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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?