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.

© 1974 Optical Society of America

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References

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  1. B. R. Frieden, Opt. Acta 16, 795 (1969).
    [CrossRef]
  2. A. Walther, J. Opt. Soc. Am. 57, 639 (1967).
    [CrossRef]
  3. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [CrossRef] [PubMed]
  4. F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).
    [CrossRef]
  5. C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).
    [CrossRef]
  6. C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
    [CrossRef]
  7. D. Slepian, J. Opt. Soc. Am. 55, 1110 (1965).
    [CrossRef]
  8. Y. Itoh, J. Opt. Soc. Am. 60, 10 (1970).
    [CrossRef]
  9. M. L. Mehta and C. L. Mehta, J. Opt. Soc. Am. 63, 826 (1973).
    [CrossRef]
  10. D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
    [CrossRef]
  11. H. J. Landau and H. O. Pollak, Bell System Tech. J. 40, 65 (1961).
    [CrossRef]
  12. H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).
    [CrossRef]
  13. D. Slepian, Bell System Tech. J. 43, 3009 (1964).
    [CrossRef]
  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).
    [CrossRef]
  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 by A. Walther, J. Opt. Soc. Am. 60, 141 (1970).
    [CrossRef]
  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).
    [CrossRef]
  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.
    [CrossRef]
  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).
    [CrossRef]
  27. F. Gori and G. Guattari, J. Opt. Soc. Am. 61, 36 (1971).
    [CrossRef]
  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).

1974 (1)

1973 (2)

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

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

1971 (1)

1970 (2)

1969 (2)

1968 (1)

1967 (2)

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

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

1966 (1)

1965 (3)

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

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

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

1964 (1)

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

1962 (1)

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

1961 (2)

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

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

Barnes, C. W.

Beran, M. J.

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

Frieden, B. R.

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

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.

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.

Guattari, G.

Harris, R. W.

Itoh, Y.

Kogelnik, H.

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

Landau, H. J.

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

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

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

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

Mehta, C. L.

Mehta, M. L.

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).
[CrossRef]

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

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

Riesz, F.

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

Rushforth, C. K.

Slepian, D.

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

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

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

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

Sonnenblick, E.

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

Thompson, B. J.

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

Toraldo di Francia, G.

Walther, A.

Acta Mathematica (1)

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

Bell System Tech. J. (5)

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

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

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

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

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

J. Opt. Soc. Am. (10)

Opt. Acta (1)

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

Opt. Commun. (1)

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

Trans. Am. Math. Soc. (1)

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

Other (12)

See, for example, Ref. 14.

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

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).

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.

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

For a rigorous treatment, see Ref. 15.

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

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.

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Figures (2)

Fig. 1
Fig. 1

Numerical determination of the minimum upper bound for an eigenvalue of fixed order (n = 11) and space–bandwidth product (a = 10).

Fig. 2
Fig. 2

Upper (○) and lower (●) bounds for the eigenvalues, versus the order index, for fixed space–bandwidth product (a = 6). Full lines show upper and lower bounds as derived from the general properties.

Tables (2)

Tables Icon

Table I Upper limit P2/P of the ratio between the sum of the squared eigenvalues and the sum of the eigenvalues, for some typical pupils.

Tables Icon

Table II Lower and upper bounds of the eigenvalue μ1(a) for some value of the space–bandwidth product.

Equations (44)

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α A ( x ) = { 1 , x A 0 , x A
u A ( x ) = 1 A α A ( y ) α A ( y + x ) d y .
f ˜ ( ν ) = f ( x ) exp ( - 2 π i ν x ) d x .
sinc ( x ) = sin ( π x ) π x , rect ( x ) = { 1 , x 1 2 0 , x > 1 2 Λ ( x ) = { 1 - x , x 1 0 , x > 1.
f 2 = f ( x ) 2 d x .
I ( x ) = A O ( y ) S ( x - y ) d y .
S ( x ) = p ( ν ) exp ( 2 π i ν x ) d ν .
A Φ ( y ) S ( x - y ) d y = μ Φ ( x ) ,
H = A A S ( x - y ) f ( x ) f * ( y ) d x d y ,
H = p ( ν ) f ˜ ( ν ) 2 d ν .
n = 0 μ n = A S ( x - x ) d x = A S ( O ) = A P 1 ,
P 1 = p ( ν ) d ν .
P 1 = u P ( ν ) d ν = 1 P α P ( ω ) α P ( ω + ν ) d ω d ν = P .
n = 0 μ n 2 = A A S ( x - y ) 2 d x d y = α A ( x ) α A ( y ) S ( x - y ) 2 d x d y .
n = 0 μ n 2 = A S ( t ) 2 u A ( t ) d t .
n = 0 μ n 2 A S ( t ) 2 d t ,
n = 0 μ n 2 A P 2 ,
P 2 = p 2 ( ν ) d ν .
r = n = 0 μ n 2 / n = 0 μ n ,
r P 2 / P = p 2 ( ν ) d ν / p ( ν ) d ν .
A Φ ( x ) 2 d x .
A Φ ( x ) 2 d x = μ - 1 A A Φ * ( x ) Φ ( y ) S ( x - y ) d x d y = μ - 1 Φ ˜ T ( ν ) 2 p ( ν ) d ν ,
Φ ( x ) 2 d x .
Φ ( x ) 2 d x = Φ ˜ ( ν ) 2 d ν = μ - 2 Φ ˜ T ( ν ) 2 p 2 ( ν ) d ν .
A Φ ( x ) 2 d x / Φ ( x ) 2 d x = μ ( Φ ˜ T ( ν ) 2 p ( ν ) d ν / Φ ˜ T ( ν ) 2 p 2 ( ν ) d ν ) .
p ( ν ) = Λ ( ν ) .
- a / 2 a / 2 Φ n ( a ) ( y ) sinc 2 ( x - y ) d y = μ n ( a ) Φ n ( a ) ( x )             ( n = 0 , 1 , ) ,
- c / 2 c / 2 v n ( c ) ( y ) sinc ( x - y ) d y = λ n ( c ) v n ( c ) ( x )             ( n = 0 , 1 , ) ,
μ n < 1 - n - 1 4 a             ( 1 n [ 2 a ] + 1 ) .
μ n sup 1 f 2 - 1 1 Λ ( ν ) f ˜ ( ν ) 2 d ν ,
- 1 1 Λ ( ν ) f ˜ ( ν ) 2 d ν < - τ τ f ˜ ( ν ) 2 d ν + ( 1 - τ ) { - 1 - τ f ˜ ( ν ) 2 d ν + τ 1 f ˜ ( ν ) 2 d ν } .
- τ τ f ˜ ( ν ) 2 d ν .
μ n < sup { + ( 1 - ) ( 1 - τ ) } .
τ = n - 1 2 a             ( 1 n [ 2 a ] + 1 ) .
μ n < sup 1 f 2 - 1 1 f ˜ ( ν ) 2 d ν .
μ n < λ n ( c ) .
μ n > 0.5 - n + 1 4 a             ( 0 n [ 2 a ] - 1 ) .
μ n > inf 1 f 2 - 1 1 Λ ( ν ) f ˜ ( ν ) 2 d ν ,
- 1 1 Λ ( ν ) f ˜ ( ν ) 2 d ν > ( 1 - τ ) - τ τ f ˜ ( ν ) 2 d ν = ( 1 - τ ) E .
τ = n + 1 2 a .
n = 2 τ a + δ ,
μ n < λ n ( 2 τ a ) + ( 1 - τ ) ( 1 - λ n ( 2 τ a ) ) .
μ n > ( 1 - τ ) λ n ( 2 τ a ) .
e 2 = i = 0 N n i 2 / μ i 2 ,