## 3 Results and Discussion  
### The Coherence Volume of \(D\) = 3 Dimensional Random Lasers  
We find a full set of self-consistent scales, e.g., the diffusion coefficient \(D\), the scattering mean-free path \(l_{s}\), the transport mean-free path or coherence length \(\xi\), the energy transport velocity \(v_{E}\), the phase velocity \(c_{p}\), the energy and intensity dependent LDOS \(N_{\omega}\), as well as the laser thresholds and the random laser gain in stationary state from the solution of the Bethe-Salpeter equation and the intensity correlator including interference effects \(\Phi_{ee}\). As the main solution from the generalized diffusion equation, Equation (13), we derive the coherence length \(\xi\) in a 3-dimensional setup with one finite dimension \(z\). The finite system is characterized by lossy boundaries, otherwise we neglect losses for our considerations here in order to derive a systematic presentation of the random lasing phase transition. The translation invariance is consequentially broken in the \(z\) direction. We present in Figure 2 results for one specific set of a possible random laser sample under three different excitation regimes with respect to the microscopic bulk matter properties of the scatterer's material, sub-threshold excitation with respect to bulk, excitation at the bulk threshold and excitation above the bulk threshold. We show results for an arrangement of diamond nano resonators of refractive index \(n\,=\,2.4\) with the radius of the single scatterer \(r_{scat}\,=\,423.0\,\) nm in a slab geometry of \(50\,\%\) filling and the extent of \(80\,r_{scat}\,=\,33.84\,\)um in z-direction whereas it is infinite in-plane. The sample is equally lossy on either of the boundaries. We assume spatially homogeneous pumping of the sample and we show the results for the excitation strengths \(P\,=\,0.2\,\gamma_{21}\), \(P\,=\,0.5\,\gamma_{21}\), \(P\,=\,1.0\,\gamma_{21}\) and \(P\,=\,9.0\,\gamma_{21}\), where \(\gamma_{21}\) is the rate for stimulated emission at the lasing transition of the bulk matter alone. We thus focus on four possible regimes of the bulk reference and we check whether the same classification still holds for the disordered arrangements where we believe that multiple scattering between otherwise independent active nano resonators will be the leading order effect in solid state random laser samples of a high filling fraction. As a result we find that large scale systems can be driven with _subcritical_ excitation strengths up to the laser transition. Here we show the case for \(P\,=\,0.2\,\gamma_{21}\)... \(P\,=\,0.5\,\gamma_{21}\). The pump power is definitely far below the microscopic laser threshold and thus is _subcritical_. For \(P\,=\,0.2\,\gamma_{21}\) we obtain a symmetrically shaped coherence volume with respect to the z-axis with a radius in the sample's center, \(Z\,=\,0\), of \(\xi\,=\,103.212\,\upmu\)m, see Figure 2. Thus the coherence volume has an extent in the sample's center of \(2\cdot\xi\,=\,206.424\,\upmu\)m. By increasing the pump rate to \(P\,=\,0.5\,\gamma_{21}\) the coherence length reduces quantitatively to \(\xi\,=\,92.343\,\upmu\)m. At the threshold value of the bulk material \(P\,=\,1.0\,\gamma_{21}\) we find qualitatively the same behavior; however, the quantity is reduced again to \(\xi\,=\,80.201\,\upmu\)m. By increasing the pump power to \(P\,=\,0.5\,\gamma_{21}\) we find the coherence length of \(\xi\,=\,37.90\,\upmu\)m. The length scale \(\xi\) is a direct measure of the random laser mode in stationary state under specific conditions. It is also a measure of the Cooperon contribution, which is the perfect interference of a photon and its time reversal in multiple scattering procedures that acts as the stimulation process  
Figure 2: Coherence volume as the solution from Equation (10) within a D = 3 dimensional random laser slab of monodisperse Mie scatterers.