Abstract

The fields diffracted by planar one- or two-dimensional periodic objects, and in particular their Fourier and Fresnel self-images, can be computed with the aid of a ray-tracing technique based on the Fermat principle. This method (geometrical self-imaging) yields accurate results for any numerical aperture and image field. An analytical study of the image formation, carried out in the fourth-order approximation for the phase, leads to the definition of selfimaging aberrations. These aberrations are strongly dependent on spatial frequency and render the well-known relationships derived by Rayleigh for the location and magnification of self-images approximate at best. The aberrations can be described graphically by a phase diagram and a magnification diagram, which permit interpretation of the properties of high-aperture, large-field self-images and the prediction of optimal imaging conditions. In the case of large magnifications (10OX and larger), we present a simple method to eliminate all fourth-order aberrations completely and even sixth-order ones partially. This method consists of introducing a compensating spherical aberration to the incident wave, e.g., by the insertion of a glass plate of appropriate index and thickness just before the object. Thus object spatial frequencies up to about 800 m<sup>-1</sup> can be imaged almost without aberration for several image periods.

© 1983 Optical Society of America

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  1. F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 403–405 (1836), Sec. 2.
  2. J. M. Cowley and A. F. Moodie, "Fourier images," Parts I-III, Proc. Phys. Soc. B 70,486–513 (1956).
  3. M. Fujiwara, "Effects of spatial coherence on Fourier imaging of a periodic object," Opt. Acta 21, 861–869 (1974).
  4. S. Szapiel and K. Patorski, "Fresnel diffraction images of periodic objects under Gaussian beam illumination," Opt. Acta, 26, 439–446 (1979).
  5. K. Patorski and G. Parfjanowicz, "Self-imaging phenomenon of a sinusoidal complex object," Opt. Acta 28, 357–367 (1981).
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  13. J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, "Spatial filters for differentiation," Opt. Commun. 9, 168–171 (1973).
  14. K. Patorski, S. Yokozeki, and T. Suzuki, "Image subtraction using Fourier imaging phenomenon," Nouv. Rev. Opt. 6, 25–31 (1975).
  15. D. E. Silva, "A simple interferometric method of beam collimation," Appl. Opt. 10, 1980–1982 (1971).
  16. R. F. Edgar, "The Fresnel diffraction images of periodic structures," Opt. Acta 16, 281–287 (1969).
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  18. D. Joyeux and Y. Cohen-Sabban, "High magnification selfimaging," Appl. Opt. 21, 625–627 (1982).
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  20. J. B. Keller, "Geometrical theory of diffraction," J. Opt. Soc. Am. 52, 116–130 (1962).
  21. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 401–407.

1982 (1)

1981 (1)

K. Patorski and G. Parfjanowicz, "Self-imaging phenomenon of a sinusoidal complex object," Opt. Acta 28, 357–367 (1981).

1979 (1)

S. Szapiel and K. Patorski, "Fresnel diffraction images of periodic objects under Gaussian beam illumination," Opt. Acta, 26, 439–446 (1979).

1978 (1)

A. Kalestynski and B. Smolinska, "Self restoration of the autoidolon of defective periodic objects," Opt. Acta 25, 125–134 (1978).

1975 (1)

K. Patorski, S. Yokozeki, and T. Suzuki, "Image subtraction using Fourier imaging phenomenon," Nouv. Rev. Opt. 6, 25–31 (1975).

1974 (1)

M. Fujiwara, "Effects of spatial coherence on Fourier imaging of a periodic object," Opt. Acta 21, 861–869 (1974).

1973 (3)

O. Bryngdahl, "Image formation using self-imaging techniques," J. Opt. Soc. Am. 63, 416–419 (1973).

J. K. T. Eu and A. W. Lohmann, "Spatial filtering effects by means of hologram copying," Opt. Commun. 8, 176–182 (1973).

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, "Spatial filters for differentiation," Opt. Commun. 9, 168–171 (1973).

1971 (2)

1969 (1)

R. F. Edgar, "The Fresnel diffraction images of periodic structures," Opt. Acta 16, 281–287 (1969).

1965 (1)

1962 (1)

1959 (1)

1956 (1)

J. M. Cowley and A. F. Moodie, "Fourier images," Parts I-III, Proc. Phys. Soc. B 70,486–513 (1956).

1836 (1)

F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 403–405 (1836), Sec. 2.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 401–407.

Breazeale, M. A.

Bryngdahl, O.

Cohen-Sabban, Y.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, "Fourier images," Parts I-III, Proc. Phys. Soc. B 70,486–513 (1956).

Damman, H.

Edgar, R. F.

R. F. Edgar, "The Fresnel diffraction images of periodic structures," Opt. Acta 16, 281–287 (1969).

Eu, J. K. T.

J. K. T. Eu and A. W. Lohmann, "Spatial filtering effects by means of hologram copying," Opt. Commun. 8, 176–182 (1973).

Eu, J. T. K.

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, "Spatial filters for differentiation," Opt. Commun. 9, 168–171 (1973).

Fujiwara, M.

M. Fujiwara, "Effects of spatial coherence on Fourier imaging of a periodic object," Opt. Acta 21, 861–869 (1974).

Groh, G.

Hiedemann, E. A.

Joyeux, D.

Kalestynski, A.

A. Kalestynski and B. Smolinska, "Self restoration of the autoidolon of defective periodic objects," Opt. Acta 25, 125–134 (1978).

Keller, J. B.

Kock, M.

Liu, C. Y. C.

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, "Spatial filters for differentiation," Opt. Commun. 9, 168–171 (1973).

Lohmann, A. W.

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, "Spatial filters for differentiation," Opt. Commun. 9, 168–171 (1973).

J. K. T. Eu and A. W. Lohmann, "Spatial filtering effects by means of hologram copying," Opt. Commun. 8, 176–182 (1973).

A. W. Lohmann and D. E. Silva, "An interferometer based on the Talbot effect," Opt. Commun. 2, 413–415 (1971); "A Talbot interferometer with circular gratings," Opt. Commun. 4, 326–328 (1972).

Montgomery, W. D.

W. D. Montgomery, "Self imaging objects of infinite aperture," J. Opt. Soc. Am. 57, 772–778 (1967); "Algebraic formulation of diffraction applied to self-imaging," J. Opt. Soc. Am. 58, 1112– 1124 (1968).

Moodie, A. F.

J. M. Cowley and A. F. Moodie, "Fourier images," Parts I-III, Proc. Phys. Soc. B 70,486–513 (1956).

Parfjanowicz, G.

K. Patorski and G. Parfjanowicz, "Self-imaging phenomenon of a sinusoidal complex object," Opt. Acta 28, 357–367 (1981).

Patorski, K.

K. Patorski and G. Parfjanowicz, "Self-imaging phenomenon of a sinusoidal complex object," Opt. Acta 28, 357–367 (1981).

S. Szapiel and K. Patorski, "Fresnel diffraction images of periodic objects under Gaussian beam illumination," Opt. Acta, 26, 439–446 (1979).

K. Patorski, S. Yokozeki, and T. Suzuki, "Image subtraction using Fourier imaging phenomenon," Nouv. Rev. Opt. 6, 25–31 (1975).

Rubinowicz, A.

A. Rubinowicz, "The Miyamoto–Wolf diffraction wave," in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1965), Vol. IV, pp. 199–240.

Silva, D. E.

D. E. Silva, "A simple interferometric method of beam collimation," Appl. Opt. 10, 1980–1982 (1971).

A. W. Lohmann and D. E. Silva, "An interferometer based on the Talbot effect," Opt. Commun. 2, 413–415 (1971); "A Talbot interferometer with circular gratings," Opt. Commun. 4, 326–328 (1972).

Smolinska, B.

A. Kalestynski and B. Smolinska, "Self restoration of the autoidolon of defective periodic objects," Opt. Acta 25, 125–134 (1978).

Suzuki, T.

K. Patorski, S. Yokozeki, and T. Suzuki, "Image subtraction using Fourier imaging phenomenon," Nouv. Rev. Opt. 6, 25–31 (1975).

Szapiel, S.

S. Szapiel and K. Patorski, "Fresnel diffraction images of periodic objects under Gaussian beam illumination," Opt. Acta, 26, 439–446 (1979).

Talbot, F.

F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 403–405 (1836), Sec. 2.

Winthrop, J. J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 401–407.

Worthington, C. R.

Yokozeki, S.

K. Patorski, S. Yokozeki, and T. Suzuki, "Image subtraction using Fourier imaging phenomenon," Nouv. Rev. Opt. 6, 25–31 (1975).

Appl. Opt. (3)

J. Opt. Soc. Am. (4)

Nouv. Rev. Opt. (1)

K. Patorski, S. Yokozeki, and T. Suzuki, "Image subtraction using Fourier imaging phenomenon," Nouv. Rev. Opt. 6, 25–31 (1975).

Opt. Acta (4)

K. Patorski and G. Parfjanowicz, "Self-imaging phenomenon of a sinusoidal complex object," Opt. Acta 28, 357–367 (1981).

R. F. Edgar, "The Fresnel diffraction images of periodic structures," Opt. Acta 16, 281–287 (1969).

A. Kalestynski and B. Smolinska, "Self restoration of the autoidolon of defective periodic objects," Opt. Acta 25, 125–134 (1978).

M. Fujiwara, "Effects of spatial coherence on Fourier imaging of a periodic object," Opt. Acta 21, 861–869 (1974).

Opt. Commun. (2)

J. K. T. Eu and A. W. Lohmann, "Spatial filtering effects by means of hologram copying," Opt. Commun. 8, 176–182 (1973).

J. T. K. Eu, C. Y. C. Liu, and A. W. Lohmann, "Spatial filters for differentiation," Opt. Commun. 9, 168–171 (1973).

Philos. Mag. (1)

F. Talbot, "Facts relating to optical science," Philos. Mag. 9, 403–405 (1836), Sec. 2.

Proc. Phys. Soc. (1)

J. M. Cowley and A. F. Moodie, "Fourier images," Parts I-III, Proc. Phys. Soc. B 70,486–513 (1956).

Other (5)

S. Szapiel and K. Patorski, "Fresnel diffraction images of periodic objects under Gaussian beam illumination," Opt. Acta, 26, 439–446 (1979).

A. W. Lohmann and D. E. Silva, "An interferometer based on the Talbot effect," Opt. Commun. 2, 413–415 (1971); "A Talbot interferometer with circular gratings," Opt. Commun. 4, 326–328 (1972).

W. D. Montgomery, "Self imaging objects of infinite aperture," J. Opt. Soc. Am. 57, 772–778 (1967); "Algebraic formulation of diffraction applied to self-imaging," J. Opt. Soc. Am. 58, 1112– 1124 (1968).

A. Rubinowicz, "The Miyamoto–Wolf diffraction wave," in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1965), Vol. IV, pp. 199–240.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), pp. 401–407.

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