Center filters are used with professional large-format wide-angle lenses to correct the natural darkening of the sides of the image. Center filters are dark in the center and clear on the outside. They aren't needed with SLR cameras.
Light falloff is the darkening of an image's corners. It happens with wide lenses because the light has to travel much farther to reach the corners of the image than the center.
I'm going to explain the difference between large format lenses and SLR lenses. Then I'm going to explain the problems with each related to light falloff.
Then I'll explain center filters and how to use them. If you don't care about the technology feel free to skip straight to the usage recommendations at the end here.
Retrofocus SLR Lenses
The SLR cameras most familiar to amateurs need special retrofocus designs to keep the back of the lens away from the swinging SLR mirror. Before retrofocus lenses were invented by Angenieux in the 1950s there was no way to use short lenses on SLR cameras.
Retrofocus lenses use a strong negative front section with a positive rear section. This gives a wide angle while keeping the back of the lens much further away from the film.
This eliminates most falloff, since the greater distance from the film helps equalize the distances the light travels between center and corner. Other clever design tricks eliminate the rest of the falloff, so falloff is something very foreign to most amateurs. Just look through the front of any SLR ultrawide lens and you'll see the clear center "eye" of the lens actually look bigger as you look at it from an angle!
Large Format Wide Angle Lenses
The reason we don't use retrofocus lenses on non-SLR cameras is because everything else about retrofocus designs are worse then ordinary short lenses. Retrofocus designs are much heavier with their huge front negative elements. Retrofocus designs also create barrel distortion which lens designers attempt to correct with varying degrees of success. These designs also have a harder time of staying sharp in the far corners. This is why rangefinder cameras like Leica M and the Mamiya 7 and Mamiya 6 have such an easy time of making excellent, compact wide lenses.
This is also why it's normal for lenses on any non-SLR camera to have falloff. The wider the lens, the more falloff. It depends on angle, not focal length or format.
Falloff is easily corrected with a filter, while there is no trivial optical way to fix a lack of sharpness or distortion.
The Technical Side
There are actually four reasons the sides get darker. That's why the law is called cosine ^ 4. Cosine is the trigonometric function which is equal to 100% at 0 degrees and drops to 0% at 90 degrees. The four reasons for darkening of the corners are:
1.) Distance. It's longer for the light to travel form the center of the lens to the side of the image than the center.
2.) Distance again. Light falls off as the square of the distance, since at twice the distance the light is spread across four times the area.
3.) Angle at the film. At the corner the light only grazes the film at an angle, which spreads the light over a larger area.
4.) Angle through the aperture of the lens. Straight on you see a circle. As you get to the side you only see a smaller ellipse.
All four of these vary as the cosine of the angle of light through a conventional lens. Retrofocus lenses avoid most of this since these apply to light coming out the rear of the lens, and the longer back focus straightens the angles involved.
In the center of the image you get 100% of your light. Cos 0º = 100%.
At 30 degrees off center you get (cos 30º ^4) = 0.87 ^ 4 = 56% = a stop of darkening.
At 45º off center (a 90º field of view) we have (cos 45º) ^ 4 = 0.71 ^ 4 = 0.25, which is two stops lost.
This very quickly starts to get very bad as the angles approach ultra-wide, while remaining negligible for normal and tele lenses.
Center filters simply are dark in the middle and clear at the sides. Put one on your wide angle lens, add as much exposure as the center is dark, and you now have even exposure throughout your image. Most center filters are 1.5 or 2 stops darker in the center.
I prefer the wide angle lens' effect of darkening the sides. Ansel Adams deliberately added this effect to many of his images by burning the edges of his prints. See Ansel's book "The Print." This artistic effect emphasizes the central point of the image and helps keep the viewers' eyes from wandering off the image.
The effect is subtle even with a 65 mm lens for real subjects. This is because of the great range from light to dark that we take for granted. A stop or two on the sides is actually quite subtle unless you're photographing blank walls.
This is one of my most famous images. It was made with a 65 mm lens on a 4 x 5" camera and no center filter. I deliberately avoided using my center filter because I wanted to emphasize the center. I even added a three-stop grad, dark on top, to dim the sky! There must be five stops more exposure in the center of the image compared to the top. This is what gives the wall its glowing effect so critical to the image. Needless technical perfection is a bad thing in creative art.
Because of this, and the loss of usually two stops and more stuff to carry around, I rarely bother with center filters.
The only time a center filter makes sense to me is with my 47 mm lens on a 6 x 12 cm camera. The extreme angle, 113º at full 16 mm of rise, computes to only 9% of the light, or 3.4 stops of darkening in the far corners. You can see that! I love the 4x (2 stop) center filter Schneider makes for my 47 mm XL lens. You can use this filter without any mechanical vignetting, and in fact, I often use another 67 mm screw-in filter between the center filter and the lens with flawless results.
I've never bothered with a center filter for my 75 mm and longer lenses in 4 x 5." In 6 x 7 cm my Mamiya 43 mm lens gives a similar angle of view and the same falloff as the 75 mm lens on 4 x 5.
How to Use a Center Filter
I do all my composition, movements and focusing without the filter. When I'm all ready I screw it on and make my exposure.
I determine exposure as I usually do, and add the appropriate exposure compensation for the filter. This is usually 1.5 or 2 stops.
I love to use colored filters screwed into the front of my lens. If you're screwing a center filter to the front of your lens you reduce your options.
You may have to use a gel or balance your color filter on the rear of the lens. Remember that focus changes if you put a glass filter on the rear of your lens.
if you're rich you can use a much larger screw in filter on the front of the center filter. Center filters have much larger front threads than rear threads. My 67 mm center filter for my 47XL has 86 mm front threads!
Personally I often cheat and simply add the colored screw in filter between the lens and center filter. Usually you'll get nasty vignetting and screw up the correction, however I get away with it with my 47XL lens since if has such enormous coverage. It's rare that you can get away with this.
If you insert a second filter between the lens and center filter you run the risk of having the sides of the image slightly too light. This is because the sides of the image now look through a lighter part of the center filter closer to its edge, giving the potential for over correction of falloff.
As all photography, try anything and everything at home before you go shoot something critical.
Schneider's Technical Info and Application Charts are here
Rodenstock and Schneider versus Hoya and Heliopan
When I'm crazy enough to use a center filter I'm crazy enough to want to use exactly the right one. That means one supplied by the lens maker. As the chart at the link above shows you Schneider makes a plethora of center filters, even with the same density and filter size. Each lens needs something a little bit different.
I haven't tried the generic ones made by Hoya and Heliopan. When I've looked at prices I didn't see much difference, which also helped make up my mind.
By all means try Hoya and Heliopan and they are probably fine, but if you're picky enough to worry about this then I'd suggest the one made by your lens maker.
Remember that I usually don't bother with a center filter unless I'm using over about 90 degrees of view.
How to Tell a Good Center Filter
The worst way is to go shoot a wall and look for perfectly even illumination. Center filters aren't designed for complete correction out to the very last millimeter of image circle. If they were you'd have three or four stops of loss. Having this dense a filter would lead to even more problems. A center filter corrects most of the falloff, but not all.
The best way to tell if you have a good one is to make comparisons and ensure that it doesn't affect the color rendition. It's difficult to make a truly neutral ND filter. Many times there will be a slight color cast. Just try out your filter and be sure there's no objectionable color casts.
Also test to see what filter factor works well for you. Personally I seem to shoot my 1.5 stop filters with a 2 stop correction, which also eases exposure calculations.
Why Center Filters are so Big
They are usually made of thick glass. They are used on very wide angle lenses. The angle is large and the light has to travel through the thick glass. The front of the filter has to be bigger than the rear thread to avoid mechanical vignetting. Otherwise you'd see the sides of the filter as black corners in your image, which would defeat the entire purpose.
Why Center Filters are so Expensive
Center filters cost a few hundred dollars each! They are expensive because it's very difficult to make a filter which is both perfectly neutral and has a uniform and correct feathering of density from the center to the sides. These cost about $350 because they are made in one of two ways:
1.) Take two pieces of glass, one clear and one solid neutral density. Grind them each as if they were lens elements to be cemented together. Grind the clear one concave and dark one convex. Now glue them together. This gives you a dark center and clear sides.
2.) Sputter a metallic coating on the center of a piece of glass. The coating must be both perfectly neutral and spread with the perfect densities towards the sides, as well as be perfectly round. Good luck! That's why these cost $350.