Thanks, I've added that article to my personal collection of links.

Forsell's article strikes me as a good attempt at balancing theory and measurement. I like his use of the "zero extension magnification" as a tool for simplifying some of the equations.

However...

I have a serious concern with the formulas listed by Forsell in that they systematically ignore separation between the principal planes. Sometimes this is OK, sometimes it's not, and there's no way to know which is which except by measurement.

Before I illustrate that comment, let me explain that there's a simple way to accurately measure the focal length of any lens or lens combo. Just measure its magnification with two different extensions behind the lens, and calculate focal length as

FL = (Ext1 - Ext2) / (Mag1 - Mag2)

As an example, my Canon 100 mm f/2.8 IS USM lens at closest setting of the focus ring and no added extension gives magnification 1.029, while adding 66.94 mm of extension takes it to 1.932. From these, we can accurately calculate that at this focus setting the lens has a focal length of (66.94 - 0) / (1.932 - 1.029) = 74.1 mm. At infinity focus, the corresponding calculation is (66.94 - 0) / (0.6657 - 0) = 100.5mm, consistent with the lens designation.

Now let me illustrate my comment that sometimes it's OK to ignore separation between principal planes and sometimes it's not.

In section 8b, Forsell gives the formula F = D/(1/M+M+2) as a way of calculating focal length given the magnification M and distance D between subject and sensor. That's equivalent to the formula I gave earlier, with separation omitted. Still, using the simplified formula with my lens and my measurements of M=1.029 and D = 298 mm gives a calculated result of 74.5 mm. This is about 0.5% accuracy, which seems great.

Unfortunately...

In section 5, Forsell gives the resultant focal length of a stacked lens system as Ftot = (Fp*Fr)/(Fp+Fr).

The more accurate formula is Ftot = (Fp*Fr)/(Fp+Fr-d), where d is the separation between corresponding principal planes of front and back lenses.

Forsell gives an example using certain lenses where his simplified formulas give the right result to within a few percent.

But I have different lenses.

I happened to choose a Canon 100 mm f/2.8L IS USM lens for the rear, and a Mamiya-Sekor 55 mm f/1.8 lens reversed in the front. With the rear lens focused at infinity, the combo gives a magnification of 1.83. That's within about 0.5% of what we would expect based on a simple calculation of mag = F_rear/F_front using just the lens specs. (That particular simple calculation happens to be exact when the rear lens is at infinity focus.)

Now, what happens when I add 66.94 mm of extension?

According to Forsell's formulas, the focal length of the combo should be (55*100)/(55+100) = 35.48 mm, so the 66.94 mm of extension should increase the magnification to (1.83*35.48+66.94)/35.48 = 3.72.

However, when I actually extend the combo and take the picture, the measured magnification increases only to 2.15, not 3.72.

The discrepancy is not due to slight errors in the lens specs or in my physical measurements. Instead, it's because Forsell's formula is too simple to be correct in this case. The principal planes are buried deep inside these lenses, so that in the physical stack they are quite far apart. Instead of having a resultant focal length of 35.48 mm, the combo actually has a resultant focal length of 206.5 mm. If you're surprised and confused that this is longer than either lens alone, then I sympathize, but that's just the way lenses work.

Use the simplified formulas at your own peril.

--Rik