Modal wavefront reconstruction from slope measurements for rectangular apertures

Li, Mengyang; Li, Dahai; Zhang, Chen; Kewei, E; Wang, Qiong‐Hua; Chen, Haiping

doi:10.1364/josaa.32.001916

Cited by 21 publications

(9 citation statements)

References 22 publications

(29 reference statements)

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“…In principle, this data set could have been reconstructed with other basis sets, such as a Zernike-basis vector set, orthogonalized over a rectangular pupil [34]. However, that does not have a straightforward closed-form relationship which makes it difficult to generate such extreme orders.…”

Section: Fidelity Check Of Mid-to-high Spatial Frequency Reconstructionmentioning

confidence: 99%

Modal Data Processing for High Resolution Deflectometry

Aftab

Burge

Smith

et al. 2019

Int. J. of Precis. Eng. and Manuf.-Green Tech.

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In this paper, we present a modal data processing methodology, for reconstructing high resolution surfaces from measured slope data, over rectangular apertures. One of the primary goals is the ability to effectively reconstruct deflectometry measurement data for high resolution and freeform surfaces, such as telescope mirrors. We start by developing a gradient polynomial basis set which can quickly generate a very high number of polynomial terms. This vector basis set, called the G polynomials set, is based on gradients of the Chebyshev polynomials of the first kind. The proposed polynomials represent vector fields that are defined as the gradients of scalar functions. This method yields reconstructions that fit the measured data more closely than those obtained using conventional methods, especially in the presence of defects in the mirror surface and physical blockers/markers such as fiducials used during deflectometry measurements. We demonstrate the strengths of our method using simulations and real metrology data from the Daniel K. Inouye Solar Telescope (DKIST) primary mirror.

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Section: Fidelity Check Of Mid-to-high Spatial Frequency Reconstructionmentioning

confidence: 99%

Modal Data Processing for High Resolution Deflectometry

Aftab

Burge

Smith

et al. 2019

Int. J. of Precis. Eng. and Manuf.-Green Tech.

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“…The novelty of our method lies in the fact that it is analytical and matrix based. The motivation for the matrix approach is that, apparently, in a number of important papers in wavefront analysis [4,5,10,11,[13][14][15] it is possible to deal with wavefront analysis problems by manipulating the expansion coefficients in the form of matrices and acquire closed-form results similar to those already published in the literature. The basis of virtually all wavefront work is that the phase can be represented as a linear combination of a discrete infinite series, which can be expressed in vector form whose dimension can take any required size.…”

Section: Discussionmentioning

confidence: 99%

“…The above literature has dealt with the characterization of wavefront slope including the use of various polynomial systems. To our knowledge, the only study that has dealt with the wavefront slope in noncircular pupils was in [15]. It is our assertion that a procedure can be found to derive vector polynomials orthonormal in any pupil.…”

Section: Introductionmentioning

confidence: 96%

“…Orthonormal Zernike-based (OZ) polynomials have been applied to pupils other than circular pupils [10][11][12][13]. Recent efforts to deal with orthonormality in wavefront slope include the work of Ye et al, who devised a matrix-based numerical orthogonal transformation method for generating numerical orthogonal gradient polynomials [14], and that of Li et al, who derived orthonormal vector polynomials orthogonal in a square pupil of unit half-length [15].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Orthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomials

Mafusire

Krüger

2018

J. Opt. Soc. Am. A

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The concept of orthonormal vector circle polynomials is revisited by deriving a set from the Cartesian gradient of Zernike polynomials in a unit circle using a matrix-based approach. The heart of this model is a closed-form matrix equation of the gradient of Zernike circle polynomials expressed as a linear combination of lower-order Zernike circle polynomials related through a gradient matrix. This is a sparse matrix whose elements are two-dimensional standard basis transverse Euclidean vectors. Using the outer product form of the Cholesky decomposition, the gradient matrix is used to calculate a new matrix, which we used to express the Cartesian gradient of the Zernike circle polynomials as a linear combination of orthonormal vector circle polynomials. Since this new matrix is singular, the orthonormal vector polynomials are recovered by reducing the matrix to its row echelon form using the Gauss-Jordan elimination method. We extend the model to derive orthonormal vector general polynomials, which are orthonormal in a general pupil by performing a similarity transformation on the gradient matrix to give its equivalent in the general pupil. The outer form of the Gram-Schmidt procedure and the Gauss-Jordan elimination method are then applied to the general pupil to generate the orthonormal vector general polynomials from the gradient of the orthonormal Zernike-based polynomials. The performance of the model is demonstrated with a simulated wavefront in a square pupil inscribed in a unit circle.

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“…This is an exciting prospect as many vector polynomial sets, derived from orthogonal scalar polynomials, do not hold orthogonality over the specified aperture and must be orthogonalized, using a procedure such as the Gram-Schmidt orthogonalization process [17]. For example, this is the case for Zernike based rectangular gradient polynomials [18]. Although these polynomials have advantages for aberration balancing, we do not require the representation of balanced aberrations, as explained in Section 1.…”

Section: Modal Basis For Surface / Wavefront Reconstructionmentioning

confidence: 99%

Chebyshev gradient polynomials for high resolution surface and wavefront reconstruction

Aftab

Burge

Smith

et al. 2018

Optical Manufacturing and Testing XII

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A new data processing method based on orthonormal rectangular gradient polynomials is introduced in this work. This methodology is capable of effectively reconstructing surfaces or wavefronts with data obtained from deflectometry systems, especially during fabrication and metrology of high resolution and freeform surfaces. First, we derived a complete and computationally efficient vector polynomial set, called G polynomials. These polynomials are obtained from gradients of Chebyshev polynomials of the first kinda basis set with many qualities that are useful for modal fitting. In our approach both the scalar and vector polynomials, that are defined and manipulated easily, have a straightforward relationship due to which the polynomial coefficients of both sets are the same. This makes conversion between the two sets highly convenient. Another powerful attribute of this technique is the ability to quickly generate a very large number of polynomial terms, with high numerical efficiency. Since tens of thousands of polynomials can be generated, mid-to-high spatial frequencies of surfaces can be reconstructed from high-resolution metrology data. We will establish the strengths of our approach with examples involving simulations as well as real metrology data from the Daniel K. Inouye Solar Telescope (DKIST) primary mirror.

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Modal wavefront reconstruction from slope measurements for rectangular apertures

Cited by 21 publications

References 22 publications

Modal Data Processing for High Resolution Deflectometry

Modal Data Processing for High Resolution Deflectometry

Orthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomials

Chebyshev gradient polynomials for high resolution surface and wavefront reconstruction

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