A Numerical Method for Modeling Light Scattering from Spherical Particles with a Probability Density in the Parametric Space
With modern computational tools, modeling light scattering from a particle with known physical parameters has become relatively trivial. Evaluation using analytical solutions is highly eficient compared to a full solution using numerical models, and such analytical solutions are known only for (homogeneous or layered) spheres, infinite cylinders and spheroids. But when given a probability density in the parametric space i.e. dimensions, refractive index and wavelength, the number of such evaluation can increase to ≈ 10^6 particles in case of a naive discretization of the parametric space into nominal particles. Uncertainty in physical parameters makes the solution of such forward or inverse problems in astrophysical, biological and atmospheric sensing applications cost prohibitive. We present a numerical method for layered and homogeneous dielectric spheres, where such cumbersome forward evaluations of the effective scattering from a distribution can be reduced to the cost of modeling a few particles. First, the scattering coefficients in the original form consisting of various Bessel functions are approximated into appropriate nested trigonometric functions, reducing the cost of evaluations by factors upto 10^2.This reduction uses a less known circular law in the complex plane constraining the possible scattering coeficients of a homogeneous and layered spheres, and it can as well be applied to numerical solution of inverse problems. Secondly, in evaluating the expected scattering from a distribution, simply replacing the naive discretization with a numerical integration of the weighted (oscillating) scattering functions using an appropriate quadrature, reduces the cost of evaluation by an additional factor larger than 10^2. Overall, using both the techniques, the total computing cost over the parametric space is reduced by factors upto ≈ 10^4. Numerical results of computing the effective scattering from a few distributions is analyzed.