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J. Eur. Opt. Society-Rapid Publ. 22, 50( 2026)
Fig. 6. Comparison of the definitions for azimuthal orientation of ± m-paired Zernike polynomials. The angular orientation of the Zernike polynomials Z m n is illustrated for different choices of h 0( black arrow: common definition h lp
0
; white arrow: proposed new definition h fp
2
0
). Left: tertiary astigmatism Z
6 with peaks at same azimuthal orientation. Middle: primary coma Z 1
3, where p = 180 ° has to be added. Right: secondary trefoil Z 3
5, where p / 3 = 60 ° must be subtracted.
side of pyramid) this last peak in radial direction at r = 1is alwayslocatedonthepositivex-axis( at x = 1, y = 0).
Here, we define another angle h fp
0 for the angular orientation, the one that points to the first local peak in the radial direction( proposed definition). The difference is illustrated in Figure 6.
If( n mod 4) =( m mod 4), then there is no difference in the definition. However, in all other cases( i. e., for every second diagonal wing pair of the Zernike pyramid, beginning with the primary coma pair), a special treatment is required. p / m must either be added or subtracted, depending on the signs of the coefficient pair, i. e.
8 h lp 0
>< h fp 0 ¼ h lp
0 þ p if ðn mod 4Þ ¼ ðm mod 4Þ m else if ðc �m h lp 0 � p n
< 0Þ _ ðc �m n
¼ 0 ^ c m n > 0Þ:
>: else if ðc �m n
> 0Þ _ ðc �m n
¼ 0 ^ c m n < 0Þ m ð8Þ
The three conditions can be combined into a compact formula suitable for implementation
h fp
0 ¼ 1 � atan2 c�m n
; c m n m � p ðn � mÞ mod 4 m f ½ Š 6¼ 0
� ð9Þ gsign atan2 c�m n
; c m n;
where the first term corresponds to equation( 7), the bracketed factor {.} represents a logical flag [ 0, 1 ] to distinguish the cases, and the sign-function determines whether p / m is added or subtracted.
The alternative definition of the azimuthal orientation of Zernike polynomials with m 6¼ 0 is motivated by the aim of enabling a more effective analysis of spoke-shaped aberrations. Consider the synthetically generated sample wavefront B( Fig. 3), which contains rotation-invariant aberration components only and is composed of spokeshaped aberrations with azimuthal orders m = 3( straight spokes) and m = 18( bended spokes). The angular orientation of the straight, spoke-shaped trefoil in sample
Fig. 7. Comparison of the two definitions for angular orientation. The vector length represents the magnitude of the Zernike coefficient jc m n j, and the angle represents its orientation.
wavefront B is compared for both definitions in Figure 7. According to the previous definition( h lp 0 as defined in equation( 7), shown as red dashed arrows), the angle is rotated by p / m for every second occurrence of the polynomial degree n( i. e. n = 5, 9, 13,...). In contrast, under the proposed definition( h fp
0 accordingtoequation( 9), shownas light blue arrows), the angular orientation always points in the same direction( here �45 °). This makes it possible to trace the radial progression of the spokes.
To analyze the bending or the shape of the spokes, representations other than the phasor plot in Figure 7 are more suitable. First, dominant azimuthal orders are identified by applying a threshold to the root-sum-square combination of the coefficients of a given order m. To plot the determined angles h fp
0 as a function of radius, or to directly visualize their variation within the circular aperture, the radial coordinate of the first local maximum is required( cf. the endpoints of the white arrows in Fig. 6). This radius is determined numerically by searching for the first extremum of the corresponding radial polynomial R m n
. The values are computed once and stored in a look-up table.
Note that the angle h 0 in both equations( 7) and( 8) is confined to a circular sector spanning the arc 0 ± p / m. With increasing m, this minor sector becomes progressively smaller. As a consequence, owing to the arctan function, the evaluated angle h 0 is wrapped modulo 2p / m. Toconsistently trace the trajectory of a spoke from the aperture rim toward the coordinate origin, it can therefore be advantageous to unwrap the initial angle h 0.
Figure 8 illustrates the proposed definition of the angle h fp 0
, shownin( a) asafunctionoftheradiusandin( b) asa polar plot over the circular aperture.
The dominant azimuthal orders( m = 3andm = 18) of sample wavefront B are presented, including both the wrapped angle( solid line with markers) and the unwrapped angle( dashed line without markers). Since, for m = 3, the