Linear quadratic regulator for laser beam shaping

- The results show images qualitatively consistent in term of root mean square error.
- LQR method require less than 30 iterations to reach a minimum difference between the reference and the output amplitude.
- The Taylor linearlization of the Fresnel propagation is adequate to describe the laser propagation.
- The LQR method can adjust the Deformable Mirror automatically in order to correct the dynamics aberrations.
- The LQR approach has better response to misalignment, perturbations and modeling errors than the Phase Retrieval Method.

The performance of an adaptive optics system depends on multiple factors, including the quality of the laser beam before being projected to the mesosphere. In general, cumbersome procedures are required to optimize the laser beam in terms of amplitude and phase. However, aberrations produced by the optics of the laser beam system are still detected during the operations due to, for example, uncertainty in the utilized models. In this paper we propose the use of feedback to overcome the presence of model uncertainty and disturbances. In particular we use a Linear Quadratic Regulator (LQR) for closed loop laser beam shaping using a setup of two deformable mirrors. The proposed method is studied and simulated to provide an automatic optimization of the Amplitude of the laser beam. The performance of the LQR control algorithm is evaluated via numerical simulations using the root mean square error (RMSE). The results show an effective amplitude correction of the laser system aberrations after 20 iterations of the algorithm, a RMSE less than 0.7 was obtained, with about 140 actuators per mirror and a separation of z=3 [m] among the mirrors.