For this study, we looked at a simple case with an aLIGO-like test mass geometry (R=0.17m, H=0.2m) plus a barrel RH with 0.02m width at 0.03m from the AR surface with FEniCSx. The irradiance profiles are constant inside the width along the longitudinal direction, and zero outside the width. For the baseline non-astigmatic actuation with constant irradiance azimuthally. We have obtained roughly equal quadratic actuations along the x and y directions, as shown in figure. The total delivered power on the entire barrel is normalized to 1 W. The actuation on the curvature per power Delta S/Delta P in this non-astigmatic case thus is 0.835 uD/W.

For the astigmatic case however, the irradiance for the regions from one diagonal is increased by a given amount, compared to the non-astigmatic case, whereas for the other diagonal regions is decreased by the same amount (thus the total power is unchanged at 1 W). The HR deformation when the power is changed by 50%, for instance, is shown in picture, where the deformation along the x direction is larger than the y direction. The deformation in each direction however remains quadratic, with different curvature per powers for the x and y components, as shown in plot. The actuation on the curvature per power for an increasing amount of astigmatism is shown in plot. In terms of Zernike polynomials, the maximum amount of Z22 (astigmatism) for 1 W of total power is 2um while the remaining curvature content (Z20) is 6nm. This is shown in plot.