J. Eur. Opt. Society-Rapid Publ. 22, 50( 2026) 495
Fig. 5. Heatmaps of Zernike aberration coefficients for a 80th-order approximation in full-pyramid representation( a) and in halfpyramid representation( b). To emphasize coefficients with small magnitudes, the coefficients are displayed on a bi-symmetric logarithmic scale [ 8 ]. In the full-pyramid representation, the signs of the coefficients are preserved, whereas in the half-pyramid representation, the magnitudes of the respective coefficient pairs are shown.
5 Representing mid- and high-spatial frequency Zernike terms
Whereas the Zernike pyramid representation with scaled bubbles is adequate for visualizing polynomial degrees up to the 10th or 12th order, it becomes impractical for highorder fits. Zernike coefficient maps( also called Zernike spectra) are ideally suited for this purpose.
Figure 5 presents heatmaps of Zernike aberration coefficients obtained from a 80th-order fit of sample wavefront A. To emphasize coefficients with small magnitudes, the data are displayed using a bi-symmetric logarithmic scale [ 8 ]. The scaling constant of the bi-symmetric transfer function, which controls the slope near the origin, was set here to j = 1 / ln( 10) nm.
Note that the heatmaps are not displayed as a conventional pixel grid, but instead as a square tiling rotated by 45 ° and rendered using patches. This representation avoids gaps in the spectrum and ensures that the pyramid is fully filled. Alternative spectral representations [ 9, 10 ] employ a modified radial index n F for the same purpose, namely to circumvent such gaps. The representation presented here simultaneously provides a fully filled pyramid while preserving the conventional ordering of the radial polynomial degree n along the ordinate.
The log-based representation clearly reveals characteristic azimuthal orders with m = 18 and their higher harmonics( m = 36, 54, 72) in the wavefront. The heatmaps therefore reveal specific mid-spatial-frequency errors, such as ripple and spoke-like structures. Rotationally variant irregularities, or spoke-shaped aberrations, represent a specific form of surface undulations characterized by regular azimuthal orders extending toward the origin of the wavefront map. Such azimuthal waviness is typically introduced by manufacturing processes involving rotational motion about the workpiece center. Examples include precision
contour grinding of lenses, where chatter marks may arise from ring tools, as well as diamond turning or direct laser beam writing using air-bearing spindles, where slight imbalances can cause run-out errors.
6 Azimuthal orientation
The angular or azimuthal orientation refers to the angle with respect to the positive x-axis at which the aberration occurs. For example, combining equal positive tilts in the x- and y-directions produces a resultant tilt oriented at + 45 °. For a tilt, the direction of the aberration is unambiguous when referenced to its maximum. However, once the radial polynomial contains more than one term, local maxima or minima arise along the radial profile. How, for example, should the orientation of primary coma( cf. Fig. 6, center) be specified? Two approaches can be considered. One may, for instance, refer to the maximum value at the edge of the aperture, i. e. at r = 1( previous definition). Alternatively, one may refer to the first local maximum in radial direction, which for primary coma occurs at the radius r = p 2 / 3 0.471( proposed definition). These two variants are discussed in more detail in the following.
In Evans et al. [ 6 ] an equation and procedure to determine the first peak in angular direction at the edge of the aperture( i. e. at radius r = 1) is derived, which further simplifies with the atan2-function to
h lp
0 ¼ 1 m
� atan2 c�m n
; c m n; ð7Þ
where m 2 N. This variant is referred to here as the previous definition. The angular offset angle is denoted hereafter by the superscript“ lp” which stands for“ last peak in radial direction”. As can be seen from Figure 1, forallZernikepolynomials with a cosine azimuthal dependence( m > 0, right