Can one focus sunlight so much that the temperature attained at the focus is higher than the temperature on the surface of the sun? Consider this question in the framework of geometrical optics (no diffraction, no solar panels or electricity). This seemingly innocent problem is actually quite tricky. For thermodynamical reasons, it is clear that the answer has to be a "no". Searching the web, it is easy to find this answer, sometimes even together with an explanation of why this is so. The explanation is that what matters for the temperature at the focus is the solid angle under which one sees the sun when sitting at the focus. If this solid angle were exactly [tex]4\pi[/tex], then the temperature at the focus is said to be the temperature on the surface of the sun. If this solid angle is smaller than [tex]4\pi[/tex], then the incoming power gets scaled down accordingly, and consequently the temperature attained is less. This reasoning seems faulty too me; for example, an elementary calculation shows that equal equilibrium temperatures endue for the case of a lens which is much smaller than the distances from the lens to the objects. (Just consider how much radiation hits the lens from each object.) Does anyone know of a reference where this problem is treated properly and thoroughly? [For I believe to have a solution, and I wonder whether it would make sense to write it up and try to get it published?]
Sounds like an interesting problem. I'm not sure how I would treat it. My first thoughts: I don't believe that a ray of light can have a temperature ascribed to it, since it is not in equilibirium with something unlike a blackbody. However, one can do power calculations of course. Theoretically there is no classical limit on focussing. Finally, to get the temperature of the body being heated you probably have to make a heat flow calculation. Ideally of of the sunlights energy is converted into heat at the given power, however heat diffuses in the object and so to get the actual temperature one has to solve thermodynamic equations. I guess the assumptions are a spot source with constant power input (given by how much of the sunlight was collected) and the thermal conductivity of the object.
Thanks for your reply. But we're talking blackbody radiation here, so while not to a single photon, we can ascribe a temperature to the photon spectrum at every point in space. This is the point! There are classical limits on focussing, although these are hard to get your hands on quantitatively. Here's my reasoning: if you take a very small lens, it's not hard to focus the sunlight on a very small spot. Now if you make the lens bigger and bigger to collect more and more sunlight, there are rays further and further from the optical axis, and abberation will spoil your focus. So the more light you try to collect, the worse the focus gets. No, the assumptions are a black body source of finite extent and vanishing thermal conductivity of the object. Equilibrium entails due to black body radiation of both the source and the object. Sorry for not stating these clearly in the OP.
Hmm, I think you are right. But we assume photons are non-interacting? So whatever path they go, they always keep the same temperature of the sun?! Absolutely true. I didn't take this into account. I have to think about that. It seems if I catch any amount of sunlight into the hole of a perfectly isolated black body radiator, this object will become 6000 degrees?
A heated object radiates itself. So the equilibrium state is attained when the radiated power is equal to the incident power. Let us imagine that the Sun spectrum is a black-body like with T_{S} and its flux is concentrated on a body. After a lens it is not parallel but converging. Then the radiated flux will be diverging. As soon as the radiation solid angle is larger (=4pi) the body temperature will be lower than T_{S} because its radiation power will be equal to the incoming power at smaller T due to larger radiation angle. When you think of a very small body, it seems that it should be very hot to provide the same radiation flux. In fact, it is a feeding power flux that is just transformed from convergent to divergent form with help of the body. The energy density is high indeed but it is incoming and outcoming, not staying at the place.
Can you suggest how to calculate that roughly? I didn't know exactly. Not sure what you mean. Do you suggest that if I construct a semi-sphere lens that concentrates light on one spot collected from the spherical angle (full sphere)/2, then the idealized body will be T_{s}?
The black-body surface power is calculated like σT^{4} but for small bodies there is another law because they radiate from volume (they are so small that become "transparent" for some wave-lengths). Worse, the body becomes transparent for the incident radiation too so the latter cannot be absorbed and converted into "additional" body temperature.
I don't get the solid angle reasoning, because I can't see how the distance (and therefore fall in intensity) is considered by this. Surely a perfectly spherical lens will focus the energy density collected from the sun to a sufficiently small volume that you can conclude that the temperature at that point is higher than some finite limit, by relating temperature to energy density using the black body laws?
True, that's why the solid angle reasoning is wrong! Do I understand your argument correctly that we can therefore, in particular, reach temperatures much higher than that on the sun? If yes, then this was exactly my very thinking initially. But as I tried to mention in the OP, this is not compatible with the second law. For if both bodies have the same temperature initially, we could use such a device to create unequal temperatures. (If you're uncomfortable with the solar system not being a closed system, just imagine a huge mirrored Dyson sphere around it.) So it seems that the assumption "ideal lenses exist" gives a result here which is qualitatively wrong. Making a lens bigger makes the focus worse. How can we formulate this observation quantitatively? In fact, I do have a mathematically precise answer to this problem. It applies to any kind of optical system, not such to single lenses. Since it uses differential geometry (the geodesic deviation equation) and properties of symplectic matrices, it's quite lengthy, and so I don't want to go into details here. I was just wondering whether anyone knows of relevant research literature about this problem? Because I'm considering writing my solution up and so I would like to know whether publishing it might be possible. Thanks to all who replied so far!
Let us be realistic. The incident waves are not only absorbed but also reflected and, in case of small body, pass through. The body itself radiates not only photons but also electrons, ions, atoms, when it is heated. So the absorption mechanism is not as perfect as we can imagine, and the loss mechanisms are more numerous that we discuss. So sigma*T^{4} is not applicable for small bodies at high temperatures. By the way, very high T are obtained in laser-irradiated pellets in some thermo-nuclear projects. There the losses include flying the target apart due to high P and T.
With regards to the solid angle stuff. There does seem to be some sort of limit when focusing a diffuse source. I think etendue is something to do with it but I've not looked into that. The problem is that the bigger the lens (or more correctly the NA) the more divergent the light and that is what limits the focus. The only example I have found where irradiance can be higher in the focussed region is Koehler illumination, but you get other losses.
You can bypass the debate about whether ideal lenses exist by using mirrors instead. At least, that eliminates chromatic aberration issues from the discussion. It doesn't eliminate the fact that the wavelength of sunlight is finite, and that limits the size of your "focus" to something much bigger than atomic scale. From a practical point of view, the radiation balance argument gives the limit. Just for interest, you can quickly and easily heat up small objects (say 1mm cubes) to well over 1000C using nothing more sophisticated than a 12V 50W halogen car headlamp bulb. That is just using the mirror built into the bulb to "focus" the light - not exactly precision optical engineering!