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of the corresponding Zernike aberration, such that the relative scaling has a clear physical meaning. These bubble charts are particularly valuable when rapid visual feedback on individual aberration terms is required, for example during aberration compensator adjustment, real-time wavefront visualization of dynamic processes, or optical alignment procedures. In addition, they can improve the clarity and interpretability of inspection reports or measurement certificates for lenses and optical systems.
Zernike heatmaps, by contrast, are well suited for representing mid- and high-spatial-frequency errors. For this purpose, the magnitude of the RMS-normalized coefficients is displayed on a logarithmic scale in order to enhance the visibility of contributions with small amplitudes. Such heatmaps reveal characteristic manufacturing artifacts, such as spoke-like structures or ripple patterns.
Furthermore, this work introduces an alternative definition of the azimuthal orientation of Zernike terms for m 6¼ 0. This approach enables the identification of the shape and bending of spoke-like aberrations. For such specific analyses, other numerical methods, such as Fourier-based filtering or multi-angle averaging, may also be employed [ 11 ]. However, given the availability of convenient numerical methods for fitting wavefronts or aberrations to very high Zernike orders [ 12 ], these features can now be directly extracted from the Zernike coefficients themselves.
Funding This research did not receive any specific funding.
Conflicts of interest The author declares no conflicts of interest.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request. The MATLAB ® code for visualizing Zernike coefficients( bubble charts, heatmaps) are publicly available on the MATLAB ® File Exchange at the following URL: https:// de. mathworks. com / matlabcentral / fileexchange / 183649-zernike-visualization-heatmapsand-bubble-plots.
References
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Author contribution statement
This work was conducted and written solely by the author, who is fully responsible for all contributions and content.