Abstract

The accuracy and the measurement range of surface profilometry by wavelength scanning interferometry applied to diffusely reflecting surfaces are investigated. The influences of surface roughness and the imaging system in the interferometer are theoretically analyzed by derivation of the autocorrelation function of interferograms arising from wavelength scanning. By using a dye laser with a tuning range of 4.2 nm to a yield resolution of 39.1 μm, we have observed interferograms and their Fourier transforms and autocorrelations to study effects of defocusing and the size ratio of speckle to the CCD pixel for a plane diffuse object positioned normal to the incident beam.

© 1998 Optical Society of America

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References

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  1. M. Takeda, K. Mutoh, “Fourier transform profilometry for automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef] [PubMed]
  2. M. Haliouna, H. C. Ling, “Optical three-dimensional sensing by phase-measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
    [CrossRef]
  3. Y. Zou, Y. Diao, Y. Peng, H. Tiziani, “Geometry for contouring by electronic speckle pattern interferometry based on shifting illumination beams,” Appl. Opt. 31, 6616–6621 (1992).
    [CrossRef] [PubMed]
  4. R. R. Vera, D. Kerr, F. M. Santoyo, “Electronic speckle contouring,” J. Opt. Soc. Am. A 9, 2000–2008 (1992).
    [CrossRef]
  5. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  6. M. Takeda, H. Yamamoto, “Fourier-transform speckle profilometry: three-dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. 33, 7829–7837 (1994).
    [CrossRef] [PubMed]
  7. T. H. Barnes, T. Eiju, K. Matsuda, “Rough surface profile measurement using speckle optical frequency domain reflectometry with an external cavity tunable diode laser,” Optik 103, 93–100 (1996).
  8. S. Kuwamura, I. Yamaguchi, “Wavelength scanning profilometry for real-time surface shape measurement,” Appl. Opt. 37, 4473–4482 (1997).
    [CrossRef]
  9. I. Yamaguchi, “Fringe loci and visibility in holographic interferometry with diffuse objects,” Opt. Acta 25, 299–314 (1978).
    [CrossRef]
  10. I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1985), Vol. 22, pp. 271–340.
    [CrossRef]
  11. J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” Rept. Progr. Phys. 19, 188–240 (1956).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), p. 441.

1997 (1)

S. Kuwamura, I. Yamaguchi, “Wavelength scanning profilometry for real-time surface shape measurement,” Appl. Opt. 37, 4473–4482 (1997).
[CrossRef]

1996 (1)

T. H. Barnes, T. Eiju, K. Matsuda, “Rough surface profile measurement using speckle optical frequency domain reflectometry with an external cavity tunable diode laser,” Optik 103, 93–100 (1996).

1994 (1)

1992 (3)

1989 (1)

M. Haliouna, H. C. Ling, “Optical three-dimensional sensing by phase-measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

1983 (1)

1978 (1)

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry with diffuse objects,” Opt. Acta 25, 299–314 (1978).
[CrossRef]

1956 (1)

J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” Rept. Progr. Phys. 19, 188–240 (1956).
[CrossRef]

Barnes, T. H.

T. H. Barnes, T. Eiju, K. Matsuda, “Rough surface profile measurement using speckle optical frequency domain reflectometry with an external cavity tunable diode laser,” Optik 103, 93–100 (1996).

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), p. 441.

Diao, Y.

Dresel, T.

Eiju, T.

T. H. Barnes, T. Eiju, K. Matsuda, “Rough surface profile measurement using speckle optical frequency domain reflectometry with an external cavity tunable diode laser,” Optik 103, 93–100 (1996).

Haliouna, M.

M. Haliouna, H. C. Ling, “Optical three-dimensional sensing by phase-measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

Häusler, G.

Kerr, D.

Kuwamura, S.

S. Kuwamura, I. Yamaguchi, “Wavelength scanning profilometry for real-time surface shape measurement,” Appl. Opt. 37, 4473–4482 (1997).
[CrossRef]

Ling, H. C.

M. Haliouna, H. C. Ling, “Optical three-dimensional sensing by phase-measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

Matsuda, K.

T. H. Barnes, T. Eiju, K. Matsuda, “Rough surface profile measurement using speckle optical frequency domain reflectometry with an external cavity tunable diode laser,” Optik 103, 93–100 (1996).

Mutoh, K.

Peng, Y.

Ratcliff, J. A.

J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” Rept. Progr. Phys. 19, 188–240 (1956).
[CrossRef]

Santoyo, F. M.

Takeda, M.

Tiziani, H.

Venzke, H.

Vera, R. R.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), p. 441.

Yamaguchi, I.

S. Kuwamura, I. Yamaguchi, “Wavelength scanning profilometry for real-time surface shape measurement,” Appl. Opt. 37, 4473–4482 (1997).
[CrossRef]

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry with diffuse objects,” Opt. Acta 25, 299–314 (1978).
[CrossRef]

I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1985), Vol. 22, pp. 271–340.
[CrossRef]

Yamamoto, H.

Zou, Y.

Appl. Opt. (5)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

I. Yamaguchi, “Fringe loci and visibility in holographic interferometry with diffuse objects,” Opt. Acta 25, 299–314 (1978).
[CrossRef]

Opt. Lasers Eng. (1)

M. Haliouna, H. C. Ling, “Optical three-dimensional sensing by phase-measuring profilometry,” Opt. Lasers Eng. 11, 185–215 (1989).
[CrossRef]

Optik (1)

T. H. Barnes, T. Eiju, K. Matsuda, “Rough surface profile measurement using speckle optical frequency domain reflectometry with an external cavity tunable diode laser,” Optik 103, 93–100 (1996).

Rept. Progr. Phys. (1)

J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” Rept. Progr. Phys. 19, 188–240 (1956).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), p. 441.

I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1985), Vol. 22, pp. 271–340.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Principles and setup of surface profilometry by wavelength scanning interferometry.

Fig. 2
Fig. 2

Typical interference signal (a) from a diffuse surface and (b) its Fourier transform.

Fig. 3
Fig. 3

Coordinate system for theoretical analysis.

Fig. 4
Fig. 4

Three-dimensional plot of the coherence factor of the interference signal against wavelength shift and defocus.

Fig. 5
Fig. 5

Coherence factor for various amounts of defocus.

Fig. 6
Fig. 6

Examples of measured profiles from a plane diffuse object positioned at distances of 1.1 and 2.6 mm from the reference plane.

Fig. 7
Fig. 7

(a) Interference signals, (b) its Fourier transform, (c) autocorrelation obtained from the correct results (1) and (3) and the erroneous ones (2) and (4) in Fig. 6.

Fig. 8
Fig. 8

(a) Mean height and (b) standard deviation of the results from a plane object at various axial positions.

Fig. 9
Fig. 9

Dependencies of (a) the average height and (b) the standard deviation on the size ratio of speckle to the pixel.

Equations (25)

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R o x ,   k = ρ   exp i 2 kh x ,
U o x ,   z : k = I o k exp ikz R o x ,   k = I o k ρ   exp ik 2 h x + z ,
U X ; z : k =   K X ; x ,   z : k U o x ,   z : k d x ,
K X ; x ,   z : k = exp ik L o + z   P p exp ik   zp 2 2 F 1 2 × exp - ikp x F 1 + X F 2 d p ,
U R k = I R k exp ikL o .
I S X ; z : k = | U R k + U X ; z : k | 2 = | U R k | 2 + | U X ; z : k | 2 + 2   Re   U R k U * X ; z : k .
I S X ; z : k I S X ; z : k = I R k I R k + I R k | U X ; z : k | 2 + I R k | U X ; z : k | 2 + | U X ; z : k | 2 | U X ; z : k | 2 + | U * X ; z : k U X ; z : k | 2 + 2   Re   U R k U R * k U * X ; z : k U X ; z : k .
U X ; z : k U * X ; z : k = I o ρ 2 exp iz k - k ×   K X ; x ,   z : k K * X ; x ,   z : k exp i 2 kh x - k h x d x d x .
exp i 2 kh x - k h x = exp - 2 σ h 2 k 2 + k 2 × exp 2 σ h 2 kk ρ h x - x ,
σ h 2 = h x 2
ρ h x =   h x h x + x d x / σ h 2
exp i 2 kh x - k h x = exp - 2 σ h 2 k - k 2 × δ x - x .
U X ; z : k U * X ; z : k = I o ρ 2 exp i k - k z × exp - 2 σ h 2 k - k 2 ×   K X ; x ,   z : k K * X ; x ,   z : k d x = I o ρ 2 exp i k - k L o + 2 z × exp - 2 σ h 2 k - k 2 ×   P p P * kp k × exp ik   zp 2 2 F 1 2 1 - k k d p .
I S X ; z : k I S X ; z : k = I R k I R k + ρ 2 I R k I o k + I R k I o k + ρ 4 I o k I o k 1 + γ 2 + 2 γ ρ 2 I R k I R k I o k I o k 1 / 2 × cos 2 k - k z + α ,
γ   exp i α = exp [ - 2 σ h 2 k   -   k 2 ]     P p P * kp k × exp ik   zp 2 2 F 1 2 1 - k k d p     | P p | 2 d p .
Δ k = k o π / z .
δ z = π / max Δ k = λ 2 / 2 max Δ λ ,
max z = π / δ k = λ 2 / 2 δ λ ,
Δ k = Δ k S π / σ h .
P p = 1   for   | p | p 0 = 0   for   | p | > p 0 .
γ   exp i α = exp - 2 σ h 2 Δ k 2 0 p 0 1 - Δ k k × exp ik   zp 2 Δ k 2 F 1 2 1 - Δ k k d p / p o .
D = 8 N F 2 λ ,
γ   exp i α = exp - 2 σ h 2 Δ k 2 C ζ + iS ζ ζ ,
ζ = 2   z D Δ k k 1 - Δ k k - 1 1 / 2 = 2   z D k k - 1 1 / 2 .
D δ z = 8 N F 2 max Δ k k .

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