The Irradiance For Two Lambert Sources

One can compute the theoretical irradiance for Lambert sources by integrating over all points on its surface visible from the illuminated area. The following curve is for a sphere of radius R and a small area normal to and a distance z along the ray from the center.


The curve is just the inverse square law. Replacing the sphere with a disk of the same radius one gets a curve which is fairly constant near the disk but begins to follow the inverse square law as one moves farther away from it.


I evaluated the curve for the sphere numerically and it is a good fit to the formula below. The formula for the disk was evaluated analytically.


In both cases the irradiance just at the surface of the source is πL.