FF, 1.6x, aperture, FL, and lenses

Dec 29, 2006

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.