## 3 Results and Discussion
### Material-Dependency of the Mie Characteristics in Multiple Scattering Random Lasers
So far we can deduce from our results that for solid state random lasers gain due to artificial but randomly formed laser cavities is not the leading order gain mechanism, since the refractive index is obviously changed and the scattering mean-free path in stationary state is increased in comparison to the same system under weak excitation, compare Section 3.2. We check this behavior theoretically in order to propose possible experiments which may be feasible to address the issue, and one of them is to compare the measurable result of the coherence volume to the crossover behavior of the scattering mean-free path for the same but only weakly excited, so non-pumped, system in a frequency range where we know the crossover of \(l_{s}\) in the weakly excited case is very pronounced. For the system paremeters \(r_{scat}\ =\ 423.0\) nm, the samples extent in z-direction of \(d\ =\ 80\,r_{scat}\ =\ 33.84\,\upmu\)m, a filling fraction of \(50\,\%\) and the refractive indices of \(n\ =\ 2.4\) (C) and \(n\ =\ 2.0041\) (ZnO) such a very clear crossover of \(l_{s}\) is found at \(\lambda\ =\ 600.0\) nm, compare Figure 4b. We show in Figure 4a the in-depth dependency of the correlation length \(\xi\) for both systems, C and ZnO, under moderate pumping \(P\ =\ 1.0\,\gamma_{21}\). The pump wavelength as well as the transport wavelength are chosen as \(\lambda\ =\ 580.0\) nm and \(\lambda\ =\ 620.0\) nm. We find two qualitatively different crossover situations. First we find an obvious crossover of \(\xi\) for the parameters \(n\ =\ 2.4\) and \(\lambda\ =\ 580.0\) nm in the outer selvedge of the sample, which is in the outer \(10\,\%\) of the samples extent on either open boundary. Second we find a qualitative change of the coherence length \(\xi\) when we consider a different material by changing the passive refractive index from \(n\ =\ 2.0041\) to \(n\ =\ 2.4\). For the parameters \(\lambda\ =\ 580.0\) nm and \(n\ =\ 2.0041\) the coherence length \(\xi\) all over the sample depth obtains almost the same characteristics and the same value as \(\xi\) for the case of \(n\ =\ 2.4\) and \(\lambda\ =\ 620.0\) nm. The direct counterparts, \(\xi\) for the parameter set of \(n\ =\ 2.0041\), \(\lambda\ =\ 620.0\) nm, and on the other hand \(\xi\) for the set of \(n\ =\ 2.4\) and \(\lambda\ =\ 580.0\) nm show a significant difference of about \(40\,\%\). This crossover becomes more obvious, when we display the coherence lengths \(\xi\), as single marked points, in comparison to the scattering mean-free path \(l_{s}\) for the weakly excited regime, marked as the solid and the dashed line in Figure 4b. Whereas the absolute value of \(l_{s}\) is rather symmetric for the refractive indices \(n\ =\ 2.0041\) and \(n\ =\ 2.4\) in the spectral positions of \(\lambda\ =\ 580.0\) nm, and of \(\lambda\ =\ 620.0\) nm, to their crossover at \(\lambda\ =\ 600.0\) nm, the coherence length by contrast shows a crossover as well, but the almost perfect point symmetry behavior of the scattering mean-free path in the weakly excited case is not present in the crossover behavior of \(\xi\) in stationary state. The crossover of \(\xi\) is also not confirmed for any crossover point of \(l_{s}\) for weakly excited Mie resonators in the spectrum.