Max temperature that can be obtained from radiation source when nonlinear interaction with matter is important? Using a black body radiation as furnace it is usually said that the max achievable $T$ is limited to the T of the source. So a source of 3000K can not be used to heat a body above 3000K.
A long wavelength source let say $3000\ \mathrm{K}$ or nearly $1$ micron is considered. Having enough power and using a lens to focus, the intensity at the focus could exceed the binding potential of any core electron in any atom or some at least. The nonlinear interactions are important. So this 3000K source is ionizing radiation in the volume where the intensity is high enough to create "high" temperature plasma. (A small scale plasma experiment similar to the national ignition facility.)


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*What can we say about the entropy and the temperature in that volume? 
Is the temperature in the system much higher than $3000\ \mathrm{K}$? Let say intensity is high enough to doubly ionize He gas and create "hot" plasma.

*Is there limit for the max temperature?

*Once the matter is completely ionized would increase in intensity lead to higher T?
My question is not strictly related to lasers but here are few numbers for a laser source showing that the energies are "relatively small" for high nonlinear interactions: For a laser with $1\ \mathrm{mJ}$ at $1000\ \mathrm{nm}$ (3000K) beam diameter $20\ \mathrm{mm}$, $40\ \mathrm{fs}$ pulse duration, using $F=15\ \mathrm{cm}$ at the focus the peak intensity and field are $Ip=6.5 10^{16}\ \frac{\mathrm{W}}{\mathrm{cm}}, E= 7*10^9\ \frac{\mathrm{V}}{\mathrm{cm}}$. This is roughly on the scale what will completely remove the electrons from He.
 A: Well a Red photon at about 650 nm wavelength is about 2 electron Volts ( 2eV), so a one micron wavelength photon is about 1.3 eV (you can google up the exact number yourself).
So good luck on doing much ionizing with such low energy photons.
If your source is about like a 3,000 kelvin thermal radiation source, then 98% of that radiant energy lies between 0.5 microns and 8.0 microns wavelength, with a 1% tail at each end, so you will get less than 1% of your energy as 2.6 eV or higher photons.
A: Some points to consider:


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*There is no helium at the surface of the sun, all of it is in the core where it formed and/or sunk to, where the temperature is on the order of megakelvin--everything is ionised already. This I believe should answer the first question--if He is ionised at equilibrium, situation is way over 3000K. 

*Suppose He did exist at the surface. 3000K blackbody radiation may ionise He to He++, but my rough guess is 1/1,000,000 collisions are ionisation events, since some photons of 3000K emission (at the extreme end) may possess 79 eV. Use Planck's Law for exact probability. 

*This small likelihood drops to zero if you use a $1\mu m$ laser, since now 100% of your photons are less than 79eV (way less even for first ionisation requiring only 25eV). Doesn't matter how intense the laser is--this would increase the rate of ionisations assuming the individual photons already meet the threshold energy which they don't. You are looking at x-ray lasers. 

