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I have been wondering about a limiting case of the focal length. Specifically, a lens with a focal length of $0$. Could that exist in real life, and what would be its meaning? Also, what does it mean theoretically?

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If a lens brings parallel rays of light to a focus, the focal length is the distance from the lens to the focal point. The distance is measured from a particular point inside the lens.

A focal length of 0 would mean that the focal point is at that point. No real lens does this.

Given a fixed diameter, lenses with short focal lengths are more complicated and have more aberrations (failures to bring light to a perfect focus) than lenses with long focal lengths.

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    $\begingroup$ No real lens brings parallel rays to a focus. $\endgroup$
    – MJC
    Jul 1, 2019 at 10:53
  • $\begingroup$ @MJC - Not to a perfect focus. If nothing else, diffraction prevents that. But many lenses do a very good job. $\endgroup$
    – mmesser314
    Jul 1, 2019 at 16:53
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    $\begingroup$ I was more talking about paralllel rays existing. $\endgroup$
    – MJC
    Jul 2, 2019 at 7:24
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If the focal length is itself 0, theoretically, the rays would or tend to focus at the optical centre. But, practically, no image can be obtained at the optical centre. It can't exist in real life, because for the focal length to tend to 0, you would practically require a lens of infinitesimally large thickness (infinitesimally thick). That is not possible. So, there will be no implication for this case.

When I talk about thickness, it is the thickness of the centre relative to the edge.

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Imagine a lens that is an inch in diameter, with a focal length of two inches. Shrink it to a half-inch diameter, with a focal length of one inch, then to a quarter-inch diameter, with a focal length of a half inch, and so on. Until the lens diameter begins to approach the wavelength of light, its behavior can be described in terms of waves or rays. The usual Lensmaker's Formula will work well.

However, when the lens diameter is smaller than half the wavelength of light, its behavior is best described in terms of waves. Light can get through the lens, but it will have two components: the evanescent field which is only observable very close to the lens, and the far field which can be observed all the way downstream to infinity. The evanescent field dies out to essentially zero within a distance corresponding to a few wavelengths. The rest of the light, though, is effectively a spherical wavefront emitted from a pinhole the size of the lens aperture. Neither the evanescent wave nor the far field wave from a single sub-wavelength lens can form an image at least in the normal sense of the term "image".

So, if the size of the lens - and its focal length - keep getting smaller and approaching zero, it will cease to act as a lens and start to act like a point source, emitting a spherical wave whose phase is directly related to the phase of the illuminating light. A pinhole will work almost exactly the same as the lens. Though a single such lens does not exhibit very interesting behavior, an array of such lenses can exhibit interesting behavior, simply because of that phase relationship between illuminating light an emitted light.

So, in direct answer to your question: Yes, for all practical purposes, a zero focal length lens can exist. Its practical and theoretical meanings are discussed above.

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