aperture - What is a "diffraction limit"?

I've seen the term used, but what is a "diffraction limit", when should I worry about it, and what undesirable effects are a result of it ?

Answer

There have been some very good answers, however there are a couple details that have not been mentioned. First, diffraction always happens, at every aperture, as light bends around the edges of the diaphragm and creates an "Airy Disk". The size of the airy disk, and the proportion of the disk that comprises the outer rings, and the amplitude of each wave in the outer rings, increases as the aperture is stopped down (the physical aperture gets smaller.) When you approach photography in the way Whuber mentioned in his answer:

Think of a scene as comprised of many small discrete points of light.

You realize that every one of those points of light, when focused by your lens, is generating its own airy disk on the imaging medium.

Regarding Image Medium

It should also be clearly noted that the diffraction limit is not actually a limitation of a lens. As noted above, lenses are always creating a diffraction pattern, only the degree and extent of that pattern changes as the lens is stopped down. The "limit" of diffraction is a function of the imaging medium. A sensor with smaller photosites, or film with smaller grain, will have a lower limit of diffraction than those with larger photosites/grains. This is due to the fact that a smaller photosite covers less of the airy disk area than a larger photosite. When the airy disk grows in size and intensity as a lens is stopped down, the airy disk affects neighboring photosites.

The diffraction limit is the point where airy disks grow large enough that they begin to affect more than a single photosite. Another way to look at it is when the airy disks from two point light sources resolvable by the sensor begin to merge. At a wide aperture, two point light sources imaged by a sensor may only affect single neighboring photosites. When the aperture is stopped down, the airy disk generated by each point light source grows, to the point where the outer rings of each airy disk begin to merge. This is the point where a sensor is "diffraction limited", since individual point light sources no longer resolve to a single photosite...they are merging and covering more than one photosite. The point at which the center of each airy disk merges is the limit of resolution, and you will no longer be able to resolve any finer detail regardless of the aperture used. This is the diffraction cutoff frequency.

It should be noted that it is possible for a lens to resolve a smaller spot the pixels in an imaging medium. This is the case when airy disks focused by a lens cover only a fraction of a photosite. In this case, even if two highly resolved point light sources generate airy disks that merge over a single photosite, the end result will be the same...the sensor will only detect a single point light regardless of the aperture. The "diffraction limit" of such a sensor would be higher (say f/16) than for a sensor that is able to distinctly resolve both point light sources (which might be diffraction limited at f/8). It is also possible, and likely that point light sources will NOT be perfectly focused onto the center of a photosite. It is entirely plausible for an airy disk to be focused at the border between two photosites, or the junction of four photosites. In a black and white sensor or foveon sensor (stacked color sensels), that would only cause softening. In a color bayer sensor, where a square junction of 4 photosites will be capturing an alternating pattern of GRGB colors, as airy disk can affect the final color rendered by those four photosites as well as cause softening or improper resolution.

My Canon 450D, a 12.2mp APS-C sensor, has a diffraction limit of f/8.4. In contrast, the Canon 5D Mark II, a 21.1mp Full Frame sensor, has a diffraction limit of f/10.3. The larger sensor, despite having nearly twice as many megapixels, can go an extra stop before it encounters its diffraction limit. This is because the physical size of the photosites on the 5D II are larger than those on the 450D. (A good example of one of the numerous benefits of larger sensors.)

Wrenches in the mix

You may often come across tables on the internet that specify a specific diffraction limited aperture for specific formats. I often see f/16 used for APS-C sensors, and f/22 for Full Frame. In the digital world, these numbers are generally useless. The diffraction limiting aperture (DLA) is ultimately a function of the relation of the size of a focused point of light (including the airy disk pattern) to the size of a single light sensing element on a sensor. For any given sensor size, APS-C or Full Frame, the diffraction limit will change depending on the size of the photosites. An example of this can be seen with Canon's EOS Rebel line of cameras over the years:

Camera   |   DLA
--------------------
350D     |   f/10.4
400D     |   f/9.3
450D     |   f/8.4
500D     |   f/7.6
550D     |   f/6.8

The story should be similar for film grain size. Films with finer grain would ultimately be more susceptible to diffraction softening at lower apertures than films with larger grains.

The Diffraction Cutoff Frequency

Diffraction is often touted as an image killer, and people talk about the "diffraction limit" as the point at which you can no longer resolve an image "usefully". On the contrary, the diffraction limit is only the point where diffraction starts to affect an image for the particular image medium you are using. The diffraction cutoff frequency is the point at which additional sharpness is impossible for a given aperture, and this is indeed a function of the lens and physical aperture.

The formula for diffraction cutoff frequency for (perfect) optical systems is as follows:

fc = 1 / (λ * f#) cycles/mm

This states that the reciprocal of the wavelength of the light being focused multiplied by the f-number of the lens is the number of cycles per millimeter that can be resolved. The diffraction cutoff frequency is generally the point where resolution reaches the wavelength of the frequencies of light itself. For visible light, λ between 380-750nm, or 0.38-0.75 microns. Until the cutoff frequency has been met for a given aperture, more resolution can be achieved.

Visual Examples

Whubers sequence of images above is a decent example of the effect of diffraction, as well as the effect of optical aberrations when the lens is wide open. I think it suffers a bit from some focus shift due to spherical aberration, so I have created an animated GIF that demonstrates the effects of changing the aperture of a Canon 50mm f/1.4 lens down from its widest aperture to its narrowest, in full stops.

(Note: The image is large, 3.8meg, so let it fully download to see the comparison of sharpness at each stop.) The image exhibits marked optical aberration when shot wide open, particularly Chromatic Aberration and some Spherical Aberration (there may be some slight purple fringing...I tried to get focus dead on.) Stopped down to f/2, CA is lessened considerably. From f/2.8 through f/8, sharpness is at its prime, with f/8 being ideal. At f/11, sharpness drops ever so slightly, due to diffraction. At f/16 and particularly f/22, diffraction visibly affects image sharpness. Note that even with diffraction blurring, f/22 is still considerably sharper than f/1.4 or f/2.