Orthogonal Polynomials which arise in the expansion of a wavefront function for optical systems with
circular pupils. The Odd and Even Zernike polynomials are given by
(1) 

(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
The Zernike polynomial is a special case of the Jacobi Polynomial with
(8) 
(9)  
(10)  
(11)  
(12) 
(13)  
(14)  
(15) 
(16) 
(17) 
(18) 
Let a ``primary'' aberration be given by
(19) 
(20) 
(21) 
Aberration  
spherical aberration  0  4  0  
coma  0  3  1  
astigmatism  0  2  2  
field curvature  1  2  0  
distortion  1  1  1 
See also Jacobi Polynomial
References
Bezdidko, S. N. ``The Use of Zernike Polynomials in Optics.'' Sov. J. Opt. Techn. 41, 425, 1974.
Bhatia, A. B. and Wolf, E. ``On the Circle Polynomials of Zernike and Related Orthogonal Sets.'' Proc. Cambridge Phil. Soc. 50, 40, 1954.
Born, M. and Wolf, E. ``The Diffraction Theory of Aberrations.'' Ch. 9 in Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. New York: Pergamon Press, pp. 459490, 1989.
Mahajan, V. N. ``Zernike Circle Polynomials and Optical Aberrations of Systems with Circular Pupils.'' In Engineering and Lab. Notes 17 (Ed. R. R. Shannon), p. S21, Aug. 1994.
Prata, A. and Rusch, W. V. T. ``Algorithm for Computation of Zernike Polynomials Expansion Coefficients.'' Appl. Opt. 28, 749754, 1989.
Wang, J. Y. and Silva, D. E. ``WaveFront Interpretation with Zernike Polynomials.'' Appl. Opt. 19, 15101518, 1980.
Zernike, F. ``Beugungstheorie des Schneidenverfahrens und seiner verbesserten Form, der Phasenkontrastmethode.'' Physica 1, 689704, 1934.
Zhang, S. and Shannon, R. R. ``Catalog of Spot Diagrams.'' Ch. 4 in Applied Optics and Optical Engineering, Vol. 11. New York: Academic Press, p. 201, 1992.
© 19969 Eric W. Weisstein