Radiance and fluence in a scattering disc under Lambertian illumination

Abstract

We investigated light propagation in a (non-)scattering disc under uniform Lambertian illumination analytically and by means of a Monte Carlo simulation. Thereby, we found that in a non-scattering disc the fluence plateaus near the center, and, starting at a critical radius determined by the refractive index, decays towards the boundary. A non-vanishing very small scattering coefficient causes an abrupt transition to a homogeneous fluence, similarly to the case of a sphere. Moreover, absorption and inhomogeneous scatterer distributions were considered, with the absorption leading to a reduced fluence near the disc center. In contrast to the homogeneously scattering case, it is shown that a non-vanishing scattering coefficient inside the critical radius does not cause a homogeneous fluence. New analytic solutions of the radiative transfer equation for radiance and fluence of unscattered and singly scattered light showed excellent agreement with their counterparts obtained via Monte Carlo simulation within its statistical noise. Additionally, we derived an analytic expression for the unscattered radiance and fluence in case of Lambertian illumination at a single point on the disc boundary, which also describes caustics.