Crop Sensors and Aperture

[Update: a more recent post gives a more accurate comparison between full-frame and crop sensor cameras.]

Earlier today I made an off-hand comment on how you need to multiply the f-number by the crop factor on a crop-sensor camera to get the equivalent aperture for a full-frame. This is just like you need to do with the focal length of a lens, a concept which most people with crop sensors are used to. Needless to say this caused a few questions and comments, so I promised to look into this a bit more. So let’s start by setting the scene: what we’re considering here is a comparison between two scenarios:

  1. A crop-sensor camera with crop factor C (say, 1.6 for Canon APS-C), fitted with a lens having focal length x (say, 50 mm) and aperture y (as an f-number, say f/3.5).
  2. A full-frame camera in the same position and looking at the same scene as scenario 1.

The question is: what focal length and aperture do you need on scenario 2 to get the same image as scenario 1? Anyone will tell you the focal length needs to be C × x, or 80 mm in the example above. However, you also need to have an aperture of C × y, or f/5.6 in the example.

My initial reasoning was very simple: since the depth of field is dictated only by the absolute aperture (i.e. how wide the diaphragm opens in mm), then if you increase the focal length, you need to increase the f-number (i.e. the relative aperture) by an equal amount. Now I haven’t looked at optics in quite a while, so I hope I remembered this right. And of course this statement requires some restrictive assumptions (e.g. lens needs to be symmetrical, etc), so it’s not exactly true in many practical cases.

So to illustrate things a bit, I set up an experiment. Now since I don’t have a digital full-frame camera, all images will be taken with my 30D, which is a crop-sensor with an approximate crop factor of 1.6. Instead, I took two sets of images:

  1. The 30D with a 50mm lens, cropping the resulting image by a factor of 1.6. This is meant to simulate the crop-sensor camera in scenario 1 earlier.
  2. The 30D with an 80mm lens, taking the image straight from the camera. This is meant to simulate the full-frame camera in scenario 2.

_MG_6744Starting with the ‘full-frame’ simulation, the image above was taken with the 80mm lens at f/5.6. I had set up a scene with a main subject (the knight) very close to the camera (about 0.5m), secondary subjects further back (the rook and the vertical bars), and LED lights at the far back to create clear circles of confusion. The camera was set up on a tripod to make sure the focal plane remains static between shots.

_MG_6745Next, I fit the 50mm lens, leaving camera and subjects untouched. I initially kept the aperture at f/5.6 and fired the first shot (next image). Clearly this had (roughly) the same shutter speed for the same exposure. I cropped this image, taking the central section with a crop factor of 1.6. You can observe a few things with this image: a) the circle of confusion is clearly smaller, b) it is now octagonal rather than circular, and c) the field of view is actually tighter than before. The first was expected. The second may be easily explained by noting that the first image was taken with the lens wide open (I know, f/5.6 doesn’t scream wide open, but then this is just a plain 28-80mm f/3.5-5.6 zoom lens) so the circle of confusion is circular. The lens in the second image is stopped down so the shape of the diaphragm shows (it’s a 50mm f/1.4 lens). The change in field of view may look odd. However, it’s worth noting a few things:

  • For the same viewpoint you need to keep the lens plane (assuming a thin lens model) fixed; I did not do that, but only kept the focal plane fixed.
  • Changing the zoom setting and focusing both move the lens elements, changing the field of view in a rather complex way.
  • Since I had the main subject really close to the camera, the subject distance is not much larger than the distances between elements, so the thin lens model becomes even less accurate.

_MG_6746Finally, I opened the aperture up to f/3.5, which is supposed to be equivalent to f/5.6 on the 80mm lens. The circle of confusion is now larger, though arguably bigger-looking than that on the first image. It is worth noting here that while the camera suggests ‘clean’ values for the aperture, one needs to realise that these are not meant to be exact. That is, besides all the field-of-view changes already mentioned. Another aspect often forgotten is that if we have opened the aperture by a full stop, we need to compensate the exposure by using a shutter speed that’s one stop faster. I have also completely ignored second order effects (such as diffraction).

You can see from this simple experiment just how complicated any reasonable comparison between different sensor sizes can be. And this is ignoring other physical differences between sensor sizes (e.g. larger sensors will have larger wells for each pixel, which capture more photons and therefore can be more sensitive). If you haven’t had enough already, you can read more on the subject at various places on the web. One option is at Cambridge in Colour.

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