Study of Zernike polynomials of an elliptical aperture obscured with an elliptical obscuration: comment José Antonio Díaz1,* and Virendra N. Mahajan2 1
Departamento de Óptica, Universidad de Granada, Granada 18071, Spain
2
The Aerospace Corporation, 2310 El Segundo Boulevard, El Segundo, California 90245, USA *Corresponding author:
[email protected] Received 11 March 2013; accepted 16 May 2013; posted 11 July 2013 (Doc. ID 186616); published 15 August 2013
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 OCIS codes: (110.0110) Imaging systems; (010.7350) Wave-front sensing; (220.1010) Aberrations (global); (120.3180) Interferometry; (220.0220) Optical design and fabrication. http://dx.doi.org/10.1364/AO.52.005962
Recently, Hasan and Shaker [1] published a set of orthonormal polynomials for an annular elliptical pupil obtained by the Gram–Schmidt orthogonalization of the Zernike circle polynomials. However, they did not mention the relevance of these polynomials by way of any practical applications. Without relevance, such work may not have utility, and thus reduce to a mathematical exercise. Upon further examination, we find that the expressions for many of the annular elliptical polynomials given by them are incorrect. This is apparently due to the wrong usage of the Gram–Schmidt orthogonalization process. The annular elliptical polynomials Ej x; y; b; ϵ for an elliptical pupil of aspect ratio b and obscuration ratio ϵ can be obtained by Gram–Schmidt orthonormalizing the Zernike circle polynomials Zj x; y over the annular elliptical pupil recursively according to [2]
1559-128X/13/245962-03$15.00/0 © 2013 Optical Society of America 5962
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Ej1 N j1 Zj1 −
j X
hZj1 Ek iEk ;
(1)
k1
where E1 1; Z Z bp 1 1−x2 1 hZj1 Ek i dx p Zj1 Ek dy 2 πb1 − ϵ −1 −b 1−x2 Zϵ Z bp 2 2 ϵ −x − dx Z E dy ; p j1 k −b ϵ2 −x2
−ϵ
(2)
(3)
and N j1 is a normalization constant so that the E-polynomials are orthonormal over the pupil, i.e., Z Z bp 1 1−x2 1 dx hEj Ej0 i p Ej Ej0 dy πb1 − ϵ2 −1 −b 1−x2 Zϵ Z bp ϵ2 −x2 − dx p Ej Ej0 dy δjj0 : −ϵ
−b ϵ2 −x2
(4)
Table 1.
Orthonormal Annular Elliptical Polynomials Ex ;y ; b; ϵ for an Elliptical Pupil with an Aspect Ratio b and Obscuration Ratio ϵ
Ej
Ex; y; b; ϵ
E1
1
E2
N2x
E3
N3y
E4
N 4 2ρ2 − 12 1 b2 1 ϵ2
E5
3−1 b4 1 ϵ4 1 − 4ρ2 ϵ2 b2 1 ϵ2 N 5 x2 − y2 14 −1 b2 1 ϵ2 − 431 ϵ4 − 2b2 1 4ϵ2 ϵ4 3b4 1 ϵ4
E6
2N 6 xy
E7
N7 1 3b2 1 ϵ2 ϵ4 y 6ρ2 − 2 2 1ϵ
E8
N8 3 b2 1 ϵ2 ϵ4 x 6ρ2 − 2 1 ϵ2
E9
a
E10 b
8 9 −3b2 −1 b2 1 2ϵ2 3ϵ4 3ϵ6 2ϵ8 ϵ10 = 4N 9 < y 3b2 x2 −1 − 2ϵ2 − 6ϵ4 − 2ϵ6 − ϵ8 3b2 1 2ϵ2 − 2ϵ4 2ϵ6 ϵ8 ; A : 2 y −5 − 10ϵ2 − 6ϵ4 − 10ϵ6 − 5ϵ8 3b2 1 2ϵ2 6ϵ4 2ϵ6 ϵ8 8 4N 10 < x B :
9 −3b2 −1 b2 1 2ϵ2 3ϵ4 3ϵ6 2ϵ8 ϵ10 = 2 4 6 8 2 2 4 6 8 2ϵ − 2ϵ 2ϵ ϵ b 1 2ϵ 6ϵ 2ϵ ϵ ; b2 x2 −31 2ϵ2 6ϵ4 2ϵ6 ϵ8 b2 5 10ϵ2 6ϵ4 10ϵ6 5ϵ8 3y2 −31
E11
1 31 ϵ2 3x2 2 ϵ2 2ϵ4 3b4 ϵ2 x2 3ϵ2 y2 3b4 y2 2 ϵ2 2ϵ4 2b2 x2 − 2y2 N 11 6ρ4 3 2b2 3b4 1 4ϵ2 ϵ4 − − 8 4 1 ϵ2 ϵ4 b2 1 ϵ2 ϵ4
E12
3−1 b2 1 ϵ2 1 ϵ4 4N 12 xy x2 − y2 2 4 81 ϵ ϵ
E13
−10.2857ρ2 18.1919ρ4 sin 2θ
E14
3.2184 − 20.4282ρ2 31.6338ρ4 13.7867ρ2 − 35.1829ρ4 cos 2θ 8.2278ρ4 cos 4θ
E15
7.5667ρ2 − 18.4404ρ4 sin 2θ 7.3881ρ4 sin 4θ
A 5 10ϵ2 6ϵ4 10ϵ6 5ϵ8 − 6b2 1 2ϵ2 6ϵ4 2ϵ6 ϵ8 9b4 1 2ϵ2 − 2ϵ4 2ϵ6 ϵ8 : B 91 2ϵ2 − 2ϵ4 2ϵ6 ϵ8 − 6b2 1 2ϵ2 6ϵ4 2ϵ6 ϵ8 b4 5 10ϵ2 6ϵ4 10ϵ6 5ϵ8 :
a b
Here, δjj0 is a Kronecker delta. Letting j0 1, we find that the mean value hEj i of a polynomial is zero (except when j 1). Similarly, letting j j0 , its mean square value hE2j i is unity. The new polynomials can also be obtained nonrecursively using a matrix approach, as discussed in [3]. The closed-form analytical expressions of the first 12 annular elliptical polynomials obtained by using the Gram–Schmidt algorithm Eqs. (1)–(3) are given in Tables 1 and 2. Of course, these polynomials reduce to (1) unobscured elliptical polynomials [4–6] as ϵ → 0; (2) annular polynomials as b → 1; and (3) circle polynomials as b → 1 and ϵ → 0. Hasan and Shaker use a different numbering of the polynomials, but without any explanation. For example, our E15 is their A13. However, their polynomials A5 and
A9–A11 given in Table 2 are incorrect. Also their normalization coefficients C4, C5, and C7–C9 in Table 3 are incorrect. Hence, all of their orthonormal polynomials Cj Aj are incorrect, except for j 1, 2, 3, 6, and 12. Even the unobscured elliptical polynomials given by them in their Table 1 have errors. For example, their pE2 should be 2x and E4 should be multiplied by 2. We have also included the numerical form of our polynomials E13 , E14 , and E15 for b 0.7 and k 0.5, because the closed-form analytical expressions for them are too long. Comparing with the corresponding numerical results of Hasan and Shaker, given by them as polynomials A12, A15, and A13 in their Table 6, we find that they are incorrect. Further examination shows that their polynomials A6, A10, 20 August 2013 / Vol. 52, No. 24 / APPLIED OPTICS
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Table 2.
Normalization Constants N j for the Annular Elliptical Polynomials Given in Table 1
Nj
Expression
N1
1
N2
21 ϵ2 −1∕2
N3
2b−1 1 ϵ2 −1∕2
N4
p 1231 ϵ4 − 2b2 1 4ϵ2 ϵ4 3b4 1 ϵ4 −1∕2
N5
91 ϵ4 9b4 1 ϵ4 − 6b2 1 4ϵ2 ϵ4 1∕2 2b4 −1 ϵ2 2 1 ϵ2 ϵ4 −1∕2
N6
p 2 6b 1 ϵ2 ϵ4 −1∕2
N7
641 ϵ2 1∕2 fb2 5 10ϵ2 6ϵ4 10ϵ6 5ϵ8 − 6b2 1 2ϵ2 6ϵ4 2ϵ6 ϵ8 9b4 1 2ϵ2 − 2ϵ4 2ϵ6 ϵ8 g−1∕2
N8
641 ϵ2 1∕2 91 2ϵ2 − 2ϵ4 2ϵ6 ϵ8 − 6b2 1 2ϵ2 6ϵ4 2ϵ6 ϵ8 b4 5 10ϵ2 6ϵ4 10ϵ6 5ϵ8 −1∕2
N9
5
N 12
6ϵ4
10ϵ6
b6 1 2ϵ2 ϵ4 1 ϵ2 1 ϵ4 1 4ϵ2 ϵ4 5ϵ8 9b4 1 2ϵ2 − 2ϵ4 2ϵ6 ϵ8 − 6b2 1 2ϵ2 6ϵ4 2ϵ6 ϵ8
3201 ϵ2 ϵ4 1∕2
451 b8 1 ϵ2 3ϵ4 − 4ϵ6 3ϵ8 ϵ10 ϵ12 − 60b2 b8 1 3ϵ2 3ϵ4 4ϵ6 3ϵ8 3ϵ10 ϵ12 −1∕2 b4 94 78ϵ2 282ϵ4 712ϵ6 282ϵ8 78ϵ10 94ϵ12
2 b 5 10ϵ2 5ϵ4 3 3ϵ4 2ϵ6 ϵ8 6b2 1 2ϵ2 3ϵ4 8ϵ6 3ϵ8 2ϵ10 ϵ12 5b4 1 2ϵ2 3ϵ4 3ϵ8 2ϵ10 ϵ12 −1∕2 2 4 1601 ϵ ϵ
and A14 are also incorrect, as may be seen by comparing with our polynomials E5 , E10 , and E12 , respectively, for the above values of b and k. References 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.
5964
−1∕2
b4 1 2ϵ2 ϵ4 1 ϵ2 1 ϵ4 1 4ϵ2 ϵ4 −1∕2 ×91 2ϵ2 − 2ϵ4 2ϵ6 ϵ8 − 6b2 1 2ϵ2 6ϵ4 2ϵ6 ϵ8 b4 5 10ϵ2 6ϵ4 10ϵ6 5ϵ8 1∕2
N 10
N 11
10ϵ2
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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.
