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## Refraction: it’s so hot right now.

Happy New Year!

A pair of interesting results on the common theme of refraction, or more specifically, refractive index, have been published in the last month. The first, reported in last year’s penultimate Nature (subscription required) is a good example of a simple physical question for which an answer is still elusive. The problem is this: what is the momentum of light in media other than the vacuum?

I think questions like these are wonderful for their simplicity, but on the other hand it’s embarrassing to realise modern physics can’t answer them. To see why this is the case, let’s examine some possible responses. We can take as given that light slows down in a non-vacuous media of refractive index n, to have speed c/n. There are two ways to proceed, the first due to Minkowski1, the second to Abraham2.

• The wave equation inside a medium of index n is λν = c/n. The Planck relation for light is E = hν, so that E(λ/h) = n/c, and as the de Broglie momentum is p = h/λ, the result is simple and even requires no Greek letters: p = (nE)/c.
• The postulates of relativity necessitate, after some thought, that E=mc2. While it isn’t sensible to talk of light having mass per se, no-one would frown if you wrote out the momentum as ‘mass’ times velocity, and this is just what Abraham did: p = (E/c2)(c/n) = E/nc.

Oh no! The two expressions differ by a factor of n2. They are only equivalent for the vacuum case, which is no help at all. For everyday transparent substances, we expect n to be just a bit larger than 1, but enough so n2 is a measurable factor. The arbiter is of course experiment, and these have been carried out. Minkowski predicts that the momentum goes up, Abraham that it goes down. Place your bets now.

The experiments all use the idea of passing light into a substance of n > 1, usually water, and measuring the momentum gained/lost by the substance (so that the system as a whole conserves momentum). This change should be evident, for example, in the water level: it will rise if the momentum goes up, or fall if it goes down. The punchline: shine a laser in a glass of water and the water level goes up.

Unfortunately, it isn’t so simple. The light induces shear stresses on the dielectric material if it irradiates it in a non-uniform manner. Tragically, these stresses also change the water level. Once this was realised, the theorists rushed off to perform a full general-relativistic calculation of the shear and the experimenters rushed off to use atom optics to try and get rid of its presence in the system. The latter technique is very nice: passing a beam of light through a condensate produces a standing atom wave. Passing two beams of light through, in opposite directions, creates two waves that can interfere. The interference pattern of the waves betrays the refractive momentum change of the light. The result of the experiment again favours the Minkowski prediction. However, general relativity is actually able to distinguish the two prediction as different types of momentum, applicable in different circumstances. I would therefore describe the result as snake eyes (from Dereli, Gratus & Tucker3):

“…it is widely recognised today that this controversy is an argument about definitions and that the relative merits of alternative definitions are undecidable without a complete (experimentally verifiable) covariant description of relativistic continuum mechanics for matter and fields…”

Cop-out? Maybe. On the other hand, the original physical arguments are so simple that it would be deeply disturbing if one were incorrect. The take-home message: always calculate in GR.

So, physics has its hands full even with standard media of refractive index n~1. In undergraduate physics courses we rarely hear about materials of substantial refractive index. Diamond is about two-and-a-half. A few years back it was fashionable to compete to make high-n materials, a bit like finding high-z galaxies. Last week’s creation is something else entirely: negative refractive indices4. While this sounds as impossible as negative probability, the physical structure is firm – just reverse all your normal intuitions about optics. Light bends out instead of in when entering the material, the optics is backward, there are reverse Doppler shifts: in short, they follow the left-hand rule.

Meta-materials, substances that take their properties from their current structure rather than composition, can be constructed to have such indices. The feasibility of such a programme was demonstrated a few years ago by inducing a reverse refraction effect with infra-red light, but last week’s demonstration used visible wavelengths, which we instinctively regard as more credible because they’re just that bit more familiar. So now all the talk is about fancy applications. Even the austere science journals can’t resist:

Such materials could be used in new kinds of lenses or even ‘invisibility cloaks’.

But the real question is: what’s the momentum of light in a material of negative refractive index? Anyone who sticks a negative sign in either of Minkowski’s or Abraham’s results gets a slap on the wrist.

References

1. Minkowski, H. (1908) Nachr. Ges. Wiss. Göttn Math.-Phys. Kl. 53
2. Abraham, M. (1909) Rend. Circ. Matem. Palermo 28, 1
3. Dereli, T., Gratus, J. & Tucker, R. W. (2007) Phys. Lett. A 361, 190–193
4. Dolling, G., et al. (2006) Opt. Lett. 32, 53–55

### 3 Responses

1. Can you even use photon models in that simple way for light travelling in a medium?

If I really wanted to answer this question I’d use classical electromagnetism.

2. Brendon – I don’t think classical electromagnetism can provide a full solution to the problem, as the stress tensor manipulation has to take place in a relativistic framework to provide a full description of the effects. While I agree that the photon model is the dodgier on paper, I would also note that the crucial point is the moment of interaction at the boundary between the media — not the traversal — and a quantised interaction is appropriate there.

3. A reader emails to say:
<blockquote>I’d say the truth is neither Minkowski nor Abraham: photon momentum doesn’t change in a medium. Consider the following thought experiment:
—————
| |
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* * * * | * * * * * * | * * * *
| |
| |
—————
So you have a stream of photons passing through a glass block. Once in a steady state, the block doesn’t move (clamp it while the laser or whatever is turned on). Inside the block, the
photons move more slowly, and are more closely spaced to compensate. The rate at which photons cross the mid-plane of the block is thus the same at which they enter the block, and at which they leave it. Since momentum is certainly transported by the photons, and since momentum is not accumulating in the block (each photon leaves with the same momentum it had to begin with), the momentum per photon must be constant in order for the rate of transport of momentum across the mid-plane of the block to be equal to the constant global value.
A minor objection is that this glass block has perfect transmission at the glass-air interface, which is not correct – but nothing seems to change if you give it an AR coating.</blockquote>