Abstract

Strongly magnified images of a periodic object can be obtained by using the Talbot effect. Two particular advantages are simplicity and compactness of the setup even for a high magnification ratio. First we recall the principle of self-imaging and the basic results derived by Rayleigh. These results were obtained from simplifying assumptions, which we show to be particularly unrealistic for high magnification ratios. A possibly more accurate and nevertheless practical approach is proposed.

© 1982 Optical Society of America

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References

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  1. F. Talbot, Philos. Mag. 9, 401 (1836).
  2. J. Eu, C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973).
    [CrossRef]
  3. O. Bryngdahl, J. Opt. Soc. Am. 63, 416 (1973).
    [CrossRef]
  4. K. Patorski, S. Yokozchi, T. Suzuki, Nouv. Rev. Optique (Paris) 6, 25 (1975).
    [CrossRef]
  5. A. Lohmann, in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman & Hall, London, 1961), p. 58.
  6. M. J. Cowley, A. F. Moodie, Proc. Phys. Soc. (London), Sect. B 70, 486 (1957).
    [CrossRef]
  7. J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), p. 19.
  8. A. Kalestynski, B. Smolinska, Opt. Acta 25, 125 (1978).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 11.7.

1978

A. Kalestynski, B. Smolinska, Opt. Acta 25, 125 (1978).
[CrossRef]

1975

K. Patorski, S. Yokozchi, T. Suzuki, Nouv. Rev. Optique (Paris) 6, 25 (1975).
[CrossRef]

1973

J. Eu, C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973).
[CrossRef]

O. Bryngdahl, J. Opt. Soc. Am. 63, 416 (1973).
[CrossRef]

1957

M. J. Cowley, A. F. Moodie, Proc. Phys. Soc. (London), Sect. B 70, 486 (1957).
[CrossRef]

1836

F. Talbot, Philos. Mag. 9, 401 (1836).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 11.7.

Bryngdahl, O.

Cowley, J. M.

J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), p. 19.

Cowley, M. J.

M. J. Cowley, A. F. Moodie, Proc. Phys. Soc. (London), Sect. B 70, 486 (1957).
[CrossRef]

Eu, J.

J. Eu, C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973).
[CrossRef]

Kalestynski, A.

A. Kalestynski, B. Smolinska, Opt. Acta 25, 125 (1978).
[CrossRef]

Liu, C.

J. Eu, C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973).
[CrossRef]

Lohmann, A.

A. Lohmann, in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman & Hall, London, 1961), p. 58.

Lohmann, A. W.

J. Eu, C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973).
[CrossRef]

Moodie, A. F.

M. J. Cowley, A. F. Moodie, Proc. Phys. Soc. (London), Sect. B 70, 486 (1957).
[CrossRef]

Patorski, K.

K. Patorski, S. Yokozchi, T. Suzuki, Nouv. Rev. Optique (Paris) 6, 25 (1975).
[CrossRef]

Smolinska, B.

A. Kalestynski, B. Smolinska, Opt. Acta 25, 125 (1978).
[CrossRef]

Suzuki, T.

K. Patorski, S. Yokozchi, T. Suzuki, Nouv. Rev. Optique (Paris) 6, 25 (1975).
[CrossRef]

Talbot, F.

F. Talbot, Philos. Mag. 9, 401 (1836).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 11.7.

Yokozchi, S.

K. Patorski, S. Yokozchi, T. Suzuki, Nouv. Rev. Optique (Paris) 6, 25 (1975).
[CrossRef]

J. Opt. Soc. Am.

Nouv. Rev. Optique (Paris)

K. Patorski, S. Yokozchi, T. Suzuki, Nouv. Rev. Optique (Paris) 6, 25 (1975).
[CrossRef]

Opt. Acta

A. Kalestynski, B. Smolinska, Opt. Acta 25, 125 (1978).
[CrossRef]

Opt. Commun.

J. Eu, C. Liu, A. W. Lohmann, Opt. Commun. 9, 168 (1973).
[CrossRef]

Philos. Mag.

F. Talbot, Philos. Mag. 9, 401 (1836).

Proc. Phys. Soc. (London), Sect. B

M. J. Cowley, A. F. Moodie, Proc. Phys. Soc. (London), Sect. B 70, 486 (1957).
[CrossRef]

Other

J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), p. 19.

A. Lohmann, in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman & Hall, London, 1961), p. 58.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 11.7.

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Figures (5)

Fig. 1
Fig. 1

Self-imaging with magnification M = 1 + zd. z and d are related to wavelength λ and grating period a by 1/z + 1/d = λ/na2 (n integer).

Fig. 2
Fig. 2

Photometric section of a magnified self-image. Parameters are: Ronchi grating, period 10-μm black/white = 1/1, z = 100 mm, n = 3, M3 = 210.

Fig. 3
Fig. 3

Geometrical limitation of the number of contributing orders. Here, only orders 0 and ±1 interfere at P; other orders do not illuminate P. Negative orders are not shown for clarity.

Fig. 4
Fig. 4

Diffraction by pupil edges. (a) Geometrical parameters are those of the self-imaging setup with a = 10 μm, z = 100 mm, n = 4, and M4 = 160. Referring to this setup, the source position y = 0 and y = 0.123 mm are, respectively, those of Fourier focuses #0 and #3. The specified section would be the central self-image period. (b) Illuminance in the photometric section defined in (a) for two values of y.

Fig. 5
Fig. 5

Computed photometric section of the on-axis image period (geometrical computation without diffraction by edges). Parameters are z = 100 mm, a = 10 μm, n = 4, M = 160. Dotted line: parabolic phase approximation (Rayleigh result), improved by limitation of the object in the Fourier series (see Fig. 3). Solid line: accurate phase computation based on the Fermat principle.

Equations (2)

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1 z + 1 d = λ n a 2             ( n , integer ; λ , wavelength ) ,
for n = 1 d 1 = 0.156 mm and M 1 = 633 , n = 2 d 2 = 0.317 mm and M 2 = 316 , n = 4 d 4 = 0.636 mm and M 4 = 158.

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