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.