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## Optics Express

• Editor: C. Martijn de Sterke
• Vol. 20, Iss. 27 — Dec. 17, 2012
• pp: 28341–28346

## Deflectometric method for surface shape reconstruction from its gradient

Antonin Miks and Jiri Novak  »View Author Affiliations

Optics Express, Vol. 20, Issue 27, pp. 28341-28346 (2012)
http://dx.doi.org/10.1364/OE.20.028341

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### Abstract

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.

## 1. Introduction

In this work we present a deflectometric approach to solving a 3D surface reconstruction problem, which is based on measurements of a surface gradient of optically smooth surfaces. It is shown that a description of this problem leads to a nonlinear partial differential equation, from which the surface shape can be reconstructed numerically. The reconstruction process is presented on an example of 3D aspheric shape reconstruction.

## 2. Differential equation for surface reconstruction

Consider an optical surface, mathematically described by the formula z=f(x,y). The principle of the gradient deflectometric method is shown in Fig. 1
Fig. 1 Principle of gradient deflectometric method
, where O is the origin of a chosen coordinate system and P(x,y,z) is a point at the surface.

Further, assume that the axis z is perpendicular to the plane ξ, which is located at the distance a from the plane xy. The ray described by its unit direction vector s, which is parallel to the z axis, intersects the plane ξ in the point C and the measured surface in the point P(x,y,z). The incidence angle ε is defined as an angle between the direction vector s and the unit vector n of the surface normal in the point P(x,y,z). After the reflection from the surface the ray intersects the plane ξ in the point Q, which is located in the distance t from the point C. The distance t = t(x,y) depends on the position of the point P(x,y,z) at the measured surface. The direction of the reflected ray is characterized by the unit vector s' and the angle ε' with respect to the unit surface normal vector n.

## 3. Example and simulation

As an example we present a numerical simulation of the described reconstruction method for the case of an aspheric surface.

We also performed an analysis of the influence of random errors in coordinates x,y using a computer simulation, where we considered the same uncertainty in both coordinates. The root-mean-square (RMS) of approximation errors for the given aspheric surface example is 0.1 μm for the uncertainty 3 μm, 0.4 μm for the uncertainty 10 μm, and 0.75 μm for the uncertainty 20 μm. These results prove the robustness of the proposed method.

## Acknowledgment

This work has been supported by the Ministry of Industry and Trade of the Czech Republic the grant FR−TI2/074.

 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. SPIE 6609, 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. SPIE 5457, 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. SPIE 5457, 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. Express 16(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. SPIE 3739, 283–290 (1999). [CrossRef] 14 I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 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. A 29(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).

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure
(220.4840) Optical design and fabrication : Testing

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: September 26, 2012
Revised Manuscript: November 29, 2012
Manuscript Accepted: November 29, 2012
Published: December 6, 2012

Citation
Antonin Miks and Jiri Novak, "Deflectometric method for surface shape reconstruction from its gradient," Opt. Express 20, 28341-28346 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28341

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### References

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).

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