Approximate P3 equation analysis in multi-layer slab media: Steady-state and time-domain based on the diffusion model

This study aimed to establish a steady-state and time-domain solutions for the approximate P3 equation in arbitrary multi-layer slab media for light propagation, based on a diffusion model with an isotropic point source in the first layer and extrapolated boundary conditions. Spatially resolved diffuse reflectance and transmittance were calculated using the steady-state approximation of the P3 equation, whereas temporally resolved diffuse reflectance and transmittance were derived from the time-domain approximation. The validity of the approximate P3 solutions was confirmed by comparing their results with Monte Carlo simulations. In steady-state analysis, the approximate P3 equation demonstrated superior accuracy to the diffusion equation for reflectance, particularly at smaller thicknesses. Accuracy further improved as the absorption coefficient and detection distance increased. For transmittance, the approximate P3 equation closely matched the diffusion equation at low thickness, but divergence occurred with higher absorption. In time-domain analysis, the approximate P3 equation aligned closely with Monte Carlo simulations at peak values, while its numerical values were close approximations of the diffusion equation away from the peak. The potential application of the approximate P3 equation in multi-layer media offers significant advancements for optical non-invasive detection and treatment techniques, enabling the extraction of optical parameters from such media.