Extending the applicability of the T-matrix method to light scattering by flat particles on a substrate via truncation of Sommerfeld integrals

The simulation of light scattering by particles on a substrate with the $T$-matrix method relies on the expansion of the scattered field in spherical waves, followed by a plane wave expansion to allow the evaluation of the reflection from the substrate. In practice, the plane wave expansion (i.e., the Sommerfeld integrals) needs to be truncated at a maximal in-plane wavenumber $\kappa_\mathrm{max}$. An appropriate selection of $\kappa_\mathrm{max}$ is essential: counter-intuitively, the overall accuracy can degrade significantly if the integrals are truncated with a too large value. In this paper, we propose an empirical formula for the selection of $\kappa_\mathrm{max}$ and discuss its application using a number of example simulations with dielectric and metallic oblate spheroids on dielectric and metallic substrates. The computed differential scattering cross sections are compared to results obtained from the discrete-sources method.