Abstract

We investigate the properties of the moiré fringes of matched radial–parallel (MRP) gratings that are useful for the fine visual indication of angular displacements. Moiré patterns formed by overlapping two different MRP gratings of angular pitches, P θ and P θ′ (< P θ), can provide a simple fringe-counting method for determining both the angle and the direction of the relative angular displacement of the gratings within the accuracy of (P θ′/M θ), where M θ is the angular magnification for radial vernier fringes.

© 1997 Optical Society of America

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References

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  1. L. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93 (1874).
  2. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
    [CrossRef]
  3. Many useful articles on moire fringes are found in G. Indebetouw, R. Czarnek, eds., Selected Papers on Optical Moiré and Applications, Vol. MS64 of SPIE Milestone Series (SPIE, Bellingham, Wash., 1992).
  4. P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969), Chap. 2, pp. 19–24; Chap. 3, pp. 112–131.
    [CrossRef]
  5. A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970), Chap. 1, pp. 7–32; Chap. 3, pp. 79–91; Chap. 4, pp. 98–106.
  6. O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1990), Chap. 2, p. 36; Chap. 5, pp. 81–86.
  7. Y. Nishijima, G. Oster, “Moiré patterns: their application to refractive index and refractive-index gradient measurements,” J. Opt. Soc. Am. 54, 1–5 (1964).
    [CrossRef]
  8. G. Oster, M. Wasserman, C. Zwerling, “Theoretical interpretation of moiré patterns,” J. Opt. Soc. Am. 54, 169–175 (1964).
    [CrossRef]
  9. L. O. Vargady, “Moiré fringes as visual position indicators,” Appl. Opt. 3, 631–636 (1964).
    [CrossRef]
  10. D. M. Meadows, W. D. Johnson, J. B. Allen, “Generation of surface contours by moiré patterns,” Appl. Opt. 9, 942–947 (1970).
    [CrossRef] [PubMed]
  11. H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
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  12. M. C. King, D. H. Berry, “Photolithographic mask alignment using moiré techniques,” Appl. Opt. 11, 2455–2459 (1972).
    [CrossRef] [PubMed]
  13. G. T. Reid, “A moiré fringe alignment aid,” Opt. Laser Eng. 4, 121–126 (1983).
    [CrossRef]
  14. O. Bryngdahl, W. H. Lee, “Shearing interferometry in polar coordinates,” J. Opt. Soc. Am. 64, 1606–1615 (1974).
    [CrossRef]
  15. A. Pirard, “Consideration sur la methode du moiré en photoelasticite,” Anal. Contraintes, Mem. GAMAC 5, 1–24 (1960).
  16. H. H. M. Chau, “Moiré pattern resulting from superposition of two zone plates,” Appl. Opt. 8, 1707–1712 (1969).
    [CrossRef] [PubMed]
  17. H. H. M. Chau, “Properties of two overlapping zone plates of different focal lengths,” J. Opt. Soc. Am. 60, 255–259 (1970).
    [CrossRef]
  18. Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
    [CrossRef]
  19. B. Sandler, E. Keren, A. Livnat, O. Kafri, “Moiré patterns of skewed radial gratings,” Appl. Opt. 26, 772–773 (1987).
    [CrossRef] [PubMed]
  20. P. Szwaykowski, “Self-imaging in polar coordinates,” J. Opt. Soc. Am. A 5, 185–191 (1988).
    [CrossRef]
  21. P. Szwaykowski, K. Patorski, “Moiré fringes by evolute gratings,” Appl. Opt. 28, 4679–4681 (1989).
    [CrossRef] [PubMed]
  22. J. S. Song, B. J. Kim, J. H. Jo, S. Chang, K. C. Yuk, “Precise measurement of linear displacements by moiré fringes of elongated circular gratings,” in 17th Congress of the International Commission for Optics for Science and New Technology, J. S. Chang, J. H. Lee, C. H. Nam, eds., Proc. SPIE2778, 1116–1117 (1996).
  23. I. Glatt, O. Kafri, “Beam direction determination by moiré deflectometry using circular gratings,” Appl. Opt. 26, 4051–4053 (1987).
    [CrossRef] [PubMed]
  24. Y. C. Park, S. W. Kim, “Determination of two-dimensional planar displacement by moiré fringes of concentric circle gratings,” Appl. Opt. 33, 5171–5176 (1994).
    [CrossRef] [PubMed]

1994 (1)

1989 (1)

1988 (2)

Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
[CrossRef]

P. Szwaykowski, “Self-imaging in polar coordinates,” J. Opt. Soc. Am. A 5, 185–191 (1988).
[CrossRef]

1987 (2)

1983 (1)

G. T. Reid, “A moiré fringe alignment aid,” Opt. Laser Eng. 4, 121–126 (1983).
[CrossRef]

1974 (1)

1972 (1)

1970 (3)

1969 (1)

1964 (4)

1960 (1)

A. Pirard, “Consideration sur la methode du moiré en photoelasticite,” Anal. Contraintes, Mem. GAMAC 5, 1–24 (1960).

1874 (1)

L. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93 (1874).

Allen, J. B.

Berry, D. H.

Bryngdahl, O.

Chang, S.

J. S. Song, B. J. Kim, J. H. Jo, S. Chang, K. C. Yuk, “Precise measurement of linear displacements by moiré fringes of elongated circular gratings,” in 17th Congress of the International Commission for Optics for Science and New Technology, J. S. Chang, J. H. Lee, C. H. Nam, eds., Proc. SPIE2778, 1116–1117 (1996).

Chau, H. H. M.

Durelli, A. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970), Chap. 1, pp. 7–32; Chap. 3, pp. 79–91; Chap. 4, pp. 98–106.

Glatt, I.

I. Glatt, O. Kafri, “Beam direction determination by moiré deflectometry using circular gratings,” Appl. Opt. 26, 4051–4053 (1987).
[CrossRef] [PubMed]

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1990), Chap. 2, p. 36; Chap. 5, pp. 81–86.

Jo, J. H.

J. S. Song, B. J. Kim, J. H. Jo, S. Chang, K. C. Yuk, “Precise measurement of linear displacements by moiré fringes of elongated circular gratings,” in 17th Congress of the International Commission for Optics for Science and New Technology, J. S. Chang, J. H. Lee, C. H. Nam, eds., Proc. SPIE2778, 1116–1117 (1996).

Johnson, W. D.

Kafri, O.

Keren, E.

Kim, B. J.

J. S. Song, B. J. Kim, J. H. Jo, S. Chang, K. C. Yuk, “Precise measurement of linear displacements by moiré fringes of elongated circular gratings,” in 17th Congress of the International Commission for Optics for Science and New Technology, J. S. Chang, J. H. Lee, C. H. Nam, eds., Proc. SPIE2778, 1116–1117 (1996).

Kim, S. W.

King, M. C.

Koops, H. W. P.

Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
[CrossRef]

Lee, W. H.

Livnat, A.

Meadows, D. M.

Nishijima, Y.

Oster, G.

Park, Y. C.

Parks, V. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970), Chap. 1, pp. 7–32; Chap. 3, pp. 79–91; Chap. 4, pp. 98–106.

Patorski, K.

Pirard, A.

A. Pirard, “Consideration sur la methode du moiré en photoelasticite,” Anal. Contraintes, Mem. GAMAC 5, 1–24 (1960).

Rayleigh, L.

L. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93 (1874).

Reid, G. T.

G. T. Reid, “A moiré fringe alignment aid,” Opt. Laser Eng. 4, 121–126 (1983).
[CrossRef]

Ronchi, V.

Sandler, B.

Song, J. S.

J. S. Song, B. J. Kim, J. H. Jo, S. Chang, K. C. Yuk, “Precise measurement of linear displacements by moiré fringes of elongated circular gratings,” in 17th Congress of the International Commission for Optics for Science and New Technology, J. S. Chang, J. H. Lee, C. H. Nam, eds., Proc. SPIE2778, 1116–1117 (1996).

Szwaykowski, P.

Takasaki, H.

Theocaris, P. S.

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969), Chap. 2, pp. 19–24; Chap. 3, pp. 112–131.
[CrossRef]

Vargady, L. O.

Vladimirsky, Y.

Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
[CrossRef]

Wasserman, M.

Yuk, K. C.

J. S. Song, B. J. Kim, J. H. Jo, S. Chang, K. C. Yuk, “Precise measurement of linear displacements by moiré fringes of elongated circular gratings,” in 17th Congress of the International Commission for Optics for Science and New Technology, J. S. Chang, J. H. Lee, C. H. Nam, eds., Proc. SPIE2778, 1116–1117 (1996).

Zwerling, C.

Anal. Contraintes, Mem. GAMAC (1)

A. Pirard, “Consideration sur la methode du moiré en photoelasticite,” Anal. Contraintes, Mem. GAMAC 5, 1–24 (1960).

Appl. Opt. (10)

J. Opt. Soc. Am. (4)

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

J. Vac. Sci. Technol. B (1)

Y. Vladimirsky, H. W. P. Koops, “Moiré method and zone plate pattern inaccuracies,” J. Vac. Sci. Technol. B 6, 2142–2146 (1988).
[CrossRef]

Opt. Laser Eng. (1)

G. T. Reid, “A moiré fringe alignment aid,” Opt. Laser Eng. 4, 121–126 (1983).
[CrossRef]

Philos. Mag. (1)

L. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93 (1874).

Other (5)

J. S. Song, B. J. Kim, J. H. Jo, S. Chang, K. C. Yuk, “Precise measurement of linear displacements by moiré fringes of elongated circular gratings,” in 17th Congress of the International Commission for Optics for Science and New Technology, J. S. Chang, J. H. Lee, C. H. Nam, eds., Proc. SPIE2778, 1116–1117 (1996).

Many useful articles on moire fringes are found in G. Indebetouw, R. Czarnek, eds., Selected Papers on Optical Moiré and Applications, Vol. MS64 of SPIE Milestone Series (SPIE, Bellingham, Wash., 1992).

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969), Chap. 2, pp. 19–24; Chap. 3, pp. 112–131.
[CrossRef]

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970), Chap. 1, pp. 7–32; Chap. 3, pp. 79–91; Chap. 4, pp. 98–106.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1990), Chap. 2, p. 36; Chap. 5, pp. 81–86.

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Figures (11)

Fig. 1
Fig. 1

Matched radial-parallel grating in which the parallel stripes of variable spacings are matched with the radial stripes of the uniform angular pitch P θ (=π/30 rad).

Fig. 2
Fig. 2

Diagram of moiré fringes produced by overlapping two identical MRP gratings of angular pitch P θ (=π/10 rad). One grating (dashed lines) is rotated clockwise by 2P θ relative to the other grating (solid lines).

Fig. 3
Fig. 3

Computer-generated moiré patterns formed by a MRP grating pair of angular pitch P θ (=π/30 rad) when the gratings are relatively rotated through (a) 0, (b) P θ/2, (c) P θ, (d) 3P θ/2, and (e) 2P θ.

Fig. 4
Fig. 4

Diagram of moiré fringes produced by overlapping two different MRP gratings. The specimen grating of angular pitch P θ′ (=π/11 rad) (dashed lines) is rotated clockwise by 2P θ′, relative to the reference grating of angular pitch P θ (=π/10 rad) (solid lines).

Fig. 5
Fig. 5

Computer-generated moiré patterns formed by overlapping two MRP gratings as the specimen grating of angular pitch P θ′ (=π/30 rad) is rotated clockwise by (a) 0, (b) P θ′/4, (c) P θ′/2, (d) 3P θ′/4, (e) P θ′, (f) 5P θ′/4, (g) 3P θ′/2, (h) 5P θ′/4, and (i) 2P θ′, relative to the reference grating of angular pitch P θ (=π/29 rad).

Fig. 6
Fig. 6

Comparison of the moiré patterns produced by a MRP grating pair of different angular pitch when the specimen grating of angular pitch P θ′ (=π/30 rad) is rotated (a) counterclockwise and (b) clockwise through P θ′, relative to the reference grating of angular pitch P θ (=π/29 rad).

Fig. 7
Fig. 7

Relations between angles θ and θ′, made by the point of observation with the radial stripes of index 0 in the reference and specimen gratings, respectively, when the center of the specimen grating is displaced by an infinitesimal value c along (a) the horizontal and (b) the vertical directions from that of the reference grating.

Fig. 8
Fig. 8

Computer-generated moiré patterns formed by overlapping two MRP gratings with the horizontal displacement, c = R 0P θ′/8, when the specimen grating of angular pitch P θ′ (=π/30 rad) is rotated clockwise by (a) 0, (b) P θ′/4, (c) P θ′/2, (d) 3P θ′/4, and (e) P θ′, relative to the reference grating of P θ (=π/29 rad). Δ0 denotes the position error of the dark vernier fringe of the zeroth order that is measured at r′ = R 0′, where R 0′ is the inner radius of the radial part in the specimen MRP grating.

Fig. 9
Fig. 9

Computer-generated moiré patterns formed by overlapping two MRP gratings with the vertical displacement, c = R 0P θ′/8. Other conditions are the same as in Fig. 8.

Fig. 10
Fig. 10

Experimental setup for observing the moiré fringes of a MRP grating pair with small angular displacements. A specimen grating of angular pitch P θ ′ (=π/30 rad) is fixed on the axis of a dc stepping motor and projected by a lens of focal length f onto the reference grating of angular pitch P θ (=π/29 rad).

Fig. 11
Fig. 11

Moiré patterns photographed at the back of the reference MRP grating that the specimen grating is projected onto, when (a) no pulse, (b) 25 pulses, and (c) 50 pulses are applied to a dc stepping motor.

Equations (13)

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

m=N-N.
δ=l+12Pθ  l=0, 1, 2, .
δ=lPθ  l=0, 1, 2, .
mδPθ,
W=PθPθPθ-Pθ.
Mθ=WPθ=PθPθ-Pθ.
m δPθ.
θ  θ-δ-c cos θr
θ  θ-δ+c sin θr
θPθ-θPθ=l+12  for l=integers.
θl  l+12 W+δMθ+Δl,
Δl  cMθr cosl+12 W+δMθ
Δl-cMθr sinl+12 W+δMθ

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