Exact expressions have been derived from the Mie equations for the angular distribution coefficients, an, in the equation, [Equation] proposed by Hartel for the fraction of randomly polarized radiation scattered by a spherical particle into a unit solid angle in the direction θ. The coefficients are functions of the wavelength, particle diameter, and physical properties, and Pn (cosθ) are the Legendre polynomials. This equation is shown to be more convenient in form for tabulation of the scattering properties and for application than the intensity functions used by Mie and most subsequent investigators. Approximate methods for evaluating the coefficients were found to be inaccurate or inconvenient. The exact expressions, although detailed, are suitable for straightforward computation.
CHIAO-MIN CHU and STUART W. CHURCHILL, "Representation of the Angular Distribution of Radiation Scattered by a Spherical Particle," J. Opt. Soc. Am. 45, 958-961 (1955)