Abstract

The relationship between the transverse and the on-axis behaviors of various pupil-plane filters is described. Expressions for general energy constraints associated with these filters are also derived. Transverse and axial diffraction properties of filters with quadratic radial transmittance are demonstrated experimentally.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Boivin, “On the theory of diffraction by concentric arrays of ring-shaped apertures,” J. Opt. Soc. Am. 42, 60–64 (1952).
    [CrossRef]
  2. B. J. Thompson, “Diffraction by semitransparent and phase annuli,” J. Opt. Soc. Am. 55, 145–149 (1965).
    [CrossRef]
  3. J. Ojeda-Castañeda, L. R. Berriel-Valdos, E. Montes, “Spatial filter for increasing the depth of focus,” Opt. Lett. 10, 520–522 (1985).
    [CrossRef] [PubMed]
  4. G. Toraldo di Francia, “Super gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
    [CrossRef]
  5. B. R. Frieden, “On arbitrary perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
    [CrossRef]
  6. R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
    [CrossRef]
  7. Z. S. Hegedus, V. Sarafis, “Superresolving filters in confocally scanned imaging systems,” J. Opt. Soc. Am. A 3, 1892–1896 (1986).
    [CrossRef]
  8. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  9. I. J. Cox, C. J. R. Sheppard, T. Wilson, “Reappraisal of arrays of concentric annuli as superresolving filters,” J. Opt. Soc. Am. 72, 1287–1291 (1982).
    [CrossRef]

1986 (1)

1985 (1)

1982 (1)

1980 (1)

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

1969 (1)

B. R. Frieden, “On arbitrary perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

1965 (1)

1964 (1)

1952 (2)

G. Toraldo di Francia, “Super gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
[CrossRef]

A. Boivin, “On the theory of diffraction by concentric arrays of ring-shaped apertures,” J. Opt. Soc. Am. 42, 60–64 (1952).
[CrossRef]

Berriel-Valdos, L. R.

Boivin, A.

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

A. Boivin, “On the theory of diffraction by concentric arrays of ring-shaped apertures,” J. Opt. Soc. Am. 42, 60–64 (1952).
[CrossRef]

Boivin, R.

R. Boivin, A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field,” Opt. Acta 27, 587–610 (1980).
[CrossRef]

Cox, I. J.

Frieden, B. R.

B. R. Frieden, “On arbitrary perfect imagery with a finite aperture,” Opt. Acta 16, 795–807 (1969).
[CrossRef]

Hegedus, Z. S.

McCutchen, C. W.

Montes, E.

Ojeda-Castañeda, J.

Sarafis, V.

Sheppard, C. J. R.

Thompson, B. J.

Toraldo di Francia, G.

G. Toraldo di Francia, “Super gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
[CrossRef]

Wilson, T.

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.


Figures (3)

Fig. 1
Fig. 1

Behavior of a three-ring filter, showing the dependence of the main parameters on the strengths of the two central rings k0 and k1. Plots represent (a) transversal GT and (b) axial GA resolution gains and (c) intensity of the focus F relative to the total power.

Fig. 2
Fig. 2

Pupils used in experiments to measure the three-dimensional diffracted intensity distributions: (a) quadratic transverse superresolving and (b) clear and (c) quadratic transverse apodizing.

Fig. 3
Fig. 3

Measured diffracted intensity distribution in five equally spaced planes along the optical axis. Each plane is perpendicular to the optical axis. The three rows of data correspond to the three pupils shown in Fig. 2.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

U ( υ , u ) = 2 0 1 P ( ρ ) J 0 ( υ ρ ) exp ( j u ρ 2 / 2 ) ρ d ρ .
υ = k r sin α ,
u = 4 k z sin 2 ( α / 2 ) ,
U ( υ , 0 ) = 2 0 1 P ( ρ ) J 0 ( υ ρ ) ρ d ρ ,
U ( 0 , u ) = 2 0 1 P ( ρ ) exp ( j u ρ 2 / 2 ) ρ d ρ .
U ( 0 , u ) = 0 1 Q ( t ) exp ( j u t / 2 ) d t .
U ( υ , 0 ) = 0 1 Q ( t ) ( 1 υ 2 4 t ) d t = I 0 ¼ υ 2 I 1 + ,
I n = 0 1 Q ( t ) t n d t
I ( υ , 0 ) = I 0 2 ½ υ 2 I 0 I 1 + .
U ( 0 , u ) = I 0 + ½ j u I 1 u 2 I 2 + ;
I ( 0 , u ) = I 0 2 u 2 4 ( I 0 I 2 I 1 2 ) + .
I ( υ , 0 ) = 1 ½ υ 2 t ¯ +
I ( 0 , u ) = 1 u 2 4 ( t 2 ¯ t 2 ¯ ) + ,
t ¯ = I 1 / I 0 ,
t 2 ¯ = I 2 / I 0 .
I 0 = 1 , I 1 = 1 / 2 , I 2 = 1 / 3 , t ¯ = 1 / 2 , t 2 ¯ = 1 / 3 .
G T = 2 t ¯ ,
G A = 12 ( t 2 ¯ t 2 ¯ ) .
S = I 0 2 / ( P max ) 2 .
E = 0 1 Q 2 ( t ) d t / ( P max ) 2 ,
F = S / E = I 0 2 / 0 1 Q 2 ( t ) d t .
P ( ρ ) = ρ 2 ,
I 0 = 1 / 2 , I 1 = 1 / 3 , I 2 = 1 / 4 , t ¯ = 2 / 3 , t 2 ¯ = 1 / 2 ,
G T = 4 / 3 , G A = 2 / 3 , S = 1 / 4 , E = 1 / 3 , F = 3 / 4 .
P ( ρ ) = 1 ρ 2 ,
I 0 = 1 / 2 , I 1 = 1 / 6 , I 2 = 1 / 12 , t ¯ = 1 / 3 , t 2 ¯ = 1 / 6 ,
G T = 2 / 3 , G A = 2 / 3 , S = 1 / 4 , E = 1 / 3 , F = 3 / 4 .
P ( ρ ) = ( ρ 2 ) 1 / n ,
G T = 2 ( 1 + n 1 + 2 n ) , G A = 12 n 2 ( 1 + n ) ( 1 + 2 n ) 2 ( 1 + 3 n ) , S = n 2 ( 1 + n ) 2 , E = n ( 2 + n ) , F = n ( 2 + n ) ( 1 + n ) 2 .
P ( ρ ) = ( 1 ρ 2 ) 1 / n ,
G T = 2 n 1 + 2 n , G A = 12 n 2 ( 1 + n ) ( 1 + 2 n ) 2 ( 1 + 3 n ) , S = n 2 ( 1 + n ) 2 , E = n 2 + n , F = n ( 2 + n ) ( 1 + n ) 2 .
P ( ρ ) = f ( ξ ) ,
P ( ρ ) = ( 2 ρ 2 1 ) 2 / n ,
G T = 1 , G A = 3 ( n + 2 ) 2 + 3 n , S = n 2 ( 2 + n ) 2 , E = n 4 + n , F = n ( 4 + n ) ( 2 + n ) 2 .
P ( ρ ) = [ 1 ( 2 ρ 2 1 ) 2 ] 1 / n = [ 4 ρ 2 ( 1 ρ 2 ) ] 1 / n .
G T = 1 , G A = 3 / 5 , S = 4 / 9 , E = 8 / 15 , F = 5 / 6 ,
G T = 2 δ , G A = δ 2 , S = δ 2 , E = δ , F = δ .
Q ( t ) = k , t < 1 / 2 = 1 , t > 1 / 2 ,
G T = 3 + k 2 ( 1 + k ) , G A = 1 + 14 k + k 2 4 ( 1 + k ) 2 , S = ( 1 + k ) 2 4 , | k | < 1 , = ( 1 + k ) 2 4 k 2 , | k | > 1 , E = 1 + k 2 2 k 2 , | k | < 1 , = 1 + k 2 2 k 2 , | k | > 1 , F = ( 1 + k ) 2 2 ( 1 + k 2 ) .
G T = 2 + k 1 1 + k 1 + k 0 , G A = 3 ( k 1 + 4 k 0 + k 1 k 0 ) ( 1 + k 1 + k 0 ) 2 , S = ( 1 + k 1 + k 0 ) 2 δ 2 L ( 1 , k 1 2 , k 0 2 ) , E = ( 1 + k 1 2 + k 0 2 ) δ L ( 1 , k 1 2 , k 0 2 ) , F = ( 1 + k 1 + k 0 ) 2 δ ( 1 + k 1 2 + k 0 2 ) ,
G A = 3 ( G T 1 ) ( 2 1 k 0 G T ) ,

Metrics