Easy General Way to Approximate f

The Method

This method does not depend on the magnification ratio, so we concentrate only on equation (5) from the previous page:

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

We start as follows:

1. Take the lens off the camera body.
2. Locate an opaque object at a relatively close distance.
3. Place a light source behind the object.
4. Position the lens in the shadow of the object.
5. Turn the front element of the lens towards the object.
6. Place a white piece of paper behind the rear element of the lens, perpendicular to the lens axis. This paper will be used as a screen that shows the image that the lens projects.
7. Move the paper back and forth until the image of the object is in focus.
8. Measure the distance a between the object and the front-most point of the lens.
9. Measure the distance d between the rear-most point of the lens mount and the projected image.

Now the only remaining unknowns in (5) are b and c. So we continue:

1. Find a dark distant object in front of a bright background (for example a distant tree in front of a sunrise or a sunset).
2. With the lens focus ring in the same position as in the previous experiment, point the front lens element towards the object.
3. Place a white piece of paper behind the rear element of the lens. Use the paper as a screen to project an image with the lens.
4. Move the paper back and forth until the image of the distant tree is in focus.
5. Measure the distance d& between the rear-most point of the
   lens and the projected image.

Figure 1 illustrates.

Figure 1.

To calculate c, we start again with equation (5):

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

Now comes the approximation — we assume that the distance a& + b from the lens to the distant object is equal to infinity. Equation (5′) then simplifies to:

        1         1     1
    -------- + ------ = -                     <=>
    infinity   c + d'   f


       1     1
    ------ = -                                <=>
    c + d'   f


    f = c + d'                                <=>


    c = f - d'                                (7)

To find b, without refocusing the lens we repeat our previous experiment except that we use the lens is backwards: the rear element is pointed towards the distant object, and the front element is pointed towards the formed image. We measure the distance d" between the front-most point of the lens filter thread and the paper.

Figure 2 illustrates.

Figure 2.

To calculate b, we start again with equation (5):

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

Now we use our approximation again — we assume that a" + c is equal to infinity. Equation (5″) then simplifies to:

        1         1     1
    -------- + ------ = -                     <=>
    infinity   b + d"   f


       1     1
    ------ = -                                <=>
    b + d"   f


    f = b + d"                                <=>


    b = f - d"                                (8)

Now we substitute (7) and (8) into (5), and solve for f:

        1             1       1
    ---------- + ---------- = -               <=>
    a + f - d"   f - d' + d   f


    1   f - d' + d + a + f - d"
    - = ------------------------              <=>
    f   (a + f - d")(f - d' + d)


        (a + f - d")(f - d' + d)
    f = ------------------------              <=>
        f - d' + d + a + f - d"


        f^2 + f(a - d" + d - d') + (a - d")(d - d')
    f = -------------------------------------------                        <=>
                   2f + a - d" + d - d'


    2f^2 + f(a - d" + d - d') = f^2 + f(a - d" + d - d') + (a - d")(d - d')      <=>


    f^2 = (a - d")(d - d')


          ________________
    f = \/(a - d")(d - d').

We can also calculate M from our measurements. We start with (6), and substitute (7) and (8) into it. Then we solve for M:

        c + d   f - d' + d   sqrt [(a - d")(d - d')] + d - d'
    M = ----- = ---------- = --------------------------------        <=>
        a + b   a + f - d"   sqrt [(a - d")(d - d')] + a - d"


        sqrt (d - d') sqrt [(a - d") + (d - d')]
    M = ----------------------------------------                     <=>
        sqrt (a - d") sqrt [(a - d") + (d - d')]


       _    ______
        |  /d - d'
    M = | / ------ .
        |/  a - d"

Method Evaluation

The two main merits of this method are:

1. Method is applicable to all lenses, not just macros.
2. Relative simplicity, both experimental and mathematical. Here we need to measure four quantities, compared to two in the “macro” method.

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

1. Because the method assumes that the distant object lies at infinity, the result is only approximate. The ill-effects of this approximation are relatively small, however. For example, for a 100 mm lens and an object located 20 meters away instead of at infinity, the maximum error in b and in c is at most 0.5 millimeters. The effect of this error is decreased if one performs the first experiment with a smaller value of a.
2. Measurement errors are more significant with this method because their values are multiplied together. However, then their square-root is taken, so the effect is still not so significant.
3. The method finds the focal length only for the chosen a. 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 (equations (5) and (5′) become linearly dependant then).