The fourier and laplace transform are pretty much the same thing except σ is nonzero when talking about "laplace" transforms . Fourier transform is laplace evaluated with σ=0.
Fourier transform is the jω axis of the laplace transform. If you "draw" the laplace space. And only take a look at the slice which goes along the jω axis then that is the fourier "space".
Fourier transformation is only a "thin slice" (the jω axis) from the s plane. Stable systems contain laplace transforms that converge on the jω axis and thus have a fourier transforms assuming dirichlet conditions are fullfilled. If you have a laplace transform for a stable system. You can simply put σ=0 and volla you have your fourier transform.
Keep in mind that S = σ + jω. So puting σ=0 you have.....jω
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You can only do this for systems that have laplace transforms that converge for the jω axis.