Formula for the focus distance scale on a manual lens

For the purpose of working out the focus distance scale, the most convenient form of the lens equation is: $$xy=f^2$$ where \$x\$ is the object distance from the front focal point of the lens, \$y\$ is the image distance from the (rear) focal point of the lens, and \$f\$ is the focal length of the lens.

For traditional manual lenses the lens is moved relative to the film plane (sensor plane) to achieve focus. At infinity focus, the film (sensor) is at the rear focal point of the lens (usually just called the focal point). If the lens is moved by the distance \$y\$ away from this position, the object distance which is in focus is given by the lens equation: $$x=\frac{f^2}{y}$$

However, the distance scale engraved on the barrel of the lens is not the object distance measured from the front focal point of the lens, but the object distance measured from the film (sensor). Call this distance \$X\$. For a thin lens, $$X=x+y+2f=\frac{f^2}{y}+y+2f$$

In this formula, \$y\$ is the distance the lens has moved from the infinity focus position, which will usually be proportional to the angle the focussing ring has been turned (assuming the focussing ring is a simple helical screw, which is very common on old manual prime lenses).

\$X\$ is the object distance (i.e. the focus distance) from the film or sensor plane. So the value of \$X\$ is the distance engraved on the lens barrel corresponding to the lens having been moved by the distance \$y\$ from its infinity focus position.

Real camera lenses are not thin lenses and this formula is inexact for most real lenses. We need to apply a correction to this formula to make it work for thick lenses: $$X=x+y+2f+h=\frac{f^2}{y}+y+2f+h $$ where \$h\$ is a constant that depends on the exact design of the lens. The easiest way to determine \$h\$ is to measure it by experiment. It may be positive or negative. There is no simple way to calculate it for yourself. However, if you know the full details of the lens design, then software such as Bill Claff's Optical Bench can work it out for you.

Note: The formulas above apply only to manual focus lenses that achieve focus by moving the complete optical unit of the lens. These formulas do not apply to lenses that have internal focussing (i.e. focus is achieved by moving just an internal part of the lens, not the whole lens). Most modern autofocus lenses use internal focussing.

Example

Suppose we have a 50mm lens. Suppose the focussing ring moves the lens by 0.1mm for each degree turned.

Now, let's work out where the mark for 10m needs to be put on the barrel. We assume that the mark for infinity focus has already been put in place (that is always the starting point).

We need to work out the value of \$y\$ that corresponds to \$X=10000\$ (in mm) and \$f=50\$. Now \$x=X-2f-y-h\$, but \$2f\$ and \$y\$ and \$h\$ are small compared to \$X\$, so \$x=X\$ is a pretty good approximation.

Hence the following formula is a good approximation: $$y=\frac{f^2}{X}=\frac{2500}{10,000}=0.25$$

So the lens has to be moved 0.25mm from the infinity focus position to the 10m focus position (approximately). Hence the focus ring has to be turned by 2.5 degrees. So the 10m distance mark must be 2.5 degrees away from the infinity distance mark.

So, to a first approximation, the focus distance is inversely proportional to the angular distance that the focus ring must be turned from the infinity focus position.

For shorter object distances, it is not a good approximation to say that \$x=X\$ and the longer formula for \$X\$ needs to be used. This makes it more complicated to work out \$y\$ from \$X\$, but it is still possible.

Update

I'm looking for how the focus distance scale numbers are typically constructed on the lens barrel, and whether there is a simple formula that can predict their distribution.

There is a simple formula that is approximately true. The error increases as the distance gets shorter, but the formula is almost exact at the larger distances.

Look at the distance between a distance mark and the infinity mark. So, for the scale on the lens illustrated, the 10m mark is twice as far from infinity as the 20m mark is. The 5m mark is twice as far from infinity as is the 10m mark.

In general, if you consider the position of the mark for the focus distance \$X\$, its distance from the infinity mark is inversely proportional to the focus distance:

$$Y=\frac{c}{X}$$

where \$Y\$ is the angular distance between the mark on the barrel for focus distance \$X\$ and the infinity mark; and \$c\$ is a constant (which depends on the focal length of the lens and also the pitch of the helical screw used to change focus).

As I have already said, this formula is close to exact for larger values of \$X\$ but it becomes less good as \$X\$ becomes smaller.