Abstract

The superposition of several periodic or quasiperiodic patterns produces moiré effects. In particular, the combination of three grid structures adds great flexibility to the use of moiré phenomena. Information about the fringe structure, given by vector addition in Fourier space, allows investigation of formations, interpretation, and expectation problems concerning moiré-pattern parameters. Any frequency and orientation of the pattern is possible, using three-line gratings. On the other hand, three quasiperiodic gratings with variations of frequency and/or orientation result in a zone-plate shape of the moiré pattern regardless of grating distortions. Both elliptical and hyperbolic zone plates occur. Several illustrations demonstrate these phenomena and indicate some potential applications.

© 1974 Optical Society of America

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References

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  1. Rayleigh, Philos. Mag. 47, 81 (1874); Philos. Mag. 47, 193 (1874).
  2. A. Righi, Nuovo Cimento 21, 203 (1887); Nuovo Cimento 22, 10 (1888).
    [Crossref]
  3. A. Schuster, Philos. Mag. 48(6), 609 (1924).
  4. V. Ronchi, Attualita scientifiche, No. 37 (N. Zanichelli, Bologna, 1925), Ch. 9.
  5. C. V. Raman and S. K. Datta, Trans. Opt. Soc. 27, 51 (1925/26).
    [Crossref]
  6. S. K. Datta, Trans. Opt. Soc. 28, 214 (1926/27).
    [Crossref]
  7. P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, England, 1969); A. J. Durelli and V. J. Parks, Moiré Analysis of Strain (Prentice–Hall, Englewood Cliffs, N. J., 1970).
  8. R. Lehmann and A. Wiener, Feingerätetechnik 2, 199 (1953).
  9. A. Pirard, Analyse des Contraintes, Mém. GAMAC 5, No. 2, 1 (1960).
  10. G. Oster, M. Wasserman, and C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
    [Crossref]
  11. G. L. Rogers, Proc. Phys. Soc. 73, 142 (1959).
    [Crossref]
  12. A. W. Lohmann, Optik 18, 514 (1961).
  13. C. A. Sciammarella, Exp. Mech. 5, 154 (1965).
    [Crossref]
  14. A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1567 (1967).
    [Crossref] [PubMed]
  15. I. Leifer, J. M. Walls, and H. N. Southworth, Opt. Acta 20, 33 (1973).
    [Crossref]
  16. J. Moser, S. Bahčevandžijev, M. Janoska, and Lj. Janićijević, God. Zborn. PMF Skopje 15A, 113 (1964).
  17. R. L. Conger, L. T. Long, and J. A. Parks, Appl. Opt. 7, 623 (1968).
    [Crossref] [PubMed]
  18. L. A. Sayce, Photogr. J. 80, 454 (1940).

1973 (1)

I. Leifer, J. M. Walls, and H. N. Southworth, Opt. Acta 20, 33 (1973).
[Crossref]

1968 (1)

1967 (1)

1965 (1)

C. A. Sciammarella, Exp. Mech. 5, 154 (1965).
[Crossref]

1964 (2)

J. Moser, S. Bahčevandžijev, M. Janoska, and Lj. Janićijević, God. Zborn. PMF Skopje 15A, 113 (1964).

G. Oster, M. Wasserman, and C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
[Crossref]

1961 (1)

A. W. Lohmann, Optik 18, 514 (1961).

1960 (1)

A. Pirard, Analyse des Contraintes, Mém. GAMAC 5, No. 2, 1 (1960).

1959 (1)

G. L. Rogers, Proc. Phys. Soc. 73, 142 (1959).
[Crossref]

1953 (1)

R. Lehmann and A. Wiener, Feingerätetechnik 2, 199 (1953).

1940 (1)

L. A. Sayce, Photogr. J. 80, 454 (1940).

1924 (1)

A. Schuster, Philos. Mag. 48(6), 609 (1924).

1887 (1)

A. Righi, Nuovo Cimento 21, 203 (1887); Nuovo Cimento 22, 10 (1888).
[Crossref]

1874 (1)

Rayleigh, Philos. Mag. 47, 81 (1874); Philos. Mag. 47, 193 (1874).

Bahcevandžijev, S.

J. Moser, S. Bahčevandžijev, M. Janoska, and Lj. Janićijević, God. Zborn. PMF Skopje 15A, 113 (1964).

Conger, R. L.

Datta, S. K.

S. K. Datta, Trans. Opt. Soc. 28, 214 (1926/27).
[Crossref]

C. V. Raman and S. K. Datta, Trans. Opt. Soc. 27, 51 (1925/26).
[Crossref]

Janicijevic, Lj.

J. Moser, S. Bahčevandžijev, M. Janoska, and Lj. Janićijević, God. Zborn. PMF Skopje 15A, 113 (1964).

Janoska, M.

J. Moser, S. Bahčevandžijev, M. Janoska, and Lj. Janićijević, God. Zborn. PMF Skopje 15A, 113 (1964).

Lehmann, R.

R. Lehmann and A. Wiener, Feingerätetechnik 2, 199 (1953).

Leifer, I.

I. Leifer, J. M. Walls, and H. N. Southworth, Opt. Acta 20, 33 (1973).
[Crossref]

Lohmann, A. W.

Long, L. T.

Moser, J.

J. Moser, S. Bahčevandžijev, M. Janoska, and Lj. Janićijević, God. Zborn. PMF Skopje 15A, 113 (1964).

Oster, G.

Paris, D. P.

Parks, J. A.

Pirard, A.

A. Pirard, Analyse des Contraintes, Mém. GAMAC 5, No. 2, 1 (1960).

Raman, C. V.

C. V. Raman and S. K. Datta, Trans. Opt. Soc. 27, 51 (1925/26).
[Crossref]

Rayleigh,

Rayleigh, Philos. Mag. 47, 81 (1874); Philos. Mag. 47, 193 (1874).

Righi, A.

A. Righi, Nuovo Cimento 21, 203 (1887); Nuovo Cimento 22, 10 (1888).
[Crossref]

Rogers, G. L.

G. L. Rogers, Proc. Phys. Soc. 73, 142 (1959).
[Crossref]

Ronchi, V.

V. Ronchi, Attualita scientifiche, No. 37 (N. Zanichelli, Bologna, 1925), Ch. 9.

Sayce, L. A.

L. A. Sayce, Photogr. J. 80, 454 (1940).

Schuster, A.

A. Schuster, Philos. Mag. 48(6), 609 (1924).

Sciammarella, C. A.

C. A. Sciammarella, Exp. Mech. 5, 154 (1965).
[Crossref]

Southworth, H. N.

I. Leifer, J. M. Walls, and H. N. Southworth, Opt. Acta 20, 33 (1973).
[Crossref]

Theocaris, P. S.

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, England, 1969); A. J. Durelli and V. J. Parks, Moiré Analysis of Strain (Prentice–Hall, Englewood Cliffs, N. J., 1970).

Walls, J. M.

I. Leifer, J. M. Walls, and H. N. Southworth, Opt. Acta 20, 33 (1973).
[Crossref]

Wasserman, M.

Wiener, A.

R. Lehmann and A. Wiener, Feingerätetechnik 2, 199 (1953).

Zwerling, C.

Analyse des Contraintes, Mém. GAMAC (1)

A. Pirard, Analyse des Contraintes, Mém. GAMAC 5, No. 2, 1 (1960).

Appl. Opt. (2)

Exp. Mech. (1)

C. A. Sciammarella, Exp. Mech. 5, 154 (1965).
[Crossref]

Feingerätetechnik (1)

R. Lehmann and A. Wiener, Feingerätetechnik 2, 199 (1953).

God. Zborn. PMF Skopje (1)

J. Moser, S. Bahčevandžijev, M. Janoska, and Lj. Janićijević, God. Zborn. PMF Skopje 15A, 113 (1964).

J. Opt. Soc. Am. (1)

Nuovo Cimento (1)

A. Righi, Nuovo Cimento 21, 203 (1887); Nuovo Cimento 22, 10 (1888).
[Crossref]

Opt. Acta (1)

I. Leifer, J. M. Walls, and H. N. Southworth, Opt. Acta 20, 33 (1973).
[Crossref]

Optik (1)

A. W. Lohmann, Optik 18, 514 (1961).

Philos. Mag. (2)

A. Schuster, Philos. Mag. 48(6), 609 (1924).

Rayleigh, Philos. Mag. 47, 81 (1874); Philos. Mag. 47, 193 (1874).

Photogr. J. (1)

L. A. Sayce, Photogr. J. 80, 454 (1940).

Proc. Phys. Soc. (1)

G. L. Rogers, Proc. Phys. Soc. 73, 142 (1959).
[Crossref]

Trans. Opt. Soc. (2)

C. V. Raman and S. K. Datta, Trans. Opt. Soc. 27, 51 (1925/26).
[Crossref]

S. K. Datta, Trans. Opt. Soc. 28, 214 (1926/27).
[Crossref]

Other (2)

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, England, 1969); A. J. Durelli and V. J. Parks, Moiré Analysis of Strain (Prentice–Hall, Englewood Cliffs, N. J., 1970).

V. Ronchi, Attualita scientifiche, No. 37 (N. Zanichelli, Bologna, 1925), Ch. 9.

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Figures (12)

Fig. 1
Fig. 1

Vector representation of superposed periodic gratings for predicting moiré beat patterns. (a) Two gratings, forming a low-frequency moiré pattern indicated by R ¯. (b) Three gratings oriented to give a zero-frequency moiré pattern.

Fig. 2
Fig. 2

Moiré fringes formed by superposing two identical Ronchi rulings. (a) Fringes resulting from a small angle between the gratings. (b) Magnified portion of the pattern in (a).

Fig. 3
Fig. 3

Moiré fringes formed by superposing three Ronchi rulings. The gratings are oriented to give the same moiré frequency in the cases shown. (a) Identical gratings; (b) grating-frequency ratios 1 : 1 : 2; (c) grating-frequency ratios 1 : 3 : 2; (b) (d), and (f) show magnified portions of (a), (c), and (e), respectively.

Fig. 4
Fig. 4

Moiré fringes formed by superposing four Ronchi rulings. (a) Grating-frequency ratios 1 : 1 : 1 + 2 : 1 + 2. (b) Magnified portion of the pattern in (a).

Fig. 5
Fig. 5

Vector representation of superposed Sayce gratings (arrow indicates direction of frequency increase) and periodic gratings. (a) and (b) show different combinations of Sayce gratings; (c)–(h) show combinations of two Sayce gratings and one periodic grating.

Fig. 6
Fig. 6

Moiré fringes formed by the combination of Sayce and periodic gratings indicated in Fig. 5. (a) and (b), (c) and (d), (e) and (f), and (g) and (h) illustrate the elliptical and corresponding hyperbolic zone plate obtained by reversing one of the Sayce gratings.

Fig. 7
Fig. 7

Moiré fringes formed by a combination of two Sayce and two periodic gratings. (a) and (b) illustrate the elliptical and corresponding hyperbolic zone plate obtained by reversing the horizontal Sayce gratings.

Fig. 8
Fig. 8

Vector representation of superposition of three identical circular gratings. (a)–(d) show different relative lateral positions among the centers of the gratings.

Fig. 9
Fig. 9

Moiré fringes formed by the combinations of circular gratings indicated in Fig. 8. The moiré patterns due to three gratings have elliptical and hyperbolic zone-plate gratings.

Fig. 10
Fig. 10

Moiré fringes formed by superposing two circular and one periodic grating. In (a) and (c) the frequency of the periodic grating is smaller than that of the circular and in (b) and (d) the frequency of the periodic grating is larger than that of the circular. In (a) and (b) the periodic grating is horizontal and in (c) and (d) it is vertical.

Fig. 11
Fig. 11

Moiré fringes formed by superposing a quasiperiodic grating and two identical periodic gratings. (a)–(f) are examples of recordings made with different relative orientations among the individual gratings.

Fig. 12
Fig. 12

Formation of a variable Fresnel-zone plate by superposition of three Sayce targets. Two extreme target positions are shown. In (a) the zone-plate moiré is formed at the high–frequency ends of the targets and in (b) at their low-frequency ends.

Equations (9)

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r 1 exp ( i φ 1 ) + r 2 exp ( i φ 2 )
r 1 exp ( i φ 1 ) + r 2 exp [ i ( φ 2 + π ) ] .
R = [ r 1 2 + r 2 2 + 2 r 1 r 2 cos ( φ 1 - φ 2 ) ] 1 2 and Φ = tan - 1 { ( r 1 sin φ 1 + r 2 sin φ 2 ) / ( r 1 cos φ 1 + r 2 cos φ 2 ) } .
R = 2 r cos ( φ 1 - φ 2 ) / 2 and Φ = ( φ 1 + φ 2 ) / 2.
R ¯ = r ¯ n = r n exp ( i φ n ) = R exp ( i Φ ) ; R = [ ( r n cos φ n ) 2 + ( r n sin φ n ) 2 ] 1 2 ; and Φ = tan - 1 ( r n sin φ n / r n cos φ n ) .
f ( u ) = f ( u 0 ) + f ( u 0 ) ( u - u 0 ) + 1 2 f ( u 0 ) ( u - u 0 ) 2 + ,
R 2 = R u 2 + R v 2 , Φ = tan - 1 ( R v / R u ) ,
R 2 = [ R u ( u 0 ) + R u ( u 0 ) ( u - u 0 ) ] 2 + [ R v ( v 0 ) + R v ( v 0 ) ( v - v 0 ) ] 2 , Φ = tan - 1 { [ R v ( v 0 ) + R v ( v 0 ) ( v - v 0 ) ] / [ R u ( u 0 ) + R u ( u 0 ) ( u - u 0 ) ] } .
R 2 = { R u ( 0 ) } 2 u 2 + { R v ( 0 ) } 2 v 2 , Φ = tan - 1 { [ R v ( 0 ) / R u ( 0 ) ] v / u } .