Analytical algorithm to determine the spherical particle size distribution from spectral absorption measurements

• A new and simple scheme is proposed to improve the accuracy of absorption in ADT through inclusion of the effect of reflection and refraction.
• A method of fast analytical inversion of spherical particle size distribution from spectral absorption measurements is implemented.
• The modified ADT transform pair and scaling relationship in retrieval are found.

A modified anomalous diffraction theory (MADT) by including the effects of reflection and refraction is proposed. With respect to MADT, an analytical technique for retrieval of spherical particle size distribution (PSD), based on absorption was developed. Also, an MADT transform pair between the size distribution and the absorption spectrum was constructed. This provides a new tool to study the related particle optical properties. By Gaver–Stehfest inversion method, the derived complex-inversion formula is finally converted into the new real-inversion formula so that the measured absorption data can be applied directly. The inversion experiments show that the use of extended precision instead of double precision arithmetic can produce more reliable results at the expense of CPU time. The effects of complex refractive index on retrieval of PSD were examined. Also it was found that an appropriate reduction of the truncation number with the smoothing technique improved the anti-noise ability for the algorithm.