Abstract

The diffraction pattern produced by an objective may be altered by applying coats to its exit pupil. If a coating is applied in such a manner as to reduce the diameter of the central bright disk, the central maximum intensity of the diffraction pattern is reduced. In this paper that diffraction pattern which has the highest central maximum intensity is shown for each fixed diameter of the central bright disk. The coating producing this diffraction pattern is such as to make a central zone of the exit pupil out of phase by one-half wave-length with the rest of the exit pupil, the size of the central zone being determined by the diameter of the central bright disk.

© 1950 Optical Society of America

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References

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  1. R. K. Luneberg, Mathematical Theory of Optics (Brown university Press, Providence, 1944), pp. 391–395.
  2. H. Osterberg and J. E. Wilkins, J. Opt. Soc. Am. 39, 553 (1949).
    [Crossref]
  3. H. Osterberg and F. Wissler, J. Opt. Soc. Am. 39, 558 (1949).
    [Crossref]

1949 (2)

J. Opt. Soc. Am. (2)

Other (1)

R. K. Luneberg, Mathematical Theory of Optics (Brown university Press, Providence, 1944), pp. 391–395.

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Figures (2)

Fig. 1
Fig. 1

The highest possible central maximum intensity for any diffraction pattern with central diameter 2r0.

Fig. 2
Fig. 2

Diffraction patterns vanishing at 0.6, 0.75, 0.8 of the Airy limit which have the highest possible central maximum intensity. Curve 1 is the classical Airy diffraction pattern.

Equations (20)

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U ( r ) = 2 π 0 ρ m ρ P ( ρ ) J 0 ( 2 π r ρ ) d ρ ,
P ( ρ ) Ψ ( ρ ) .
U ( r 0 ) / 2 π = 0 ρ m ρ P ( ρ ) J 0 ( 2 π r 0 ρ ) d ρ = 0.
r 0 < 0.6098 / ρ m .
U ( 0 ) 2 = 4 π 2 | 0 ρ m ρ P ( ρ ) d ρ | 2 ,
| 0 ρ m ρ P ( ρ ) d ρ | .
P 0 ( ρ ) = - Ψ ( ρ ) when     0 ρ ρ 1 , P 0 ( ρ ) = + Ψ ( ρ ) when     ρ 1 < ρ ρ m ,
0 ρ 1 ρ Ψ ( ρ ) J 0 ( 2 π r 0 ρ ) d ρ = ρ 1 ρ m ρ Ψ ( ρ ) J 0 ( 2 π r 0 ρ ) d ρ .
0 ρ 1 ρ J 0 ( 2 π r 0 ρ ) d ρ = ρ 1 ρ m ρ J 0 ( 2 π r 0 ρ ) d ρ .
2 x J 1 ( x ) = y J 1 ( y ) .
Q ( r 0 ) = ( 1 - 2 ρ 1 2 ρ m - 2 ) 2 = ( 1 - 2 x 2 y - 2 ) 2 .
U ( r ) 2 = 4 π 2 ρ m 4 u - 2 [ J 1 ( u ) - 2 x y - 1 J 1 ( x y - 1 u ) ] 2 ,
0 ρ 1 ρ η ( ρ ) J 0 ( 2 π r 0 ρ ) d ρ J 0 ( 2 π r 0 ρ 1 ) 0 ρ 1 ρ η ( ρ ) d ρ , ρ 1 ρ m ρ η ( ρ ) J 0 ( 2 π r 0 ρ ) d ρ J 0 ( 2 π r 0 ρ 1 ) ρ 1 ρ m ρ η ( ρ ) d ρ .
0 J 0 ( 2 π r 0 ρ 1 ) 0 ρ m ρ η ( ρ ) d ρ , J 0 ( 2 π r 0 ρ 1 ) 0 ρ m ρ P ( ρ ) d ρ J 0 ( 2 π r 0 ρ 1 ) ρ m ρ P 0 ( ρ ) d ρ .
0 J 0 ( 2 π r 0 ρ 1 ) 0 ρ m ρ ζ ( ρ ) d ρ , - J 0 ( 2 π r 0 ρ 1 ) 0 ρ m ρ P ( ρ ) d ρ J 0 ( 2 π r 0 ρ 1 ) 0 ρ m ρ P 0 ( ρ ) d ρ .
J 0 ( 2 π r 0 ρ 1 ) 0 ρ m ρ P 0 ( ρ ) d ρ J 0 ( 2 π r 0 ρ 1 ) | 0 ρ m ρ P ( ρ ) d ρ | ,
| 0 ρ m ρ P 0 ( ρ ) d ρ | | 0 ρ m ρ P ( ρ ) d ρ | ,
0 ρ 1 ρ Ψ ( ρ ) J 0 ( 2 π r 0 ρ ) d ρ 0 ρ 1 ρ m ρ Ψ ( ρ ) J 0 ( 2 π r 0 ρ ) d ρ ,
0 ρ m ρ [ a P 2 ( ρ ) + b P 1 ( ρ ) ] d ρ = 0.
| 0 ρ m ρ P 0 ( ρ ) d ρ | | 0 ρ m ρ P * ( ρ ) d ρ | = | 0 ρ m ρ P ( ρ ) d ρ | .