Abstract

The concept of invariance of information capacity is discussed and applied to the resolution of an optical system. Methods of obtaining superresolution in microscopy are discussed, and scanning microscopy has many distinct advantages for such applications.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. B. Fellgett, E. H. Linfoot, Proc. R. Soc. London Ser. A 247, 369–407 (1955).
  2. W. Lukosz, J. Opt. Soc. Am. 56, 1463–1472 (1966).
    [CrossRef]
  3. W. Lukosz, J. Opt. Soc. Am. 57, 932–941 (1967).
    [CrossRef]
  4. G. Toraldo di Francia, Nuovo Cimento Suppl. 3 IX, 426–438 (1952).
    [CrossRef]
  5. A. W. Lohmann, D. P. Paris, Appl. Opt. 3, 1037–1043 (1964).
    [CrossRef]
  6. A. I. Kartashev, Opt. Spectrosc. (USSR) 9, 204–206 (1960).
  7. J. D. Armatage, A. W. Lohmann, D. P. Paris, Jpn. J. Appl. Phys. Suppl. 1, 273–275 (1965).
  8. M. A. Grimm, A. W. Lohmann, J. Opt. Soc. Am. 56, 1151–1156 (1966).
    [CrossRef]
  9. A. Bachl, W. Lukosz, J. Opt. Soc. Am. 57, 163–169 (1967).
    [CrossRef]
  10. C. E. Shannon, Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
    [CrossRef]
  11. C. E. Shannon, Proc. IRE 37, 10–21 (1948).
    [CrossRef]
  12. G. W. King, A. G. Emslie, J. Opt. Soc. Am. 41, 405–412 (1951).
    [CrossRef] [PubMed]
  13. C. W. King, A. G. Emslie, J. Opt. Soc. Am. 43, 664–668 (1953).
    [CrossRef]
  14. A. Blanc-Lapierre, Ann. Inst. Henri Poincaré 13, 245–296 (1953).
  15. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497–501 (1955).
    [CrossRef]
  16. G. Toraldo di Francia, Opt. Acta 2, 5–8 (1955).
    [CrossRef]
  17. M. Bendinelli, A. Consortini, L. Ronchi, B. R. Frieden, J. Opt. Soc. Am. 64, 1498–1502 (1974).
    [CrossRef]
  18. N. J. Bershad, J. Opt. Soc. Am. 59, 157–163 (1969).
    [CrossRef]
  19. D. L. Fried, J. Opt. Soc. Am. 69, 399–406 (1979).
    [CrossRef]
  20. R. V. L. Hartley, Bell Syst. Tech. J. 3, 535–563 (1928).
    [CrossRef]
  21. J. L. Harris, J. Opt. Soc. Am. 54, 606, 931–936 (1964).
    [CrossRef]
  22. A. Rose, in Advances in Electronics, L. Marton, ed. (Academic, New York, 1948), Vol. 1, pp. 131–166.
  23. C. K. Rushworth, R. W. Harris, J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  24. S. J. Howard, J. Opt. Soc. Am. 71, 95–98 (1981).
    [CrossRef]
  25. M. Bertero, E. R. Pike, Opt. Acta 29, 6, 727–746 (1982).
    [CrossRef]
  26. B. Saleh, J. Opt. Soc. Am. 67, 71–76 (1977).
    [CrossRef]
  27. W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushford, J. Opt. Soc. Am. A 1, 241–250 (1984).
    [CrossRef]
  28. C. J. R. Sheppard, A. Choudhury, Opt. Acta 24, 1051–1073 (1977).
    [CrossRef]
  29. M. D. Egger, M. Petrán, Science 157, 305–307 (1967).
    [CrossRef] [PubMed]
  30. M. Petrán, M. Hadravský, M. D. Egger, R. Galambos, J. Opt. Soc. Am. 58, 661–664 (1968).
    [CrossRef]
  31. E. A. Ash, G. Nicholls, Nature 237, 510–512 (1972).
    [CrossRef] [PubMed]
  32. G. A. Massey, J. A. Davis, S. M. Katanik, E. Omon, J. Opt. Soc. Am. A 1, 1259 (1984).
  33. G. A. Massey, Appl. Opt. 23, 658–660 (1984).
    [CrossRef]
  34. A. Harootunian, K. Betzig, A. Muray, A. Lewis, M. Isaacson, J. Opt. Soc. Am. A 1, 1293 (1984).
  35. C. W. McCutchen, J. Opt. Soc. Am. 57, 1190–1192 (1967).
    [CrossRef] [PubMed]
  36. I. J. Cox, C. J. R. Sheppard, T. Wilson, Optik 60, 391–396 (1982).
  37. I. J. Cox, J. Microsc. 133, 149–154 (1984).
    [CrossRef] [PubMed]
  38. R. D. Allen, N. S. Allen, J. Microsc. 129, 3–17 (1983).
    [CrossRef] [PubMed]
  39. D. K. Hamilton, C. J. R. Sheppard, Opt. Acta 29, 1573–1577 (1982).
    [CrossRef]

1984 (5)

W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushford, J. Opt. Soc. Am. A 1, 241–250 (1984).
[CrossRef]

G. A. Massey, J. A. Davis, S. M. Katanik, E. Omon, J. Opt. Soc. Am. A 1, 1259 (1984).

G. A. Massey, Appl. Opt. 23, 658–660 (1984).
[CrossRef]

A. Harootunian, K. Betzig, A. Muray, A. Lewis, M. Isaacson, J. Opt. Soc. Am. A 1, 1293 (1984).

I. J. Cox, J. Microsc. 133, 149–154 (1984).
[CrossRef] [PubMed]

1983 (1)

R. D. Allen, N. S. Allen, J. Microsc. 129, 3–17 (1983).
[CrossRef] [PubMed]

1982 (3)

D. K. Hamilton, C. J. R. Sheppard, Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

I. J. Cox, C. J. R. Sheppard, T. Wilson, Optik 60, 391–396 (1982).

M. Bertero, E. R. Pike, Opt. Acta 29, 6, 727–746 (1982).
[CrossRef]

1981 (1)

1979 (1)

1977 (2)

B. Saleh, J. Opt. Soc. Am. 67, 71–76 (1977).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

1974 (1)

1972 (1)

E. A. Ash, G. Nicholls, Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

1969 (1)

1968 (2)

1967 (4)

1966 (2)

1965 (1)

J. D. Armatage, A. W. Lohmann, D. P. Paris, Jpn. J. Appl. Phys. Suppl. 1, 273–275 (1965).

1964 (2)

A. W. Lohmann, D. P. Paris, Appl. Opt. 3, 1037–1043 (1964).
[CrossRef]

J. L. Harris, J. Opt. Soc. Am. 54, 606, 931–936 (1964).
[CrossRef]

1960 (1)

A. I. Kartashev, Opt. Spectrosc. (USSR) 9, 204–206 (1960).

1955 (3)

P. B. Fellgett, E. H. Linfoot, Proc. R. Soc. London Ser. A 247, 369–407 (1955).

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497–501 (1955).
[CrossRef]

G. Toraldo di Francia, Opt. Acta 2, 5–8 (1955).
[CrossRef]

1953 (2)

C. W. King, A. G. Emslie, J. Opt. Soc. Am. 43, 664–668 (1953).
[CrossRef]

A. Blanc-Lapierre, Ann. Inst. Henri Poincaré 13, 245–296 (1953).

1952 (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 3 IX, 426–438 (1952).
[CrossRef]

1951 (1)

1948 (2)

C. E. Shannon, Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

C. E. Shannon, Proc. IRE 37, 10–21 (1948).
[CrossRef]

1928 (1)

R. V. L. Hartley, Bell Syst. Tech. J. 3, 535–563 (1928).
[CrossRef]

Allen, N. S.

R. D. Allen, N. S. Allen, J. Microsc. 129, 3–17 (1983).
[CrossRef] [PubMed]

Allen, R. D.

R. D. Allen, N. S. Allen, J. Microsc. 129, 3–17 (1983).
[CrossRef] [PubMed]

Armatage, J. D.

J. D. Armatage, A. W. Lohmann, D. P. Paris, Jpn. J. Appl. Phys. Suppl. 1, 273–275 (1965).

Ash, E. A.

E. A. Ash, G. Nicholls, Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Bachl, A.

Bendinelli, M.

Bershad, N. J.

Bertero, M.

M. Bertero, E. R. Pike, Opt. Acta 29, 6, 727–746 (1982).
[CrossRef]

Betzig, K.

A. Harootunian, K. Betzig, A. Muray, A. Lewis, M. Isaacson, J. Opt. Soc. Am. A 1, 1293 (1984).

Blanc-Lapierre, A.

A. Blanc-Lapierre, Ann. Inst. Henri Poincaré 13, 245–296 (1953).

Cathey, W. T.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Consortini, A.

Cox, I. J.

I. J. Cox, J. Microsc. 133, 149–154 (1984).
[CrossRef] [PubMed]

I. J. Cox, C. J. R. Sheppard, T. Wilson, Optik 60, 391–396 (1982).

Davis, J. A.

G. A. Massey, J. A. Davis, S. M. Katanik, E. Omon, J. Opt. Soc. Am. A 1, 1259 (1984).

Egger, M. D.

Emslie, A. G.

Fellgett, P. B.

P. B. Fellgett, E. H. Linfoot, Proc. R. Soc. London Ser. A 247, 369–407 (1955).

Fried, D. L.

Frieden, B. R.

Galambos, R.

Grimm, M. A.

Hadravský, M.

Hamilton, D. K.

D. K. Hamilton, C. J. R. Sheppard, Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

Harootunian, A.

A. Harootunian, K. Betzig, A. Muray, A. Lewis, M. Isaacson, J. Opt. Soc. Am. A 1, 1293 (1984).

Harris, J. L.

J. L. Harris, J. Opt. Soc. Am. 54, 606, 931–936 (1964).
[CrossRef]

Harris, R. W.

Hartley, R. V. L.

R. V. L. Hartley, Bell Syst. Tech. J. 3, 535–563 (1928).
[CrossRef]

Howard, S. J.

Isaacson, M.

A. Harootunian, K. Betzig, A. Muray, A. Lewis, M. Isaacson, J. Opt. Soc. Am. A 1, 1293 (1984).

Kartashev, A. I.

A. I. Kartashev, Opt. Spectrosc. (USSR) 9, 204–206 (1960).

Katanik, S. M.

G. A. Massey, J. A. Davis, S. M. Katanik, E. Omon, J. Opt. Soc. Am. A 1, 1259 (1984).

King, C. W.

King, G. W.

Lewis, A.

A. Harootunian, K. Betzig, A. Muray, A. Lewis, M. Isaacson, J. Opt. Soc. Am. A 1, 1293 (1984).

Linfoot, E. H.

P. B. Fellgett, E. H. Linfoot, Proc. R. Soc. London Ser. A 247, 369–407 (1955).

Lohmann, A. W.

Lukosz, W.

Massey, G. A.

G. A. Massey, J. A. Davis, S. M. Katanik, E. Omon, J. Opt. Soc. Am. A 1, 1259 (1984).

G. A. Massey, Appl. Opt. 23, 658–660 (1984).
[CrossRef]

McCutchen, C. W.

Muray, A.

A. Harootunian, K. Betzig, A. Muray, A. Lewis, M. Isaacson, J. Opt. Soc. Am. A 1, 1293 (1984).

Nicholls, G.

E. A. Ash, G. Nicholls, Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Omon, E.

G. A. Massey, J. A. Davis, S. M. Katanik, E. Omon, J. Opt. Soc. Am. A 1, 1259 (1984).

Paris, D. P.

J. D. Armatage, A. W. Lohmann, D. P. Paris, Jpn. J. Appl. Phys. Suppl. 1, 273–275 (1965).

A. W. Lohmann, D. P. Paris, Appl. Opt. 3, 1037–1043 (1964).
[CrossRef]

Petrán, M.

Pike, E. R.

M. Bertero, E. R. Pike, Opt. Acta 29, 6, 727–746 (1982).
[CrossRef]

Rhodes, W. T.

Ronchi, L.

Rose, A.

A. Rose, in Advances in Electronics, L. Marton, ed. (Academic, New York, 1948), Vol. 1, pp. 131–166.

Rushford, C. K.

Rushworth, C. K.

Saleh, B.

Shannon, C. E.

C. E. Shannon, Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

C. E. Shannon, Proc. IRE 37, 10–21 (1948).
[CrossRef]

Sheppard, C. J. R.

I. J. Cox, C. J. R. Sheppard, T. Wilson, Optik 60, 391–396 (1982).

D. K. Hamilton, C. J. R. Sheppard, Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Toraldo di Francia, G.

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497–501 (1955).
[CrossRef]

G. Toraldo di Francia, Opt. Acta 2, 5–8 (1955).
[CrossRef]

G. Toraldo di Francia, Nuovo Cimento Suppl. 3 IX, 426–438 (1952).
[CrossRef]

Wilson, T.

I. J. Cox, C. J. R. Sheppard, T. Wilson, Optik 60, 391–396 (1982).

Ann. Inst. Henri Poincaré (1)

A. Blanc-Lapierre, Ann. Inst. Henri Poincaré 13, 245–296 (1953).

Appl. Opt. (2)

Bell Syst. Tech. J. (2)

R. V. L. Hartley, Bell Syst. Tech. J. 3, 535–563 (1928).
[CrossRef]

C. E. Shannon, Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

J. Microsc. (2)

I. J. Cox, J. Microsc. 133, 149–154 (1984).
[CrossRef] [PubMed]

R. D. Allen, N. S. Allen, J. Microsc. 129, 3–17 (1983).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (16)

J. Opt. Soc. Am. A (3)

W. T. Cathey, B. R. Frieden, W. T. Rhodes, C. K. Rushford, J. Opt. Soc. Am. A 1, 241–250 (1984).
[CrossRef]

A. Harootunian, K. Betzig, A. Muray, A. Lewis, M. Isaacson, J. Opt. Soc. Am. A 1, 1293 (1984).

G. A. Massey, J. A. Davis, S. M. Katanik, E. Omon, J. Opt. Soc. Am. A 1, 1259 (1984).

Jpn. J. Appl. Phys. Suppl. (1)

J. D. Armatage, A. W. Lohmann, D. P. Paris, Jpn. J. Appl. Phys. Suppl. 1, 273–275 (1965).

Nature (1)

E. A. Ash, G. Nicholls, Nature 237, 510–512 (1972).
[CrossRef] [PubMed]

Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 3 IX, 426–438 (1952).
[CrossRef]

Opt. Acta (4)

G. Toraldo di Francia, Opt. Acta 2, 5–8 (1955).
[CrossRef]

D. K. Hamilton, C. J. R. Sheppard, Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

M. Bertero, E. R. Pike, Opt. Acta 29, 6, 727–746 (1982).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

A. I. Kartashev, Opt. Spectrosc. (USSR) 9, 204–206 (1960).

Optik (1)

I. J. Cox, C. J. R. Sheppard, T. Wilson, Optik 60, 391–396 (1982).

Proc. IRE (1)

C. E. Shannon, Proc. IRE 37, 10–21 (1948).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

P. B. Fellgett, E. H. Linfoot, Proc. R. Soc. London Ser. A 247, 369–407 (1955).

Science (1)

M. D. Egger, M. Petrán, Science 157, 305–307 (1967).
[CrossRef] [PubMed]

Other (1)

A. Rose, in Advances in Electronics, L. Marton, ed. (Academic, New York, 1948), Vol. 1, pp. 131–166.

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 (5)

Fig. 1
Fig. 1

Information capacity of an optical system as a function of SNR.

Fig. 2
Fig. 2

Resolution improvement against SNR, for analytic continuation with a large space–bandwidth product.

Fig. 3
Fig. 3

Resolution improvement against SNR for analytic continuation with a restricted object field, that is, small space–bandwidth product.

Fig. 4
Fig. 4

Optical system of a general scanning microscope.

Fig. 5
Fig. 5

Signal and signal–flare ratio in a scanning microscope with detector normalized radius υd.

Tables (2)

Tables Icon

Table 1 Signal-to-Flare Ratio for a Point Source

Tables Icon

Table 2 Signal-to-Flare Ratio for a Point Detector

Equations (23)

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

N F = 2 ( 1 + L x B x ) ( 1 + L y B y ) ( 1 + T B T ) .
N = 2 B x L x 2 B y L y log [ ( s + n ) / n ] 1 / 2
N = 2 B x L x 2 B y L y log [ ( s + n ) / n ]
M = 2 T B T + 1 ,
M = 2 ( 2 L x B x + 1 ) ( 2 L y B y + 1 ) ( 2 L z B z + 1 ) ( 2 T B T + 1 ) ,
[ ( s + n ) / n ] 1 / 2 .
m = [ ( s + n ) / n ] M / 2 .
N = log m ,
N = log { ( s + n ) / n ] } M / 2 = ( 2 L z B z + 1 ) ( 2 L y B y + 1 ) × ( 2 L z B z + 1 ) ( 2 T B T + 1 ) log ( 1 + s / n ) .
N = ( 2 L x B x + 1 ) ( 2 L y B y + 1 ) 2 L z B z + 1 ) × ( 2 T B T + 1 ) log ( 1 + s / n ) ,
B x > B x , B y > B y , B T < B T ,
( 2 L x B x + 1 ) ( 2 L y B y + 1 ) ( 2 T B T + 1 ) = ( 2 L x B x + 1 ) ( 2 L y B y + 1 ) ( 2 T B T + 1 ) .
B x B y B x B y = B T B T .
SR = Δ μ 2 Δ μ 1 .
SR = log ( 1 + SNR 1 ) log ( 1 + SNR 2 )
SR = [ log ( 1 + SNR 1 ) ( log ( 1 + SNR 2 ) ] 1 / 2 ,
SR = SR L + 1 2 n 1 ( SR L 1 ) ,
I = [ 2 J 1 ( υ ) υ ] ,
υ = 2 π r sin α / λ ,
I ( υ d ) = 0 υ d [ 2 J 1 ( υ ) υ ] 2 2 π υ d υ ,
I ( υ d ) = 1 J 0 2 ( υ d ) J 1 2 ( υ d ) .
R = 4 [ 1 J 0 2 ( υ d ) J 1 2 ( υ d ) ] / υ d 2 .
υ p = 2 π ( λ 8 s i n α ) sin α λ = π 4 .

Metrics