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Mid-spatial frequency structure on an optical surface induces small-angle scatter in the transmitted wavefront. Freeform
surfaces are particularly susceptible to mid-spatial frequency errors due to the sub-aperture nature of the fabrication
processes. Several surface metrology methods that work for freeform surfaces use an indirect principle, recon...
Contexts in source publication
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... single number evaluation is useful for assessing the approximate length-scale of the spurious MSF structure studied, and for assessing trends and sensitivities to input parameters like noise in the slope measurements and the integration algorithm used. Figure 2 shows a simulation example of surface reconstruction from the noisy slope data integrated by the NSA. Here, the underlying shape is assumed to be a flat surface. ...
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... the noise-free slope data across the whole aperture is a constant. The noise is taken to be pixel-independent random noise and modeled with a Gaussian probability distribution with zero mean and 10 -5 rad standard deviation, shown in Figure 2a. We then use the NSA to reconstruct the surface height map from the noisy slope data, followed by subtraction of the nominal flat surface shape, resulting in the error map shown in Figure 2b. ...
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... noise is taken to be pixel-independent random noise and modeled with a Gaussian probability distribution with zero mean and 10 -5 rad standard deviation, shown in Figure 2a. We then use the NSA to reconstruct the surface height map from the noisy slope data, followed by subtraction of the nominal flat surface shape, resulting in the error map shown in Figure 2b. We see that, while the input slope noise is pixel-independent and random, the surface error map shows a clustered, large length-scale spatial pattern that is significantly larger than even the integration kernel of the NSA algorithm. ...
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... uncertainty of the correlation length, 0.005 mm, is taken to be the standard deviation of the 900 correlation lengths, normalized by the square root of 900. The examples shown in Figures 2 and 3 are for the case of slope noise residing on the slope values for a planar surface. We also used the same slope noise data to add to a spherical surface whose radius is 80mm. ...
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... can understand this by considering the finite size nature of the data maps and slope values. Our input slope map shown in Figure 2a contains 512 × 512 pixel samples. The random values of noise on each will not be exactly zero on average for this finite set, but rather converges to zero only as the number of data points becomes infinite. ...
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... a finite sample of noise, the average is not zero, meaning there is a non-zero piston term in the slope map. Piston in a gradient dataset integrates to tilt in the height map, an example of which can be seen as the background tilt in Figure 2b. This idea can be extended to consideration of a sub-aperture in the data set. ...
Citations
... The measurement noise in the derivatives is a serious problem for surface reconstruction (Freischlad, 1992;Legarda-Sáenz et al., 2000;Elster et al., 2002;Karaçali and Karaçali, 2004;Lowitzsch et al., 2005;Petz and Tutsch, 2005;Bonfort, 2006;Legarda-Sáenz, 2007;Kolhe and Agrawal, 2009;Fischer et al., 2011;Li et al., 2012a;Höfer et al., 2013a;Patel and Chellappa, 2013;Dong et al., 2014;Pak, 2014;DeMars et al., 2019) due to the low sensitivity of measured slopes to absolute surface height (Falconi, 1964). Today, only scanning techniques allow uncertainties commensurate with the tolerances of demanding optical surfaces (e.g., synchrotron mirrors) (Weingaertner et al., 2001;Elster and Weingärtner, 2002;Lammert et al., 2004;Xiao et al., 2011). ...
... The state of the art sensitivity in DM is currently at the level of a few nm for small-scale defects (Kugimiya, 1988;Hahn et al., 1990;Beyerer and Pérard, 1997;Bothe et al., 2004;Jüptner and Bothe, 2009;Burke, 2019), several tens of nm for mid-frequency errors (Su et al., 2012a;Dong et al., 2014;Su, 2014;Coniglio et al., 2021), and about 100-200 nm for flat or low-curvature surfaces (Graves et al., 2007;Ettl et al., 2008;Sandner et al., 2011;Li et al., 2012a;Liu et al., 2012;Hofbauer et al., 2013;Olesch and Häusler, 2014;Ren et al., 2015;Li et al., 2017), with smaller parts yielding even better results (Su et al., 2012b;Huang R. et al., 2013;Su et al., 2013c;Bergmann et al., 2015). As mentioned above, for best results one may replace lenses with an actual pinhole aperture . ...
Deflectometry as a technique to assess reflective surfaces has now existed for some 40 years. Its different aspects and variations have been studied in multiple theses and research articles; reviews are available for certain subtopics. Still a field of active development with many unsolved problems, deflectometry now encompasses a large variety of application domains, hardware setup types, and processing workflows for different purposes, and spans a range from qualitative defect inspection of large vehicles to precision measurements of microscopic optics. Over these years, many exciting developments have accumulated in the underlying theory, in the systems design, and in the implementation specifics. This diversity of topics is difficult to grasp for experts and non-experts alike and may present an obstacle to a wider acceptance of deflectometry as a useful tool for research and industrial applications. This paper presents an attempt to summarize the status of deflectometry and to map relations between its notable branches. Its aim is to provide a communication basis for experienced practitioners and also to offer a convenient entry point for those interested in learning about the method. The list of references introduces some prominent trends and established research groups in order to facilitate further self-directed exploration.
... The measurement noise in the derivatives is a serious problem for surface reconstruction [245,274,279,284,308,335,356,[362][363][364][365][366][367][368][369][370] due to the low sensitivity of measured slopes to absolute surface height [45]. Today, only scanning techniques allow uncertainties commensurate with the tolerances of demanding optical surfaces (e.g., synchrotron mirrors) [41,[371][372][373]. ...
... The state of the art sensitivity in DM is currently at the level of a few nm for small-scale defects [121,123,190,[374][375][376], several tens of nm for mid-frequency errors [28,48,64,369], and about 100 nm to 200 nm for flat or low-curvature surfaces [30,264,309,321,335,[377][378][379][380], with smaller parts yielding even better results [29,217,381,382]. As mentioned above, for best results one may replace lenses with an actual pinhole aperture [217]. ...
Deflectometry as a technical approach to assessing reflective surfaces has now existed for almost 40 years. Different aspects and variations of the method have been studied in multiple theses and research articles, and reviews are also becoming available for certain subtopics. Still a field of active development with many unsolved problems, deflectometry now encompasses a large variety of application domains, hardware setup types, and processing workflows designed for different purposes, and spans a range from qualitative defect inspection of large vehicles to precision measurements of microscopic optics. Over these years, many exciting developments have accumulated in the underlying theory, in the systems design, and in the implementation specifics. This diversity of topics is difficult to grasp for experts and non-experts alike and may present an obstacle to a wider acceptance of deflectometry as a useful tool in other research fields and in the industry. This paper presents an attempt to summarize the status of deflectometry, and to map relations between its notable "spin-off" branches. The intention of the paper is to provide a common communication basis for practitioners and at the same time to offer a convenient entry point for those interested in learning and using the method. The list of references is extensive but definitely not exhaustive, introducing some prominent trends and established research groups in order to facilitate further self-directed exploration by the reader.
... However, the fabrication techniques involved leave tooling marks on the surface because of the sub-aperture nature of the process and the tooling marks are difficult to remove. Mid-spatial frequency error leads to small angle scatter in the transmitted optical beam [2] and is a concern in many applications [3]. ...
In this manuscript we investigate a Zernike polynomial representation for quantifying the mid-spatial frequency (MSF) content of surfaces and how this representation captures certain characteristics of the MSF. Filtering aspect of these polynomials is also explored.
