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I would like to be able to focus with my eye on something that is closer than I can normally focus without optics. I wanna put my eye up to a hole in a box and be able to see the thing in the box clearly and in focus. The thing in the box is approximately 60 x 60 mm. And the maximum distance the lens could be from it is approximately 50 mm, and the lens maximum diameter could not exceeded 70 mm. also my eyeball will be behind protective glasses so there will be a limit to how close I can get my eye to the lens.

Do I want a double convex lens or do I want a Plano lens? Or some other kind of lens. Do I want a focal length of 50 mm because that's the distance away the object will be from the lens? Where is the focal length on the lens measured from? I assume the thickness of the lens needs to be accounted for. And what about the field of view I want to be able to see all of the 60 x 60 object, but I don't want it to be terribly distorted.

I have been experimenting with some lenses I have on hand, but none of them have an appropriate focal length or diameter so I would like to calculate the proper lens required so I can try and find that.

I hope this is clear, I have been trying to educate myself on the physics of lenses, but I can't find the specific information I need so far, so hopefully a human out there can help.

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  • $\begingroup$ Is there a requirement for your eye to be close to the box? If not then some multi lens relay/magnifier/objective would work. If your eye needs to be close then you may need an optic with a custom focal length. $\endgroup$ Commented Jan 27 at 8:40
  • $\begingroup$ My eye needs to be fairly close so that I can clearly see what's in the box, with high fidelity. $\endgroup$ Commented Jan 27 at 10:43

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To get a rough focal length on your own, here are the three required pieces of information:

  1. The eye's minimum focus distance is ~250 mm.
  2. The thin lens equation: $1/s + 1/s' = 1/f$
  3. The angular magnification, i.e. how much detail the eye can see, is generally the highest when the eye is very close to the lens.

Beyond that, without professional optics design software, I believe experimentation will be required.

For the eye to be able to focus, the image needs to be more than 250 mm away from it. If $s$ is the object to lens distance, $s'$ image to lens distance (positive if on the opposite side than the object) and $s_e$ lens to eye distance, then $s_e-s' = s_{ei}$ is your eye to image distance. Since you probably want $s_e$ to be very small, $s'$ will need to be negative to keep your $s_{ei}$ above 250 mm. That is, the image will need to be on the object side, i.e. virtual.

I suggest you play around with the thin lens equation a bit. Pick a feasible $s$ and $s'$, then calculate the focal distance $f$ you'd need. Then see which focal distances are feasible and calculate some other $s$ or $s'$ which would work with that. Iterate until you get somewhere or realize you'll need a different approach.

Keep in mind that this system may be more fiddly in practice than you expect, even if a mathematical solution exists. With cheap off-the-shelf optics, you might get more detail by increasing the lens to object distance $s$, if possible.

Edit: Forgot about the field of view.

The lens is like a window through which you see the image. By sketching out this window and using the triangle similarity we can determine how large the lens needs to be for the whole object to be visible: $D_l / s_e > D_o / s_{ei}$, where $D_l$ is the lens diameter and $D_o$ object diameter or diagonal.

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  • $\begingroup$ Thank you very much, this is exactly what I was looking for: an equation! And what of the kind of convex lens? I see a lot of double convex lenses available in a price range that would facilitate experiment. Plano convex lenses seem to be more expensive and it would be a non-starter to have to buy more than one or two of them in the experimenting phase. $\endgroup$ Commented Jan 27 at 22:13
  • $\begingroup$ Honestly, I don't know anything about optics design or about individual lens elements. Gaussian optics solves the problems I have, which are similar to your setup, but not as space-constrained. I only ever use off-the-shelf lens assemblies. $\endgroup$ Commented Jan 28 at 8:47

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