# Light, up to 11

Whew! Back from a very successful wedding and honeymoon, moving into a new apartment, writing thank-you notes, and all the fun jazz that comes with being newly married.

But hey, we’ve got to get this blog cranking again at some point, and now’s as good a time as any. We’ll kick things back off with a letter from a reader,

Scott writes in with a question:

If you have a cubic meter of nothing but highly condensed photons, what would the upper limit on its energy density be? (If there is even a limit.)

Classically there’s no theoretical limit on the field strength, though radiation pressure will eventually be too large for any physical box to contain. It would be an entertaining calculation to look at the stress-strain curves of something like solid steel to see just how much light we might be able to stuff into a magic 100% reflective box before we broke the yield strength.

But the physical reality in which we live is not an entirely classical one, and Scott’s question in fact presupposes that we’re dealing with the full quantized description of photons of light. Quantum Electrodynamics (QED for short) is a tricky subject that I’m actually in the midst of learning formally as part of the required Standard Model class. We don’t need to actually do hard-core QED to answer the question though, an order-of magnitude estimate will work fine.

It’ll turn out that if you stuff enough energy into the vacuum you’ll eventually start creating matter (electron/positron pairs in this type of circumstance) via Einstein’s famous E = mc^2 relationship. In nuclear weapons we’re used to seeing the m turn into E very dramatically, but of course the other direction works just as well. Get enough E and you’ll start making m.

We might estimate that if we let m be the mass of the electron, that if we stuff E = mc^2 worth of electromagnetic energy into one Compton wavelength of the electron that we might have a good estimate for the maximum amount of energy we can fit into a given volume before we start pair production and therefore don’t have (just) a box full of light anymore.

The Compton wavelength of the electron is:

Which works out to be about 3.96e-13 meters. We might take the cube of this to get the “Compton volume” of the electron, which is about 5.76e-38 cubic meters. The E=mc^2 energy of the electron works out to be about 8.19e-14 joules, so the total energy density of works out to be, drumroll…

1.4e24 joules/meter^3.

Which is pretty hefty. We’re talking “giant asteroid impact per cubic meter” hefty. But how does that compare to the light intensity we can generate with current laser technology?

Well, we’ve calculated an energy density, not an intensity. Intensity is watts per square meter – or if you prefer, power per area. We calculated energy per volume. So we have to do a bit of unit conversion. If we imagine that we turn on a flashlight for 1 second, we’ve created a column of light with a length of about 186,000 miles containing a total energy equal to the power of the flashlight times 1 second. The relationship between energy density and intensity is thus (E/V)*c = I, where I is the intensity, watts per meter^2. Which is good because the units work out. It comes out to be something like 4.2e32 watts per square meter, or about 4.2e28 W/cm^2.

Right now our best lasers are generally in the 10^20 W/cm^2 range, so we have a ways to go before we can start stuffing boxes to their limits with photons.

Still, when you’re observing interactions between light fields and electrons that are already moving relativistically fast you can actually get these wacky QED effects at reachable intensities. Chad wrote a bit about this a while back, in fact.

Will we ever get to the point when we just can’t make our lasers any more intense without turning them into particle beams? Well, I’m not holding my breath. But it would be cool.

What the heck, one more picture from the reception:

1. #1 raghuvir jha ( india )
September 24, 2010

it is very good to find you again here. i speculating that you may not come so quickly. but you are very professional. you are looking very smart and handsome in the above picture. god was very kind on you , he given you both look and mind.

now i come into right track ,i have a very silly doubt, how can you confine a photon into a box? as far as i know, no sooner photon creates it moves with the very high speed. i have another doubt, suppose there is nothing but a single photon or single electron in the universe that is to say a isolated photon then what will be the behaviour of photon ? will it be in rest or will it move with 1,86000 mile per sec?

2. #2 Jesse
September 24, 2010

Physics Buzz had a post about the maximum strength of lasers about a month back while back. They claimed that new calculations show pair production will occur in current generation lasers like the ELI and XFEL.

3. #3 Jesse
September 24, 2010

“about a month back while back”

Apparently I can’t edit my posts. Ah well. I’m sure you get the idea.

4. #4 Uncle Al
September 24, 2010

A 1 gigatesla field has 4×10^24 J/m^3, E/c^2 mass density 1000 times that of lead. Don’t drop the bottle – same EOS as Ideal Gas.

Real photons are inconvenient for not staying put. Separated charge is a pain in the patootie for sparking: ionizing a gas fill, cold emission from surfaces, and atomic nuclei with Z greater than 137 (to first order. Rather more than that with QED corrections). The choice way to put ten pounds of photons in a five pound vacuum bag is magnetic field.

Magnetars pull ~10 gigateslas and above – more than enough to foment intense vacuum birefringence and spontaneous pair formation,

The hard work is done. All one need do is look.

5. #5 Carl Brannen
September 24, 2010

Congratulations on becoming married!

The steel box problem depends on the thickness of the steel, of course. But if you assume the thickness is proportionate to the diameter, the maximum pressure you can contain is approximately equal to the tensile strength of the steel. Say 40,000 psi.

Light carries momentum proportional to its energy. So it doesn’t matter what frequency light you use, the maximum energy for 40,000 psi will be the same. I’ll leave it to you to work ou the details.

I’m looking at going to grad school next fall and I’m thinking that my primary objective will be UT Austin. I take the Physics GREs in two weeks.

6. #6 Anonymous
September 24, 2010

what if several dozen lasers were focused into a single beam?
would that increase the power?

7. #7 Tercel
September 26, 2010

I’m still convinced that Uncle Al is some sort of spam-bot that spits out pseudo-coherent text related to the post on which it is commenting. I mean seriously, what the hell sort of sentence structure is that? It’s certainly not a human.

8. #8 complex field
September 26, 2010

@ #6 — don’t cross the beams…..