Abstract

The author’s previous work on the computation of algebraic higher order aberration coefficients and their derivatives has been illustrated numerically by providing explicit calculations, relating to a certain simple triplet operating over a relatively small field and at low aperture. Doubts have been expressed as to the usefulness of using such aberration coefficients to describe the (geometrical) behavior of systems operating at higher apertures and over wider fields. In this paper, therefore, all the primary, secondary, and tertiary aberration coefficients, and the coefficient of quaternary spherical aberration are given for two further systems, viz. (i) a wide angle objective operating at an aperture of f/4.5 and over a field of 70°; and (ii) an objective operating at an aperture of f/2.3 over a field of 40°. A number of diagrams compare the predicted displacements, relative to the ideal image point, of the intersection points with the ideal image plane of certain selected (a) tangential pencils, (b) sagittal pencils, and (c) general skew pencils of rays, with the actual displacements as obtained from trigonometrical traces. The agreement is very satisfactory. Even in cases, however, in which the agreement—for the largest apertures or fields considered—leaves something to be desired a knowledge of the coefficients and their derivatives provides a powerful tool for the control over the performance of the system. A detailed elementary example illustrating this assertion is included.

© 1959 Optical Society of America

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References

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  1. H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, London, 1954). All references to this work will hereafter be distinguished by the letter M.
  2. C. G. Wynne, Proc. Phys. Soc. (London) B70, 1013 (1957).
  3. A. Reuschel, Intern. Math. News (April, 1956).
  4. The relevant theory and computing schemes are developed in the following papers: H. A. Buchdahl, J. Opt. Soc. Am. 48, 563, 747, 757 (1958). These are hereafter referred to as II, III, IV, respectively.
    [CrossRef]
  5. W. Merté, Das Pholographische Objektiv (Verlag Julius Springer, Berlin, 1932), p. 315.

1958 (1)

1957 (1)

C. G. Wynne, Proc. Phys. Soc. (London) B70, 1013 (1957).

1956 (1)

A. Reuschel, Intern. Math. News (April, 1956).

Buchdahl, H. A.

The relevant theory and computing schemes are developed in the following papers: H. A. Buchdahl, J. Opt. Soc. Am. 48, 563, 747, 757 (1958). These are hereafter referred to as II, III, IV, respectively.
[CrossRef]

H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, London, 1954). All references to this work will hereafter be distinguished by the letter M.

Merté, W.

W. Merté, Das Pholographische Objektiv (Verlag Julius Springer, Berlin, 1932), p. 315.

Reuschel, A.

A. Reuschel, Intern. Math. News (April, 1956).

Wynne, C. G.

C. G. Wynne, Proc. Phys. Soc. (London) B70, 1013 (1957).

Intern. Math. News (1)

A. Reuschel, Intern. Math. News (April, 1956).

J. Opt. Soc. Am. (1)

Proc. Phys. Soc. (London) (1)

C. G. Wynne, Proc. Phys. Soc. (London) B70, 1013 (1957).

Other (2)

H. A. Buchdahl, Optical Aberration Coefficients (Oxford University Press, London, 1954). All references to this work will hereafter be distinguished by the letter M.

W. Merté, Das Pholographische Objektiv (Verlag Julius Springer, Berlin, 1932), p. 315.

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Figures (7)

F. 1
F. 1

Σ2: The displacements of eight tangential pencils of rays, for which U1 = 0°, 5.7°, 11.3°, 16.7°, 21.8°, 26.6°, 31.0°, 35.0°.⋯·· primary displacement,– – – – – primary+secondary displacement,– · – · – · – · primary+secondary+tertiary displacement, ————— actual displacement, given by trigonometrical traces.

F. 2
F. 2

Σ2: Spherical aberration. – · · – · · – · · – · · – primary+secondary+tertiary+quatemary displacements, meaning of other curves as in Fig. 1.

F. 3
F. 3

Σ2: The displacements of seven sagittal pencils of rays. Inclinations of pencils, and meaning of curves, as in Fig. 1.

F. 4
F. 4

Σ3: The displacements of five tangential pencils of rays, for which U1=0°, 5.7°, 11.3°, 16.7°, 19.3°. Meaning of curves as in Fig. 1.

F. 5
F. 5

Σ3: Spherical aberration. Meaning of curves as in Fig. 2.

F. 6
F. 6

Σ3: The displacements of three families of skew rays corresponding to circular zones, U1=11.3°, ρ = 0.18; U1=16.7°, ρ = 0.16; U1=19.3°, ρ = 0.15.

F. 7
F. 7

Σ3: Spherical aberration of (a) original system, (b) varied system.

Tables (5)

Tables Icon

Table I Specifications of system Σ2.

Tables Icon

Table II Σ2: The final aberration coefficients of orders three, five, and seven, and the coefficient of ninth-order spherical aberration.

Tables Icon

Table III Specifications of system Σ3.

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Table IV Σ3: The final aberration coefficients of orders three, five, and seven, and the coefficient of ninth-order spherical aberration.

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Table V Σ3: First separation and curvature derivatives of υ, A, S.

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

Σ j = 1 k c j ( G / c j ) Σ j = 2 k d j ( G / d j ) = s G , ( c j c 0 j ) ,
δ ε = 0.136 ρ 3 + 1.16 ρ 5 + 0 ( ρ 7 ) ,
T ¯ 1
B ¯
T ¯ 2
C ¯
T ¯ 3
S ¯ 1
T ¯ 4
S ¯ 2
T ¯ 5
S ¯ 3
T ¯ 6
S ¯ 4
T ¯ 7
S ¯ 5
T ¯ 8
S ¯ 6
T ¯ 9
T ¯ 10
T ¯ 1
B ¯
T ¯ 2
C ¯
T ¯ 3
S ¯ 1
T ¯ 4
S ¯ 2
T ¯ 5
S ¯ 3
T ¯ 6
S ¯ 4
T ¯ 7
S ¯ 5
T ¯ 8
S ¯ 6
T ¯ 9
T ¯ 10
4 N [ 1 4 ( 3 r 5 + r 6 ) t 9 r 4 t 81 ] = A p k / d j
k t 3 r 1 ( 3 2 k t 3 t 2 ) = a p / c 0
4 [ 1 4 ( 3 r 5 * + r 6 * ) r 1 r 2 r 4 * ] + r 9 = A p k / c 0 j
4 [ 1 4 ( 3 t 16 + t 20 ) t 9 + t 15 t 81 ] = α υ ( d ) / N
N { 6 [ 1 2 ( r 4 r 10 + r 5 r 11 ) r 7 t 81 ] + r 8 t 9 } = S 1 p k / d j
[ k ( k t 3 t 2 * ) + t 2 * + t 3 ] 1 4 t 1 t 10
4 [ 1 4 ( 3 t 16 * + t 20 * ) r 1 + t 15 * r 2 ] + r 9 = α υ ( c )
1 3 r 1 r 8 * + r 16 r 4 * + r 17 r 5 * + t 34 r 9 + r 14