure 7 and the baseline trace. (**B**) The time differentiation of the three traces shown in (**A**) with the addition of model fit. with which the traces received reasonably good fits contains a hyperbolic function that is multiplied by an exponentially decaying function. Since the hyperbolic function can be expanded to a series of exponentially decaying functions, the model form of Equation (13) is effectively the summation of a series of exponentially decaying functions differing in the rate of temporal decay. This decomposition of a hyperbolic decay trace of a delayed luminescence to multiple phases of exponential decay has been theorized in [3]. One can thus expect that the pattern of the later phase of the luminescence, as depicted in Figure 8B, will be enveloped by an exponential decay with the common or the slowest time constant of the model form of Equation (13). That common time constant may be referred to as a terminal-phase time constant. The following observations may be made regarding Figure 8B. The bottom-most trace in Figure 8B rests at an apparent residual level of 0.50 (A.U.) after an initial decay of a terminal-phase time constant of 15.32 s. This trace would correspond to the spontaneous photon emission of the yeast when reaching a steady residual value. Comparatively, the medium trace grossly resembles the bottom trace; however, the medium trace reveals a much slower decay with a terminal-phase time constant of 977.90 s and a terminal or residual level not greater than the residual level of the bottom trace. This would correspond to the afterglow of the fiber and the background glow of the chamber excited by the prior fiber-delivered irradiation containing an ultraviolet band. The topmost trace in Figure 8B reveals a decay of a terminal-phase time constant of 407.89 s that descends to the level of the residual value of the bottom-most trace. This topmost trace would correspond to the delayed luminescence of the yeast.