New near-analytical path to the threshold size of spherical random lasing via geometric-distribution-probability weighting of the diffusive photon fluence rate

We demonstrate a new near-analytical path to the threshold-size of random-lasing for the case of a uniform and isotropic-scattering sphere. We assess a geometric-distribution-probability (GDP) weighted integration of the diffusion-equation derived time-dependent photon fluence-rate at a spherical boundary, in response to uniform, synchronous, and Delta-functional photon generations within the sphere. The GDP weights the contribution of the modeled Delta-functional photon sources to the temporal behavior of the photon fluence rate at the spherical boundary-domain based on the line-of-sight distance between the modeled-photon source and the same field point. The integral manifests a bi-phasic pattern versus time with a global minimum followed by an exponential growth. The line-of-sight length that corresponds to the time of global minimum decreases monotonically as the size of the sphere increases. The condition that this line-of-sight length equaling the radius of the sphere is hypothesized to indicate a threshold whereby the medium can sustain the growth of the photon fluence-rate at the boundary over time. This threshold line-of-sight length is assessed over a gain/scattering ratio of [0.001, 10000] covering the diffusive to quasi-ballistic regimes. The threshold line-of-sight length applied with a simple empirical gain/scattering ratio predicts the threshold size over the diffusive region and outperforms the threshold size given by the eigen-mode-decomposition in the semi-ballistic region, when compared to the radiative transfer approach. The method sheds new insights to amplified diffusion process in a scattering medium with gain.

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