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…
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: