Abstract

The different ways of combining optical patterns with spatial periodicities are examined. The characteristics of and differences between additive, subtractive, and multiplicative pattern superposition are presented. The micro- and macro- (Moiré) structures in these cases are different. Examples are shown for two-pattern and multiple-pattern superpositions.

© 1976 Optical Society of America

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References

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  1. J. Guild, Diffraction Gratings as Measuring Scales (Oxford University Press, London, 1960); P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969); A. J. Durelli and V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).
  2. Moiré Pattern in Printing (Research and Engineering Council of the Graphic Arts Industry, Inc., Washington, D. C., 1960).
  3. O. Bryngdahl, J. Opt. Soc. Am. 62, 839 (1972).
    [Crossref]
  4. O. Bryngdahl, J. Opt. Soc. Am. 64, 1287 (1974).
    [Crossref]
  5. O. Bryngdahl, J. Opt. Soc. Am. 65, 685 (1975).
    [Crossref]

1975 (1)

1974 (1)

1972 (1)

Bryngdahl, O.

Guild, J.

J. Guild, Diffraction Gratings as Measuring Scales (Oxford University Press, London, 1960); P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969); A. J. Durelli and V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

J. Opt. Soc. Am. (3)

Other (2)

J. Guild, Diffraction Gratings as Measuring Scales (Oxford University Press, London, 1960); P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, Oxford, 1969); A. J. Durelli and V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Moiré Pattern in Printing (Research and Engineering Council of the Graphic Arts Industry, Inc., Washington, D. C., 1960).

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Figures (12)

FIG. 1
FIG. 1

Illustration of different types of superposition of two sinusoidal structures α and β.

FIG. 2
FIG. 2

Illustration of two inclined periodic line structures. The beat frequencies are formed by two interlaced patterns.

FIG. 3
FIG. 3

Illustration of different types of superposition of two binary grating structures α and β.

FIG. 4
FIG. 4

Different types of superposition of two grating structures. Macrostructures are shown to the left and the corresponding microstructures to the right. (a) and (b): additive superposition. (c) and (d): subtractive superposition. (e) and (f): multiplicative superposition. (g) and (h): opposite polarity of the distributions in (e) and (f).

FIG. 5
FIG. 5

Superposition of two zone plate structures displaced less than the diameter of the central zone apart. (a) shows additive superposition, (b) subtractive superposition, and (c) multiplicative superposition.

FIG. 6
FIG. 6

Superposition of two zone plate structures displaced more than the diameter of the central zone apart. (a) shows additive superposition, (b) subtractive superposition, and (c) multiplicative superposition.

FIG. 7
FIG. 7

Additive and multiplicative superposition of a circular grating and a one dimensional zone plate structure. (a) illustrates an elliptical configuration of macrostructure and (b) hyperbolic configuration.

FIG. 8
FIG. 8

Different types of superposition of three grating structures α, β, and γ. Macrostructures are shown to the left and the corresponding symmetrical microstructures to the right. (a) and (b): α+β+γ. (c) and (d): αβ+γ. (e) and (f): α(β+γ). (g) and (h): |α+βγ|. (i) and (j): |αβγ|. (k) and (l): |αβγ|. (m) and (n): |α(βγ)|. (o) and (p):αβγ. (q) and (r): opposite polarity of the distributions in (o) and (p).

FIG. 9
FIG. 9

Different types of superposition of three grating structures α, β and γ. Macrostructures are shown to the left and the corresponding asymmetrical microstructures to the right. (a) and (b): αβ+γ. (c) and (d):α(β+γ). (e) and (f): |α+βγ|. (g) and (h): |αβγ|. (i) and (j): |α(β−,γ|.

FIG. 10
FIG. 10

Different types of superposition of three zone plate structures α, β and γ. (a) illustrates α+β+γ. (b) αβ+γ, (c) α(β+γ), and (d) αβγ.

FIG. 11
FIG. 11

Displays of interaction between a one-dimensional zone plate structure and different harmonics of a regular grating. (a) shows additive, (b) subtractive, and (c) multiplicative types of superposition.

FIG. 12
FIG. 12

Higher harmonic interactions from two regular grating structures. (a) and (b) show 2nd harmonic moiré with additive and multiplicative types of superposition and (c) and (d) corresponding 3rd harmonic moiré.

Equations (12)

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1 2 ( 1 + cos 2 π ν 1 x )
1 2 ( 1 + cos 2 π ν 2 x ) .
1 2 ( 1 + cos 2 π ν 1 x ) + 1 2 ( 1 + cos 2 π ν 2 x ) = 1 + cos π ( ν 1 + ν 2 ) x cos π ( ν 1 - ν 2 ) x ;
1 2 ( 1 + cos 2 π ν 1 x ) - 1 2 ( 1 + cos 2 π ν 2 x ) = - sin π ( ν 1 + ν 2 ) x sin π ( ν 1 - ν 2 ) x ;
1 2 ( 1 + cos 2 π ν 1 x ) 1 2 ( 1 + cos 2 π ν 2 x ) = cos 2 π ν 1 x cos 2 π ν 2 x = 1 4 [ cos π ( ν 1 + ν 2 ) x + cos π ( ν 1 - ν 2 ) x ] 2 .
1 2 { 1 + cos 2 π ν ( y cos φ + x sin φ ) }
1 2 { 1 + cos 2 π ν ( y cos φ - sin φ ) }
1 2 { 1 + cos 2 π ν ( y cos φ + x sin φ ) } + 1 2 { 1 + cos 2 π ν ( y cos φ - x sin φ ) = 1 + cos 2 π ( ν y cos φ ) cos 2 π ( ν x sin φ ) ;
1 2 { 1 + cos 2 π ν ( ν cos φ + x sin φ } - 1 2 { 1 + cos 2 π ν ( ν cos φ - x sin φ ) } = - sin 2 π ( ν y cos φ ) sin 2 π ( ν x sin φ ) ;
1 2 { 1 + cos 2 π ν ( y cos φ + x sin φ ) } × 1 2 { 1 + cos 2 π ν ( y cos φ - x sin φ ) } = 1 4 { 1 + 2 cos 2 π ( ν y cos φ ) cos 2 π ( ν x sin φ ) + 1 2 cos 2 π ( 2 ν y cos φ ) + 1 2 cos 2 π ( 2 ν x sin φ ) } .
ν x = 2 ν sin φ and ν y = 2 ν cos φ .
α + β + γ , α β + γ , α ( β + γ ) , α + β - γ , α - β - γ , α β - γ , α ( β - γ ) , α β γ .