Average Optical Path Length Estimation in a Slab of Arbitrary Finite Thickness

Timothy Kindervatter^1*Wei Li¹Nan Chen¹Yuping Huang¹Yongxiang Hu²Snorre Stamnes²Xiaomei Lu²Børge Hamre³Jakob J. Stamnes³Tomonori Tanikawa⁴Jennifer Lee⁵Carl Weimer⁵Xubin Zeng⁶Charles K. Gatebe⁷Knut Stamnes¹

• ¹Stevens Institute of Technology, Hoboken, United States
• ²Langley Research Center, National Aeronautics and Space Administration, Hampton, Virginia, United States
• ³University of Bergen, Bergen, Hordaland, Norway
• ⁴Meteorological Research Institute, Japan Meteorological Agency, Tsukuba, Ibaraki, Japan
• ⁵BAE Systems (United States), Arlington, Virginia, United States
• ⁶University of Arizona, Tucson, Arizona, United States
• ⁷Ames Research Center, National Aeronautics and Space Administration, Moffet Field, California, United States

A method for determining the average photon path length in a slab of multiple scattering material is presented. Radiances can be obtained from the radiative transfer equation and subsequently differentiated to obtain the average photon path length. These radiances can be obtained via multiple methods including Monte Carlo simulations, analytic two-stream approximations, and multi-stream numerical solutions such as the AccuRT computational tool. Average path lengths obtained via numerical differentiation of these radiances are found to agree closely with path length estimates predicted by existing methods found in the literature. The average photon path length is further considered for a slab of finite physical thickness. It was found that for a slab consisting of non-absorbing material there is a linear relationship between the slab thickness and the average photon path length, but that for materials with nonzero absorption, this linear relationship breaks down as the slab thickness increases. Average path lengths may be converted to time spans to determine the amount of time a photon spends in a multiple scattering medium, which may be used to quantify the impact of multiple scattering on pulse stretching in lidar/radar applications.