Fig. 1. Simulated examples for the ring characterization in ZEHBRO for a Zernike coefficient of λ/20 (according to the Zernike normalization mentioned in the text) of astigmatism (left), coma (center) and trefoil (right). The WF map is shown in the top row and the corresponding FF below. The black line indicates the position for the ring intensity extraction, which is shown in the third row. The last row shows the absolute values of the first four FFT coefficients of the ring intensity, which can be rearranged to the descriptor vector (bottom).

Fig. 2. The descriptor values for simulated donut foci under a varying amount of each of the six pure Zernike aberrations considered by ZEHBRO. The gray dashed line shows the slope in the origin, belonging to the descriptor associated with the aberration whose amplitude varies along the X-axis.

Fig. 3. for different SNR values, considered at three different input aberration amplitudes. The ring diameter was set to 15 pixels and to 0.0125λ.

Fig. 4. over the ring diameter in pixels for scenarios featuring different amounts of disturbance defined by combinations of varying SNRs and values.

Fig. 5. for different total input aberration amplitudes under the presence of a varying of random higher order aberrations.

Fig. 6. Experimental setup at the ELI-NP facility for the validation of ZEHBRO.

Fig. 7. Screenshot of the ZEHBRO module in WOMBAT: (a) camera view (live) with overlays indicating the sampling positions; (b) extracted ring intensity; (c) reproduced FF view; (d) retrieved Zernike coefficients.

Fig. 8. FF distributions at the compressor sensor. The images were created by centering and averaging 300 images, equal to 30 seconds of operation: (a) the FF prior to the insertion of the spiral phase plate; (b) after insertion of the phase plate; (c) after conventional manual optimization; (d) after optimization using ZEHBRO.

Fig. 9. The RMS of the deviation from an ideal, homogeneous ring intensity, calculated using Equation (3), after manual optimization and optimization using ZEHBRO, corresponding to Figures 8(c) and 8(d), respectively. The intensity of the fluctuations becomes apparent both in the RMS and the shape of the individual focal rings (pictures on the top).

Fig. 10. Statistics of the ring intensity of the FF distributions as shown in Figures 8(c) and 8(d).

Fig. 11. Statistics of the Zernike coefficients as returned by ZEHBRO after the manual optimization and the optimization based on the output of ZEHBRO, using the Zernike-projection technique. The dataset is identical to the one used in Figures 8–10.