Let us say I have a single (converging lens) could I use the Abbe diffraction limit (for a microscope) to find its resolution or do I have to use the Rayleigh criterion. (i.e. Is the Abbe diffraction limit strictly for a microscope with more then one lens, or can it be used with only one lens?)


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The Abbe diffraction limit and the Rayleigh criterion (first zero in Bessel function) describe the same reality at the same level of abstraction.

If optics are well-corrected for aberrations, we say they are diffraction limited, meaning that the geometrical and color aberrations are so small, that they don't matter in comparison to the diffraction limit (wave optics).

For an object and image at small field angles (close to optical axis) and weak lenses (low refractive power) with almost monochromatic light (no chromatic aberrations) there will be a range of imaging conditions where the paraxial approximation is valid enough (i.e aberrations are small enough) even for a single biconvex lens with spherical surfaces to speak of the limitation being the diffraction limit. Once you allow an aspheric surface or two, this becomes a reality for more cases (still monochromatic).

An example for a single optical surface that routinely fulfills this condition look at parabolic mirrors for astronomical imaging (very small field angles, no chromatic effects since it is a mirror, object at infinity for optical purposes).


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