Fourier transforms of Gaussians

first let's do a mean-zero gaussian (i.e., it's centered at x=0)

gaussian[x_,s_] =1/(s Sqrt[2 Pi ])  Exp[-x^2/(2s^2)]
Integrate[gaussian[x,s],{x,-∞,∞},Assumptions->{Im[s] ==0,Re[s]>0}]

f[k_,s_] = 1/Sqrt[2 Pi]Integrate[gaussian[x,s]Exp[I k x],{x,-∞,∞},Assumptions->{Im[s] ==0,Re[s]>0}]

<<FourierSeries`

FourierTransform[gaussian[x,s],x,k]

next let's put in a nonzero mean

gaussian[x_,s_,m_] =1/(s Sqrt[2 Pi ])  Exp[-(x-m)^2/(2s^2)]
Integrate[gaussian[x,s],{x,-∞,∞},Assumptions->{Im[s] ==0,Re[s]>0}]

FourierTransform[gaussian[x,s,m],x,k]

notice the complex term.  Thus, in general the Fourier transform of a Gaussian is NOT a gaussian, unless the gaussian is zero mean

superposition of two gaussian pulses:  spectral interferometry

gaussian0[x_,s_] =1/(s Sqrt[2 Pi ])  Exp[-x^2/(2s^2)]
gaussian1[x_,s_,deltax_] =1/(s Sqrt[2 Pi ])  Exp[-(x-deltax)^2/(2s^2)]

superposition[x_,deltax_]  = gaussian0[x,1] + gaussian1[x,1,deltax]

SI[deltax_,k_]= FourierTransform[superposition[x,deltax],x,k]

deltax = 5
Plot[superposition[x,deltax],{x,-5,15}]
Plot[Abs[SI[deltax,k]],{k,-10,10},PlotRange->All]

deltax = 15
Plot[superposition[x,deltax],{x,-5,25},PlotRange->All]
Plot[Abs[SI[deltax,k]],{k,-10,10},PlotRange->All]