Abstract

A new method for automatic lens design is described. Instead of finding the approximate solution to the system of simultaneous nonlinear equations, the solution is found to the system of simultaneous nonlinear inequalities. Thus, the aberrations are driven into the domains which have been set up around target values. The bandwidth of the domain is the allowable tolerance associated with each aberration. A relaxation method is used to solve the system of linearized inequalities and iterative cycles are used to find the solution to the system of nonlinear ones. This method allows a high degree of control of individual performance functions. Mechanical conditions can be treated in the same way as aberrations. The program was written for the IBM 7090 computer and satisfactory designs were obtained in general within a few minutes for gauss-type photographic lenses.

© 1966 Optical Society of America

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References

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  1. D. P. Feder, J. Opt. Soc. Am. 52, 177 (1962).
    [Crossref] [PubMed]
  2. J. Merion and H. M. Loebenstein, J. Opt. Soc. Am. 47, 1104 (1957).
    [Crossref]
  3. B. Brixner, Appl. Opt. 2, 1281 (1963).
    [Crossref]
  4. C. G. Wynne and P. M. J. H. Wormell, Appl. Opt. 2, 1233 (1963).
    [Crossref]
  5. R. E. Hopkins and D. P. Feder, Appl. Opt. 2, 1227 (1963).
    [Crossref]
  6. G. H. Spencer, Appl. Opt. 2, 1257 (1963).
    [Crossref]
  7. R. E. Hopkins and G. Spencer, J. Opt. Soc. Am. 52, 172 (1962).
    [Crossref] [PubMed]
  8. A. V. Fiacco, Operations. Res. 9, 184 (1961).
    [Crossref]
  9. T. Suzuki and et al., Proceedings of the Conference on Photographic and Spectroscopic Optics (Tokyo and Kyoto, 1964), [J. Appl. Phys. (Japan) Suppl. 4, 74 (1965)].
  10. S. Agmon, Can. J. Math. 6, 382 (1954).
    [Crossref]
  11. Yu. I. Merzlyakov, USSR Computational Math. Math. Phys. 3, 504 (1964).
  12. D. P. Feder, Appl. Opt. 2, 1209 (1963).
    [Crossref]
  13. J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33 (1955).
    [Crossref]
  14. D. P. Feder, J. Opt. Soc. Am. 47, 902 (1957).
    [Crossref]
  15. H. W. Kuhn and A. W. Tucker, Annals of Mathematical Studies (Princeton University Press, Princeton, New Jersey, 1956), No. 38, p. 99.

1964 (1)

Yu. I. Merzlyakov, USSR Computational Math. Math. Phys. 3, 504 (1964).

1963 (5)

1962 (2)

1961 (1)

A. V. Fiacco, Operations. Res. 9, 184 (1961).
[Crossref]

1957 (2)

1955 (1)

J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33 (1955).
[Crossref]

1954 (1)

S. Agmon, Can. J. Math. 6, 382 (1954).
[Crossref]

Agmon, S.

S. Agmon, Can. J. Math. 6, 382 (1954).
[Crossref]

Brixner, B.

Chernoff, H.

J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33 (1955).
[Crossref]

Crockett, J. B.

J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33 (1955).
[Crossref]

Feder, D. P.

Fiacco, A. V.

A. V. Fiacco, Operations. Res. 9, 184 (1961).
[Crossref]

Hopkins, R. E.

Kuhn, H. W.

H. W. Kuhn and A. W. Tucker, Annals of Mathematical Studies (Princeton University Press, Princeton, New Jersey, 1956), No. 38, p. 99.

Loebenstein, H. M.

Merion, J.

Merzlyakov, Yu. I.

Yu. I. Merzlyakov, USSR Computational Math. Math. Phys. 3, 504 (1964).

Spencer, G.

Spencer, G. H.

Suzuki, T.

T. Suzuki and et al., Proceedings of the Conference on Photographic and Spectroscopic Optics (Tokyo and Kyoto, 1964), [J. Appl. Phys. (Japan) Suppl. 4, 74 (1965)].

Tucker, A. W.

H. W. Kuhn and A. W. Tucker, Annals of Mathematical Studies (Princeton University Press, Princeton, New Jersey, 1956), No. 38, p. 99.

Wormell, P. M. J. H.

Wynne, C. G.

Appl. Opt. (5)

Can. J. Math. (1)

S. Agmon, Can. J. Math. 6, 382 (1954).
[Crossref]

J. Opt. Soc. Am. (4)

Operations. Res. (1)

A. V. Fiacco, Operations. Res. 9, 184 (1961).
[Crossref]

Pacific J. Math. (1)

J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33 (1955).
[Crossref]

USSR Computational Math. Math. Phys. (1)

Yu. I. Merzlyakov, USSR Computational Math. Math. Phys. 3, 504 (1964).

Other (2)

T. Suzuki and et al., Proceedings of the Conference on Photographic and Spectroscopic Optics (Tokyo and Kyoto, 1964), [J. Appl. Phys. (Japan) Suppl. 4, 74 (1965)].

H. W. Kuhn and A. W. Tucker, Annals of Mathematical Studies (Princeton University Press, Princeton, New Jersey, 1956), No. 38, p. 99.

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Figures (6)

Fig. 1
Fig. 1

Initial and final configurations of a gauss-type photographic lens under test. Initial: f/1.66, focal length 31.12, stop diameter 10.88. Final: f/1.75, focal length 32.34, stop diameter 10.88.

Fig. 2
Fig. 2

Spherical aberration of the initial and final designs.

Fig. 3
Fig. 3

Coma (16 deg) of initial and final designs.

Fig. 4
Fig. 4

Coma (24 deg) of initial and final designs.

Fig. 5
Fig. 5

Astigmatism of initial and final designs.

Fig. 6
Fig. 6

Distortion of initial and final designs.

Tables (4)

Tables Icon

Table I Initial configuration data for f/1.7 gauss-type photographic lens.

Tables Icon

Table II Configuration data for f/1.7 gauss-type photographic lens at the first solution.

Tables Icon

Table III Configuration data for f/1.7 gauss-type photographic lens at the second solution.

Tables Icon

Table IV Lower and upper boundary conditions and image errors at the initial, the first, and the second stages. No. 1, 2, ⋯, 26, 27, represent the boundaries and the image errors at the 1st stage and No. 1, 2, ⋯, 26, 27′ represent the boundaries and the image errors at the 2nd stage. The tolerances of 5% of bandwidths themselves were allowed at both sides of the respective boundaries.

Equations (26)

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f i ( x 1 , x 2 , , x n ) = μ i ,             i = 1 , 2 , , m ,
ϕ = i = 1 m w i 2 ( f i - μ i ) 2 ,
x * = ( x 1 * , x 2 * , , x n * )
ϕ * = i = 1 m ( w i δ i * ) 2 ,
max μ i Ω i δ i α i - β i ,             i = 1 , 2 , , m ,
α i f i ( x ) β i ,             i = 1 , 2 , , m .
f i ( x 1 , x 2 , , x n ) ,             i = 1 , 2 , , m .
α i f i ( x 1 , x 2 , , x n ) β i ,             i = 1 , 2 , , m ,
α i f i ( x 1 , x 2 , , x n ) β i
= i = 1 m i
α i [ grad f i ( x 0 ) ] ( x - x 0 ) + f i ( x 0 ) β i ,
S 0 = i = 1 m S i 0
α i [ grad f i ( x 0 ) ] ( x - x 0 ) + f i ( x 0 ) β i ,             i = 1 , 2 , , m .
α i [ grad f i ( x 1 ) ] ( x - x 1 ) + f i ( x 1 ) β i ,             i = 1 , 2 , , m .
x j = c j y j ,             j = 1 , 2 , , n ,
x = C y ,
α i [ grad f i ( x 0 ) ] C ( y - y 0 ) + f i ( x 0 ) β i ,             i = 1 , 2 , , m ,
a i = { [ grad f i ( x 0 ) ] C , i = 1 , 2 , , m , - [ grad f i ( x 0 ) ] C , i = m + 1 , m + 2 , , 2 m , b i = { f i ( x 0 ) - [ grad f i ( x 0 ) ] C y 0 - α i , i = 1 , 2 , , m , - f i ( x 0 ) + [ grad f i ( x 0 ) ] C y 0 + β i , i = m + 1 , m + 2 , , 2 m ,
T i ( y 1 , y 2 , , y n ) = j = 1 n a i j y j + b i 0 ,             i = 1 , 2 , , 2 m ,
T i ( y ) = a i y + b i 0 ,             i = 1 , 2 , , 2 m .
y ( k + 1 ) - y < y ( k ) - y .
v = i I λ i a i ,
0 < < 2 v 2 i I λ i T i ( y ( k ) ) ,
y ( k + 1 ) = y ( k ) + v ,             k = 0 , 1 , 2 , ,
y ( k + 1 ) - y 2 = ( y ( k ) - y ) + v 2 = y ( k ) - y 2 + 2 ( y ( k ) - y , v ) + 2 v 2 < y ( k ) - y 2 ,
2 ( y ( k ) - y , v ) + 2 v 2 < { 2 ( y ( k ) - y , i I λ i a i ) + i I λ i T i ( y ( k ) ) } = 2 { i I λ i [ T i ( y ( k ) ) - T i ( y ) ] - i I λ i T i ( y ( k ) ) } = - 2 i I λ i T i ( y ) 0 .