I am currently reading this paper on lasers by G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu.
The author considered various noise sources present in the laser experiment and obtained the power spectral density using Wiener Khinchin theorem as Voigt profile in Eq. \eqref{2} reproduced below $$ I_E(\omega)= E_0^2\frac{\sqrt{\pi}}{\sigma} K(X,Y) \label{2}\tag{10} $$ The general definition of Voigt profile is the convolution of a Gaussian and a Lorentzian function. But the author uses Wiener Khinchin theorem to obtain Voigt from Eq. \eqref{1} (again reproduced below). $$ I_E(\omega)= \frac{E^2}{2}\int_{0}^{\infty} e^{i(\omega-\omega_0)\tau+\Gamma\tau-(\sigma\tau/2)}d\tau+ \text{c.c.} \label{1}\tag{9} $$ The author somehow modifies Eq. \eqref{1} which is an integral of a Gaussian function to Eq. \eqref{2} which is Voigt. I don't understand the mathematical steps involved in going from Eq. \eqref{1} to Eq. \eqref{2}.
My attempt: I tried to apply convolution theorem on Eq. \eqref{1} which can written as product of a Gaussian and an exponentially decaying function. Fourier transform of Gaussian is a Gaussian and exponentially decaying function is a Lorentzian. From convolution theorem Eq. \eqref{1} is simply the convolution of Gaussian and Lorentizian which gives a Voigt? Is this correct way to understand the derivation?