# Photons Travelling at Less than c in "free space"

nitsuj
BBC has a small article on how Prof Daniele Faccio of Heriot-Watt University and Prof Miles Padgett of Glasgow slowed a photon to less than c in "free space", which I pressume to mean a vacuum.

Done by "changing the photon's shape" via some medium. When the photon returned to "free space" it continued it's retarded velocity.

Is this pucky? Or is there some known reason that a photons "shape" can effect what it's maximum velocity is? I pressume that the retarded velocity is still the maximum for the particular photon since they're massless.

article

Edit: reading about light through mediums maybe this means the photon maintains it's slowed group velocity after leaving the medium.

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## Answers and Replies

Mentor
I'm pretty sure we had a recent thread on this but I can't find it right now. IIRC, the key point is that yes, the "shape" of a photon can affect its velocity. Strictly speaking, photons only travel at ##c## if they are plane waves in free space. The photons coming out of the medium in this experiment, even though they emerge into free space, are not plane waves.

Science Advisor
Gold Member
I think this is the paper the BBC article was referencing: http://arxiv.org/ftp/arxiv/papers/1411/1411.3987.pdf

I think this phenomenon is not so much different from a wave-guide, except they've found a (quite intricate) way to keep the "shape" of the wave-packet in the absence of a waveguide. But I've only made a cursory glance at the paper, so probably an optics guy can answer any questions better. Peter's statement that "photon's only travel at c in plane waves in free space" actually appears to be the main point of this paper. The relation ##v_p v_g = c^2## (where ##v_p## is the phase velocity and ##v_g## is the group velocity of a wave packet) is a quite general dispersion relation for light though, as far as I know. So if you can make a wave packet where ##v_p \neq v_g## then you necessarily create a situation where ##v_g<c## and ##v_p >c##.

nitsuj
Thanks for the replies guys!

And thanks for the link to the paper!

waves are pretty interesting :)

Mentor
I think this phenomenon is not so much different from a wave-guide, except they've found a (quite intricate) way to keep the "shape" of the wave-packet in the absence of a waveguide.

This is my understanding as well, and IIRC was also the understanding of others in the previous thread I mentioned.

Science Advisor
I used to be an optics guy, although I'm a bit rusty these days. I've only read the article Matterwave linked, and don't know any deep background.

They're using Spatial Light Modulators (SLM - programmable diffraction gratings, basically) to manipulate a single photon wave into a form that spreads out, rather than remaining collimated. They chose Gaussian and Bessel function beam profiles because they are easy to produce and the maths to describe beam width as a function of distance is relatively straightforward in those cases. They used parametric down-conversion to produce two in-phase photons, fed one through their apparatus and one past it, then used a form of interferometry I'm not familiar with to measure the delay introduced by the structuring. Using SLMs to introduce the structuring is kind of neat because the optical path length isn't changed by turning them on, so the interferometer stays calibrated when the structuring element is introduced, which it would not do if they used the equivalent "real" optical elements.

What's also kind of neat is that they can use a "simple ray geometric model" to estimate the slow-down in the cases they used. In other words, they calculate the radius, r, of the photon at a distance, D, then calculate the delay by arguing that the photon has "actually" travelled ##\sqrt{r^2+D^2}##, instead of just D that a plane-wave photon would have done.

At least, that's my reading.

Lord Crc
They're using Spatial Light Modulators (SLM - programmable diffraction gratings, basically) to manipulate a single photon wave into a form that spreads out, rather than remaining collimated.

As a layman, what I'm trying to wrap my head around is how exactly you can collimate a single photon, or what exactly they've "spread out". Are we talking about the wave function here or? Would it affect what you see if you put a spatial detector (camera) at the half-way mirror in figure 2C in their preprint? After adjusting the coincidence counting of course.

Mentor
Are we talking about the wave function here

Yes.

Would it affect what you see if you put a spatial detector (camera) at the half-way mirror in figure 2C in their preprint?

I believe so, yes.

Lord Crc
Science Advisor
As a layman, what I'm trying to wrap my head around is how exactly you can collimate a single photon, or what exactly they've "spread out". Are we talking about the wave function here or? Would it affect what you see if you put a spatial detector (camera) at the half-way mirror in figure 2C in their preprint? After adjusting the coincidence counting of course.
You may have read about the single-photon double-slit experiment, where you can repeatedly send single photons through two slits and build up a pattern showing that each photon interferes with its own wave-function. In some senses, this experiment is a variation on that. It's just that they are using a diffraction grating that is a bit more complex than the double slit, and the diffracted wave turns out to have some interesting properties.

In terms of what you'd see if you stuck a camera in the middle of 2C, they seem to be pushing a single photon through each arm of the interferometer. That means that your camera would see what any camera sees when it receives a single photon: a dot somewhere in the field. However, if you left the shutter open and repeated the experiment, there would be a pattern to the way the dots build up. There's a sequence of images showing this kind of pattern build-up at Wikipedia (although this particular example is electrons through a double slit). In this experiment you would get the same effect, except that the pattern that built up would be uniform (with the SLMs off), a Gaussian dot (with the Gaussian beam) or an Airy disc (with the Bessel beam). Or possibly the Fourier transforms thereof (I'm not sure without thinking a lot harder about this) - there would be structure anyway.

I'll repeat my health warning from earlier - I am qualified to talk about optics, but it's been a few years and I haven't read any more about this than the BBC article and the paper Matterwave linked.

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Lord Crc
In this experiment you would get the same effect, except that the pattern that built up would be uniform (with the SLMs off), a Gaussian dot (with the Gaussian beam) or an Airy disc (with the Bessel beam).

Ok yes I was thinking of the build-up ala the single photon double slit experiment, so that's what I expected if they modified the shape of the wavefunction.

Now to wrap my head around this altered wavefunction somehow travels slower. I'll read the preprint again tomorrow.

Thanks for the replies to the both of you.

Science Advisor
More or less. In a plane wave, all points on the surface of a wavefront are moving in the same direction at speed c. In an expanding wave, not all parts of the beam are pointing in the same direction. The part of the wave front that is pointing an angle $\theta$ off axis has a forward velocity of $c\cos\theta$. Averaged over the wavefront, then, the forward velocity is less than c. The lateral velocities average out - for every point at an angle $\theta$, there's a point at $-\theta$.

That's a very simple-minded analysis that really only works for certain beam profiles like the Gaussian and Bessel beams. For other structures you actually need to look at the field equations. You'll have to look for someone else to get an explanation at that level. :D

nitsuj and joeyshmowe
Gold Member
I, for one, am confused. How is that not in contradiction with say, Born & Wolf page 15 where they simply solved the wave equation and sought spherical waves solutions and found out that the speed of propagation is what would be "c" for light?
By that I mean the claim of the paper
even in free space, the invariance of the speed of light only applies to plane waves.
.

Lord Crc
In a plane wave, all points on the surface of a wavefront are moving in the same direction at speed c. In an expanding wave, not all parts of the beam are pointing in the same direction. The part of the wave front that is pointing an angle $\theta$ off axis has a forward velocity of $c\cos\theta$. Averaged over the wavefront, then, the forward velocity is less than c.

Right, I mostly had managed to confuse myself because based on the video I thought the slow speed continued after they "undid" the spatial modification, but from what I can gather from the paper that's not their claim.

So my mental model of their work is that if one imagines a slinky stretched along the x-axis on a table, with both ends fixed but with one end moving in the x direction in a cyclic manner, then each compression wave is kinda like a regular phonton flying along, and what they've done is manipulate the slinky so that the each compression wave also moves in the y-plane. Then the path length has changed and of course it takes the wave longer to reach the other end, assuming it moves with a fixed speed.

Is this roughly it, or is it way off in the fields?

nitsuj
Then the path length has changed and of course it takes the wave longer to reach the other end, assuming it moves with a fixed speed.

That's how I'm understanding this from reading about waves in this thread and on wiki; my intuition was what is being measured? I don't understand how a photon relates to a wave and from a wave to group & phase velocities.

This wiki page has a great visual. So with this experiment, is it the group velocity is slowed to less than c, but the phase velocity is c?

zen loki
In which case, what is the theoretical minimum vacuum group velocity of light?
Can we just slow it down arbitrarily?

Mentor
is it the group velocity is slowed to less than c, but the phase velocity is c?

I believe the phase velocity is faster than c. The product of group velocity and phase velocity is c, so if group velocity is slower, phase velocity is faster.

Staff Emeritus
Science Advisor
Gold Member
In an anomalously dispersive material, we can have ##v_g > c##.

We can even set up a situation where ##v_g < 0##

zen loki
Of course we can slow light down in a medium, but it is intriguing that it can be done in a vacuum.
If the light is NOT in a medium, just how much can we slow it down?

Staff Emeritus
Science Advisor
Gold Member
I suspect it is a very similar principle, i.e., combining wavelengths and phases to get the desired result. It is interesting that this done at the photon level, but I don`t find it to be very surprising theoretically (experimentally is a different story). Maybe I haven't looked at it deeply enough, though

Science Advisor
Gold Member
In an anomalously dispersive material, we can have ##v_g > c##.

Are you sure about this statement? It is my understanding that the group velocity being greater than c would mean one can deliver energy faster than c in violation of SR.

Staff Emeritus
Science Advisor
Gold Member
Are you sure about this statement? It is my understanding that the group velocity being greater than c would mean one can deliver energy faster than c in violation of SR.

SR is not violated; take a look at this cool animation by Greg Egan,

http://gregegan.customer.netspace.net.au/APPLETS/20/20.html

Science Advisor
I, for one, am confused. How is that not in contradiction with say, Born & Wolf page 15 where they simply solved the wave equation and sought spherical waves solutions and found out that the speed of propagation is what would be "c" for light?
By that I mean the claim of the paper
even in free space, the invariance of the speed of light only applies to plane waves.
.
Interesting point. It might be that they are talking about beams only - if you enclose a spherical wave source in a box with a small hole to make a beam, you'll get diffraction effects that I guess (haven't done any maths to prove it) will introduce dispersion. It might also be that this is a pre-print, and some reviewer will pick up on that. :) I'm really not sure.

As a side note, I can't find my copy of Born and Wolf. How do I lose a book that size...?