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Zernike Polynomials: a Guide See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/241585467 Zernike polynomials: A guide Article in Journal of Modern Optics · April 2011 DOI: 10.1080/09500340.2011.633763 CITATIONS READS 41 12,492 2 authors: Vasudevan Lakshminarayanan Andre Fleck University of Waterloo Grand River Hospital 399 PUBLICATIONS 1,806 CITATIONS 40 PUBLICATIONS 1,210 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Theoretical and bench color and optics work View project Quantum Relational Databases View project All content following this page was uploaded by Vasudevan Lakshminarayanan on 13 May 2014. The user has requested enhancement of the downloaded file. Journal of Modern Optics Vol. 58, No. 7, 10 April 2011, 545–561 TUTORIAL REVIEW Zernike polynomials: a guide Vasudevan Lakshminarayanana,b*y and Andre Flecka,c aSchool of Optometry, University of Waterloo, Waterloo, Ontario, Canada; bMichigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI, USA; cGrand River Hospital, Kitchener, Ontario, Canada (Received 5 October 2010; final version received 9 January 2011) 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. Keywords: optical aberrations; Zernike polynomials; special functions; aberrometry 1. Introduction the Zernikes will not be orthogonal over a discrete set The Zernike polynomials are a sequence of polyno- of points within a unit circle [11]. However, Zernike mials that are continuous and orthogonal over a unit polynomials offer distinct advantages over other poly- circle. A large fraction of optical systems in use today nomial sequences. Using the normalized Zernike employ imaging elements and pupils which are circu- expansion to describe aberrations offers the advantage lar. As a result, Zernike polynomials have been that the coefficient or value of each mode represents adopted as a mathematical description of optical the root mean square (RMS) wavefront error attrib- wavefronts propagating through such systems. An utable to that mode. The Zernike coefficients used to optical wavefront can be thought of as the surface of mathematically describe a wavefront are independent equivalent phase for radiation produced by a mono- of the number of polynomials used in the sequence. chromatic light source. For a point source at infinite This condition of independence or orthogonality, distance this surface is a plane wave. An example is means that any number of additional terms can be shown in Figure 1 with an aberrated wavefront for added without impact on those already computed. comparison. The mathematical description, offered by Coefficients of larger magnitude indicate greater con- Zernike polynomials, is useful in defining the magni- tribution of that particular mode to the total RMS tude and characteristics of the differences between the wavefront error of the system and thus greater image formed by an optical system and the original negative impact on the optical performance of the object. These optical aberrations can be a result of system. optical imperfections in the individual elements of an Having stated the advantages of Zernike polyno- optical system and/or the system as a whole. First mials in describing wavefront aberrations over a unit employed by F. Zernike, in his phase contrast circle it should also be clearly noted that Zernike method for testing circular mirrors [3], they have polynomials are not always the best polynomials for since gained widespread use due to their orthogonality fitting wavefront test data. In certain cases, Zernike and their balanced representation of classical aberra- polynomials may provide a poor representation of the tions yielding minimum variance over a circular pupil wavefront. Some of the effects of air turbulence in [4–9]. astronomy and effects of fabrication errors in the The Zernike polynomials are but one of infinite production of optical elements may not be well number of complete sets of polynomials, with two represented by even a large expansion of the Zernike variables, that are orthogonal and continuous over the sequence. In the testing of conical optical elements interior of a unit circle [10]. The condition of being or irregular ocular optics, such as a condition known continuous is important to note because, in general, as keratoconus (literally a cone shaped cornea), *Corresponding author. Email: [email protected] yVL is also with the departments of Physics and Electrical Engineering at the University of Waterloo. ISSN 0950–0340 print/ISSN 1362–3044 online ß 2011 Taylor & Francis DOI: 10.1080/09500340.2011.554896 http://www.informaworld.com 546 V. Lakshminarayanan and A. Fleck Figure 1. A plane wave (perfect wavefront) and an aberrated wavefront subdivided into smaller wavefronts where the local slope determines the location of a ray tracing projection of the sublet (subdivided wavelet). This is the operating principle of wavefront sensors [1]. (The color version of this figure is included in the online version of the journal.) additional terms must be added to Zernike polyno- mials to accurately represent alignment errors and even in so doing, the description of the optical quality may be incomplete [12]. In any system with noncircular pupils, the Zernike circle polynomials will not be orthogonal over the whole of the pupil area and thus may not be the ideal polynomial sequence. 2. Mathematical basis In general, the function describing an arbitrary wavefront in polar coordinates (r, ), denoted by W(r, ), can be expanded in terms of a sequence of polynomials Z that are orthonormal over the entire surface of the circular pupil:X m m Wðr, Þ¼ Cn Zn ðr, Þ, ð1Þ n, m where C denotes the Zernike amplitudes or coefficients Figure 2. Cartesian (x, y) and polar (r, ) coordinates of a and Z the polynomials. The coordinate system is point Q in the plane of a unit circle representing the circular shown in Figure 2. exit pupil of an imaging system [6]. The Zernike polynomials expressed in polar coor- dinates (X ¼ r sin , Y ¼ r cos ) are given by the complex combination: clockwise from the y-axis. This is consistent with Zmðr, ÞiZÀmðr, Þ¼VÀmðr cos , r sin Þ n n n aberration theory definitions, but different from the m ¼ Rn ðrÞ expðimÞ, conventional mathematical definition of polar coordi- which leads to nates. The convention employed is at the discretion of Zmðr, Þ¼RmðrÞ cos m for m 0, the author and may differ depending on the applica- n n m Àm m tion. The radial function, Rn ðrÞ, is described by: Zn ðr, Þ¼Rn ðrÞ sin m for m 5 0, ð2Þ where r is restricted to the unit circle (0 r 1), ðnXÀmÞ=2 ð1Þl ðn À lÞ! RmðrÞ¼ ÂÃÂÃrnÀ2l: ð3Þ meaning that the radial coordinate is normalized by n l! 1 ðn þ mÞl ! 1 ðn À mÞl ! the semi-diameter of the pupil, and is measured l¼0 2 2 Journal of Modern Optics 547 visual/ophthalmic optics; it is nothing but the ordinary spectacle correction that is commonly prescribed by optometrists. Furthermore, the radial order n ¼ 1is equivalent to prism terms that are prescribed in spectacles as well. The normalization has been chosen to satisfy Æm Rn ð1Þ¼1 for all values of n and m. This gives a normalization constant described by: 1=2 m 2ðn þ 1Þ Nn ¼ , ð4Þ 1 þ m0 where m0 is the Kronecker delta (m0 ¼ 0 for m 6¼ 0). The algebraic form of the first 10 orders of Zernike polynomials which correspond to the surface plots of Figure 4 are shown in Tables 1 and 2, in both polar and Cartesian form, for comparison. In Cartesian coordi- nates, the 0 r 1 limit becomes x2 þ y2 1, and the recursion relationship for the sequence in Cartesian form is given by [13]: ! ! Xq XM MXÀJ n À 2M M À j m iþj Zn ðx, yÞ¼ ð1Þ i¼0 j¼0 k¼0 2i þ p k ðn À j Þ!  xy, ð5Þ j!ðM À j Þ!ðn À M À j Þ! where n n jsj M ¼ À m À , p ¼ ðs þ 1Þ, 2 2 2 s q ¼ðd À sMod½n,2Þ , s ¼ sgnðd Þ, d ¼ n À 2m, 2 ¼ 2ði þ kÞþp, ¼ n À 2ði þ j þ kÞp: As can be seen from the relationship above and the expanded form shown in Tables 1 and 2, representa- tion of the polynomials in polar form offers some advantage in efficiency by nature of its circular symmetry. Tables of Zernikes are given, for example, Figure 3. (a) The Zernike radial function for m ¼ 0, n ¼ 2 (defocus), 4 (spherical aberration), 6, 8 [6]. (b) The Zernike by Born and Wolf [4] and in the article by Noll [9]. radial function for m ¼ 1, n ¼ 1, 3, 5, 7. (c) The Zernike radial However, there is a difference between the two: function for m ¼ 2, n ¼ 2, 4, 6, 8 [6]. each of Noll’s polynomials should be multiplied by a factor 1/(2(n þ 1))1/2 to get the terms given by Born and Wolf. Although different indexing schemes exist for the The radial function is plotted in Figure 3 for the Zernike polynomial sequence, the recommended first four radial orders for the angular frequency 1, 2 double indexing scheme is portrayed [14,15]. and 3. The first 10 orders of the Zernike polynomial Occasionally a single indexing scheme is used for sequence are shown as surface plots in Figure 4. describing the Zernike expansion coefficients.
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