Abstract

The main features of eigenfunctions and eigenvalues of integral equations connected with incoherent imaging through clear pupils are studied. As in the coherent imaging, both the object and the image can be expanded in eigenfunction series. On the contrary, the eigenvalue-step-function behavior typical of coherent imaging is not preserved when passing to the incoherent case. Upper and lower bounds for the eigenvalues are established in the one-dimensional case. They show that the eigenvalues, roughly speaking, decrease almost linearly with respect to the order index.

PDF Article

References

  • View by:
  • |
  • |

  1. B. R. Frieden, Opt. Acta 16, 795 (1969).
  2. A. Walther, J. Opt. Soc. Am. 57, 639 (1967).
  3. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
  4. F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).
  5. C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).
  6. C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
  7. D. Slepian, J. Opt. Soc. Am. 55, 1110 (1965).
  8. Y. Itoh, J. Opt. Soc. Am. 60, 10 (1970).
  9. M. L. Mehta and C. L. Mehta, J. Opt. Soc. Am. 63, 826 (1973).
  10. D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
  11. H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).
  12. H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).
  13. D. Slepian, Bell System Tech. J. 43, 3009 (1964).
  14. B. R. Frieden, in Progress in Optics, Vol. IX, edited by E. Wolf (North-Holland, Amsterdam, 1971). This paper contains a review of the properties of the prolate spheroidal wave functions as well as a review of optical applications up to 1971.
  15. H. J. Landau, Acta Mathematica 117, 37 (1967).
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Ch. 6.
  17. F. Riesz and B. Sz. Nagy, Functional Analysis (Ungar, New York, 1955), Ch. 6.
  18. This was already noted in a different connection byA. Walther, J. Opt. Soc. Am. 60, 141 (1970).
  19. For a rigorous treatment, see Ref. 15.
  20. J. W. Goodman, in Progress in Optics, Vol. VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970).
  21. F. Gori and G. Guattari, J. Opt. Soc. Am. 64, 453 (1974).
  22. See first theorem on p. 238 of Ref. 17.
  23. H. J. Landau, Trans. Am. Math. Soc. 115, 242 (1965). Theorem 1 and the considerations that follow the proof of theorem 2.
  24. See, for example, Ref. 14.
  25. See Ref. 21, theorem 2 and the considerations that follow it.
  26. D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).
  27. F. Gori and G. Guattari, J. Opt. Soc. Am. 61, 36 (1971).
  28. B. J. Thompson, in Progress in Optics, Vol. VII, edited by E. Wolf (North-Holland, Amsterdam, 1969).
  29. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N. J., 1964), Ch. 8.
  30. We refer to the analysis of Ref. 6 as being pertinent to the coherent case, because the main results of this analysis are based on the hypothesis that the pupil function equals rect(ν).
  31. H. Kogelnik, in Lasers, Vol. I, edited by A. K. Levine (Dekker, New York, 1966).

Barnes, C. W.

C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N. J., 1964), Ch. 8.

di Francia, G. Toraldo

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).

Frieden, B. R.

B. R. Frieden, in Progress in Optics, Vol. IX, edited by E. Wolf (North-Holland, Amsterdam, 1971). This paper contains a review of the properties of the prolate spheroidal wave functions as well as a review of optical applications up to 1971.

B. R. Frieden, Opt. Acta 16, 795 (1969).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Ch. 6.

J. W. Goodman, in Progress in Optics, Vol. VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970).

Gori, F.

F. Gori and G. Guattari, J. Opt. Soc. Am. 64, 453 (1974).

F. Gori and G. Guattari, J. Opt. Soc. Am. 61, 36 (1971).

F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).

Guattari, G.

F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).

F. Gori and G. Guattari, J. Opt. Soc. Am. 61, 36 (1971).

F. Gori and G. Guattari, J. Opt. Soc. Am. 64, 453 (1974).

Harris, R. W.

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

Itoh, Y.

Y. Itoh, J. Opt. Soc. Am. 60, 10 (1970).

Kogelnik, H.

H. Kogelnik, in Lasers, Vol. I, edited by A. K. Levine (Dekker, New York, 1966).

Landau, H. J.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).

H. J. Landau, Trans. Am. Math. Soc. 115, 242 (1965). Theorem 1 and the considerations that follow the proof of theorem 2.

H. J. Landau, Acta Mathematica 117, 37 (1967).

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).

Mehta, C. L.

M. L. Mehta and C. L. Mehta, J. Opt. Soc. Am. 63, 826 (1973).

Mehta, M. L.

M. L. Mehta and C. L. Mehta, J. Opt. Soc. Am. 63, 826 (1973).

Nagy, B. Sz.

F. Riesz and B. Sz. Nagy, Functional Analysis (Ungar, New York, 1955), Ch. 6.

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N. J., 1964), Ch. 8.

Pollak, H. O.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).

Riesz, F.

F. Riesz and B. Sz. Nagy, Functional Analysis (Ungar, New York, 1955), Ch. 6.

Rushforth, C. K.

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

Slepian, D.

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).

D. Slepian, J. Opt. Soc. Am. 55, 1110 (1965).

Sonnenblick, E.

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).

Thompson, B. J.

B. J. Thompson, in Progress in Optics, Vol. VII, edited by E. Wolf (North-Holland, Amsterdam, 1969).

Walther, A.

This was already noted in a different connection byA. Walther, J. Opt. Soc. Am. 60, 141 (1970).

A. Walther, J. Opt. Soc. Am. 57, 639 (1967).

Other

B. R. Frieden, Opt. Acta 16, 795 (1969).

A. Walther, J. Opt. Soc. Am. 57, 639 (1967).

G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).

F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).

C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

D. Slepian, J. Opt. Soc. Am. 55, 1110 (1965).

Y. Itoh, J. Opt. Soc. Am. 60, 10 (1970).

M. L. Mehta and C. L. Mehta, J. Opt. Soc. Am. 63, 826 (1973).

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).

B. R. Frieden, in Progress in Optics, Vol. IX, edited by E. Wolf (North-Holland, Amsterdam, 1971). This paper contains a review of the properties of the prolate spheroidal wave functions as well as a review of optical applications up to 1971.

H. J. Landau, Acta Mathematica 117, 37 (1967).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Ch. 6.

F. Riesz and B. Sz. Nagy, Functional Analysis (Ungar, New York, 1955), Ch. 6.

This was already noted in a different connection byA. Walther, J. Opt. Soc. Am. 60, 141 (1970).

For a rigorous treatment, see Ref. 15.

J. W. Goodman, in Progress in Optics, Vol. VIII, edited by E. Wolf (North-Holland, Amsterdam, 1970).

F. Gori and G. Guattari, J. Opt. Soc. Am. 64, 453 (1974).

See first theorem on p. 238 of Ref. 17.

H. J. Landau, Trans. Am. Math. Soc. 115, 242 (1965). Theorem 1 and the considerations that follow the proof of theorem 2.

See, for example, Ref. 14.

See Ref. 21, theorem 2 and the considerations that follow it.

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).

F. Gori and G. Guattari, J. Opt. Soc. Am. 61, 36 (1971).

B. J. Thompson, in Progress in Optics, Vol. VII, edited by E. Wolf (North-Holland, Amsterdam, 1969).

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N. J., 1964), Ch. 8.

We refer to the analysis of Ref. 6 as being pertinent to the coherent case, because the main results of this analysis are based on the hypothesis that the pupil function equals rect(ν).

H. Kogelnik, in Lasers, Vol. I, edited by A. K. Levine (Dekker, New York, 1966).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.