Optical Engineering Science. Stephen Rolt

39 39 32 10 2 61 57 44 5 1 18 16 13 8 0 40 37 24 10 4 62 59 55 5 3 19 18 18 8 2 41 38 31 10 6 63 61 68 5 5 20 20 25 8 4 42 40 40 10 8 64 63 83 6 −6 21 27 37 8 6 43 42 51 10 10 65 65 100

      It must be stated, at this point, that the 36 polynomials used, in this instance, are not those that would be ordered as in Table 5.1. That is to say, they are not the first 36 ANSI standard polynomials. As mentioned earlier, there are, unfortunately, a number of competing conventions for the numbering of Zernike polynomials. The convention used in determining the P to Vr figure is the so called Zernike Fringe polynomial convention. The logic of ordering the polynomials in a different way is that this better reflects, in the case of the fringe polynomial set, the spatial frequency content of the polynomial and its practical significance in real optical systems.

      5.3.5 Other Zernike Numbering Conventions

The ordering convention adopted by the Fringe polynomials expresses, to a significant degree, the spatial frequency content of the polynomial. As a consequence, the polynomials are ordered by the sum of their radial and polar orders, rather than primarily by the radial order. That is to say, the polynomials are ordered by the sum n + m, as opposed to n alone. For polynomials of equal ‘fringe order’ they are then ordered by descending values of the modulus of m, i.e. |m|, with the positive or cosine term presented first.

Another convention that is very widely used is the Noll convention. The Noll convention proceeds in a broadly similar way to the ANSI convention, in that it uses the radial order, n, as the primary parameter for sorting. However, there are a number of key differences. Firstly, the sequence starts with the number one, as opposed to zero, as is the case for the other conventions. Secondly, the ordering convention for the polar order, m, as in the case of the fringe polynomials, follows the modulus of m rather its absolute value. However, the ordering is in ascending sequence of |m|, unlike the fringe polynomials. The ordering of the sine and cosine terms is presented in such a way that all positive m (cosine terms) are allocated an even number. In consequence, sometimes the sine term occurs before the cosine term in the sequence and sometimes after. Table 5.4 shows a comparison of the different numbering systems up to ANSI number 65.

Further Reading

      1 American National Standards Institute (2017). Methods for Reporting Optical Aberrations of Eyes, ANSI Z80.28:2017. Washington DC: ANSI.

      2 Born, M. and Wolf, E. (1999). Principles of Optics, 7e. Cambridge: Cambridge University Press. ISBN: 0-521-642221.

      3 Fischer, R.E., Tadic-Galeb, B., and Yoder, P.R. (2008). Optical System Design, 2e. Bellingham: SPIE. ISBN: 978-0-8194-6785-0.

      4 Hecht, E. (2017). Optics, 5e. Harlow: Pearson Education. ISBN: 978-0-1339-7722-6.

      5 Noll, R. (1976). Zernike polynomials and atmospheric turbulence. J. Opt. Soc. Am. 66 (3): 207.

      6 Zernike, F. (1934). Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode. Physica 1 (8): 689.