Abstract

A method of approximating the homogeneous second derivatives of a merit function with respect to a set of design variables is presented. If these derivatives are used in an automatic lens design program, the familiar problem of stagnation may usually be avoided.

© 1978 Optical Society of America

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References

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  1. J. Meiron, H. M. Loebenstein, J. Opt. Soc. Am. 47, 1104 (1957).
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  2. D. P. Feder, J. Opt. Soc. Am. 47, 902 (1957).
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  3. J. Meiron, J. Opt. Soc. Am. 49, 293 (1959).
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  4. G. H. Spencer, Appl. Opt. 2, 1257 (1963).
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  5. D. P. Feder, Appl. Opt. 2, 1209 (1963).
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  6. J. Meiron, J. Opt. Soc. Am. 55, 1105 (1965).
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  7. D. R. Buchele, Appl. Opt. 7, 2433 (1968).
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  8. N. J. Kidger, C. G. Wynne, Opt. Acta 14, 279 (1967).
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  9. C. G. Wynne, P. M. Wormell, Appl. Opt. 2, 1233 (1963).
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  10. D. S. Grey, J. Opt. Soc. Am. 53, 672 (1963).
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1968 (1)

1967 (1)

N. J. Kidger, C. G. Wynne, Opt. Acta 14, 279 (1967).
[CrossRef]

1965 (1)

1963 (4)

1959 (1)

1957 (2)

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Figures (7)

Fig. 1
Fig. 1

The optical system used to define four simple problems.

Fig. 2
Fig. 2

Summary of the dependence of two aberrations upon two variables for four cases for which the damped least squares (DLS) and pseudo second derivative (PSD) methods of optimization were compared. For all cases, variable V1 was the entrance aperture diameter, and V2 was the lens thickness. The aberrations corrected in each case are indicated, and the symbols within the boxes indicate whether that aberration is I, independent of; L, linearly dependent upon; or N, nonlinearly dependent upon that variable.

Fig. 3
Fig. 3

Comparison of the rates of convergence of the DLS and PSD methods for optimizing case A. In Figs. 37, curves labeled A refer to the number of times the aberration array had been calculated by the program by the time it had reduced the merit function ϕ to the level indicated. Curves labeled L refer to the number of times the matrix L * had been inverted.

Fig. 4
Fig. 4

Comparison of the rates of convergence of the DLS and PSD methods for optimizing case B.

Fig. 5
Fig. 5

Comparison of the rates of convergence of the DLS and PSD methods for optimizing case C.

Fig. 6
Fig. 6

Comparison of the rates of convergence of the DLS and PSD methods for optimizing case D.

Fig. 7
Fig. 7

Comparison of the rates of convergence of the DLS and PSD methods for optimizing a germanium triplet.

Equations (4)

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0 = G j + G j X k Δ X k .
G + L * Δ X = 0 , Δ X = L 1 * G .
G j = w i f i f i X j , L j k = G j X k = w i f i X j f i X k + w i f i 2 f i X j X k .
2 f i X j 2 = f i X j | X j + Δ X j f i X j | X j Δ X j

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