This is an applied physics question for an engineering problem. The well known experiment we have all done--burn paper and wood with a magnifying glass and the sun's rays. My question is, if one were to focus to a point that is in the center of a block of clear plastic (with suitably low melt point), would the plastic at the focus point melt? Or would there have to be impurities in the plastic for heat to be generated? If there are impurities, would the decreasing intensity of the light as a function of distance be linear or exponential?
To get the local heating you will need some measure of the optical density of the material and an estimate of the local intensity of the light.
- You can probably look-up an estimate of the optical density, or if you need more precision measure it yourself.
- Starting from a known intensity, ray optics will give you an easy estimate of the position dependent intensity. Don't forget that focusing often occurs in both transverse direction, nor that there will be refraction at the boundary of your clear material.
To determine if the material will melt you will need to solve the inhomogeneous heat equation with the heat input as given in the above computation as see if the local temperature rises above the melting point anywhere in the medium.
There is a complication: both the optical and thermal properties may be temperature dependent. If they are, you may need to employ an iterative solution.
By the way, if you want to ask a question about implementing the iterative solution you should use the Scientific Computation site, as or FAQ disavows those questions on this site.