From time to time, questions are asked here about the best set up for photographing art works or copying documents.
Answers generally discuss lighting, color balance, etc. Getting most of this right is a matter of technique (using the right white balance, getting the object square in the frame, maybe needing polarizers, ...).
However, for the best results, you also should have uniform (“flat”) lighting. How do you get that?
Conventional wisdom tends to be something on the order of: put your lights (what ever they are!) at 45 degree angles on either side of your piece, and that bigger/more diffuse lights are better.
Well, 45 degrees may not be the optimum angle, and you can achieve remarkably flat lighting with just a few plain old light bulbs.
How? Well, let me tell you.
I have created a spread sheet that allows me to place up to 4 light bulbs at arbitrary (x,y,z) positions and calculate the resultant light pattern in the (y,z) plane (i.e. parallel to the plane of the light bulb arrangement which is at a distance of x from the area of interest). The bulbs are assumed to be “isotropic radiators” - they radiate the same amount of light in ALL directions. This is probably not too far from reality for ordinary light bulbs (LEDs, CFLs, incandescents) for the hemisphere surrounding the top of the bulb, which is all that counts. This will not be the case for floodlights or other directional lights. I also assume that all the bulbs have the same brightness.
A square arrangement of the bulbs, at the proper distance, creates an exceedingly uniform light distribution over an area that is half as big as the size of the square of bulbs. The magic distance is a factor of 0.695 the size of the square out from the plane of the light bulbs.
In other words, if you have bulbs at the corners of a square that is 2 meters on each side, the uniform plane of illumination is the central square meter area that is 1.39 meters ( = 2 times 0.695) out in front of the bulbs (where the tops of the bulbs are assumed to point out from the corners of the square). (Strictly speaking, all measurements should be with respect to the center of the bulb’s light producing region.)
Over this area, assuming my ideal light bulbs, the light is uniform with a maximum deviation of less than +/- 0.2% . That is less than 0.003 f-stop!! This is probably way better than whatever flat field calibration is built in to your camera.
The angle of the bulbs with respect to the center of the uniform area is actually quite close to 45 degrees in this case, but see below for the 2-bulb case.
If you have bigger pieces to photograph, you can still get very good flatness over an area as big as your square of bulbs. You have to move the light array a bit closer: best uniformity is achieved with the distance of the bulbs about a factor of 0.61 the size of the square from the plane of interest (1.22 meters for our 2 meter square of bulbs). In this case, the uniformity of lighting is within +/- 3% (less than +/- 0.05 f-stop) over the entire 2 meter square area.
I can readily calculate the flatness of illumination across the center of other rectangular arrangements of the bulbs, such as might be relevant for non-square art work or documents. In the picture is a shot of a set up for the case where the bulbs are one meter apart across the area of interest, but are only 0.8 meters apart top-to-bottom. In this case, the center half rectangle of best illumination is 0.5 by 0.4 meters in size, and the bulbs are optimally a factor of 0.705 of the total width out from the plane of interest.
If you have only two bulbs, where should they go?
The best you can do is to get a strip of almost uniform illumination (which will be parallel to the line between the bulbs). Not surprisingly, above or below this line, the light falls off. If, once again, the (now 2) bulbs are two meters apart, the center one meter strip will be most uniformly lit when the bulbs are a factor of about 0.83 of total separation out from the line (1.66 meters for bulbs separated by 2 meters). In this case, the uniformity is better than 0.1% along the entire one meter line length. If you want to think angles instead, each bulb should be at an angle of about 31 degrees with respect to the normal (the line out from the center) of the line of interest. This is far from the canonical 45 degrees!
My spreadsheet is in QuattroPro (I’m rather anti-Microsoft for word processing and spreadsheet, similar to the antipathy by some here to Adobe products), and I would be glad to provide it to any interested users. (I’ll have to provide a bit more internal documentation!!)
Originally posted by AstroDave 
Post #1 by AstroDave
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