In this work we provide a theoretical analysis of gradient deflectometric method for 3D topography measurements of optically smooth surfaces. It is shown that the surface reconstruction problem leads to a nonlinear partial differential equation. A shape of a surface can be calculated by solution of a derived equation. An advantage of the presented method is a noncontact character and no need for a reference surface.

© 2012 OSA

1. J. A. Bosch, Coordinate Measuring Machines and Systems (CRC Press, 1995).
2. D. J. Whitehouse, Handbook of Surface and Nanometrology (Institute of Physics Publishing, 2003).
3. T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC Press, 2009).
4. F. M. Santoyo, Handbook of Optical Metrology (CRC Press, 2008).
5. D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).
6. J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photon.3(2), 128–160 (2011). [CrossRef]
7. R. Leach, Optical Measurement of Surface Topography (Springer, 2011).
8. A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE6609, 66090U (2007).
9. T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004). [CrossRef]
10. M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE5457, 366–376 (2004). [CrossRef]
11. Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express16(19), 15090–15096 (2008). [CrossRef] [PubMed]
12. M. Rosete-Aguilar and R. Díaz-Uribe, “Profile testing of spherical surfaces by laser deflectometry,” Appl. Opt.32(25), 4690–4697 (1993). [CrossRef] [PubMed]
13. W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE3739, 283–290 (1999). [CrossRef]
14. I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE3782, 306–317 (1999). [CrossRef]
15. M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ.5, 10026 (2010).
16. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
17. A. Mikš and P. Novák, “Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays: comment,” J. Opt. Soc. Am. A29(7), 1356–1357, discussion 1358 (2012). [CrossRef] [PubMed]
18. K. Rektorys, Survey of Applicable Mathematics (M.I.T. Press, 1969).
19. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Courier Dover Publications, 2000).
20. E.Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen (B.G.Teubner GmbH, 1977).
21. L. E. Scales, Introduction to Non-linear Optimization (Springer-Verlag, 1985).
22. M. Aoki, Introduction to Optimization Techniques: Fundamentals and Applications of Nonlinear Programming (Maxmillian, 1971).

Adv. Opt. Photon.

• J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photon.3(2), 128–160 (2011). [CrossRef]

Appl. Opt.

• M. Rosete-Aguilar and R. Díaz-Uribe, “Profile testing of spherical surfaces by laser deflectometry,” Appl. Opt.32(25), 4690–4697 (1993). [CrossRef] [PubMed]

J. Eur. Opt. Soc. Rap. Publ.

• M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ.5, 10026 (2010).

J. Opt. Soc. Am. A

• A. Mikš and P. Novák, “Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays: comment,” J. Opt. Soc. Am. A29(7), 1356–1357, discussion 1358 (2012). [CrossRef] [PubMed]

Opt. Express

• Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express16(19), 15090–15096 (2008). [CrossRef] [PubMed]

Proc. SPIE

• W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE3739, 283–290 (1999). [CrossRef]
• I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE3782, 306–317 (1999). [CrossRef]
• A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE6609, 66090U (2007).
• T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004). [CrossRef]
• M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE5457, 366–376 (2004). [CrossRef]

Other

• R. Leach, Optical Measurement of Surface Topography (Springer, 2011).
• J. A. Bosch, Coordinate Measuring Machines and Systems (CRC Press, 1995).
• D. J. Whitehouse, Handbook of Surface and Nanometrology (Institute of Physics Publishing, 2003).
• T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC Press, 2009).
• F. M. Santoyo, Handbook of Optical Metrology (CRC Press, 2008).
• D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).
• M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
• K. Rektorys, Survey of Applicable Mathematics (M.I.T. Press, 1969).
• G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Courier Dover Publications, 2000).
• E.Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen (B.G.Teubner GmbH, 1977).
• L. E. Scales, Introduction to Non-linear Optimization (Springer-Verlag, 1985).
• M. Aoki, Introduction to Optimization Techniques: Fundamentals and Applications of Nonlinear Programming (Maxmillian, 1971).

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