Here are the results in the annulus thickness vs annulus diameter space ...
I added a COMSOL model of the aLIGO ITM being heated by an axicon-formed annulus to the 40m SVN. The model assumes a fixed input beam size into an axicon pair and then varies the distance between the axicons. The output is imaged onto the ITM with varying magnitudes. The thermal lens is determined in the ITM and added to the self-heating thermal lens (assuming 1W absorption, I think - need to check). The power in the annulus is varied until the sum of the two thermal lenses scatters the least amount of power out of the TEM00 mode of the IFO.
The results across the parameter space (axicon separation and post-axicon-magnification) are attached. These were then mapped from this space to the space of annulus thickness vs annulus diameter, also attached.