## 3 Experimental Technique
### Sample Preparation and Characterization
As was previously established [32], the relationship (11) can be used to evaluate the diffuse transmittance of the multiple-scattering media with the slab geometry in the case that the slab thickness is at least 5-6 times larger than the transport mean free path. Therefore, this relationship can be fairly applied to estimate \(l^{*}\) for the TiO\({}_{2}\)-based system (Figure 3). On the contrary, following from the data presented in Figure 3 for the SiO\({}_{2}\)-based system, it can be concluded that its diffuse transmittance cannot be adequately described in terms of the diffusion approximation (\(l^{*}\) occurs larger than \(L/5\div L/6\)). Therefore, the spectral dependence \(l^{*}(\lambda)\) was recovered using the inverse Monte Carlo (MC) modeling with the core MC algorithm repeatedly described in [33]. Figure 4 displays the recovered spectral dependencies of the transport mean free path for the examined water-impregnated matrix media in the spectral range from 500 nm to 900 nm. Note that the titania-based system exhibits a dramatically larger scattering strength compared to the silica-based system (the values of the transport mean free path in the former case are at least two and a half orders of magnitude lower than in the latter case). A remarkable decrease in \(T_{d}(\lambda)\) (and, correspondingly, in \(l^{*}(\lambda)\)) in the short-wavelength region (below 600 nm) for the titania-based system is obviously caused by an abruptly increasing refractive index of TiO\({}_{2}\) particles near the fundamental absorption band). Figure 3: The diffuse transmittance spectra of the examined samples in the wet state. 1—TiO\({}_{2}\)-based system; 2—SiO\({}_{2}\)-based system. Selectively shown error bars correspond to a confidence level of 0.9. Note that a characteristic size of the scattering sites in the prepared samples can differ from the particle sizes in raw powders declared by the manufacturers due to the particle aggregation. Actually, only the volume fractions of the hard phase and the chemical compositions of the prepared samples are the well-established characteristics. It is interesting to compare these sizes. The calculations of \(l^{*}(\lambda)\) for the various sizes of the scattering sites, providing a reasonable fit of the recovered spectral dependencies, can be used to estimate the characteristic sizes of scattering sites. These calculations can be carried out using an appropriate wavelength-dependent effective medium model. For this purpose, we applied a wavelength-dependent effective medium model developed in the framework of the coherent potential approximation [34], that was successfully used for calculations of the optical transport parameters (\(l^{*}\) and \(n_{ef}\)) in the various strongly scattering systems consisting of densely packed dielectric particles (see, e.g., Refs. [35; 36] for a detailed description of the model and examples of the applications). The simulation showed that the characteristic sizes of the scattering sites, which correspond to the experimentally observed spectral dependencies \(l^{*}(\lambda)\) are several times (5-8) larger than the declared particle sizes in the raw powders.