Does a lens always act as a Fourier transform?

Question

I understand that putting a lens behind an aperture at the distance 1f, it will "get" the diffraction pattern to appear in the back focal plane. In this case the FT of the aperture plane happens within the realm of using Fraunhofer diffraction.

Does a lens generally Fourier transform its front focal plane into its back focal plane or is that just the special case with Fraunhofer diffraction and the idea of the lens as a FT is kind of misleading?

Answer 1

An ideal lens takes a point source that is located on its focal plane, say $\mathcal {F}$ emitting monochromatic homocentric rays/spherical waves and transforms them into parallel rays/plane waves whose direction (propagation phase) depends on the location of the point source in the focal plane relative to the symmetry axis.

By reciprocity it also goes the other way around, that is an ideal lens converts incident parallel rays/plane waves into a point located on the focal plane on the other side of the lens. All this means that as far as the focal planes are concerned, say $\mathcal F$ on one side or $\mathcal F'$ on the other side, homocentric (congruent) rays on $\mathcal F$ are converted into plane waves and vice versa, and thus is a 2D Fourier transform. Point sources are just "Dirac delta", and their FT is constant, hence, plane waves, only their phases vary according to the location of the source. To show this you still need to consider that the system itself is linear shift invariant and linear superposition holds for rays as well as for wavefronts.

Note though that the result of the transformation is not a pure Fourier Transform for two reasons:

1. there is no ideal lens whose focus is uniformly point-like across its "focal plane"
2. even if it had ideal foci the resulting transformed parallel rays/planar wavefronts have an excess position dependent quadratic phase shift (spatial phase modulation). The FT magnitude is correct but its phase is off. Also, the result is not the FT of the aperture field (ie., the field at the exit aperture of the lens) itself but rather the aperture field also carries a spatial quadratic phase modulation and then we take its FT after the modulation.