Sensitivity tests on the convergence tendency of the scattering order formulation of the discrete dipole approximation

In this study, we performed a series of sensitivity tests in order to elucidate the convergence tendency of the scattering order formulation (SOF) of the discrete dipole approximation (DDA). Using both the original formulation of the SOF and a new marching SOF, the progression of orders of scattering marches, along with the propagation of the incident plane wave through the scatterer, allow dipoles that come into steady-state oscillation with the incident wave earlier to more quickly advance to the next order of scattering that is local to them. Using the original SOF, we found that for cases in which the simulations converge (rods and very small spheres), there are a number of different possible convergence tendencies, among them convergence behavior that resembles the decaying oscillations of a damped harmonic oscillator. For the cases in which the original SOF does not converge, we did not find an indication that the lack of convergence is due to a numerical issue, such as round-off error, or that the divergence could be alleviated by increasing the dipole resolution or by decreasing the size of the marching step in the marching SOF. For cases in which the original SOF does not converge, with both the original SOF and the marching SOF, we found that the calculated extinction cross section exhibits oscillations about the correct value, but with increasing amplitude rather than with decreasing amplitude.