Diffraction, focal length, and the price of beans
Let me try and cut through some of the palaver on the way different format sizes, or pixel pitches, or focal lengths, or focal plane magnifications, influence the "aperture at which diffraction effects become significant".
Firstly, let me suggest the following (which are all arbitrary; it will turn out that the conclusion isn't really affected by what we choose here!):
1. For any given resolution value, R, (in, for example, cycles/mm), we can simplistically think of the resolution limitation being equivalent to the result of blurring by a blur figure of diameter d.
2. Adopting Rayleigh's criterion, it is reasonable to say that:
d=2/R.
3. The inherent "geometric" resolution of a sensel array is:
R=1/p
where p is the sensel pitch. ["Geometric resolution" ignores the Kell factor.]
4. We can thus think of the resolution of the sensel array as being the result of a blur figure whose diameter is given by:
d=2p
5. We can consider the limitation of resolution produced by diffraction as being equivalent to the result caused by blurring with a blur circle whose diameter can be thought of as the diameter of the Airy disk (the central "dark" portion of the blur figure created by diffraction).
6. We can arbitrarily decide that the impact of diffraction upon resolution is "significant" when the diameter of the Airy disk (A) is the same as the diameter of the equivalent blur circle corresponding to the geometric resolution limit based on sensel pitch. That is, when:
A=d
and thus when
A=2p
Now, the diameter of the Airy disk, A, can be reckoned as:
A=2.44 LN
where L is the wavelength of light of interest and N is the f/number of the lens. (This assumes the Airy disk is created on a focal plane which lies at the rear focal point of the lens; that is, lies the focal length behind the second principal point when the lens is focused on an object at infinity).
Note that the focal length of the lens does not enter into this result.
We can combine the various equations and solve for N to determine the f/number at which (based on our huge pile of assumptions) the impact of diffraction would be considered significant:
N=0.82 (p/L)
Now, just for kicks, what value would that give for some sensel pitch of interest to us?
Well, for the Canon EOS 40D, the sensel pitch is about 5.8 um. We will assume "midband" light whose wavelength is 550 nm. Then, the relationships above would suggest a critical aperture of f/10.5. (Well, fancy that!)
Now my point here is not to suggest that this value will match the results of empirical testing. For, one things, the assumptions on which it is based are very naïve. For example, using a Kell factor of 0.75 would make the critical aperture come out to f/14.
It is to point out that, using whatever assumptions we feel are appropriate, the result as to the "critical" f/number will not depend on focal length or sensor size or anything else but sensel pitch (and of course the wavelength of light we assume).
