WHAT IS BOKEH?
Bokeh describes the rendition of out-of-focus points of light.
Bokeh is different from sharpness. Sharpness is what happens at the point of best focus. Bokeh is what happens away from the point of best focus.
Bokeh describes the appearance, or "feel," of out-of-focus areas. Bokeh is not how far something is out-of-focus, bokeh is the character of whatever blur is there.
Unfortunately good bokeh doesn't happen automatically in lens design. Perfect lenses render out-of-focus points of light as circles with sharp edges. Ideal bokeh would render each of these points as blurs, not hard-edged circles. Mathematicians would say the intensity distribution of the blur circles are rectangular in perfect lenses, and good bokeh would prefer a Gaussian distribution. This is one area in which physics doesn't mirror what we want artistically.
Differing amounts of spherical aberration alter how lenses render out-of-focus points of light, and thus their bokeh. The word "bokeh" comes from the Japanese word "boke" (pronounced bo-keh) which literally means fuzziness or dizziness.
A technically perfect lens has no spherical aberration. Therefore a perfect lens focuses all points of light as cones of light behind the lens. The image is in focus if the film is exactly where the cone reaches its finest point. The better the lens, the tinier this point gets.
If the film is not exactly where that cone of light reaches its smallest point, then that point of the image is not in focus. Then that point is rendered on film as a disk of light, instead instead of as a point. This disc is also called the "blur circle," or "circle of confusion" by people calculating depth-of-field charts. In a lens with no spherical aberration this blur circle is an evenly illuminated disc. Out of focus points all look like perfect discs with sharp edges. (OK, at smaller apertures where the image is in pretty good focus you may see additional "Airy" rings around the circle, but that's a diffraction pattern we're not discussing here.) This isn't optimal for bokeh, since as you can imagine the sharp edge of these discs can start to give definition to things intended to be out-of-focus.
There are no perfect lenses, so one usually does not see these perfect discs.
Real lenses have some degree of spherical aberration. This means that in practice, even though all the light coming through the lens from a point on the subject may meet at a nice, tiny point on the film, that the light distribution within the cone itself may be uneven. Yes, we are getting abstract here, which is why some denser photographers refuse to try understand bokeh.
Example of Bad Bokeh. The blurs of the out-of-focus twigs still have sharp edges. Good bokeh would have soft-edged blurs.
Spherical aberration means that the discs made by out-of-focus points on the subject will not be evenly illuminated. Instead they tend to have more of the light collect in the middle of the disc or towards the edges. Here are some illustrations:
As you may have gathered, if the light tends to collect towards the middle of the out-of-focus discs on one side of the cone, then it will collect on the outsides of the discs on the other side of the cone. Under-corrected spherical aberration causes the light to collect in one way, overcorrected spherical aberration causes it to collect in the other. Therefore, a lens with great bokeh for backgrounds has awful bokeh for foregrounds, and vice-versa.
Things get weirder from here. Another big factor is how sharply the outside of the blur circle is rendered. Even if we have a poor signature similar to fig. 1), if the outer edge of this is rendered softly, as it is in the AF-S Nikkor 80-200 f/2.8, we have good bokeh.
Artistically most people tend to prefer sharper foregrounds and softer backgrounds. Fuzzy foregrounds tend to make people crazy, and fuzzy backgrounds are fine. Therefore I classify lenses with good bokeh as those with good background bokeh. Personally I avoid anything fuzzy in my foregrounds by moving the camera or the foreground object.
The reason bokeh is discussed in photography is because we prefer soft out-of-focus areas to hard ones that seem to take on texture, even though everything is out-of-focus. Because of this, it is preferable to those who want soft out-of-focus areas to have the distribution of the light within each blur circle to be concentrated more towards the center of the blur circle. That way each blur circle tends to be a bright spot that gets dimmer gradually towards the edges. This way all the blur circles blend nicely.
On the other hand, if one is trying to keep everything as sharp as possible, these bokeh effects will work differently where your image is close to being in focus. If in doubt, try it out. Lens design very quickly gets very weird.
There is no measurement for bokeh, since scientists aim for the mediocre example of figure 2.) as their "perfect" lens. Like everything else in art, you gauge bokeh by looking at the image.
Nikon's Defocus Control, or DC, Lenses
Nikon's Defocus Control, or DC, lenses for their popular 35mm SLR cameras actually allow one to manipulate the nature of the spherical aberration correction to allow one to locate the region of good bokeh to be either in the foreground or background. It also allows one to change the amount of spherical aberration for total control.
Reflex and Mirror Lenses
Mirror, or reflex lenses, have awful bokeh. This is because they have a relay mirror in the front of the lens that blocks the central part of the lens' aperture. Therefore all the out-of-focus highlights are represented as doughnuts, which looks unnatural and awful.
Leitz 90 mm f/2.2 Thambar
This was a 1940s soft-focus lens with a twist. Spherical aberration was deliberately left uncorrected at the sides. The softening is most obvious at full aperture. The lens becomes sharp as you stop it down. Leitz pulled a clever trick and included a removable front filter with an opaque central circle. The central stop eliminates the contribution from the lens' highly corrected central portion and let you get a soft central image as desired.
The shape and number of a lens' diaphragm blades has little to do with bokeh. They define the shape of the blur circle, but they don't define how the light is distributed within that circle. These circles are no longer circles, but shapes with as many sides as there are blades. For instance, with five blades as most Hasselblad and Mamiya lenses one gets five-sided pentagons as the shapes of out-of-focus highlights instead of circles. This isn't too great. With six blades, most common in discount lenses for 35mm SLRs, one gets hexagons. With seven blades (most Nikkor SLR lenses) things really start to improve, since the seven-sided heptagons start looking like circles instead of recognizable shapes. Nine blades (common on Nikkor telephotos) are great, and lately they are being designed with curved blades to give a close approximation of a circle.
Odd numbers of blades will give diffraction and reflection stars around very bright points of light that have double the number of points as the number of blades. For instance, a seven-blade diaphragm will give a lovely 14-pointed star. Even numbers of blades will give stars with the same number of points as you have blades. An eight-bladed diaphragm will give a boring eight pointed star.
Again, how well one approximates a circle is only a small part of the equation. The important part is how the light is distributed. Obviously at full aperture where most people worry about this the diaphragm plays no part.
The reason some manufacturers attempt to draw a correlation between bokeh and numbers of diaphragm blades is because it's easy to see how many blades there are at the sales counter, but almost impossible to see bokeh.
How to See Bokeh
In Existing Images
Look for points of light in the background. Perfect subjects for this are distant points of light at night or sometimes light shining through leaves or specular reflections in daylight.
If they all blend together nicely, that's nice bokeh. If they are perfect little circles, then that's neutral bokeh. If they are all swimmy and look little little rolled up condoms or donuts, then that's bad bokeh.
If they all are regular polygons that tells you the shape of the lens diaphragm. Yes, you can actually tell how many blades the lens's diaphragm had!
If they are perfectly round in the middle of the image and oval or lentil shaped at the sides that tells you the image was probably shot at full aperture.
If they are all flattened ellipsoids (vertical ovals about twice as tall as they are wide) then that tells you that the image was shot with an anamorphic lens. You'll see this in cinemascope motion pictures, not in still photographs.
In Your Lens
Find a point of light in the distance. You can do this easily at night by finding a distant point streetlight, or you can do it indoors by taking the reflector off of a Mag-Light flashlight and just setting it up on the other side of the room.
Now look at the ground glass as you focus. If you see perfect round disks your lens has neutral bokeh, if you see soft-edged shapes you have good bokeh, and if you see doughnuts you have bad bokeh.
If you see something other than neutral bokeh you'll see the quality of the bokeh change as you focus both in front of and in back of the point of light. Of course you can't usually focus beyond a distant point of light, unless you have a view camera or a lens that allows focusing beyond infinity.
This is tough, since I don't really have any glaringly bad or good images to show, and I won't copy other's work to show you.
For bad bokeh have a look at the July 2004 Fuji flyer that has green squares on the cover that says "In the Right Hands, Technology can be a Beautiful Thing." Inside the cover there's a great shot of an Indian playing a flute in Zion by Tim Meyer. See how the little bright dots from all the leaves in the background get busy and almost look like little rolled condoms? That's an example of the bad kind of bokeh. Yes, technical items have little to do with how good a shot is, and yes, for all we know this is exactly the look Tim wanted. Just know that you have control over this part of the look of the image based on your choice of lenses.
Also check out Zeiss' lengthy article on Depth-of-Field and Bokeh.