Abstract

The eigenfunctions of the integral equation describing the image-formation process are used for studying the degrees of freedom of coherent images from point-like-element pupils. The possibility of determining the number of effective degrees of freedom without solving the integral equation is demonstrated for an important class of cases. Finally, the number of effective degrees of freedom is explicitly calculated for cases of practical interest.

© 1974 Optical Society of America

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References

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  1. F. Gori and G. Guattari, J. Opt. Soc. Am. 61, 36 (1971).
    [Crossref]
  2. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [Crossref] [PubMed]
  3. J. T. Winthrop, J. Opt. Soc. Am. 61, 15 (1971).
    [Crossref]
  4. C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
    [Crossref]
  5. J. W. Goodman, in Progress in Optics, VIII, edited by E. Wolf (North–Holland, Amsterdam, 1970).
  6. B. R. Frieden, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971).
  7. A. Walther, J. Opt. Soc. Am. 57, 639 (1967).
    [Crossref]
  8. D. Slepian and H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).
    [Crossref]
  9. H. J. Landau and H. O. Pollak, Bell Syst. Tech. J. 40, 65 (1961).
    [Crossref]
  10. H. J. Landau and H. O. Pollak, Bell Syst. Tech. J. 41, 1295 (1962).
    [Crossref]
  11. D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
    [Crossref]
  12. D. Slepian and E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965).
    [Crossref]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), Ch. 6.
  14. F. Riesz and B. Sz.- Nagy, Functional Analysis (Ungar, New York, 1955), Ch. 6.
  15. The function R(x) can be considered as a solution of Eq. (3) corresponding to the eigenvalue λ = 0. In what follows, however, we will mean by eigenvalues only the λ’s that differ from zero.
  16. F. Gori and G. Guattari, Opt. Commun. 7, 163 (1973).
    [Crossref]

1973 (1)

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

1971 (2)

1969 (1)

1968 (1)

1967 (1)

1965 (1)

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

1964 (1)

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
[Crossref]

1962 (1)

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

1961 (2)

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

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

Frieden, B. R.

B. R. Frieden, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971).

Goodman, J. W.

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

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

Gori, F.

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

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

Guattari, G.

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

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

Harris, R. W.

Landau, H. J.

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

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

Nagy, B. Sz.-

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

Pollak, H. O.

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

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

D. Slepian and H. O. Pollak, Bell Syst. 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 and E. Sonnenblick, Bell Syst. Tech. J. 44, 1745 (1965).
[Crossref]

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
[Crossref]

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

Sonnenblick, E.

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

Toraldo di Francia, G.

Walther, A.

Winthrop, J. T.

Bell Syst. Tech. J. (5)

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

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

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

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
[Crossref]

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

J. Opt. Soc. Am. (5)

Opt. Commun. (1)

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

Other (5)

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

B. R. Frieden, in Progress in Optics, IX, edited by E. Wolf (North–Holland, Amsterdam, 1971).

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.

The function R(x) can be considered as a solution of Eq. (3) corresponding to the eigenvalue λ = 0. In what follows, however, we will mean by eigenvalues only the λ’s that differ from zero.

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Figures (6)

Fig. 1
Fig. 1

Hypothetical behavior of the normalized eigenvalues of a point-like-element pupil (full line) and equivalent rectangular behavior (broken line).

Fig. 2
Fig. 2

Normalized eigenvalues of a 12 equally spaced-element pupil vs the ratio between the object length and the impulse-response period.

Fig. 3
Fig. 3

Normalized eigenvalues of a 50 equally spaced-element pupil vs their order index. The ratio between the object length and the impulse-response period is 1/π.

Fig. 4
Fig. 4

Normalized eigenvalues vs their order index for (□) a 50 equally spaced-element pupil and for (●) a 50-element pupil, in which the distance between two adjacent elements decreases linearly from one end of the pupil to the other.

Fig. 5
Fig. 5

Eigenfunctions, normalized to unit energy, for a 7 equally spaced-element pupil. The corresponding eigenvalues are λ1 = 1; λ2 = 0.85; λ3 = 0.35; λ4 = 0.42 × 10−1; λ5 = 0.17 × 10−2 λ6 = 0.28 × 10−4; λ7 = 0.16 × 10−6.

Fig. 6
Fig. 6

Upper and lower limits of the ratio between the numbers of effective and structural degrees of freedom for a 12 equally spaced-element pupil vs the ratio between the object length and the impulse-response period.

Equations (27)

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I ( x ) = - L / 2 L / 2 O ( y ) S ( x - y ) d y ;
S ( x ) = n = 1 N exp ( 2 π i ν n x ) ,
- L / 2 L / 2 Φ n ( y ) S ( x - y ) d y = λ n Φ n ( x ) .
O ( x ) = n = 1 N b n Φ n ( x ) + R ( x ) ,
b n = - L / 2 L / 2 O ( x ) Φ n * ( x ) d x .
I ( x ) = n = 1 N λ n b n Φ n ( x ) .
N e = 2 ( n = 1 N λ n ) / λ max ,
m = 1 N φ m ( n ) α l m = λ n φ l ( n )             ( l = 1 , , N ) ,
φ m ( n ) = - L / 2 L / 2 Φ n ( y ) exp ( - 2 π i ν m y ) d y
α l m = - L / 2 L / 2 exp [ - 2 π i ( ν l = ν m ) x ] d x .
n = 1 N λ n = N L .
R = ( - L / 2 L / 2 f ( x ) 2 d x ) / n = 1 N a n 2 ,
f ( x ) = n = 1 N a n exp ( 2 π i ν n x )
R = P - L / 2 L / 2 f ( x ) 2 d x / - P / 2 P / 2 f ( x ) 2 d x .
N e = 2 n = 1 N λ n / λ max = 2 N L λ max L P N s .
λ max P / r ,
r = - P / 2 P / 2 f ( x ) 2 d x / - L / 2 L / 2 f ( x ) 2 d x .
N e = 2 n = 1 N λ n / λ max = 2 N L λ max N s L P r .
max ( 2 , N s L P ) N e N s L P r ,
N e max [ 2 , N s ( L / P ) ] .
α l m = L sin [ π ( ν l - ν m ) L ] π ( ν l - ν m ) L ,
( ν l - ν m ) L = h ,
ν k + 1 - ν k = ν 0 - k ( ν 0 / 100 )             ( k = 1 , , 49 ) .
max ( 1 / N , ν ¯ / ν 0 ) N e / N s ν ¯ r / ν 0 .
f ( x ) = sin ( N π ν ¯ x ) sin ( π ν ¯ x ) .
max ( 1 / N , ν ¯ / ν 0 ) N e / N s ( ν ¯ / ν 0 ) ( 1 - 1 / N ) + 1 / N .
N e / N s ν ¯ / ν 0 .