Abstract

Talbot interferometry is used to study the surface profile of a transparent object. Periodic patterns are produced by illuminating a grating with a collimated laser beam. The object is placed on the self-image plane of the grating. The deformed grating image, which interferes with another grating, results in the Talbot interferometric fringes. The fringe pattern is recorded on a CCD camera for subsequent analysis, and the phase variation is achieved by a linear translation stage. In this application two specimens are tested to demonstrate the validity of the method; one is a transparent object with a spherical shape with a height of less than 350 µm, and the other is a transparent object with an uneven surface of 50-µm average height. The experimental results are compared with the test results obtained with the mechanical stylus method.

© 2005 Optical Society of America

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References

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  1. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–403 (1836).
  2. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
    [CrossRef]
  3. A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–329 (1972).
    [CrossRef]
  4. M. P. Kothiyal, R. S. Sirohi, “Improved collimation testing using Talbot interferometry,” Appl. Opt. 26, 4056–4057 (1987).
    [CrossRef] [PubMed]
  5. M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry with modified gratings,” Opt. Laser Technol. 23, 361–365 (1991).
    [CrossRef]
  6. C. Shakher, S. Prakash, D. Nand, R. Kumar, “Collimation testing with circular gratings,” Appl. Opt. 40, 1175–1179 (2001).
    [CrossRef]
  7. M. Takeda, S. Kobayashi, “Lateral aberration measurement with a digital Talbot interferometer,” Appl. Opt. 23, 1760–1764 (1984).
    [CrossRef]
  8. Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24, 3162–3166 (1985).
    [CrossRef] [PubMed]
  9. L. M. Bernardo, O. D. D. Soares, “Evaluation of the focal distance of a lens by Talbot interferometry,” Appl. Opt. 27, 296–301 (1988).
    [CrossRef] [PubMed]
  10. Y. Nakano, K. Murata, “Measurement of phase objects using Talbot effect and moiré techniques,” Appl. Opt. 23, 2296–2299 (1984).
    [CrossRef]
  11. Y. Nakano, “Measurement of the small tilt angle variation of an object surface using moiré interferometry and digital image processing,” Appl. Opt. 26, 911–914 (1987).
    [CrossRef]
  12. S. Prakash, S. Upadhyay, C. Shakher, “Real-time out-of-plane vibration measurement using Talbot interferometry,” Opt. Laser Eng. 34, 251–259 (2000).
    [CrossRef]
  13. C. Shakher, A. J. P. Daniel, “Talbot interferometer with circular gratings for the measurement of temperature in axi-symmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
    [CrossRef] [PubMed]
  14. C. Shakher, A. J. P. Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (premixed laminar flow slot-burner) using Talbot interferometry with circular gratings,” Opt. Eng. 33, 2663–2669 (1994).
    [CrossRef]
  15. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Raid, eds. (Institute of Physics, Bristol, 1993), Chap. 4.

2001 (1)

2000 (1)

S. Prakash, S. Upadhyay, C. Shakher, “Real-time out-of-plane vibration measurement using Talbot interferometry,” Opt. Laser Eng. 34, 251–259 (2000).
[CrossRef]

1995 (1)

1994 (2)

C. Shakher, A. J. P. Daniel, “Talbot interferometer with circular gratings for the measurement of temperature in axi-symmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
[CrossRef] [PubMed]

C. Shakher, A. J. P. Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (premixed laminar flow slot-burner) using Talbot interferometry with circular gratings,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

1991 (1)

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry with modified gratings,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

1988 (1)

1987 (2)

M. P. Kothiyal, R. S. Sirohi, “Improved collimation testing using Talbot interferometry,” Appl. Opt. 26, 4056–4057 (1987).
[CrossRef] [PubMed]

Y. Nakano, “Measurement of the small tilt angle variation of an object surface using moiré interferometry and digital image processing,” Appl. Opt. 26, 911–914 (1987).
[CrossRef]

1985 (1)

1984 (2)

1972 (1)

A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–329 (1972).
[CrossRef]

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–403 (1836).

Angra, S. K.

C. Shakher, A. J. P. Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (premixed laminar flow slot-burner) using Talbot interferometry with circular gratings,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

Bernardo, L. M.

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Raid, eds. (Institute of Physics, Bristol, 1993), Chap. 4.

Daniel, A. J. P.

C. Shakher, A. J. P. Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (premixed laminar flow slot-burner) using Talbot interferometry with circular gratings,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

C. Shakher, A. J. P. Daniel, “Talbot interferometer with circular gratings for the measurement of temperature in axi-symmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
[CrossRef] [PubMed]

de Bougrenet de la Tocnaye, J. L.

Hamam, H.

Kobayashi, S.

Kothiyal, M. P.

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry with modified gratings,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

M. P. Kothiyal, R. S. Sirohi, “Improved collimation testing using Talbot interferometry,” Appl. Opt. 26, 4056–4057 (1987).
[CrossRef] [PubMed]

Kumar, R.

Lohmann, A. W.

A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–329 (1972).
[CrossRef]

Murata, K.

Nakano, Y.

Nand, D.

Prakash, S.

C. Shakher, S. Prakash, D. Nand, R. Kumar, “Collimation testing with circular gratings,” Appl. Opt. 40, 1175–1179 (2001).
[CrossRef]

S. Prakash, S. Upadhyay, C. Shakher, “Real-time out-of-plane vibration measurement using Talbot interferometry,” Opt. Laser Eng. 34, 251–259 (2000).
[CrossRef]

Shakher, C.

C. Shakher, S. Prakash, D. Nand, R. Kumar, “Collimation testing with circular gratings,” Appl. Opt. 40, 1175–1179 (2001).
[CrossRef]

S. Prakash, S. Upadhyay, C. Shakher, “Real-time out-of-plane vibration measurement using Talbot interferometry,” Opt. Laser Eng. 34, 251–259 (2000).
[CrossRef]

C. Shakher, A. J. P. Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (premixed laminar flow slot-burner) using Talbot interferometry with circular gratings,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

C. Shakher, A. J. P. Daniel, “Talbot interferometer with circular gratings for the measurement of temperature in axi-symmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
[CrossRef] [PubMed]

Silva, D. E.

A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–329 (1972).
[CrossRef]

Sirohi, R. S.

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry with modified gratings,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

M. P. Kothiyal, R. S. Sirohi, “Improved collimation testing using Talbot interferometry,” Appl. Opt. 26, 4056–4057 (1987).
[CrossRef] [PubMed]

Soares, O. D. D.

Sriram, K. V.

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry with modified gratings,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

Takeda, M.

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–403 (1836).

Upadhyay, S.

S. Prakash, S. Upadhyay, C. Shakher, “Real-time out-of-plane vibration measurement using Talbot interferometry,” Opt. Laser Eng. 34, 251–259 (2000).
[CrossRef]

Appl. Opt. (8)

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

Opt. Commun. (1)

A. W. Lohmann, D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun. 4, 326–329 (1972).
[CrossRef]

Opt. Eng. (1)

C. Shakher, A. J. P. Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (premixed laminar flow slot-burner) using Talbot interferometry with circular gratings,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

Opt. Laser Eng. (1)

S. Prakash, S. Upadhyay, C. Shakher, “Real-time out-of-plane vibration measurement using Talbot interferometry,” Opt. Laser Eng. 34, 251–259 (2000).
[CrossRef]

Opt. Laser Technol. (1)

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry with modified gratings,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

Philos. Mag. (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–403 (1836).

Other (1)

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Raid, eds. (Institute of Physics, Bristol, 1993), Chap. 4.

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Figures (13)

Fig. 1
Fig. 1

Experimental arrangement.

Fig. 2
Fig. 2

Fringe pattern on the spherical shape.

Fig. 3
Fig. 3

Wrapped phase map.

Fig. 4
Fig. 4

Unwrapped phase map.

Fig. 5
Fig. 5

Contour map.

Fig. 6
Fig. 6

A 3-D plot of the spherical shape.

Fig. 7
Fig. 7

Comparison of surface profile along the D–D section between the present and the mechanical stylus method.

Fig. 8
Fig. 8

Fringe pattern on an uneven transparent object.

Fig. 9
Fig. 9

Wrapped phase map.

Fig. 10
Fig. 10

Unwrapped phase map.

Fig. 11
Fig. 11

Contour map.

Fig. 12
Fig. 12

A 3-D plot of the surface profile.

Fig. 13
Fig. 13

Comparison of the surface profile along the E–E section between the present and the mechanical stylus method.

Equations (8)

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Z k = ± k d 2 λ ,
I ( x , y ) = B ( x , y ) { 1 + γ cos [ 2 k h ( x , y ) n T / L + δ ( t 21 ) ] } ,
I ( x , y ) = B ( x , y ) { 1 + cos [ ϕ ( x , y ) + δ ( t 21 ) ] } .
I 1 ( x , y ) = a ( x , y ) + b ( x , y ) cos ϕ ( x , y ) ,
I 2 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + π / 2 ] ,
I 3 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + π ] ,
ϕ ( x , y ) = tan 1 { [ I 1 ( x , y ) 2 I 2 ( x , y ) + I 3 ( x , y ) ] I 1 ( x , y ) I 3 ( x , y ) } .
h ( x , y ) = λ L 4 π · n T ϕ ( x , y ) .

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