It is of great significance to study the optical properties of biological tissue and simulate the distribution of light in tissue for light therapy and diagnosis. Therefore, a theoretical model of optical properties needs to be determined. Currently, the model commonly used in biological tissue is the radiative transfer equation. The most widely used one is the first-order approximation of the radiative transfer model, namely the diffusion equation. However, in the case of small detection distance and large absorption, the diffusion equation is not accurate. Therefore, some people have studied the third-order approximation P3 equation of the radiative transfer model. Compared with the diffusion equation, the P3 equation is more accurate and more widely applied, but it is more complicated to establish a mathematical model. At present, several P3 equation models are studied in one layer medium. In fact, biological tissue is a multilayer medium, so it is necessary to establish the P3 model of multilayer tissue. At present, there are several models in the diffusion equation that can solve the problem in a multilayer medium. In particular, Kienle et al. obtained the exact solutions of the diffusion equation in the steady state and frequency and time domains of light transmission in two semi-infinite thick media through the inverse Fourier transform. It is necessary to establish the P3 equation of light transmission in two or more layers. In this paper, the P3 time domain equation of light transmission in a two-layer slab medium is given and compared with Monte Carlo simulation and diffusion equation.

On the basis of the radiative transfer theory, the P3 equation is given. According to Fourier transform method, the frequency domain solution is established. Based on the Fourier transform in the frequency domain, the time domain solution of the P3 equation in a two-layer slab medium is given. Monte Carlo simulation is a statistical verification method, which can replace experiments to verify the correctness of the theoretical model. The P3 time domain equation and diffusion equation are calculated, and the Monte Carlo simulation program for the multi-layer medium is written. The P3 time domain equation of light transmission in a two-layer slab medium is verified by Monte Carlo simulation. The advantages of the P3 time domain equation and the diffusion equation in the case of low absorption coefficient at a long distance and high absorption coefficient at a short distance are compared.

The P3 time domain equation of the two-layer slab medium is compared with the Monte Carlo simulation. The results show that the reflectance and transmittance of the P3 time domain equation for light transmission in the two-layer slab medium are in good agreement with the Monte Carlo simulation results, which indicates that the P3 time domain equation for light transmission in the two-layer slab medium correctly reflects the light migration in the medium. The P3 time domain equation is compared with the time domain diffusion equation of a multi-layer medium. When the absorption coefficient is low, and the detection distance is large, the results of the P3 equation are consistent with those of the diffusion equation. Near the peak value, the reflectance error of the P3 equation is about 3%, and that of the diffusion equation is about 7%. The transmittance error of the P3 equation is about 7%, and that of the diffusion equation is about 13%. In other words, when the diffusion equation is satisfied, the P3 equation is more accurate than the diffusion equation at the peak value. When the absorption coefficient is high, and the detection distance is small, the reflectance error of the diffusion equation is about 30%. The transmittance error of the P3 equation is about 10%, and that of the diffusion equation is about 20%. It is further indicated that the reflectance and transmittance of the P3 equation are more accurate than those of the diffusion equation when the absorption coefficient is larger near the source. So the P3 time domain equation in a two-layer medium has an advantage over the diffusion equation.

The P3 time domain equation of light transmission in a two-layer slab medium is given. The P3 time domain equation is consistent with the Monte Carlo simulation results and is more accurate than the diffusion equation. So the diffusion equation of the two-layer medium can be replaced by the P3 time domain equation. When the optical parameters of the two layers are the same, a one-layer model can be derived. Therefore, the P3 equation in a two-layer slab medium not only includes the P3 equation in a one-layer slab medium but also lays the foundation for the P3 equation in a multi-layer slab medium. At present, the diffusion equation is used to extract the optical parameters of biological tissue, and the P3 equation in the one-layer medium is used to extract the optical parameters of biological tissue. Therefore, the P3 time domain diffusion equation in a two-layer medium can be used to extract the optical parameters of multi-layer media.