Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces

Tanabe, Takao; Shibuya, Masato; Maehara, Kazuhisa

doi:10.1117/1.oe.55.3.035101

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…As proven in Ref. 6, the Zernike expansion of monomial t α is given as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 ; 3 2 6 ; 3 4 7…”

Section: Practical Impossibility Of Expansion Of Odd-ordermentioning

confidence: 99%

“…Tanabe et al 6 have derived the expansion formula for monomial and described the possibility of the close approximation of odd-order aspherical terms by a finite number of evenorder aspherical terms. As proven in Ref.…”

Section: Practical Impossibility Of Expansion Of Odd-ordermentioning

confidence: 99%

“…Also, the fact that odd-order surfaces can be exactly represented by Zernike polynomials has not been proven. To resolve these problems, Tanabe et al 6 proved that not only the displacement but also slopes of odd-order surfaces are exactly represented by a finite number of Zernike polynomials. As a result, odd-order surfaces are exactly represented by a finite number of even-order polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Aberration properties of odd-order surfaces in optical design

Tanabe

Shibuya

2016

Opt. Eng

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Abstract. We theoretically represent the effectiveness of the odd-order surface for optical designs via an aberration theory. The theory employs an expression of the odd-order surface with aberration coefficients, which are derived by power-series expansion of wavefront aberration with respect to the coordinates of optical surfaces. This expression allows us to understand the aberration characteristics of odd-order surfaces. By applying this aberration theory to the design of extreme ultraviolet lithography optics, we show that odd-order surfaces are effective in reducing higher-order aberrations with fewer coefficients than even-order surfaces do. © The Authors.Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

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“…As proven in Ref. 6, the Zernike expansion of monomial t α is given as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 ; 3 2 6 ; 3 4 7…”

Section: Practical Impossibility Of Expansion Of Odd-ordermentioning

confidence: 99%

Section: Practical Impossibility Of Expansion Of Odd-ordermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Aberration properties of odd-order surfaces in optical design

Tanabe

Shibuya

2016

Opt. Eng

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“…In our investigations of the characteristics of odd-order surfaces, 20,21 we have analogically predicted that steep normal distribution can be sufficiently represented by a small finite number of power terms including odd-order terms which can be used in many kinds of optical design software. Actually, we confirmed the fulfillment of this prediction numerically.…”

mentioning

confidence: 99%

Practical method for evaluating optical image defects caused by center artifacts

2017

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Abstract. In modern optical element manufacturing, center artifacts are a common problem. A center artifact is a shape error that is rotationally symmetrical, steep, and localized at the center. These properties cause characteristic image defects different from those caused by ordinary irregularities. However, tolerancing center artifacts has not been fully discussed or properly carried out. We propose a simple mathematical model for center artifacts using normal distribution function as a figure model and showing that this function can be represented by a polynomial including odd-order terms. Our method enables appropriate optical simulation and tolerancing for center artifacts using general optical design software.

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Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates

Andersen

2018

Opt. Express

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For some time it has been known and recommended that the calculation of Zernike polynomials to radial orders higher than 8 to 10 should be performed using recurrence relations rather than explicit expressions due increasingly large cancellation errors. This paper presents a set of simple recurrence relations that can be used for the unit-normalized Zernike polynomials in polar coordinates and easily adapted to Cartesian coordinates as well. The recurrence relations are also well suited for the calculation of the Cartesian derivatives of the Zernike polynomials. The recurrence relations are easily extended to arbitrarily high orders. Assessments of the precision achievable with standard 64-bit floating point arithmetic show that Zernike polynomials up to radial order 30 can be calculated over the unit disc with errors not exceeding 5E-14, and up to radial order 50 with errors not exceeding 1.2E-13. Comparison with the Zernike capability in OpticStudio (Zemax) shows that the recurrence relations are superior in performance (both speed and precision) over the existing algorithm implemented in the software. General pseudo-code for the calculation of Zernike polynomials and their derivatives is also presented.

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Convergence and differentiation of Zernike expansion: application for an analysis of odd-order surfaces

Cited by 4 publications

References 14 publications

Aberration properties of odd-order surfaces in optical design

Aberration properties of odd-order surfaces in optical design

Practical method for evaluating optical image defects caused by center artifacts

Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates

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