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.

中文翻译：

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多层板介质中的近似 P3 方程分析：基于扩散模型的稳态和时域


本研究旨在基于第一层各向同性点源的扩散模型和外推边界条件，为任意多层板介质中的近似 P3 方程建立用于光传播的稳态和时域解。空间分辨漫反射率和透射率是使用 P3 方程的稳态近似计算的，而时间分辨漫反射率和透射率是从时域近似推导出来的。通过将近似 P3 解的结果与 Monte Carlo 模拟进行比较来证实其有效性。在稳态分析中，近似 P3 方程的反射率精度优于扩散方程，尤其是在较小厚度下。随着吸收系数和检测距离的增加，准确性进一步提高。对于透射率，近似 P3 方程与低厚度下的扩散方程紧密匹配，但吸收率较高时会出现发散。在时域分析中，近似 P3 方程在峰值处与蒙特卡洛模拟紧密一致，而其数值是远离峰值的扩散方程的近似值。近似 P3 方程在多层介质中的潜在应用为光学非侵入式检测和处理技术提供了重大进步，能够从此类介质中提取光学参数。