## Measuring the ‘kick’ of a photon leaving a fiber!

In physics, there are a number of fundamental and seemingly simple questions which have remained a source of controversy for years, even decades. Last month, a paper was published by a Chinese research group describing an experiment which throws new light on one of these controversies, the so-called Abraham-Minkowski controversy.  For nearly a century, theorists and experimentalists have struggled to answer the question A-M raised: how does the momentum of a photon change as it enters a material?

Though, as we will note, the new experiment does not completely put Abraham-Minkowski to rest (as we note below, there may not in fact be a controversy anymore at all), it does illustrate quite nicely that light carries momentum, and does so in a way which can be recorded on camera!  We take a look at the controversy and the recent experiment below.

It is easy to forget that light waves (or photons, if you have a quantum-mechanical bias) possess momentum.  The ‘kick’ of a photon is so small that it doesn’t have an observable influence in our day to day activities: obviously, we don’t get ‘beaten down’ by sunlight when we go outside!  In space, however, this momentum can be quite important: several years ago researchers demonstrated that absorption/emission of photons by asteroids can significantly alter the spin of an asteroid, an effect known as the YORP effect (after scientists Yarkovsky-O’Keefe-Radzievskii-Paddack).

It has also been hypothesized that spacecraft could be entirely propelled by the radiation pressure from the sun: a spacecraft sporting a giant reflective ‘solar sail‘ could in principle be blown through space by photons much like a sailboat is propelled by the winds.  Such solar sails have been a staple of science fiction for years, as well as fodder for other sorts of speculation.  Another device known as the Crookes radiometer, a collection of vanes enclosed in a glass bulb which rotate when light is shined upon them, is often mistakenly said to be a demonstration of the radiation pressure of light:

The rotation is actually caused, however, by a thermodynamic effect: the black side of each vane gets heated more than the reflective side, and the air around the vane gets heated asymmetrically.  (Funny story: My old department physics building had a Crookes radiometer in an ‘optics’ display case.   I’m not sure what it was intended to show, however, because the radiometer was broken and stood completely still in the face of a bright bulb!)

An individual photon has an energy $E$ in vacuum proportional to its frequency (color) $\nu$, the constant of proportionality being Planck’s constant $h=6.626068 \times 10^{-34}$ m2 kg / s:

$E = h\nu$.

The momentum $p$ of the same photon in vacuum is given by:

$p = h\nu/c= h/\lambda$,

where $c= 3 \times 10^8$ m/s is the speed of light in vacuum and $\lambda = c/\nu$ is the wavelength of the photon.  A photon of wavelength 500 nanometers (which would be green) would have a momentum of $p = 1.3 \times 10^{-27}$ kg m/s.  For comparison, a proton moving at a paltry $v =1$ m/s (human walking speed) would have a momentum of $p = mv = 1.6 \times 10^{-27}$ kg m/s.

None of the preceding discussion is in any way controversial.  The controversy arises when one asks how the momentum of a photon changes when it enters a medium with a refractive index $n$.  (Refractive index is a rough measure of the fraction by which the speed of light is reduced on entering a material.  A photon moving at speed $c$ in vacuum moves at speed $c/n$ in a material of refractive index $n$.)  In a paper written in 19081, Hermann Minkowski proposed one expression for the momentum of a photon in a medium, and in 19092, Max Abraham countered with a seemingly contradictory expression.  In modern parlance, the difference is usually expressed in terms of the momentum of a photon in a medium.  Minkowski proposed that the momentum increases by a factor of $n$, i.e.

$p_M = n h/\lambda$,

while Abraham proposed that the momentum decreases by a factor of $n$, i.e.

$p_A = h/(n\lambda)$.

The original papers of M and A involve a detailed analysis of Maxwell’s electromagnetic equations, and we will return to them in a future post (after I find the time to translate the lengthy German and Italian articles).  We can, however, offer at least a ‘plausibility’ argument for each of their conclusions, with the caveat that neither scientist may have endorsed these arguments!

Looking first at the Minkowski momentum, we have already stated that the momentum of a photon in vacuum is given by $p = h/\lambda$.  Let us assume this formula holds even for photons in a material medium.  When a wave enters a medium of refractive index $n$, its wavelength is shortened to $\lambda' = \lambda/n$.   This results in the momentum of the photon being increased by a factor of $n$.

To justify the Abraham result, we note that the momentum of a particle with mass $m$ and velocity $v$ is given by $p = mv$.  Though this expression does not apply directly to the massless photon, we could imagine that the momentum of a photon is proportional to its speed in a similar manner.  Then, since the speed of a photon is given by $v = c/n$ in a medium, we can argue that the momentum must be decreased by a factor of $n$.

This argument is similar to the original controversy concerning the wave/particle nature of light that arose in Newton’s time, as noted in my post of Arago’s 1810 experiment.

A number of experiments have been performed since A-M’s time to try and determine the ‘true’ form of photon momentum in a medium.  These experiments are difficult for a number of reasons.  First, as we have noted, the momentum of light is generally very small and attempts to measure it are often swamped by other effects (as in the Crookes radiometer).  Second, it is difficult to isolate the momentum of light within a medium in a way in which it can be measured.  One example is an experiment performed by Jones and Richards3 in 1954.  To quote the abstract,

Light falls asymmetrically on a metal vane mounted on a torsional suspension.  The normal pressure of the radiation on the vane gives rise to a mechanical couple, resulting in a small deflexion which is observed by means of an optical lever amplifier.  The suspended system is mounted in a container which can be filled with various liquids, and the pressure on the vane when it is immersed in a liquid is compared with the pressure on the vane in air.

The Jones and Richards result was initially thought to support the Minkowski theory, though a careful analysis suggested the Abraham theory was true.  Other experiments have produced contradictory results, and the A-M controversy has remained open.

We now flash forward to the recent experiment:  Researchers She, Yu and Feng, from Sun Yat-Sen University in China looked at measuring the momentum change in a different manner: by looking at the ‘kick’ an optical fiber receives as a laser pulse exits it!

A nanofiber silica filament was allowed to freely hang in air or vacuum.  A pulse from an unpolarized semiconductor laser of 10 mW peak power is sent through the filament.  The idea behind the experiment is quite simple: In the Minkowski interpretation, the momentum of light will decrease on exiting the fiber, while in the Abraham interpretation, the momentum of light will increase on exiting the fiber.  Because the total momentum of the fiber and optical pulse should be conserved, the fiber will either experience a recoil (in the Abraham interpretation) or a pull (in the Minkowski interpretation).  Even if the pulse exits the fiber asymmetrically (at an angle to the vertical), there will be a measurable difference between the response of the fiber, as the figure (adapted and exaggerated from the paper) below illustrates:

By comparing the actual behavior of the fiber with a careful computational simulation of the fiber’s behavior, the researchers determined that the results were consistent with the Abraham prediction.  Effects due to temperature (such as in the Crookes radiometer) or due to unusual scattering were analyzed and determined to not influence the measured results.

Perhaps the neatest thing: the researchers were able to video the bending of the fiber  (h/t Swans on Tea):

Okay, so the video is a bit repetitive and may give you a seizure if you watch it.  It is nevertheless very exciting to see a terrestrial demonstration of the momentum of light.

Does this resolve the controversy?  I’m not an expert on this topic, and there are a lot of subtleties, but I would venture to say: probably not.  In fact, there are convincing arguments that suggest that there isn’t really a controversy at all!

The difficulty lies in the fact that any discussion of the momentum of light in a medium must properly account for the total momentum of the system, which includes the momentum of the medium itself.  When traveling into a medium of refractive index much greater than unity, the light is strongly interacting with the material and it becomes almost arbitrary to distinguish between the momentum of the photon and that of the matter: the two are completely intertwined.  With this perspective, one would say that the designation of ‘light momentum’ and ‘medium momentum’ are completely arbitrary, merely different ways to slice ‘total momentum pie’.  Differences in experimental results can be explained away as a failure to completely account for the interaction between the light and the medium.

In fact, a recent 2007 review article (h/t Swans on Tea, again) by Pfeifer, Nieminen and Heckenberg4 argues quite convincingly that this is the case.  A lot of controversies in physics end this way: with the participants realizing that, in fact, the question they originally asked was ill-posed.  With my relatively superficial understanding of the controversy, this seems like the most sensible resolution of the controversy.

Nevertheless, the work by the group at Sun Yat-Sen is a really neat demonstration of the momentum of light and light/matter momentum transfer.

1 H. Minkowski, “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern,” Nachr. Kön. Ges. Wiss. Gött. Math.-Phys. Kl. (1908), 53.
2 M. Abraham, “Sull’elettrodinamica di Minkowski,” Rend. Circ. Mat. Palermo 28 (1909), 1.
3 R.V. Jones and J.C.S. Richards, “The pressure of radiation in a refracting medium,” Proc. Roy. Soc. Lond. A 221 (1954), 480.
4 R.N.C.Pfeifer, T.A. Nieminen and N.R. Heckenberg, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” 79 (2007), 1197.

Weilong She, Jianhui Yu, Raohui Feng (2008). Observation of a Push Force on the End Face of a Nanometer Silica Filament Exerted by Outgoing Light Physical Review Letters, 101 (24) DOI: 10.1103/PhysRevLett.101.243601

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### 5 Responses to Measuring the ‘kick’ of a photon leaving a fiber!

1. stuwat says:

I haven’t read the She, Yu, Feng paper and would really like to know how they ruled out thermal effects. It seems like there should be a more obvious explanation of why the fibre behaves the way it does, one which accounts for it based on the repeated absorption of light and dissipation of heat, in what is a very fine fibre. The movement may just happen to resemble the way it would move according to the Abraham prediction. It’s a lovely demonstration and fuel for curious minds.

2. IronMonkey says:

It also appears to me that the She et al. experiment does not resolve the issue. The observation made by stuwat is very interesting. A quick search on Wikipedia for the speed of light in a dense medium suggests:

“As light propagates through dielectric material it undergoes continuous absorption and re-radiation. Therefore when the speed of light in a medium is said to be less than c, this should be read as the speed of energy propagation at the macroscopic level. At the microscopic level electromagnetic waves always travel at c.”

It thus appears to me that the issue of light momentum could be similarly treated as above. Namely, that some experiments may measure the a “macroscopic light momentum” as M or A, however the momentum itself may never change and remain p=h/lambda as in vacuum… Hence, our interpretation might depend on the frame of reference used.

In any case, this is a very interesting physical question and I’m looking forward to see if it will be resolved (or has already been done).

3. stuwat: They do address the possibility of thermal expansion in the fiber in some detail. Basically, they adhered a colophony filament to their fiber and ran cw power through it for a period of time. The colophony decomposes at 300 C, and no decomposition was observed, putting an upper limit on the fiber temperature and thermal expansion. Of course, there could be subtler effects at play.

IronMonkey: My interpretation is very similar to yours! In fact, there seems to be little question about the microscopic definition of electromagnetic momentum. The problem comes in when one tries to apply the macroscopic equations, which are in a very real sense an oversimplification of the problem.

4. IronMonkey says:

Today I made a rapid reading of the colloquium by R. N. C. Pfeifer et al. (skipping the tensors’ stuff). Unsurprisingly, my initial understanding of the issue was oversimplifying the A-M problem which requires a rigorous account of all material, internal and external effects on the system. I must say the authors made a nice historical review. They also point out that the A-M controversy has already been resolved: each version of the electromagnetic momentum tensor is not adequate on its own, we must include the complete material momentum tensor such that the total momentum tensor is correctly evaluated. Hence our choice of A or M in the equation would be more a question of aesthetics.

5. Blake Stacey says:

A lot of controversies in physics end this way: with the participants realizing that, in fact, the question they originally asked was ill-posed. With my relatively superficial understanding of the controversy, this seems like the most sensible resolution of the controversy.

So, we should denote the momentum of the photon by [tex\mu[/tex]?

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