Abstract

The information inside each subcell of a two-dimensional periodic object is replicated throughout all the subcells of the unit cell at certain planes. An explicit expression describing the relative phase relationship among the replicated information is derived. From this expression, the wave amplitude at all the subcells caused by the interaction among the information coming from different subcells in the original object is obtained. A computer simulation of gray-level image synthesis using binary substructures and image differentiation is also given.

© 1994 Optical Society of America

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References

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  1. W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 403–405 (1836).
  2. A. W. Lohmann, “Array illuminator based on Talbot effect,” Optik 79, 41–45 (1988).
  3. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optic phase plates at fractional-Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]
  4. H. Dammann, G. Groh, M. Kock, “Restoration of faulty images of periodic objects by means of self-imaging,” Appl. Opt. 10, 1454–1455 (1971).
    [CrossRef] [PubMed]
  5. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  6. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
    [CrossRef]
  7. J. C. Bhattacharya, A. K. Aggarwal, “Measurement of the focal length of a collimating lens using the Talbot effect and moire technique,” Appl. Opt. 30, 4479–4480 (1991).
    [CrossRef] [PubMed]
  8. K. V. Sriram, M. P. Kothiyal, R. S. Sirohi, “Talbot interferometry in noncollimated illumination for curvature and focal length measurements,” Appl. Opt. 31, 75–79 (1992).
    [CrossRef] [PubMed]
  9. E. Tepichin, J. Ojeda-Castaneda, “Talbot interferometer with simultaneous dark and bright fields,” Appl. Opt. 28, 1517–1520 (1989).
    [CrossRef] [PubMed]
  10. 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]
  11. B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
    [CrossRef]
  12. A. Kolodziejczyk, “Contrast reversal of a binary periodic object,” Opt. Acta 33, 867–875 (1986).
    [CrossRef]
  13. Y. S. Cheng, R. C. Chang, “Image addition and subtraction using Talbot effect,” in Optical Computing and Neural Networks, K. Y. Hsu, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1812, 268–273 (1992).
  14. J. D. Gaskill, Linear System, Fourier Transform, and Optics (Wiley, New York, 1978), Chap. 6.
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

1992 (1)

1991 (1)

1990 (1)

1989 (1)

1988 (1)

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

1986 (2)

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

A. Kolodziejczyk, “Contrast reversal of a binary periodic object,” Opt. Acta 33, 867–875 (1986).
[CrossRef]

1973 (1)

1971 (1)

1967 (1)

1965 (1)

1836 (1)

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 403–405 (1836).

Aggarwal, A. K.

Bhattacharya, J. C.

Bryngdahl, O.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
[CrossRef]

Chang, R. C.

Y. S. Cheng, R. C. Chang, “Image addition and subtraction using Talbot effect,” in Optical Computing and Neural Networks, K. Y. Hsu, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1812, 268–273 (1992).

Cheng, Y. S.

Y. S. Cheng, R. C. Chang, “Image addition and subtraction using Talbot effect,” in Optical Computing and Neural Networks, K. Y. Hsu, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1812, 268–273 (1992).

Dammann, H.

Eschbach, R.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear System, Fourier Transform, and Optics (Wiley, New York, 1978), Chap. 6.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

Groh, G.

Kock, M.

Kolodziejczyk, A.

A. Kolodziejczyk, “Contrast reversal of a binary periodic object,” Opt. Acta 33, 867–875 (1986).
[CrossRef]

Kothiyal, M. P.

Leger, J. R.

Lohmann, A. W.

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

Montgomery, W. D.

Ojeda-Castaneda, J.

Packross, B.

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Sirohi, R. S.

Sriram, K. V.

Swanson, G. J.

Talbot, W. H. F.

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 403–405 (1836).

Tepichin, E.

Winthrop, J. T.

Worthington, C. R.

Appl. Opt. (4)

J. Opt. Soc. Am. (3)

Opt. Acta (1)

A. Kolodziejczyk, “Contrast reversal of a binary periodic object,” Opt. Acta 33, 867–875 (1986).
[CrossRef]

Opt. Commun. (1)

B. Packross, R. Eschbach, O. Bryngdahl, “Image synthesis using self imaging,” Opt. Commun. 56, 394–398 (1986).
[CrossRef]

Opt. Lett. (1)

Optik (1)

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

Philos. Mag. (1)

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 403–405 (1836).

Other (3)

Y. S. Cheng, R. C. Chang, “Image addition and subtraction using Talbot effect,” in Optical Computing and Neural Networks, K. Y. Hsu, H. Liu, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1812, 268–273 (1992).

J. D. Gaskill, Linear System, Fourier Transform, and Optics (Wiley, New York, 1978), Chap. 6.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 60.

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Figures (5)

Fig. 1
Fig. 1

Gij is a subcell with area D2 in the unit cell of the object with periods MD and ND in the orthogonal directions.

Fig. 2
Fig. 2

Relative phase relationship among all the replicated subcells for an object with unit cell 4D × 3D at the first S plane.

Fig. 3
Fig. 3

Phase distribution of Fig. 2 along (a) Y = D/2, (b) Y = 3D/2, (c) Y = 5D/2, (d) Y = 7D/2.

Fig. 4
Fig. 4

(a) Four-subcell distribution (G11, G13, G31, and G33) inside an object with unit cell 4D × 4D for gray-level image synthesis. (b) Intensity distribution at the first S plane for the object of (a). There are four subcells showing four gray levels.

Fig. 5
Fig. 5

(a) Four identical squares slightly displaced in the orthogonal directions. (b) Intensity distribution at the first S plane showing edge detection.

Tables (4)

Tables Icon

Table 1 Planes (ZSxy)a Within One Talbot Distance from the Object Plane on Which the Information Inside Each Subcell (D × D) of the Two-Dimensional Object (MD × ND) Is Replicated throughout All the Subcells of the Unit Cellb

Tables Icon

Table 2 Possible Relative Phasesa π[(2m + M)2/M + (2n + N)2/N]/4 for Replicated Subcell Information with the Number of Occurrences within One Unit Cell of the Two-Dimensional Objects

Tables Icon

Table 3 Possible Relative Phasesa π[(2mM)2/M − (2nN)2/N]/4 for Replicated Subcell Information with the Number of Occurrences for Two-Dimensional Objects

Tables Icon

Table 4 Wave Amplitude at the First S Plane of the (4D × 4D) Objecta

Equations (27)

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f ( x , y ) = G ( x , y ) * m = n = δ ( x m M D ) δ ( y n N D ) ,
f ( x , y ) = G ( x , y ) * [ 1 M N D 2 m = n = exp ( i 2 π m M D x ) × exp ( i 2 π n N D y ) ] .
u ( x , y ) = { G ( x , y ) * [ 1 M N D 2 m = n = exp ( i 2 π m M D x ) × exp ( i 2 π n N D y ) ] } * { exp ( i k Z ) i λ Z exp [ i π λ Z ( x 2 + y 2 ) ] } ,
u ( x , y ) = G ( x , y ) * { m = exp ( i 2 π m M D x ) × exp [ i πλ Z ( m M D ) 2 ] n = exp ( i 2 π n N D y ) × exp [ i πλ Z ( n N D ) 2 ] } ,
Z S x 1 = K 1 Z T x 2 + Z T x 2 M ,
Z S x 2 = K 2 Z T x 2 Z T x 2 M ,
Z S y 1 = K 3 Z T y 2 + Z T y 2 N ,
Z S y 2 = K 4 Z T y 2 Z T y 2 N ,
K 1 = K 3 N 2 + N M M 2 ,
K 1 = K 4 N 2 N M M 2 .
K 2 = K 3 N 2 + N + M M 2 ,
K 2 = K 4 N 2 N + M M 2 .
u ( x , y ) = G ( x , y ) * { m = exp ( i 2 π m M D x ) × exp [ i πλ Z S x y ( m M D ) 2 ] n = exp ( i 2 π n N D y ) × exp [ i πλ Z S x y ( n N D ) 2 ] } .
u ( x , y ) = m = n = G [ x 1 2 ( K 1 + 1 ) M D m D , y 1 2 ( K 3 + 1 ) N D n D ] × exp { i π 4 [ ( 2 m + M ) 2 M + ( 2 n + N ) 2 N ] } ,
m = exp ( i 2 π m M D x ) exp ( i π m 2 M ) = m = δ ( x 1 2 M D m D ) exp [ i π 4 ( 2 m + M ) 2 M ] .
u ( x , y ) = m = n = G [ x 1 2 ( K 1 + 1 ) M D m D , y 1 2 ( K 4 1 ) N D n D ] × exp { i π 4 [ ( 2 m + M ) 2 M ( 2 n N ) 2 N ] } .
u ( x , y ) = m = n = G [ x 1 2 ( K 2 1 ) M D m D , y 1 2 ( K 3 + 1 ) N D n D ] × exp { i π 4 [ ( 2 m M ) 2 M ( 2 n + N ) 2 N ] } .
u ( x , y ) = m = n = G [ x 1 2 ( K 2 1 ) M D m D , y 1 2 ( K 4 1 ) N D n D ] × exp { i π 4 [ ( 2 m M ) 2 M + ( 2 n N ) 2 N ] } .
u ( x , y ) = m = n = G ( x m D , y n D ) × exp { ± i π 4 [ ( 2 m ± M ) 2 M + ( 2 n ± N ) 2 N ] } .
u ( x , y ) = m = n = G ( x m D , y n D ) × exp { ± i π 4 [ ( 2 m ± M ) 2 M ( 2 n N ) 2 N ] } .
u m , n ( x , y ) = m = 1 M n = 1 N G m n ( x , y ) × exp ( ± i π 4 { [ 2 ( m m + 1 ) ± M ] 2 M + [ 2 ( n n + 1 ) ± N ] 2 N } )
u m , n ( x , y ) = m = 1 M n = 1 N G m n ( x , y ) × exp ( ± i π 4 { [ 2 ( m m + 1 ) ± M ] 2 M [ 2 ( n n + 1 ) N ] 2 N } )
m = exp ( i 2 π m M D x ) exp ( i π m 2 M ) = m = δ ( x 1 2 M D m D ) exp [ i π ( 2 m + M ) 2 4 M ] .
m = exp ( i 2 π m M D x ) exp ( i π m 2 M ) = m 0 = m 1 = 0 M 1 exp ( i 2 π m 0 M + m 1 M D x ) × exp [ i π ( m 0 M 2 + 2 m 0 m 1 M + m 1 2 ) M ] = m 0 = exp [ i 2 π m 0 D ( x M 2 D ) ] × m 1 = 0 M 1 exp [ i 2 π m 1 M D ( x m 1 2 D ) ] = m 0 = 0 δ ( x M 2 D m 0 D ) × m 1 = 0 M 1 exp [ i 2 π m 1 M D ( x m 1 2 D ) ] = m 0 = δ ( x M 2 D m 0 D ) exp [ i π ( 2 m 0 + M ) 2 4 M ] × m 1 = 0 M 1 exp [ i π M ( m 1 m 0 M 2 ) 2 ] .
exp { i π M [ ( m 1 + h 1 M ) m 0 M 2 ] 2 } = exp [ i π M ( m 1 m 0 M 2 ) 2 ] × exp [ i 2 h 1 π ( m 1 m 0 M 2 ) ] exp ( i h 1 2 π M ) = exp [ i π M ( m 1 m 0 M 2 ) 2 ] × exp [ i 2 h 1 π ( m 1 m 0 ) ] exp [ i π h 1 ( 1 + h 1 ) M ] = exp [ i π M ( m 1 m 0 M 2 ) 2 ] .
m 1 = 0 M 1 exp { i π M [ m 1 ( m 0 + h 0 ) M 2 ] 2 } = m 1 = 0 M 1 exp { i π M [ ( m 1 h 0 ) m 0 M 2 ] 2 } = m 1 = h 0 M h 0 1 exp [ i π M ( m 1 m 0 M 2 ) 2 ] ,
m 1 = 0 M 1 exp { i π M [ m 1 ( m 0 + h 0 ) M 2 ] 2 } = m 1 = 0 M 1 exp [ i π M ( m 1 m 0 M 2 ) 2 ] .

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