J. Eur. Opt. Society-Rapid Publ. 22, 50( 2026) 493
Fig. 1. Pyramid representation of the non-normalized Zernike polynomials Z m n up to the 4th order.
radial polynomial and m the azimuthal order, is essential for unambiguously describing the Zernike polynomials.
The radial polynomials R m n ðrÞ are explicitly defined by
R m n
2 ðrÞ ¼Xn�jmj s¼0 ð�1Þ s ðn � sÞ! s! ð nþm � sÞ! ð n�m � sÞ! rn�2s; ð2Þ
2 2
Fig
. 2. Sample wavefront A: measurement data of a lens surface. whereas the azimuthal function M m( h) isgivenby 1
� sinðmhÞ for m < 0 M m ðhÞ ¼: ð3Þ cosðmhÞ for m 0
Optionally, a normalization factor N m n may be applied, defined as
N m n ¼ 1 for no normalization pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 � d m0 Þðn þ 1Þ for RMS normalization ð4Þ
where d m0 is the Kronecker delta function. 2 If the normalization factor N m n is set to 1, the Zernike polynomials are referred to as non-normalized, implying that either their maximum is 1 or that their maximum and minimum are ± 1, respectively. In contrast, for RMS-normalized polynomials, the normalization factor is defined such that the root-mean-square( RMS) value of each polynomial is 1. This ensures that, when a polynomial fit isappliedtoadata set, the absolute value jc m n j of the fitted Zernike coefficients directly represent the RMS values of the associated aberration terms. For numerical reasons, it is advantageous to apply the RMS normalization not to the Zernike polynomials themselves, but to the coefficients obtained from the fit.
Note that each Zernike polynomial pair, Z m �m n and Z n
( for m 6¼ 0), has the same normalization factor N m n ¼ N �m n as well as the same radial polynomial
R m n ðrÞ ¼R�m n ðrÞ.
1 Note that the sine function is an odd function where sin( �x)=
�sin( x). For m < 0, the negative sign of the sine function is necessary for a correct notation. 2 Kronecker delta is defined as d m0 ¼ 1 for m ¼ 0
0 for m 6¼ 0:
Fig. 3. Sample wavefront B: synthetic aberration data.
3 Sample wavefront data
Two representative wavefronts are employed to illustrate the visualization methods discussed in this paper. Wavefront A( Fig. 2) corresponds to the measured aberrations of a lens surface, containing both low- and mid-spatial frequency contributions, while wavefront B( Fig. 3) is synthetically generated to highlight the orientation of individual azimuthal orders.