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Light Transport in Tissue Reflection and transmission of thin layers Starting the adding-doubling method requires knowledge of the reflection and transmission operators for a thin slab. Several methods exist for obtaining these operators: diamond initialization [66], infinitesimal generator [20], and successive scattering [28]. The first two methods have been compared by Wiscombe who found that the more complicated diamond initialization technique was better about two-thirds of the time [67]. The successive scattering technique was the first method used for calculating the reflection and transmission operators of a thin slab [63]. With this method initial optical depths up to $\tau \sim 1$ are possible although Hansen and Travis indicate that an optical thickness of $\tau \sim 2^{-10}$ is optimal for this starting method [24]. The infinitesimal generator technique assumes that the tissue is sufficiently thin that single scattering accurately estimates the reflection and transmission for the slab. This is the method implemented for calculations in this chapter because of its simplicity. Single scattering equations for the reflection and transmission functions are given by van de Hulst for isotropic scattering [60]. For azimuthally independent anisotropic scattering the redistribution function must be included. The single scattering reflection function for thin layers is $$R(a,\tau,\mu,\mu_0) = {a \pi h(\mu,-\mu_0) \over \mu+\mu_0} ... ...ft[1-\exp\left(-{\tau\over\mu}-{\tau\over\mu_0}\right) \right]$$ (3.30) The slight difference from van de Hulst results from differences in phase function normalization. At grazing angles ($\mu_0=\mu=0$), the reflection has a singularity $$R(a,\tau,0,0) = \infty$$ (3.31) When the angle of incidence is not equal to the angle of transmission ($\mu_0\ne\mu$), then the transmission function is given by $$T(a,\tau,\mu,\mu_0) = {a \pi h(\mu,-\mu_0) \over \mu_0-\mu} ... ...u\over\mu_0}\right) -\exp\left(-{\tau\over\mu}\right) \right]$$ (3.32) When $\mu_0=\mu$ then the primary beam (with an incident cone of $\mu$) must be included, $$T(a,\tau,\mu,\mu) = {a \pi h(\mu,\mu) \over \mu} \exp\left(... ...er\mu_0}\right)+ {1\over2\mu}\exp\left(-{\tau\over\mu}\right)$$ (3.33) The factor of $2\mu$ in the unscattered beam results from the use of star multiplication. Finally, for grazing incidence ( $\mu= \mu_0 = 0$), the transmission is zero $$T(a,\tau,0,0) = 0$$ (3.34)

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.