Abstract

Recently, Hasan and Shaker published a set of orthonormal polynomials for an annular elliptical pupil obtained by the Gram–Schmidt orthogonalization of the Zernike circle polynomials [Appl. Opt. 51, 8490 (2012)]. However, the expressions for many of the polynomials are incorrect, apparently due to wrong usage of the Gram–Schmidt orthogonalization process. We provide the correct equations for the orthogonalization process and the expressions for the orthonormal polynomials obtained by applying them.

© 2013 Optical Society of America

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References

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  1. S. Y. Hasan and A. S. Shaker, “Study of Zernike polynomials of an elliptical aperture obscured with and elliptical obscuration,” Appl. Opt. 51, 8490–8497 (2012).
  2. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968), p. 454.
  3. G.-m. Dai and V. N. Mahajan, “Nonrecursive orthonormal polynomials with matrix formulation,” Opt. Lett. 32, 74–76 (2007).
  4. V. N. Mahajan and G.-m. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007).
  5. V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: analytical solution: errata” J. Opt. Soc. Am. A 29, 1673–1674 (2012).
  6. V. N. Mahajan, “Orthonormal polynomials in wavefront analysis,” in Handbook of Optics, V. N. Mahajan and E. V. Stryland, eds., 3rd ed., Vol. II (McGraw-Hill, 2010), pp. 11.3–11.41.

2012 (2)

2007 (2)

Dai, G.-m.

Hasan, S. Y.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968), p. 454.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968), p. 454.

Mahajan, V. N.

Shaker, A. S.

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Tables (2)

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Table 1. Orthonormal Annular Elliptical Polynomials E(x,y;b;ϵ) for an Elliptical Pupil with an Aspect Ratio b and Obscuration Ratio ϵ

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Table 2. Normalization Constants Nj for the Annular Elliptical Polynomials Given in Table 1

Equations (4)

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Ej+1=Nj+1[Zj+1k=1jZj+1EkEk],
E1=1,
Zj+1Ek=1πb(1ϵ2)[11dxb1x2b1x2Zj+1Ekdyϵϵdxbϵ2x2bϵ2x2Zj+1Ekdy],
EjEj=1πb(1ϵ2)[11dxb1x2b1x2EjEjdyϵϵdxbϵ2x2bϵ2x2EjEjdy]=δjj.

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