Abstract

The one-dimensional case of the image of a sinusoidal transmittance distribution in partially coherent illumination (with the quasimonochromatic approximation) is described analytically, and shown generally to bear a nonlinear relation to the object. It is shown that the significant parameter is the ratio of coherence interval to the diameter of the Airy disk (or diffraction spot) of the imaging lens. It is further shown that since the spatial frequency of the object is related to coherence interval, typical nonlinear effects can take place at low frequencies. Since the transfer function is defined only for the incoherent limit without ambiguity, an apparent transfer function, dealing only with the image component which existed in the object, is used for comparison. The harmonics generated by the nonlinear behavior are ignored, and the variation of transfer function is observed to be a function of coherence and input modulation. It becomes apparent that the transfer function, as currently defined and measured, is inadequate to describe optical-system performance under all conditions of illumination.

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References

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  1. G. Parrent, Jr., and R. Becherer, J. Opt. Soc. Am. 56, 548A (1966).
  2. R. Lamberts, J. Opt. Soc. Am. 51, 932 (1961).
  3. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, N. Y., 1964), pp. 510, 532–533.
  4. G. Reynolds and J. Ward, J. Soc. Phot. Instr. Eng. 5, 3 (1966).
  5. J. Altman, J. Phot. Sci. Eng. 10, 156 (1966).
  6. D. Falconer, J. Phot. Sci. Eng. 10, 133 (1966).
  7. E. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1963), pp. 122–127.
  8. M. Beran and G. Parrent, Jr., Theory of Partial Cohzerelce (Prentice Hall, Englewood Cliffs, N. J., 1964), pp. 106–113.
  9. All integrals in this paper, unless otherwise noted, are evaluated between - ∞ and ∞.
  10. Reference 8, p. 111.
  11. Reference 3, pp. 528–530.
  12. Reference 3, Appendix IV, Eq. 12.
  13. This is based on the generalization that the larger the relative aperture (or the smaller the ƒ-number), the better the general image quality.
  14. K. Knopp, Theory and Application of Infinite Series (Blackie & Son, Ltd., London and Glasgow, 1947), p. 424.
  15. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Co., New York, 1965).

Altman, J.

J. Altman, J. Phot. Sci. Eng. 10, 156 (1966).

Becherer, R.

G. Parrent, Jr., and R. Becherer, J. Opt. Soc. Am. 56, 548A (1966).

Beran, M.

M. Beran and G. Parrent, Jr., Theory of Partial Cohzerelce (Prentice Hall, Englewood Cliffs, N. J., 1964), pp. 106–113.

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, N. Y., 1964), pp. 510, 532–533.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Co., New York, 1965).

Falconer, D.

D. Falconer, J. Phot. Sci. Eng. 10, 133 (1966).

Knopp, K.

K. Knopp, Theory and Application of Infinite Series (Blackie & Son, Ltd., London and Glasgow, 1947), p. 424.

Lamberts, R.

R. Lamberts, J. Opt. Soc. Am. 51, 932 (1961).

O’Neill, E.

E. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1963), pp. 122–127.

Parrent, Jr., G.

M. Beran and G. Parrent, Jr., Theory of Partial Cohzerelce (Prentice Hall, Englewood Cliffs, N. J., 1964), pp. 106–113.

G. Parrent, Jr., and R. Becherer, J. Opt. Soc. Am. 56, 548A (1966).

Reynolds, G.

G. Reynolds and J. Ward, J. Soc. Phot. Instr. Eng. 5, 3 (1966).

Ward, J.

G. Reynolds and J. Ward, J. Soc. Phot. Instr. Eng. 5, 3 (1966).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, N. Y., 1964), pp. 510, 532–533.

Other

G. Parrent, Jr., and R. Becherer, J. Opt. Soc. Am. 56, 548A (1966).

R. Lamberts, J. Opt. Soc. Am. 51, 932 (1961).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, N. Y., 1964), pp. 510, 532–533.

G. Reynolds and J. Ward, J. Soc. Phot. Instr. Eng. 5, 3 (1966).

J. Altman, J. Phot. Sci. Eng. 10, 156 (1966).

D. Falconer, J. Phot. Sci. Eng. 10, 133 (1966).

E. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1963), pp. 122–127.

M. Beran and G. Parrent, Jr., Theory of Partial Cohzerelce (Prentice Hall, Englewood Cliffs, N. J., 1964), pp. 106–113.

All integrals in this paper, unless otherwise noted, are evaluated between - ∞ and ∞.

Reference 8, p. 111.

Reference 3, pp. 528–530.

Reference 3, Appendix IV, Eq. 12.

This is based on the generalization that the larger the relative aperture (or the smaller the ƒ-number), the better the general image quality.

K. Knopp, Theory and Application of Infinite Series (Blackie & Son, Ltd., London and Glasgow, 1947), p. 424.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill Book Co., New York, 1965).

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