# FF, 1.6x, aperture, FL, and lenses

Started Dec 29, 2006 | Discussions thread
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FF, 1.6x, aperture, FL, and lenses

THE PREMISE:

A lens on a 1.6x camera will produce the same image in terms of DOF, shutter speed, and noise as the same scene from the same position as a FF camera will produce with a lens that has 1.6 times the FL, 1 1/3 stops down, and 1 1/3 stops higher ISO.

For example: a 17-55 / 2.8 IS on 1.6x is equivalent to a 27-88 / 4.5 IS on FF.

First, focal length (FL). Since a FF sensor is 1.6x wider than a 1.6x sensor, you need 1.6x the FL on FF to achieve the same FOV (field of view) from the same position .

Secondly, because you are using a 1.6 times longer FL from the same position to achieve the same FOV on FF, the DOF will be 1 1/3 stops more shallow for the same aperture . So, if you stop the lens down 1 1/3 stops, you'll also achieve the same DOF.

Where does 1 1/3 stops come from? Mathematically, "stops" are powers of sqrt 2 ( 1.4). So, (sqrt 2) ^ (1 1/3) = 1.6. However, you can also verify this here: http://www.dofmaster.com/dofjs.html . Select a 1.6x camera, a focal length, an f-ratio, and a distance and compute the DOF. Then repeat for a FF camera, but use an FL that is 1.6 times longer, an f-ratio that is 1 1/3 stops higher, and the same distance and you will see the DOF is essentially the same. By "essentially" I mean that, due to rounding errors, the difference is not exactly 1 1/3 stops, but very close.

Lastly is exposure and noise. Just as subject-camera distance is the only factor in determining perspective, absolute aperture (not the f-ratio) and sensor size are the only factors in determining exposure.

Allow me to explain. The aperture is the opening that allows light onto the sensor. You can compute its diameter with the f-ratio. For example, the aperture of an 85 / 1.2L at f / 1.2 is 85mm / 1.2 = 71mm. The aperture of a 50 / 1.2L also at f / 1.2 is 50mm / 1.2 = 42mm.

So, let's consider this scenario: a 50 / 1.2 on a 1.6x DSLR and an 85 / 1.2 on a FF DSLR, both taking a pic of the same scene from the same position at f / 1.2. Since 85mm ~ 50mm x 1.6, the FOV is basically the same, so the two cameras should produce basically the same image, right?

Yes. The 85mm lens, at f / 1.2, allows in more light (since its aperture is larger), but that light is distributed on a larger sensor (1.6 times larger which has 1.6^2 = 2.56 times more area). However, that's exactly the the same ratio as the ratio of the areas of the apertures!

Let me explain further. Actually, to make the math perfect, I have to use 80mm (50mm x 1.6) instead of 85mm which give an aperture of 80mm / 1.6 = 67mm. The ratio of the areas of the apertures from the 80mm and 50mm lenses, both at f / 1.2 is (67/42)^2 = 2.54 (not exactly 2.56 due to round off errors on the absolute apertures).

Thus, the two cameras will expose exatly the same! That is, they will have the same shutter speed for the same f-ratio on the same scene. However, the DOFs will be different. So, if we stop the FF lens down by 1 1/3 stops, we will achieve the same DOF, but we will have to up the ISO 1 1/3 stops to keep the same shutter speed.

This brings up noise. Assuming that the two sensors have the same noise characteristics , then the FF sensor, in an equivalent framing situation, such as that above, gets 2.56 times as much light for the same exposure, thus is 2.56 times (1 1/3 stops) more sensitive ISO. However, when you stop the lens on the FF camera down to match the DOF, you get the same amount of light, so you get the same noise!

Thus, a 24-105 / 4L IS on FF is equivalent to a 15-65 / 2.5L IS on 1.6x -- that is, the two different lenses on the two different formats will take, for all intents and purposes, the same pictures. Likewise, a 17-55 / 2.8 IS on 1.6x will take, for all intents and purposes, the same pictures as a 27-88 / 4.5 IS on FF.

Anyway, I hope this makes perfect sense to all.

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--joe

Please feel free to criticize, make suggestions, and edit my photos. If you wish to use any of my photos for any purpose other than editing in these forums, please ask.

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