Abstract

The diffraction pattern which represents the image of a luminous point produced by an instrument free from aberrations and having a circular aperture, when observed out of focus, has been calculated by A. E. Conrady, A. Buxton, and G. Lansraux. Conrady and Buxton used numerical integration. Such a process yields a sequence of values of the complex amplitude at a particular point in the diffraction pattern for a particular lack of focus, and for any given integration, the distance W of the point under consideration from the center of the pattern, and the lack of focus ψ, are functions of the upper limit of integration. If ψ is plotted against W, the path obtained is a parabola, ψ = aW2.

The author has found a method whereby any desired path can be followed instead of a parabola. In particular, if the point of observation is near the edge of the purely geometrical diffusion disk and remains at a constant distance from it as ψ changes, then, for large values of ψ, the form of the diffraction fringes is approximately independent of ψ and their intensity is inversely proportional to ψ2.

© 1949 Optical Society of America

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References

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  1. A. E. Conrady, M. N. R. A. S. 79, 575 (1919)
  2. A. Buxton, M. N. R. A. S. 81, 547 (1921).
  3. G. Lansraux, Revue d’Optique 26, 24 (1947).

1947 (1)

G. Lansraux, Revue d’Optique 26, 24 (1947).

1921 (1)

A. Buxton, M. N. R. A. S. 81, 547 (1921).

1919 (1)

A. E. Conrady, M. N. R. A. S. 79, 575 (1919)

Buxton, A.

A. Buxton, M. N. R. A. S. 81, 547 (1921).

Conrady, A. E.

A. E. Conrady, M. N. R. A. S. 79, 575 (1919)

Lansraux, G.

G. Lansraux, Revue d’Optique 26, 24 (1947).

M. N. R. A. S. (2)

A. E. Conrady, M. N. R. A. S. 79, 575 (1919)

A. Buxton, M. N. R. A. S. 81, 547 (1921).

Revue d’Optique (1)

G. Lansraux, Revue d’Optique 26, 24 (1947).

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Figures (3)

Fig. 1
Fig. 1

Wave front emerging from the exit pupil of an instrument, and reference sphere.

Fig. 2
Fig. 2

Parabolic paths of integration and path followed by the edge of the purely geometrical diffusion disk.

Fig. 3
Fig. 3

Plot of Γ1 and Γ2 against W for ψ = 4π (above) and ψ = 8π (below). The dotted curves above give an approximation to the curves for ψ = 4π derived from those for ψ = 8π with the aid of formula (12d).

Equations (17)

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Γ = ( 2 / W 2 ) 0 W exp ( - i v 2 ψ / W 2 ) J 0 ( v ) v d v = ,
= 2 0 1 exp ( - i ψ u 2 ) J 0 ( W u ) u d u ,
Γ = exp ( - i ψ ) p = 1 2 p ( i ψ ) p - 1 J p ( W ) / W p .
ψ = a W 2 .
Γ / W = i W Γ / 2 ψ - i exp ( - i ψ ) J 1 ( W ) / ψ .
Γ = ( i / ψ ) [ exp ( - i ψ ) - 1 ] ,
Γ = exp ( i W 2 / 4 ψ ) { ( i / ψ ) [ exp ( - i ψ ) - 1 ] - 0 W ( i / ψ ) exp ( - i ψ - i W 2 / 4 ψ ) J 1 ( W ) d W } .
Γ / W = - 2 0 1 exp ( - i ψ u 2 ) J 1 ( W u ) u 2 d u ,
Γ / ψ = - 2 i 0 1 exp ( - i ψ u 2 ) J 0 ( W u ) u 3 d u .
Γ / ψ = ( 1 / ψ ) exp ( - i ψ u 2 ) J 0 ( W u ) u 2 | 0 1 - ( 1 / ψ ) 0 1 exp ( - i ψ u 2 ) [ 2 u J 0 ( W u ) ] - W u 2 J 1 ( W u ) ] d u = = ( 1 / ψ ) exp ( - i ψ ) J 0 ( W ) - Γ / ψ - ( W / 2 ψ ) ( Γ / W ) .
Γ / ψ = [ exp ( - i ψ ) / ψ ] [ J 0 ( W ) + ( i W / 2 ψ ) J 1 ( W ) ] - [ 1 / ψ + i W 2 / 4 ψ 2 ] Γ .
d Γ / d ψ = Γ / ψ + ( Γ / W ) ( d W / d ψ ) ,
W = 2 ψ + b .
d Γ / d ψ = ( i / 2 ψ 2 ) ( - 2 ψ + b ) exp ( - i ψ ) J 1 ( W ) + ( 1 / ψ ) exp ( - i ψ ) J 0 ( W ) + [ i ( 4 ψ 2 - b 2 ) / 4 ψ 2 - 1 / ψ ] Γ .
Γ = ( 1 / ψ ) exp ( i ψ + i b 2 / 4 ψ ) { C + ( i / 2 - i b / 4 ψ ) × J 0 ( W ) exp ( - 2 i ψ - i b 2 / 4 ψ ) + [ b / 2 ψ - i b / 4 ψ 2 + b 2 / 8 ψ 2 - b 3 / 16 ψ 3 ] J 0 ( W ) × exp ( - 2 i ψ - i b 2 / 4 ψ ) d ψ } ,
Γ ( 1 / ψ ) exp ( i ψ ) [ C + 1 2 i exp ( - 2 i ψ ) J 0 ( W ) ] .
Γ ( 1 / ψ ) exp ( i ψ ) C