Diffraction from a set of concentric rings with random phase

• A
Gezstarski
TL;DR Summary
Looking for a treatment of the focussing properties of a set of concentric rings with random phase
I have been considering the properties of a Diffractive Optical Element (DOE) consisting of a very large number of concentric rings of equal (small) width, where the thicknessses of the rings are such as to produce random phase shifts in the range 0 to 2pi. I think I understand the behaviour of such a DOE - it acts as a diffraction limited lens that is achromatic but of exceedingly low efficiency. I am sure that this problem must have been tackled before but I have not been able to find a reference or a textbook treatment. Can anyone help?