## 1 Introduction
\(R_{\mathit{LIPE}}(t)\), however, differs from the random noise in that the magnitude of time integration of \(R_{\mathit{LIPE}}(t)\) is proportional to the length of the time of integration as is the following:  
\[\int_{t}^{t+\Delta t}R_{\mathit{LIPE}}(t^{\prime})dt=\langle R_{\mathit{LIPE}} \rangle\cdot\Delta t \tag{5}\]  
Now, the acquired photon luminescence practically contains three components as the following:  
\[R_{\mathit{DL/UPE}}^{\mathit{noise}}(t)=R_{\mathit{DL}}(t)+R_{\mathit{LIPE}}( t)+R_{\mathit{noise}}(t) \tag{6}\]  
of which the time integration becomes  
\[I_{\mathit{DL/UPE}}^{\mathit{noise}}(0\Longleftrightarrow t)=\int_{0}^{t}R _{\mathit{DL}}\big{(}t^{\prime}\big{)}dt^{\prime}+\int_{0}^{t}R_{\mathit{LIPE}} \big{(}t^{\prime}\big{)}dt^{\prime}+\langle R_{\mathit{noise}}(tt)\rangle \tag{7}\]  
\[=\int_{0}^{t}R_{\mathit{DL}}(t^{\prime})dt+\langle R_{\mathit{LIPE}}\rangle \cdot t\]  
Then, if the following measurement is available:  
\[I_{\mathit{UPE}}^{\mathit{noise}}(0\Longleftrightarrow t)=\langle R_{ \mathit{LIPE}}\rangle\cdot t, \tag{8}\]  
one may obtain the ultraweak photon emission via time differentiation of Equation (8):  
\[\int_{t}^{t+AI}R_{noise}(t^{\prime})dt=\left\langle R_{noise}(t^{\prime}) \right\rangle=0 \tag{2}\]0  
and similarly, the pure-traced delayed luminescence can be obtained by time differentiation of the difference between Equations (7) and (8):  
\[\int_{t}^{t+AI}R_{noise}(t^{\prime})dt=\left\langle R_{noise}(t^{\prime}) \right\rangle=0 \tag{2}\]1  
Therefore, by taking advantage of the time-integrated photon emission, which can be a combination of random noise, baseline ultraweak photon emission, and delayed luminescence, it may be possible to resolve the time integration of delayed luminescence alone. From the time integration of delayed luminescence alone, the delayed luminescence can be resolved via simple time differentiation. The resulting exclusive delayed luminescence may be assessable at the level of the noise-laden ultraweak photon emission. If we denote that delayed luminescence consists of various phases of decay [3] as the following, then  
\[\int_{t}^{t+AI}R_{noise}(t^{\prime})dt=\left\langle R_{noise}(t^{\prime}) \right\rangle=0 \tag{2}\]2  
where \(\tau_{n}\) denotes the time constant of the n-th phase of the delay if assuming simultaneous initiation by the same irradiation and taking an ascending order according to the magnitude of the time constant, i.e., the slower the decay, the greater the number of n. The integration of Equation (11) leads to the following pattern with multiple shoulders or phases of saturation that will occur for substantially different time constants:  
\[\int_{t}^{t+AI}R_{noise}(t^{\prime})dt=\left\langle R_{noise}(t^{\prime}) \right\rangle=0 \tag{2}\]3  
The time integration of UPE-only will be a linear function of time, whereas the time integration of the delayed luminescence only will reveal patterns of saturation, and the patterns of saturation are likely tapered by the linear increment of the time integration of the spontaneous ultraweak photon emission. Any afterglow of the irradiation introduced to excite delayed luminescence may also leave traces in the time-integrated luminescence.