Abstract

We present an easy, simple, and inexpensive technique for checking the quality of the collimation of optical beams using the Lau effect combined with moiré readout. The experimental arrangement consists of a modified Lau-based interferometer in which a white-light incoherent source illuminates a set of two gratings. A collimating lens is placed between the two gratings such that the self-images of the second grating are formed. The third grating is positioned at one of the self-imaging planes forming moiré fringes. The type of the moiré fringe demonstrates the quality of collimation of the optical beam. The necessary theoretical background is presented and the results of our experimental investigation are reported. The technique can also be used for accurate determination of the focal length of a collimating lens using low-cost components.

© 2006 Optical Society of America

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  1. D. Crespo, J. Alonso, and E. Bernabeu, "Generalized grating imaging using an extended monochromatic light source," J. Opt. Soc. Am. A 17, 1231-1240 (2000).
    [CrossRef]
  2. D. Crespo, J. Alonso, and E. Bernabeu, "Experimental measurements of generalized grating images," Appl. Opt. 41, 1223-1228 (2002).
    [CrossRef] [PubMed]
  3. C. S. Avudainayagam, K. V. Avudainayagam, and D. C. Atchison, "Lau fringes and subjective refraction," Opt. Lett. 27, 476-478 (2002).
    [CrossRef]
  4. M. Thakur and C. Shakher, "Evaluation of focal distance of lenses by white-light phase interferometry," Appl. Opt. 41, 1841-1845 (2002).
    [CrossRef] [PubMed]
  5. C. J. Tay, M. Thakur, L. Chen, and C. Shakher, "Measurement of focal length of lens using phase shifting Lau phase interferometry," Opt. Commun. 248, 339-345 (2005).
    [CrossRef]
  6. M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature and temperature profile of an axisymmetric gaseous flame using Lau phase interferometer with linear gratings," Opt. Lasers Eng. 36, 373-380 (2001).
    [CrossRef]
  7. M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature profile of a gaseous flame with a Lau phase interferometer that has circular gratings," Appl. Opt. 41, 654-656 (2002).
    [CrossRef] [PubMed]
  8. M. Thakur, S. K. Angra, and C. Shakher, "Measurements of the temperature profile of a two-dimensional slot burner using a Lau phase interferometer with linear gratings," Opt. Eng. 42, 86-91 (2003).
    [CrossRef]
  9. S. Prakash, S. Singh, and S. Rana, "Automated small tilt-angle measurement using Lau interferometry," Appl. Opt. 44, 5905-5909 (2005).
    [CrossRef] [PubMed]
  10. D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
    [CrossRef]
  11. H. Hammam, "Lau array illuminator," Appl. Opt. 43, 2888-2894 (2004).
    [CrossRef]
  12. S. Prakash, A. Sharma, and S. Rana, "Array illuminator based on self-imaging in white light," Opt. Laser Technol. (to be published).
  13. E. Lau, "Beugungserscheinung an Doppelrastern," Ann. Phys. (Leipzig) 6, 417-427 (1948).
    [CrossRef]
  14. J. Jahns and A. W. Lohmann, "The Lau effect (a diffraction experiment with incoherent illumination)," Opt. Commun. 28, 263-267 (1979).
    [CrossRef]
  15. F. Gori, "Lau effect and coherence theory," Opt. Commun. 31, 4-8 (1979).
    [CrossRef]
  16. G. J. Swanson and E. N. Leith, "Lau effect and grating imaging," J. Opt. Soc. Am. 72, 552-555 (1982).
    [CrossRef]
  17. K. Patorski, "Incoherent superposition of multiple self-imaging under plane wavefront illumination," Appl. Opt. 25, 2396-2403 (1986).
    [CrossRef] [PubMed]
  18. D. E. Silva, "A simple interferometric method of beam collimation," Appl. Opt. 10, 1980-1982 (1971).
    [CrossRef]
  19. J. C. Fouere and D. Malacara, "Focusing errors in a collimating lens or mirror: use of a moiré technique," Appl. Opt. 13, 1322-1326 (1974).
    [CrossRef] [PubMed]
  20. J. Choi, G. M. Perera, M. D. Agarwal, R. P. Shukla, and M. V. Mantravadi, "Wedge-plate shearing interferometers for collimation testing: use of a moiré technique," Appl. Opt. 34, 3628-3638 (1995).
    [CrossRef] [PubMed]
  21. K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Self-referencing collimation testing techniques," Opt. Eng. 32, 94-100 (1993).
    [CrossRef]
  22. A. Anand, V. K. Chhaniwal, and C. S. Narayanamurthy, "Collimation testing with optically active materials," Appl. Opt. 44, 4244-4247 (2005).
    [CrossRef] [PubMed]
  23. K. V. Avudainayagam and S. Chitralekha, "Lau effect and beam collimation," J. Mod. Opt. 44, 175-178 (1997).
    [CrossRef]
  24. J. O. Castaneda and E. E. Sicre, "Theta modulation decoder based on Lau effect," Opt. Commun. 59, 87-91 (1986).
    [CrossRef]
  25. J. M. Cowley and A. F. Moodie, "Fourier images. 1. The point sources," Proc. Phys. Soc. B 70, 486-496 (1957).
    [CrossRef]
  26. S. Yokozeki, K. Patorski, and K. Ohnishi, "Collimation method using Fourier imaging and moiré techniques," Opt. Commun. 14, 401-405 (1975).
    [CrossRef]
  27. D. E. Silva, "Talbot interferometer for radial and lateral derivatives," Appl. Opt. 11, 2613-2624 (1972).
    [CrossRef] [PubMed]
  28. H. O. Bartelt and J. Jahns, "Interferometry based on Lau effect," Opt. Commun. 30, 268-274 (1979).
    [CrossRef]

2005 (3)

2004 (1)

2003 (1)

M. Thakur, S. K. Angra, and C. Shakher, "Measurements of the temperature profile of a two-dimensional slot burner using a Lau phase interferometer with linear gratings," Opt. Eng. 42, 86-91 (2003).
[CrossRef]

2002 (4)

2001 (1)

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature and temperature profile of an axisymmetric gaseous flame using Lau phase interferometer with linear gratings," Opt. Lasers Eng. 36, 373-380 (2001).
[CrossRef]

2000 (2)

D. Crespo, J. Alonso, and E. Bernabeu, "Generalized grating imaging using an extended monochromatic light source," J. Opt. Soc. Am. A 17, 1231-1240 (2000).
[CrossRef]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

1997 (1)

K. V. Avudainayagam and S. Chitralekha, "Lau effect and beam collimation," J. Mod. Opt. 44, 175-178 (1997).
[CrossRef]

1995 (1)

1993 (1)

K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Self-referencing collimation testing techniques," Opt. Eng. 32, 94-100 (1993).
[CrossRef]

1986 (2)

J. O. Castaneda and E. E. Sicre, "Theta modulation decoder based on Lau effect," Opt. Commun. 59, 87-91 (1986).
[CrossRef]

K. Patorski, "Incoherent superposition of multiple self-imaging under plane wavefront illumination," Appl. Opt. 25, 2396-2403 (1986).
[CrossRef] [PubMed]

1982 (1)

1979 (3)

J. Jahns and A. W. Lohmann, "The Lau effect (a diffraction experiment with incoherent illumination)," Opt. Commun. 28, 263-267 (1979).
[CrossRef]

F. Gori, "Lau effect and coherence theory," Opt. Commun. 31, 4-8 (1979).
[CrossRef]

H. O. Bartelt and J. Jahns, "Interferometry based on Lau effect," Opt. Commun. 30, 268-274 (1979).
[CrossRef]

1975 (1)

S. Yokozeki, K. Patorski, and K. Ohnishi, "Collimation method using Fourier imaging and moiré techniques," Opt. Commun. 14, 401-405 (1975).
[CrossRef]

1974 (1)

1972 (1)

1971 (1)

1957 (1)

J. M. Cowley and A. F. Moodie, "Fourier images. 1. The point sources," Proc. Phys. Soc. B 70, 486-496 (1957).
[CrossRef]

1948 (1)

E. Lau, "Beugungserscheinung an Doppelrastern," Ann. Phys. (Leipzig) 6, 417-427 (1948).
[CrossRef]

Agarwal, M. D.

Alonso, J.

Anand, A.

Angra, S. K.

M. Thakur, S. K. Angra, and C. Shakher, "Measurements of the temperature profile of a two-dimensional slot burner using a Lau phase interferometer with linear gratings," Opt. Eng. 42, 86-91 (2003).
[CrossRef]

Atchison, D. C.

Avudainayagam, C. S.

Avudainayagam, K. V.

C. S. Avudainayagam, K. V. Avudainayagam, and D. C. Atchison, "Lau fringes and subjective refraction," Opt. Lett. 27, 476-478 (2002).
[CrossRef]

K. V. Avudainayagam and S. Chitralekha, "Lau effect and beam collimation," J. Mod. Opt. 44, 175-178 (1997).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt and J. Jahns, "Interferometry based on Lau effect," Opt. Commun. 30, 268-274 (1979).
[CrossRef]

Bernabeu, E.

Castaneda, J. O.

J. O. Castaneda and E. E. Sicre, "Theta modulation decoder based on Lau effect," Opt. Commun. 59, 87-91 (1986).
[CrossRef]

Chen, L.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, "Measurement of focal length of lens using phase shifting Lau phase interferometry," Opt. Commun. 248, 339-345 (2005).
[CrossRef]

Chhaniwal, V. K.

Chitralekha, S.

K. V. Avudainayagam and S. Chitralekha, "Lau effect and beam collimation," J. Mod. Opt. 44, 175-178 (1997).
[CrossRef]

Choi, J.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, "Fourier images. 1. The point sources," Proc. Phys. Soc. B 70, 486-496 (1957).
[CrossRef]

Crespo, D.

Fouere, J. C.

Gori, F.

F. Gori, "Lau effect and coherence theory," Opt. Commun. 31, 4-8 (1979).
[CrossRef]

Hammam, H.

Jahns, J.

J. Jahns and A. W. Lohmann, "The Lau effect (a diffraction experiment with incoherent illumination)," Opt. Commun. 28, 263-267 (1979).
[CrossRef]

H. O. Bartelt and J. Jahns, "Interferometry based on Lau effect," Opt. Commun. 30, 268-274 (1979).
[CrossRef]

Kothiyal, M. P.

K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Self-referencing collimation testing techniques," Opt. Eng. 32, 94-100 (1993).
[CrossRef]

Lau, E.

E. Lau, "Beugungserscheinung an Doppelrastern," Ann. Phys. (Leipzig) 6, 417-427 (1948).
[CrossRef]

Leith, E. N.

Lohmann, A. W.

J. Jahns and A. W. Lohmann, "The Lau effect (a diffraction experiment with incoherent illumination)," Opt. Commun. 28, 263-267 (1979).
[CrossRef]

Malacara, D.

Mantravadi, M. V.

Moodie, A. F.

J. M. Cowley and A. F. Moodie, "Fourier images. 1. The point sources," Proc. Phys. Soc. B 70, 486-496 (1957).
[CrossRef]

Morlanes, T.

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

Narayanamurthy, C. S.

Ohnishi, K.

S. Yokozeki, K. Patorski, and K. Ohnishi, "Collimation method using Fourier imaging and moiré techniques," Opt. Commun. 14, 401-405 (1975).
[CrossRef]

Patorski, K.

K. Patorski, "Incoherent superposition of multiple self-imaging under plane wavefront illumination," Appl. Opt. 25, 2396-2403 (1986).
[CrossRef] [PubMed]

S. Yokozeki, K. Patorski, and K. Ohnishi, "Collimation method using Fourier imaging and moiré techniques," Opt. Commun. 14, 401-405 (1975).
[CrossRef]

Perera, G. M.

Prakash, S.

S. Prakash, S. Singh, and S. Rana, "Automated small tilt-angle measurement using Lau interferometry," Appl. Opt. 44, 5905-5909 (2005).
[CrossRef] [PubMed]

S. Prakash, A. Sharma, and S. Rana, "Array illuminator based on self-imaging in white light," Opt. Laser Technol. (to be published).

Rana, S.

S. Prakash, S. Singh, and S. Rana, "Automated small tilt-angle measurement using Lau interferometry," Appl. Opt. 44, 5905-5909 (2005).
[CrossRef] [PubMed]

S. Prakash, A. Sharma, and S. Rana, "Array illuminator based on self-imaging in white light," Opt. Laser Technol. (to be published).

Shakher, C.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, "Measurement of focal length of lens using phase shifting Lau phase interferometry," Opt. Commun. 248, 339-345 (2005).
[CrossRef]

M. Thakur, S. K. Angra, and C. Shakher, "Measurements of the temperature profile of a two-dimensional slot burner using a Lau phase interferometer with linear gratings," Opt. Eng. 42, 86-91 (2003).
[CrossRef]

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature profile of a gaseous flame with a Lau phase interferometer that has circular gratings," Appl. Opt. 41, 654-656 (2002).
[CrossRef] [PubMed]

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

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature and temperature profile of an axisymmetric gaseous flame using Lau phase interferometer with linear gratings," Opt. Lasers Eng. 36, 373-380 (2001).
[CrossRef]

Sharma, A.

S. Prakash, A. Sharma, and S. Rana, "Array illuminator based on self-imaging in white light," Opt. Laser Technol. (to be published).

Shukla, R. P.

Sicre, E. E.

J. O. Castaneda and E. E. Sicre, "Theta modulation decoder based on Lau effect," Opt. Commun. 59, 87-91 (1986).
[CrossRef]

Silva, D. E.

Singh, S.

Sirohi, R. S.

K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Self-referencing collimation testing techniques," Opt. Eng. 32, 94-100 (1993).
[CrossRef]

Sriram, K. V.

K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Self-referencing collimation testing techniques," Opt. Eng. 32, 94-100 (1993).
[CrossRef]

Swanson, G. J.

Tay, C. J.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, "Measurement of focal length of lens using phase shifting Lau phase interferometry," Opt. Commun. 248, 339-345 (2005).
[CrossRef]

Thakur, M.

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, "Measurement of focal length of lens using phase shifting Lau phase interferometry," Opt. Commun. 248, 339-345 (2005).
[CrossRef]

M. Thakur, S. K. Angra, and C. Shakher, "Measurements of the temperature profile of a two-dimensional slot burner using a Lau phase interferometer with linear gratings," Opt. Eng. 42, 86-91 (2003).
[CrossRef]

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature profile of a gaseous flame with a Lau phase interferometer that has circular gratings," Appl. Opt. 41, 654-656 (2002).
[CrossRef] [PubMed]

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

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature and temperature profile of an axisymmetric gaseous flame using Lau phase interferometer with linear gratings," Opt. Lasers Eng. 36, 373-380 (2001).
[CrossRef]

Vyas, A. L.

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature profile of a gaseous flame with a Lau phase interferometer that has circular gratings," Appl. Opt. 41, 654-656 (2002).
[CrossRef] [PubMed]

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature and temperature profile of an axisymmetric gaseous flame using Lau phase interferometer with linear gratings," Opt. Lasers Eng. 36, 373-380 (2001).
[CrossRef]

Yokozeki, S.

S. Yokozeki, K. Patorski, and K. Ohnishi, "Collimation method using Fourier imaging and moiré techniques," Opt. Commun. 14, 401-405 (1975).
[CrossRef]

Ann. Phys. (1)

E. Lau, "Beugungserscheinung an Doppelrastern," Ann. Phys. (Leipzig) 6, 417-427 (1948).
[CrossRef]

Appl. Opt. (11)

H. Hammam, "Lau array illuminator," Appl. Opt. 43, 2888-2894 (2004).
[CrossRef]

K. Patorski, "Incoherent superposition of multiple self-imaging under plane wavefront illumination," Appl. Opt. 25, 2396-2403 (1986).
[CrossRef] [PubMed]

D. E. Silva, "A simple interferometric method of beam collimation," Appl. Opt. 10, 1980-1982 (1971).
[CrossRef]

J. C. Fouere and D. Malacara, "Focusing errors in a collimating lens or mirror: use of a moiré technique," Appl. Opt. 13, 1322-1326 (1974).
[CrossRef] [PubMed]

J. Choi, G. M. Perera, M. D. Agarwal, R. P. Shukla, and M. V. Mantravadi, "Wedge-plate shearing interferometers for collimation testing: use of a moiré technique," Appl. Opt. 34, 3628-3638 (1995).
[CrossRef] [PubMed]

D. Crespo, J. Alonso, and E. Bernabeu, "Experimental measurements of generalized grating images," Appl. Opt. 41, 1223-1228 (2002).
[CrossRef] [PubMed]

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

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature profile of a gaseous flame with a Lau phase interferometer that has circular gratings," Appl. Opt. 41, 654-656 (2002).
[CrossRef] [PubMed]

S. Prakash, S. Singh, and S. Rana, "Automated small tilt-angle measurement using Lau interferometry," Appl. Opt. 44, 5905-5909 (2005).
[CrossRef] [PubMed]

A. Anand, V. K. Chhaniwal, and C. S. Narayanamurthy, "Collimation testing with optically active materials," Appl. Opt. 44, 4244-4247 (2005).
[CrossRef] [PubMed]

D. E. Silva, "Talbot interferometer for radial and lateral derivatives," Appl. Opt. 11, 2613-2624 (1972).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

K. V. Avudainayagam and S. Chitralekha, "Lau effect and beam collimation," J. Mod. Opt. 44, 175-178 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (6)

H. O. Bartelt and J. Jahns, "Interferometry based on Lau effect," Opt. Commun. 30, 268-274 (1979).
[CrossRef]

J. O. Castaneda and E. E. Sicre, "Theta modulation decoder based on Lau effect," Opt. Commun. 59, 87-91 (1986).
[CrossRef]

S. Yokozeki, K. Patorski, and K. Ohnishi, "Collimation method using Fourier imaging and moiré techniques," Opt. Commun. 14, 401-405 (1975).
[CrossRef]

C. J. Tay, M. Thakur, L. Chen, and C. Shakher, "Measurement of focal length of lens using phase shifting Lau phase interferometry," Opt. Commun. 248, 339-345 (2005).
[CrossRef]

J. Jahns and A. W. Lohmann, "The Lau effect (a diffraction experiment with incoherent illumination)," Opt. Commun. 28, 263-267 (1979).
[CrossRef]

F. Gori, "Lau effect and coherence theory," Opt. Commun. 31, 4-8 (1979).
[CrossRef]

Opt. Eng. (3)

K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, "Self-referencing collimation testing techniques," Opt. Eng. 32, 94-100 (1993).
[CrossRef]

D. Crespo, J. Alonso, T. Morlanes, and E. Bernabeu, "Optical encoder based on the Lau effect," Opt. Eng. 39, 817-824 (2000).
[CrossRef]

M. Thakur, S. K. Angra, and C. Shakher, "Measurements of the temperature profile of a two-dimensional slot burner using a Lau phase interferometer with linear gratings," Opt. Eng. 42, 86-91 (2003).
[CrossRef]

Opt. Laser Technol. (1)

S. Prakash, A. Sharma, and S. Rana, "Array illuminator based on self-imaging in white light," Opt. Laser Technol. (to be published).

Opt. Lasers Eng. (1)

M. Thakur, A. L. Vyas, and C. Shakher, "Measurement of temperature and temperature profile of an axisymmetric gaseous flame using Lau phase interferometer with linear gratings," Opt. Lasers Eng. 36, 373-380 (2001).
[CrossRef]

Opt. Lett. (1)

Proc. Phys. Soc. B (1)

J. M. Cowley and A. F. Moodie, "Fourier images. 1. The point sources," Proc. Phys. Soc. B 70, 486-496 (1957).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of the experimental arrangement for collimation testing using the Lau effect combined with moiré readout. EWLS, extended white-light source.

Fig. 2
Fig. 2

Image recorded via CCD camera at (a) in-focus, (b) at-focus, and (c) out-of-focus positions of a collimating lens of focal length 250 mm.

Equations (5)

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1 / Z J + 1 / Z W = 1 / 2 J Z T ,
M = 1 + Z J / Z W .
P = d 2 d 3 / | d 2 d 3 | ,
P = M d / | M 1 | .
| 1 Z W | ( 2 N Z J ) 1 ,

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