A Polarizing Intuition

In The Feynman Lectures on Physics, the chosen method of clear, mostly intuitive presentation very rarely leaves anything as an exercise to the reader, so when it did that with polarizing filters it kind of stood out.

An interesting paradox is presented by the following situation. We know that it is not possible to send a beam of light through two polaroid sheets with their axes crossed at right angles. But if we place a third polaroid sheet between the first two, with its pass axis at $45^{\circ}$ to the crossed axes, some light is transmitted. We know that polaroid absorbs light, it does not create anything. Nevertheless, the addition of a third polaroid at $45^{\circ}$ allows more light to get through. The analysis of this phenomenon is left as an exercise for the student.

My first thought was something about quantum something or other that simply hadn’t been covered yet. This is a book that contains the word “refrangibilities” two pages before…. I wouldn’t put it past it to challenge the enthusiastic reader with quantum mechanics after just learning about polarized light.

My thinking went straight to Bell’s Theorem that there can be no local hidden variables explaining such behavior. There must be something quantum and spooky going on here to explain why adding an additional filter would increase the amount of light passing through! How does this clearly-quantum-mechanical stuff work? I didn’t really know.

Turns out, this is just another case of bad reasoning by analogy. The idea I had in my head of a filter is something that simply blocks light from going through. That is, I assumed any portion of the light not very nearly aligned with the direction of polarization would be blocked completely.

A polarizing filter does something different, removing one of the directional components of incoming light and changing the light in the process. This becomes clear when you look at a filter itself, in normal light. Does it block almost all light from getting through? I own a pair of polarized sunglasses, and I can see through them just fine. Somehow I never put the two ideas together.

Of course, we see a similar transmission effect when we pass photons through our filter setups one at a time, so there is something wonderfully quantum mechanical going on there. But it doesn’t explain the main thrust of how this setup actually works to pass through 50% of light.

Even a cursory googling of this problem quickly leads to this classical explanation of the phenomenon, so I’m somewhat amazed at my own previous ignorance (a common experience, to be sure). Live and learn, I guess.