Abstract

A lens system may be judged by its ability to relay entropy from object to image. The pertinent criterion of optical quality is <i>h</i>, the loss of entropy between corresponding sampling points in the object and image planes. Since <i>h</i> is a unique function of the optical pupil, depending on the object only through its cutoff frequency Ω, by the proper choice of a pupil function it is possible to maximize <i>h</i> at each given Ω. Physically, the optimum pupil function is an absorbing film applied to a diffraction-limited lens system. A numerical procedure is established for determining, with arbitrary accuracy, the optimum pupil function, the resulting transfer function, and the maximum <i>h</i>, all at a given Ω. These quantities are determined, both for the one-dimensional pupil and the circular pupil, in the approximation that the optimum pupil function may be represented as a Fourier (Bessel) series of five terms. The computed values of <i>h</i><sub>max</sub>, at a sequence of Ω values, are estimated to be correct to 0.2% for the 1-D pupil, and to 0.5% for the circular pupil. The optimum pupil functions are apodizers at small Ω and superresolvers at large Ω.

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  1. P. M. Duffieux, L’integrale de Fourier et ses applications á l’optique, (Besançon, 1947), privately printed.
  2. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963).
  3. C. E. Shannon, Bell System Tech. J. 27, 379, 623 (1948).
  4. P. B. Fellgett and E. H. Linfoot, Proc. Roy. Soc. (London) 247A, 369 (1955), §4.2.
  5. See, e.g., D. Middleton, Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), p. 315, Eq. (6.78). Its derivation proceeds on pp. 308, 309, 315,
  6. P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), III, p. 32.
  7. B. Dossier, Rev. Opt. 33, 67 (1954).
  8. The use of such expansions is suggested in Ref. 6.
  9. See, for example, F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Company, New York, 1956), p. 447, where it is called the Newton-Raphson method.
  10. See, for example, R. Courant, Differential and Integral Calculus (Blackie and Son, Ltd., London, 1942), Vol. II, pp. 188-199.
  11. Ref. 9, p. 343.
  12. Ref. 6, p. 122.
  13. R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill Book Co., New York, 1941), p. 164.

Churchill, R. V.

R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill Book Co., New York, 1941), p. 164.

Courant, R.

See, for example, R. Courant, Differential and Integral Calculus (Blackie and Son, Ltd., London, 1942), Vol. II, pp. 188-199.

Dossier, B.

B. Dossier, Rev. Opt. 33, 67 (1954).

Duffieux, P. M.

P. M. Duffieux, L’integrale de Fourier et ses applications á l’optique, (Besançon, 1947), privately printed.

Fellgett, P. B.

P. B. Fellgett and E. H. Linfoot, Proc. Roy. Soc. (London) 247A, 369 (1955), §4.2.

Hildebrand, F. B.

See, for example, F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Company, New York, 1956), p. 447, where it is called the Newton-Raphson method.

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), III, p. 32.

Linfoot, E. H.

P. B. Fellgett and E. H. Linfoot, Proc. Roy. Soc. (London) 247A, 369 (1955), §4.2.

Middleton, D.

See, e.g., D. Middleton, Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), p. 315, Eq. (6.78). Its derivation proceeds on pp. 308, 309, 315,

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963).

Roizen-Dossier, B.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), III, p. 32.

Shannon, C. E.

C. E. Shannon, Bell System Tech. J. 27, 379, 623 (1948).

Other

P. M. Duffieux, L’integrale de Fourier et ses applications á l’optique, (Besançon, 1947), privately printed.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963).

C. E. Shannon, Bell System Tech. J. 27, 379, 623 (1948).

P. B. Fellgett and E. H. Linfoot, Proc. Roy. Soc. (London) 247A, 369 (1955), §4.2.

See, e.g., D. Middleton, Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), p. 315, Eq. (6.78). Its derivation proceeds on pp. 308, 309, 315,

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), III, p. 32.

B. Dossier, Rev. Opt. 33, 67 (1954).

The use of such expansions is suggested in Ref. 6.

See, for example, F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Company, New York, 1956), p. 447, where it is called the Newton-Raphson method.

See, for example, R. Courant, Differential and Integral Calculus (Blackie and Son, Ltd., London, 1942), Vol. II, pp. 188-199.

Ref. 9, p. 343.

Ref. 6, p. 122.

R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill Book Co., New York, 1941), p. 164.

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