Abstract

The influence of tilted surfaces on the measurement of shape by spatial coherence profilometry is investigated. Based on theoretical analysis and experimental results, the systematic measurement error caused by surface tilt is determined. The systematic measurement error depends not only on the tilt angle but also on the parameters of the experimental setup. The theoretical analysis and the experiments show the similarities and differences between spatial coherence profilometry and white-light inter ferometry. We also suggest the conditions to obtain correct measurements by use of spatial coherence profilometry.

© 2009 Optical Society of America

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References

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  1. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107-4111(2000).
    [CrossRef]
  2. M. Takeda, J. Rosen, and Z. Duan, “Space-time analogy in synthetic coherence functions applied to optical tomography and profilometry,” in Proceedings of the International Conference on Laser Applications and Optical Metrology (ICLAOM-03), C. Shakher and D. S. Mehta, eds. (Anamaya Publisher, 2003), pp. 1-8.
    [PubMed]
  3. W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence function by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt. 41, 1962-1971 (2002).
    [CrossRef] [PubMed]
  4. M. Gokhler, Z. Duan, J. Rosen, and M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830-836 (2003).
    [CrossRef]
  5. M. Gokhler and J. Rosen, “Synthesis of a multipeak spatial degree of coherence for imaging through absorbing media,” Appl. Opt. 44, 2921-2927 (2005).
    [CrossRef] [PubMed]
  6. M. Gokhler and J. Rosen, “General configuration for using the longitudinal spatial coherence,” Opt. Commun. 252, 22-28(2005).
    [CrossRef]
  7. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free absolute interferometry based on angular spectrum scanning,” Opt. Express 14, 655-663 (2006).
    [CrossRef] [PubMed]
  8. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14, 12109-12121 (2006).
    [CrossRef] [PubMed]
  9. V. Ryabukho, D. Lyakin, and M. Lobachev, “Influence of longitudinal spatial coherence on the signal of a scanning interferometer,” Opt. Lett. 29, 667-669 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. V. Ryabukho, D. Lyakin, and M. Lobachev, “Longitudinal pure spatial coherence of a light field with wide frequency and angular spectra,” Opt. Lett. 30, 224-226 (2005).
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    [CrossRef]
  14. P. Pavliček and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766-770 (2004).
    [CrossRef] [PubMed]
  15. V. P. Ryabukho and D. V. Lyakin, “The effects of longitudinal spatial coherence of light in interference experiments,” Opt. Spectrosc. 98, 273-283 (2005).
    [CrossRef]
  16. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919-925 (1992).
    [CrossRef] [PubMed]
  17. R. Gross, P. Ettl, X. Laboureux, and C. Richter, “White-light interferometry in high precision industrial applications,” VDI-Ber. 1860, 123-130 (2004).
  18. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2003).
  19. I. S. Mc Lean, Electronic Imaging in Astronomy: Detectors and Instrumentation (Springer, 2008).
  20. P. Hlubina, “White-light spectral interferometry to measure the effective thickness of optical elements of known dispersion,” Acta Phys. Slov. 55, 387-393 (2005).

2006

2005

M. Gokhler and J. Rosen, “Synthesis of a multipeak spatial degree of coherence for imaging through absorbing media,” Appl. Opt. 44, 2921-2927 (2005).
[CrossRef] [PubMed]

M. Gokhler and J. Rosen, “General configuration for using the longitudinal spatial coherence,” Opt. Commun. 252, 22-28(2005).
[CrossRef]

V. Ryabukho, D. Lyakin, and M. Lobachev, “Longitudinal pure spatial coherence of a light field with wide frequency and angular spectra,” Opt. Lett. 30, 224-226 (2005).
[CrossRef] [PubMed]

V. P. Ryabukho and D. V. Lyakin, “The effects of longitudinal spatial coherence of light in interference experiments,” Opt. Spectrosc. 98, 273-283 (2005).
[CrossRef]

P. Hlubina, “White-light spectral interferometry to measure the effective thickness of optical elements of known dispersion,” Acta Phys. Slov. 55, 387-393 (2005).

2004

P. Pavliček and J. Soubusta, “Measurement of the influence of dispersion on white-light interferometry,” Appl. Opt. 43, 766-770 (2004).
[CrossRef] [PubMed]

R. Gross, P. Ettl, X. Laboureux, and C. Richter, “White-light interferometry in high precision industrial applications,” VDI-Ber. 1860, 123-130 (2004).

V. Ryabukho, D. Lyakin, and M. Lobachev, “Influence of longitudinal spatial coherence on the signal of a scanning interferometer,” Opt. Lett. 29, 667-669 (2004).
[CrossRef] [PubMed]

V. P. Ryabukho, D. V. Lyakin, and M. I. Lobachev, “Manifestation of longitudinal correlations in scattered coherent fields in an interference experiment,” Opt. Spectrosc. 97, 299-304(2004).
[CrossRef]

2003

M. Gokhler, Z. Duan, J. Rosen, and M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830-836 (2003).
[CrossRef]

2002

2001

2000

1992

1990

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2003).

Dresel, T.

Duan, Z.

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free absolute interferometry based on angular spectrum scanning,” Opt. Express 14, 655-663 (2006).
[CrossRef] [PubMed]

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express 14, 12109-12121 (2006).
[CrossRef] [PubMed]

M. Gokhler, Z. Duan, J. Rosen, and M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830-836 (2003).
[CrossRef]

M. Takeda, J. Rosen, and Z. Duan, “Space-time analogy in synthetic coherence functions applied to optical tomography and profilometry,” in Proceedings of the International Conference on Laser Applications and Optical Metrology (ICLAOM-03), C. Shakher and D. S. Mehta, eds. (Anamaya Publisher, 2003), pp. 1-8.
[PubMed]

Ettl, P.

R. Gross, P. Ettl, X. Laboureux, and C. Richter, “White-light interferometry in high precision industrial applications,” VDI-Ber. 1860, 123-130 (2004).

Gokhler, M.

M. Gokhler and J. Rosen, “Synthesis of a multipeak spatial degree of coherence for imaging through absorbing media,” Appl. Opt. 44, 2921-2927 (2005).
[CrossRef] [PubMed]

M. Gokhler and J. Rosen, “General configuration for using the longitudinal spatial coherence,” Opt. Commun. 252, 22-28(2005).
[CrossRef]

M. Gokhler, Z. Duan, J. Rosen, and M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830-836 (2003).
[CrossRef]

Gross, R.

R. Gross, P. Ettl, X. Laboureux, and C. Richter, “White-light interferometry in high precision industrial applications,” VDI-Ber. 1860, 123-130 (2004).

Häusler, G.

Hlubina, P.

P. Hlubina, “White-light spectral interferometry to measure the effective thickness of optical elements of known dispersion,” Acta Phys. Slov. 55, 387-393 (2005).

Kozaki, H.

Laboureux, X.

R. Gross, P. Ettl, X. Laboureux, and C. Richter, “White-light interferometry in high precision industrial applications,” VDI-Ber. 1860, 123-130 (2004).

Lee, B. S.

Lobachev, M.

Lobachev, M. I.

V. P. Ryabukho, D. V. Lyakin, and M. I. Lobachev, “Manifestation of longitudinal correlations in scattered coherent fields in an interference experiment,” Opt. Spectrosc. 97, 299-304(2004).
[CrossRef]

Lyakin, D.

Lyakin, D. V.

V. P. Ryabukho and D. V. Lyakin, “The effects of longitudinal spatial coherence of light in interference experiments,” Opt. Spectrosc. 98, 273-283 (2005).
[CrossRef]

V. P. Ryabukho, D. V. Lyakin, and M. I. Lobachev, “Manifestation of longitudinal correlations in scattered coherent fields in an interference experiment,” Opt. Spectrosc. 97, 299-304(2004).
[CrossRef]

Mc Lean, I. S.

I. S. Mc Lean, Electronic Imaging in Astronomy: Detectors and Instrumentation (Springer, 2008).

Miyamoto, Y.

Pavlicek, P.

Pförtner, A.

Richter, C.

R. Gross, P. Ettl, X. Laboureux, and C. Richter, “White-light interferometry in high precision industrial applications,” VDI-Ber. 1860, 123-130 (2004).

Rosen, J.

M. Gokhler and J. Rosen, “General configuration for using the longitudinal spatial coherence,” Opt. Commun. 252, 22-28(2005).
[CrossRef]

M. Gokhler and J. Rosen, “Synthesis of a multipeak spatial degree of coherence for imaging through absorbing media,” Appl. Opt. 44, 2921-2927 (2005).
[CrossRef] [PubMed]

M. Gokhler, Z. Duan, J. Rosen, and M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830-836 (2003).
[CrossRef]

W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence function by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt. 41, 1962-1971 (2002).
[CrossRef] [PubMed]

J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107-4111(2000).
[CrossRef]

M. Takeda, J. Rosen, and Z. Duan, “Space-time analogy in synthetic coherence functions applied to optical tomography and profilometry,” in Proceedings of the International Conference on Laser Applications and Optical Metrology (ICLAOM-03), C. Shakher and D. S. Mehta, eds. (Anamaya Publisher, 2003), pp. 1-8.
[PubMed]

Ryabukho, V.

Ryabukho, V. P.

V. P. Ryabukho and D. V. Lyakin, “The effects of longitudinal spatial coherence of light in interference experiments,” Opt. Spectrosc. 98, 273-283 (2005).
[CrossRef]

V. P. Ryabukho, D. V. Lyakin, and M. I. Lobachev, “Manifestation of longitudinal correlations in scattered coherent fields in an interference experiment,” Opt. Spectrosc. 97, 299-304(2004).
[CrossRef]

Schwider, J.

Soubusta, J.

Strand, T. C.

Takeda, M.

Venzke, H.

Wang, W.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2003).

Acta Phys. Slov.

P. Hlubina, “White-light spectral interferometry to measure the effective thickness of optical elements of known dispersion,” Acta Phys. Slov. 55, 387-393 (2005).

Appl. Opt.

Opt. Commun.

M. Gokhler and J. Rosen, “General configuration for using the longitudinal spatial coherence,” Opt. Commun. 252, 22-28(2005).
[CrossRef]

Opt. Eng.

M. Gokhler, Z. Duan, J. Rosen, and M. Takeda, “Spatial coherence radar applied for tilted surface profilometry,” Opt. Eng. 42, 830-836 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

V. P. Ryabukho, D. V. Lyakin, and M. I. Lobachev, “Manifestation of longitudinal correlations in scattered coherent fields in an interference experiment,” Opt. Spectrosc. 97, 299-304(2004).
[CrossRef]

V. P. Ryabukho and D. V. Lyakin, “The effects of longitudinal spatial coherence of light in interference experiments,” Opt. Spectrosc. 98, 273-283 (2005).
[CrossRef]

VDI-Ber.

R. Gross, P. Ettl, X. Laboureux, and C. Richter, “White-light interferometry in high precision industrial applications,” VDI-Ber. 1860, 123-130 (2004).

Other

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2003).

I. S. Mc Lean, Electronic Imaging in Astronomy: Detectors and Instrumentation (Springer, 2008).

M. Takeda, J. Rosen, and Z. Duan, “Space-time analogy in synthetic coherence functions applied to optical tomography and profilometry,” in Proceedings of the International Conference on Laser Applications and Optical Metrology (ICLAOM-03), C. Shakher and D. S. Mehta, eds. (Anamaya Publisher, 2003), pp. 1-8.
[PubMed]

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Figures (12)

Fig. 1
Fig. 1

Schematic of spatial coherence profilometry.

Fig. 2
Fig. 2

Schematic of spatial coherence profilometry used to measure the shape of tilted surfaces.

Fig. 3
Fig. 3

Numerically calculated envelope of the interferogram of spatial coherence profilometry for f = 160 mm , R = 12.5 mm , x A = 0 , d R = 100 μm ; (a)  α = 0 and (b)  α = 8 mrad .

Fig. 4
Fig. 4

Simulation of the height profile of a tilted surface measured by spatial coherence profilometry (solid line) compared with the actual position (dashed line). The simulation was performed for f = 160 mm , R = 12.5 mm , and α = 8 mrad .

Fig. 5
Fig. 5

Numerically calculated error δ as a function of distance d R for f = 160 mm , R = 7.5 mm (solid line), R = 12.5 mm (dashed line), and α = 4 and 8 mrad.

Fig. 6
Fig. 6

Schematic of the experimental setup used to measure the influence of the tilted surface.

Fig. 7
Fig. 7

Measured height profile of a tilted mirror for α = 8 mrad , R = 12.5 mm , f = 160 mm , and three values of d R .

Fig. 8
Fig. 8

Measured interferogram of spatial coherence profilometry for α = 8 mrad , R = 12.5 mm , f = 160 mm , and d R = 0 .

Fig. 9
Fig. 9

Systematic error δ caused by surface tilt as a function of distance d R for f = 160 mm and (a)  R = 12.5 mm and (b)  R = 7.5 mm . The symbols indicate the measured values; the dotted lines represent the numerical calculations.

Fig. 10
Fig. 10

Petzval surface and reference mirror.

Fig. 11
Fig. 11

Numerically calculated error δ as a function of lateral coordinate r for R C = 118 mm , d R A = 180 μm , R = 12.5 mm , and α = 8 mrad .

Fig. 12
Fig. 12

Height profile of the tilted surface ( α = 8 mrad ) measured with an achromatic lens in an imaging system. (a) Expected behavior for R = 12.5 mm and f = 160 mm , (b) measured by spatial coherence profilometry with R = 12.5 mm and f = 160 mm , (c) measured by white-light interferometry.

Tables (1)

Tables Icon

Table 1 Numerically Calculated Values of Coefficients A and B for f = 160 mm

Equations (18)

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D L 2 r O + 2 R f + z f ,
D I 2 r O + 2 R d 1 f ,
l R = S , A ¯ 2 f + z A + 2 d R x S x A + y S y A f x S 2 + y S 2 2 f 2 ( z A + 2 d R ) .
l O = S , A ¯ 2 f + z A + 2 d O + Δ z A x S x A + y S y A f x S Δ x A f x S 2 + y S 2 2 f 2 ( z A + 2 d O + Δ z A ) ,
Δ Φ = 2 π λ ( l O l R ) 2 π λ [ ( 2 d O 2 d R + Δ z A ) ( 1 x S 2 + y S 2 2 f 2 ) x S Δ x A f ] .
I ( A 0 ) = 0 2 π 0 R I 0 ( 1 + cos Δ Φ ) r S d r S d φ ,
I ( A 0 ) = 0 2 π 0 R I 0 ( 1 + cos { 2 π λ [ ( 2 d O 2 d R + Δ z A ) ( 1 r S 2 2 f 2 ) Δ x A r S cos φ f ] } ) r S d r S d φ .
I ( A 0 ) = 2 π I 0 0 R { 1 + cos [ 2 π λ ( 2 d O 2 d R + Δ z A ) ( 1 r S 2 2 f 2 ) ] J 0 ( 2 π Δ x A r S λ f ) } r S d r S ,
Δ x A = 2 ( d O + x A tan α ) cos α sin α ,
Δ z A = 2 ( x A d O tan α ) cos α sin α ,
I ( A 0 ) = π R 2 I 0 { 1 + sinc [ π λ R 2 f 2 ( d O d R ) ] cos [ 4 π λ ( d O d R ) ( 1 R 2 4 f 2 ) ] } .
FWHM 1.21 λ f 2 R 2 .
δ = d O M + x A tan α d R .
δ = d O M + x A tan α d R ( A 1 + B d R 2 1 ) d R ,
d R ( x , y ) = R C R C 2 x 2 y 2 + d R A ,
δ ( x , y ) ( A 1 + B ( R C R C 2 x 2 y 2 + d R A ) 2 1 ) ( R C R C 2 x 2 y 2 + d R A ) .
d O M = d R x A tan α + δ ,
Δ = d O M WLI d O M SCP = [ ( N 1 ) + ( 1 1 n ) ] D ( n 1 n ) D ,

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