Convergence of the discrete dipole approximation. I. Theoretical analysis

TL;DR

This paper provides a rigorous theoretical analysis of the convergence behavior of the discrete dipole approximation, showing error bounds and the influence of particle shape and size on accuracy.

Contribution

It offers the first detailed convergence proof for DDA, highlighting the impact of shape and size on error bounds and convergence rates.

Findings

Errors are bounded by linear and quadratic terms in dipole size d.

Cubically shaped particles exhibit significantly smaller linear errors.

Convergence is quadratic for large scatterers within DDA applicability.

Abstract

We performed a rigorous theoretical convergence analysis of the discrete dipole approximation (DDA). We prove that errors in any measured quantity are bounded by a sum of a linear and quadratic term in the size of a dipole d, when the latter is in the range of DDA applicability. Moreover, the linear term is significantly smaller for cubically than for non-cubically shaped scatterers. Therefore, for small d errors for cubically shaped particles are much smaller than for non-cubically shaped. The relative importance of the linear term decreases with increasing size, hence convergence of DDA for large enough scatterers is quadratic in the common range of d. Extensive numerical simulations were carried out for a wide range of d. Finally we discuss a number of new developments in DDA and their consequences for convergence.