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 Optical Society of America

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References

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  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).

2012 (1)

2011 (1)

2010 (1)

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).

2008 (1)

2007 (1)

A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE 6609, 66090U (2007).

2004 (2)

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]

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]

1999 (2)

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE 3739, 283–290 (1999).
[Crossref]

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306–317 (1999).
[Crossref]

1993 (1)

Bäumer, S. M.

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE 3739, 283–290 (1999).
[Crossref]

Bothe, T.

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]

Díaz-Uribe, R.

Ehret, G.

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).

Elster, C.

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306–317 (1999).
[Crossref]

Fitzenreiter, A.

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).

Geng, J.

Häusler, G.

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]

Horijon, J. L.

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE 3739, 283–290 (1999).
[Crossref]

Jing, H.

Juptner, W.

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]

Kaminski, J.

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]

Knauer, M.

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]

Li, W. S.

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]

Liu, Y.

Mikš, A.

Novák, J.

A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE 6609, 66090U (2007).

Novák, P.

Rosete-Aguilar, M.

Schulz, M.

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).

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306–317 (1999).
[Crossref]

Su, X.

Tang, Y.

van Amstel, W. D.

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE 3739, 283–290 (1999).
[Crossref]

von Kopylow, C.

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]

Weingärtner, I.

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306–317 (1999).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

J. Eur. Opt. Soc. Rap. Publ. (1)

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 (1)

Opt. Express (1)

Proc. SPIE (5)

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE 3739, 283–290 (1999).
[Crossref]

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306–317 (1999).
[Crossref]

A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE 6609, 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. SPIE 5457, 411–422 (2004).
[Crossref]

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]

Other (12)

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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Figures (3)

Fig. 1
Fig. 1 Principle of gradient deflectometric method

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Fig. 2
Fig. 2 Principal scheme of measuring sensor.

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Fig. 3
Fig. 3 Approximation error of shape of reconstructed aspheric surface

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Equations (22)

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s ′ =s−2n(sn)
cosε=−sn,   ε ′ =ε.
n= s− s ′ 2[ 1−(s s ′ ) ] .
t PC ¯ = t a−z =tan2ε= 2tanε 1− tan 2 ε .
p= ∂z ∂x ,  q= ∂z ∂y .
n=[ − p N , − q N ,  1 N ],
s=(0,0,−1).
cosε=−sn=1/N.
tanε=± 1− cos 2 ε cosε =± p 2 + q 2 ,
tan2ε=± 2 p 2 + q 2 1− p 2 − q 2 .
K= t a−z =± 2 p 2 + q 2 1− p 2 − q 2 .
K(1− p 2 − q 2 )=±2 p 2 + q 2 .
(1− p 2 − q 2 ) 2 − 4 K 2 ( p 2 + q 2 )=0.
( p 2 + q 2 ) 2 −2b( p 2 + q 2 )+1=0,
b=(1+2/ K 2 ).
p 2 + q 2 =b± b 2 −1 .
( ∂z ∂x ) 2 + ( ∂z ∂y ) 2 =F(x,y,z),
F(x,y,z)=b± b 2 −1 =1+2/ K 2 ±2 1+1/ K 2 /K.
z=f(x,y)= ∑ m=0 M ∑ n=0 N c mn x m y n .
M= ( ∂z ∂x ) 2 + ( ∂z ∂y ) 2 −F(x,y,z).
Φ= ∑ i=1 I ∑ k=1 K [ M ik ( c 00 , c 10 , c 01 , c 20 , c 11 , c 02 ,.....) ]   2 .
z=f(x,y)= c( x 2 + y 2 ) 1+ 1− c 2 ( x 2 + y 2 ) + ∑ s=1 S A s+1 ( x 2 + y 2 ) s+1 ,

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