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I'm not the best at math, but I'm wondering if there is a formula to figure out the thickness of a lens from only knowing the focal length and diameter.

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You need to know more than those values, but they're necessary. The other thing you need to know is the refractive index of the material the lens is made from.

Knowing these three values, ray tracing computations can give you the exact profile of both surfaces of a single element, which will necessarily tell the minimum thickness. However, within reason, you can add an arbitrary amount of extra thickness to the lens without affecting its focal length (though you will change the curvature of at least one of the surfaces because the extra thickness is extra distance separating the first surface from the second -- and the two surfaces are independent refracting elements).

The computation isn't really a simple formula, it's a set of rules for how light refracts at an interface, and except for very simple cases, it requires software to get from a set of parameters to a final lens prescription. Ray tracing software of a general sort isn't available for free, but I recall reading about a limited version that was freely available (limited to something like four or five elements and only a few glasses -- it also handles dispersion, so you could calculate chromatic aberration corrections).

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  • Thanks for your answer. I was specifically curious about plano convex lenses from: surplusshed.com/search_lenses.php and trying to figure out thicknesses of different lenses. So ray tracing computation software would be able to do it? I'd guess that most of these lenses are a refractive index of 1.5... – Curiousmarble Apr 10 '20 at 18:13
  • More or less. Real lenses are seldom thicker than they need to be -- edge thickness tends to run in the 2-3 mm range, at most. That index, however, is a pretty low precision estimate, so your end result isn't going to be very precise, either. Might be easier to contact Surplus Shed and ask how thick the lenses you're interested in are. – Zeiss Ikon Apr 10 '20 at 18:44
  • My predicament is that I'm trying to find a lens with a focal length equal to it's length(thickness?), imgur.com/a/P9IXx3W to function as a larger Stanhope lens. (was hoping for a diameter of 15-30mm) There are a lot to choose from in this range on Surplusshed....I am aware that a Stanhope has a short focal length, so theres that. I'm not sure which ones would be likely the one I'm after. Just didn't want to ask them for multiple lens thicknesses, they might find that annoying – Curiousmarble Apr 10 '20 at 18:51
  • This is the selection I'm given, im not sure if any of these have the same thickness as their length: surplusshed.com/… – Curiousmarble Apr 10 '20 at 18:52
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    Might just ask them if they have any that focus on their own plano surface, or any that act as Stanhopes. They might not know, but that's an example of "abitrary extra thickness" as noted in my answer above. – Zeiss Ikon Apr 10 '20 at 19:02

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