Abstract

For many optical systems with zero obscuration, it has been found that, for extraaxial pencils, the periphery of the vignetted pupil shape may be represented by an ellipse with good approximation. A computer program is described which first determines the rim rays and then fits the ray coordinates to an ellipse by the method of least squares. Some examples are included showing the accuracy of the elliptical pupil.

© 1968 Optical Society of America

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References

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  1. H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).
  2. H. H. Hopkins, Opt. Acta 13, 343 (1966).
    [Crossref]

1966 (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[Crossref]

1965 (1)

H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[Crossref]

H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).

Japan. J. Appl. Phys. (1)

H. H. Hopkins, Japan. J. Appl. Phys. 4, Suppl. 1, 31 (1965).

Opt. Acta (1)

H. H. Hopkins, Opt. Acta 13, 343 (1966).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

The first estimate of the center (0, c0) of the pupil and its semidiameter b0 in the meridian section.

Fig. 2
Fig. 2

An example of the plot of the periphery points Bi with R = 5, Δϕ = π/4.

Fig. 3
Fig. 3

B1ĒBR′ represents the sectional view of the projected elliptical pupil, with Ē as the effective center of the entrance pupil.

Fig. 4
Fig. 4

The scaled pupil coordinates (x0/a1, y0/b1) of the rim rays plotted against a unit circle for a 16-mm microscope objective.

Fig. 5
Fig. 5

The scaled pupil coordinates (x0/a1, y0/b1) of the rim rays plotted against a unit circle for a 61-cm (24-in.), f/5.6, 24° field system.

Fig. 6
Fig. 6

The scaled pupil coordinates (x0/a1, y0/b1) of the rim rays plotted against a unit circle for a 15-cm (6-in.), f/2, 50° field system.

Fig. 7
Fig. 7

The scaled pupil coordinates (x0/a1, y0/b1) of the rim rays plotted against a unit circle for a 15-cm (6-in.), f/6.3, 90° field system.

Fig. 8
Fig. 8

The scaled pupil coordinates (x0/a1, y0/b1) of the rim rays plotted against a unit circle for a Roossinov 7-cm (2.75-in.), f/8, 122° field system.

Tables (1)

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Table I The Residual Errors in Approximating the Vignetted Pupil Periphery to the Best Fitting Ellipse

Equations (10)

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( x s / a s ) 2 + ( y s / b s ) 2 = 1 ,
x 2 + y 2 = 1 ,
x = x s / a s ,             y = y s / b s .
c 0 = [ y 0 ( 1 ) + y 0 ( R ) ] / 2 , b 0 = [ y 0 ( 1 ) - y 0 ( R ) ] / 2 ,
δ r 1 ( k ) = [ ρ max ( k ) - ρ 1 ( k ) ] / h ( k ) ,
( x 0 ) 2 i / a 2 ) + { ( y 0 ) i - c } 2 / b 2 = 1 ,
{ 2 a [ ( x 0 ) i a ] 2 } δ a + { 2 b [ ( y 0 ) i - c b ] 2 } δ b + { 2 [ ( y 0 ) i - c ] b 2 } δ c + { 1 - [ ( ( x 0 ) i a ) 2 + ( ( y 0 ) i - c b ) 2 ] } = 0.
{ ( x 0 ) i / a 1 } 2 + { [ ( y 0 ) i - c 1 ] / b 1 } 2 = 1.
( x 0 ) i = r i sin ϕ i , ( y 0 ) i = c + r i cos ϕ i .
L = - u 1 ( x 0 ) i { 1 + ( u ¯ 1 τ ) 2 } 1 2 , M = 1 u 1 τ + u 1 ( y 0 ) i { 1 + ( u ¯ 1 τ ) 2 } 1 2 , N = + { 1 - ( L 1 2 + M 1 2 ) } 1 2 ,

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