Reflection efficiency of a phase reversal zone plate antenna made of thin conducting film

Conventionally, when dealing with the electromagnetic reflection from a conducting plate, one assumes the plate to be much thicker than the skin depth δ_S so that the concept of surface impedance applies. For the phase reversal zone plate antennas studied jointly at University of Bradford and Mawzones Ltd, the conducting rings are printed with ink and its thickness together with its conductivity determines the cost. A problem of what happens if the conducting film is thinner than the skin depth, and how the reflection efficiency is influenced by the film thickness is solved by studying the reflection from a multi-layered structure