Abstract

Distributions of irradiance H in the sharply focused diffraction images of self-radiant line elements are examined for objectives of Sonine type S0, S1, and S2 with the view to finding a common characteristic that can be applied to the systematic measurement of length of line elements. With respect to the distribution H(x) along the length x of the diffraction image, the only outstanding, common characteristic is the point of steepest slope, at which dH(x)/dx assumes its greatest value. The relationships between the distance xa from the center of the image to the point of steepest slope and the half-length K of the geometrical image are derived and tabulated for types S0, S1, and S2. Distances xa have a strong tendency to equal K. It is found that objectives of type S0 are more sensitive than the classical Airy type S1 for measuring the shortest measurable line elements.

© 1964 Optical Society of America

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References

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  1. R. W. Burnham and J. E. Jackson, Science 122, 951–953 (1955).
    [CrossRef] [PubMed]
  2. G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), p. 373.
  3. H. Osterberg and J. E. Wilkins, J. Opt. Soc. Am. 39, 553–557 (1949).
    [CrossRef]
  4. A. Gray, G. Mathews, and T. MacRobert, Bessel Functions (Macmillan and Company Ltd., London, 1931), pp. 300–302.

1955 (1)

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

1949 (1)

Burnham, R. W.

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

Gray, A.

A. Gray, G. Mathews, and T. MacRobert, Bessel Functions (Macmillan and Company Ltd., London, 1931), pp. 300–302.

Jackson, J. E.

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

MacRobert, T.

A. Gray, G. Mathews, and T. MacRobert, Bessel Functions (Macmillan and Company Ltd., London, 1931), pp. 300–302.

Mathews, G.

A. Gray, G. Mathews, and T. MacRobert, Bessel Functions (Macmillan and Company Ltd., London, 1931), pp. 300–302.

Osterberg, H.

Watson, G. N.

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), p. 373.

Wilkins, J. E.

J. Opt. Soc. Am. (1)

Science (1)

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

Other (2)

G. N. Watson, Bessel Functions (Cambridge University Press, London, 1944), p. 373.

A. Gray, G. Mathews, and T. MacRobert, Bessel Functions (Macmillan and Company Ltd., London, 1931), pp. 300–302.

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Figures (4)

Fig. 1
Fig. 1

Method of realizing objectives of type S0. In principle the annulus has infinitesimal width, but narrow annuli are permitted. The pupil function P(ρ) is substantially constant over the annulus. It is convenient to take ρm as the numerical aperture corresponding to the central circle of the annulus.

Fig. 2
Fig. 2

The normalized irradiances Hν(xa), ν=0, 1, 2, as functions of xa for line elements having the half-length K=0.261. Both xa and K appear in Airy units.

Fig. 3
Fig. 3

The normalized irradiances Hν(xa), ν=0, 1, 2, as functions of xa for line elements having half-length K=1.305. Both xa and K are stated in Airy units.

Fig. 4
Fig. 4

Positions xa of steepest slope vs the half-length K of line elements in images formed by objectives of type S0, S1, and S2. Both K and xa are in Airy units.

Tables (3)

Tables Icon

Table I Positions xa of steepest slope vs the half-length K of the line element for objectives of type S0. Both K and xa are given in Airy units.

Tables Icon

Table II Positions xa of steepest slope vs the half-length K of the line element for objectives of type S1, the Airy type.

Tables Icon

Table III Positions of steepest slope for objectives of type S2.

Equations (37)

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U ( r ) = 0 ρ m [ 1 - ( ρ / ρ m ) 2 ] ν - 1 J 0 ( 2 π r ρ ) ρ d ρ , = ρ m 2 2 ν - 1 Γ ( ν ) J ν ( 2 π r ρ m ) / ( 2 π r ρ m ) ν ,
P ( ρ ) = [ 1 - ( ρ / ρ m ) 2 ] ν - 1 ,
U ν ( r ) = C ν J ν ( 2 π ρ m r ) / ( 2 π ρ m r ) ν
U 0 ( r ) = ρ m - δ ρ m + δ J 0 ( 2 π r ρ ) ρ d ρ = 2 δ ρ m J 0 ( 2 π ρ m r ) .
- W W - z z U ν 2 { [ ( u - x ) 2 + ( v - y ) 2 ] 1 2 } d u d v ,
2 W - z z U ν 2 ( u - x ) d u .
- z z J ν 2 [ 2 π ρ m ( u - x ) ] / [ 2 π ρ m ( u - x ) ] 2 ν d u .
ζ = 2 π ρ m u ,             ζ m = 2 π ρ m z ,
H ν ( x ) = - ζ m ζ m J ν 2 ( ζ - 2 π ρ m x ) / ( ζ - 2 π ρ m x ) 2 ν d ζ .
2 π ρ m r a = 3.831706 β ,
x a = x / r a ,             z a = K = z / r a .
H ν ( x a ) = - ζ m ζ m J ν 2 ( t - β x a ) / ( t - β x a ) 2 ν d t ,
H 0 ( x a ) = - ζ m ζ m J 0 2 ( β x a - t ) d t .
d H 0 ( x a ) / d x a = - 2 β - ζ m ζ m J 0 ( β x a - t ) J 1 ( β x a - t ) d t = β [ J 0 2 ( β x a + ζ m ) - J 0 2 ( β x a - ζ m ) ] ,
d 2 H 0 ( x a ) / d x a 2 = 2 β 2 [ J 0 ( β x a - ζ m ) J 1 ( β x a - ζ m ) - J 0 ( β x a + ζ m ) J 1 ( β x a + ζ m ) ] .
J 0 ( b ) J 1 ( b ) = J 0 ( h ) J 1 ( h ) ,
b = β x a - ζ m ,             h = β x a + ζ m ,
H 0 ( x a ) J 0 2 ( β x a ) .
β x a = 1.082 ,             x a = 0.2824 Airy unit .
J 0 ( h ) J 1 ( h ) = 0 ,             h = 2 ζ m .
2 ζ m is a root of { J 0 ( 2 ζ m ) = 0 or J 1 ( 2 ζ m ) = 0.
B 0 = J 0 ( h ) J 1 ( h ) ,
B 0 , 0 = J 0 ( 2 ζ m ) J 1 ( 2 ζ m ) .
b = β x a - ζ m = 2 B 0 , 0 ,
x a = K + 2 B 0 , 0 / β = K + 0.522 B 0 , 0 .
J 1 ( b ) J 2 ( b ) / b 2 = J 1 ( h ) J 2 ( h ) / h 2 ,
β x a = 1.488 ,             x a = 0.3883 Airy unit .
2 ζ m is a root of { J 1 ( 2 ζ m ) = 0 or J 2 ( 2 ζ m ) = 0.
B 1 = J 1 ( h ) J 2 ( h ) / h 2 ,
B 1 , 0 = J 1 ( 2 ζ m ) J 2 ( 2 ζ m ) / ( 4 ζ m 2 ) .
x a = K + 16 B 1 , 0 / β = K + 4.176 B 1 , 0 .
J 2 ( b ) J 3 ( b ) / b 4 = J 2 ( h ) J 3 ( h ) / h 4 ,
β x a = 1.797 ,             x a = 0.4690 Airy unit .
2 ζ m is a root of { J 2 ( 2 ζ m ) = 0 or J 3 ( 2 ζ m ) = 0.
B 2 = J 2 ( h ) J 3 ( h ) / h 4 ,
B 2 , 0 = J 2 ( 2 ζ m ) J 3 ( 2 ζ m ) / ( 16 ζ m 4 ) .
x a = K + 100.2 B 2 , 0 .