Approximate P3 solution for the semi-infinite medium: steady state and time domain

The steady-state solution of the Green's function obtained by the P3 equation in a semi-infinite medium is presented, the proposed solution is a diffusion-based model. Two time-domain solutions are established: one is the solution under extrapolation boundary condition, which we call the optical parameter method, and the other corresponds to the diffusion equation, which we call the double-diffusion coefficient method. The spatial-resolved reflectance and the time-resolved reflectance are calculated. The Monte Carlo simulation is used to verify the P3 equation. The results show that the P3 steady-state equation and the two time-domain equations are in good agreement with the Monte Carlo simulation. In the steady state, when the distance between the detector and the light source is less than several free paths, the P3 equation is more accurate than the diffusion equation. In other cases, the P3 model and the diffusion model have similar results. However, when the absorption coefficient is large, P3 is more accurate. In the time domain, the optical parameter method is more accurate, and the double-diffusion coefficient method is more consistent with the diffusion equation.