Abstract

Some superresolving pupils, featuring a central maximum narrower than Airy’s core and a first secondary maximum of a few percent, have been computed, following Frieden’s formula and his criterion of maximum superresolution for circular symmetry. A superresolution factor of about 28% may be achieved without much sacrifice of illumination. Other pupils have been investigated where the SF reaches 35%. The behavior of the sidelobes in the point impulse image is examined. The corresponding encircled energy and point impulse response are given.

© 1976 Optical Society of America

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References

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  1. P. Jacquinot, B. Roizen-Dossier, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1964), Vol. 3.
    [CrossRef]
  2. D. Slepian, J. Opt. Soc. Am. 55, 1110 (1965).
    [CrossRef]
  3. M. Mino, Y. Okano, Appl. Opt. 10, 2219 (1971).
    [CrossRef] [PubMed]
  4. B. Roy Frieden, Opt. Acta 16, 795 (1969).
    [CrossRef]
  5. B. Roy Frieden, Appl. Opt. 9, 2489 (1970).
    [CrossRef]
  6. B. Roy Frieden, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1971), Vol. 9.
  7. D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
  8. D. Slepian, H. O. Pollak, Bell Syst. Tech. J. 40, 43 (1961).
  9. H. J. Landau, H. O. Pollak, Bell Syst. Tech. J. 40, 65 (1961).
  10. J. C. Heurtley, W. Streifer, J. Opt. Soc. Am. 55, 1472 (1965).
    [CrossRef]
  11. G. Toraldo di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [CrossRef] [PubMed]

1971 (1)

1970 (1)

1969 (2)

1965 (2)

1964 (1)

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

1961 (2)

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

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

Heurtley, J. C.

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1964), Vol. 3.
[CrossRef]

Landau, H. J.

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

Mino, M.

Okano, Y.

Pollak, H. O.

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

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

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1964), Vol. 3.
[CrossRef]

Roy Frieden, B.

B. Roy Frieden, Appl. Opt. 9, 2489 (1970).
[CrossRef]

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

B. Roy Frieden, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1971), Vol. 9.

Slepian, D.

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

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

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

Streifer, W.

Toraldo di Francia, G.

Appl. Opt. (2)

Bell Syst. Tech. J. (3)

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

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

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

J. Opt. Soc. Am. (3)

Opt. Acta (1)

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

Other (2)

P. Jacquinot, B. Roizen-Dossier, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1964), Vol. 3.
[CrossRef]

B. Roy Frieden, Progress in Optics, E. Wolf, Ed. (North Holland, Amsterdam, 1971), Vol. 9.

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Figures (8)

Fig. 1
Fig. 1

Encircled energy ratio vs m, the number of terms defining the pupil filter.

Fig. 2
Fig. 2

Behavior of the superresolving factor vs m, the number of terms defining the pupil filter.

Fig. 3
Fig. 3

Log intensity of the impulse response of superresolving pupil filters (c = 5). The dotted curve is for the Airy disk, m = 1, 2, and 3.

Fig. 4
Fig. 4

Log intensity of the impulse response of superresolving pupil filters (c = 10). The dotted curve is for the Airy disk; m = 1, 2, 3, 4, and 5.

Fig. 5
Fig. 5

Log intensity of the impulse response of superresolving pupil filters (c = 20). The dotted curve is for the Airy disk; m = 1, 3, 8, and 9.

Fig. 6
Fig. 6

Unnormalized transmittance of superresolving pupil filters for c = 5. The SF is 37%, 53%, 35%, 32% for A, B, C, D, respectively. The first side lobe is 4.5%, 31%, 3.1%, 4.0%, respectively, for W3 = 0.15, 0.5, 0.1, 0.05. n is the number of terms defining the filter.

Fig. 7
Fig. 7

Normalized transmittance of the superresolving pupils filters with c = 10. n is the number of terms defining the filter.

Fig. 8
Fig. 8

Unnormalized transmittance of superresolving pupil filters for c = 20. n is the number of terms defining the filter.

Tables (1)

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Table I Values of T(%)

Equations (29)

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0 r 0 J N ( ω r ) Φ N , n ( r ) rdr = ( - 1 ) n ( r 0 / Ω ) λ N , m 1 / 2 Φ N , n ( ω r 0 / Ω ) ,
0 r 0 Φ N , n ( r ) Φ N , m ( r ) rdr = λ N , n δ m , n
n = 0 λ N , n - 1 Φ N , n ( r ) Φ N , n ( r ) = δ ( r - r ) / r for 0 r ; r r 0 ,
0 J N ( ω r ) Φ N , n ( r ) rdr = { ( - 1 ) n ( r 0 / Ω ) λ N , n - 1 / 2 Φ N , n ( ω r 0 / Ω ) for 0 ω Ω , for ω Ω .
0 Φ N , m ( r ) Φ N , n ( r ) rdr = δ m , n .
f ( x ) = n = 0 a N , n Φ N , n ( x ) .
a N , n = 1 λ N , n 0 x 0 f ( x ) Φ N , n ( x ) xdx ,
0 Ω J 0 ( ω r ) V ( ω ) ω d ω = δ ( r ) / r , for 0 r .
V ( ω ) = n = 0 a 0 , n Φ 0 , n ( ω r 0 Ω ) .
a 0 , n = ( - 1 ) n r 0 Ω λ 0 , n - 3 / 2 Φ 0 , n ( 0 ) .
Φ N , n ( c , r ) = [ λ N , n / r 0 0 1 Φ N , n ( c , t ) 2 d t ] 1 / 2 ψ N , n ( c , r / r 0 ) r 1 / 2 .
Φ 0 , n ( 0 ) = λ 0 , n + 1 / 2 ( r 0 L 0 , n ) - 1 .
ψ N , m ( c , x ) / x 1 , x 0
L 0 , n 2 = 0 1 ψ 0 , n ( c , t ) 2 d t ;
a 0 , n = ( - 1 ) n ( Ω L 0 , n λ 0 , n ) - 1 .
V M ( ω ) = n = 0 M a 0 , n Φ 0 , n ( ω r 0 Ω ) ,
0 Ω J 0 ( ω r ) V M ( ω ) ω d ω = g M ( r ) = 1 r 0 n = 0 M L 0 , n - 1 λ 0 , n - 1 / 2 Φ 0 , n ( r ) ;
g M ( r ) = 1 r 0 2 n = 0 M L 0 , n - 2 ψ 0 , n ( r / r 0 ) ( r / r 0 ) 1 / 2 .
g ( r ) = k 0 Ω J 0 ( ω r ) V ( ω ) ω d ω ,
E = n = 0 M a 0 , n 2 λ 0 , n / n = 0 M a 0 , n 2 ;
V M ( ω ) = V ( ω ) / V MAX ,
T = 2 Ω 2 0 Ω V M ( ω ) V M * ( ω ) ω d ω .
T = 2 V 2 MAX ( Ω r 0 ) 2 n = 0 M a 0 , n 2 λ 0 , n .
T = 2 V 2 MAX n = 0 M λ 0 , n - 1 L 0 , n - 2 ,
V MAX = V ( α ) = 1 c n = 0 M ( - 1 ) n λ 0 , n - 1 / 2 L 0 , n - 2 ψ 0 , n ( α Ω ) ( α Ω ) 1 / 2 ,
T = 2 n = 0 M λ 0 , n - 1 L 0 , n - 2 [ n = 0 M ( - 1 ) n λ 0 , n - 1 / 2 L 0 , n - 2 c - 1 ψ o , n ψ 0 , n ( α / Ω ) / ( α / Ω ) 1 / 2 ] 2 .
V M = n = 0 M a 0 , n W n Φ 0 , n ( ω r 0 / Ω ) .
g M ( r ) = Ω r 0 n = 0 M ( - 1 ) n a 0 , n W n λ 0 , n 1 / 2 ϕ 0 , n ( r ) ,
g M ( r ) = 1 r 0 2 n = 0 M W n L 0 , n - 2 ψ o , n ( r / r 0 ) / ( r / r 0 ) 1 / 2 ,

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