Fast and non-approximate methodology for calculation of wavelength-modulated Voigt lineshape functions suitable for real-time curve fitting

Abstract

Wavelength modulation (WM) produces lock-in signals that are proportional to various Fourier coefficients of the modulated lineshape function of the molecular transition targeted. Unlike the case for the Lorentzian lineshape function, there is no known analytical expression for the Fourier coefficients of a modulated Voigt lineshape function; they consist of nested integrals that have to be solved numerically, which is often time-consuming and prevents real-time curve fitting. Previous attempts to overcome these limitations have so far consisted of approximations of the Voigt lineshape function, which brings in inaccuracies. In this paper we demonstrate a new means to calculate the lineshape of nf-WM absorption signals from a transition with a Voigt profile. It is shown that the signal can conveniently be expressed as a convolution of one or several Fourier coefficients of a modulated Lorentzian lineshape function, for which there are analytical expressions, and the Maxwell–Boltzmann velocity distribution for the system under study. Mathematically, the procedure involves no approximations, wherefore its accuracy is limited only by the numerical precision of the software used (in this case ∼10−16) while the calculation time is reduced by roughly three orders of magnitude (10−3) as compared to the conventional methodology, i.e. typically from the second to the millisecond range. This makes feasible real-time curve fitting to lock-in output signals from modulated Voigt profiles.