Fourier Optics: Why Does a Lens Perform a Fourier Transform?

[X-Kirk]

I have been studying Fourier Optics and I have a basic conceptual question. I understand the mathematics of how to perform Fourier Transforms however the part of this topic I seem to have missed is why the action of a lens on light is the same as performing a Fourier Transform on the functional form of the input wave?
Other than having been told by my lecturers that the output of a lenses is the Fourier Transform of the input I have no idea why this happens?

I hope this question is clear. The whole concept is rather confusing to me.

 

[orthogonal]

Contrary to popular belief (Goodman even leads his readers this direction) a lens does not perform a Fourier Transform. A lens is only helpful in displaying and manipulating the Fourier transform.

What creates the Fourier transform is the coherent light interacting with the fine structure of the object illuminated. Do you need a lens to perform a Fourier transform? No one simply need to go into the far field to observe it (Fraunhofer distance). By using a lens, one can move the far-field into the focal plane of the lens.

Incidentally, even with a lens, without coherent light, the Fourier transform is not formed.