Abstract

Resolution of shape of self-radiant line elements by objectives of Sonine types Sν is stated quantitatively as the ratio Rs of length to width of a uniquely chosen contour of constant irradiance in the diffraction image. This contour passes through the point xs of steepest slope of the irradiance H(x) along the length x of the geometrical image. Computations are included of the resolution factor Rs as a function of the half-length K (in Airy units) of line elements imaged by objectives of types S0, S1, and S2. Type S0 excels type S1 (identical with the Airy type) for resolving shape. In turn, the apodized type S2 is inferior to type S1.

© 1964 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Objects that scatter radiation into the objective can be considered self-radiant.
  2. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 406 (1964).
    [CrossRef]
  3. R. W. Burnham and J. E. Jackson, Science 122, 951–953 (1955).
    [CrossRef] [PubMed]
  4. L. F. Rowe developed the numerical method on a basis of finite differences and used an IBM 7070 computer.
  5. G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), p. 147.
  6. See Ref. 2.
  7. See Eqs. (8) and (10) and Ref. 5, p. 417.
  8. See Ref. 5, pp. 666–697.

1964 (1)

1955 (1)

R. W. Burnham and J. E. Jackson, Science 122, 951–953 (1955).
[CrossRef] [PubMed]

Burnham, R. W.

R. W. Burnham and J. E. Jackson, Science 122, 951–953 (1955).
[CrossRef] [PubMed]

Jackson, J. E.

R. W. Burnham and J. E. Jackson, Science 122, 951–953 (1955).
[CrossRef] [PubMed]

Osterberg, H.

Smith, L. W.

Watson, G. N.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), p. 147.

J. Opt. Soc. Am. (1)

Science (1)

R. W. Burnham and J. E. Jackson, Science 122, 951–953 (1955).
[CrossRef] [PubMed]

Other (6)

L. F. Rowe developed the numerical method on a basis of finite differences and used an IBM 7070 computer.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), p. 147.

See Ref. 2.

See Eqs. (8) and (10) and Ref. 5, p. 417.

See Ref. 5, pp. 666–697.

Objects that scatter radiation into the objective can be considered self-radiant.

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 (4)

F. 1
F. 1

Description of objectives of Sonine types S0, S1, and S2. P(ρ) is the usual pupil function with respect to the axial bundle of rays from P to P′. The numerical aperture with respect to the image space is ρm. With objectives of type S0, a narrow annulus of “width” 2δ restricts the aperture and the numerical aperture ρm corresponds to the central circle of the annulus.

F. 2
F. 2

Normalized irradiances H0, H1, and H2 as functions of distance xa in Airy units along the line y = 0. The half-length of the object is K = 0.913 Airy unit.

F. 3
F. 3

Normalized irradiances H0, H1, and H2 as functions of distance ya in Airy units along the line x = 0. The half-length K of the object is 0.913 Airy unit as in Fig. 2.

F. 4
F. 4

The resolution factor Rs as a function of the half-length K of the line element for objectives of type S0, S1, and S2.

Tables (2)

Tables Icon

Table I The integrals I ν ( y , Z ) = 0 z J ν 2 ( ω ) / ω 2 ν d t, for ω = (y2+t2)1/2, ν=0, 1, 2, y = 0(0.5)8.0, and Z = 0.5(0.5)25. The last decimal place may not always be justified for values of y and Z near the upper limits of their respective ranges.

Tables Icon

Table II The resolution factor Rs as a function of the half-length K of the line element for objectives of Sonine type S0, S1, and S2.

Equations (29)

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

H ( 0 , y s ) = H ( x s , 0 ) .
R s = x s / y s .
U ν ( r ) = J ν ( 2 π ρ m r ) / ( 2 π ρ m r ) ν .
H ν ( x , y ) = z z w w U ν 2 ( r ) d υ d u ,
H ν ( x , y ) = 2 w z z U ν 2 ( r ) d u ,
x a = x / r a , y a = y / r a , K = z / r a ,
2 π ρ m r a = 3.8317 β .
H ν ( x a , y a ) = β K β K J ν 2 ( r ) r 2 ν d t ,
r = [ ( t x a ) 2 + β 2 y a 2 ] 1 2 .
I ν ( y , Z ) = 0 Z J ν 2 [ ( y 2 + t 2 ) 1 2 ] ( y 2 + t 2 ) ν d t .
H ν ( x a , y a ) = I ν ( β y a , Z 1 ) + I ν ( β y a , Z 2 ) ,
Z 1 = β ( K x a ) , Z 2 = β ( K + x a ) .
I 0 ( y , Z ) = ν = 0 ( 1 ) ν h ν ,
h ν = h ν 1 2 ν 1 ν 2 ( 2 ν + 1 ) y 2 + P ν ,
P ν = P ν 1 ( 2 ν 1 ) 2 ν 3 ( 2 ν + 1 ) y 2 + Z 2 2 , h 0 = P 0 = Z .
I 1 ( y , Z ) = 1 4 ν = 0 ( 1 ) ν h ν ,
h ν = h ν 1 y 2 ( ν + 1 ) ( ν + 2 ) + P ν ,
P ν = P ν 1 2 ν 1 ν ( ν + 1 ) ( ν + 2 ) y 2 + Z 2 2 ,
h 0 = P 0 = Z .
I 2 ( y , Z ) = 1 16 ν = 0 ( 1 ) ν h ν ,
h ν = h ν 1 2 ν + 3 ( 2 ν + 1 ) ( ν + 2 ) ( ν + 4 ) y 2 + P ν ,
P ν = P ν 1 2 ν + 3 ( 2 ν + 1 ) ( ν + 2 ) ( ν + 4 ) ( 2 1 ν ) y 2 + Z 2 2 ,
h 0 = P 0 = Z / 4 .
I 0 ( 0 , Z ) = 0 Z J 0 2 ( t ) d t = 2 n = 0 ( 1 ) n { J 0 ( Z ) J 2 n + 1 ( Z ) + Z 2 n + 1 [ J 1 ( Z ) J 2 n + 1 ( Z ) J 0 ( Z ) J 2 n + 2 ( Z ) ] } .
I 1 ( 0 , Z ) = 0 Z t 2 J 1 2 ( t ) d t = 2 Z 3 [ J 0 2 ( Z ) + J 1 2 ( Z ) ] J 1 2 ( Z ) 3 Z 2 3 J 0 ( Z ) J 1 ( Z ) .
I 2 ( 0 , Z ) = 0 Z t 4 J 2 2 ( t ) d t = [ 16 Z 315 + 2 21 Z ] × [ J 1 2 ( Z ) + J 2 2 ( Z ) ] [ 8 105 Z + 1 7 Z 3 ] J 2 2 ( Z ) [ 16 105 + 2 21 Z 2 ] J 1 ( Z ) J 2 ( Z ) .
0 t 2 J 1 2 ( t ) = 4 ( 3 π ) .
r 0 = 0.61 λ / N . A . ,
H 1 ( 0 , y ) H 1 ( 2 y ) / y 2 ,