# Formula for polarized “light” transmission through close filters?

I'm still trying to understand photons (or polarized electromagnetic radiation).
This question is similar to one of my previous questions, but different:
Consider the famous demonstration of crossed polarizers.
Two polarizers at right angles block all light. Insert a polarizer at 45 degree angle between the two crossed polarizers, and 25% of the light is transmitted (neglecting absorption).

Now perform the experiment with "light" of wavelength on the order of centimeters to meters, ie radar or radio band.
(Let's just choose 10 centimeters to be specific.)
Use parallel wire filters as the polarizing filters.
Now place the two crossed filters a meter apart, say at postions 0 and 100 centimeters.
Consider inserting the 45 degree filter in the middle at position 50 centimeters.
We should get 25% transmission.

What happens if we move the 45 degree filter from position 0 to position 100?
I expect zero percent transmission at positions 0 and 100.
I expect 25% transmission at all positions between 10 and 90, but I can't prove it.
I expect transmission intermediate betwween 0% and 25% as the filter moves from position 0 to position 10, and the mirror image as the filter moves from position 90 to position 100.
What formula describes this transition?
More generally what formula describes the transmission as a function of position all the way from 0 to 100?

## 1 Answer

[I'm not 100% sure, but here's my guess.]

Let's talk about semi-real wire grids, with infinite-conductivity wires but space L between the wires. (The "ideal" wire grid would be L --> 0.) If the spacing between the two polarizers is much greater than L, they act sequentially following Malus' law. If the spacing is much less than L, they act like a single unit, which (as Jim says) blocks all light. (Why? Because a double-wire grid, with wires running in two directions, can conduct net current in any direction in the plane, so therefore it can screen any electric field.)

For the "ideal" wire grid (L=0), the light component with that polarization goes immediately to zero after crossing the grid. No matter what the wavelength is. It's as true for visible light (small wavelength) as it is for screening DC fields (infinite wavelength).

For the transition region where L is comparable to the separation between filters, I doubt there's a simple formula. In fact, there isn't even a simple formula for how they would screen DC fields. See this paper. It's probably wavelength-dependent but I don't know.

• Thanks for the reference. I will try to get it from my library. You are at least understanding my question correctly. – Jim Graber Sep 19 '11 at 19:03