Forward scattering formula of Tarn and Zardecki evaluated by use of cubic sections of spherical hypersurfaces

To evaluate the series expansion in scattering orders derived by Tarn and Zardecki for multiple forward scattering irradiance, the multidimensional integrals describing the contributions of the individual scattering orders n have been transformed into 1-D integrals by the use of a weighting function F(n). The function F(n) represents that section of a spherical hypersurface centered at the origin which is enclosed within the unit hypercube. For the calculation of the F(n), two approaches are proposed. The first one starts from a combinatorial consideration and yields a complete mathematical expression for the F(n) in the form of multidimensional integrals which, however, can be computed recursively from one another in order of increasing n by a 1-D analytical or numerical integration. For higher scattering orders a second approach is developed which allows direct analytical calculation of the Fourier transforms of the F(n) and, in a second step, the representation of the F(n) in the form of a series of functions being nearly identical with the eigenfunctions of the quantum mechanical harmonic oscillator. This function series is used further to expand the contributions of the individual scattering orders directly in a fast converging series in negative powers of n which reveals general features of multiple forward scattering.