# Finding *f* with a Macro Lens

## The Method

We start with equations (5) and (6) as derived on the leading page.

1 1 1 ----- + ----- = - (5) a + b c + d f c + d M = ----- (6) a + b

Then we express *c* + *d* from (6), substitute into (5), and solve for *f*:

c + d = M (a + b) 1 1 1 - = ----- + ----- <=> f a + b c + d 1 1 1 - = ----- + --------- <=> f a + b M (a + b) 1 M + 1 - = --------- => f M (a + b) f (M + 1) = M (a + b) (7)

The unknowns here are *M*, *a*, and *b*. *M* is easy — we use a macro lens, and read it right off the lens barrel.

To find out *a*, we proceed as follows:

- Mount the lens on a camera body.
- Turn the lens focusing ring to a freely chosen magnification ratio
*M*. - Focus on an object by moving the lens-camera combo back and forth.
- Measure the distance
*a*between the object and the front-most point of the lens.

For *b* we perform these further steps:

- Find a dark distant object in front of a bright background (for example a distant tree in front of a sunrise or a sunset).
- Without refocusing the lens, take it off the camera, and point its
**rear**element towards the object. - Place a white piece of paper behind the
**front**element of the lens. Use the paper as a screen to project an image with the lens. - Move the paper back and forth until the image of the distant tree is in focus.
- Measure the distance
*d*between the front-most point of the lens filter thread and the paper.

Figure 1 illustrates.

Figure 1.

Now we see how to calculate *b* from *d*. We start again with equation (5), but because the lens is used backwards, and because of the equation’s symmetry, *b* and *c* swap their positions. So (5) becomes:

1 1 1 ----- + ----- = - (8) a + c b + d f

Now comes the approximation — we assume that the distance to the object (the tree) is equal to infinity. Equation (8) then simplifies to:

1 1 1 -------- + ----- = - <=> infinity b + d f 1 1 ----- = - <=> b + d f f = b + d <=> b = f - d (9)

Now we substitute (9) into (7), and solve for *f*:

f (M + 1) = M (a + f - d) <=> fM + f = fM + M (a - d) <=> f = M (a - d).

## Method Evaluation

The two main merits of this method are:

- Simplicity, both experimental and mathematical.
- High tolerance to measurement errors. We measure the quantities
*a*and*d*, and their dependance to*f*is linear. Second, we are free to choose*M*, and for smaller values, the effect of any measurement errors decreases.

At the same time, three main drawbacks should be pointed out:

- Because the method assumes that the distant object lies at infinity, the result is only approximate. The ill-effects of this approximation are very small, however. For example, for a 100 mm lens and an object located 20 meters away instead of at infinity, the maximum error in
*f*is 0.5 millimeters (for*M*= 1:1, smaller for smaller values of*M*). - The method is directly applicable only for lenses that have the magnification ratio marked on the barrel. It can be applied to any lens, but then
*M*must be calculated in a separate experiment. - The method finds the focal length only for the chosen
*M*. In general this is no limitation, as most lenses retain a constant*f*as they are refocused. For those that do change*f*, the method can be applied a number of times to get an idea of how*f*changes. The method fails for one important case, however, when the lens is focused at infinity (*M*= 0). This is so because*M*is present in the final expression for*f*.