Scaling property of the diffusion equation for light in a turbid medium with varying refractive index

A spatially varying refractive index leads to the bending of photon paths in a medium, which complicates the Monte Carlo modeling of a photon random walk. We show that the process of photon diffusion in a turbid medium with varying refractive index and curved photon paths can be mapped to the diffusion process in a medium with straight photon paths and modified optical properties. Specifically, the diffusion coefficient, the absorption, and the refractive index of the second medium should differ from the corresponding properties of the first medium by the factor of the squared refractive index of the first medium. The specific intensity of light in the second medium will then be equal to the specific intensity in the first medium divided by the same factor, which also means that the photon density distributions in the two media will be identical. In a Monte Carlo simulation the scaling property suggests that two different algorithms can be used to obtain the photon density distribution, namely, the algorithm with curved photon paths and given optical properties and the algorithm with straight photon paths and modified optical properties.