Abstract

We report a technique to determine the 3D contour of objects with dimensions of at least 4 orders of magnitude larger than the illumination optical wavelength. Our proposal is based on the numerical reconstruction of the optical wave field of digitally recorded holograms. The required modulo 2π phase map in any contouring process is obtained by means of the direct subtraction of two phase-contrast images under different illumination angles to create a phase-difference image of a still object. Obtaining the phase-difference images is only possible by using the capability of numerical reconstruction of the complex optical field provided by digital holography. This unique characteristic leads us to a robust, reliable, and fast procedure that requires only two images. A theoretical analysis of the contouring system is shown, with verification by means of numerical and experimental results.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. C. Williams, Optical Methods in Engineering Metrology (Chapman & Hall, 1993).
  2. L. Chen and C. Quan, "Fringe projection profilometry with nonparallel illumination: a least-squares approach," Opt. Lett. 30, 2101-2103 (2005).
    [CrossRef] [PubMed]
  3. H. Cline, W. Lorensen, and A. Holik, "Automatic moire contouring," Appl. Opt. 23, 1454-1459 (1984).
    [CrossRef] [PubMed]
  4. O. Kafri and I. Glatt, The Physics of Moire Metrology (Wiley, 1990).
  5. B. Hildebrand and K. Haines, "Multiple-wavelength and multiple-source holography applied to contour generation," J. Opt. Soc. Am 57, 155-162 (1967).
  6. J. Zelenka and J. Varner, "Multiple-index holographic contouring," Appl. Opt. 8, 1431-1434 (1969).
    [CrossRef] [PubMed]
  7. P. DeMattia and V. Fossati-Bellani, "Holographic contouring by displacing the object and the illumination beam," Opt. Commun. 26, 17-21 (1978).
  8. R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, 1983).
  9. K. Creath and J. Wyant, "Absolute measurement of surface roughness," Appl. Opt. 29, 3823-3827 (1990).
    [CrossRef] [PubMed]
  10. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  11. L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1989), translated from Russian by D. Parsons.
  12. U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002) and the reference therein.
  13. U. Schnars and W. P. Jueptner, Digital Holography (Springer-Verlag, 2005).
  14. Y. Zou, G. Pedrini, and H. J. Tiziani, "Surface contouring in a video frame by changing the wavelength of a diode laser," Opt. Eng. 35, 1074-1079 (1996).
  15. C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).
  16. I. Yamaguchi, S. Ohta, and J. Kato, "Surface contouring by phase-shifting digital holography," Opt. Lasers Eng. 36, 417-428 (2001).
  17. T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23, 1221-1223 (1998).
    [CrossRef]
  18. G. Pedrini, P. Froning, H. J. Tiziani, and F. Mendoza Santoyo, "Shape measurement of microscopic structures using digital holograms," Opt. Commun. 164, 257-268 (1999).
  19. L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and a virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).
  20. E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
    [CrossRef]
  21. J. Gass, A. Dakoff, and M. K. Kim, "Phase imaging without 2π ambiguity by multiwavelength digital holography," Opt. Lett. 28, 1141-1143 (2003).
    [CrossRef] [PubMed]
  22. M. Sebesta and M. Gustafsson, "Object characterization with refractometric digital Fourier holography," Opt. Lett. 30, 471-473 (2005).
    [CrossRef] [PubMed]
  23. R. Colier, C. Burckhardt, and L. Lin, Optical Holography (Academic, 1971).
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

2005 (2)

2003 (1)

1999 (1)

1998 (1)

1990 (1)

1984 (1)

1969 (1)

Bevilacqua, F.

Burckhardt, C.

R. Colier, C. Burckhardt, and L. Lin, Optical Holography (Academic, 1971).

Cai, L. Z.

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and a virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).

Chen, L.

Cline, H.

Colier, R.

R. Colier, C. Burckhardt, and L. Lin, Optical Holography (Academic, 1971).

Creath, K.

Cuche, E.

Dakoff, A.

DeMattia, P.

P. DeMattia and V. Fossati-Bellani, "Holographic contouring by displacing the object and the illumination beam," Opt. Commun. 26, 17-21 (1978).

Depeursinge, C.

Fossati-Bellani, V.

P. DeMattia and V. Fossati-Bellani, "Holographic contouring by displacing the object and the illumination beam," Opt. Commun. 26, 17-21 (1978).

Froning, P.

G. Pedrini, P. Froning, H. J. Tiziani, and F. Mendoza Santoyo, "Shape measurement of microscopic structures using digital holograms," Opt. Commun. 164, 257-268 (1999).

Gass, J.

Glatt, I.

O. Kafri and I. Glatt, The Physics of Moire Metrology (Wiley, 1990).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gustafsson, M.

Haines, K.

B. Hildebrand and K. Haines, "Multiple-wavelength and multiple-source holography applied to contour generation," J. Opt. Soc. Am 57, 155-162 (1967).

Hildebrand, B.

B. Hildebrand and K. Haines, "Multiple-wavelength and multiple-source holography applied to contour generation," J. Opt. Soc. Am 57, 155-162 (1967).

Holik, A.

Jones, R.

R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, 1983).

Jueptner, W. P.

U. Schnars and W. P. Jueptner, Digital Holography (Springer-Verlag, 2005).

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002) and the reference therein.

Kafri, O.

O. Kafri and I. Glatt, The Physics of Moire Metrology (Wiley, 1990).

Kato, J.

I. Yamaguchi, S. Ohta, and J. Kato, "Surface contouring by phase-shifting digital holography," Opt. Lasers Eng. 36, 417-428 (2001).

Kim, M. K.

Lin, L.

R. Colier, C. Burckhardt, and L. Lin, Optical Holography (Academic, 1971).

Liu, Q.

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and a virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).

Lorensen, W.

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Merzlyakov, N. S.

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1989), translated from Russian by D. Parsons.

Ohta, S.

I. Yamaguchi, S. Ohta, and J. Kato, "Surface contouring by phase-shifting digital holography," Opt. Lasers Eng. 36, 417-428 (2001).

Osten, W.

C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).

Pedrini, G.

G. Pedrini, P. Froning, H. J. Tiziani, and F. Mendoza Santoyo, "Shape measurement of microscopic structures using digital holograms," Opt. Commun. 164, 257-268 (1999).

Y. Zou, G. Pedrini, and H. J. Tiziani, "Surface contouring in a video frame by changing the wavelength of a diode laser," Opt. Eng. 35, 1074-1079 (1996).

Quan, C.

Santoyo, F. Mendoza

G. Pedrini, P. Froning, H. J. Tiziani, and F. Mendoza Santoyo, "Shape measurement of microscopic structures using digital holograms," Opt. Commun. 164, 257-268 (1999).

Schnars, U.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002) and the reference therein.

U. Schnars and W. P. Jueptner, Digital Holography (Springer-Verlag, 2005).

Sebesta, M.

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Tiziani, H. J.

Y. Zou, G. Pedrini, and H. J. Tiziani, "Surface contouring in a video frame by changing the wavelength of a diode laser," Opt. Eng. 35, 1074-1079 (1996).

G. Pedrini, P. Froning, H. J. Tiziani, and F. Mendoza Santoyo, "Shape measurement of microscopic structures using digital holograms," Opt. Commun. 164, 257-268 (1999).

Varner, J.

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).

Wang, Y. R.

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and a virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).

Williams, D. C.

D. C. Williams, Optical Methods in Engineering Metrology (Chapman & Hall, 1993).

Wyant, J.

Wykes, C.

R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, 1983).

Yamaguchi, I.

T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23, 1221-1223 (1998).
[CrossRef]

I. Yamaguchi, S. Ohta, and J. Kato, "Surface contouring by phase-shifting digital holography," Opt. Lasers Eng. 36, 417-428 (2001).

Yang, X. L.

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and a virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).

Yaroslavskii, L. P.

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1989), translated from Russian by D. Parsons.

Zelenka, J.

Zhang, T.

Zou, Y.

Y. Zou, G. Pedrini, and H. J. Tiziani, "Surface contouring in a video frame by changing the wavelength of a diode laser," Opt. Eng. 35, 1074-1079 (1996).

Appl. Opt. (3)

Opt. Lett. (5)

Other (16)

R. Colier, C. Burckhardt, and L. Lin, Optical Holography (Academic, 1971).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

D. C. Williams, Optical Methods in Engineering Metrology (Chapman & Hall, 1993).

O. Kafri and I. Glatt, The Physics of Moire Metrology (Wiley, 1990).

B. Hildebrand and K. Haines, "Multiple-wavelength and multiple-source holography applied to contour generation," J. Opt. Soc. Am 57, 155-162 (1967).

G. Pedrini, P. Froning, H. J. Tiziani, and F. Mendoza Santoyo, "Shape measurement of microscopic structures using digital holograms," Opt. Commun. 164, 257-268 (1999).

L. Z. Cai, Q. Liu, X. L. Yang, and Y. R. Wang, "Sensitivity adjustable contouring by digital holography and a virtual reference wavefront," Opt. Commun. 221, 49-54 (2003).

P. DeMattia and V. Fossati-Bellani, "Holographic contouring by displacing the object and the illumination beam," Opt. Commun. 26, 17-21 (1978).

R. Jones and C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, 1983).

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1989), translated from Russian by D. Parsons.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002) and the reference therein.

U. Schnars and W. P. Jueptner, Digital Holography (Springer-Verlag, 2005).

Y. Zou, G. Pedrini, and H. J. Tiziani, "Surface contouring in a video frame by changing the wavelength of a diode laser," Opt. Eng. 35, 1074-1079 (1996).

C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).

I. Yamaguchi, S. Ohta, and J. Kato, "Surface contouring by phase-shifting digital holography," Opt. Lasers Eng. 36, 417-428 (2001).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Schematic representation of the experimental setup; see text for details.

Fig. 2
Fig. 2

Phase-contrast images directly obtained from the reconstructed holograms: A, numerical modeling; B, experimental result. In these images we have illuminated the object with OB 1 . The phase-contrast images in both cases are random intensity distributions due to the surface roughness of the sphere. The results for OB 2 look alike.

Fig. 3
Fig. 3

Phase-difference image numerical modeling results of subtracting the phase-contrast maps obtained by illuminating with OB 1 and OB 2 : A, before and B, after subtraction of the linear carrier term in Eq. (5). C and D show the same information for the experimental results.

Fig. 4
Fig. 4

Three-dimensional reconstruction via phase-difference images: The original object, a semisphere 7.80 ± 0.05 mm in radius, is successfully reconstructed using A. numerical modeling and B. experimental setup.

Fig. 5
Fig. 5

Experimental 3D reconstruction via phase-difference images. The original object, a pyramid 11.50 ± 0.05   mm in height, is successfully reconstructed. The phase-difference images, A, before and B, after the subtraction of the linear carrier, show the need for using phase-unwrapping techniques. The resulting unwrapped phase difference C leads us to the D, 3D representation and E, the side view.

Tables (1)

Tables Icon

Table 1 Experimental Study of the Effect of the Sensitivity on the Method of 3D Reconstruction by Phase-Difference Images a

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

E ( x i , y i , z ) = i E 0 λ z exp [ i π λ z ( x i 2 + y i 2 ) ] × I ( x h , y h ) exp [ i π λ z ( x h 2 + y h 2 ) ] × exp [ i 2 π λ z ( x h x i + y h y i ) ] d x h d y h ,
E ( m , n , z ) = i E 0 λ z exp [ i π λ z ( m 2 N x 2 Δ x h 2 + n 2 N y 2 Δ y h 2 ) ] × k = 0 N x 1 l = 0 N y 1 I ( k , l ) exp [ i π λ z ( k 2 Δ x h 2 + l 2 Δ y h 2 ) ] exp [ i 2 π ( k m N x + l n N y ) ] .
I ( m , n , z ) = E ( m , n , z ) E * ( m , n , z ) = Re 2 [ E ( m , n , z ) ] + Im 2 [ E ( m , n , z ) ] ,
ϕ ( m , n ) = arctan 2 { Im [ E ( m , n ) ] Re [ E ( m , n ) ] } ,
Δ ϕ ( x , y ) = 2 π λ 2 sin Δ α 2 [ x cos ( α + Δ α 2 ) h ( x , y ) sin ( α + Δ α 2 ) ] .
Δ ϕ ( x , y ) = { ϕ 2 ( x , y ) ϕ 1 ( x , y ) , if   ϕ 2 ( x , y ) ϕ 1 ( x , y ) , ϕ 2 ( x , y ) ϕ 1 ( x , y ) + 2 π , if   ϕ 2 ( x , y ) < ϕ 1 ( x , y ) ,
Δ h ( x , y ) = λ 2 sin Δ α 2 sin ( α + Δ α 2 ) ,

Metrics