## Deflectometric method for surface shape reconstruction from its gradient |

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.

© 2012 OSA

## 1. Introduction

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]

## 2. Differential equation for surface reconstruction

*x*,

*y*,

*z*) is a point at the surface.

*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**.

**s**,

**s**' and

**n**as well as the points P, C and Q lie in one plane. It holds due to the reflection law [16,17

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]

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]

**s**of the incident ray is identical with the

*z*axis direction, thenIt holdsUsing the following relationship [18,19]we obtain from Eqs. (4) and (6)By substitution of previous formula into Eq. (4) we haveEquation (11) can be also written in the formBy squaring the previous equation we obtainAfter simplification of Eq. (13) we can write the final equationwhere we denotedThe solution of Eq. (14) has the following formEquation (16) can be generally written in the formwhereThe nonlinear partial differential equation of the first order (17) is a general solution of the problem of the reconstruction of the surface

## 3. Example and simulation

*t*. The ray with the direction vector

**s**propagates through the semitransparent mirror

*M*and reflects at the measured surface

**s′**, is reflected by the mirror

*M*and intersects the plane of the two dimensional array CCD sensor in the point Q. The position of the point Q is evaluated and the distance

*t*of the point Q from the center of the sensor C can be determined (see Fig. 1). During the measurement the surface is scanned and the distance

*t*(

*x,y*) is measured for different scanned points at the surface. Consider now the case of the shape reconstruction of a rotationally symmetric aspheric surface, given by the following formula [5]where

*R*is the vertex radius, and

^{−13}mm and the proposed mathematical technique for the solution of Eq. (17) is working very well. It is evident from the previous example that a very good surface shape approximation can be found from an initial optimization point (approximation coefficients), which is really different from the optimal solution. We performed the analysis on several examples of spherical and aspheric surfaces with similar results, which prove the robustness of the described method. The proposed measurement method and the approach to find the solution of the partial differential Eq. (17), which describes theoretically the problem of surface reconstruction, give very good results and can be principally used for measurements of flat, spherical and aspheric surfaces.

*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.

## 4. Conclusion

## Acknowledgment

## References and links

1. | J. A. Bosch, |

2. | D. J. Whitehouse, |

3. | T. Yoshizawa, |

4. | F. M. Santoyo, |

5. | D. Malacara, |

6. | J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photon. |

7. | R. Leach, |

8. | A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE |

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 |

10. | M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE |

11. | Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express |

12. | M. Rosete-Aguilar and R. Díaz-Uribe, “Profile testing of spherical surfaces by laser deflectometry,” Appl. Opt. |

13. | W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE |

14. | I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE |

15. | M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ. |

16. | M. Herzberger, |

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 |

18. | K. Rektorys, |

19. | G. A. Korn and T. M. Korn, |

20. | E.Kamke, |

21. | L. E. Scales, |

22. | M. Aoki, |

**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

- 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).
- J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photon.3(2), 128–160 (2011). [CrossRef]
- R. Leach, Optical Measurement of Surface Topography (Springer, 2011).
- 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]
- 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]
- M. Rosete-Aguilar and R. Díaz-Uribe, “Profile testing of spherical surfaces by laser deflectometry,” Appl. Opt.32(25), 4690–4697 (1993). [CrossRef] [PubMed]
- 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]
- 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).
- M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
- 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]
- 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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