Analytical inversion of the absorption spectrum to determine non-spherical particle size distribution

•We propose the analytical transform techniques to retrieve the non-spherical PSD.•Inversions are based on absorption spectra.•Some ADT transform pairs are found.•Fast inversions are implemented with the help of Gaver-Stehfest׳s method.

For a variety of non-spherical particles (oriented spheroids, cuboids, triangular prisms, and hexagonal prisms), analytical transform techniques are proposed to retrieve the particle size distribution (PSD) from measured absorption spectra. The absorption efficiency of particles is calculated using the anomalous diffraction theory (ADT). We find that for each type of non-spherical particles, there exists an ADT transform pair between the size distribution and the complex absorption spectrum, which provides the physical basis for solving the inverse problem. Furthermore, the relation between the size distribution and real absorption spectrum is established by using Gaver-Stehfest׳s method. The numerical calculations show that the use of extended precision instead of double precision arithmetic can produce more reliable results at the expense of computational efficiency. Also it is shown that a small Stehfest number (standing for truncation number) tends to enhance the anti-noise level of inversion.