Modified analytic expression for the single-scattering phase function
Authors: Shi Ze-Lin  , Cui Sheng-Cheng  , Xu Qing-Shan  ,   Key Laboratory of Atmospheric Composition and Optical Radiation, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, China;,   Science Island Branch of Graduate School, University of Science and Technology of China, Hefei 230031, China;,   Shenyang Institute of Automation Chinese Academy of Sciences, Shenyang 110016, China
Published: December 2020

In electromagnetic radiative transfer calculation, the accuracy and the computation time are usually determined by the representation of single-scattering phase function. Accurate calculation is time consuming even for spherical particle, thus, an analytic representation is commonly adopted to approximate the exact phase function and then accelerate the calculation. Most widely used single-scattering phase functions are the Henyey-Greenstein phase function and modified Henyey-Greenstein phase function (Henyey-Greenstein*). Although the Henyey-Greenstein phase function and the Henyey-Greenstein* phase function can represent the forward-scattering peak of Mie-scattering phase function well, they fail to reproduce the backscattering behavior, limiting the accuracy of the calculation. In order to better fit exact calculations and simulate the backward-scattering peak, we develop a new analytic expression based on the fundamental theory of electromagnetic scattering and radiation transmission. This phase function is an algebraic expression with one single free parameter (asymmetry factor), and can be expanded into Legendre polynomials. The new phase function converges to the Rayleigh phase function when the asymmetry factor approximates to 0, and it can approach to the Henyey-Greenstein phase function as the asymmetry factor is about 1. We compare the Henyey-Greenstein phase function, the Henyey-Greenstein* phase function, and the new phase function for different asymmetry factors, and find that the new phase function provides a more realistic description for the unpolarized light scattering from small particles. Furthermore, the calculated value for the ratio of the scattering intensity at 90 degree to that in the backward direction is more reasonable. We also investigate the effectiveness by approximating the scattering from polydispersed particles through comparing the new phase function, the Henyey-Greenstein* phase function, and the Mie-scattering phase function for three types of Derimendjian's polydispersions. Results show that the new phase function fits the Mie-scattering phase function much better than the Henyey-Greenstein* phase function. For the new phase function, the root-mean-square error is small for 73.3% data. By contrast, only 26.7% data fit the Mie-scattering phase function well for the Henyey-Greenstein* phase function. Similarly, the effectiveness of new function is most significant when calculating the ratio of the scattering intensity at 90 degree to that in the backward direction. In summary, the new Henyey-Greenstein* phase function provides a more accurate calculation for the scattering intensity in the backward direction, and is conducive to electromagnetic radiative transfer calculation. Furthermore, because the proposed phase function has the same basic form as the Heny-Greenstein phase function, reformatting radiative transfer model in terms of the new phase function should require relatively little effort.