Comparative assessment of orthogonal polynomials for wavefront reconstruction over the square aperture
01 Oct 2014-Journal of The Optical Society of America A-optics Image Science and Vision (J Opt Soc Am A Opt Image Sci Vis)-Vol. 31, Iss: 10, pp 2304-2311
TL;DR: Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.
Abstract: Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials, 2D Legendre polynomials, Zernike square polynomials and Numerical polynomials. They are all orthogonal over the full unit square domain. 2D Chebyshev polynomials are defined by the product of Chebyshev polynomials in x and y variables, as are 2D Legendre polynomials. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness. Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.
Citations
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TL;DR: A detailed review of the different types of optical freeform surface representation techniques and their applications and discuss their properties and differences is presented.
Abstract: Modern advanced manufacturing and testing technologies allow the application of freeform optical elements. Compared with traditional spherical surfaces, an optical freeform surface has more degrees of freedom in optical design and provides substantially improved imaging performance. In freeform optics, the representation technique of a freeform surface has been a fundamental and key research topic in recent years. Moreover, it has a close relationship with other aspects of the design, manufacturing, testing, and application of optical freeform surfaces. Improvements in freeform surface representation techniques will make a significant contribution to the further development of freeform optics. We present a detailed review of the different types of optical freeform surface representation techniques and their applications and discuss their properties and differences. Additionally, we analyze the future trends of optical freeform surface representation techniques.
63 citations
TL;DR: This work derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror, and extends this work to elliptical, rectangular, and square pupils.
Abstract: This paper derives closed-form orthonormal polynomials over noncircular apertures using the Gram-Schmidt orthogonalization process. Isometric plots, interferograms, and point-spread functions are illustrated. Their use in wavefront analysis is discussed.
29 citations
TL;DR: The mathematical apparatus of orthogonal polynomials defined over a square aperture, which was developed before for the tasks of wavefront reconstruction, is used to describe shape of a mirror surface.
Abstract: In the recent years a significant progress was achieved in the field of design and fabrication of optical systems based on freeform optical surfaces. They provide a possibility to build fast, wide-angle and high-resolution systems, which are very compact and free of obscuration. However, the field of freeform surfaces design techniques still remains underexplored. In the present paper we use the mathematical apparatus of orthogonal polynomials defined over a square aperture, which was developed before for the tasks of wavefront reconstruction, to describe shape of a mirror surface. Two cases, namely Legendre polynomials and generalization of the Zernike polynomials on a square, are considered. The potential advantages of these polynomials sets are demonstrated on example of a three-mirror unobscured telescope with F/# = 2.5 and FoV = 7.2x7.2°. In addition, we discuss possibility of use of curved detectors in such a design.
27 citations
TL;DR: The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified, indicating that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.
Abstract: Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.
24 citations
TL;DR: In this article, the mathematical apparatus of orthogonal polynomials defined over a square aperture was used to describe shape of a mirror surface, which was developed before for the tasks of wavefront reconstruction.
Abstract: In the recent years a significant progress was achieved in the field of design and fabrication of optical systems based on freeform optical surfaces. They provide a possibility to build fast, wide-angle and high-resolution systems, which are very compact and free of obscuration. However, the field of freeform surfaces design techniques still remains underexplored. In the present paper we use the mathematical apparatus of orthogonal polynomials defined over a square aperture, which was developed before for the tasks of wavefront reconstruction, to describe shape of a mirror surface. Two cases, namely Legendre polynomials and generalization of the Zernike polynomials on a square, are considered. The potential advantages of these polynomials sets are demonstrated on example of a three-mirror unobscured telescope with F/#=2.5 and FoV=7.2x7.2°. In addition, we discuss possibility of use of curved detectors in such a design.
23 citations
References
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TL;DR: In this paper, a Zernike representation of the Kolmogoroff spectrum of turbulence is given that provides a complete analytical description of the number of independent corrections required in a wave-front compensation system.
Abstract: This paper discusses some general properties of Zernike polynomials, such as their Fourier transforms, integral representations, and derivatives. A Zernike representation of the Kolmogoroff spectrum of turbulence is given that provides a complete analytical description of the number of independent corrections required in a wave-front compensation system.
2,741 citations
"Comparative assessment of orthogona..." refers methods in this paper
...Zernike circle polynomials [1] have been widely applied in the reconstruction of wavefront or surface topography across the circular aperture....
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TL;DR: In this paper, a completely rewritten chapter was added to cover wavefront fitting and evaluation as well as holographic and Moire methods, and an appendix was added suggesting appropriate tests for typical optical surfaces.
Abstract: Fringe scanning techniques, now renamed heterodyning or phase shift interferometry, are covered in a completely rewritten chapter. New chapters have been added to cover wavefront fitting and evaluation as well as holographic and Moire methods. The chapter on parameter measurements has been completely rewritten and an appendix added suggesting appropriate tests for typical optical surfaces.
2,372 citations
"Comparative assessment of orthogona..." refers methods in this paper
...2D (two-dimensional) Chebyshev polynomials [6,7], 2D Legendre polynomials [7–9], Zernike orthogonal square polynomials [4,5] and Numerical orthogonal polynomials [10] can be used as the base polynomials for the decomposition of a wavefront across the square aperture....
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TL;DR: In this paper, the problem of wavefront estimation from wave-front slope measurements has been examined from a least-squares curve fitting model point of view, and a new zonal phase gradient model is introduced and its error propagator, which relates the mean square wavefront error to the noisy slope measurements, has been compared with two previously used models.
Abstract: The problem of wave-front estimation from wave-front slope measurements has been examined from a least-squares curve fitting model point of view. It is shown that the slope measurement sampling geometry influences the model selection for the phase estimation. Successive over-relaxation (SOR) is employed to numerically solve the exact zonal phase estimation problem. A new zonal phase gradient model is introduced and its error propagator, which relates the mean-square wave-front error to the noisy slope measurements, has been compared with two previously used models. A technique for the rapid extraction of phase aperture functions is presented. Error propagation properties for modal estimation are evaluated and compared with zonal estimation results.
958 citations
"Comparative assessment of orthogona..." refers background in this paper
...Reconstruction techniques of wavefront or surface shape can be classified approximately as zonal and modal methods [2,3]....
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TL;DR: In this paper, the authors derived closed-form polynomials that are orthogonal over a hexagonal pupil, such as the hexagonal segments of a large mirror.
Abstract: Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.
140 citations
TL;DR: In this paper, the residual error of wave-front reconstruction with Zernike polynomials and Karhunen-Loeve functions from average slope measurements with circular and annular apertures is investigated.
Abstract: Modal wave-front reconstruction by use of Zernike polynomials and Karhunen–Loeve functions from average slope measurements with circular and annular apertures is discussed because of its practical applications in astronomy. A new error source, referred to as the remaining error, is formulated theoretically and evaluated numerically. The total reconstruction error is found to be the sum of the uncompensated wave-front residual error, the measurement error, and the remaining error. Numerical calculation shows that modal wave-front reconstruction with atmospheric Karhunen–Loeve functions results in a smaller residual error than with Zernike polynomials.
118 citations
"Comparative assessment of orthogona..." refers background in this paper
...A nonrecursive and fast method to determine the conversion matrix, M, in an arbitrary integrable domain rather than the classical Gram–Schmidt orthogonalization process has been discussed by Dai and Mahajan [14]....
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...As noted by Dai [16], the total wavefront reconstruction error by the modal method is the sum of the uncompensated wavefront residual error, the remaining error, and the measurement error....
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