Abstract

An analytical study of the influence of the diffraction process aberrations on the self-imaging phenomenon is presented. It is found that in a nonparabolic approximation of spherical waves there exists a range of self-imaging configurations of nonunity magnifications with a considerably reduced field of view that is free of image-line deformations and contrast variations. An experimental verification of the principles derived is given.

© 1988 Optical Society of America

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References

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  1. F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).
  2. W. D. Montgomery, “Self-imaging objects of infinite aperture,”J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  3. J. M. Cowley, A. F. Moodie, “Fourier images. I. The point sources,” Proc. Phys. Soc. B 70, 486–496 (1956); “Fourier images. II. The out-of-focus patterns,” 70, 497–504 (1956); “Fourier images. III. Finite sources,” 70, 505–513 (1956).
    [CrossRef]
  4. 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]
  5. S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
    [CrossRef]
  6. S. Yokozeki, K. Patorski, K. Ohnishii, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
    [CrossRef]
  7. R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
    [CrossRef]
  8. J. Jahns, A. W. Lohmann, J. Ojeda-Castañeda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
    [CrossRef]
  9. D. Joyeux, Y. Cohen-Sabban, “High magnification self-imaging,” Appl. Opt. 21, 625–627 (1982).
    [CrossRef] [PubMed]
  10. Y. Cohen-Sabban, D. Joyeux, “Aberration-free nonparaxial self-imaging,”J. Opt. Soc. Am. 73, 707–719 (1983).
    [CrossRef]
  11. J. E. Harvey, R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
    [CrossRef] [PubMed]
  12. See, for example, D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1979).
  13. K. Patorski, “Production of binary amplitude gratings with arbitrary opening ratio and variable period,” Opt. Laser Technol. 12, 267–270 (1980).
    [CrossRef]
  14. P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–260 (1985).
    [CrossRef]
  15. J. M. Cowley, A. F. Moodie, “Fourier images. IV. The phase grating,” Proc. Phys. Soc. London Ser. B 76, 378–384 (1960).
    [CrossRef]
  16. K. Patorski, G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
    [CrossRef]
  17. K. Patorski, “Incoherent superposition of multiple self-imaging: Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
    [CrossRef]

1985 (1)

P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–260 (1985).
[CrossRef]

1984 (1)

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

1983 (3)

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging: Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Y. Cohen-Sabban, D. Joyeux, “Aberration-free nonparaxial self-imaging,”J. Opt. Soc. Am. 73, 707–719 (1983).
[CrossRef]

1982 (1)

1981 (1)

K. Patorski, G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
[CrossRef]

1980 (1)

K. Patorski, “Production of binary amplitude gratings with arbitrary opening ratio and variable period,” Opt. Laser Technol. 12, 267–270 (1980).
[CrossRef]

1979 (1)

S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

1978 (1)

1975 (1)

S. Yokozeki, K. Patorski, K. Ohnishii, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

1967 (1)

1965 (1)

1960 (1)

J. M. Cowley, A. F. Moodie, “Fourier images. IV. The phase grating,” Proc. Phys. Soc. London Ser. B 76, 378–384 (1960).
[CrossRef]

1956 (1)

J. M. Cowley, A. F. Moodie, “Fourier images. I. The point sources,” Proc. Phys. Soc. B 70, 486–496 (1956); “Fourier images. II. The out-of-focus patterns,” 70, 497–504 (1956); “Fourier images. III. Finite sources,” 70, 505–513 (1956).
[CrossRef]

1836 (1)

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Cohen-Sabban, Y.

Cowley, J. M.

J. M. Cowley, A. F. Moodie, “Fourier images. IV. The phase grating,” Proc. Phys. Soc. London Ser. B 76, 378–384 (1960).
[CrossRef]

J. M. Cowley, A. F. Moodie, “Fourier images. I. The point sources,” Proc. Phys. Soc. B 70, 486–496 (1956); “Fourier images. II. The out-of-focus patterns,” 70, 497–504 (1956); “Fourier images. III. Finite sources,” 70, 505–513 (1956).
[CrossRef]

Harvey, J. E.

Jahns, J.

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

Joyeux, D.

Jozwicki, R.

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

Lohmann, A. W.

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

Montgomery, W. D.

Moodie, A. F.

J. M. Cowley, A. F. Moodie, “Fourier images. IV. The phase grating,” Proc. Phys. Soc. London Ser. B 76, 378–384 (1960).
[CrossRef]

J. M. Cowley, A. F. Moodie, “Fourier images. I. The point sources,” Proc. Phys. Soc. B 70, 486–496 (1956); “Fourier images. II. The out-of-focus patterns,” 70, 497–504 (1956); “Fourier images. III. Finite sources,” 70, 505–513 (1956).
[CrossRef]

Ohnishii, K.

S. Yokozeki, K. Patorski, K. Ohnishii, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

Ojeda-Castañeda, J.

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

Parfjanowicz, G.

K. Patorski, G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
[CrossRef]

Patorski, K.

K. Patorski, “Incoherent superposition of multiple self-imaging: Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

K. Patorski, G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
[CrossRef]

K. Patorski, “Production of binary amplitude gratings with arbitrary opening ratio and variable period,” Opt. Laser Technol. 12, 267–270 (1980).
[CrossRef]

S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

S. Yokozeki, K. Patorski, K. Ohnishii, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

Shack, R. V.

Szapiel, S.

S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

Szwaykowski, P.

P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–260 (1985).
[CrossRef]

Talbot, F.

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Winthrop, J. T.

Worthington, C. R.

Yokozeki, S.

S. Yokozeki, K. Patorski, K. Ohnishii, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

Opt. Acta (5)

K. Patorski, G. Parfjanowicz, “Self-imaging phenomenon of a sinusoidal complex object,” Opt. Acta 28, 357–367 (1981).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging: Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

S. Szapiel, K. Patorski, “Fresnel diffraction images of periodic objects under Gaussian beam illumination,” Opt. Acta 26, 439–446 (1979).
[CrossRef]

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase corrections of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

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

Opt. Commun. (1)

S. Yokozeki, K. Patorski, K. Ohnishii, “Collimation method using Fourier imaging and moiré techniques,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

Opt. Laser Technol. (2)

K. Patorski, “Production of binary amplitude gratings with arbitrary opening ratio and variable period,” Opt. Laser Technol. 12, 267–270 (1980).
[CrossRef]

P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–260 (1985).
[CrossRef]

Philos. Mag. (1)

F. Talbot, “Facts relating to optical science. IV,” Philos. Mag. 9, 401–407 (1836).

Proc. Phys. Soc. B (1)

J. M. Cowley, A. F. Moodie, “Fourier images. I. The point sources,” Proc. Phys. Soc. B 70, 486–496 (1956); “Fourier images. II. The out-of-focus patterns,” 70, 497–504 (1956); “Fourier images. III. Finite sources,” 70, 505–513 (1956).
[CrossRef]

Proc. Phys. Soc. London Ser. B (1)

J. M. Cowley, A. F. Moodie, “Fourier images. IV. The phase grating,” Proc. Phys. Soc. London Ser. B 76, 378–384 (1960).
[CrossRef]

Other (1)

See, for example, D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1979).

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Figures (3)

Fig. 1
Fig. 1

Three diffraction orders of a sinusoidal-transmittance diffraction grating G illuminated by a point source S0. The axial and lateral distances of the sources of the first diffraction orders with respect to S0 are indicated as δx and Δ, respectively.

Fig. 2
Fig. 2

Moiré fringes obtained by placing the detecting grating in the Fresnel field of a self-imaging grating with a spatial frequency of 40 lines/mm; z0 = 75 mm, z ≈ 120 mm. The detecting grating lines (frequency, 25 lines/mm) are (a) parallel and (b) slightly inclined with respect to the lines of a self-imaging grating.

Fig. 3
Fig. 3

Moiré fringes in the plane of detecting gratings with spatial frequencies of 20 lines/mm and diameters of 85 mm, placed in the Fresnel field of a 40-line/mm self-imaging grating. z0 = 40 mm, z ≈ 80 mm. (a) Parallel grating lines; (b) slightly rotated grating lines. See the text for further explanations.

Equations (23)

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T G ( x , y ) = A 0 + 2 A 1 cos ( 2 π x / d ) ,
U 0 ( x , y , z ) = A 0 exp { i 2 π λ [ x 2 + y 2 2 z 1 8 z 3 ( x 2 + y 2 ) 2 ] } ,
U + 1 ( x , y , z ) = A 1 exp ( i 2 π λ { ( x Δ ) 2 + y 2 2 z 1 8 z 3 [ ( x Δ ) 2 + y 2 ] 2 δ x } ) ,
U 1 ( x , y , z ) = A 1 exp ( i 2 π λ { ( x + Δ ) 2 + y 2 2 z 1 8 z 3 [ ( x + Δ ) 2 + y 2 ] 2 δ x } ) ,
| Δ | = z 0 sin θ = z 0 ( λ / d ) ,
δ x = | z 0 ( 1 cos θ ) | .
δ x = | z 0 ( 1 cos θ ) | | z 0 [ 1 2 ( λ d ) 2 + 1 8 ( λ d ) 4 ] | .
E ( x , y , z ) = U 0 ( x , y , z ) + U 1 ( x , y , z ) + U 1 ( x , y , z )
I ( x , y , z ) = A 0 2 + 2 A 1 2 + 2 A 0 A 1 cos 2 π λ [ 2 x Δ + Δ 2 2 z 4 x Δ ( x 2 + y 2 + Δ 2 ) + 2 Δ 2 ( 3 x 2 + y 2 ) + Δ 4 8 z 3 δ x ] + 2 A 0 A 1 cos 2 π λ [ 2 x Δ Δ 2 2 z 4 x Δ ( x 2 + y 2 + Δ 2 ) 2 Δ 2 ( 3 x 2 + y 2 ) Δ 4 8 z 3 + δ x ] + 2 A 1 2 cos 2 π λ [ 2 x Δ z x Δ ( x 2 + y 2 + Δ 2 ) z 3 ] = A 0 2 + 2 A 1 2 + 4 A 0 A 1 cos 2 π λ [ Δ 2 2 z 2 Δ 2 ( 3 x 2 + y 2 ) + Δ 4 8 z 3 δ x ] cos 2 π λ [ x Δ z 4 x Δ ( x 2 + y 2 + Δ 2 ) 8 z 3 ] + 2 A 1 2 cos 2 π λ [ 2 x Δ z x Δ ( x 2 + y 2 + Δ 2 ) z 3 ] .
I ( x , y , z ) = A 0 2 + 2 A 1 2 + 4 A 0 A 1 cos 2 π [ z 0 2 2 z λ d 2 z 0 λ ( 1 cos θ ) z 0 2 4 z 3 λ d 2 ( 3 x 2 + y 2 ) z 0 4 λ 3 8 z 3 d 4 ] × cos 2 π d ( z / z 0 ) { x x [ x 2 + y 2 + ( λ z 0 / d ) 2 ] 2 z 2 } + 2 A 1 2 cos 2 π d ( z / z 0 ) { 2 x x [ x 2 + y 2 + ( λ z 0 / d ) 2 ] z 2 } .
δ x = | z 0 ( 1 cos θ ) | = 1 2 z 0 ( λ d ) 2 .
I P ( x , y , z ) = A 0 2 + 2 A 1 2 + 4 A 0 A 1 cos π λ d 2 ( z z 0 ) z 0 z × cos 2 π d ( z / z 0 ) x + 2 A 1 2 cos 2 π d ( z / z 0 ) 2 x ,
z 0 ( z z 0 ) z λ d 2 = M
cos 2 π d ( z / z 0 ) { x x [ x 2 + y 2 + ( λ z 0 / d ) 2 ] 2 z 2 } = cos 2 π d ( z / z 0 ) ( 1 Δ 2 2 z 2 ) × [ x 1 2 z 2 ( 1 Δ 2 / 2 z 2 ) x ( x 2 + y 2 ) ] = cos 2 π d ( z / z 0 ) [ x 1 2 z 2 x ( x 2 + y 2 ) ]
u ( x , y ) = 1 2 z 2 x ( x 2 + y 2 ) .
2 π [ z 0 2 2 z λ d 2 z 0 λ ( 1 cos θ ) z 0 4 λ 3 8 z 3 d 4 ] = M π ,
2 π { λ z 0 2 d 2 ( z 0 z ) z 1 8 z 0 λ 3 d 4 [ 1 + ( z 0 z ) 3 ] } = M π .
2 π z 0 2 ( 3 x 2 + y 2 ) 4 z 3 λ d 2 = N π 2 ,
3 x 2 + y 2 = N z 3 z 0 2 d 2 λ ,
2 x e = | ± 2 ( z / z 0 ) d ( N z / 3 λ ) 1 / 2 | ,
2 y e = | ± 2 ( z / z 0 ) d ( N z / λ ) 1 / 2 | ,
2 x e = | ± 2 β [ N ( d 2 / λ ) z / 3 ] 1 / 2 | ,
2 y e = | ± 2 β [ N ( d 2 / λ ) z ] 1 / 2 | ,

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