# Optical element is heated by laser: is it possible to get oscillating heat distribution?

Imagine that we have thin optical element, which is irradiated by laser. Laser heats element, so there is some heat distribution in element. There is a heat sink through upper and lower element faces proportional to temperature. There is also a heat conduction in element.

The most important characteristic of the model is that heat consumption depends not only on power of laser, but on heat of the element too and relation between current heat and heat change may be very complex. With given parameters after some time heat distribution in element usually sets to some stable state (with equal heat income and outcome in all element points). Stable heat distribution can vary depending on initial heat distribution.

So, the question is: if laser radiation power distribution depends on element coordinates and heat consumption is related to temperature in current part of element, but all the parameters do not change in time, is it possible that heat distribution will not come to some stable state but instead will oscillate, or behave in more complex way?

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There is not really enough information to answer this but if the "relation between current heat and heat change may be very complex" of course you can have oscillating behavior or steady state. This is comparable to a externally driven oscillator. –  Alexander Nov 16 '11 at 0:01
I think with a bit of editing this question may have a better chance at a complete answer on math.SE... I took a shot at an answer, and if that's the type of answer the OP is looking for math.SE might be better. (Of course we might have someone here who can give a great answer too). –  Kyle Jul 3 '13 at 20:07
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## 2 Answers

Not exactly what you are asking, but you can see this kind of oscillatory behavior in microtoroid optical resonators: if you couple a resonator to a laser tuned to one of its resonances, it will absorb power, leading to an increase in temperature. This causes expansion and also changes the refractive index, which shifts the resonance, leading to a decrease in absorbed power... oscillation.

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If the only heat transfer inside the optical element is due to conduction, then the heat distribution is fully described by the heat equation:

$\dfrac{\partial u}{\partial t} - \alpha\nabla^2u=0$

This is a parabolic PDE, so its solutions are diffusive and will always evolve toward a steady state provided the boundary conditions are constant, and you said "all the parameters do not change in time" (though I'm afraid I'm at a loss to prove this; if anyone can prove it or provide a reference to a proof, please do). Unless you're actively driving an oscillation by varying the boundary conditions, you're stuck with a steady state solution (or a singular one, though you couldn't actually create a physical "singularity of heat" in your system so I think it's safe to ignore that class of solutions).

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