Zernike polymomials for rectangular area and applications in off-axis asphere testing

He, Pinjiang; Zong, Xin; Fu, Ruimin

doi:10.1117/12.865486

Cited by 1 publication

(2 citation statements)

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“…It is necessary to construct a novel set of Zernike terms. According to reference [25], the original Zernike terms could project into a new defined region by Gram-Schmidt orthogonalization. The procedure could be expressed as…”

Section: 2 Simulation Discussion and Data Analysismentioning

confidence: 99%

“…In addition, it is worth noting that since the traditional Zernike polynomials are defined within the unit circle, it would lose the original favorable properties in a non-circular region. Fortunately, He et al proposed an effective method to construct novel Zernike polynomials in non-circular regions, which extends their application [25]. However, none of the aforementioned studies have considered the performance of high-order Zernikes.…”

Section: Introductionmentioning

confidence: 99%

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An inline surface measurement method for membrane mirror fabrication using two-stage trained Zernike polynomials and elitist teaching–learning-based optimization

Liu

Chen

Yang

et al. 2016

Meas. Sci. Technol.

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The accuracy of surface measurement determines the manufacturing quality of membrane mirrors. Thus, an efficient and accurate measuring method is critical in membrane mirror fabrication. This paper formulates this measurement issue as a surface reconstruction problem and employs two-stage trained Zernike polynomials as an inline measuring tool to solve the optical surface measurement problem in the membrane mirror manufacturing process. First, all terms of the Zernike polynomial are generated and projected to a non-circular region as the candidate model pool. The training data are calculated according to the measured values of distance sensors and the geometrical relationship between the ideal surface and the installed sensors. Then the terms are selected by minimizing the cost function each time successively. To avoid the problem of ill-conditioned matrix inversion by the least squares method, the coefficient of each model term is achieved by modified elitist teaching–learning-based optimization. Subsequently, the measurement precision is further improved by a second stage of model refinement. Finally, every point on the membrane surface can be measured according to this model, providing more the subtle feedback information needed for the precise control of membrane mirror fabrication. Experimental results confirm that the proposed method is effective in a membrane mirror manufacturing system driven by negative pressure, and the measurement accuracy can achieve 15 µm.

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Section: 2 Simulation Discussion and Data Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%