Mathematical Explanation of Faraday Cage Theory - Reflected/Transmitted Waves

[MisterX]

I was looking for a mathematical explanation of a Faraday cage. In particular I was seeking something that relates hole size to the reflected and transmitted waves, using electromagnetic theory. This might also relate to a wire grid polarizer. Would anyone be able to help me?

 

[Born2bwire]

There are several ways of doing it, depending on how much you want to simplify and approximate.

For example, you can treat the cage as an infinite sheet with periodic holes. There are many methods, some analytic but generally more computational, that can solve for an infinite periodic mesh and find the Floquet modes that describe the transmission and reflection characteristics (EDIT: Remembered the word!).

You could also treat the holes as rectangular waveguides of a very small thickness. You could find the evanescent mode at your frequency for your given hole size and then approximate the power loss in the transmission. This would ignore the coupling between the holes that the above method takes into account. The loss of the evanescent mode depends upon the size of the hole.

Then there is the hand-wavy explanation. The gridded mesh, if the size of the holes are electrically small, behaves like a solid PEC sheet. This is realized from the fact that for an infinite PEC sheet, the induced currents from a plane wave are constant magnitude, lie in the same direction, and only differ in phase. A rectangular mesh therefore can support a superposition of currents that run in normal directions, so by decomposing any incident wave into the superposition of two plane waves we can see how the induced currents can be supported. The rule for the holes comes about because the ideal PEC sheet has a continuous distribution of currents whereas the mesh has a discrete stepping in the phase shift between current sources. So the hole size is dictated by the minimum phase shift that we can allow before we consider the reflection of the plane wave to be unsupportable.