## 5 Discussion of the Results
### Comparison with Other Experimental Data  
It is interesting to compare the estimated optical properties of the examined systems with the characteristics of the previously studied similar systems applied in the random lasing experiments. In particular, in one of the early works on random lasing, [PERSON] et al. [37] studied the radiative properties of the laser dye (Rhodamine 640 perchlorate in methanol) dispersed in strongly scattering matrix media (colloidal suspensions of titanium dioxide particles). The Al\({}_{2}\)O\({}_{3}\)-coated titania particles with an average diameter \(\langle d\rangle\) of 250 nm were used to obtain the dye-containing random media with the particle densities \(\rho\) varying between 1.4\(\cdot\)10\({}^{9}\) cm\({}^{-3}\) and 8.6\(\cdot\)10\({}^{11}\) cm\({}^{-3}\). Accordingly, the particle volume fractions \(f\approx\pi\langle d\rangle^{3}\rho/6\) varied between \(\approx\) 1.1\(\cdot\)10\({}^{-5}\) and \(\approx\) 7.0\(\cdot\)10\({}^{-3}\). These small values of \(f\) allow for estimation of the scattering mean free path \(l\) and the transport mean free path \(l^{*}\) for the examined systems in the absence of absorption using the weak scattering approximation: \(l\approx 1/\sigma_{s}\rho\) and \(l^{*}\approx 1/\sigma_{s}\rho(1-g)\), where \(\sigma_{s}\) is the scattering cross-section of particles and \(g\) is the scattering anisotropy parameter. Application of the Mie scattering calculator [38] to these systems gives the values of \(l\) and \(l^{*}\) at the lowest and highest particle concentrations and \(\lambda=600\) nm equal to \(l_{\rm max}\approx 4000\) um, \(l^{*}_{\rm max}\approx\)7630 um, and \(l_{\rm min}\approx 6.5\) um, \(l^{*}_{\rm min}\approx 12.4\) um. Note that in accordance with the experimental data presented in [37], the evident transition from spontaneous amplification of fluorescence radiation to the random lasing in the studied titania-based systems with an increasing pump intensity occurs when the particle concentration is equal to \(\approx\)2.8 \(\times\) 10\({}^{10}\) cm\({}^{-3}\) (\(l\approx\) 200 um, \(l^{*}\approx\) 382 um) or more. It should be mentioned that a non-linear optical behavior of this system (in particular, broadening of the fluorescence emission spectrum due to the scattering-enhanced self-absorption of the dye resulting in re-emission) was only observed in the cuvettes thicker than 100 um. Thus, the critical ratio of the system thickness to the transport mean free path \(\Psi_{cr}=L_{cr}/l^{*}\), which is necessary for a non-linear optical behavior of the examined system under pulse laser pumping, can be estimated at \(\lambda=600\) nm as \(\Psi_{cr}\approx\) 0.262. In our case, the titania- and silica-based fluorescent random systems are continuously irradiated by the laser light at significantly lower pump intensities in comparison with [37] (at least seven decades below the peak intensity used by [PERSON]'s group, taking into account the laser beam parameters and a large oblique incidence angle in our case). In addition, the examined systems are specified by a larger value of the scattering strength (in the case of the TiO\({}_{2}\)-based system) and lower dye concentration. Nevertheless, the applied CW pumping of TiO\({}_{2}\)-based system stimulates its non-linear behavior. The \(L/l^{*}\) ratios in the spectral range between 600 nm and 700 nm vary between \(\approx\)1.5 and \(\approx\)1.8 for the silica-based system and between \(\approx\)200 and \(\approx\)350 for the titania-based system. The primary effect observed in both systems examined in this work is a remarkable elongation of the average propagation path \(\langle s\rangle\) of fluorescence radiation (and, consequently, the average path-length difference \(\langle\Delta s\rangle\)) in the spectral region corresponding to the dye fluorescence band).