Birefringe Benefits

This week, I set a plastic container on our kitchen counter in the sun, and noticed an interesting effect:

The reflection of the sunlight off the counter gave a rainbow pattern. I recalled seeing a similar effect in demonstrations of polarized light. The stresses frozen into plastic cause a change in the light's polarization, which you can observe by putting it between two polarizing filters:

Wikipedia

I've mentioned polarizing filters before, in the context of reflection, but as I thought about this situation more, a problem occurred to me: The plastic is birefringent, which means it rotates the polarization of light passing through it, but the light from the sun is unpolarized, so rotating it should have no effect. The light only gets polarized afterward when it reflects off the countertop. To understand why, I have to get into the weeds a bit, so I suggest reading that earlier post I linked to before continuing.

First we should talk about what exactly a birefringent material does. Recall that materials have a property called the index of refraction, which tells how quickly light moves through them. In birefringent materials though, there are different indexes of refraction for different directions of the light's electric field. Depending on the thickness of the material, this can change, for example, horizontally polarized light to vertical, or to circularly polarized.

To really understand what was going on here, I decided to try working out some math on how the light's polarization changed as it passed through the plastic and reflected off the counter. One technique for keeping track of polarizations is Jones calculus. This is a linear algebra technique that represents polarization states as 2-element complex vectors:

These represent horizontal, vertical, right-circular, and left-circular polarizations. In this notation, the effect of a birefringent material is given by the matrix

where η is an overall phase delay, and Γ is given by

where Δn is the difference in indexes of refraction, L is the thickness of the plastic, and λ is the wavelength of light. For the case with the utensils between polarizers above, we could use this to see how the difference in indexes was changing: The difference determines which wavelength of color could pass through. This still doesn't answer the question about unpolarized light though. For that, we need Mueller calculus.

Now instead of a 2-element complex vector, we use a 4-element real vector. The components represent the total intensity, horizontal/vertical linear polarization, +/-45° linear polarization, and left-/right-circular polarization. Using the matrices given in the article above, we can construct the system we have in this case, a birefringent material, followed by a horizontal polarizer:

If we pass in unpolarized light, represented by the vector [1, 0, 0, 0], we'll get out a constant intensity, confirming my suspicion: In order to get the colors shown in the picture above, the light must have has some amount of polarization before passing through the container. Then the vector would pick up some dependence on Γ, which in turn depends on the wavelength.

I had never seen Mueller calculus previously, so this was a really interesting problem to work through. I just hope it was interesting for you too!