Kattawar G W, Plass G N
Appl Opt. 1976 Dec 1;15(12):3166-78. doi: 10.1364/AO.15.003166.
Deep in a homogeneous medium that both scatters and absorbs photons, such as a cloud, the ocean, or a thick planetary atmosphere, the radiance decreases exponentially with depth, while the angular dependence of the radiance and polarization is independent of depth. In this diffusion region, the asymptotic radiance and polarization are also independent of the incident distribution of radiation at the upper surface of the medium. An exact expression is derived for the asymptotic radiance and polarization for Rayleigh scattering. The approximate expression for the asymptotic radiance derived from the scalar theory is shown to be in error by as much as 16.4%. An exact expression is also derived for the relation between the diffusion exponent k and the single scattering albedo. A method is developed for the numerical calculation of the asymptotic radiance and polarization for any scattering matrix. Results are given for scattering from the haze L and cloud C3 distributions for a wide range of single scattering albedos. When the absorption is large, the polarization in the diffusion region approaches the values obtained for single scattered photons, while the radiance approaches the value calculated from the expression: phase function divided by (1 + kmicro), where micro is the cosine of the zenith angle. The asymptotic distribution of the radiation is of interest since it depends only on the inherent optical properties of the medium. It is, however, difficult to observe when the absorption is large because of the very low radiance values in the diffusion region.
在诸如云层、海洋或浓厚行星大气层等既散射又吸收光子的均匀介质深处，辐射亮度随深度呈指数下降，而辐射亮度和偏振的角度依赖性与深度无关。在这个扩散区域，渐近辐射亮度和偏振也与介质上表面的入射辐射分布无关。推导出了瑞利散射的渐近辐射亮度和偏振的精确表达式。结果表明，从标量理论导出的渐近辐射亮度的近似表达式误差高达16.4%。还推导出了扩散指数k与单次散射反照率之间关系的精确表达式。开发了一种用于数值计算任意散射矩阵的渐近辐射亮度和偏振的方法。给出了针对广泛的单次散射反照率，从霾L和云C3分布进行散射的结果。当吸收很大时，扩散区域内的偏振接近单次散射光子所获得的值，而辐射亮度接近由以下表达式计算得出的值：相函数除以(1 + kmicro)，其中micro是天顶角的余弦。辐射的渐近分布很重要，因为它仅取决于介质的固有光学特性。然而，当吸收很大时，由于扩散区域内的辐射亮度值非常低，很难进行观测。
