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## Sep 9 Rangefinder Design - Coupling

Rangefinder coupling is the process of linking a rangefinder adjustment to the focusing of a lens. If the system uses interchangeable coupled lenses, we call the the focal length of this lens initial lens the nominal focal length, if other lenses are used, they will need a cam or a system of helicoids that emulate the movement of the nominal focal length.

## Focusing extension

Lens focus is traditionally achieved by moving the entire lens assembly forward to focus closer. Nowadays floating element designs are common, where only certain optical groups are moved independently to correct for aberrations caused when focusing and yield better images. Here we will only cover the former type.

You will often find that the thin lens equation is used (1/f = 1/o + 1/i), but this will not yield great results. A better formula is given in Josef Stüper & Norman Goldberg’s books :

Δ = f^2/E-2f

where Δ is the movement of the lens

f is the focal length

E is the focusing distance

This formula is accurate up to 10•f.

ex : for a 50mm lens, focusing at 1m.

Δ = f^2/E-2f

Δ = 50^2/1000-250

Δ = 2.778mm

## Coupling

So to couple a rangefinder one has to link the lens displacement distance with the angle needed to focus the rangefinder to that same distance. There are two ways of doing this, either with a straight cam, or with a shaped one. Shaped cams are more expensive and complicated, thus straight cams are often used. Since we are matching a linear movement to a rotational one, there will be some approximation. So the question arises what are the requirements within which errors due to the approximation are negligible?

If we say that there are two focusing distances Eo & Er, for the lens and rangefinder respectively.

Δ = f^2/Eo-2f

because of the small angle approximation :

θ = Δ/h

therefore:

Er = b/α = b/2θ = bh/2Δ

so

Er = (Eo - 2f) bh/2f^2

As long as 2f is negligible there is a value h0 = 2f^2/b where a straight cam can be used.

In most cases there will not be space for a single long arm, multiple smaller arms are used to link the cam to the mirror. If L is the length of the mirror linkage, a1 and a2 the lengths of the arms of the lens linkage then it is necessary that:

h = (a2/a1) • L

For accurate focus to be achieved.

Of course, in real life there is a difference between Eo and Er:

Er = (Eo - 2f) bh/2f^2

Er = (Eo - 2f) h/h0

the value of h can thus be chosen to mitigate the effect of 2f at a certain focusing point. The trick is to chose a value of h so that Eo is within the DoF of the lens.

DoF of 50mm f/2.8 lens

Here is a chart showing the rangefinder coupling accuracy of the second version of the Holmium, if the linkage is straight (non adjusted, red) the system would front focus between 3 and 0.8m, if h is adjusted it can be within the DoF limit for the whole range.

Now if a shaped linkage is used instead, like the one I designed for the Holmium, the coupling accuracy is much higher, and is even better when h has not been adjusted. The difference is quite notable, while the staight linkage system struggles to keep everything in focus even at f/2.8, the shaped linkage would have no problem with a f/0.7 lens.

When adapting lenses different from the nominal focal length to focus with a rangefinder, one has to use a system of helicoids or a shaped cam to emulate the nominal focal length.

As the focal length decreases, so does the displacement and vice versa. So for lenses shorter than the nominal focal length the cam has to move more than the lens, and for long lenses it should move less. Essentially you have to ensure that for each focusing distance of the focal length you are using, the rangefinder believes that the nominal focal length would be at that same focusing distance