## 5 Discussion of the Results  
### The Wavelength-Averaged Dispersion of the Third-Order Statistical Moments of Intensity Fluctuations  
The differences between the recovered \(\left\langle I^{3}(\lambda)\right\rangle/\left\langle I(\lambda)\right\rangle^{3}\) spectra obtained for the examined systems under various conditions can be quantified using the estimations of the wavelength-averaged dispersion of the normalized third-order moment \(M^{3}(\lambda)\):  
\[\overline{D_{\langle I\rangle/\langle I\rangle^{3}}}=\frac{\int\limits_{ \lambda_{1}}^{\lambda_{2}}\frac{d\left\langle(I^{3})/\langle I\rangle^{3} \right\rangle}{d\lambda}d\lambda}{\lambda_{2}-\lambda_{1}} \tag{12}\]  
The dispersion values \(\overline{D_{\langle I\rangle/\langle I\rangle^{3}}}\) estimated for various experimental conditions in the spectral region from \(\lambda_{1}=600\) nm to \(\lambda_{2}=750\) nm are presented in Table 1.  
Figure 10 displays the noise-corrected spectral dependencies \(M^{3}(\lambda)\) for the examined fluorescent systems in the case of the scanning mode; the noise reduction was carried out in accordance with Equation (5) using the \"static\" reference data (the time-averaged values \(\left\langle I^{2}\right\rangle/\langle I\rangle^{2},\left\langle I^{3} \right\rangle/\langle I\rangle^{3}\) obtained without scanning). Comparing these data with the previously reported results for Rhodamine-6G-doped coarse-grained silica layers, which were obtained using the Horiba Jobin \begin{table}
\begin{tabular}{c c c c c c} \hline \hline (I), Dye Solution & \begin{tabular}{c} **(II), SiO\({}_{2}\)-Based** \\ **System, Static** \\ \end{tabular} & \begin{tabular}{c} **(III), SiO\({}_{2}\)-Based** \\ **System, Scan** \\ \end{tabular} & \
\begin{tabular}{c} **(IV), TiO\({}_{2}\)-Based** \\ **System, Static** \\ \end{tabular} &
\begin{tabular}{c} **(V), TiO\({}_{2}\)-Based** \\ **System, Scan** \\ \end{tabular} \\ \hline \(\approx\)1.0 \(\times\) 10\({}^{-6}\) & \(\approx\)1.53 \(\times\) 10\({}^{-6}\) & \(\approx\)97.33 \(\times\) 10\({}^{-6}\) & \(\approx\)2.40 \(\times\) 10\({}^{-6}\) & \(\approx\)240.27 \(\times\) 10\({}^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The wavelength-averaged dispersion of the third normalized moment \(\left\langle I^{3}\right\rangle/\langle I\rangle^{3}\), nm\({}^{-1}\). Figure 10: The noise-corrected wavelength-dependent values of the normalized third-order moment of intensity fluctuations in the spectral region of interest for TiO\({}_{2}\)-based (1) and SiO\({}_{2}\)-based (2) systems. microscope as a detection unit [29], we clearly see the abovementioned suppression effect due to mode excitation in the light-transferring multimode channel. Typically, the ratios \((M_{3}-1)_{i}/(M_{3}-1)_{ii}\) are in the range from \(\approx\)1.0 \(\times\) 10\({}^{-3}\) to \(\approx\) 5.0 \(\times\) 10\({}^{-3}\) for the wavelengths changing from 600 nm to 700 nm (\(i\) corresponds to the data for the examined silica-based system presented in Figure 8, \(ii\) corresponds to the data taken from [29]). Despite a certain difference in the optical transport properties of the examined systems and fluorescence excitation conditions, the wavelength-averaged value \(\beta=(M_{3}-1)_{i}/(M_{3}-1)_{ii}\) can be used to characterize the \"random generation\" of uncorrelated radiation sources due to mode excitation in the patch-cord (in the case of application of the Horiba Jobin Ivon Lab Ram HR800 microscope this effect is absent). The estimated value \(\overline{\beta}=\int\limits_{600}^{700}\beta(\lambda)d\lambda/100\) is approximately equal to \(2\times 10^{-3}\), which is in a reasonable agreement with the abovementioned estimation of the number of excited modes (\(N_{m}\sim\beta^{-1}\)). Note that such rapid increase \(M_{3}\) for the examined fluorescent systems with the increasing wavelength (see Figure 10) cannot be explained only in terms of the decreasing number of excited modes.