Abstract

The conditions for and formation of low-spatial-frequency beat patterns between superposed grid structures are studied. The application of moiré techniques for Fourier analysis of incoherently illuminated distributions is demonstrated. The different multiples of the fundamental spatial frequency of a grating are used to form distinguishable moiré patterns. Two particular applications are treated: increased sensitivity from fringe multiplication in a moiré display and examination of the element-profile shape of gratings. Moiré procedures with two, and with multiple, gratings are discussed. Indications of moiré pattern microstructures and moiré fringe profiles are indicated.

© 1975 Optical Society of America

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References

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  1. Rayleigh, Phil. Mag. 47, 81, 193 (1874).
  2. J. Guild, The Interference Systems of Crossed Diffraction Gratings (Clarendon, Oxford, England, 1956).
  3. O. Bryngdahl, J. Opt. Soc. Am. 64, 1287 (1974).
    [Crossref]
  4. C. A. Sciammarella, Exp. Mech. 5, 154 (1965).
    [Crossref]
  5. G. L. Rogers, Proc. Phys. Soc. 73, 142 (1959).
    [Crossref]
  6. R. Lehmann and A. Wiemer, Feingerätetechnik 2, 199 (1953).
  7. O. Bryngdahl and A. W. Lohmann, J. Opt. Soc. Am. 58, 141 (1968); O. Bryngdahl, J. Opt. Soc. Am. 59, 142 (1969).
    [Crossref]
  8. D. Post, Appl. Opt. 6, 1938 (1967); Appl. Opt. 10, 901 (1971).
    [Crossref] [PubMed]
  9. J. M. Burch, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 73.
    [Crossref]
  10. J. Ebbeni, Nouv. Rev. d’Opt. Appl. 1, 333 (1970).
    [Crossref]
  11. G. A. R. Foster, J. Text. Inst. 21, T18 (1930).
    [Crossref]
  12. M. Born, R. Fürth, and R. W. Pringle, Nature 156, 756 (1945).
    [Crossref]
  13. I. Leifer, G. L. Rogers, and N. W. F. Stephens, Opt. Acta 16, 535 (1969).
    [Crossref]
  14. K. Murata in Progress in Optics, V, edited by E. Wolf (North-Holland, Amsterdam, 1966), p. 201.
  15. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  16. R. H. Katyl, Appl. Opt. 11, 2278 (1972).
    [Crossref] [PubMed]

1974 (1)

1972 (1)

1970 (1)

J. Ebbeni, Nouv. Rev. d’Opt. Appl. 1, 333 (1970).
[Crossref]

1969 (1)

I. Leifer, G. L. Rogers, and N. W. F. Stephens, Opt. Acta 16, 535 (1969).
[Crossref]

1968 (1)

1967 (1)

1965 (1)

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

1959 (1)

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

1953 (1)

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

1945 (1)

M. Born, R. Fürth, and R. W. Pringle, Nature 156, 756 (1945).
[Crossref]

1930 (1)

G. A. R. Foster, J. Text. Inst. 21, T18 (1930).
[Crossref]

1874 (1)

Rayleigh, Phil. Mag. 47, 81, 193 (1874).

Born, M.

M. Born, R. Fürth, and R. W. Pringle, Nature 156, 756 (1945).
[Crossref]

Bryngdahl, O.

Burch, J. M.

J. M. Burch, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 73.
[Crossref]

Ebbeni, J.

J. Ebbeni, Nouv. Rev. d’Opt. Appl. 1, 333 (1970).
[Crossref]

Foster, G. A. R.

G. A. R. Foster, J. Text. Inst. 21, T18 (1930).
[Crossref]

Fürth, R.

M. Born, R. Fürth, and R. W. Pringle, Nature 156, 756 (1945).
[Crossref]

Guild, J.

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Clarendon, Oxford, England, 1956).

Katyl, R. H.

Lehmann, R.

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

Leifer, I.

I. Leifer, G. L. Rogers, and N. W. F. Stephens, Opt. Acta 16, 535 (1969).
[Crossref]

Lohmann, A. W.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Murata, K.

K. Murata in Progress in Optics, V, edited by E. Wolf (North-Holland, Amsterdam, 1966), p. 201.

Post, D.

Pringle, R. W.

M. Born, R. Fürth, and R. W. Pringle, Nature 156, 756 (1945).
[Crossref]

Rayleigh,

Rayleigh, Phil. Mag. 47, 81, 193 (1874).

Rogers, G. L.

I. Leifer, G. L. Rogers, and N. W. F. Stephens, Opt. Acta 16, 535 (1969).
[Crossref]

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

Sciammarella, C. A.

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

Stephens, N. W. F.

I. Leifer, G. L. Rogers, and N. W. F. Stephens, Opt. Acta 16, 535 (1969).
[Crossref]

Wiemer, A.

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

Appl. Opt. (2)

Exp. Mech. (1)

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

Feingerätetechnik (1)

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

J. Opt. Soc. Am. (2)

J. Text. Inst. (1)

G. A. R. Foster, J. Text. Inst. 21, T18 (1930).
[Crossref]

Nature (1)

M. Born, R. Fürth, and R. W. Pringle, Nature 156, 756 (1945).
[Crossref]

Nouv. Rev. d’Opt. Appl. (1)

J. Ebbeni, Nouv. Rev. d’Opt. Appl. 1, 333 (1970).
[Crossref]

Opt. Acta (1)

I. Leifer, G. L. Rogers, and N. W. F. Stephens, Opt. Acta 16, 535 (1969).
[Crossref]

Phil. Mag. (1)

Rayleigh, Phil. Mag. 47, 81, 193 (1874).

Proc. Phys. Soc. (1)

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

Other (4)

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Clarendon, Oxford, England, 1956).

J. M. Burch, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 73.
[Crossref]

K. Murata in Progress in Optics, V, edited by E. Wolf (North-Holland, Amsterdam, 1966), p. 201.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

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Figures (13)

FIG. 1
FIG. 1

Fourier-space representations of moiré formations using (a) two gratings and (b) three gratings. Two situations are shown in each diagram: moiré formed with the fundamental r 0 and the second harmonic 2 r 0 of a grid structure. The dashed circle centered at the origin indicates the area with visible spatial frequencies.

FIG. 2
FIG. 2

Two-grating moiré displays of information contained in different harmonics of a quasiperiodic grid structure. The different moiré patterns are obtained by combining the structure with periodic gratings the frequencies of which are low multiples of the fundamental.

FIG. 3
FIG. 3

Three-grating moiré displays of information contained in different harmonics of a quasiperiodic grid structure. The different moiré patterns are obtained by variation of the relative orientations of two periodic gratings.

FIG. 4
FIG. 4

Predicted moiré-fringe profile in the case of superposition of (a) two and (b) three Ronchi rulings.

FIG. 5
FIG. 5

Photographs of (a) two- and (b) three-grating moiré patterns that correspond to the illustrations in Fig. 4.

FIG. 6
FIG. 6

Two- and three-grating moiré patterns formed by superposition of identical binary gratings. Each column in the figure represents a different opening ratio of the gratings.

FIG. 7
FIG. 7

Microstructure configurations of the moiré patterns in Fig. 2.

FIG. 8
FIG. 8

Microstructure configurations of the moiré patterns in Fig. 3.

FIG. 9
FIG. 9

Moiré displays of spectral content in gratings with rectangular element profiles.

FIG. 10
FIG. 10

Calculated Fourier coefficients of gratings corresponding to the ones studied in (b)–(e) of Fig. 9.

FIG. 11
FIG. 11

Moiré displays of spectral content in gratings with binary asymmetric element profiles.

FIG. 12
FIG. 12

Illustration of spectral content investigation using the three-grating moiré technique. The patterns are formed by superposition of a quasiperiodic structure and two Ronchi rulings. Five harmonics of five different opening ratios h/d of a structure are shown.

FIG. 13
FIG. 13

Predicted three-grating moiré fringe profiles. Illustration of the influence of the opening ratio of a binary grating superimposed on two Ronchi rulings.

Equations (23)

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T ( x ) = n = - c n exp { i 2 π n x / d } ,
c n = ( 1 / d ) - d / 2 d / 2 T ( x ) exp { - i 2 π n x / d } d x ;
R = r n = R exp ( i ϕ ) ; R = { ( r n cos φ n ) 2 + ( r n sin φ n ) 2 } 1 / 2 ; ϕ = tan - 1 ( r n sin φ n / r n cos φ n ) .
x / d + f ( x , y ) = p ,
x / D 1 = q ,
x ( 1 / d - 1 / D 1 ) + f ( x , y ) = p - q , x ( 1 / d + 1 / D 1 ) + f ( x , y ) = p + q .
x / D n = q .
x / d + f ( x , y ) = p
x / D n = n q
x ( n / d - 1 / D n ) + n f ( x , y ) = n ( p - q ) .
D n = d D 1 / { d + ( n - 1 ) D 1 } ,
x ( 1 / d - 1 / D 1 ) + n f ( x , y ) = s ,
x / d + f ( x , y ) = p , ( x cos φ 1 + y sin φ 1 ) / D 1 = q 1 for φ = φ 1 , ( - x cos φ 1 + y sin φ 1 ) / D 1 = q 2 for φ = π - φ 1
2 x cos φ 1 / D 1 = q 1 - q 2 = q .
x ( 1 / d - 2 cos φ 1 / D 1 ) + f ( x , y ) = p - q .
x / d + f ( x , y ) = p
2 x cos φ n / D 1 = n q
x ( n / d - 2 cos φ n / D 1 ) + n f ( x , y ) = n ( p - q ) .
x ( 1 / d - 2 cos φ 1 / D 1 ) + n f ( x , y ) = s ,
cos φ n = ( n - 1 ) D 1 / 2 d + cos φ 1 .
T m ( x ) = lim A 1 A A T 1 ( x ) T 2 ( x - x ) d x ,
T m ( x ) = lim A 1 A A T 1 ( x ) T 2 ( x ) T 3 ( x - x - x ) d x d x .
c n = d - 1 - h / 2 h / 2 exp { - i 2 π n x / d } d x = ( n π ) - 1 sin { π n h / d } .