# Spectral Separation of Up-Conversion Luminescence Processes
## 4 Results  
Here, we demonstrate the spectral components separation for four spectral regions discussed in the previous Section, namely 1-4. In Figure 3, it is illustrated why the choice of the approach Equations (5)-(7) is reasonable for our case. Figure 3 shows the dependence of UCL J(\(\lambda,I\)) on pump power I in log-log scale for several wavelengths corresponding to \(\lambda=365\) nm, 350 nm, 338 nm, and 370 nm and marked, respectively, with dots, squares, diamonds, and triangles. One can see that even in the region of overlapping spectral lines, it is rather straightforward to estimate the log-log derivatives as in Equation (6). Nevertheless, due to experimental imperfections estimation of log-log derivatives in a close vicinity of particular pump power for different wavelengths might give considerably different results. To moderate effects of experimental imperfections, for estimating derivatives Equation (5), we used averaging over the sufficiently large regions of variation in pump power near each used pump value: for I\({}_{1}=980\) mW, it is from 750 mW to 1250 mW; for I\({}_{2}=1435\) mW, it is from 1250 mW to 1550 mW; and for I\({}_{3}=1825\) mW, it is from 1250 mW to 1830 mW. That is, as can be seen in Figure 3, to calculate derivatives, in each interval (say, 1-2, 2-3), the dependence of I\(\eta\) J(\(\lambda,I\)) on I I was approximated with a linear dependence; the intervals were chosen in such a way as to provide for a good linear fit in all considered experimental cases. In Figure 4, the procedure of the spectral components separation is demonstrated for the spectral interval 335-380 nm. The solid line in Figure 4 corresponds to the measured total up-conversion signal power J normalized to its largest value J\({}_{max}\) at the pump power 1825 mW. To demonstrate the possibility of self-testing, we use three values the pump power marked by the numbered vertical lines in Figure 3. Large dots, circles, and stars show the estimated \(n(\lambda,I_{j})\) with Equation (5) for these pump power values (I\({}_{1,2,3}\)). Notice that decrease in the average values of \(n(\lambda,I_{j})\) for all \(\lambda\) with the growing pump power I\({}_{j}\) probably indicates UCL saturation [16]. However, due to relatively low concentration (0.1%) of Tm\({}^{3+}\) ions, the effects of cross-relaxation exchange between Tm\({}^{3+}\) ions are not observed. Thus, the factorization condition (2) seems holding and the spectral inference results are close for all the considered intensities. In Figure 4, small dots, circles, and stars for each component spectral shapes (denoted as \(\rm J_{1,2}\) in Figure 4) demonstrate the results of the inference described by Equation (7) for the \(n(\lambda,\rm I_{j})\) shown by large dots for \(\rm I_{1}\), circles for \(\rm I_{2}\), and stars for \(\rm I_{3}\). Figure 4: The spectral separation in the interval 338–380 nm. The solid line corresponds to the measured total up-conversion signal J normalized to its largest value \(\rm J_{max}\); this signal is for the pump power 1825 mW. Large dots, circles, and stars show \(n(\lambda,\rm I_{1,2,3})\) estimated with Equation (5) near the following pump power values: \(\rm I_{1}=980\) mW, \(\rm I_{2}=1435\) mW, and \(\rm I_{3}=1825\) mW, marked by the numbers 1, 2, 3 in Figure 3. Correspondingly, small dots, circles, and stars for each component’s spectral shape (denoted as \(\rm J(\lambda,\rm I_{j})\)) demonstrate the results of the inference described by Equation (7) for \(n(\lambda,\rm I_{j})\) shown by large dots, circles, and stars. Figure 3: The measured values of UCL intensity as a function of the pump power \(\rm J(\lambda,\rm I)\) on a log-log scale for several wavelengths corresponding to \(\lambda=365\) nm, 350 nm, 338 nm, and 370 nm are marked with dots, squares, diamonds, and triangles, respectively.