Zernike polynomials: a guide

  title={Zernike polynomials: a guide},
  author={Vasudevan Lakshminarayanan and Andrea Fleck},
  journal={Journal of Modern Optics},
  pages={545 - 561}
In this paper we review a special set of orthonormal functions, namely Zernike polynomials which are widely used in representing the aberrations of optical systems. We give the recurrence relations, relationship to other special functions, as well as scaling and other properties of these important polynomials. Mathematica code for certain operations are given in the Appendix. 
Zernike functions, rigged Hilbert spaces, and potential applications
We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1,1) + su(1,1). We show how they induce discrete as well continuous bases that coexist in the framework of rigged
Recurrence relations for the Cartesian derivatives of the Zernike polynomials.
  • Philip Stephenson
  • Mathematics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2014
A recurrence relation for the first-order Cartesian derivatives of the Zernike polynomials is derived. This relation is used with the Clenshaw method to determine an efficient method for calculating
Change of Basis from Bernstein to Zernike
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known
Comprehensive Study of Continuous Orthogonal Moments—A Systematic Review
This article provides a comprehensive and comparative review for continuous orthogonal moments along with their applications.
Engineering structured light with optical vortices
In this work, we demonstrate the possibility of generating and controlling any given kind of structured radially symmetric intensity profile with an embedded optical vortex. This is achieved with the
Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering.
This study proposes a simple and systematic process to derive Zernike like functions that are applicable to all types of spherical cap and can serve as essential tools for surface characterization required for a wide range of applications like large-angle lenses description in illumination design.
Topographic synthesis of arbitrary surfaces with vortex Jinc functions.
The higher-order Bessel functions with a vortex azimuthal factor are introduced to propose a family of functions to generalize the function defining the Airy pattern, and these functions happen to form an orthogonal set, making them suitable applications in visual optics or analysis of aberrations of optical systems.
Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates.
A set of simple recurrence relations that can be used for the unit-normalized Zernike polynomials in polar coordinates and easily adapted to Cartesian coordinates are presented and are superior in performance and precision over the existing algorithm implemented in the software.
Non-imaging optics
In this report a semi-analytic solution to the Laplacian magic window is proposed. The Laplacian magic window is a term recently introduced in 2017[2]. When a uniform wavefront hits a refractive
Unitarily invariant strictly positive definite kernels on spheres
We present a Fourier characterization for the continuous and unitarily invariant strictly positive definite kernels on the unit sphere in $${\mathbb {C}}^{q}$$Cq, thus adding to a celebrated work of


Zernike polynomials and atmospheric turbulence
This paper discusses some general properties of Zernike polynomials, such as their Fourier transforms, integral representations, and derivatives. A Zernike representation of the Kolmogoroff spectrum
Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape.
The Gram-Schmidt orthogonalization technique presented can be extended to both apertures of arbitrary shape and other basis functions.
On the circle polynomials of Zernike and related orthogonal sets
The paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect
Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform
An algorithm in the form of a discrete cosine transform is presented for the computation of Zernike polynomials of arbitrary degree n which comes with advantages over other methods in terms of computation time, accuracy and transparancy.
Robust and fast computation for the polynomials of optics.
Applications to arbitrarily high orders are shown to be enabled by remarkably simple and robust algorithms that are derived from well known recurrence relations.
Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula.
  • G. Dai
  • Physics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2006
A more intuitive derivation of a simpler, nonrecursive formula, which is used to calculate the instantaneous refractive power, is described.
Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation.
  • G. Dai
  • Physics, Mathematics
    Optics letters
  • 2006
The set of Fourier series is discussed following some discussion of Zernike polynomials, which are derived that allow for relating Fourierseries expansion coefficients to Zernik polynomial expansion coefficients.
Algorithm for computation of Zernike polynomials expansion coefficients.
A numerically efficient algorithm for expanding a function in a series of Zernike polynomials is presented and shows that typically at least a fourfold improvement in computational speed can be expected in practical use.
A New Method for Describing the Aberrations of the Eye Using Zernike Polynomials
  • C. Campbell
  • Physics
    Optometry and vision science : official publication of the American Academy of Optometry
  • 2003
The standard Zernike polynomial functions are reformulated in a way so that the number of functions (or terms) needed to describe an arbitrary wavefront surface to a given Zernike radial order is