In order to improve the computational efficiency of the traditional approach to Monte Carlo simulations, we implement the semi-analytical approach (also known as the partial photon technique).8,27,28,42,43 Using this method, a portion of scattered intensity is collected after a photon reaches each new position within the medium. This “partial photon” intensity is calculated by multiplying the probability that the photon is scattered in the direction of the surface by the probability that it will reach surface according to the Beer-Lambert law : Display Formula
(34)where is the angle between and the surface normal vector and is the distance to the surface. Modifying Eq. (11) under the semi-analytical approach, we find function for different the different polarization channels as: Display Formula
(35)where the index of summation now represents the number of scattering events. The product of functions for the orthogonal polarization channels can be found in analogy with Eq. (36) as: Display Formula
(36)Scoring photons in this way enables both computational efficiency through collection of information at each scattering event and accuracy through collection of intensity in the exact backscattering direction.