# Measuring the Angular Momentum of Light

I'm teaching a junior/senior level elective this term on quantum mechanics. We're using Townsend's A Modern Approach to Quantum Mechanics, which starts with spin-1/2 and develops the whole theory in terms of state vectors and matrices. This is kind of an uneasy fit for me, as I'm very much a swashbuckling experimentalist, and not as comfortable with formal mathematics.

This occasionally leads to good things, though, such as Monday's class, on photon polarizations. the book uses some vector arithmetic to show that circularly polarized photons have spin angular momentum of one unit of h-bar. Being a formal and mathematical book, it pretty much leaves the subject there, but my immediate reaction is to look for an experiment that proves the angular momentum is real. So I did a little Googling, and turned up a paper from 1936(!) that does just that. And I talked about it in class, because I think experiments are way cool, and like to bring them in whenever possible. Having looked this up and read it carefully, I figure I might as well write it up for ResearchBlogging while I'm at it. The Q&A format worked pretty well last time, so we'll stick with that.

What's the paper? The paper is "Mechanical Detection and Measurement of the Angular Momentum of Light" by Richard A. Beth, who is the sole author listed, though he does single out a Mr. Wilbur Harris who "deserves much credit for the thoroughness and enthusiasm with which he carried out the tedious observations." Much credit, but not co-authorship, evidently...

What does the paper describe? The paper reports on a series of experiments looking for angular momentum in light. Angular momentum, as the name suggests, is related to the rotational motion of objects, and circularly polarized light is predicted to have angular momentum. The experiments in the paper looked for, and found, evidence of this angular momentum by measuring the twisting of a quartz plate when circularly polarized light was sent through it. The apparatus is shown at right.

Back up a minute-- circular polarization? Yeah, circular polarization. Normally, when people talk about the polarization, they refer to the direction of the electric field associated with the classical light wave. The electric field oscillates up and down along some direction, changing its magnitude all the time.

There's another way to make polarized light, though, which is to keep the magnitude of the electric field constant, and make the direction change all the time. In this case, the electric field starts out pointing up (say), then some time later points to the left, then down, then to the right, then up again. It completes one full revolution in the same time that it takes the light wave to complete an oscillation. There are two different circular polarization states, corresponding to the two different directions of rotation.

And this is real? Absolutely. You can make circularly polarized light using properly cut calcite, or a variety of other materials. It's even got technological applications-- some 3-d projection systems use circular polarizers as the lenses of the glasses, because they look less dorky than colored filters, and don't require you to hold your head at a particualr angle to get the 3-d effect.

OK, so they used this to make a glass plate spin? Sort of. The plate in question was a "half-wave plate," a specially cut piece of glass that shifts the polarization of light. If you send in linear polarization, it emerges with a polarization 90 degrees away from what came in. If you send in circular polarization, it comes out with the opposite handedness-- left-hand circular polarization is turned into right-hand, or vice versa.

What they did in the experiment (I'll use the plural pronoun in honor of Mr. Wilbur Harris) was to suspend a half-wave plate (marked "M" in the figure) from a thin quartz fiber, and hang it inside a vacuum chamber, to eliminate any effects of air currents. Above the plate, they put a second quarter-wave plate (which turns linear polarization into circular, or vice versa), with a reflective coating on its top surface ("T" in the figure). This had the effect of reversing the polarization of any light reflecting off it.

Below the hanging wave plate, they had a light source, with a quarter-wave plate above it ("B" in the figure-- it was outside the vacuum chamber, so the light also passed through a window marked "W"). When this quarter-wave plate was set properly, it produced a beam of circularly polarized light that passed up through the hanging plate, reflected off the upper plate, and passed through the hanging plate again on the way down.

How did this make anything spin? Well, if you look at the light coming up from below, it's initially right-hand circular polarization. After it passes through the plate, it emerges as left-hand circular polarization. Its angular momentum has thus changed sign-- it starts off positive, and becomes negative after passing through the plate.

Wait, isn't angular momentum conserved? Exactly. In the absence of some external force producing a rotation, the angular momentum of a system has to be a constant. So if the light comes in with positive angular momentum, and leaves with negative angular momentum, the plate has to pick up positive angular momentum to compensate. So it starts to turn.

What about the second pass? on the downward path, the light starts off with left-hand circular polarization, which corresponds to angular momentum in the same (positive) direction as the first beam. Again, it emerges with the opposite circular polarization, and thus negative angular momentum, so the plate must pick up positive angular momentum to compensate.

So it spins twice as fast? Sort of. The quartz fiber the plate is hung from resists turning, so the plate will twist some distance, then stop. They put markings on the outer edge of the plate, and used a telescope outside the vacuum chamber to measure the amount that it twisted for various polarizations of incoming light.

Sounds tedious. Mr. Harris deserves much credit.

Isn't this all really classical, though? In the limit where they were working, you can get the same result from a classical theory. It works out the same way, though, if you assume that each individual photon of light carries a discrete amount of angular momentum, given by Planck's constant divided by 2π. Their measurements weren't sensitive enough to see a single-photon effect, but, hey, cut them some slack. It was 1935.

True. And people make use of the single-photon angular momentum all the time-- it's one of the tricks that makes laser cooling work, for example. By choosing the appropriate polarizations for your lasers, you can make sure that they only interact with atoms in some states, and not others. That lets you do selective cooling without accidentally heating the same, and lets you reach absurdly low temperatures. It's also important for a lot of work in quantum computing.

That's pretty cool. Especially the 1935 part. Yeah, it's amazing what they managed to get done, back in the day. Even if it didn't rate an author credit for Mr. Harris. And the math you need to describe photon polarization is the same as the math you use for spin-1/2. Which makes it a great example for my current class, hence looking up this paper, and hence this post.

Beth, R. (1936). Mechanical Detection and Measurement of the Angular Momentum of Light Physical Review, 50 (2), 115-125 DOI: 10.1103/PhysRev.50.115

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Nicely put! It's probably worth noting that the concept for the experiment goes back even further, to 1909, and a theoretical paper by Poynting.

Most of the early papers on the angular momentum of light were compiled into the nice book, "Optical Angular Momentum", L. Allen, S.M. Barnett and M.J. Padgett, eds. (IoP Press, 2003).

Building on gg's reference, that particular book goes into depth on the 'other' angular momentum of light. Orbital angular momentum, which is similar to the orbital angular momentum of an an electron in the hydrogen atom (though I have heard some say Jackson derives it, despite the fact I can't find such a derivation anywhere in the 3rd edition).

Padgett et al. performed an experiment where circularly polarized (CP) light with orbital angular momentum of '1' to turn a micro-triangle in an optical tweezers setup at a frequency corresponding to 2 units of h bar per photon! They could also get the micro-thingy to stop spinning by reversing the handedness of the CP beam. Pretty neat stuff, though still waiting on a truly quantum consequence of it.

It's amazing that the original macroscopic version of this experiment works. There is a cool way to perform the experiment with optical tweezers and microscopic objects. It's written up in "Beth's experiment using optical tweezers," D. N. Moothoo, J. Arlt, R. S. Conroy, F. Akerboom, A. Voit, and K. Dholakia, American Journal of Physics, Volume 69, Issue 3, pp. (March 2001) 271-276.

one point you made here - that the electric field of a circularly polarized light beam doesn't oscillate through a value of zero - never clicked with me before. That the mean value of E is zero but if never actually is zero.

thanks

As long as we're talking about orbital angular momentum, I have to point out this 2004 paper, which I just got a kick out of when it first came out. (Optics Express is open access, so everyone should be able to see it and the associated video.)

The basic idea is to fashion a light-driven "microoptomechanical pump" using the orbital angular momentum of light to spin things around.

gg,

that is a cool paper. I wonder what would happen if one would to include a pumping mechanism due to the ponder-motive force.

I'm not sure if this means there are two ways to get circularly polarized light then. Radio waves generated via a dipole antenna separate into two circularly polarized modes in magnetized plasma. It's still birefringence, but the modes separate -- the wave speed of left-handed waves, the "ordinary mode", depends on electron number density only, while the speed of a right-handed wave also depends on magnetic field strength.

The teams associated with the Radio Plasma Imager on the ill-fated IMAGE satellite used this to good effect to study the magnetosphere and plasmasphere since this mode separation is easy to see due to waveguiding when radio waves propagate parallel to magnetic field in a magnetized plasma. The birefringence of radio waves in magnetized plasma is also used by astronomers and astrophyscists, who call it Faraday rotation.

PS. Sorry for geeking out. I've switched fields completely and pretty much never get to talk physics, let alone have time to think about my old work or similar stuff.

@agm
That's pretty awesome.