Optimal Scaling of the Inverse Fraunhofer Diffraction Particle Sizing Problem: The linear system produced by quadrature

Solution of the linear system of equations obtained by discretization and numerical quadrature of the Fredholm integral equation describing Fraunhofer diffraction by a distribution of particles is considered. The condition of the resulting system of equations depends on the discretization strategy. However, the specific set of equations is shown to depend on the discretization scheme used for the scattering angle domain (the number, positions and apertures of the detectors) and for the size domain (the number and extent of the discrete size classes). The term scaling is used here to describe particular formulations or configurations of the scattering angles and size classes, and a method for optimally scaling the systems is presented. Optimality is determined using several measures of the condition (stability) of the resulting system of linear equations. The results provide design rules for specifying an optimal photodetector configuration of a Fraunhofer diffraction particle sizing instrument.