Question

How does the luminance of a real image depend on diameter D of a thin converging lens if that image is observed 

(a) directly; 

(b) on a white screen backscattering according to Lambert's law?

Answer 1

(a) If s = object distance, s' = average distance, L = luminance of the sounce, ΔS = area of the source as sumed to be a plane surface held normal to the principal axis, then we find for the flux ΔΦ incident on the lens

Here we are assuming D « s, and ignoring the variation of L since a is small

Then if L' is the luminance of the image, and  is the area of the image then similarly

(b) In this case the image on the white screen from a Lambert source. Then if its luminance is L₀ its luminosity will be the πL₀ and

since s' depends on f, s but not on D.