Determination of refractive indices of Mie scatterers from Kramers–Kronig analysis of spectral extinction data

Kramers–Kronig relations for semiconducting particles are established for a set of functions related to the scattering amplitude function in the forward direction. It is shown that some of these relationships allow subtractive Kramers–Kronig relations to be formed that are independent of any material constants. As a consequence, the refractive-index spectra can be determined without requiring the value of the index to be known at any frequency a priori. The approach is to use the optical theorem to establish the spectrum of one part of a complex function from the measured data. Its Kramers–Kronig counterpart is then computed, thereby establishing the values of two functions at each frequency. Since the dependence of each of the functions on the refractive index is known from the Mie theory, the real and imaginary parts of the index are then determined by simultaneous solution of the two functions at each frequency. Calculations are presented based on simulated data for combustion-generated hydrocarbon soot particles, and the influence of various types of experimental error on data inversion is investigated.