J. Eur. Opt. Society-Rapid Publ. 2026, 22, 50 Ó The Author( s), published by EDP Sciences, 2026 https:// doi. org / 10.1051 / jeos / 2026034 Available online at: https:// jeos. edpsciences. org
EOSAM 2025 Guest editors: Omar El Gawhary, Stefan Witte, Ignacio Moreno
Journal of the European Optical Society-Rapid Publications
RESEARCH ARTICLE Visualization of wavefront aberrations by Zernike polynomials
Stephan Reichelt * Institute of Applied Optics( ITO), University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany Received 28 January 2026 / Accepted 31 March 2026
Abstract. Zernike aberration coefficients are typically presented in tabular form or as simple bar graphs, making it difficult to intuitively interpret their meaning and relate them to the specific aberration term. In this work, we present intuitive and rapidly interpretable visual representations of Zernike aberration terms that highlight the magnitude and orientation of their dominant contributions. Depending on the aberration order, we recommend different graphical formats – such as bubble plots and heatmaps – for visualizing low-order and mid-spatial frequency Zernike terms. These graphical representations are particularly valuable when quick visual feedback on individual aberration terms is needed, such as during aberration compensator adjustment, real-time wavefront visualization of dynamic processes, or optical alignment procedures. They can also enhance the clarity and interpretability of inspection reports or measurement certificates. Furthermore, an alternative definition of the azimuthal orientation of Zernike terms with m 6¼ 0 is proposed, enabling a more effective analysis of the shape and bending of spoke-shaped aberrations.
Keywords: Zernike polynomials, Visualization, Wavefront analysis, Wavefront aberrations, Optical testing.
1 Introduction
Polynomial sets such as Zernike polynomials [ 1 – 4 ] are widely used in various applications to approximate optical aberrations in a compact and sufficiently accurate manner. For example, in precision optical manufacturing, Zernike polynomials are used to characterize and approximate surface figure errors of optical components( e. g., lenses, flats) or wavefront aberrations of optical systems measured by interferometers or wavefront sensors. Freeform surfaces, which can also be described by Zernike polynomials, play an important role in optical design. In addition, Zernike polynomials are typically used in adaptive optics to compensate for dynamic aberrations, such as those caused by atmospheric turbulence.
The present work is motivated by the aim of providing a representation of Zernike approximation results that is as simple and intuitive as possible. Conventional formats such as tabular listings or bar plot visualizations, while widely used, often fail to adequately convey structural relationships and limit the direct interpretability of the underlying wavefront or surface characteristics. For users working with Zernike polynomial representations in optical design, metrology, or ophthalmology, the interpretation of coefficient sets can be non-trivial, particularly in the presence of multiple interacting modes or when spatial patterns must be inferred indirectly. This can make it difficult to extract
* Corresponding author: reichelt @ ito. uni-stuttgart. de relevant insights efficiently, even for experienced practitioners. To address this, the present work introduces a novel representation that is specifically designed to make structurally relevant features more directly accessible, thereby facilitating interpretation and supporting more effective analysis of Zernike-based wavefront or surface descriptions.
2 Definition of Zernike polynomials
Figure 1 illustrates the first 15 Zernike polynomials, arranged in a pyramid up to the 4th order.
Zernike polynomials are expressed either in Cartesian( x, y) or conveniently in polar coordinates( r, h). They are defined within the unit circle, where r is the normalized radial distance from the origin, and h is the angle from the x axis( common right-handed coordinate system). Apart from a preceding normalization factor N m n
, Zernike polynomials are the product of a radial polynomial R m n ðrÞ and an azimuthal function M m( h) Z m n ðr; hÞ ¼N m n Rm n ðrÞM mðhÞ; ð1Þ
where n is the degree of the polynomial and m is the azimuthal dependence parameter [ 2, 5 ]. The azimuthal order m is an integer( m 2 Z), whereas the polynomial degree or radial order n is a non-negative integer( n 2 N 0). For a particular Zernike polynomial Z m n
, the integral values n and m are either both even or both odd. The double-indexing scheme, in which n denotes the highest power of the
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