Asymptotic behavior of the reflectance of a narrow beam by a plane-parallel slab

We consider the radiative transfer of a finite width collimated beam incident normally on a plane-parallel slab composed of a uniform absorbing and scattering medium. This problem is fundamental for modeling and interpreting non-invasive measurements of light backscattered by a multiple scattering medium. Assuming that the beam width is the smallest length scale in the problem, we introduce a perturbation method to determine the asymptotic expansion for the solution of this problem. Using this asymptotic expansion, we determine the leading asymptotic behavior of the reflectance. This result includes the influence integral, which gives the influence of the phase function on the leading asymptotic behavior of the reflectance. We validate this asymptotic theory using a novel implementation of the Monte Carlo method that is fully vectorized to run efficiently in MATLAB. We evaluate the usefulness of this asymptotic behavior for different phase functions and show that it provides valuable insight into the influence of the phase function on spatially resolved non-invasive measurements of light backscattered by a multiple scattering medium.