Lab : FF vs APS : Real Crop Factor explained.

With the arrival of DSLRs (Digital Single Lens Reflex) on the market, a new term has appeared in the photographic jargon : the crop Factor ! One can thus read here and there that a body with an APS sensor transforms a 100mm lens into 160mm ! How is this possible, and what does really ?
I will consider in this article the 35mm format as standard. Not that I forget the medium/large formats, but crop factor referring to 35mm, let us avoid spreading. Also, let me set out something : crop factor, Magnification Factor, Focal Length Multiplier, all refers to the same thing.

Crop factor and 35mm
Crop factor and depth of field (DoF)
Aparté on perspective.
Crop factor and pixel density.
Motion blur
Aparté on diffraction.
It gets even more complex ...
Conclusion

35mm format relies on an image circle of 43mmdiameter projected by the lens, in which a sensor or film of 24mm on 36mm is included, the famous 24x36.
At the beginning of the digital era, early in the 90s, the manufacturers have been confronted with a problem of size. Indeed, today the manufacturing prices and thus sale prices of sensors have strongly fallen, but at that time, producing a 24x36 sensor was extremely expensive, and it would have been unsaleable. Canon and Nikon thus proposed smaller sensors, around 15x23mm (APS format). And there, problem of size, once again : the framing of the lenses changes in proportion. The lens is indeed the same, and throws the same image into the reflex cage, but only a portion of the 35mm format is used. Indeed, the image obtained is cropped, as if we had taken a 35mm format and cut the edges out. See beside in which proportions. The angle of view is thus modified.
So, if you want to keep the same framing with an APS sensor, you have to go away from the subject, or to change lens for a shorter one (wider angle of view). But in what measure? It is there that crop factor takes place ; it has been established to give a mark regarding to the 35mm, so that the photographers who were used to this format can easier regain with digital.
Note that since 2002, Canon has been releasing some EOS DSLRs with 24x36 sensors, thus without any crop factor as they correspond to the standard 35mm size. It is the case of 1Ds line (1Ds, markII, markIII). The reductions of manufacturing costs indeed allowed to propose such sensors, even if they remain more expensive than APS type sensors.

Crop factor and 35mm

As we can see above, the diagonal of a 35mm format is of 43mm. APS one is of 27mm. We could also take the length or the width, we would find the same : 43 / 27=1.6.
So, our crop factor for an APS is 1.6 (1.5 for Nikons). This means that if we want a lens to frame like a 50mm, we need a 50 / 1.6=31mm lens. And vice-versa, our 50mm screw on an APS will give us the framing of a 80mm lens.

At first sight, one could say we just need that in mind and habit will do the rest ..That's forgetting the existing lenses line-up ! Indeed, for example Canon (like any other 35mm line-up) based his line-up on 16-35, 24-70, 70-200 zooms and 24, 35, 50, 85 primes. These zooms and primes correspond to different uses, but also to technical constraints, otherwise we would have 8-800 f/1 L IS for a long time !
So our 24-70L, which corresponds to a wide angle at 24 and small tele at 70, meets many purposes without changing lens. Now, with an APS sensor, our 24-70 behaves like a 38-115 !! No more wide angle, and max focal lenght of 115 which does not correspond to much, because when working in this range, a 70-200 is more indicated ! Also our 85 1.8, fabulous for portrait photography, becomes a 136 ! Even worse, no more ultra wide angle, since our 16-35 becomes a banal 26-60, but without the performances of a 24-70 (wide angles are more difficult to conceive, especially for Canon ... ), and more expensive ! On the other hand, our 400mm now frames like a 640mm, without having gained 10kg and increased by 10k€  ; this can be handy, especially for wildlife photographers (birders and so).

It's really necessary to understand that the visible change of focal length is due to a crop, and absolutely not to any form of increase of the focal length. This one follows the optical laws : it is the distance which separates the optical center of the focal plane (in other words, the film or the sensor). If I cut my 35mm format to keep 1mm at the center, I won't have a 100mm lens transformed into 3000 mm ! We indeed speak about focal "equivalence".

Canon since brought out a range of EF-S lenses (DX by Nikon) which can only on be mounted on APS bodies, covering a smaller image circle. This allows to produce smaller, lighter and less expensive optics, at least in theory. There is in particular a 10-22 f / 3.5-4.5 to compensate for the loss of ultra wide angle. However, that doesn't solve completely the problem. Indeed, the diameter of EF-S lenses is divided by 1.6, but not the distance of the bayonet lens mount to the focal plane ! This one decreases thanks to the mirror which is smaller on APS bodies, authorizing EF-S lenses to move backwards into the reflex cage, but not by a factor of 1.6. Therefore is it almost impossible to build a 10-22 f / 2.8 even for APS, unless building a heavy and extremely expensive monster, with not really interesting performances. Especially since to get our 16-35 f / 2.8 back, we would not be need a 10-22 f / 2.8, but a 10-22 f / 1.8, because the crop factor does not only affect the angle of view.

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Crop factor and depth of field (DoF)

Depth of field corresponds to the sharpness zone along optical axis. If a subject is forward or at the back of this zone, it will appear blurred. The demarcation is indeed not clear, but progressive, more and more blurry further from this zone. This zone is defined by focal length, aperture, subject distance, circle of confusion, and the size of the sensor. The circle of confusion is, in brief, the size of the details which we are capable to distinguish, and allows to differentiate the sharp zone from the blured zone. This parameter depends on many things, among others print size and the viewing distance. The simple fact of including the circle of confusion in the relation makes that we could write a whole book on the definition of depth of field, especially since the advent of digital sensors and their tiny photosites.

As seen above, the APS cropping behavior has an immediate consequence on the angle of view. So, to get same framing, we have to get further of the subject, or shoot with a shorter focal lenght lens, than with a 35FF. This one will thus have a shorter DoF than APS. It will then be necessary to compensate on the APS with wider aperture (if we can!). A shallow DoF is appreciated for portrait photography for example. Conversely, for landscape photography, it will be necessary to stop down more on 35FF to get same DoF as on APS.

	35FF, 80mm, f/2.8, 5m from subject	DoF is 0.7m
	APS, 50mm, f/2.8, 5m from subject	DoF is 1.16m (Same framing)
	APS, 80mm, f/2.8, 8m from subject	DoF is 1.15m (Same framing)
	APS, 80 mm, f/2.8, 5m from subject	DoF is 0.4m (Different framing)
	35FF, 80mm, f/4, 5m from subject	DoF is 1.02m (Same framing)

We see here that to have exact same picture , we must compare a 35FF at 80mm f/4.5 to an APS at 50mm f/2.8. The DoF difference is indeed around 1.5 stop.