Are hyperbolic lenses free of abberation?
They still have chromatic abberations, but the spherical abberations are virtually non existant.
Theoretically speaking, what shape of a convergent lens (that focuses rays onto a single point) is free of abberation?
Does such shape/surface exist?
There are five 3rd order geometric aberrations; Spherical, Coma, Astigmatism, Distortion and Curvature of Field. A parabolic lens will eliminate spherical aberration, but not the others.
You can get lenses whose aberrations are less than the diffraction limited spot, in other words the lenses are so good that imperfections are washed out by diffraction. Such lenses are only optimised to a particular object distance and focal length, so you can't get lenses that are aberration free for an arbitrary object distance.
Chromatic aberrations are still present in the lens, you can get lenses called achromats (or achromatic lenses) that are virtually chromatic aberration free within a particular wavelength range.
To summarise, you can get diffraction-limited lenses, but only for a particular object distance and a particular wavelength range.
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