Abstract

We report on a method that can be used to improve the result of multiwavelength contouring in the case of objects with rough surface. It is based on the combined evaluation of multiple measurements with varying direction of illumination. While the individual measurements share the same systematics with respect to the shape of the investigated object, the noise arising from speckle decorrelation fluctuates statistically and hence can be reduced by means of averaging. For the case of three illumination directions we show that weighted averaging of the measured phase distributions enhances the signal-to-noise ratio by approximately 3 dB.

© 2012 Optical Society of America

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References

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  1. S. Seebacher, W. Osten, T. Baumbach, and W. Jüptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
    [CrossRef]
  2. I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
    [CrossRef]
  3. K. Haines and B. P. Hildebrand, “Contour generation by wavefront reconstruction,” Phys. Lett. 19, 10–11 (1965).
    [CrossRef]
  4. C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85 (2000).
    [CrossRef]
  5. N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. A 4, 201–212 (1974).
    [CrossRef]
  6. C. Wykes, “De-correlation effects in speckle-pattern interferometry. 1. wavelength change dependent de-correlation with application to contouring and surface roughness measurement,” J. Mod. Opt. 24, 517–532 (1977).
    [CrossRef]
  7. C. Towers, D. Towers, and J. Jones, “Absolute fringe order calculation using optimised multi-frequency selection in full-field profilometry,” Opt. Lasers Eng. 43, 788–800 (2005).
    [CrossRef]
  8. J. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and related Phenomena, J. Dainty, ed. (Springer, 1975), pp. 9–75.
  9. D. Tichenor and J. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. A 62, 293–295 (1972).
    [CrossRef]
  10. D. P. Kelly, J. T. Sheridan, and W. T. Rhodes, “Fundamental diffraction limitations in a paraxial 4-f imaging system with coherent and incoherent illumination,” J. Opt. Soc. Am. A 24, 1911–1919 (2007).
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  15. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
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  16. J. W. Goodman, Introduction To Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
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2007 (1)

2006 (1)

2005 (1)

C. Towers, D. Towers, and J. Jones, “Absolute fringe order calculation using optimised multi-frequency selection in full-field profilometry,” Opt. Lasers Eng. 43, 788–800 (2005).
[CrossRef]

2003 (1)

2001 (2)

S. Seebacher, W. Osten, T. Baumbach, and W. Jüptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
[CrossRef]

2000 (1)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

1999 (1)

1982 (1)

1977 (1)

C. Wykes, “De-correlation effects in speckle-pattern interferometry. 1. wavelength change dependent de-correlation with application to contouring and surface roughness measurement,” J. Mod. Opt. 24, 517–532 (1977).
[CrossRef]

1974 (1)

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. A 4, 201–212 (1974).
[CrossRef]

1972 (1)

D. Tichenor and J. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. A 62, 293–295 (1972).
[CrossRef]

1967 (1)

1965 (1)

K. Haines and B. P. Hildebrand, “Contour generation by wavefront reconstruction,” Phys. Lett. 19, 10–11 (1965).
[CrossRef]

Aspert, N.

Baumbach, T.

S. Seebacher, W. Osten, T. Baumbach, and W. Jüptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

Charrière, F.

Colomb, T.

Coppola, G.

Cuche, E.

Depeursinge, C.

Ferraro, P.

Finizio, A.

George, N.

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. A 4, 201–212 (1974).
[CrossRef]

Goodman, J.

D. Tichenor and J. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. A 62, 293–295 (1972).
[CrossRef]

J. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and related Phenomena, J. Dainty, ed. (Springer, 1975), pp. 9–75.

Goodman, J. W.

J. W. Goodman, Introduction To Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Grilli, S.

Haines, K.

K. Haines and B. P. Hildebrand, “Contour generation by wavefront reconstruction,” Phys. Lett. 19, 10–11 (1965).
[CrossRef]

Hildebrand, B. P.

K. Haines and B. P. Hildebrand, “Contour generation by wavefront reconstruction,” Phys. Lett. 19, 10–11 (1965).
[CrossRef]

Ina, H.

Jain, A.

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. A 4, 201–212 (1974).
[CrossRef]

Jones, J.

C. Towers, D. Towers, and J. Jones, “Absolute fringe order calculation using optimised multi-frequency selection in full-field profilometry,” Opt. Lasers Eng. 43, 788–800 (2005).
[CrossRef]

Jüptner, W.

S. Seebacher, W. Osten, T. Baumbach, and W. Jüptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

U. Schnars and W. Jüptner, Digital Holography (Springer, 2005).

Kato, J.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
[CrossRef]

Kelly, D. P.

Kobayashi, S.

Kühn, J.

Magro, C.

Marquet, P.

Nicola, S. D.

Ohta, S.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
[CrossRef]

Osten, W.

S. Seebacher, W. Osten, T. Baumbach, and W. Jüptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Pierattini, G.

Rhodes, W. T.

Schnars, U.

U. Schnars and W. Jüptner, Digital Holography (Springer, 2005).

Seebacher, S.

S. Seebacher, W. Osten, T. Baumbach, and W. Jüptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Sheridan, J. T.

Sherman, G. C.

Takeda, M.

Tichenor, D.

D. Tichenor and J. Goodman, “Coherent transfer function,” J. Opt. Soc. Am. A 62, 293–295 (1972).
[CrossRef]

Towers, C.

C. Towers, D. Towers, and J. Jones, “Absolute fringe order calculation using optimised multi-frequency selection in full-field profilometry,” Opt. Lasers Eng. 43, 788–800 (2005).
[CrossRef]

Towers, D.

C. Towers, D. Towers, and J. Jones, “Absolute fringe order calculation using optimised multi-frequency selection in full-field profilometry,” Opt. Lasers Eng. 43, 788–800 (2005).
[CrossRef]

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Wykes, C.

C. Wykes, “De-correlation effects in speckle-pattern interferometry. 1. wavelength change dependent de-correlation with application to contouring and surface roughness measurement,” J. Mod. Opt. 24, 517–532 (1977).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. A (1)

N. George and A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. A 4, 201–212 (1974).
[CrossRef]

J. Mod. Opt. (1)

C. Wykes, “De-correlation effects in speckle-pattern interferometry. 1. wavelength change dependent de-correlation with application to contouring and surface roughness measurement,” J. Mod. Opt. 24, 517–532 (1977).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

C. Wagner, W. Osten, and S. Seebacher, “Direct shape measurement by digital wavefront reconstruction and multi-wavelength contouring,” Opt. Eng. 39, 79–85 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (3)

S. Seebacher, W. Osten, T. Baumbach, and W. Jüptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417–428 (2001).
[CrossRef]

C. Towers, D. Towers, and J. Jones, “Absolute fringe order calculation using optimised multi-frequency selection in full-field profilometry,” Opt. Lasers Eng. 43, 788–800 (2005).
[CrossRef]

Phys. Lett. (1)

K. Haines and B. P. Hildebrand, “Contour generation by wavefront reconstruction,” Phys. Lett. 19, 10–11 (1965).
[CrossRef]

Other (3)

J. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and related Phenomena, J. Dainty, ed. (Springer, 1975), pp. 9–75.

U. Schnars and W. Jüptner, Digital Holography (Springer, 2005).

J. W. Goodman, Introduction To Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (4)

Fig. 1.
Fig. 1.

Geometry model of contouring: the phase value at the observation point B depends on the path QP¯+PB¯ and therefore enables determination of the surface profile. The extent of the measurement volume V is assumed to be small compared to its distance from the light source Q (image not to scale), so that the illumination can be approximated by a plane wave with wave vector k⃗. The inset describes the geometric relations between the illumination direction e⃗k, the observation direction e⃗B, and the sensitivity vector s⃗. For further details please refer to the text.

Fig. 2.
Fig. 2.

Holographic setup: The key element of the experimental setup is a long-distance microscope objective (LDM). The distance between the image plane and the camera sensor is 100 mm. Multiple fibers are used to illuminate the object consecutively from different directions and to provide a spherical reference wave. In our experiments, a dye laser was employed to generate the two wavelengths λ1=580nm and λ2=583nm required for the contouring process. The sensor of the camera has 2048 by 2048 pixels with a pixel pitch of 3.45 μm.

Fig. 3.
Fig. 3.

Measurement of a plane sheet of steel: (a) Phase distribution obtained from a single illumination direction and (b) Average phase distribution obtained from three illumination directions. The standard error in the phase measurement is decreased by approximately 3 dB from σ1=0.60rad to σ3=0.31rad due to the weighted averaging.

Fig. 4.
Fig. 4.

Surface profile of a 2 Euro coin: (a) Photograph of the coin. (b) Phase distribution obtained from a single illumination direction, and (c) Average phase distribution obtained from three illumination directions. (d) Surface profile of the coin after unwrapping. The arrows indicate the projected directions of illumination.

Equations (11)

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ϕP(r⃗P)=(k⃗,r⃗P)+ϕ0.
ϕ(r⃗B)=ϕP(r⃗P)+2πλ(e⃗B,r⃗Pr⃗B),
ϕ(r⃗B)=(k⃗,r⃗B)+2πλ(s⃗,h⃗)+ϕ0.
ϕ˜(r⃗B)=2πλ(s⃗,h⃗)=2πλzP(1+cosα),
Λ=λ1λ2|λ2λ1|.
Δu=a1a2exp(i2πΛ[zP(1+cosα)+(e⃗k,r⃗B)]+iϕ0+iϵϕ),
Δu˜=a1a2exp(i2πΛzP(1+cosα)+iϵϕ).
n=1NΔu˜n=exp(i2πΛzP(1+cosα))1Nn=1Na1,na2,nexp(iϵϕ,n).
ϕl=πλ(|x⃗|2a+|u⃗|2b).
F{uI(u⃗)}=F{uS(y⃗)}·Hd(ν⃗),
Hd(ν⃗)=exp[i2πdλ1λ2|ν⃗|2].

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