Abstract

The incoherent illumination moiré phenomenon between a periodic and a quasiperiodic structure is examined by the use of the spatial frequency analysis of the resultant transmittance. This approach has an advantage in that it completely explains all the fringe parameters without considering the averaging effect of the detecting system. We obtain the basic equation of the original moiré pattern due to the beat phenomenon exclusively, without any external factors like, for example, the influence of the observing system. From the basic equation, we derive the fringe profile equation representing the moiré fringe shape and the fringe equation expressing the spacing, the orientation, and the local displacement of the moiré fringes. The fringe profile equation is experimentally verified. The sensitivity enhancement by the fringe multiplication method is readily interpreted by the use of the derived fringe equation.

© 1976 Optical Society of America

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References

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  1. D. Post, Exp. Mech. 7, 154 (1967).
    [Crossref]
  2. R. H. Katyl, Appl. Opt. 11, 2278 (1972).
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  3. P. H. Langenbeck, Appl. Opt. 8, 543 (1969).
    [Crossref]
  4. O. Bryngdahl, J. Opt. Soc. Am. 65, 685 (1975).
    [Crossref]
  5. A. W. Lohmann, Optik 18, 514 (1961).
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    [Crossref]
  7. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1567 (1967).
    [Crossref] [PubMed]
  8. J. D. Hovanesian, Y. Y. Hung, Appl. Opt. 10, 2734 (1971).
    [Crossref] [PubMed]
  9. G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
    [Crossref]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 101.
  11. S. Yokozeki, Opt. Laser Technol. 5, 229 (1973).
    [Crossref]
  12. S. Yokozeki, Opt. Commun. 11, 378 (1974).
    [Crossref]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 423.
  14. S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 9, 1011 (1970).
    [Crossref]
  15. D. M. Meadows, W. O. Johnson, J. B. Allen, Appl. Opt. 9, 942 (1970).
    [Crossref] [PubMed]
  16. H. Takasaki, Appl. Opt. 9, 1467 (1970).
    [Crossref] [PubMed]

1975 (2)

O. Bryngdahl, J. Opt. Soc. Am. 65, 685 (1975).
[Crossref]

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[Crossref]

1974 (1)

S. Yokozeki, Opt. Commun. 11, 378 (1974).
[Crossref]

1973 (1)

S. Yokozeki, Opt. Laser Technol. 5, 229 (1973).
[Crossref]

1972 (1)

1971 (1)

1970 (3)

1969 (1)

P. H. Langenbeck, Appl. Opt. 8, 543 (1969).
[Crossref]

1967 (2)

1965 (1)

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

1961 (1)

A. W. Lohmann, Optik 18, 514 (1961).

Allen, J. B.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 423.

Bryngdahl, O.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 101.

Harburn, G.

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[Crossref]

Hovanesian, J. D.

Hung, Y. Y.

Johnson, W. O.

Katyl, R. H.

Langenbeck, P. H.

P. H. Langenbeck, Appl. Opt. 8, 543 (1969).
[Crossref]

Lohmann, A. W.

Meadows, D. M.

Paris, D. P.

Post, D.

D. Post, Exp. Mech. 7, 154 (1967).
[Crossref]

Sciammarella, C. A.

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

Suzuki, T.

S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 9, 1011 (1970).
[Crossref]

Takasaki, H.

Welberry, T. R.

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[Crossref]

Williams, R. P.

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 423.

Yokozeki, S.

S. Yokozeki, Opt. Commun. 11, 378 (1974).
[Crossref]

S. Yokozeki, Opt. Laser Technol. 5, 229 (1973).
[Crossref]

S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 9, 1011 (1970).
[Crossref]

Appl. Opt. (6)

Exp. Mech. (2)

D. Post, Exp. Mech. 7, 154 (1967).
[Crossref]

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

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

S. Yokozeki, T. Suzuki, Jpn. J. Appl. Phys. 9, 1011 (1970).
[Crossref]

Opt. Acta (1)

G. Harburn, T. R. Welberry, R. P. Williams, Opt. Acta 22, 409 (1975).
[Crossref]

Opt. Commun. (1)

S. Yokozeki, Opt. Commun. 11, 378 (1974).
[Crossref]

Opt. Laser Technol. (1)

S. Yokozeki, Opt. Laser Technol. 5, 229 (1973).
[Crossref]

Optik (1)

A. W. Lohmann, Optik 18, 514 (1961).

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 101.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 423.

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Figures (2)

Fig. 1
Fig. 1

Two grid structures.

Fig. 2
Fig. 2

Computed and experimental moiré fringe profiles. The opening-ratio pairs: (a) (0.8, 0.8), (b) (0.5, 0.5), (c) (0.2, 0.2), (d) (0.8, 0.2). The solid lines illustrate the computed moiré fringe profiles and the dashed lines show the changes of the computed fringe profiles due to the opening-ratio error. The cross marks are for experimental results.

Equations (18)

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T 1 ( x , y ) = a 0 + m = 1 { b m cos 2 π m d 1 [ x - f ( x , y ) ] + c m sin 2 π m d 1 [ x - f ( x , y ) ] } ,
( 2 π / d 1 ) [ x - f ( x , y ) ] = 2 π p ,
x = p d 1 + f ( x , y ) .
T 2 ( x , y ) = e 0 + n = 1 [ f n cos 2 π n d 2 ( x cos θ - y sin θ ) + g n sin 2 π n d 2 ( x cos θ - y sin θ ) ] ,
x cos θ - y sin θ = q d 2 ,
T ( x , y ) = T 1 ( x , y ) T 2 ( x , y ) , = a 0 e 0 + i ( ( b M i f i 2 + c M i g i 2 ) cos 2 π i { M d 1 [ x - f ( x , y ) ] - 1 d 2 ( x cos θ - y sin θ ) } + ( c M i f i 2 - b M i g i 2 ) sin 2 π i × { M d 1 [ x - f ( x , y ) ] - 1 d 2 ( x cos θ - y sin θ ) } ) + e 0 m { b m cos 2 π m d 1 [ x - f ( x , y ) ] + c m sin 2 π m d 1 × [ x - f ( x , y ) ] } + a 0 m [ f m cos 2 π m d 2 × ( x cos θ - y sin θ ) + g m sin 2 π m d 2 × ( x cos θ - y sin θ ) ] + m m n M n ( ( b m f n 2 + c m g n 2 ) × cos 2 π { m d 1 [ x - f ( x , y ) ] - n d 2 ( x cos θ - y sin θ ) } + ( c m f n 2 - b m g n 2 ) sin 2 π { m d 1 [ ( x - f ( x , y ) ] - n d 2 ( x cos θ - y sin θ ) } ) + m n ( ( b m f n 2 - c m g n 2 ) × cos 2 π { m d 1 [ x - f ( x , y ) ] + n d 2 ( x cos θ - y sin θ ) } + ( b m g n 2 + c m f n 2 ) sin 2 π { m d 1 [ x - f ( x , y ) ] + n d 2 ( x cos θ - y sin θ ) } ) .
d 1 M d 2 ,
( d 1 2 + M 2 d 2 2 - 2 M d 1 d 2 cos θ ) 1 / 2 / d 1 d 2 < 1 / d 1 ,
T m ( x , y ) = a 0 e 0 + i = 1 ( ( b M i f i 2 + c M i g i 2 ) cos 2 π i × { M d 1 [ x - f ( x , y ) ] - 1 d 2 ( x cos θ - y sin θ ) } + ( c M i f i 2 - b M i g i 2 ) sin 2 π i { M d 1 [ x - f ( x , y ) ] - 1 d 2 ( x cos θ - y sin θ ) } ) .
T p ( α ) = a 0 e 0 + i = 1 [ ( b M i f i 2 + c M i g i 2 ) cos i α + ( c M i f i 2 - b M i g i 2 ) sin i α ] .
y = d 2 sin θ r + d 1 cos θ - d 2 M d 1 sin θ x + d 2 M d 1 sin θ f ( x , y ) ,             for θ 0 ,
x = d 1 d 2 d 2 M - d 1 r + d 2 d 2 M - d 1 M f ( x , y ) ,             for θ = 0 and d 1 M d 2 ,
r = - f ( x , y ) d 1 M , for θ = 0 and d 1 = M d 2 ,
D r = f ( x , y ) d 1 M , for θ 0.
D r = f ( x , y ) d 1 M , for θ = 0 and d 1 M d 2 .
a 0 = h 1 d ,             b m = 2 sin ( π m h 1 / d ) π m ,             and c m = 0.
e 0 = h 2 d ,             f m = 2 sin ( π m h 2 / d ) π m ,             and g m = 0.
T p ( α ) = h 1 h 2 d 2 + 2 i = 1 sin [ π i ( h 1 / d ) ] π i · sin [ π i ( h 2 / d ) ] π i cos i α .

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