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J. Eur. Opt. Society-Rapid Publ. 22, 50( 2026)
Fig. 4. Bubble chart plots of low-order Zernike aberration coefficients in full-pyramid representation( a) and in half-pyramid representation( b). The area of the depicted aberration terms( bubble size) is proportional to the magnitude of its RMS-normalized Zernike coefficient c m n
. The blue circles on the upper right represent the areas of the total RMS-deviation of the wavefront error( dark blue) and its fitted RMS-deviation by the shown low-order aberrations fit of a 10th order polynomial( light blue). The numbers within the pyramid represent the RMS-normalized Zernike coefficients [ nm ], whereas the numbers in the blue reference circles( upper right corner) are the RMS-values [ nm ] of the original wavefront and its fitted 10th-order counterpart. Primary coma and trefoil are the main aberration terms.
4 Representing low-order Zernike aberrations
Often, the coefficients resulting from a least-squares Zernike fit are presented in tabular form or as simple bar graphs using a single-indexing scheme. However, Zernike polynomials depend on two integer parameters, n and m, which makes any single-indexing scheme arbitrary. This makes it difficult to interpret their meaning and relate them to the specific aberration term, especially since different single-index ordering schemes are in use, such as ANSI / ISO 24157, Born / Wolf, Standard, or Fringe.
Evansetal.[ 6 ] proposed a graphical representation, showing the magnitude of the various Zernike orders in dependence of the azimuthal order m and the degree n of the radial polynomial, a method that was adopted by the author years ago for plotting the aberration coefficients as two-dimensional bar plots, e. g. in [ 7 ].
Inspired by this, an even more intuitive way for representing low-order Zernike aberration coefficients is proposed here. It makes use of the Zernike pyramid and the fact, that the RMS-normalized Zernike coefficients are uncorrelated to each other. Hence, the root-sum-of-squares( rss) level of individual RMS-normalized Zernike coefficients c m n yields the total RMS-value of the aberration sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X 1
X 1 � RMS total ¼ RMS 2 i
¼: ð5Þ i¼1 i¼1
c m 2 n i
The basic idea is just use a bubble plot depiction of the Zernike pyramid and scale the size of each individual bubble( or individual Zernike polynomial) with the magnitude of the RMS-normalized Zernike coefficient c m n
. There is a physical meaning in this scaling as the sum of all individual bubble areas corresponds to the total RMS-value. By way of example this is shown in Figure 4a for an X / Y representation of the full Zernike pyramid.
Here, the total RMS deviation of sample wavefront A having 7.41 nm is taken as the reference area( area / RMS 2 total / r2 total
, cf. dark blue circle in the upper right corner). The individual radii scale then with c m n i
= RMS total, i. e. the total area of all shown bubbles
( 10th order fit with 66 polynomials) therefore corresponds to the light blue dotted area / RMS 2
10th
¼ ð 6: 52 nm Þ2. It can be easily seen in Figure 4a that primary Y – coma and primary Y – trefoil are the main aberration contributors. An advantage of the full pyramid or X / Y representation is that the signs of the Zernike coefficients c m n are preserved, either through the positions of the maxima and minima of the individual Zernike terms or through additional numerical annotation.
Another, even more convenient depiction is a Mag / Angle representation utilizing only half of the Zernike pyramid, see Figure 4b. The magnitude of the paired X / Y coefficients is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jc m n j¼
� c �m 2 � n þ c þm 2 n
; ð6Þ
�whereas the angularorientation results just from adding up c �m n
Z �m n þ c þm n
Z þm n pairwise. Magnitude and orientation can be then interpreted intuitively.
Note that in Figure 4, the terms piston, tilt, and defocus have been set to zero intentionally, since they describe alignment parameters rather than intrinsic surface or wavefront aberrations.