How does a converging lens create the Fourier transform of an aperture?

[fisico30]

Hello,

does anyone know how a converging lens forms the Fourier transform of an aperture when the obs. screen is at distance=f?
If each point emits a spherical wave, the lens should make it then parallel and the FT should be the interference resulting from that.
However, if we decompose the amperture in plane waves, each plane wave will leave be focused to a point in the focal plane.
The latter explanation (plane wave decomp.) is clear but I think the first (spherical wave) is more physically true. Any clarifications or correction to these views? How does this spherical wave interference match the plane wave result?

 

[lpfr]

I think that the easiest way to understand this is to compute de Fraunhofer (that is: at infinity) diffraction pattern of the aperture. Then, if you realize that a converging lens concentrates all rays coming from a direction to a point in the focal plane, you will see that the lens just brings the diffraction pattern from infinity to the focal plane.

If you use your cornea as converging lens, you can see the diffraction pattern at infinity with your retina. Try to see a distant point source (plane waves) through a very small hole (0.1 to 0.2 mm in diameter).

You are right; you can decompose the aperture in spherical waves but not in plane waves. A plane wave is a complete plane and not a bit of a plane.