Dependence of the neodymium random laser threshold on the diameter of the pumped spot

We experimentally studied the dependence of the threshold energy density E-th/S in Nd0.5La0.5Al3 (BO3)(4) random laser on the diameter of the pumped spot d and found that at d >= 130 mu m, E-th/S is proportional to 1/d + const. This functional dependence is different from the one commonly expected in the case of diffusion, proportional to 1/d(2) + const [1], [2]. However, the obtained experimental dependence does not mean the failure of the diffusion model. Calculating the mean photon's residence time 7, (which photons, making their diffusion-like random walks, spend inside the gain volume) as the function of d and further assuming that E-th/S proportional to (tau(p)(res))(-1), we predicted the experimentally obtained functional dependence, proportional to 1/d + const. The major difference between our model and that of [1] and [2] was in the boundary conditions.