J. Eur. Opt. Society-Rapid Publ. 22, 50( 2026) 497
Fig. 8. Visualization of the proposed definition for azimuthal orientation.( a) Angle h fp
0 ðrðnÞ; mÞ over the radius where the first peak in radial direction occurs.( b) Polar plot of the same data.
wrapped angle never reaches the limit of uniqueness( indicated by the dotted line), the wrapped and unwrapped angles are here identical. The representation in Figure 8a reveals the radial dependence of the spoke bending. For m = 3, the bending is constant( straight spokes), whereas for m = 18 it exhibits a linear dependence on the radius. The polar plot in Figure 8b provides a convenient representation for tracing the trajectory of a single spoke of a given azimuthal order.
Finally, Figure 9 illustrates the fitted dominant orders( i. e., the sum of all Zernike terms with the same m), overlaid with the trajectory of the unwrapped angle h fp
0. Analyzing the shape and curvature of these spoke-like aberrations can therefore provide valuable insights into the underlying machine kinematics and support process optimization.
Fig. 9. Zernike-fit decomposition of the azimuthal orders of sample wavefront B, overlaid with the trajectory of the unwrapped angle h fp
0
. A 100th-order Zernike fit was applied, with panel( a) showing the azimuthal order m = 3 and panel( b) showing m = 18.
7 Summary
This paper presents a set of graphical visualization methods for results obtained from a Zernike approximation of wavefront aberrations or surface figure errors. These representations are intended to provide an intuitive visualization of the extensive information contained in the RMS-normalized Zernike coefficient vector. In particular, dominant characteristics are intended to be rapidly recognized and extracted.
Zernike bubble charts are well suited for representing low-order aberrations up to the 10th or 12th order. The area of each bubble is scaled according to the RMS value