Abstract

The Fresnel diffraction of periodic objects at rational fractions of the Talbot distance is described in terms of the Wigner distribution function (WDF). The analysis provides a heuristic model for understanding the formation of the diffraction patterns as well as for evaluating the complex amplitude at any fractional Talbot plane. Furthermore, certain symmetry properties of the Fresnel-diffracted wave field can be derived directly from the WDF. Additionally, a discussion is given on how periodic signals and information about the phase are encoded in the WDF.

© 1996 Optical Society of America

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References

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  1. 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]
  2. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).
  3. J. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates as fractional Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]
  4. Xiao-Yi Da, “Talbot effect and the array illuminators based on it,” Appl. Opt. 31, 2983–2986 (1992).
    [CrossRef] [PubMed]
  5. V. Arrizón, J. Ojeda-Castaneda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [CrossRef] [PubMed]
  6. J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
    [CrossRef]
  7. J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
    [CrossRef]
  8. J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
    [CrossRef]
  9. M. J. Bastiaans, “Wigner distribution function and its application to first order optics,” J. Opt. Soc. Am. 69, 1710–1717 (1979).
    [CrossRef]
  10. V. Arrizón, J. Ojeda-Castaneda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
    [CrossRef]
  11. H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  12. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  13. M. R. Schroeder, Number Theory in Science and Communication, 2nd ed. (Springer, New York, 1990), Chap. 7, pp. 95–110.
  14. V. Arrizón, J. Ojeda-Castaneda, “Fresnel diffraction of substructured gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
    [CrossRef] [PubMed]

1995 (1)

1994 (1)

1992 (2)

1990 (1)

1988 (1)

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

1985 (1)

J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

1984 (1)

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

1980 (1)

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

1979 (1)

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1971 (1)

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

1965 (1)

Arrizón, V.

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Wigner distribution function and its application to first order optics,” J. Opt. Soc. Am. 69, 1710–1717 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Da, Xiao-Yi

Guigay, J. P.

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

Jahns, J.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

Leger, J.

Lohmann, A. W.

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

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Ojeda-Castaneda, J.

Schroeder, M. R.

M. R. Schroeder, Number Theory in Science and Communication, 2nd ed. (Springer, New York, 1990), Chap. 7, pp. 95–110.

Sicre, E. E.

J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

Swanson, G. J.

Winthrop, J. T.

Worthington, C. R.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

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

Opt. Acta (3)

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

Opt. Commun. (2)

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

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

Other (1)

M. R. Schroeder, Number Theory in Science and Communication, 2nd ed. (Springer, New York, 1990), Chap. 7, pp. 95–110.

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Figures (5)

Fig. 1
Fig. 1

Wigner distribution function (WDF) of a comb function and its projections. The symbols + and − indicate the discrete locations of delta functions with positive and negative signs, respectively.

Fig. 2
Fig. 2

WDF of the Fresnel pattern of a comb function at 1/3 of the Talbot distance. The dotted lines indicate columns of delta peaks, which correspond to nonzero intensity. The dotted–dashed line illustrates the shear parallel to the ν axis that produce the WDF pattern of a Fresnel propagated comb function from the WDF of a comb function.

Fig. 3
Fig. 3

Same as Fig. 2, but at 1/4 of the Talbot distance.

Fig. 4
Fig. 4

Same as Fig. 2, but at 1/6 of the Talbot distance.

Fig. 5
Fig. 5

Graphical display for obtaining the non-Cartesian distributions of the WDF describing the wave propagation of a comb function from a Cartesian distribution. The crossing points of the lines indicate the location of the delta functions.

Equations (26)

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W u ( x , ν ) = - u ( x + x / 2 ) u * ( x - x / 2 ) exp ( - i 2 π ν x ) d x ,
u ( x ) 2 = - W u ( x , ν ) d ν ,             u ˜ ( ν ) 2 = - W u ( x , ν ) d x ,
g ( x ) = n = - g n exp ( i 2 π n d x ) .
W p , q ( x , ν ) = g p 2 δ ( ν - p d ) + g q 2 δ ( ν - q d ) + g p g q * exp [ i 2 π x d ( p - q ) ] δ ( ν - p + q 2 d ) + g q g p * exp [ i 2 π x d ( q - p ) ] δ ( ν - q + p 2 d ) .
u ( x ) = exp ( i ϕ ) - W u ( x / 2 , ν ) exp ( i 2 π x ν ) d ν [ - W u ( 0 , ν d ν ) ] 1 / 2 ,
W g ( x , ν ) = n = - m = - g n g m * exp [ i 2 π x d ( n - m ) ] × δ ( ν - n + m 2 d ) .
comb ( x ) = n = - δ ( x - n d ) ,
W comb ( x , ν ) = 2 d n = - exp ( i 2 π x 2 d n ) 1 2 d × n = - exp ( i 2 π x d n ) δ ( ν - n 2 d ) .
W comb ( x , ν ) = 1 2 d n = - n = - ( - 1 ) n n δ ( x - n d 2 ) × δ ( ν - n 2 d ) .
g ( x , z ) = g 0 ( x ) comb ( x ) exp ( i π λ z x 2 ) ,
W ( x , ν , z ) = W ( x - λ ν z , ν , 0 ) .
λ ( 1 / 2 d ) z T = d ,
W comb ( x , ν , z T M N ) = 1 2 d n = - n = - ( - 1 ) n n δ ( x - n d 2 - n M d N ) × δ ( ν - n 1 2 d ) .
comb ( x , z T M N ) = 1 N n = - n = 0 N - 1 δ ( x - n d - 2 M d N n ) × exp ( i 2 π M N n 2 ) .
comb ( x , z T M N ) = 1 N n = - k = 0 N - 1 δ ( x - n d - d N k ) × exp [ i θ ( k ) ] .
k 2 M n             ( mod N ) .
θ k = { π k 2 / ( 2 N ) ; k even π k 2 / ( 2 N ) - π / 2 ; k odd , ( N - 1 ) / 2 even π k 2 / ( 2 N ) + π / 2 ; k odd , ( N - 1 ) / 2 odd .
comb ( x , z T M N ) = 2 N n = - k = 0 N / 2 - 1 δ ( x - n d - 2 d N k ) × exp ( i 2 π M N n 2 ) .
k M n ( mod N 2 ) ,
comb ( x , z T M N ) = 2 N n = - k = 0 N / 2 - 1 δ ( x - n d - 2 d N k - d N ) × exp ( i 2 π M n 2 + n N ) ,
u ( x ) = u ( x ) exp ( i π a x 2 ) W u ( x , ν ) = W u ( x , ν - a x ) .
comb ( x , z T M N ) = 1 N n = - k = 0 N - 1 δ ( x - n d - d N k ) × exp ( i π a x 2 ) .
2 M n + N n = 1 ,
comb ( x , z T N ) = 1 N n = - k = 0 N - 1 δ ( x - n d - d N k ) × exp ( i π n k 2 N ) .
W ( x , ν ) = W ( - x , - ν ) comb ( x , z ) = exp ( i ϕ ) comb ( - x , z ) .
z z T - z comb ( x , z ) comb ( x , z T - z ) = comb * ( - x , z ) .

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