Analysis of optical spectra by Fourier methods: filtering and least-squares regression in reciprocal space

Abstract

We describe a method for accurately determining critical point parameters from optical spectra in which digital filtering in real (energy) and reciprocal (Fourier-coefficient) space is treated on an equivalent basis. Experimental and theoretical line shapes are also filtered in parallel, thereby eliminating systematic errors that can arise in the standard approach in which only the data are processed. Real-space filtering is done using false data to isolate individual or groups of critical points in complicated spectra, to provide a more accurate representation of the data in reciprocal space, and to minimize the effects of end-point discontinuities and truncation errors on the Fourier coefficients calculated from these spectra. Reciprocal-space filtering is done by numerically differentiating the data to maximize the amplitudes of the Fourier coefficients carrying the critical point information, followed by truncating low- and high-order coefficients to minimize artifacts that are due to baseline effects and noise. The optimum order of differentiation (not necessarily integral) is determined from the coefficients themselves. We show that a least-squares regression (LSR) analysis of a restricted interval of equally weighted points in reciprocal space is equivalent to the LSR analysis of all data points equally weighted in real space, making LSR particularly useful for analyzing higher-derivative spectra, where the real-space line shapes rapidly approach zero outside the central structure. For a specific example discussed here, maximum accuracy is obtained if the data are analyzed in the form of a third derivative, as was previously concluded empirically from numerical processing in real space.