Abstract

A joint Talbot effect with two separated and crossed gratings of substructures is proposed. The mutual coupling in the joint self-imaging leads to a mechanism for the formation of the new moiré patterns. The conditions are studied for generating the logic-operated moiré fringes. Both theoretical analyses and experiments are given.

© 1990 Optical Society of America

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References

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  1. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401 (1836).
  2. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  3. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  4. W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [CrossRef]
  5. Y. Cohen-Sabban, D. Joyeux, “Aberration-free nonparaxial self-imaging,” J. Opt. Soc. Am. 73, 707–719 (1983).
    [CrossRef]
  6. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).
  7. O. Bryngdahl, “Orthogonal-states-grating moiré,” Opt. Commun. 41, 249–254 (1982).
    [CrossRef]
  8. S. Yokozeki, K. Patorski, “Moiré fringe profile prediction method and its application to fringe sharpening,” Appl. Opt. 17, 2541–2547 (1978).
    [PubMed]
  9. R. H. Katyl, “Moiré screens coded with pseudo-random sequences,” Appl. Opt. 11, 2278–2285 (1972).
    [CrossRef] [PubMed]
  10. J. Zhang, L. Liu, “Optical logical operation and moiré pattern,” Opt. Commun. 66, 179–182 (1988).
    [CrossRef]
  11. L. Liu, “Theory for Lau effect of plane objects,” Acta Opt. Sinica 6, 807–814 (1986).
  12. O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. 66, 87–94 (1976).
    [CrossRef]
  13. H. Bartelt, A. W. Lohmann, E. E. Sicre, “Optical logical processing in parallel with theta modulation,” J. Opt. Soc. Am. A 1, 944–951 (1984).
    [CrossRef]

1988 (2)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

J. Zhang, L. Liu, “Optical logical operation and moiré pattern,” Opt. Commun. 66, 179–182 (1988).
[CrossRef]

1986 (1)

L. Liu, “Theory for Lau effect of plane objects,” Acta Opt. Sinica 6, 807–814 (1986).

1984 (1)

1983 (1)

1982 (1)

O. Bryngdahl, “Orthogonal-states-grating moiré,” Opt. Commun. 41, 249–254 (1982).
[CrossRef]

1978 (1)

1976 (1)

1972 (1)

1968 (1)

1967 (1)

1965 (1)

1836 (1)

F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401 (1836).

Bartelt, H.

Bryngdahl, O.

O. Bryngdahl, “Orthogonal-states-grating moiré,” Opt. Commun. 41, 249–254 (1982).
[CrossRef]

O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. 66, 87–94 (1976).
[CrossRef]

Cohen-Sabban, Y.

Joyeux, D.

Katyl, R. H.

Liu, L.

J. Zhang, L. Liu, “Optical logical operation and moiré pattern,” Opt. Commun. 66, 179–182 (1988).
[CrossRef]

L. Liu, “Theory for Lau effect of plane objects,” Acta Opt. Sinica 6, 807–814 (1986).

Lohmann, A. W.

Montgomery, W. D.

Patorski, K.

Sicre, E. E.

Talbot, F.

F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401 (1836).

Winthrop, J. T.

Worthington, C. R.

Yokozeki, S.

Zhang, J.

J. Zhang, L. Liu, “Optical logical operation and moiré pattern,” Opt. Commun. 66, 179–182 (1988).
[CrossRef]

Acta Opt. Sinica (1)

L. Liu, “Theory for Lau effect of plane objects,” Acta Opt. Sinica 6, 807–814 (1986).

Appl. Opt. (2)

J. Opt. Soc. Am. (5)

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

Opt. Commun. (2)

J. Zhang, L. Liu, “Optical logical operation and moiré pattern,” Opt. Commun. 66, 179–182 (1988).
[CrossRef]

O. Bryngdahl, “Orthogonal-states-grating moiré,” Opt. Commun. 41, 249–254 (1982).
[CrossRef]

Optik (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

Philos. Mag. (1)

F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401 (1836).

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Figures (11)

Fig. 1
Fig. 1

Setup for observing the joint Talbot effect with two separated gratings.

Fig. 2
Fig. 2

(a) Moiré structural element, (b) moiré array function.

Fig. 3
Fig. 3

Matrix representation of the original structural element.

Fig. 4
Fig. 4

Matrix representation of the in-between structural element.

Fig. 5
Fig. 5

Matrix representation of the final structural element.

Fig. 6
Fig. 6

XOR ¯ (xor) logic-operated moiré pattern.

Fig. 7
Fig. 7

Ternary-logic-operated moiré patterns with two gratings coded by 1, 0, 0: (a) the difference beat, (b) the sum beat.

Fig. 8
Fig. 8

Quarternary-logic-operated moiré patterns with two gratings coded by 1, 0, 0, 0: (a) the difference beat, (b) the sum beat.

Fig. 9
Fig. 9

Quarternary-logic-operated moiré patterns with two gratings coded by 1, 1, 0, 0 and 1, 0, 0, 0: (a) the difference beat, (b) the sum beat.

Fig. 10
Fig. 10

(a) Photograph of the original moiré pattern with two binary gratings, (b) photograph of the XOR ¯ logic-operated moiré pattern.

Fig. 11
Fig. 11

Photographs of the quarternary-logic-operated moiré patterns with two gratings coded by 1, 1, 0, 0 and 1, 0, 0, 0: (a) the original moiré pattern, (b) the difference beat, (c) the sum beat.

Equations (40)

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g ( x , y ) = rect ( x h ) * n δ ( x n T ) ,
e ( x , y ) = g ( x , y ) * * 1 j λ z exp ( j π x 2 + y 2 λ z ) = rect ( x h ) * L ( x : α β ) ,
z = α β T 2 λ ,
L ( x : α β ) = n R n δ ( x n T β ) ,
R n = 1 β k = 0 , 1 , ( β 1 ) exp [ j ( 2 π k n β + π α β k 2 ) ] , for α β even ,
L ( x : α β ) = n R n δ [ x ( n 1 2 ) T β ] ,
R n = 1 2 β k = 0 , 1 , ( β 1 ) exp { j [ π n ( 2 k + 1 ) β + π α β ( 2 k + 1 ) 2 ] } , for α β odd .
g 1 ( x , y ) = g 1 ( x ) = n = 1 N a n rect [ x ( N n N ) T h ] * n δ ( x n T )
g 2 ( x , y ) = g 2 ( x ) = n = 1 N b n rect [ x + ( N n N ) T h ] * n δ ( x n T )
z 1 = α 1 β 1 T 2 λ ,
z 2 = α 2 β 2 T 2 λ ,
z 3 = α 3 β 3 T 2 λ .
e 2 ( x , y ) = g 12 ( x ) = n = 1 N a n rect [ x ( N n N ) T h ] * L ( x : α 1 β 1 ) .
e 2 + ( x , y ) = g 12 ( x ) g 2 ( x ) .
e 3 ( x , y ) = [ g 12 ( x , y ) g 2 ( x , y ) ] * * [ 1 j λ z 2 exp ( j π x 2 + y 2 λ z 2 ) ] .
e 3 ( x , y ) = [ E 2 ( ξ , η ) * * G 2 ( ξ , η ) ] exp [ j π λ z 2 ( ξ 2 + η 2 ) ] × exp [ j 2 π ( ξ x + η y ) ] d ξ d η ,
E 2 ( ξ , n ) = n C n ( 1 ) exp ( j π λ z 1 n 2 T 2 ) δ ( ξ n T x ) δ ( η n T y )
G 2 ( ξ , n ) = n C 2 ( 2 ) δ ( ξ n T x ) δ ( η n T y ) ,
T x = T cos θ ,
T y = T sin θ ;
E 2 ( ξ , η ) * * G 2 ( ξ , η ) = n m C n ( 1 ) C m ( 2 ) × exp ( j π λ z 1 n 2 T 2 ) δ ( ξ n + m T x ) δ ( η n m T y ) .
e 3 ( x , y ) = n m C n ( 1 ) exp [ j π λ ( z 1 + z 2 ) n 2 T 2 ] × exp [ j 2 π ( cos θ x + sin θ y ) n T ] C m ( 2 ) exp ( j π λ z 2 m 2 T 2 ) × exp [ j 2 π ( cos θ x sin θ y ) m T ] exp ( j 2 π λ z 2 n m cos 2 T 2 ) .
g 13 ( x ) = n = 1 N rect [ x ( N n N ) T h ] * L ( x : α 3 β 3 ) .
g 23 ( x ) = m = 1 N rect [ x + ( N m N ) T h ] * L ( x : α 2 β 2 ) .
e 3 ( x , y ) = m g 13 ( x ) * δ ( x m λ z 2 cos 2 θ T ) C m ( 2 ) × exp ( j π λ z 2 m 2 T 2 ) exp ( j 2 π x m T ) .
λ z 2 cos 2 θ = ( p + N N ) T 2 ,
e 3 ( x , y ) = K = 0 N 1 g 13 ( x ) * δ ( x N N k T ) m C k + m N ( 2 ) × exp [ j π λ z 2 ( k + m N ) 2 T 2 ] exp [ j 2 π ( k + m N ) x T ] = K = 0 N 1 [ g 13 ( x ) * δ ( x N N k T ) ] × [ g 23 ( x ) * 1 N n ( x n T N ) exp ( j 2 π k n N x ) ] .
g 0 ( x , y ) = g 1 ( x ) g 2 ( x ) = S ( x , y : c n m ) * * A ( x , y ) .
S ( x , y : c n m ) = n = 1 N a n rect ( cos θ x + sin θ y N n N T h ) × m = 1 N b m rect ( cos θ x sin θ y + N m N T h )
A ( x , y ) = n m δ ( x n T x ) δ ( y m T y ) + n m δ ( x n T x T x 2 ) δ ( y m T y T y 2 ) .
e 2 ( x , y ) = n = 1 N α n rect ( x N n N h ) * n δ ( x n T ) ,
e 2 + ( x , y ) = S ( x , y : d n m ) * * A ( x , y ) ,
e 3 ( x , y ) = S ( x , y : f n m ) * * A ( x , y ) .
z 2 = p T 2 λ = p 1 T x 2 λ = p 2 T y 2 λ .
z 2 = α 2 β 2 T x 2 λ = p T y 2 λ .
f n m = k = 0 β 2 1 R k d [ n + k ( N / β 2 ) ] [ m k ( N / β 2 ) ] ,
z 2 = α 2 β 2 T y 2 λ = p T y 2 λ .
f n m = k = 0 β 2 1 R k d ( n + k N β 2 ) ( m + k N β 2 ) .
z 2 = α 2 x β 2 x T x 2 λ = α 2 y β 2 y T y 2 λ .
f n m = k = 0 β 2 x 1 l = 0 β 2 y 1 R k ( x ) R l ( y ) d ( n + k N β 2 x + l N β 2 y ) ( m k N β 2 x + l N β 2 y ) .

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