Abstract

This work studies both theoretically and experimentally the formation of the contour interference patterns generated by a two-wavelength real-time holographic interferometer. The resulting contour interference fringes are due to the intersection of the measured surface with parallel, equally spaced planes of constant elevation. The theoretical analysis describes how the spatial frequency of the elevation planes, their angular position, and the localization of the fringes depend on parameters of the optical setup. A theoretical model for fringe localization is developed and confirmed by the experiments, showing a strong dependence of the interferogram position on the slope of the studied surface. Due to the thick Bi12TiO20 crystal employed as the storage medium the Bragg selectivity of the holographic readout is also considered.

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  1. K. Creath, Phase Measurement Techniques: Progress in Optics, vol. XXVI, (Elsevier Science Publishers 1988)
  2. D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method. for phase unwrapping,” J. Opt. Soc. Am. 4(1), 267–280 (1987).
    [CrossRef]
  3. A. Spik and W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
    [CrossRef]
  4. M. Salvador, J. Prauzner, S. Köber, K. Meerholz, J. J. Turek, K. Jeong, and D. D. Nolte, “Three-dimensional holographic imaging of living tissue using a highly sensitive photorefractive polymer device,” Opt. Express 17(14), 11834–11849 (2009).
    [CrossRef] [PubMed]
  5. P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. 23(18), 3079–3081 (1984).
    [CrossRef] [PubMed]
  6. J. E. Millerd and N. J. Brock, “Holographic profilometry with a rhodium-doped barium titanate crystal and a diode laser,” Appl. Opt. 36(11), 2427–2431 (1997).
    [CrossRef] [PubMed]
  7. E. Hack, B. Frei, R. Kästle, and U. Sennhauser, “Additive-Subtractive Two-Wavelength ESPI Contouring by Using a Synthetic Wavelength Phase Shift,” Appl. Opt. 37(13), 2591–2597 (1998).
    [CrossRef]
  8. E. A. Barbosa and A. C. L. Lino, “Multiwavelength electronic speckle pattern interferometry for surface shape measurement,” Appl. Opt. 46(14), 2624–2631 (2007).
    [CrossRef] [PubMed]
  9. B. Breuckmann and W. Thieme, “Computer-Aided Analysis of Holographic Interferograms Using the Phase-Shift Method,” Appl. Opt. 24(14), 2145–2149 (1985).
    [CrossRef] [PubMed]
  10. I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36(5), 417–428 (2001).
    [CrossRef]
  11. G. Pedrini, P. Fröning, H. J. Tiziani, and F. M. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. 164(4-6), 257–268 (1999).
    [CrossRef]
  12. I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
    [CrossRef]
  13. E. A. Barbosa, A. A. V. Filho, M. R. R. Gesualdi, B. G. Curcio, M. Muramatsu, and D. Soga, “Single-exposure, photorefractive holographic surface contouring with multiwavelength diode lasers,” J. Opt. Soc. Am. A 22(12), 2872–2879 (2005).
    [CrossRef]
  14. D. Carl, M. Fratz, M. Pfeifer, D. M. Giel, and H. Höfler, “Multiwavelength digital holography with autocalibration of phase shifts and artificial wavelengths,” Appl. Opt. 48(34), H1–H8 (2009).
    [CrossRef] [PubMed]
  15. E. A. Barbosa and J. F. Carvalho, “Surface analysis by two-diode laser photorefractive holography,” Appl. Phys. B 87(3), 417–423 (2007).
    [CrossRef]
  16. E. A. Barbosa, and A. O. Preto, Optical Metrology: Optical Measurement Systems for Industrial Inspection, Peter Lehman (Ed.) Proc. SPIE 7389 (2009).
  17. E. A. Barbosa, C. B. F. de Sousa, and W. M. Maffei, “Measurement of low-derivative surface lenses by two-laser holography with Bi12TiO20 crystals,” Appl. Opt. 48(27), 5114–5120 (2009).
    [CrossRef] [PubMed]
  18. C. M. Vest, Holographic Interferometry, Wiley, New York (1979).
  19. J. Blanco-Garoía, J. L. Fernández, and M. Pérez-Amor, “Fringe localization control in holographic interferometry,” Appl. Opt. 31(4), 488–496 (1992).
    [CrossRef] [PubMed]
  20. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  21. A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53(1), 23–26 (1985).
    [CrossRef]

2009 (3)

2007 (2)

E. A. Barbosa and J. F. Carvalho, “Surface analysis by two-diode laser photorefractive holography,” Appl. Phys. B 87(3), 417–423 (2007).
[CrossRef]

E. A. Barbosa and A. C. L. Lino, “Multiwavelength electronic speckle pattern interferometry for surface shape measurement,” Appl. Opt. 46(14), 2624–2631 (2007).
[CrossRef] [PubMed]

2006 (1)

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[CrossRef]

2005 (1)

2001 (1)

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36(5), 417–428 (2001).
[CrossRef]

1999 (1)

G. Pedrini, P. Fröning, H. J. Tiziani, and F. M. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. 164(4-6), 257–268 (1999).
[CrossRef]

1998 (1)

1997 (1)

1992 (1)

1991 (1)

A. Spik and W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

1987 (1)

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method. for phase unwrapping,” J. Opt. Soc. Am. 4(1), 267–280 (1987).
[CrossRef]

1985 (2)

B. Breuckmann and W. Thieme, “Computer-Aided Analysis of Holographic Interferograms Using the Phase-Shift Method,” Appl. Opt. 24(14), 2145–2149 (1985).
[CrossRef] [PubMed]

A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53(1), 23–26 (1985).
[CrossRef]

1984 (1)

1969 (1)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Balboa, I.

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[CrossRef]

Barbosa, E. A.

Blanco-Garoía, J.

Breuckmann, B.

Brock, N. J.

Carl, D.

Carvalho, J. F.

E. A. Barbosa and J. F. Carvalho, “Surface analysis by two-diode laser photorefractive holography,” Appl. Phys. B 87(3), 417–423 (2007).
[CrossRef]

Curcio, B. G.

de Sousa, C. B. F.

Fernández, J. L.

Filho, A. A. V.

Ford, H. D.

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[CrossRef]

Fratz, M.

Frei, B.

Fröning, P.

G. Pedrini, P. Fröning, H. J. Tiziani, and F. M. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. 164(4-6), 257–268 (1999).
[CrossRef]

Gaskill, J. D.

Gesualdi, M. R. R.

Ghiglia, D. C.

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method. for phase unwrapping,” J. Opt. Soc. Am. 4(1), 267–280 (1987).
[CrossRef]

Giel, D. M.

Hack, E.

Höfler, H.

Jeong, K.

Kamshilin, A. A.

A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53(1), 23–26 (1985).
[CrossRef]

Kästle, R.

Kato, J.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36(5), 417–428 (2001).
[CrossRef]

Köber, S.

Kogelnik, H.

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Lam, P. S.

Lino, A. C. L.

Maffei, W. M.

Mastin, G. A.

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method. for phase unwrapping,” J. Opt. Soc. Am. 4(1), 267–280 (1987).
[CrossRef]

Meerholz, K.

Millerd, J. E.

Muramatsu, M.

Nolte, D. D.

Ohta, S.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36(5), 417–428 (2001).
[CrossRef]

Pedrini, G.

G. Pedrini, P. Fröning, H. J. Tiziani, and F. M. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. 164(4-6), 257–268 (1999).
[CrossRef]

Pérez-Amor, M.

Petrov, M. P.

A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53(1), 23–26 (1985).
[CrossRef]

Pfeifer, M.

Prauzner, J.

Robinson, W.

A. Spik and W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

Romero, L. A.

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method. for phase unwrapping,” J. Opt. Soc. Am. 4(1), 267–280 (1987).
[CrossRef]

Salvador, M.

Santoyo, F. M.

G. Pedrini, P. Fröning, H. J. Tiziani, and F. M. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. 164(4-6), 257–268 (1999).
[CrossRef]

Sennhauser, U.

Soga, D.

Spik, A.

A. Spik and W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

Tatam, R. P.

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[CrossRef]

Thieme, W.

Tiziani, H. J.

G. Pedrini, P. Fröning, H. J. Tiziani, and F. M. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. 164(4-6), 257–268 (1999).
[CrossRef]

Turek, J. J.

Wyant, J. C.

Yamaguchi, I.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36(5), 417–428 (2001).
[CrossRef]

Appl. Opt. (8)

P. S. Lam, J. D. Gaskill, and J. C. Wyant, “Two-wavelength holographic interferometer,” Appl. Opt. 23(18), 3079–3081 (1984).
[CrossRef] [PubMed]

J. E. Millerd and N. J. Brock, “Holographic profilometry with a rhodium-doped barium titanate crystal and a diode laser,” Appl. Opt. 36(11), 2427–2431 (1997).
[CrossRef] [PubMed]

E. Hack, B. Frei, R. Kästle, and U. Sennhauser, “Additive-Subtractive Two-Wavelength ESPI Contouring by Using a Synthetic Wavelength Phase Shift,” Appl. Opt. 37(13), 2591–2597 (1998).
[CrossRef]

E. A. Barbosa and A. C. L. Lino, “Multiwavelength electronic speckle pattern interferometry for surface shape measurement,” Appl. Opt. 46(14), 2624–2631 (2007).
[CrossRef] [PubMed]

B. Breuckmann and W. Thieme, “Computer-Aided Analysis of Holographic Interferograms Using the Phase-Shift Method,” Appl. Opt. 24(14), 2145–2149 (1985).
[CrossRef] [PubMed]

D. Carl, M. Fratz, M. Pfeifer, D. M. Giel, and H. Höfler, “Multiwavelength digital holography with autocalibration of phase shifts and artificial wavelengths,” Appl. Opt. 48(34), H1–H8 (2009).
[CrossRef] [PubMed]

E. A. Barbosa, C. B. F. de Sousa, and W. M. Maffei, “Measurement of low-derivative surface lenses by two-laser holography with Bi12TiO20 crystals,” Appl. Opt. 48(27), 5114–5120 (2009).
[CrossRef] [PubMed]

J. Blanco-Garoía, J. L. Fernández, and M. Pérez-Amor, “Fringe localization control in holographic interferometry,” Appl. Opt. 31(4), 488–496 (1992).
[CrossRef] [PubMed]

Appl. Phys. B (1)

E. A. Barbosa and J. F. Carvalho, “Surface analysis by two-diode laser photorefractive holography,” Appl. Phys. B 87(3), 417–423 (2007).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Opt. Soc. Am. (1)

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method. for phase unwrapping,” J. Opt. Soc. Am. 4(1), 267–280 (1987).
[CrossRef]

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

Meas. Sci. Technol. (1)

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[CrossRef]

Opt. Commun. (2)

G. Pedrini, P. Fröning, H. J. Tiziani, and F. M. Santoyo, “Shape measurement of microscopic structures using digital holograms,” Opt. Commun. 164(4-6), 257–268 (1999).
[CrossRef]

A. A. Kamshilin and M. P. Petrov, “Continuous reconstruction of holographic interferograms through anisotropic diffraction in photorefractive crystals,” Opt. Commun. 53(1), 23–26 (1985).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (2)

A. Spik and W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36(5), 417–428 (2001).
[CrossRef]

Other (3)

K. Creath, Phase Measurement Techniques: Progress in Optics, vol. XXVI, (Elsevier Science Publishers 1988)

C. M. Vest, Holographic Interferometry, Wiley, New York (1979).

E. A. Barbosa, and A. O. Preto, Optical Metrology: Optical Measurement Systems for Industrial Inspection, Peter Lehman (Ed.) Proc. SPIE 7389 (2009).

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Figures (12)

Fig. 1
Fig. 1

Incidence of the collimated beam onto the surface.

Fig. 2
Fig. 2

Angle θ of the planes of constant elevation as a function of λ1 for different values of χ and λ2 = 661.75 nm.

Fig. 3
Fig. 3

Dependence of vS as a function of λ1, for β = 0, λ2 = 661.75 nm and χ = 0.98.

Fig. 4
Fig. 4

Scheme of incident collimated beams and beams scattered by the surface at point P.

Fig. 5
Fig. 5

Position Y´ of the interference pattern for α = 0.01 rad (blue), α = 0.05 rad (red) with respect to the edged surface (black).

Fig. 6
Fig. 6

Y ´ / H as a function of the wavelength λ1, with λ2 =661.75 nm and χ = 1, 0.9 and 0.5, for a - M = H / x ´ = 0.04 and b - H / x ´ = 0.012 .

Fig. 7
Fig. 7

Experimental setup: M1 to M5, mirrors, L1 and L2, lenses, BE, beam expander, P1 and P2, polarizers, BS, beam splitter, BTO, Bi12TiO20 crystal, CCD, camera, PC, computer.

Fig. 8
Fig. 8

a - Illumination scheme for α > 0 showing the vector Δ k ; b - contour interferogram of the surface obtained with positive α, showing excentrical contour fringes; c - illumination scheme for α < 0; d – resulting contour interferogram of the surface with circular concentric fringes.

Fig. 9
Fig. 9

Measured values of the interferogram spatial frequency ν S with maximal visibility as a function of λ 1, for λ 2 = 661.4 nm. The solid curve is the fitting of the experimental data with the theoretical expression of ν S.

Fig. 10
Fig. 10

Two-wavelength holographic imaging of the edge-shaped surface by focusing the fringe pattern of a –the left half (x´ > 0 and H / x ´ < 0 ) of the surface, and b - the right half of the surface (x´ < 0 and H / x ´ > 0 ); two-wavelength holographic imaging of the spherical surface by focusing the fringe pattern of c – the left and d – the right sides of the surface.

Fig. 11
Fig. 11

Contour fringe pattern of the whole edge-shaped surface achieved for θ = 0 . The arrow separates the negative from the positive slope of the surface.

Fig. 12
Fig. 12

Measured loci of the contour interferogram generated by a flat plate as a function of its inclination.

Equations (12)

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R = R 0 { exp [ i ( k 1 Γ R 1 + φ 1 ) + exp [ i ( k 2 Γ R 2 + φ 2 ) }
S = S 0 { exp [ i ( k 1 Γ S 1 + φ 1 ) + exp [ i ( k 2 Γ S 2 + φ 2 ) }
I D | E D 1 | 2 + | E D 2 | 2 R 0 2 η 0 [ 1 + χ 2 + 2 | χ | cos ( k 1 Γ S 1 k 1 Γ R 1 k 2 Γ S 2 + k 2 Γ R 2 ) ] = R 0 2 η 0 ( 1 + χ 2 ) [ 1 + V cos ( k 1 Γ S 1 k 1 Γ R 1 k 2 Γ S 2 + k 2 Γ R 2 ) ]
φ = k 1 Γ S 1 B k 2 Γ S 2 B k 1 Γ S 1 A + k 2 Γ S 2 A ,
φ = 2 π ( k ^ 1 λ 1 k ^ 2 λ 2 k ^ λ S ) . L
t g θ = sin α λ S δ α λ 2 cos α 2 cos 2 ( α 2 ) + λ S δ α λ 2 sin α ,
λ S δ α λ 2 6 ( 1 χ ) 2 π l 0 sin γ λ S tan γ n 0
( k ^ 1 λ 1 k ^ 2 λ 2 k ^ λ S ) . R = 1
R = | λ S sin ( θ α ) 2 cos ( α 2 ) cos ( θ α 2 ) |
δ = 2 π ( k ^ 2 ´ λ 2 k ^ 1 ´ λ 1 k ^ 3 ´ λ S ) . H + 2 π λ S ( k ^ 3 ´ k ^ 4 ´ ) . r P
δ 2 π ( k ^ 2 ´ λ 2 k ^ 1 ´ λ 1 k ^ 3 ´ λ S ) . H
Y ´ = { 1 + sin 2 θ 2 ( sin θ H x ´ cos θ ) [ sin ( α θ ) + 2 cos ( α 2 ) cos ( θ α 2 ) ] H x ´ } H ( x ´ )

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