Abstract

Phase-shifting in an achromatic configuration of moiré interferometry is explained. The rigorous diffraction theory defines the phase of diffracted beams in terms of the pitch and the relative position of a compensator grating. A numerical analysis proceeds to determine the total phase change between two diffracted beams that produce moiré fringes.

© 2007 Optical Society of America

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Phase-shifting and logical moiré

A. Asundi and K. H. Yung
J. Opt. Soc. Am. A 8(10) 1591-1600 (1991)

References

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  1. D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials, Mechanical Engineering Series, (Springer-Verlag, NY, 1994, Student ed. 1997).
    [Crossref]
  2. L. Salbut and K. Patorski, “Polarization phase shifting method for moiré interferometry and flatness testing,” Appl. Opt. 29, 1471–1473 (1990).
    [Crossref] [PubMed]
  3. M. Kujawinska, L. Salbut, and K. Patorski, “Three-channel phase stepped system for moiré interferometry,” Appl. Opt. 30, 1633–1635 (1991).
    [Crossref] [PubMed]
  4. B. Han, “Higher sensitivity moiré interferometry for micromechanics studies,” Opt. Eng. 31, 1517–1526 (1992).
    [Crossref]
  5. B. Han, “Interferometric methods with enhanced sensitivity by optical/digital fringe multiplication,” Appl. Opt. 32, 4713–4718 (1993).
    [Crossref] [PubMed]
  6. X. He, D. Zou, and S. Liu, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
    [Crossref]
  7. M. R. Miller, I. Mohammed, X. Dai, N. Jiang, and P. S. Ho, “Analysis of flip chip packages using high resolution moiré interferometry,” in Proceedings of 49th Electronic Components and Technology Conference, (1999), pp. 979–986.
  8. H. Liu, A. N. Cartwright, and C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
    [Crossref]
  9. B. Han, D. Columbus, Z. Wu, and J. Lu, “Mechanical fringe shifting in moiré interferometry,” Exp. Tech. 18, 1, 16–19 (1999).
    [Crossref]
  10. D. Post, “Moiré interferometry in white light,” Appl. Opt. 18, 4163–4167 (1979).
    [Crossref] [PubMed]
  11. D. H. Mollenhauer, P. Ifju, and B. Han, “A compact, robust, and versatile moiré interferometer,” Opt. Lasers Eng. 23, 29–40 (1995).
    [Crossref]
  12. D. MaystreE. Wolf, “Rigorous vector theories of diffraction gratings,” ed. (North-Holland, Amsterdam, 1989).
  13. B. Han and D. Post, “Immersion interferometer for microscopic moiré interferometry,” Exp. Mech. 32, 38–41 (1992).
    [Crossref]
  14. C.-W. Han, S. Cho, and B. Han, “Transmission microscopic moiré interferometry,” in Proceedings of the SEM Annual Conference, (Charlotte, NC, June2–4, 2003), No 194.

2003 (1)

H. Liu, A. N. Cartwright, and C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[Crossref]

1999 (1)

B. Han, D. Columbus, Z. Wu, and J. Lu, “Mechanical fringe shifting in moiré interferometry,” Exp. Tech. 18, 1, 16–19 (1999).
[Crossref]

1998 (1)

X. He, D. Zou, and S. Liu, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[Crossref]

1995 (1)

D. H. Mollenhauer, P. Ifju, and B. Han, “A compact, robust, and versatile moiré interferometer,” Opt. Lasers Eng. 23, 29–40 (1995).
[Crossref]

1993 (1)

1992 (2)

B. Han, “Higher sensitivity moiré interferometry for micromechanics studies,” Opt. Eng. 31, 1517–1526 (1992).
[Crossref]

B. Han and D. Post, “Immersion interferometer for microscopic moiré interferometry,” Exp. Mech. 32, 38–41 (1992).
[Crossref]

1991 (1)

1990 (1)

1979 (1)

Basaran, C.

H. Liu, A. N. Cartwright, and C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[Crossref]

Cartwright, A. N.

H. Liu, A. N. Cartwright, and C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[Crossref]

Cho, S.

C.-W. Han, S. Cho, and B. Han, “Transmission microscopic moiré interferometry,” in Proceedings of the SEM Annual Conference, (Charlotte, NC, June2–4, 2003), No 194.

Columbus, D.

B. Han, D. Columbus, Z. Wu, and J. Lu, “Mechanical fringe shifting in moiré interferometry,” Exp. Tech. 18, 1, 16–19 (1999).
[Crossref]

Dai, X.

M. R. Miller, I. Mohammed, X. Dai, N. Jiang, and P. S. Ho, “Analysis of flip chip packages using high resolution moiré interferometry,” in Proceedings of 49th Electronic Components and Technology Conference, (1999), pp. 979–986.

Han, B.

B. Han, D. Columbus, Z. Wu, and J. Lu, “Mechanical fringe shifting in moiré interferometry,” Exp. Tech. 18, 1, 16–19 (1999).
[Crossref]

D. H. Mollenhauer, P. Ifju, and B. Han, “A compact, robust, and versatile moiré interferometer,” Opt. Lasers Eng. 23, 29–40 (1995).
[Crossref]

B. Han, “Interferometric methods with enhanced sensitivity by optical/digital fringe multiplication,” Appl. Opt. 32, 4713–4718 (1993).
[Crossref] [PubMed]

B. Han, “Higher sensitivity moiré interferometry for micromechanics studies,” Opt. Eng. 31, 1517–1526 (1992).
[Crossref]

B. Han and D. Post, “Immersion interferometer for microscopic moiré interferometry,” Exp. Mech. 32, 38–41 (1992).
[Crossref]

C.-W. Han, S. Cho, and B. Han, “Transmission microscopic moiré interferometry,” in Proceedings of the SEM Annual Conference, (Charlotte, NC, June2–4, 2003), No 194.

D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials, Mechanical Engineering Series, (Springer-Verlag, NY, 1994, Student ed. 1997).
[Crossref]

Han, C.-W.

C.-W. Han, S. Cho, and B. Han, “Transmission microscopic moiré interferometry,” in Proceedings of the SEM Annual Conference, (Charlotte, NC, June2–4, 2003), No 194.

He, X.

X. He, D. Zou, and S. Liu, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[Crossref]

Ho, P. S.

M. R. Miller, I. Mohammed, X. Dai, N. Jiang, and P. S. Ho, “Analysis of flip chip packages using high resolution moiré interferometry,” in Proceedings of 49th Electronic Components and Technology Conference, (1999), pp. 979–986.

Ifju, P.

D. H. Mollenhauer, P. Ifju, and B. Han, “A compact, robust, and versatile moiré interferometer,” Opt. Lasers Eng. 23, 29–40 (1995).
[Crossref]

D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials, Mechanical Engineering Series, (Springer-Verlag, NY, 1994, Student ed. 1997).
[Crossref]

Jiang, N.

M. R. Miller, I. Mohammed, X. Dai, N. Jiang, and P. S. Ho, “Analysis of flip chip packages using high resolution moiré interferometry,” in Proceedings of 49th Electronic Components and Technology Conference, (1999), pp. 979–986.

Kujawinska, M.

Liu, H.

H. Liu, A. N. Cartwright, and C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[Crossref]

Liu, S.

X. He, D. Zou, and S. Liu, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[Crossref]

Lu, J.

B. Han, D. Columbus, Z. Wu, and J. Lu, “Mechanical fringe shifting in moiré interferometry,” Exp. Tech. 18, 1, 16–19 (1999).
[Crossref]

Maystre, D.

D. MaystreE. Wolf, “Rigorous vector theories of diffraction gratings,” ed. (North-Holland, Amsterdam, 1989).

Miller, M. R.

M. R. Miller, I. Mohammed, X. Dai, N. Jiang, and P. S. Ho, “Analysis of flip chip packages using high resolution moiré interferometry,” in Proceedings of 49th Electronic Components and Technology Conference, (1999), pp. 979–986.

Mohammed, I.

M. R. Miller, I. Mohammed, X. Dai, N. Jiang, and P. S. Ho, “Analysis of flip chip packages using high resolution moiré interferometry,” in Proceedings of 49th Electronic Components and Technology Conference, (1999), pp. 979–986.

Mollenhauer, D. H.

D. H. Mollenhauer, P. Ifju, and B. Han, “A compact, robust, and versatile moiré interferometer,” Opt. Lasers Eng. 23, 29–40 (1995).
[Crossref]

Patorski, K.

Post, D.

B. Han and D. Post, “Immersion interferometer for microscopic moiré interferometry,” Exp. Mech. 32, 38–41 (1992).
[Crossref]

D. Post, “Moiré interferometry in white light,” Appl. Opt. 18, 4163–4167 (1979).
[Crossref] [PubMed]

D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials, Mechanical Engineering Series, (Springer-Verlag, NY, 1994, Student ed. 1997).
[Crossref]

Salbut, L.

Wu, Z.

B. Han, D. Columbus, Z. Wu, and J. Lu, “Mechanical fringe shifting in moiré interferometry,” Exp. Tech. 18, 1, 16–19 (1999).
[Crossref]

Zou, D.

X. He, D. Zou, and S. Liu, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[Crossref]

Appl. Opt. (4)

Exp. Mech. (1)

B. Han and D. Post, “Immersion interferometer for microscopic moiré interferometry,” Exp. Mech. 32, 38–41 (1992).
[Crossref]

Exp. Tech. (1)

B. Han, D. Columbus, Z. Wu, and J. Lu, “Mechanical fringe shifting in moiré interferometry,” Exp. Tech. 18, 1, 16–19 (1999).
[Crossref]

Opt. Eng. (3)

H. Liu, A. N. Cartwright, and C. Basaran, “Sensitivity improvement in phase-shifted moiré interferometry using 1-D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[Crossref]

X. He, D. Zou, and S. Liu, “Phase-shifting analysis in moiré interferometry and its application in electronic packaging,” Opt. Eng. 37, 1410–1419 (1998).
[Crossref]

B. Han, “Higher sensitivity moiré interferometry for micromechanics studies,” Opt. Eng. 31, 1517–1526 (1992).
[Crossref]

Opt. Lasers Eng. (1)

D. H. Mollenhauer, P. Ifju, and B. Han, “A compact, robust, and versatile moiré interferometer,” Opt. Lasers Eng. 23, 29–40 (1995).
[Crossref]

Other (4)

D. MaystreE. Wolf, “Rigorous vector theories of diffraction gratings,” ed. (North-Holland, Amsterdam, 1989).

D. Post, B. Han, and P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials, Mechanical Engineering Series, (Springer-Verlag, NY, 1994, Student ed. 1997).
[Crossref]

M. R. Miller, I. Mohammed, X. Dai, N. Jiang, and P. S. Ho, “Analysis of flip chip packages using high resolution moiré interferometry,” in Proceedings of 49th Electronic Components and Technology Conference, (1999), pp. 979–986.

C.-W. Han, S. Cho, and B. Han, “Transmission microscopic moiré interferometry,” in Proceedings of the SEM Annual Conference, (Charlotte, NC, June2–4, 2003), No 194.

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of phase-shifting in moiré interferometry; (a) mechanical phase-shifting in a conventional configuration and (b) phase-shifting in an achromatic configuration

Fig. 2.
Fig. 2.

Compensator grating with a pitch of gc and a profile of f(x) is translated linearly to change the phase of diffracted beams.

Fig. 3.
Fig. 3.

Phase change as a function of linear translation of the compensator grating: (a) the first order and (b) the second order diffraction beams.

Fig. 4.
Fig. 4.

Geometrical relationship between the diffraction order and the wave front is shown for (a) the first and (b) the second order diffraction. The corresponding wave front movements caused by translation of the compensator grating are shown in (c) and (d).

Fig. 5.
Fig. 5.

Two diffracted beams in an achromatic system; the directions of phase shift are opposite.

Equations (14)

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

δ = 2 Δ sin α
E i = exp ( i 2 π λ ( x sin α z cos α ) )
2 E + k 2 E = 0
E ( x , z ) = n E n ( x , z ) = n B n Φ n ( x , y ) = n B n exp ( i 2 π λ ( x sin θ n + z cos θ n ) )
n B n Φ n ( x , f ( x ) ) + E i ( x , f ( x ) ) = 0
n = N + N B n ( N ) Φ n ( x , f ( x ) ) + E i ( x , f ( x ) ) = 0
f ( x ) = h 2 cos ( 2 π f c ( x Δ ) ) h 2
ϕ = 2 π n f c Δ
sin θ n = n λ f c
sin θ n = n λ + Λ Δ + 1 f c = n λ f c Λ = n λ f c Δ
ϕ 1 n = 2 π n f c Δ and ϕ 2 n = 2 π n f c Δ
I ( x , z ) = I 1 ( x , z ) + I 2 ( x , z ) + 2 I 1 ( x , z ) I 2 ( x , z ) cos ( 2 π · 2 m f g U ( x , z ) + ϕ 2 ϕ 1 )
= I 1 ( x , z ) + I 2 ( x , z ) + 2 I 1 ( x , z ) I 2 ( x , z ) cos 2 π ( 2 m f g U ( x , z ) + 2 n f c Δ )
Δ 2 π = 1 2 n f c

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