develop an algorithm for implementing the Kerr effect

[bruhwiler]

• [x] how does wavefront intensity change the local index of refraction
• [x] what does this mean in a laser pulse slice (presumably average relevant physical quantities)
• [x] probably create a 2D mesh of the index of refraction
• [ ] possibly fit this to a central value plus simple quadratic radial variation
• [ ] ...what else?

[bnash]

Here is a good article about the non-linear index of refraction, with values for Sapphire (how different for Ti-Sapphire?) https://www.rp-photonics.com/nonlinear_index.html Δn = n₂ I

Note that this quantity (variation of index of refraction with intensity) is written n₂ and this was an issue we discussed with the BELLA people. This convention is why it is better not to use n₂ to represent the variation of index of refraction with transverse offset from the optical axis.

In the case that the wavefront is somewhat Gaussian, the resulting change to the index of refraction will have a constant term from the center and a quadratic term (and higher order). We need to decide whether we need to fit these two quantities every pass, or whether we include the Kerr term into the non-linear perturbation and apply it along with the other 𝛿n propagation.

Regarding values and units, sapphire has a value of 2.8 e−20 m2/W at 1550 nm. We need to get the intensity value in the same units to find out when this becomes an appreciable effect.

[bnash]

This is not a high priority for now, since this will not be needed in the early stages of experimental comparison with BELLA Ti-Sapphire experiments. Thus, we close for now, to be revisited at a later time.