Fig. 1. Sketch of the Laser AMplification area (LAM) at Apollon. The last amplifier (Amp300) was known for causing beam instabilities due to air movement in the beam path.

Fig. 2. The Bode plot of the feedback transfer function (left) and the error transfer function (right) according to Equation (1), using the parameter values from the ‘Target’ column in Table 1.

Fig. 3. Schematic of the test-bench setup, to scale.

Fig. 4. Step response of two DM modes over time (open loop). The dashed lines indicate the 10%–90% levels and corresponding settling times. Top: first mode, featuring severe ringing due to the mechanical DM properties. Bottom: fourth mode with a regular settling behavior. All modes from this order upwards feature a comparable settling behavior (see Figure 5).

Fig. 5. Step responses of a full set of mirror modes over time.

Fig. 6. The RTAO loop gain over the frequency for different feedback gains. The dashed curves are the gains of the corresponding model, that is, the magnitude of the error transfer function.

Fig. 7. Schematic setup of the ARTAO system in the Apollon laser chain.

Fig. 8. Sketch of the diagnostic setup prior to the 1 PW compressor from the side (a) and top views (b). The main beam path is shown in red, while the pilot beam path is indicated in orange.

Fig. 9. Bottom: the sample-wise correlation matrix between the WFS of the main beam and the pilot beam over the recorded sequence without tilt and mean WF. Top: example correlation of the main beam WF to a randomly picked location of the pilot WF (left) and vice versa (right). Point pairs for a transformation fit can be extracted from the locations of maximum correlation.

Fig. 10. Example of a mapped WF between the main beam WFS (top row) and the pilot beam WFS (bottom row), where the first column is the raw WF, the second one is the mapped WF from the other WFS, and the last column is the difference between the two. Note that the main beam is smaller than the pilot beam, which is why its mapped WF is smaller in the pilot WFS space.

Fig. 11. RMS of the main and the pilot beam WF, as well as the difference between the two, over a timeframe of 1 minute. The beam was actively disturbed using a hot air source for this measurement.

Fig. 12. Top: time series measurement of the Strehl ratio (compared to the reference WF, calculated via the FFT of the measured NF) of the pilot beam, where the ARTAO system is activated at t = 0. Bottom: the corresponding data series.

Fig. 13. Recorded gain curve of the ARTAO system on the pilot beam WF in the Apollon beamline under regular operation conditions, compared with two theoretical curves (dashed lines) with parameters from Table 1. The dashed red curve uses the parameters of the real-world loop, while we tweaked the feedback gain for the blue curve in order to match the data.

Fig. 14. Time series measurement of the beam pointing of the pilot beam, where the ARTAO system is activated at t = 0. The tilt X (red) and tilt Y (blue) curves represent the tilt-portion of the recorded WF, relative to the reference, and are given in peak-to-valley in terms of the central wavelength of Apollon laser system, roughly corresponding to the movement of the focal spot in focal spot diameters. The inserts show a zoomed-in portion to illustrate the fast oscillations in the beam pointing.

Fig. 15. Measured RMS of the closed-loop pilot WF during a pump event on the amplifiers. The inserts are the WFs at the times indicated by the arrows.

Fig. 16. The WF RMS of the pilot beam under closed-loop operation over an extended timeframe. The insert plots are three selected WF frames from stable conditions at the beginning (left) and the end of the recording (center), as well as from a period of instability (right).

Fig. 17. Imprints of artificially large DM strokes onto the NF fluence in a non-conjugate image plane, where each NF corresponds to a different set of random actuator positions.