Abstract

The phenomenon of Talbot self-image shift by changing the wavelength of the illuminating light is described and demonstrated experimentally. A periodic grating is illuminated by light with wavelengths λ1 and λ2 generated by two lasers, and the Talbot self-images are recorded along the longitudinal direction at individual wavelengths. The Talbot self-image shift due to the change in the wavelength of light is implemented for the measurement of the three-dimensional step height of a large discontinuous object without any phase ambiguity problem. Fourier-transform fringe analysis was used to determine the maximum contrast of the high-visibility bands for the measurement of the step height of the object. The main advantages of the proposed system are nonmechanical scanning, high stability because of its common path geometry, compactness, and a wide range of measurement as compared to interferometric three- dimensional profilers.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. F. Talbot, "Facts relating to optical science. No. IV," Philos. Mag. 9, 401-407 (1836).
  2. Lord Rayleigh, "On copying diffraction gratings, and on some phenomena connected therewith," Philos. Mag. 11, 196-201 (1881).
  3. K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North Holland, 1989), pp. 1-108.
    [CrossRef]
  4. P. Chavel and T. C. Strand, "Range measurement using Talbot diffraction imaging of gratings," Appl. Opt. 23, 862-871 (1984).
    [CrossRef] [PubMed]
  5. J. R. Leger and M. A. Snyder, "Real-time depth measurement and display using Fresnel diffraction and white-light processing," Appl. Opt. 23, 1655-1670 (1984).
    [CrossRef] [PubMed]
  6. M. Takeda and S. Kobayashi, "Lateral aberration measurements with a digital Talbot interferometer," Appl. Opt. 23, 1760-1764 (1984).
    [CrossRef] [PubMed]
  7. K. Engelhardt and G. Hausler, "Acquisition of 3D data by focus sensing," Appl. Opt. 27, 4684-4689 (1988).
    [CrossRef] [PubMed]
  8. R. Rodriguez-Vera, D. Kerr, and F. Mendoza-Santoyo, "3D contouring of diffuse objects by Talbot-projected fringes," J. Mod. Opt. 38, 1935-1945 (1991).
    [CrossRef]
  9. B. F. Oreb and R. G. Dorsch, "Profilometry by phase-shifted Talbot images," Appl. Opt. 33, 7955-7962 (1994).
    [CrossRef] [PubMed]
  10. S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Opt. Eng. 44, 013601 (2005).
    [CrossRef]
  11. L. V. Bourmborde, A. O. Tonso, C. M. V. Colautti, and E. E. Sicre, "Real-time measurement of the meniscus shape using the Talbot effect," Opt. Commun. 102, 397-401 (1993).
    [CrossRef]
  12. G. S. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A , Pure Appl. Opt. 4, S376-S380 (2002).
    [CrossRef]
  13. P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
    [CrossRef] [PubMed]
  14. H. L. Kung, A. Bhatnagar, and D. A. B. Miller, "Transform spectrometer based on measuring the periodicity of Talbot self-images," Opt. Lett. 26, 1645-1647 (2001).
    [CrossRef]
  15. N. Guérineau, E. Di Mambro, and J. Primot, "Talbot experiment re-examined: study of the chromatic regime and application to spectrometry," Opt. Express 11, 3310-3319 (2003).
    [CrossRef] [PubMed]
  16. N. Guerineau, B. Harchaoui, and J. Primot, "Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
    [CrossRef]
  17. C. Wyant, "Testing aspherics using two-wavelength holography," Appl. Opt. 10, 2113-2118 (1971).
    [CrossRef] [PubMed]
  18. C. Polhemus, "Two-wavelength interferometry," Appl. Opt. 12, 2071-2074 (1973).
    [CrossRef] [PubMed]
  19. Y. Y. Cheng and J. C. Wyant, "Two-wavelength phase-shifting interferometry," Appl. Opt. 23, 4539-4543 (1984).
    [CrossRef] [PubMed]
  20. Y. Y. Cheng and J. C. Wyant, "Multiple-wavelength phase-shifting interferometry," Appl. Opt. 24, 804-807 (1985).
    [CrossRef] [PubMed]
  21. K. Creath, Y. Y. Cheng, and J. C. Wyant, "Contouring aspheric surfaces two-wavelength phase shifting interferometry," Opt. Acta 32, 1455-1464 (1985).
  22. H. Matsumoto, "Synthetic interferometric distance-measuring system using a CO2 laser," Appl. Opt. 25, 493-498 (1986).
    [CrossRef] [PubMed]
  23. H. Tiziani, A. Rothe, and N. Maier, "Dual-wavelength heterodyne differential interferometer for high-precision measurements of reflective aspherical surfaces and step-heights," Appl. Opt. 35, 3525-3533 (1986).
    [CrossRef]
  24. K. Creath, "Step height measurement using two-wavelength phase-shifting interferometry," Appl. Opt. 26, 2810-2816 (1987).
    [CrossRef] [PubMed]
  25. R. Dandliker, R. Thalmann, and D. Pronge, "Two-wavelength laser interferometry using superheterodyne detection," Opt. Lett. 13, 339-341 (1988).
    [CrossRef] [PubMed]
  26. Y. Ishii and R. Onodera, "Two-wavelength laser-diode interferometry that uses phase-shifting technique," Opt. Lett. 16, 1523-1525 (1991).
    [CrossRef] [PubMed]
  27. P. J. de Groot, "Extending the unambiguous range of two-color interferometers," Appl. Opt. 33, 5948-5953 (1994).
    [CrossRef] [PubMed]
  28. R. Onodera and Y. Ishii, "Two-wavelength interferometry that uses a Fourier-transform method," Appl. Opt. 37, 7988-7994 (1998).
    [CrossRef]
  29. 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).
    [CrossRef]
  30. M. Takeda, T. Aoki, Y. Miyamoto, H. Tanaka, R. Gu, and Z. Zhang, "Absolute three-dimensional shape measurements using coaxial and coimage plane optical systems and Fourier fringe analysis for focus detection," Opt. Eng. 39, 61-68 (2000).
    [CrossRef]
  31. D. S. Mehta, M. Sugai, H. Hinosugi, S. Saito, M. Takeda, T. Kurokawa, H. Takahashi, M. Ando, M. Shishido, and T. Yoshizawa, "Simultaneous three-dimensional step-height measurement and high-resolution tomographic imaging using spectral interferometric microscope," Appl. Opt. 41, 3874-3885 (2002).
    [CrossRef] [PubMed]
  32. D. S. Mehta, H. Hinosugi, S. Saito, M. Takeda, T. Kurokawa, H. Takahashi, M. Ando, M. Shishido, and T. Yoshizawa, "A spectral interferometric microscope using tandem liquid-crystal Fabry-Perot interferometers for the extension of the dynamic range in three-dimensional step-height measurement," Appl. Opt. 42, 682-690 (2003).
    [CrossRef] [PubMed]
  33. D. S. Mehta, S. Saito, H. Hinosugi, M. Takeda, and T. Kurokawa, "Spectral interference Mirau microscope for three-dimensional surface profilometry with an acousto-optic tunable filter," Appl. Opt. 42, 1296-1305 (2003).
    [CrossRef] [PubMed]

2005 (2)

S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Opt. Eng. 44, 013601 (2005).
[CrossRef]

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

2003 (3)

2002 (2)

2001 (1)

2000 (2)

N. Guerineau, B. Harchaoui, and J. Primot, "Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

M. Takeda, T. Aoki, Y. Miyamoto, H. Tanaka, R. Gu, and Z. Zhang, "Absolute three-dimensional shape measurements using coaxial and coimage plane optical systems and Fourier fringe analysis for focus detection," Opt. Eng. 39, 61-68 (2000).
[CrossRef]

1998 (1)

1994 (2)

1993 (1)

L. V. Bourmborde, A. O. Tonso, C. M. V. Colautti, and E. E. Sicre, "Real-time measurement of the meniscus shape using the Talbot effect," Opt. Commun. 102, 397-401 (1993).
[CrossRef]

1991 (2)

Y. Ishii and R. Onodera, "Two-wavelength laser-diode interferometry that uses phase-shifting technique," Opt. Lett. 16, 1523-1525 (1991).
[CrossRef] [PubMed]

R. Rodriguez-Vera, D. Kerr, and F. Mendoza-Santoyo, "3D contouring of diffuse objects by Talbot-projected fringes," J. Mod. Opt. 38, 1935-1945 (1991).
[CrossRef]

1988 (2)

1987 (1)

1986 (2)

1985 (1)

1984 (4)

1982 (1)

1973 (1)

1971 (1)

1881 (1)

Lord Rayleigh, "On copying diffraction gratings, and on some phenomena connected therewith," Philos. Mag. 11, 196-201 (1881).

1836 (1)

H. F. Talbot, "Facts relating to optical science. No. IV," Philos. Mag. 9, 401-407 (1836).

Ambrosini, D.

G. S. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A , Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

Ando, M.

Aoki, T.

M. Takeda, T. Aoki, Y. Miyamoto, H. Tanaka, R. Gu, and Z. Zhang, "Absolute three-dimensional shape measurements using coaxial and coimage plane optical systems and Fourier fringe analysis for focus detection," Opt. Eng. 39, 61-68 (2000).
[CrossRef]

Bhatnagar, A.

Bourmborde, L. V.

L. V. Bourmborde, A. O. Tonso, C. M. V. Colautti, and E. E. Sicre, "Real-time measurement of the meniscus shape using the Talbot effect," Opt. Commun. 102, 397-401 (1993).
[CrossRef]

Chavel, P.

Cheng, Y. Y.

Colautti, C. M. V.

L. V. Bourmborde, A. O. Tonso, C. M. V. Colautti, and E. E. Sicre, "Real-time measurement of the meniscus shape using the Talbot effect," Opt. Commun. 102, 397-401 (1993).
[CrossRef]

Creath, K.

Dandliker, R.

de Groot, P. J.

Di Mambro, E.

Dorsch, R. G.

Engelhardt, K.

Faridi, M. S.

Gu, R.

M. Takeda, T. Aoki, Y. Miyamoto, H. Tanaka, R. Gu, and Z. Zhang, "Absolute three-dimensional shape measurements using coaxial and coimage plane optical systems and Fourier fringe analysis for focus detection," Opt. Eng. 39, 61-68 (2000).
[CrossRef]

Guerineau, N.

N. Guerineau, B. Harchaoui, and J. Primot, "Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

Guérineau, N.

Harchaoui, B.

N. Guerineau, B. Harchaoui, and J. Primot, "Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

Hausler, G.

Hinosugi, H.

Ina, H.

Ishii, Y.

Kerr, D.

R. Rodriguez-Vera, D. Kerr, and F. Mendoza-Santoyo, "3D contouring of diffuse objects by Talbot-projected fringes," J. Mod. Opt. 38, 1935-1945 (1991).
[CrossRef]

Kobayashi, S.

Kung, H. L.

Kurokawa, T.

Leger, J. R.

Mag., Philos.

Lord Rayleigh, "On copying diffraction gratings, and on some phenomena connected therewith," Philos. Mag. 11, 196-201 (1881).

Maier, N.

Matsumoto, H.

Mehta, D. S.

Mendoza-Santoyo, F.

R. Rodriguez-Vera, D. Kerr, and F. Mendoza-Santoyo, "3D contouring of diffuse objects by Talbot-projected fringes," J. Mod. Opt. 38, 1935-1945 (1991).
[CrossRef]

Miller, D. A. B.

Mirza, S.

S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Opt. Eng. 44, 013601 (2005).
[CrossRef]

Miyamoto, Y.

M. Takeda, T. Aoki, Y. Miyamoto, H. Tanaka, R. Gu, and Z. Zhang, "Absolute three-dimensional shape measurements using coaxial and coimage plane optical systems and Fourier fringe analysis for focus detection," Opt. Eng. 39, 61-68 (2000).
[CrossRef]

Onodera, R.

Oreb, B. F.

Paoletti, D.

G. S. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A , Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

Patorski, K.

K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North Holland, 1989), pp. 1-108.
[CrossRef]

Polhemus, C.

Primot, J.

N. Guérineau, E. Di Mambro, and J. Primot, "Talbot experiment re-examined: study of the chromatic regime and application to spectrometry," Opt. Express 11, 3310-3319 (2003).
[CrossRef] [PubMed]

N. Guerineau, B. Harchaoui, and J. Primot, "Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

Pronge, D.

Rayleigh, Lord

Lord Rayleigh, "On copying diffraction gratings, and on some phenomena connected therewith," Philos. Mag. 11, 196-201 (1881).

Rodriguez-Vera, R.

R. Rodriguez-Vera, D. Kerr, and F. Mendoza-Santoyo, "3D contouring of diffuse objects by Talbot-projected fringes," J. Mod. Opt. 38, 1935-1945 (1991).
[CrossRef]

Rothe, A.

Saito, S.

Shakher, C.

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Opt. Eng. 44, 013601 (2005).
[CrossRef]

Shishido, M.

Sicre, E. E.

L. V. Bourmborde, A. O. Tonso, C. M. V. Colautti, and E. E. Sicre, "Real-time measurement of the meniscus shape using the Talbot effect," Opt. Commun. 102, 397-401 (1993).
[CrossRef]

Singh, P.

Sirohi, R. S.

Snyder, M. A.

Spagnolo, G. S.

G. S. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A , Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

Strand, T. C.

Sugai, M.

Takahashi, H.

Takeda, M.

Talbot, H. F.

H. F. Talbot, "Facts relating to optical science. No. IV," Philos. Mag. 9, 401-407 (1836).

Tanaka, H.

M. Takeda, T. Aoki, Y. Miyamoto, H. Tanaka, R. Gu, and Z. Zhang, "Absolute three-dimensional shape measurements using coaxial and coimage plane optical systems and Fourier fringe analysis for focus detection," Opt. Eng. 39, 61-68 (2000).
[CrossRef]

Thalmann, R.

Tiziani, H.

Tonso, A. O.

L. V. Bourmborde, A. O. Tonso, C. M. V. Colautti, and E. E. Sicre, "Real-time measurement of the meniscus shape using the Talbot effect," Opt. Commun. 102, 397-401 (1993).
[CrossRef]

Wyant, C.

Wyant, J. C.

Yoshizawa, T.

Zhang, Z.

M. Takeda, T. Aoki, Y. Miyamoto, H. Tanaka, R. Gu, and Z. Zhang, "Absolute three-dimensional shape measurements using coaxial and coimage plane optical systems and Fourier fringe analysis for focus detection," Opt. Eng. 39, 61-68 (2000).
[CrossRef]

Appl. Opt. (18)

P. Chavel and T. C. Strand, "Range measurement using Talbot diffraction imaging of gratings," Appl. Opt. 23, 862-871 (1984).
[CrossRef] [PubMed]

J. R. Leger and M. A. Snyder, "Real-time depth measurement and display using Fresnel diffraction and white-light processing," Appl. Opt. 23, 1655-1670 (1984).
[CrossRef] [PubMed]

M. Takeda and S. Kobayashi, "Lateral aberration measurements with a digital Talbot interferometer," Appl. Opt. 23, 1760-1764 (1984).
[CrossRef] [PubMed]

Y. Y. Cheng and J. C. Wyant, "Two-wavelength phase-shifting interferometry," Appl. Opt. 23, 4539-4543 (1984).
[CrossRef] [PubMed]

Y. Y. Cheng and J. C. Wyant, "Multiple-wavelength phase-shifting interferometry," Appl. Opt. 24, 804-807 (1985).
[CrossRef] [PubMed]

H. Matsumoto, "Synthetic interferometric distance-measuring system using a CO2 laser," Appl. Opt. 25, 493-498 (1986).
[CrossRef] [PubMed]

K. Creath, "Step height measurement using two-wavelength phase-shifting interferometry," Appl. Opt. 26, 2810-2816 (1987).
[CrossRef] [PubMed]

P. J. de Groot, "Extending the unambiguous range of two-color interferometers," Appl. Opt. 33, 5948-5953 (1994).
[CrossRef] [PubMed]

B. F. Oreb and R. G. Dorsch, "Profilometry by phase-shifted Talbot images," Appl. Opt. 33, 7955-7962 (1994).
[CrossRef] [PubMed]

R. Onodera and Y. Ishii, "Two-wavelength interferometry that uses a Fourier-transform method," Appl. Opt. 37, 7988-7994 (1998).
[CrossRef]

H. Tiziani, A. Rothe, and N. Maier, "Dual-wavelength heterodyne differential interferometer for high-precision measurements of reflective aspherical surfaces and step-heights," Appl. Opt. 35, 3525-3533 (1986).
[CrossRef]

D. S. Mehta, M. Sugai, H. Hinosugi, S. Saito, M. Takeda, T. Kurokawa, H. Takahashi, M. Ando, M. Shishido, and T. Yoshizawa, "Simultaneous three-dimensional step-height measurement and high-resolution tomographic imaging using spectral interferometric microscope," Appl. Opt. 41, 3874-3885 (2002).
[CrossRef] [PubMed]

D. S. Mehta, H. Hinosugi, S. Saito, M. Takeda, T. Kurokawa, H. Takahashi, M. Ando, M. Shishido, and T. Yoshizawa, "A spectral interferometric microscope using tandem liquid-crystal Fabry-Perot interferometers for the extension of the dynamic range in three-dimensional step-height measurement," Appl. Opt. 42, 682-690 (2003).
[CrossRef] [PubMed]

D. S. Mehta, S. Saito, H. Hinosugi, M. Takeda, and T. Kurokawa, "Spectral interference Mirau microscope for three-dimensional surface profilometry with an acousto-optic tunable filter," Appl. Opt. 42, 1296-1305 (2003).
[CrossRef] [PubMed]

C. Wyant, "Testing aspherics using two-wavelength holography," Appl. Opt. 10, 2113-2118 (1971).
[CrossRef] [PubMed]

C. Polhemus, "Two-wavelength interferometry," Appl. Opt. 12, 2071-2074 (1973).
[CrossRef] [PubMed]

K. Engelhardt and G. Hausler, "Acquisition of 3D data by focus sensing," Appl. Opt. 27, 4684-4689 (1988).
[CrossRef] [PubMed]

P. Singh, M. S. Faridi, C. Shakher, and R. S. Sirohi, "Measurement of focal length with phase-shifting Talbot interferometry," Appl. Opt. 44, 1572-1576 (2005).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

R. Rodriguez-Vera, D. Kerr, and F. Mendoza-Santoyo, "3D contouring of diffuse objects by Talbot-projected fringes," J. Mod. Opt. 38, 1935-1945 (1991).
[CrossRef]

J. Opt. A (1)

G. S. Spagnolo, D. Ambrosini, and D. Paoletti, "Displacement measurement using the Talbot effect with a Ronchi grating," J. Opt. A , Pure Appl. Opt. 4, S376-S380 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

N. Guerineau, B. Harchaoui, and J. Primot, "Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime," Opt. Commun. 180, 199-203 (2000).
[CrossRef]

L. V. Bourmborde, A. O. Tonso, C. M. V. Colautti, and E. E. Sicre, "Real-time measurement of the meniscus shape using the Talbot effect," Opt. Commun. 102, 397-401 (1993).
[CrossRef]

Opt. Eng. (2)

S. Mirza and C. Shakher, "Surface profiling using phase shifting Talbot interferometric technique," Opt. Eng. 44, 013601 (2005).
[CrossRef]

M. Takeda, T. Aoki, Y. Miyamoto, H. Tanaka, R. Gu, and Z. Zhang, "Absolute three-dimensional shape measurements using coaxial and coimage plane optical systems and Fourier fringe analysis for focus detection," Opt. Eng. 39, 61-68 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Philos. Mag. (1)

H. F. Talbot, "Facts relating to optical science. No. IV," Philos. Mag. 9, 401-407 (1836).

Other (3)

Lord Rayleigh, "On copying diffraction gratings, and on some phenomena connected therewith," Philos. Mag. 11, 196-201 (1881).

K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, Vol. XXVII, E. Wolf, ed. (North Holland, 1989), pp. 1-108.
[CrossRef]

K. Creath, Y. Y. Cheng, and J. C. Wyant, "Contouring aspheric surfaces two-wavelength phase shifting interferometry," Opt. Acta 32, 1455-1464 (1985).

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 (7)

Fig. 1
Fig. 1

Schematic diagram of the two-wavelength Talbot self-imaging system. MO, microscope objective; BS, beam splitter.

Fig. 2
Fig. 2

Images of the grating recorded at a longitudinal distance of 49.3   cm . (a) Red laser with a wavelength of λ 1 = 632.8   nm only switched on, (b) green laser with a wavelength of λ 2 = 532   nm only switched on, and (c), (d) images of the grating recorded at a longitudinal distance of 58 .66   cm corresponding to wavelengths λ 1 and λ 2 , respectively.

Fig. 3
Fig. 3

Line profiles (a) and (b) correspond to the images shown in Figs. 2(a) and 2(b), respectively, and line profiles (c) and (d) correspond to the images shown in Figs. 2(c) and 2(d), respectively.

Fig. 4
Fig. 4

Schematic of the experimental setup for the measurement of the step height of a discontinuous object using the two-wavelength Talbot effect.

Fig. 5
Fig. 5

Schematic of the steplike object with grating patterns projected on the object (a) when only the red laser is switched on and (b) when only the green laser is switched on.

Fig. 6
Fig. 6

Two-dimensional Fourier spectra of the grating projected on the top of the discontinuous object at a longitudinal distance of 49.36   cm . (a) Red laser only switched on, (c) green laser only switched on, and (b), (d) Fourier spectra grating projected at the bottom of the discontinuous object corresponding to the red laser and the green laser, respectively.

Fig. 7
Fig. 7

Reconstructed 3D step height.

Equations (8)

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

Δ Z = λ 1 1 λ 2 / d 2 ,
Z T = 2 d 2 λ .
Z ( n ) = n Z T = 2 n d 2 λ ,
Z 1 ( n ) = n Z T 1 = 2 n d 2 λ 1 ,
Z 2 ( n ) = n Z T 2 = 2 n d 2 λ 2 .
Δ Z ( n ) = n ( Z T 1 Z T 2 ) = 2 n d 2 λ 1 2 n d 2 λ 2 = 2 n d 2 [ λ 2 λ 1 λ 1 λ 2 ] .
g ( x , y ) = a ( x , y ) + b ( x , y ) cos [ 2 π f 0 x + ϕ ( x , y ) ] ,
c ( x , y ) = b ( x , y ) 2 exp { i [ 2 π f 0 x + ϕ ( x , y ) ] } .

Metrics