# How to determine the strength of a laser beam needed to cut through a specific material?

How can it be determined how strong a laser beam needs to be in order to cut through specific material?

For example, if I have a 5mm thick piece of brown cardboard, is there a way to determine how strong the laser has to be (how many Watt) in order to cut a hole in it?

Firstly, I am a little worried by your inclusion of the "eye" tag for this question. If, in the small likelihood, that your question is probing the threshold level for eye safety by finding the minimum power that will cut the retina and then declare smaller powers safe, then I need to warn you that you cannot do this by calculation but must instead refer to the ISO 60825 laser safety standard to find out whether a beam is safe for viewing.

Assuming this is not so, then let's answer you question. You not only need its power but also:

1. An assessment of its beam quality, i.e. the wavefront aberration of its output wave. Laser beams can in particular suffer astigmatism (See Wiki page of this name);
2. The numerical aperture of the focussing optics (let this be $\eta$);
3. The heat transfer properties of the material to be cut (in particular the diffusion length (See the Example"solution in one dimension: diffusion length" heading on the "Fick's laws of diffusion" Wiki page) as well as the conductivity and density of the material;
4. The Kindling (or Autoignition) Temperature or equivalent temperature at which a material mechanically breaks down.

Then, the spotsize $s$ of a perfectly focussed, unaberrated beam is of the order of $\frac{\lambda}{\eta}$. For maximum efficiency, you need the beam to focus to a point of diameter smaller than the heat diffusion length. Note that this implies a minimum numerical aperture. Then you can calculate the steady state temperature distribution assuming a point source of the power in question from the heat diffusion equation. You then calculate the power you need to raise the local temperature about the kindling or equivalent breakdown temperature.

If the beam is aberrated, then you will need to calculate the point spread function for the wave and calculate whether the spotsize is still less than the heat diffusion length, then you will need to solve the heat diffusion equation in detail given the intensity distribution implied by the PSF and then calculate the power needed to raise the local temperature over the breakdown temperature.

If you want to use an unfocussed beam, then you will need to use the beam's profile and do the calculation of the last paragraph.

This is all very complicated, which is why questions such as yours are often settled emprirically.

Another means of cutting is to use a highly focussed beam to make little "explosions" in the material, causing it to spit little packets of material from the surface in a process called ablation. The power needed for this is again often found empirically, although one can roughly calculate the magnitude of the shock wave induced by swift heating to see whether it exceeds the material's strength. If not, then you have no hope of inducing ablation.

• Thank you for your answer, I did not need the "eye" tag here and have now corrected the tags. – coderworks Jan 19 '15 at 3:51

Here are some sites that do the hard math supporting the expert information documented.