Reduction of mid-spatial frequency errors on aspheric and freeform optics by circular-random path polishing
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Pseudo-random paths are a useful tool to reduce mid-spatial frequency errors created in the processing of optical surfaces by sub-aperture polishing tools. Several types of patterns have been proposed, including hexagonal, square and circular, but prior literature has largely focused on flat and gently curved surfaces. Here, an extension of the circular-random path to strongly curved aspheric and freeform surfaces is proposed. The main feature of the algorithm is to cover the entire surface to be polished with a uniformly distributed tool path. Aspheric condenser lenses are then polished with a regular raster and circular-random path. Analysis of the optical performance shows that the random path can reduce the amplitude of mid-spatial frequency errors and relative intensity of satellite images. These features are particularly desirable in short wavelength applications, such as mirrors for EUV and X-ray.
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The general approach to minimizing MSF errors in the final workpiece during fabrication is to develop processes that minimize the creation of MSF errors during surface figure correction and/or to reduce MSF errors post surface figure correction using a smoothing polishing process.
“The general approach to minimizing MRF errors in the final workpiece during fabrication is to: 1) develop processes that minimize the creation of MSF errors during surface figure correction (such as optimized rastering patterns[12-14], tool size selection, and tool process parameters[15, 16]); and/or 2) reduce MSF errors post surface figure correction using a smoothing polishing process[17-20].”
From Smoothing tool design and performance during subaperture glass polishing (2023) · cites this paper as [14]
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Research works printed but not linked: 17
- [1]1. W. Zhu and A. Beaucamp, “Compliant grinding and polishing: A review,” Int. J. Mach. Tools Manuf. 158, 103634 (2020). Crossref
- [2]2. R. A. Jones, “Optimization of computer controlled polishing,” Appl. Opt. 16(1), 218–224 (1977). Crossref
- [3]3. Y. Han, W. Zhu, L. Zhang, and A. Beaucamp, “Region adaptive scheduling for time-dependent processes with optimal use of machine dynamics,” Int. J. Mach. Tools Manuf. 156, 103589 (2020). Crossref
- [4]4. Y. Han, F. Duan, W. Zhu, L. Zhang, and A. Beaucamp, “Analytical and stochastic modeling of surface topography in time-dependent sub-aperture processing,” Int. J. Mech. Sci. 175, 105575 (2020). Crossref
- [5]5. C. R. Dunn and D. D. Walker, “Pseudo-random tool paths for CNC sub-aperture polishing and other applications,” Opt. Express 16(23), 18942–18949 (2008). Crossref
- [6]6. C. Wang, Z. Wang, and Q. Xu, “Unicursal random maze tool path for computer-controlled optical surfacing,” Appl. Opt. 54(34), 10128–10136 (2015). Crossref
- [7]7. K. Takizawa and A. Beaucamp, “Comparison of tool feed influence in CNC polishing between a novel circular-random path and other pseudo-random paths,” Opt. Express 25(19), 22411–22424 (2017). Crossref
- [8]8. Q. Zhao, L. Zhang, and C. Fan, “Six-directional pseudorandom consecutive unicursal polishing path for suppressing mid-spatial frequency error and realizing consecutive uniform coverage,” Appl. Opt. 58(31), 8529–8541 (2019). Crossref
- [9]9. H. Li, D. Walker, X. Zheng, X. Su, L. Wu, C. Reynolds, G. Yu, T. Li, and P. Zhang, “Mid-spatial frequency removal on aluminum free-form mirror,” Opt. Express 27(18), 24885–24899 (2019). Crossref
- [10]10. Z. Dong and W. Nai, “Surface ripple suppression in subaperture polishing with fragment-type tool paths,” Appl. Opt. 57(19), 5523–5532 (2018). Crossref
- [11]11. Y. Han, L. Zhanga, M. Guoa, C. Fanb, and F. Lianga, “Tool paths generation strategy for polishing of freeform surface with physically uniform coverage,” Int J Adv Manuf Technol 95(5-8), 2125–2144 (2018). Crossref
- [12]12. J. Chaves-Jacob, J.M. Linares, and M. Sprauel, “Improving tool wear and surface covering in polishing via toolpath optimization,” J. Mater. Process. Technol. 213(10), 1661–1668 (2013). Crossref
- [13]13. H. Y. Tam, H. Cheng, and Z. Dong, “Peano-like paths for subaperture polishing of optical aspherical surfaces,” Appl. Opt. 52(15), 3624–3636 (2013). Crossref
- [14]14. Y. Li, X. Zhou, and Q. Liu, “Two-dimensional vibration actuated polishing of small surfaces by generating random-like Lissajous trajectories,” Appl. Opt. 60(4), 851–863 (2021). Crossref
- [16]16. C. Brecher, S. Lange, M. Merz, F. Niehaus, C. Wenzel, M. Winterschladen, and M. Weck, “NURBS based ultra-precision free-form machining,” CIRP Ann. 55(1), 547–550 (2006). Crossref
- [17]17. S. Wan, C. Wei, C. Hu, G. Situ, Y. Shao, and J. Shao, “Novel magic angle-step state and mechanism for restraining the path ripple of magnetorheological finishing,” Int. J. Mach. Tools Manuf. 161, 103673 (2021). Crossref Previous Article Next Article Cited By Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here. Alert me when this article is cited. Figures (12) Thumbnail for Fig. 1. Fig. 1. Thumbnail for Fig. 2. Fig. 2. Thumbnail for Fig. 3. Fig. 3. Thumbnail for Fig. 4. Fig. 4. Thumbnail for Fig. 5. Fig. 5. Thumbnail for Fig. 6. Fig. 6. Thumbnail for Fig. 7. Fig. 7. Thumbnail for Fig. 8. Fig. 8. Thumbnail for Fig. 9. Fig. 9. Thumbnail for Fig. 10. Fig. 10. Thumbnail for Fig. 11. Fig. 11. Thumbnail for Fig. 12. Fig. 12. Fig. 1. Element of circular-random path. View in Article | Download Full Size Fig. 2. Procedural generation of icosphere. View in Article | Download Full Size Fig. 3. Tool path generation procedure for near-spherical surface. View in Article | Download Full Size Fig. 4. Tool path trimming procedure. View in Article | Download Full Size Fig. 5. Example of freeform surface expressed in the NURBS format. View in Article | Download Full Size Fig. 6. Tool path generation procedure for NURBS surface. View in Article | Download Full Size Fig. 7. FJP footprint. View in Article | Download Full Size Fig. 8. Tool paths used in polishing simulations. View in Article | Download Full Size Fig. 9. Surface waviness induced on surface, normalized to 1 µm of material removal depth. View in Article | Download Full Size Fig. 10. 2D Fourier transform and 1D power spectral density of simulated surface waviness. View in Article | Download Full Size Fig. 11. Experimental equipment used for polishing the lens and testing its performance. View in Article | Download Full Size Fig. 12. Normalized intensity of light collected by the aspheric condensers in the focal plane, with associated diffraction satellites order. View in Article | Download Full Size Tables (2) Tables Icon Table 1. Parameters of FJP footprint. View Table | View all tables in this article Tables Icon Table 2. Fraction of total light intensity in regions of interest. View Table | View all tables in this article Equations (2) Equations on this page are rendered with MathJax. Learn more.
- [1](1) (2) Optics Express Siddharth Ramachandran, Editor-in-Chief Journal Home About Issues in Progress Current Issue All Issues Feature Issues Table 1. Parameters of FJP footprint. Abrasives 1.5µm CeO 2 (20g/L) Jet pressure 0.6 MPa Nozzle diameter 1.0 mm Nozzle distance 2 mm Table 2. Fraction of total light intensity in regions of interest. Raster polished lens Random polished lens Central spot 53.9% 51.0% 1 st order satellites 8.9% 1.3% 2 nd order satellites 1.4% 0.7% 3 rd order satellites <0.1% <0.1% 4 th order satellites 0.6% <0.1% Tables (2) Table 1. Parameters of FJP footprint. Abrasives 1.5µm CeO 2 (20g/L) Jet pressure 0.6 MPa Nozzle diameter 1.0 mm Nozzle distance 2 mm Table 2. Fraction of total light intensity in regions of interest. Raster polished lens Random polished lens Central spot 53.9% 51.0% 1 st order satellites 8.9% 1.3% 2 nd order satellites 1.4% 0.7% 3 rd order satellites <0.1% <0.1% 4 th order satellites 0.6% <0.1% Confirm Citation Alert Please login to set citation alerts. MathJax Help Equations displayed with MathJax. Right click equation to reveal menu options. Field Error Select as filters Select Topics Cancel Publishing Home Journals Conferences Preprints (Optica Open) Information for Authors Reviewers Librarians Open Access Information Open Access Statement and Policy Terms for Journal Article Reuse Other Resources Optica AI Optica Open Optica Publishing Group Bookshelf Optics ImageBank Optics & Photonics News Spotlight on Optics Optica Home Accessibility Information OPG Accessibility Statement About About Optica Publishing Group About My Account Sign up for Alerts Contact Us Send Us Feedback My Favorites Go to My Account Login to access favorites Recent Pages Reduction of mid-spatial frequency errors on aspheric and freeform optics by circular-random pa... Clear History © Copyright 2026 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies. Privacy | Terms of Use Login or Create Account Sign in by: Please wait... Don't have an account? Create an Account. Institutional Login (Optica Publishing Group participates in eduGAIN) China CARSI Member Access China CAoS Member Access Privacy Policy Need help? Our website uses cookies In order to remember your preferences (“Functional”), to measure traffic to our site (“Analytics”), and to personalise content and ads and enable social media features (“Marketing & Social Media”), we use cookies and similar technologies. We also share information about your use of our website with our social media, analytics, and advertising partners, who may combine it with other information they have collected from your use of their services. You can change or withdraw your consent at any time. More information about the use of cookies on our website can be found in our privacy policy. Opg.optica.org Functional Cookies and similar technologies that are strictly necessary for the website to work, such as remembering your preferences, security, enabling log-in or processing forms. Consent Studio Consent Studio consent-studio__consent-id ID of the proof of consent storage within the Consent Studio platform.
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Smoothing tool design and performance during subaperture glass polishing
Suratwala, T., Tham, G., Steele, R., Wong, L., Menapace, J., Ray, N., & Bauman, B. (2023) · Applied Optics · doi:10.1364/ao.482739 · Cited by 11 · see citing works
- Authors
- Beaucamp, A., Takizawa, K., Han, Y., & Zhu, W.
- Affiliations
- Kyoto University (JP)
- Kyoto University
- Venue
- Optics Express
- Publisher
- Optica Publishing Group
- Published
- 2021-08-31
- Volume
- 29
- Issue
- 19
- Pages
- 29802
- ISSN
- 1094-4087 (ISSN-L)
- OpenAlex
- W3191961621
- Cited by
- 46
- References
- 0
- Citations by year
- 2026: 11 · 2025: 8 · 2024: 8 · 2023: 11 · 2022: 8
- FWCI
- 2.05
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Locations
- Optics Express (publishedVersion)
- PubMed (publishedVersion)
- Funders
- Japan Society for the Promotion of Science (20K04192)
- Topics
- Physical Sciences › Engineering › Biomedical Engineering › Advanced Surface Polishing Techniques
- Physical Sciences › Engineering › Computational Mechanics › Surface Roughness and Optical Measurements
- Physical Sciences › Engineering › Biomedical Engineering › Advanced optical system design
- Keywords
- mid-spatial frequency error, aspheric optics, freeform optics, sub-aperture polishing, satellite imagery, EUV mirrors, X-ray mirrors
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