On the optical power of a lens and the effect of additional optical elements

Albert Ng remarked in recent comment:

I've been calculating lately, the relation between minimum focus distance, focal length and magnification factor; if you can focus as close as your focal length, you should get 1:1 reproduction. Hence, a 100mm macro can focus to 10cm near. Knowing this, I could get a +10 closeup to do the same thing on a standard 100mm (since +10 = 1000mm / 10 = 100mm focus when lens is actually set at infinity). However, how would I know where the focus will be at when the lens is NOT set on infinity?


Firstly, an unnessecary introduction to the units:

Dioptre: a measurement unit for the optical power of a lens. It is the inverse of the lens’ focal length. The net power of optical assemblies consisting of thin lenses placed closed together can be closely approximated by summing the optical powers (measured in Dioptre) of all the lenses. This is the advantage of using Dioptre instead of focal length.


Let’s say you use a 50mm lens. When its focus is set to infinity, its power is 1/0.050m, or 20 Dioptre. (After approximating the 50mm lens as a thin lens located 50 mm away from the film plane.)

Now, suppose you want to focus on something 50 mm away. Instead of turning the lens’ focus ring, you put another 50 mm lens in front of the first, but positioned back to front. The focus of this second lens is also set to infinity.

In this double-lens set up, light from the target enters the reverse-mounted lens, and is projected as parallel rays (since it is focused to infinity). The parallel rays then enter the second lens and converge onto the focal plane.

The second lens is identical to the first, so it would be rated at 20 Dioptre too. The total optical power of the system is 20 + 20 = 40 Dioptre.

If you used a 10 Dioptre lens instead of a reverse-mounted lens, light from 0.1 m away would be projected as parallel rays towards the first lens. These parallel rays would then be focused by the 20 lens onto the focusing screen. Thus a 50 mm lens that focuses on a subject 0.1m is effectively a 20 + 10 = 30 Dioptre lens


By using this argument, it appears that you can estimate the dioptre of a lens when it is not set to infinity. Working backwards, you can then estimate the subject distance.

Have fun.