A lens’s diopter is just the reciprocal of the focal length of the lens, with units of 1/(meters); equivalently, the focal length of a lens is just the reciprocal of the diopter value, in meters. So a +1 diopter corresponds to a focal length of 1 meter; +2 diopter corresponds to 0.5 m (500 mm); +10 diopter corresponds to 0.1 m (100 mm); etc.
Diopter units are particularly convenient to use when combining lenses together: their equivalent diopter value is just the sum of the individual diopters. For example, suppose a 50 mm lens has a +2 diopter close-up lens mounted on the front. The diopter value of the lens is 1/50mm = 1/0.05m = 20. So the total equivalent diopter value is 20 + 2 = 22, which corresponds to a focal length of about 0.045 m, or 45 mm.
But wait, if the diopter of the combined lens plus close-up lens is just the sum of the diopters, does that work for more than two lenses combined? Absolutely. Adding another +3 diopter to the previous combination results in a total diopter of 25, which corresponds to a focal length of 0.04 m, i.e., 40 mm.
How can I calculate its effect on close focusing distance?
Roughly speaking (see note below), the close focusing distance will be reduced according to:
xnew = xold / ( D ∙ xold + 1) ,
- xnew is the close-focusing distance with the close-up lens applied;
- xold is the original close-focusing distance (without the close-up lens);
- D is the diopter of the close-up lens.
For example, suppose a lens can focus as closely as 500 mm. With a +4 diopter close-up lens attached, the combination can now focus as close as 500 mm / (4 m-1 ∙ 0.5 m + 1) = (500/3) mm ≈ 167 mm.
Note: The reason I say “roughly speaking” is that these distances are based on distances from the center of an idealized single thin lens with negligible thickness, such that the thin lens formula can be used. Real-world optical lens elements have thickness, and of course real-world photographic lens systems consist of several individual lens elements and lots of air space in between many of them. Additionally, when talking about a camera lens’s Minimum Focus Distance (MFD), the MFD is measured from the camera’s film/sensor plane. Thus, the MFD includes the image-side focus object-side focus distances in the thin lens formula, as well as the substantial length of most of the lens assembly (which is usually not constant as a lens focuses). But the “close focus distance” discussed above is merely the mathematically-described object-distance side of the then lens. Thus, this formula is a guide to judge what diopter you might need.
With a close-up lens, you will no longer be able to focus far away. The furthest you will be able to focus is simply the focal length of the close-up lens – i.e., the reciprocal of the diopter, in meters. So with a +4 diopter you cannot focus any further away than 0.25 meters (250 mm).
How much magnification can I get from the combination?
Assuming you’re working at the closest focus distance possible, both before using the close-up lens, and with the new closer focus distance with the close-up lens, then the image will be magnified (D ∙ xold + 1) larger than it was without the close-up lens.
For example, if a lens’s closest focus distance is 500 mm, then with a +4 diopter close-up lens and focus at the new close focus distance of xnew, the image will be (4 m-1 ∙ 0.5 m + 1) = 3 times larger than without the close-up lens.
Note that the imaging system’s magnification is not 3:1. Rather, the ratio of magnification after vs. before, Mnew / Mold, is 3.