Besides the 2nd law of thermodynamics, what laws of optics prevent the temperature of the focal point of lens from being hotter than the light source?

I'm pretty sure that you can't take a magnifying glass and make it focus to a point that is hotter than the surface of your light source. For example, when you're outside trying to fry ants with your magnifying glass, it's impossible to get it hotter than 5,000 C (the temperature of the surface of the sun). My dad was arguing with me about it because he didn't see why this was true. My easiest argument was that the 2nd law of thermodynamics prevents this from happening because heat can't flow passively from a place of lower energy/entropy to a place of higher energy/entropy. He didn't buy it, saying that there wasn't anything preventing the light from focusing to a hotter point.

So I was wondering, are there some laws of optics that prevent this from happening? Or alternatively, is there a way to show that you could build a perpetual motion machine from this? Any help is appreciated.

• conservation of etendue Jun 12 '17 at 6:47
• See, for example, K.M. Browne's "Focused radiation, the second law of thermodynamics and temperature measurements," J. Phys. D: Appl. Phys. 26 (1993) 16-19 for a detailed ray-tracing refutation of one claimed Second-Law-violating system of mirrors. If you don't invoke broad concepts such as the Second Law and conservation of étendue, it can be necessary to solve the ray equations for each new configuration. This is what keeps perpetual motion crackpots going, because they can always make a slight change and demand that their audience reanalyze the system. Jun 12 '17 at 17:49
• @Chemomechanics That's a good point. You can invoke Hamiltonian dynamics, but then you need to prove that each new "abrupt" device which can't be modelled by smooth raytracing imparts a symplectomorphism on the light's state to make this all hold together (you need to broaden the list referred to in my answer). And, apparently, there are some grating systems that don't do this, although they still conserve étendue (they don't conserve the optical invariant - i.e. the optical symplectic form, though, which is why the Hamiltonian approach won't work for them). Jun 13 '17 at 0:46

"... is there a way to show that you could build a perpetual motion machine from this?"

Yes. Focus the radiant heat from a thermal reservoir onto a spot that is hypothesized to be raised to a higher temperature through its concentration into a smaller area. Now connect heat engine - a Carnot engine - between the hot spot as the engine's heat intake and the original reservoir as the heat exhaust. Now the engine will run, outputting work. Your hypothesis means that you have an heat engine system spontaneously converting the heat in the thermal reservoir to work and there's your perpetual motion machine (of the so-called second kind).

Obligatory in any conversation of this kind is Randal Munroe's Fire From Moonlight article.

One way to understand all this is to note that optical systems are reversible, so that if light can pass from point A at the input to point B at the output, light can equally well go the other way. So if a hot body directs its radiant heat at another object through a lens system, the temperature of the latter will naturally begin to rise. That means that the second body will radiate back to towards the first body. If the second body became hotter than the first, it would be returning a higher heat power to the first along the reverse paths whence the incident heat came. Therefore, heat transfer will stop before the second body reaches the temperature of the first.

The second law of thermodynamics in optics is equivalent to the non-decreasing of étendue, which is the volume of a system of rays representing a light field in optical phase space and thus a measure of entropy. If étendue cannot be decreased, this means that density of rays in phase space cannot be increased; in turn this means that the divergence angles of a set of rays must increase if the area they pass through is shrunken down. This means that the light from any point on a hot body cannot be made brighter at the point where it reaches the target body.

This also is why a laser works differently if we try to reason as above. If energy reaches a body through a laser, the incident light paths taken have near to zero étendue - there's hardly any beam spreading at all. The second body will get hotter and hotter, but the radiant heat from the hot second body is all spread out in all directions (this is fundamental to blackbody radiation - there's no such thing as collimated blackbody radiation). So hardly any of the radiated light is accepted back along the extremely narrow range of paths back to the laser. Laser light is highly nonequilibrium light - it is the optical equivalent of thermodynamic work, rather than heat.

As well as by thermodynamic arguments, one can show that étendue is conserved very generally in passive optical systems using the Hamiltonian / symplectic geometry formulation of Fermat's principle. I discuss this in more detail in this answer here. Fermat's principle means that propagation through inhomogeneous mediums wherein the refractive index (whether the material be isotropic or otherwise) varies smoothly with position corresponds to Hamiltonian flows in optical phase space; mirrors, lenses and other "abrupt" transformations as well as smooth Hamiltonian flows can all be shown to impart symplectomorphisms on the state of the light in phase space, which means that they conserve certain differential forms, including the volume form. All these things mean that the volume of any system of rays in optical phase space is always conserved when the rays are transformed by these systems. This is the celebrated Liouville Theorem.

There is a clunkier but more perhaps accessible way to understand all this in optics. We linearize a system's behavior about any reference ray through the system, and write matrices that describe the linear transformation of all building block optical systems. It may seem that linearization involves approximation and thus something not generally true, but hold off with this thought - this is not the case. This is the Ray Transfer Matrix method and these linear transformations describe the action of the system on rays that are near to the reference (the "chief ray") ray of the light field in optical phase space. These matrices act on the state $X$ of a ray at the input plane of an optical subsystem:

$$X = \left(\begin{array}{c}x\\y\\n\,\gamma_x\\n\,\gamma_y\end{array}\right)\tag{1}$$

where $(x,\,y)$ is the position in the input plane of the ray, $(\gamma_x,\,\gamma_y)$ are the $x$ and $y$ components of the direction cosines of the ray's direction and $n$ is the refractive index at the input plane at the reference ray's position. The quantities $n\,\gamma_x$ and $n\,\gamma_y$ are the optical momentums conjugate (in the sense of Hamiltonian mechanics) to the positions $x$ and $y$; interestingly, they are indeed equivalent (modulo scaling by the constant $\hbar\,\omega/c$) to the $x$ and $y$ components of the photonic momentum $\hbar\,\vec{k}$, where $\vec{k}$ is the wavevector, but this fact is an aside. (1) describes our points in optical phase space.

Now we write down the matrices that represent the linearized action of every optical component we can think of; for example, a thin lens (representing the paraxial behavior of an optical surface) will impart the matrix:

$$\left(\begin{array}{cccc}1&0&0&0\\0&1&0&0\\-\frac{1}{f}&0&1&0\\0&-\frac{1}{f}&0&1\end{array}\right)$$

If you study this matrix's action, you'll see that it transforms a collimated beam into one that converges to a point a distance $f$ from the input plane.

A key point to take heed of is that this matrix has a determinant of 1. If you go through the list of every possible passive optical component, you'll find that the matrices that describe their paraxial behavior all have unity determinant (they are unimodular). So they all multiply together to give a unimodular ray transfer matrix of the overall system built from these subsystems chained together.

This determinant is the Jacobian of the general, non-linearized, non approximate transformation that the system imparts on any system of rays. We can imagine recalculating a matrix from every neighborhood of every chief ray in an arbitrary, noninfinitessimal volume of rays in phase space. These matrices will all be unimodular, so what we've shown is the key idea:

The Jacobian $J(X)$ of the transformation wrought by any passive optical system is unity at all points $X$ in phase space.

This means that if we work out the volume $\int\mathrm{d}V$ of a system of rays in phase space, then the volume of their images $\int\,J(X)\,\mathrm{d}V$ will be exactly the same for any passive optical component. So we've shown the exact version of the law of conservation of étendue for optics without needing the full machinery of symplectic geometry and Hamiltonian mechanics.

• Unless we are staying in the realm of idealized ray optics I do not believe that "optical systems are reversible" where reversibility is meant in a thermodynamic sense. The moment you allow for diffraction no wave propagating system is really reversible, at least in my exeprience. Jun 12 '17 at 14:45
• @hyportnex I don't think that's right. Maxwell's equations, and the diffraction operators, are all perfectly reversible. What is true, though, is that light interacting with randomizing surfaces picks up an "imprint" of those surfaces, thus contributing to its entropy. As you seem to imply, that effect seems more marked with diffraction, but I don't know how to quantify that statement. Certainly dust on optics when looked at with coherent light seems to scramble things up to a frightening degree. Jun 13 '17 at 0:34
• @hyportnex: The other important thing to keep in mind is that in most optics experiments we lose relative phase information when we take measurements. This doesn't matter for experiments that only require ray optics for analysis, but it matters critically for diffraction. You can always reconstruct a wave's history if you know both its phase and amplitude at some boundary (like a screen), but not necessarily if you only know its amplitude. If you're clever, though, you can record the phase information on a surface as well — that's what makes holography possible. Jun 13 '17 at 2:07
• I am not thinking of randomizing effects such as dusts. Instead think of any scheme that would actually reverse a diffracted beam, say e.g., in the case of a circular hole in a screen with just idealized metal surfaces and a plane wave scattering off it. What idealized equipment of a finite size could actually reverse the main beam and diffracted orders without introducing further diffraction somewhere? I cannot think of any. Jun 13 '17 at 10:35