Equivalence and its limitations (long post containing basic maths)
Equivalence and its limitations (long post containing basic maths)
Dec 12, 2011

Equivalence has become widely accepted as a theory for understanding the different results equal settings have on camera's with different sensor size, and for predicting which lenses and settings should be used if one would like to obtain similar results despite sensor size differences. There has been a kind of polarisation between the believers and the nonbelievers, which has led to statements that equivalence is a fact of life, a true reality. Well, things are not so simple.
For those willing to go one step further I will give one example to illustrate this. The popular belief is that the following settings are equivalent (at all times) :
Full frame : 100mm f/10 iso 1600
4x crop : 25mm f/2.5 iso 100
Well, the fact is, that if the optical systems behave according to the thin lens theory, the above is only true when focussing at infinity. When focussing at closer distance, the settings giving the same field of view (at the infocus plane) are different.
I will now show that for focussing at 1 meter, the equivalent settings are these :
Full frame : 100mm f/10 iso 1600
4x crop : 27mm f/2.7 iso 117
First of all, when focussing a 100mm lens at 1000mm, the distance between the sensor (or film plane) and the optical center of the lens is 111mm.
Indeed : 1/100mm = 1/1000mm + 1/111mm (thin lens formula)
For obtaing the same FOV on a 4x crop camera this distance should be reduced to 111mm/4 = 28 mm, and the focal length equivalent to 100mm is then 27mm.
Indeed : 1/27mm = 1/1000mm + 1/28mm (thin lens formula)
The DOF formulas for thin lenses indicate that the physical aperture diameter should be the same for same DOF, in this case 10mm, hence the f/2.7.
The factor to apply (once for the focal length or Fnumber, twice for the iso) is not the crop factor 4, but a lower value : 3.7
For the freaks : the calculations above can also be rewritten in fractions.
1/100 = 1/1000 + 9/1000
37/1000 = 1/1000 + 36/1000
The figure 3.7 is hence an exact result from the thin lens formulas.
The ratio 3.7/4 is equal to 0,925 which can be rewritten as 1  0,075
The deviation from the popular theory in this case is thus 7,5 % .
The nice thing of this deviation is that it also applies in the opposite sense. If one uses the popular formulas, applying a ratio of 4 instead of 3.7 when focussing at a distance of 1 meter, the DOF's between the two setups will also show a mismatch of 7.5% (it is possible to prove this mathematically and I have done so).
This deviation can be shown to tend to zero in 2 cases :
1) crop factor tending to 1 (ok this is rather predictable)
2) focussing distance tending to infinity
So one can say the popular version of the theory of equivalence tends to be true for thin lenses when focussed far away. For close focussing distances, however, the theory starts to fail in two ways :
1) real lenses will typically no longer behave like the thin lens model
2) even if they do so, there is the fact of the deviation being introduced.
These deviations are important when conducting 'scientific' tests, but let there be no doubt : I for one continue to use the formulas in their popular form when shooting in the field (with camera's with different sensor sizes) or when considering lens purchases.
With this example I hope to have provided some more food for thought ... for both believers and nonbelievers
Regards, P.
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