## 3 Experimental Technique
### Sample Preparation and Characterization  
The examined samples were prepared as the layers of oxide powders impregnated by the water solution of the dye with a high value of the quantum yield of fluorescence (Rhodamine 6G). The matrix media (powder layers) consisted of TiO\({}_{2}\) (anatase) and SiO\({}_{2}\) spheroidal nanoparticles; the layers of anatase particles (the product # 637254 from the Sigma Aldrich Inc., the average size is less than 25 nm) were characterized by the volume fraction \(f_{TiO_{2}}\) of particles approximately equal to 0.20 (in the dye-impregnated state). The prepared silica powder layers (the product #MKBV0382V) had the volume fraction of the silica \(f_{SiO_{2}}\) in the impregnated state approximately equal to 0.1. The values of \(f_{TiO_{2}}\) and \(f_{SiO_{2}}\) were obtained using the volumetric measurements of both systems in the dry and water-impregnated states taking into account the degree of contraction due to impregnation. The Rhodamine 6G solution was prepared with the dye molar fraction equal to \(2.5\times 10^{-4}\) M/l. The measurement of the absorbance of the prepared dye solution in 1-mm thick cell at the wavelength of laser pumping (532 nm) allowed for evaluation of the absorption coefficient \(\mu_{a}\) as equal to \(\approx\)3.7 mm\({}^{-1}\); this magnitude fairly agrees with the expected value, estimated using the previously published absorption spectra results for the Rhodamine 6G water solutions [30]. Consequently, the estimated value of the absorption length \(l_{a}=\mu_{a}^{-1}\) of the pumping light in the dye solution is equal to \(\approx\)270 \(\upmu\)m. The prepared samples were placed in cylindrical containers with a height of 2 mm and an inner diameter of 20 mm; 1.5 mm thick glass substrates were used as undersides of the containers. The thicknesses \(L\) of the examined dye-saturated TiO\({}_{2}\)- and SiO\({}_{2}\)-based layers were approximately equal to 1.46 mm and 1.0 mm. The matrix media in the water-saturated state were characterized using the measurements of the diffuse transmittance spectra \(T_{d}(\lambda)\) within the spectral range from 500 nm to 900 nm. The spectra were obtained using the integrating sphere (Thorlabs IS200-4) with a coupled fiber-optic patchcor (Ocean Optics P100 UV-VIS) as the light-transfer channel from the integrating sphere to the spectrometer (Ocean Optics QE65000). The Ocean Optics DH2000 unit was used as the broadband light source in the spectral measurements. All the necessary pre-calibration and reference-normalization procedures were provided. The obtained \(T_{d}\) values were averaged over five independent spectral runs. Figure 3 displays smoothed diffuse transmittance spectra \(T_{d}(\lambda)\) for the examined titania- and silica-based water-impregnated systems. The obtained spectra were used for the recovery of spectral dependencies of the transport mean free path \(l^{*}(\lambda)\) as one of the most important parameters characterizing the light transport in strongly scattering media. The necessity to characterize the examined specimens using the wavelength-dependent values \(l^{*}(\lambda)\) evidently follows from the relationship between the average path-length difference \(\left<\Delta s\right>\) as the key parameter affecting the stochastic interference of multiple-scattered partial waves in the specimens and the transport mean free path (\(\left<\Delta s\right>\sim l^{*}\), see above). In the diffusion mode of the light propagation through a strongly scattering slab, the diffuse transmittance of the slab can be expressed as [31; 32]  
\[T_{d}(\lambda)=\frac{\{1+Z_{1}(\lambda)\}l^{*}(\lambda)}{L+[Z_{1}(\lambda)+Z_{2}( \lambda)]l^{*}(\lambda)}, \tag{11}\]  
where \(Z_{1}(\lambda),Z_{2}(\lambda)\) are dimensionless parameters dependent on the effective refractive index \(n_{ef}\) of the slab.