Another post inspired by my book on Falling Felines and Fundamental Physics! I talk about geometric phases in the book in the context of falling cats, but here I focus on the polarization of light.
I regularly argue that most physics isn’t as scary and complicated as most people think. Once you get past the mathematics, which is analogous to a foreign language for the non-fluent, many of the concepts and ideas are intuitive, and even logical. This is, in fact, the motivation behind all the physics posts on this blog!
But some concepts are resistant to easy explanation, and can be quite difficult to understand, even when you are familiar with all the math involved! One topic that has vexed me, from an intuitive perspective, for a number of years is the concept of geometric phase. Broadly recognized as a general phenomenon in physics due to the groundbreaking work of Michael Berry in the 1980s, the basic idea is as follows. Some physical systems can be brought from an initial condition, or “state,” changed through a variety of intermediate states and back into its original condition, yet nevertheless have something different about it.
The easiest example of this to visualize is Foucault’s pendulum, a free-hanging pendulum on the Earth, which I have discussed in detail before. Because the pendulum is oscillating on the Earth, and the Earth is effectively turning underneath it, the pendulum changes the direction of its swing during the day.
For a pendulum at the North Pole, the Earth spins 360º underneath it during a day, making the pendulum appear to change direction by 360º. A pendulum at the equator doesn’t change direction at all over the course of a day. But this means that a pendulum at some intermediate latitude, such as Paris, changes direction by less than 360º during the course of the day. Although the Earth has rotated back to its starting position, the pendulum has not ended up swinging in the same direction — that discrepancy is what we call the geometric phase. It is “geometric” because its unusual behavior is related to the spherical geometry of the Earth.
So this case is somewhat easy to understand, but there is also a geometric phase associated with the behavior of light! This phase, called the Pancharatnam phase for reasons which we explain below, is a bit trickier to explain. Recently, however, I found a very nice way to visualize this change, and how it connects to “geometry.” This is what I will (hopefully) show in this post!