Well you might ask, don't we have enough shrinks and how do those factor in
Well, in ye olde times lenses did not need shrink-factors, because they simply did not shrink. That were the times where a lens was simply extended away from the sensor to focus closer than infinity. Some the modern lenses still work like this: a 50mm lens has to be shifted 5mm forward to focus at 60.5 cm distance which in turn gives a magnification of 1:10 (= 5mm/50mm).
Many modern designs (esp. zooms) though do their focussing by shifting only some lens(groups) within the lens. This may include some extension of the front-lens when focusing closer. Those design are normally called IF designs, the IF standing for "internal focusing". The funny thing now is, that those designs tend to vary their focal length: Typically when focusing closer they also reduce their focal length. This in turn has the following consequence: If you focus a non-IF lens at say 1.6m the magnification is larger than the IF-lens of the same nominal focal length focused at 1.6m!
Let's have a look at a practical example: My trusty old Sigma 400/5.6 APO macro gives a magnification of 1:3 at a distance of 1.6m. It is clearly an IF-design: the front lens does not move while focusing.
Let's calculate what a non-IF design 400mm lens would yield at 1.6m focusing distance.
If you mount a standard 400mm on a large bellow and extend that to 400mm your focal plane is at 1.6m distance from the sensor and you have a reproduction ratio (or magnification) of 1:1! Well, that's much better than the IF-design performs.
Now you can ask the other way round: What is the
effective focal length of my Sigma, when focused to 1.6m. Or: What focal length must a non-IF-design have to give a magnification of 1:3 at 1.6m distance.
That would be a 300mm lens at an extension of 100mm.
How do you get there?
For the sake of making the formulas more readable let's call the inverse of the magnification
i. In this case
i=3. Now do the following:
1. Divide the distance (=1.6m) by
i+1: 1.6m/(3+1)=40cm. This gives you the sum of effective focal length
f plus the extension
e of this lens to focus at 1.6m. So
f+
e=40cm
2. To get the effective focal length
f from that just divide
f+
e (=40cm) by the
i+1 and multiply by
i: 40cm/4x3 = 30cm
Easy, isn't it?
Now we have calculated that the Sigma 400mm/5.6 APO macro
shrinks to a 300mm lens when focusing to 1.6m. And what would be best to express this in terms of the ratio 300mm/400mm=0.75 (or 75% if you like). So the shrink-factor of this lens is 0.75x!
Now, what is when you focus this lens somewhere between infinity and 1.6m?
Well, I'm pretty sure that the shrink factor then is somewhere between 0.75x and 1x, but I have not tested this. And one thing is clear: focused at infinity the lens behaves perfectly like a 400mm lens. Thus the shrink-factor is 1.0x at infinity. Because this is trivial, let's just ignore that any lens at infinity has no shrink. So when we talk about
the shrink factor of a lens we refer to the shrink factor the lens exhibits at its closest focusing distance and (in case we are looking at a zoom) and its longest focal length.
So let's do another example, because we need it for this other article about
How to calculate magnification with close-up lenses. The new
Nikon 70-200/2.8 VRII.
This lens has a very weak magnification of 0,12x or 1:8.3 at a closest focusing distance of 1.4m. This magnification can be reached only if the lens is set to its nominal focal length of 200mm.
Now, what is the shrink factor?
1.
i = 8.3
2.
f+
e = 1.4m/(8.3+1) = 15cm
3.
f = 15cm/9.3x8.3 = 133mm
Whoa, shocking! This brand-new design from Nikon effectively behaves like a 133mm at its nominal focal length of 200mm when focused to its closest distance. That marks a new all-time low in shrink-factors: 133/200 = 0.67x
(just kidding)
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For the (even) more technically inclined readers here is the
disclaimer that no article about such calculations should be without:
All calculations are based on a simplified model of a lens, even for non-IF-design "normal" lenses.
So take all calculations with a grain of salt and don't delve into too many decimal places.
Apply at your own risk 
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If you read my lens-reviews you can find my calculations of the shrink-factor for a many lenses.
Would be fun, if you add your own calculated shrink-factors here so we get sort of a database and can see which company creates e.g. the "most incredibly shrinking zoom"
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