Abstract

Phase-shifting interferometry with a Fourier fringe analysis technique is implemented to analyze Talbot interferometric fringes and to evaluate the focal length of a lens. A four-step algorithm is used to obtain the phase map of the lens. The slope of the phase map is related to the focal length, and it is from this relationship that we evaluate the focal length. Experimental results are presented. Our experimental study suggests that phase-shifting Talbot interferometry combined with a Fourier fringe analysis technique can be advantageously used to improve the accuracy of measurement.

© 2005 Optical Society of America

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References

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  1. R. Kingslake, ed., Applied Optics and Optical Engineering (Academic, New York, 1965), Vol. 1, Chap. IV, p. 208.
  2. I. Geatt, O. Kafri, “Determination of the focal length of nonparaxial lenses by Moiré deflectometry,” Appl. Opt. 26, 2507–2508 (1987).
    [CrossRef]
  3. E. Keren, M. K. Kreske, O. Kafri, “Universal method for determining the focal length of optical systems by moiré deflectometry,” Appl. Opt. 27, 1383–1385 (1988).
    [CrossRef] [PubMed]
  4. S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattin, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
    [CrossRef]
  5. Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24, 3162–3166 (1985).
    [CrossRef] [PubMed]
  6. 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]
  7. K. V. Sriram, M. P. Kothiyal, R. S. Sirohi, “Direct determination of focal length by using Talbot interferometry,” Appl. Opt. 31, 5984–5987 (1992).
    [CrossRef] [PubMed]
  8. C. Shakher, “Talbot interferometry and its applications, presented at the National Symposium on Perspectives in Engineering Optics,” Chaudhary Charan Singh University, Meerut, India, 4–6 April2003.
  9. S. Mirza, C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Opt. Eng., to be published.
  10. L. Angel, M. Tenaldi, R. Henao, “Phase stepping in Lau interferometry,” Opt. Commun. 164, 247–255 (1999).
    [CrossRef]
  11. M. Thakur, C. Shakher, “Evaluation of the focal distance of lenses by white-light Lau phase interferometry,” Appl. Opt. 41, 1841–1845 (2002).
    [CrossRef] [PubMed]
  12. D. Malacara, ed., “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), Vol. 2, Chap. 14, p. 23.
  13. H. Canabal, J. A. Quigoga, E. Bernabeu, “Improved phase-shifting method for automatic processing of moiré deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
    [CrossRef]
  14. I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurement,” Opt. Eng. 35, 2930–2937 (1996).
    [CrossRef]
  15. K. Omura, T. Yatagai, “Phase measuring Ronchi test,” Appl. Opt. 27, 523–528 (1988).
    [CrossRef] [PubMed]
  16. P. Picart, J. C. Pascal, J. M. Breteau, “Systematic error of phase-shifting speckle interferometry,” Appl. Opt. 40, 2107–2116 (2001).
    [CrossRef]
  17. L. Mertz, “Real-time fringe pattern analysis,” Appl. Opt. 22, 1535–1539 (1983).
    [CrossRef]
  18. G. Bonsch, H. Bohme, “Phase determination of Fizeau interference by phase-shifting interferometry,” Optik (Stuttgart) 82, 161 (1989).
  19. G. Molesini, M. V. P. Desouza, F. Quercioli, M. Trivi, “Digital phase-shifting interferometry applied to partially developed speckle fields,” Opt. Commun. 75, 14–17 (1990).
    [CrossRef]
  20. L. Wansong, S. Xianyu, “Real time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
    [CrossRef]
  21. T. M. Kreis, “Quantitative evaluation of the interference pattern,” in Industrial Optoelectronic Measurement Systems Using Coherent Light, W. F. Fagan, ed., Proc. SPIE863, 68–77 (1987).
    [CrossRef]
  22. Y. Nakano, K. Murata, “Measurement of phase objects using the Talbot effect and a moiré technique,” Appl. Opt. 23, 2296–2299 (1984).
    [CrossRef]

2002 (1)

2001 (2)

P. Picart, J. C. Pascal, J. M. Breteau, “Systematic error of phase-shifting speckle interferometry,” Appl. Opt. 40, 2107–2116 (2001).
[CrossRef]

L. Wansong, S. Xianyu, “Real time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

1999 (1)

L. Angel, M. Tenaldi, R. Henao, “Phase stepping in Lau interferometry,” Opt. Commun. 164, 247–255 (1999).
[CrossRef]

1998 (1)

1996 (2)

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurement,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattin, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

1992 (1)

1990 (1)

G. Molesini, M. V. P. Desouza, F. Quercioli, M. Trivi, “Digital phase-shifting interferometry applied to partially developed speckle fields,” Opt. Commun. 75, 14–17 (1990).
[CrossRef]

1989 (1)

G. Bonsch, H. Bohme, “Phase determination of Fizeau interference by phase-shifting interferometry,” Optik (Stuttgart) 82, 161 (1989).

1988 (3)

1987 (1)

1985 (1)

1984 (1)

1983 (1)

Angel, L.

L. Angel, M. Tenaldi, R. Henao, “Phase stepping in Lau interferometry,” Opt. Commun. 164, 247–255 (1999).
[CrossRef]

Bernabeu, E.

Bernardo, L. M.

Bohme, H.

G. Bonsch, H. Bohme, “Phase determination of Fizeau interference by phase-shifting interferometry,” Optik (Stuttgart) 82, 161 (1989).

Bonsch, G.

G. Bonsch, H. Bohme, “Phase determination of Fizeau interference by phase-shifting interferometry,” Optik (Stuttgart) 82, 161 (1989).

Breteau, J. M.

Canabal, H.

Desouza, M. V. P.

G. Molesini, M. V. P. Desouza, F. Quercioli, M. Trivi, “Digital phase-shifting interferometry applied to partially developed speckle fields,” Opt. Commun. 75, 14–17 (1990).
[CrossRef]

Ferraro, P.

S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattin, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

Finizio, A.

S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattin, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

Geatt, I.

Henao, R.

L. Angel, M. Tenaldi, R. Henao, “Phase stepping in Lau interferometry,” Opt. Commun. 164, 247–255 (1999).
[CrossRef]

Kafri, O.

Kato, J.

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurement,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Keren, E.

Kothiyal, M. P.

Kreis, T. M.

T. M. Kreis, “Quantitative evaluation of the interference pattern,” in Industrial Optoelectronic Measurement Systems Using Coherent Light, W. F. Fagan, ed., Proc. SPIE863, 68–77 (1987).
[CrossRef]

Kreske, M. K.

Liu, J.

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurement,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Mertz, L.

Mirza, S.

S. Mirza, C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Opt. Eng., to be published.

Molesini, G.

G. Molesini, M. V. P. Desouza, F. Quercioli, M. Trivi, “Digital phase-shifting interferometry applied to partially developed speckle fields,” Opt. Commun. 75, 14–17 (1990).
[CrossRef]

Murata, K.

Nakano, Y.

Nicola, S. D.

S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattin, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

Omura, K.

Pascal, J. C.

Picart, P.

Pierattin, G.

S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattin, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

Quercioli, F.

G. Molesini, M. V. P. Desouza, F. Quercioli, M. Trivi, “Digital phase-shifting interferometry applied to partially developed speckle fields,” Opt. Commun. 75, 14–17 (1990).
[CrossRef]

Quigoga, J. A.

Shakher, C.

M. Thakur, C. Shakher, “Evaluation of the focal distance of lenses by white-light Lau phase interferometry,” Appl. Opt. 41, 1841–1845 (2002).
[CrossRef] [PubMed]

S. Mirza, C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Opt. Eng., to be published.

C. Shakher, “Talbot interferometry and its applications, presented at the National Symposium on Perspectives in Engineering Optics,” Chaudhary Charan Singh University, Meerut, India, 4–6 April2003.

Sirohi, R. S.

Soares, O. D. D.

Sriram, K. V.

Tenaldi, M.

L. Angel, M. Tenaldi, R. Henao, “Phase stepping in Lau interferometry,” Opt. Commun. 164, 247–255 (1999).
[CrossRef]

Thakur, M.

Trivi, M.

G. Molesini, M. V. P. Desouza, F. Quercioli, M. Trivi, “Digital phase-shifting interferometry applied to partially developed speckle fields,” Opt. Commun. 75, 14–17 (1990).
[CrossRef]

Wansong, L.

L. Wansong, S. Xianyu, “Real time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

Xianyu, S.

L. Wansong, S. Xianyu, “Real time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurement,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Yatagai, T.

Appl. Opt. (11)

L. Mertz, “Real-time fringe pattern analysis,” Appl. Opt. 22, 1535–1539 (1983).
[CrossRef]

Y. Nakano, K. Murata, “Measurement of phase objects using the Talbot effect and a moiré technique,” Appl. Opt. 23, 2296–2299 (1984).
[CrossRef]

Y. Nakano, K. Murata, “Talbot interferometry for measuring the focal length of a lens,” Appl. Opt. 24, 3162–3166 (1985).
[CrossRef] [PubMed]

K. Omura, T. Yatagai, “Phase measuring Ronchi test,” Appl. Opt. 27, 523–528 (1988).
[CrossRef] [PubMed]

H. Canabal, J. A. Quigoga, E. Bernabeu, “Improved phase-shifting method for automatic processing of moiré deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
[CrossRef]

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]

P. Picart, J. C. Pascal, J. M. Breteau, “Systematic error of phase-shifting speckle interferometry,” Appl. Opt. 40, 2107–2116 (2001).
[CrossRef]

M. Thakur, C. Shakher, “Evaluation of the focal distance of lenses by white-light Lau phase interferometry,” Appl. Opt. 41, 1841–1845 (2002).
[CrossRef] [PubMed]

K. V. Sriram, M. P. Kothiyal, R. S. Sirohi, “Direct determination of focal length by using Talbot interferometry,” Appl. Opt. 31, 5984–5987 (1992).
[CrossRef] [PubMed]

E. Keren, M. K. Kreske, O. Kafri, “Universal method for determining the focal length of optical systems by moiré deflectometry,” Appl. Opt. 27, 1383–1385 (1988).
[CrossRef] [PubMed]

I. Geatt, O. Kafri, “Determination of the focal length of nonparaxial lenses by Moiré deflectometry,” Appl. Opt. 26, 2507–2508 (1987).
[CrossRef]

Opt. Commun. (3)

S. D. Nicola, P. Ferraro, A. Finizio, G. Pierattin, “Reflective grating interferometer for measuring the focal length of lens by digital moiré effect,” Opt. Commun. 132, 432–436 (1996).
[CrossRef]

L. Angel, M. Tenaldi, R. Henao, “Phase stepping in Lau interferometry,” Opt. Commun. 164, 247–255 (1999).
[CrossRef]

G. Molesini, M. V. P. Desouza, F. Quercioli, M. Trivi, “Digital phase-shifting interferometry applied to partially developed speckle fields,” Opt. Commun. 75, 14–17 (1990).
[CrossRef]

Opt. Eng. (2)

L. Wansong, S. Xianyu, “Real time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

I. Yamaguchi, J. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurement,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Optik (Stuttgart) (1)

G. Bonsch, H. Bohme, “Phase determination of Fizeau interference by phase-shifting interferometry,” Optik (Stuttgart) 82, 161 (1989).

Other (5)

T. M. Kreis, “Quantitative evaluation of the interference pattern,” in Industrial Optoelectronic Measurement Systems Using Coherent Light, W. F. Fagan, ed., Proc. SPIE863, 68–77 (1987).
[CrossRef]

R. Kingslake, ed., Applied Optics and Optical Engineering (Academic, New York, 1965), Vol. 1, Chap. IV, p. 208.

D. Malacara, ed., “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed. (Wiley, New York, 1992), Vol. 2, Chap. 14, p. 23.

C. Shakher, “Talbot interferometry and its applications, presented at the National Symposium on Perspectives in Engineering Optics,” Chaudhary Charan Singh University, Meerut, India, 4–6 April2003.

S. Mirza, C. Shakher, “Surface profiling using phase shifting Talbot interferometric technique,” Opt. Eng., to be published.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the experimental setup.

Fig. 2
Fig. 2

Moiré fringes when the test lens is in between the gratings (f = 240 mm).

Fig. 3
Fig. 3

Fringes after Fourier filtering (f = 240 mm).

Fig. 4
Fig. 4

Three-dimensional phase map for a lens with a focal length of 240 mm.

Fig. 5
Fig. 5

Two-dimensional phase map for a lens with a focal length of 240 mm.

Tables (3)

Tables Icon

Table 1 Measurement of a Lens with a Focal Length of 240 mm

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Table 2 Measurement of a Lens with a Focal Length of 210 mm

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Table 3 Measurement of a Lens with a Focal Length of 150 mm

Equations (7)

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

I ( x ,     y ) = I ( x ,     y ) + I ( x ,     y ) cos [ θ ( x ,     y ) + ϕ ] ,
θ ( x ,     y ) = tan - 1 I 4 ( x ,     y ) - I 2 ( x ,     y ) I 1 ( x ,     y ) - I 3 ( x ,     y ) .
θ ( x ) = k = 1 1 2 k - 1 2 k - 1 φ ( x ) x 2 k - 1 Δ 2 k - 1 ,
θ ( x ) = [ ϕ ( x ,     y ) / x ] Δ .
ϕ = ( π / λ f T ) ( x 2 + y 2 ) .
ϕ ( x ,     y ) / x = ( π / λ f T ) 2 x .
I ( x ,     y ) = I ( x ,     y ) + I ( x ,     y ) cos { [ ϕ ( x ,     y ) / x ] Δ + ϕ } .

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