Abstract

The sampling theorem is applied to optical diffraction theory as a computational tool. Formulas are developed in terms of sampled values of the point spread function for the transfer function, total illuminance, line spread function and cumulative line spread function. Typical numerical examples are presented. The theory is presented for general point spread functions for slit and square apertures, but only for rotationally symmetric point spread functions for circular apertures. In the case of the slit aperture, the following question is answered: Given the real part of the incoherent transfer function, determine its imaginary part and vice versa. The theory of conjugate Fourier series and finite Hilbert transforms is introduced in order to answer this question.

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  1. D. Gabor, Progr. Opt. 1, 109 (1961).
  2. H. Gamo, J. Appl. Phys. Japan 25, 431 (1956) 26, 102 (1957); 26, 414 (1957); 27, 577 (1958).
  3. T. di Francia, J. Opt. Soc. Am. 45, 497 (1955).
  4. M.Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), P). 483.
  5. R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.
  6. E. T. Whittaker, Proc. Roy. Soc. (Edinburgh) 35, 181 (1915).
  7. E. H. Linfoot, J. Opt. Soc. Am. 16, 740 (1956).
  8. R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).
  9. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, London, 1948), 2nd ed.
  10. The theory of conjugate Fourier series and finite Hilbert transforms is employed extensively in aerodynamics and just about the only readable accounts are given in aerodynamic treatise. An excellent discussion is contained in: A. Robinson and J. Laurman, Wing Theory (Cambridge University Press, Cambridge, England, 1956), p. 101.
  11. E. L. O'Neill and A. Walther, Opt. Acta 10, 33 (1962).
  12. No Hilbert transform equivalent to conjugate series in two dimensions is known to the author. Howvever, some progress along these lines is contained in the recent paper: S. Goldman, J. Opt. Soc. Am. 52, 1131 (1962).
  13. The cardinal series (6.10) is quite old and in fact is implicitly contained in: J. M. Whittaker, Interpolatory Function Theory (Cambridge University Press, Cambridge, England, 1935), p. 7
  14. Footnote added in proof: The problem of determining t in terms of the sampled value of τ has been solved and will appear in a sequel to the present paper.
  15. R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 1964).

Barakat, R.

R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).

R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 1964).

Born, M.

M.Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), P). 483.

Francia, T. di

T. di Francia, J. Opt. Soc. Am. 45, 497 (1955).

Gabor, D.

D. Gabor, Progr. Opt. 1, 109 (1961).

Gamo, H.

H. Gamo, J. Appl. Phys. Japan 25, 431 (1956) 26, 102 (1957); 26, 414 (1957); 27, 577 (1958).

Houston, A.

R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 1964).

R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).

Linfoot, E. H.

E. H. Linfoot, J. Opt. Soc. Am. 16, 740 (1956).

O’Neill, E. L.

E. L. O'Neill and A. Walther, Opt. Acta 10, 33 (1962).

Paley, R.

R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.

Titchmarsh, E. C.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, London, 1948), 2nd ed.

Walther, A.

E. L. O'Neill and A. Walther, Opt. Acta 10, 33 (1962).

Whittaker, E. T.

E. T. Whittaker, Proc. Roy. Soc. (Edinburgh) 35, 181 (1915).

Whittaker, J. M.

The cardinal series (6.10) is quite old and in fact is implicitly contained in: J. M. Whittaker, Interpolatory Function Theory (Cambridge University Press, Cambridge, England, 1935), p. 7

Wiener, N.

R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.

Wolf, E.

M.Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), P). 483.

Other

D. Gabor, Progr. Opt. 1, 109 (1961).

H. Gamo, J. Appl. Phys. Japan 25, 431 (1956) 26, 102 (1957); 26, 414 (1957); 27, 577 (1958).

T. di Francia, J. Opt. Soc. Am. 45, 497 (1955).

M.Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), P). 483.

R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.

E. T. Whittaker, Proc. Roy. Soc. (Edinburgh) 35, 181 (1915).

E. H. Linfoot, J. Opt. Soc. Am. 16, 740 (1956).

R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, London, 1948), 2nd ed.

The theory of conjugate Fourier series and finite Hilbert transforms is employed extensively in aerodynamics and just about the only readable accounts are given in aerodynamic treatise. An excellent discussion is contained in: A. Robinson and J. Laurman, Wing Theory (Cambridge University Press, Cambridge, England, 1956), p. 101.

E. L. O'Neill and A. Walther, Opt. Acta 10, 33 (1962).

No Hilbert transform equivalent to conjugate series in two dimensions is known to the author. Howvever, some progress along these lines is contained in the recent paper: S. Goldman, J. Opt. Soc. Am. 52, 1131 (1962).

The cardinal series (6.10) is quite old and in fact is implicitly contained in: J. M. Whittaker, Interpolatory Function Theory (Cambridge University Press, Cambridge, England, 1935), p. 7

Footnote added in proof: The problem of determining t in terms of the sampled value of τ has been solved and will appear in a sequel to the present paper.

R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 1964).

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