D.2     Gaussian Function

D.2.1      Definition

The Gaussian function Gaus is defined as

This function centered at x₀ has a height of unity and its area is equal to |b|.

D.2.2     Properties

Gaussian functions are often used for distributions like spectral density of a light beam. In this case, an important parameter is the distribution bandwidth found at mid-height Dx_FWHM, where FWHM means full width at half maximum. It is interesting to connect this value of Dx_FWHM with the Gaussian parameters b and x₀ and to find the corresponding points x_1,2 where the Gaussian is 0.5 (half the maximum)

The Fourier transform of a Gaussian is also a Gaussian

where (D-9) is a consequence of equations (D-3) and (D-4). The Fourier transform is complex with a maximum amplitude of |b| at x=0. The position x_1,2 where the Fourier transform amplitude reach the half of its maximum and the FWHM Dx_FWHM are given by