Tuesday, December 01, 2015

Seeing Photons in a New Light

If I asked you to picture a photon, an electromagnetic wave, I’d expect the image that popped into your head to look something like the one below, with the characteristic intertwined sine curves of the electric and magnetic field vectors.
Image Credit: Wiki user P.wormer, CC BY-SA 3.0

This model of electromagnetic radiation dates back to the work of James Clerk Maxwell, whose differential equations laid the groundwork for the pre-Einsteinian understanding of electricity and magnetism, and to this day, this is still how most students first learn to think about the quantum of light—as a plane-polarized wave, composed of perpendicular 2D components.

Pictured this way, however, the photon behaves unlike any familiar 3D system, with its components shrinking out of existence before popping back in, upside-down. As for analogies, waves on the surface of a pond fall short by a full dimension, leaving the student wondering which way is “up”, and what provides the restoring force equivalent to gravity. Once a student understands the derivative relationship between electricity and magnetism—that one is induced by a change in the other—the nature of light starts to make more sense mathematically, but this doesn’t make it any more intuitively palatable, and accepting the peculiar undulations of the sine wave as a real, physical thing can be a difficult stumbling block for people to surmount. But there’s an easier, more fundamental way to think of photons—it’s surprising at first, but ultimately a great relief to learn that the plane-polarized electromagnetic wave pictured above is not one photon, but two! In truth, neither part of the photon ever vanishes, or even changes in amplitude.

At the quantum-mechanical level, every photon—regardless of wavelength and energy—is circularly polarized. All this means is that, rather than oscillating up and down, like the tip of a bird’s wing, it spirals through space, like the blades of an airplane’s propeller. The electric field vector still oscillates perpendicular to its direction of propagation, fitting the definition of a transverse wave, but the crucial thing to note is that it's moving in both perpendicular directions. If we were flying alongside a plane with lights on the tips of its propellers, matching its speed, the lights would trace out sine waves against the sky behind them, oscillating straight up and down, changing direction when they reach their maximum amplitude. However, this fact arises simply from the choice of a perspective that ignores one of the propeller’s degrees of freedom. From any other angle, we’d see that the propeller’s tips don’t slow to a stop and change direction twice per cycle, they simply continue on their path around the circle.
To an observer situated in the plane of rotation, the electromagnetic field vector appears to oscillate sinusoidally, as below.

Like the classic "Spinning Dancer" illusion, this could be seen as a side-view of either clockwise or counterclockwise rotation.
Image Credit: Wiki user ikaxer, CC BY-SA 3.0 

Ordinarily, students learn about the linearly-polarized sine wave first, in part because of the mathematical simplicity, and in part because it makes for a convenient analogy between polarizing filters and the slats of a picket fence—only light oscillating vertically can make it through the gaps between the pickets. When they’re advanced enough in their understanding to handle the concept of superposition, the teacher can introduce the idea of circularly-polarized light as a superposition of perpendicular plane-polarized states, a quarter-cycle out of phase.
Image Credit: Wiki user Averse, CC BY-SA 2.0

However, it’s possible—and perhaps even preferable—to teach it the other way around. If two circularly polarized photons are in superposition, but spinning opposite directions, they’ll appear as a single plane-polarized photon, as their components cancel out at one point in the cycle and add together a quarter-cycle later. Not only is it more correct from a quantum-mechanical point of view, thinking of the linearly-polarized photon as a superposition of two circularly-polarized photons with opposite handedness also offers a better intuitive understanding of why a linear polarizer inserted at 45 degrees between two perpendicular linear polarizers allows some light to pass through all three.

When I was in school, I was taught to think of a "ray" of light as containing linearly-polarized photons oscillating at all different angles, each staying within its plane. Ultimately, my displeasure with this description stems from the fact that, to generate a linearly-polarized photon, you need a charge moving straight up-and-down, or side-to-side, which is a very unnatural thing for electrons to do outside of an antenna: far more natural is the kind of circular motion displayed in the Bohr atom, which can generate a circularly-polarized photon with ease. It may be the easiest to teach from a mathematical perspective, but describing the fundamental quantum of light as a linearly-polarized sine wave strikes me as counterintuitive and, ultimately, counterproductive.


  1. A = √ ( L / 4 )^2 - ( λ / 4 )^2

    A = amplitude
    L = tense length of wave
    λ = wave length

    1. I love you thanks for this amazing post

  2. It doesn't seem correct to say there are two photons. Rather there are two circularly polarized eigenstates (L and R as per the Wikipedia article). A single photon exists in a superposition of these two eigenstates; it is not two individual photons propagating together while interfering with each other.

  3. Just stop delaying real use of complex numbers 'til upper division and we'll stop having problems like this. There's no excuse for making students do everything the hard way instead of showing them proper (complex) mathematics from the get-go.

  4. This is the image in my mind is at http://www.animatedphysics.com/photons/maxwell_photon_320x240.gif

  5. It helps me more understand now. Thanks.