Using Fast Fourier Transform to compute the line shape of frequency-modulated spectral profiles

An expression for the shape of spectral lines recorded by sinusoidal frequency modulation and harmonic detection is derived. It can be fast computed by means of the Fast Fourier Transform method and it holds for any modulation depth and modulation frequency. Different line shape functions, such as Voigt, Galatry, and speed-dependent Voigt, may be included in the model. In this work, the performance of the computing method is tested with experimental second harmonic rotational profiles, but it can be applied to higher harmonic spectra as well.