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I’m doing high speed scientific imaging. I have a camera mounted in a fixed location with a Micro Nikkor 105mm lens. It’s viewing a fixed subject that has a small mirror oriented 45 degrees underneath it. This allows the camera to simultaneously view the subject from the side and the bottom. Unfortunately, I can’t get both the side view and bottom view in sharp focus at the same time, because the optical path length for the two views is different due to the mirror.

I’ve thought of a few possible solutions to bring both views into focus at the same time, and I’d like to ask about one of them.

If I put a small, separate convex lens in the camera’s view so the side view light passes from the subject, through the convex lens, then into the camera lens, but the underside view does not pass though the separate convex lens. would that, given the right lens and placement, allow me to bring both views into sharp focus at the same time?

Also, if it would work, can anyone point me to the math required to estimate the right focal length for the convex lens?

If there is some other solution you can think of that would solve this, I’d be interested in hearing about that too.

Thank you!

Congratulations — you've invented bifocals! Or, in other words, sure, there's no reason this wouldn't work.

The math should be the same as that from using a front-of-the-lens macro adapter. These are usually given in units called "diopters", the same as for glasses or contact lenses. And because they're simple lenses (as opposed to complex or compound lenses), their "power" and focus distance are inherently linked. Specifically, you find the focus distance for a given lens by dividing 1m (1000mm) by the power in diopters. For example, a +8 diopter lens has a focus distance of 125mm. This is assuming that your primary lens is focused at infinity.

Of course, your "bifocal" arrangement adds some complication, because you can't just focus at infinity. What follows is theory and I don't know how it works in practice. In theory, to figure out the right distance when the lens isn't focused at infinity, you would convert the actual focus distance to diopter power — divide 1000mm by the focus distance. So, if you're working at 500mm, that'd be +2 diopters. In combination with the +8 add on lens above that'd be +10 diopters, so now that'd have a working distance of 100mm.

Cinematography solved this problem long ago with the split diopter. It's literally just half of a screw-on diopter, just like so-called "close up filter" (although it doesn't filter anything, it's called that because it mounts to the lens like other front-mounted actual filters).

Split diopters don't see much use in photography, but they used to be used in film a lot, when the director wanted to draw the viewer's focus simultaneously to a near subject and a far subject. They're still used in film, but with today's CGI and compositing, the effect is now usually done in software with multiple takes, rather than in-camera in the same shot.

This article talks about several classic cinema shots achieved with split diopters.

^{split-diopter shot from All The President's Men}

^{split-diopter shot from Jaws}