Let's say I'm doing an optical experiment. I focus a laser on a crystal with a certain amount of power. The crystal's temperature is regulated to a certain temperature but it is localy heated by the laser. How can I calculate the local temparature at the laser spot's location? I know in Raman spectroscopy you can use the Anti-stokes/stokes intensity ratio but here I want to use a different method. I think I could do it knowing the thermal conductivity of the crystal but I don't know how.
Let us assume that you have continuous laser.
You need to solve stationary heat diffusion equation given in wiki, which is basically Poisson equation. For accuracy you will need to know penetration depth of your laser. It is better if you solve equation for exponential with depth source but Gaussian shape will give you good approximation. In this case solution is given by Look here.
Overall you will need to know - penetration depth, reflection coefficient and thermal diffusivity (you typically search literature for specific heat, thermal conductivity and then calculate).
Practically you can measure for instance reflectivity as a function of temperature and then measure reflectivity as a function of power. This will give you a good estimate as well.