Abstract

An adaptive technique for optical design with computers is described Its distinctive features are compared with those of the well known damped least squares (DLS) method. The system of adaptive control ensures a particularly certain and rapid convergence, often of an order of magnitude quicker than the DLS method. The convergence is illustrated for a number of examples. The adaptive method enables the designer to learn the fundamental potentialities and limitations of a given optical system.

© 1968 Optical Society of America

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References

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  1. E. Glatzel, Optik 18, 577 (1961).
  2. R. E. Hopkins, G. Spencer, J. Opt. Soc. Am. 52, 172 (1962).
    [CrossRef]
  3. C. A. McCarthy, J. Opt. Soc. Am. 45, 1087 (1955).
    [CrossRef]
  4. K. Levenberg, Quart. Appl. Math. 2, 164 (1944).
  5. A. Girard, Rev. d’Optique 37, 225, 397 (1958).
  6. C. G. Wynne, Proc. Phys. Soc. London 73, 777 (1959).
    [CrossRef]
  7. J. Meiron, J. Opt. Soc. Am. 55, 1105 (1965).
    [CrossRef]
  8. D. P. Feder, J. Opt. Soc. Am. 47, 904 (1957).
  9. C. G. Wynne, P. Wormell, Appl. Opt. 2, 1233 (1963).
    [CrossRef]
  10. H. H. Hopkins, Proceedings of the International Conference on Lens Design with Large Computers, (Institute of Optics, Rochester, 1967).
  11. H. A. Unvala, Ref. 10.
  12. R. J. Pegis, T. P. Vogl, A. K. Rigler, R. Walters, Ref. 10.

1965 (1)

1963 (1)

1962 (1)

1961 (1)

E. Glatzel, Optik 18, 577 (1961).

1959 (1)

C. G. Wynne, Proc. Phys. Soc. London 73, 777 (1959).
[CrossRef]

1958 (1)

A. Girard, Rev. d’Optique 37, 225, 397 (1958).

1957 (1)

D. P. Feder, J. Opt. Soc. Am. 47, 904 (1957).

1955 (1)

1944 (1)

K. Levenberg, Quart. Appl. Math. 2, 164 (1944).

Feder, D. P.

D. P. Feder, J. Opt. Soc. Am. 47, 904 (1957).

Girard, A.

A. Girard, Rev. d’Optique 37, 225, 397 (1958).

Glatzel, E.

E. Glatzel, Optik 18, 577 (1961).

Hopkins, H. H.

H. H. Hopkins, Proceedings of the International Conference on Lens Design with Large Computers, (Institute of Optics, Rochester, 1967).

Hopkins, R. E.

Levenberg, K.

K. Levenberg, Quart. Appl. Math. 2, 164 (1944).

McCarthy, C. A.

Meiron, J.

Pegis, R. J.

R. J. Pegis, T. P. Vogl, A. K. Rigler, R. Walters, Ref. 10.

Rigler, A. K.

R. J. Pegis, T. P. Vogl, A. K. Rigler, R. Walters, Ref. 10.

Spencer, G.

Unvala, H. A.

H. A. Unvala, Ref. 10.

Vogl, T. P.

R. J. Pegis, T. P. Vogl, A. K. Rigler, R. Walters, Ref. 10.

Walters, R.

R. J. Pegis, T. P. Vogl, A. K. Rigler, R. Walters, Ref. 10.

Wormell, P.

Wynne, C. G.

C. G. Wynne, P. Wormell, Appl. Opt. 2, 1233 (1963).
[CrossRef]

C. G. Wynne, Proc. Phys. Soc. London 73, 777 (1959).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Optik (1)

E. Glatzel, Optik 18, 577 (1961).

Proc. Phys. Soc. London (1)

C. G. Wynne, Proc. Phys. Soc. London 73, 777 (1959).
[CrossRef]

Quart. Appl. Math. (1)

K. Levenberg, Quart. Appl. Math. 2, 164 (1944).

Rev. d’Optique (1)

A. Girard, Rev. d’Optique 37, 225, 397 (1958).

Other (3)

H. H. Hopkins, Proceedings of the International Conference on Lens Design with Large Computers, (Institute of Optics, Rochester, 1967).

H. A. Unvala, Ref. 10.

R. J. Pegis, T. P. Vogl, A. K. Rigler, R. Walters, Ref. 10.

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Figures (13)

Fig. 1
Fig. 1

Typical flow diagram for damped least squares.

Fig. 2
Fig. 2

General flow diagram for adaptive automatic correction.

Fig. 3
Fig. 3

Significance of factors x.

Fig. 4
Fig. 4

Twisted valley—rapid convergence of the adaptive method.

Fig. 5
Fig. 5

Flow diagram for adaptive control.

Fig. 6
Fig. 6

The effect of ill-conditioned equations.

Fig. 7
Fig. 7

Reduction of factors for a square matrix of order 2.

Fig. 8
Fig. 8

Stagnation points in the DLS method.

Fig. 9
Fig. 9

Distagon with a thick lens.

Fig. 10
Fig. 10

From triplet to planar.

Fig. 11
Fig. 11

From triplet to a new type.

Fig. 12
Fig. 12

From plane parallel plates to triplet.

Fig. 13
Fig. 13

Course of correction from plane parallel plates to triplet.

Tables (2)

Tables Icon

Table I Comparison of Basic Aspects of the Damped Least Squares Method with the Adaptive Method of Glatzel

Tables Icon

Table II Reduction of the Largest Factor (Schematic)

Equations (24)

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( x ) T ( x ) = min ,
( A ) ( x ) = ( f s - f o )
( x ) T ( x ) + ( λ ) T [ ( A ) ( x ) + ( f o - f s ) ] = min ,
( x ) = ( A ) T [ ( A ) ( A ) T ] - 1 ( f s - f o ) .
n m
- x + 10 y = 3 , x + 10 y = 1 , x = 2 ,
[ ( A ) T ( A ) ] = ( 3 0 0 200 ) .
[ ( A ) T ( A ) + K ( I ) ] = ( 103 0 0 300 ) ,
x + 10 y = 3 - x + 10 y = 1 , x = 2 ,
10 y = 3 , 10 y = 1 , x = 2 ,
- x + 10 y = 3 = f s - f o , x = 2 = f s - f o ,
- 0.5 x + 10 y = f 1 - f o = f 1 - ( - 3 ) , x = f 1 - f o = f 1 - ( - 2 ) ,
- x + 10 y = f 1 - f o =     - 2 - ( - 3 ) = 1 , x = f 1 - f o = - 1.5 - ( - 2 ) = 0.5.
x 1 = x o - Δ x 1 = 2 - 0.5 = 1.5 , y 1 = y o - Δ y 1 = 0.5 - 0.15 = 0.35.
x 2 = x 1 + Δ x = 0.5 + 0.5 = 1.0 , c = x 1 / Δ x = 1.5 / 0.5 = 3 , y 2 = y 1 + ( y 1 / c ) = 0.125 + 0.1167 = 0.2417.
10 y = f 2 - f o = f 2 + 3 , x = f 2 - f o = f 2 + 2 ,
- x + 10 y = f 2 - f 1 = - 0.583 + 2 = 1.417 , x = f 2 - f 1 = - 1 + 1.5 = 0.5.
x 2 = x 1 - Δ x 2 = 1.5 - 0.5 = 1.0 , y 2 = y 1 - Δ y 2 = 0.35 - 0.1917 = 0.1583.
x 3 = x 2 + 0.5 = 1.5 , c = 2 , y 3 = y 2 + ( y 2 / c ) = 0.3209.
x 5 = x 4 + x 4 = 2.0 + 0 = 2.0 , y 5 = y 4 + y 4 = 0.2750 - 0.1250 = 0.15.
f 5 = 0.75 x + 10 y - 3 = 1.5 + 1.5 - 3 = 0 , f 5 = x - 2 = 2 - 2 = 0.
- x + 10 y = f 5 - f 4 = 0 - 1.250 = - 1.250 , x = f 5 - f 4 = 0.
x 5 = x 4 - Δ x 5 = 0 , y 5 = y 4 - Δ y 5 = 0 ,
0.75 x + 10 y = 3 , x = 2 ,

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