Abstract

Summation formulas are derived for the precise calculation of certain quality parameters of the image space: (a) radius of gyration of the point spread function, (b) O’Neill’s sharpness criterion, and (c) the Linfoot quality parameters. These formulas are applicable to any optical system that exists either as a design or as a fabricated object. Also, in the case of a bandwidth-limited object spectrum it is shown that the image may be exactly represented by a summation—in place of the usual convolution integral—over the object space. All derivations employ the sampling theorem.

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  1. R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).
  2. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
  3. R. Barakat, J. Opt. Soc. Am. 55, 538 (1965).
  4. B. Tatian, J. Opt. Soc. Am. 55, 1014 (1965).
  5. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1132 (1965).
  6. R. Barakat, J. Opt. Soc. Am. 55, 1217 (1965).
  7. B. R. Frieden, J. Opt. Soc. Am. 55, 1696 (1965).
  8. Ref. 1, Tables II, III, and VII.
  9. Ref. 3, pp. 538, 539.
  10. E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963).
  11. E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).
  12. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), 481–483.
  13. D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 137.
  14. P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1964), p. 136.
  15. J. B. DeVelis, J. Opt. Soc. Am. 55, 169 (1965).
  16. E. L. O'Neill, Selected Topics in Optics and Coimmunication Theory (Itek Corporation, Boston, Mass., 1957), p. 169.
  17. A standard text on this subject is by F. B. Hildebrand, Inltroduction to Numerical Analysis (McGraw-Hill Book Co., Inc., New York, 1956).
  18. A drawback to definitions (4.1) and (4.4) is that they are singular in the case of diffraction-limited optics. In Ref. 15, general conditions are found for which ρ and x converge.
  19. Reference 16, pp. 168, 169.
  20. D. Malacara, thesis, Institute of Optics, University of Rochester, 1965.
  21. Because the notation of this section closely parallels that of Ref. 10, the latter is used as the general reference for this section.
  22. Reference 10, p. 106.
  23. Reference 10, p. 107.
  24. Reference 1, p. 922.
  25. Reference 10, p. 108.
  26. Reference 10, p. 10.

Barakat, R.

R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1132 (1965).

R. Barakat, J. Opt. Soc. Am. 55, 538 (1965).

R. Barakat, J. Opt. Soc. Am. 55, 1217 (1965).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), 481–483.

DeVelis, J. B.

J. B. DeVelis, J. Opt. Soc. Am. 55, 169 (1965).

Frieden, B. R.

B. R. Frieden, J. Opt. Soc. Am. 55, 1696 (1965).

Gabor, D.

D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 137.

Harris, J. L.

J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).

Hildebrand, F. B.

A standard text on this subject is by F. B. Hildebrand, Inltroduction to Numerical Analysis (McGraw-Hill Book Co., Inc., New York, 1956).

Houston, A.

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1132 (1965).

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1964), p. 136.

Linfoot, E. H.

E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).

Malacara, D.

D. Malacara, thesis, Institute of Optics, University of Rochester, 1965.

O’Neill, E. L.

E. L. O'Neill, Selected Topics in Optics and Coimmunication Theory (Itek Corporation, Boston, Mass., 1957), p. 169.

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 III, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1964), p. 136.

Tatian, B.

B. Tatian, J. Opt. Soc. Am. 55, 1014 (1965).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), 481–483.

Other (26)

R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).

J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).

R. Barakat, J. Opt. Soc. Am. 55, 538 (1965).

B. Tatian, J. Opt. Soc. Am. 55, 1014 (1965).

R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 1132 (1965).

R. Barakat, J. Opt. Soc. Am. 55, 1217 (1965).

B. R. Frieden, J. Opt. Soc. Am. 55, 1696 (1965).

Ref. 1, Tables II, III, and VII.

Ref. 3, pp. 538, 539.

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

E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), 481–483.

D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 137.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publishing Company, Amsterdam, 1964), p. 136.

J. B. DeVelis, J. Opt. Soc. Am. 55, 169 (1965).

E. L. O'Neill, Selected Topics in Optics and Coimmunication Theory (Itek Corporation, Boston, Mass., 1957), p. 169.

A standard text on this subject is by F. B. Hildebrand, Inltroduction to Numerical Analysis (McGraw-Hill Book Co., Inc., New York, 1956).

A drawback to definitions (4.1) and (4.4) is that they are singular in the case of diffraction-limited optics. In Ref. 15, general conditions are found for which ρ and x converge.

Reference 16, pp. 168, 169.

D. Malacara, thesis, Institute of Optics, University of Rochester, 1965.

Because the notation of this section closely parallels that of Ref. 10, the latter is used as the general reference for this section.

Reference 10, p. 106.

Reference 10, p. 107.

Reference 1, p. 922.

Reference 10, p. 108.

Reference 10, p. 10.

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