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Most lenses are spherical, but...

Are there elliptical, parabolic or hyperbolic microscope lenses?

(I asked a similar question, about telescopes, on Astronomy S.E.; I hope this is not considered 'double-posting').

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Yes, there are lenses of many types for different applications. In particular there a lenses which a specifically called aspherical lenses. These can be used to reduce certain types of optical aberrations, such as spherical aberration which arises specifically because spherical isnt't the best shape for lenses.

There are also lenses which aren't even radially symmetrical such as cylindrical lenses which can be used to adjust a beam of image in one transverse axis but not the other.

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In my answer to your related but different question in Astronomy SE I mention that it's so much easier to make spherical surfaces than aspherical that one only uses at most an occasional aspherical surface, and only when it solves a problem that an additional spherical surface can't for some reason.

Reasons that adding more spherical surfaces is unattractive include

  • additional reflecting surfaces (antireflection coatings are imperfect)
  • attenuation of light if the wavelength is unusual and lens material is absorbing
  • additional mass, complexity, or thermal issues are a problem in exotic conditions (in space for example)
  • additional length; if you are for example trying to put microscope-like performance in a cell phone camera and don't want to use mirrors to make the optical path longer.

The only routine uses of aspherical surfaces in microscope lenses that I'm aware of are

  1. Tiny microscopes, either designs for cell phones or devices that need to go into tiny places (cf. 1)
  2. Lenses that are called microscope objectives, but are really used for coupling in/out of single mode optical fibers or spatial filters. They can have terrible off-axis and chromatic aberration as long as they can focus a single wavelength to a diffraction-limited spot with high NA (cf. 1)
  3. Reflective microscope objectives are used when lenses won't do. This can be for pulsed lasers, or UV or IR light where glass causes a problem, or when you have very high NA but need low weight or to minimize heating and thermal drift. (cf. 1, 2, 3)
  4. Condenser lenses are in a microscope and are part of the imaging optics because the illumination of a sample is a critical parameter. For more on that search for "illumination partial coherence" and don't worry about the use of the word coherence, the term applies to incoherent treatments as well.
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