Abstract

The position of the first minimum or stationary value which appears in the irradiance beyond the end of the geometrical image in the image plane of an objective with a narrow annular aperture when a line element of uniform radiance lies in the object plane is examined for its relation to the length of the line element. The line element is assumed to be self-radiant or its equivalent by scattering. A table of the relation between line length and the position of the first minimum or stationary value is presented for lengths 2K in the range 0≦2K≦5.728 Airy units. Line elements as short as 0.4 Airy unit appear measurable by this method.

© 1964 Optical Society of America

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References

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  1. An objective of Sonine type S0.
  2. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 408 (1964) (accompanying paper).
  3. Similar curves of Hν,(xa) for other objectives of the Sonine family Sν do not exhibit well-developed minima along x except for lines having K values so small that the diffraction image is highly circular.
  4. See Ref. 2.
  5. R. Burnham and J. E. Jackson, Science 122, 951–953 (1955).
    [CrossRef] [PubMed]

1964 (1)

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 408 (1964) (accompanying paper).

1955 (1)

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

Burnham, R.

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

Jackson, J. E.

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

Osterberg, H.

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 408 (1964) (accompanying paper).

Smith, L. W.

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 408 (1964) (accompanying paper).

J. Opt. Soc. Am. (1)

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 408 (1964) (accompanying paper).

Science (1)

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

Other (3)

An objective of Sonine type S0.

Similar curves of Hν,(xa) for other objectives of the Sonine family Sν do not exhibit well-developed minima along x except for lines having K values so small that the diffraction image is highly circular.

See Ref. 2.

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Figures (2)

F. 1
F. 1

Distribution of normalized irradiance along the center line of the image formed of a uniformly self-radiant line by an objective with a narrow annular aperture. The number attached to each curve gives the half-length K in Airy units of the corresponding geometrical image. Distance xa along the center line is also in Airy units.

F. 2
F. 2

Location of the first few maxima and minima in the irradiance beyond the end of the geometrical image of a self-radiant line element of length K Airy units. The straight lines are given by the equations xa = K+225%, xa=K+1.441, and xa =K+0.6276 in order from top to bottom of the figure. The cuspate curves on each side of the straight lines are plotted from computed points only in the region 0≦K≦0.4.

Tables (1)

Tables Icon

Table I Solutions of Eq. (4) to be used for finding the half-length K in Airy units of a line element from the position xa, also in Airy units, of the first minimum or stationary value in image irradiance beyond the end of the geometrical image. The starred values are also solutions to Eq. (14).

Equations (16)

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H 0 ( x a ) = ζ m ζ m J 0 2 ( β x a t ) d t ,
ζ m = β K , β = 3.8317 ,
2 π ρ m r a = β ,
2 β ζ m ζ m J 0 ( β x a t ) J 1 ( β x a t ) d t = 0 .
J 0 ( b ) = ± J 0 ( h ) ; b = β x a ζ m , h = β x a ζ m .
b = γ n , h = γ ν ,
x a = K + γ n / β ,
K = ( γ ν γ n ) / ( 2 β ) .
J 0 ( u + υ ) = ν = 0 ( 1 ) ν ν J ν ( u ) J ν ( υ ) ,
J 0 ( u υ ) = ν = 0 ν J ν ( u ) J ν ( υ ) ,
β x a ζ m = b γ 1 , β x a + ζ m = h γ 1 + 2 ζ m .
H 0 ( x a ) = β h / β b / β J 0 2 ( β t ) d t ,
= β [ 0 2 K + b / β J 0 2 ( β t ) d t 0 b / β J 0 2 ( β t ) d t ] ,
H 0 ( x a ) / β = 0 2 K J 0 2 ( β t ) d t 0 b / β J 0 2 ( β t ) d t .
K = x a γ 1 / β = x a 0.6276 .
0 Δ K < 0.005 .