Question

Using thin-lens theory, determine a formula involving only the refractive index, n, for the maximum “theoretical” aperture of an imaging lens. Express the answer as an f/number. (ii) Explaining your reasoning, make a reasonable estimate of the numerical value of this maximum aperture. (iii) Comment on the practical and theoretical limitations of your solution. (iv) In addition, briefly comment on the advantages and disadvantages of large-aperture imaging lenses.

Answer 1

An aperture can be closed which is approximately an infinitely large f-stop number since no light gets through. The quickest possible (smallest f number) is somewhat harder. The speed of a lens is limited by the ratio of the entrance pupil to the focal length of the lens. The longer the focal length, the bigger the entrance pupil must be. In theory you can make one very very large, but eventually the amount of glass is going to make it so you physically would lose more light than you were gaining.

There "record" for fastest lens is arguably the f/.33 Super-Q-Gigantar 40mm, but it was really just a marketing stunt and only one was ever made. It isn't actually for use. There is a functional f/.7 lens of which 10 were made. Six were purchased by Nasa, Carl Zeiss kept one for himself and 3 of them were purchased by Stanley Kubrick and used in the film Barry Lyndon.

In theory, it should be possible to design lenses faster than this, but the cost and benefit are simply not the effort. The lenses become too expensive and complex and don't offer any actual benefit for the effort since the difficulty goes up fast. (Since each f/stop requires a doubling of the size and physical issues make it more that twice as complicated for each additional f-stop.)