If I think of a photon as a particle, I think a parallel wire filter should transmit proportionally to the uncovered area. (and reflect proportionally to the covered area). Obviously polarization adds complexity by introducing a second factor to be considered.
On the other hand, if I consider the photons as waves, I think the transmission should be less sensitive to the area covered. Both a fairly thick and a fairly thin filter should transmit most of the light in one polarization and reflect most of the light in the opposite polarization.
Which is correct?
To be more precise, consider a spectrum of filters with thick and thin wires. (on the same spacing). At one extreme end of the spectrum the thinest wires approach an empty space or vacuum, which transmits everything and reflects nothing. At the other end the thickest wires merge together and form a continuous surface, in effect a mirror which reflects everything and transmits nothing. My question is what happens in between these two extremes? Of course there are two answers, one for parallel polarization and one for perpendicular polarization. What are the two formulas for transmission as a function of wire thickness, or percent of space blocked?
My guess is the answer for the transmitted polarization is close to a step function, only rounded off in the corners. Other possibilities is a linear function decreasing from one hundred percent to zero, or from fifty percent to zero. What are the two correct equations for the two complementary polarizations? Or more specifically, what is the correct formula for the transmitted polarization?
For part credit, just give a reference or a convincing derivation showing whether this function is closer to a step function or closer to a linear approximation. For full credit, provide a reasonably accurate formula with source or derivation showing the full transition from zero to one hundred percent wire coverage. It should be accurate enough to get the round off in the corners right to a percent or two (at least 5%) or prove that there is no significant round off. A good answer will have either a derivation or a reference to a reputable source.