Optimum Aperture - Format size and diffraction

The optimum aperture of a lens, i.e. the aperture at which it is sharpest, varies from lens to lens, but as a general rule it's between 1 and 3 stops down from the maximum aperture for the center of the field. Why is this?

First you have to understand that no lens is perfect. They all have aberrations which reduce their performance. Classically there are five so called "Seidel" aberrations. They are sometimes called third order aberrations based on the mathematics used to model them. They are:

• Spherical aberration
• Coma
• Astigmatism
• Field curvature
• Distortion

All lenses have these aberrations and they are worse in fast lenses. Stopping down a lens greatly reduces Spherical aberration and to a lesser extent reduced the effects of Coma, Astigmatism and Field curvature on image sharpness. Distortion is unaffected by aperture. A 6th aberration, Chromatic aberration, is to a first approximation unaffected by aperture too.

So you might think that if aberrations are reduced as you stop down the image would get sharper and sharper as the aperture got smaller and smaller. But you'd be wrong. That's because of a phenomenon called "Diffraction". Diffraction reduces image sharpness and as you stop down more and more, diffraction effects get larger and larger. Without getting technical, diffraction is the spreading out of a light beam when it's "squeezed" though a small aperture. The smaller the aperture the more the light spreads out. One analogy is that of a garden hose. When the end is open water flows out in a narrow stream, but if you squeeze the end to form a small opening, the water fans out. The two phenomena aren't really related (different mechanisms apply), but the analogy helps to "get the picture" of what's happening.

So if stopping down reduces the Seidel aberrations but also increases diffraction, I think you can see that maybe there's a "best" aperture which is a balance between reducing the optical aberrations and increasing the effects of diffraction. Typically this happens when a lens is stopped down a few stops from wide open. That's typically enough to significantly reduce spherical aberration, but not enough to make diffraction a problem. Most good lenses will peak in sharpness in the center of the imge field somewhere between f4 and f8.

While sharpness may peak (at least in the center of the image) between f4 and f8, stopping down to smaller apertures may still improve image quality in the corners of the image. That's because Coma and Astigmatism don't affect the center of the image, so the main effect of stopping down in the center of the frame is the reduction in Spherical aberration and spherical aberration goes way very quickly for a small decrease in aperture (It goes down by the square or cube of the aperture, depending on exactly how you quantify it). The effects of Coma and Astigmatism are reduced slower with decreasing aperture, so stopping down past f8 may further reduce their effects at the edges and corners of the image. But how far down can you go before the image really starts to suffer from an overall loss of sharpness? The next section of this article deals with that in more detail.

Diffraction - how low can you go

From time to time you see statements posted on web forums complaining about the performance of lenses on DSLRs when they are stopped down to f22 or smaller apertures. Such apertures are often used in macro work in order to maximize depth of field.

So why the complaints? Are these just bad lenses? Are the results of stopping down worse on DSLRs than on film SLRs, and if so why?

Circle of Confusion

The first concept to understand is that of the circle of confusion value. This is dealt with in some depth in the article on Digital Depth of Field, but to briefly recap:

The circle of confusion value is the maximum diameter of the image of a point source which will allow a reasonably sharp 8x10 print to be made from the image. It basically states that if you want a sharp image, it has to be made up of well focused (i.e.small) points (makes sense, no?).

There are two things which affect the size of the image of a point source. The first is focus. Obviously an in-focus image will be sharper (smaller) than an out-of-focus image. The second is diffraction. Diffraction is the name given to the observation that when light squeezes through a small opening it tends to spread out. An analogy (though one based on entirely different physics!) is water running through a garden hose. If the diameter of the hose is large, the water flows out in a narrow stream. However if you squeeze the end of the hose and make it small, the water sprays out in a fan. Light acts the same way. If you force it through a small hole, it spreads out on the other side of the hole.

The size of the image of a point formed by a perfect lens (i.e. one with no other aberrations) can be calculated and is shown in the table below. The image of a point is known as the diffraction limited spot size and the value is given in microns (1 micron is 1/1000 mm).

So what has this to do with sharpness and stopping down? Well we've already seen that the COF defines the spot size which is acceptable for reasonably sharp 8x10 prints. In depth of field calculations the COF relates to the maximum amount of defocus that's acceptable (i.e. the range over which images of points look sharp). In diffraction calculations the COF defines the maximum allowable value of the diffraction limited spot size which will result in acceptably sharp prints in the in focus plane. Below is a table which results from comparing the COF with the spot size for a given aperture. The aperture listed is the smallest setting which will not result in an unacceptable loss of sharpness.

So the table indicates that for an APS-C DSLR (like the EOS 20D, or in fact any Canon, Nikon, Minolta, Pentax or Sigma DSLR except for the Canon EOS 1D and 1Ds series), the smallest aperture that should be used is f16, assuming you want a reasonably sharp 8x10 print. If you stop down more and still want a sharp print, you may be limited to 5x7 or 4x6 print size.

You can also see why digicams (typically using 1/1.8" or 1/1.5" sensors) don't let you stop down past around f8. For an 8x10 print from them, you really don't want to stop down past about f4 or f5. For smaller prints f8 will be OK, but the camera makers don't let you stop down to f16 or f22 because the results would be pretty bad.

Note that these restrictions have nothing to do with the images being digital. It's purely a consequence of the physical size of the sensor. If you used film with a format size equal to that of the digital sensor, the results would be exactly the same.

Practical Examples

First let's look at a series of images shot using an EOS 20D DSLR (APS-C format sensor) with a 75-300/4-5.6 IS lens at 300mm and minimum focus distance:

You can see here that sharpness increases as you stop down (an indication that aberrations are present at wider apertures). Maximum sharpness seems to be at around f16 under these conditions. Stopping down to f22 gives a small, but acceptable sharpness loss if you need the extra depth of field that f22 can provide. At f32 there's a distinct sharpness loss, though still possibly acceptable if you're not making large prints. It's very clear that stopping down to minimum aperture, f45 in this case, gives a big drop in sharpness and should probably be avoided.

Close focusing a non-macro lens tends to increase basic aberrations (like spherical aberration), so what do we see then the focus is at infinity, when aberrations may be better controlled? The next set of images show this. Again the same lens and camera, but this time focused on a distant object (actually a tree trunk).

In this case it's clear that stopping down from f5.6 to f8 results in a gain in sharpness (due to lessening of aberrations). Sharpness is better at f8 and perhaps just slightly better still at f11. There's a small drop in sharpness at f16, a further drop at f22, f32 and f45. Again f32 and f45 should probably be avoided, and f16 and f22 only used if necessary to give greater depth of field or slow shutter speeds.

What about a lens that has less aberrations to start with? This time an EOS 20D was used in conjunction with an EF 300/4L lens, again focused on the same distant tree trunk.

It's evident that the 300/4L lens is sharper than the 75-300/4-5.6 IS, but that's really no surprise. f4 is sharp, but f5.6 is very slightly sharper. Sharpness slowly drops as the lens is stopped down past f8. f16 is OK and f22 may be acceptable, but f32 is pretty bad.

Conclusion

If you want to keep your images sharp, don't use f32 with an APS-C DSLR. The effects of diffraction are clearly visible at f32 and significantly degrade the image. Use f22 only if you have no choice. Optimal sharpness depends on the lens. For a lens with significant aberrations (e.g. a consumer zoom at maximum focal length and minimum focus distance), stopping down to f16 may give optimum results. For a lens with less aberrations (e.g. a consumer zoom used at infinity focus), optimum performance is around f11, though both f8 and f16 are very similar. For a really good lens like the EF 300/4L, with well corrected aberrations, performance may peak at f5.6, but be good from f4 to f11. f16 is acceptable, but f22 and smaller apertures should be avoided.