Abstract

Automatic computing methods are being increasingly applied to optical design, and the development of programs for this purpose forms an interesting chapter in optical history. Mathematically, the problem consists of solving sets of simultaneous nonlinear equations in a space of thirty or more variables limited by prescribed boundaries. Although these boundary conditions do not basically alter the mathematics, they greatly complicate the resulting program, and a specific example reveals how intricate such programs can become. The full impact of automatic methods has not yet been felt, but one result should be to shift the attention of the lens designer from the detailed correction of aberrations to the problem of securing a proper compromise between the system requirements and the conditions for sharp imagery so that better balanced optical instruments may result.

© 1963 Optical Society of America

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References

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  1. A. Cauchy, Compt. rend. 25, 536 (1847).
  2. D. E. Smith, Source Book in Mathematics (Macmillan, New York, 1929), pp. 576–579.
  3. E. T. Whittaker, G. Robinson, The Calculus of Observations (Blackie and SonsLondon, 1932), p. 214.
  4. K. Levenberg, Quart. Appl. Math. 2, 164 (1944).
  5. H. B. Curry, Quart. Appl. Math. 2, 258 (1944).
  6. J. G. Baker, Design and Development of an Automatically Focusing 40-inch f/5.0 Distortionless Telephoto and Related Lenses for High-altitude Aerial Reconnaissance, N.D.R.C., Section 16.1, Optical Instruments (1944).
  7. H. R. J. Grosch, J. Opt. Soc. Am. 35, 803 (1945).
  8. H. R. J. Grosch, J. Opt. Soc. Am. 39, 1059 (1949).
  9. D. P. Feder, B. F. Handy, Optical Ray Tracing Problems and The Card Programed Calculator, Paper presented to the Association for Computing Machinery, Rutgers Univ. (March 1950).
  10. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [CrossRef]
  11. M. L. Stein, J. Research Natl. Bur. Standards 48, 407 (1952).
    [CrossRef]
  12. M. R. Hestenes, E. Stiefel, J. Research Natl. Bur. Standards 49, 409 (1952).
    [CrossRef]
  13. J. G. Baker, Technical Reports 1–12, and a Supplement, prepared by the Perkin-Elmer Corp., under Air Force Contract AF 33(038)-10836 Declassified 1959 (issued at intervals from 1951–1955).
  14. O. N. Stavroudis, D. P. Feder, J. Opt. Soc. Am. 44, 163 (1954).
    [CrossRef]
  15. S. Rosen, C. Eldert, J. Opt. Soc. Am. 44, 250 (1954).
    [CrossRef]
  16. G. Black, Nature 175, 164 (1955); also, an expansion of this theme in Proc. Phys. Soc. (London) B68, 729 (1955).
    [CrossRef]
  17. J. B. Crockett, H. Chernoff, Pacific J. Math. 5, 33 (1955).
    [CrossRef]
  18. R. E. Hopkins, C. A. McCarthy, R. Walters, J. Opt. Soc. Am. 45, 363 (1955).
    [CrossRef]
  19. F. Wachendorf, Optik 12, 329 (1955).
  20. C. A. McCarthy, J. Opt. Soc. Am. 45, 1087 (1955).
    [CrossRef]
  21. D. P. Feder, J. Opt. Soc. Am. 47, 902 (1957).
    [CrossRef]
  22. D. P. Feder, J. Opt. Soc. Am. 47, 913 (1957).
    [CrossRef]
  23. J. Meiron, H. M. Loebenstein, J. Opt. Soc. Am. 47, 1104 (1957).
    [CrossRef]
  24. A. Girard, Rev. opt. 37, 225, 397 (1958).
  25. C. G. Wynne, Proc. Phys. Soc. (London) 73, 777 (1959).
    [CrossRef]
  26. M. Nunn, C. G. Wynne, Proc. Phys. Soc. (London) 74, 316 (1959).
    [CrossRef]
  27. J. Meiron, J. Opt. Soc. Am. 49, 293 (1959).
    [CrossRef]
  28. W. C. Davidon, Variable Metric Method for Minimization, Argonne National Laboratory, Rept. ANL-5990 (May1959).
    [CrossRef]
  29. H. D. Korones, R. E. Hopkins, J. Opt. Soc. Am. 49, 869 (1959).
    [CrossRef]
  30. J. Meiron, G. Volinez, J. Opt. Soc. Am. 50, 207 (1960).
    [CrossRef]
  31. W. P. Hennessy, G. H. Spencer, J. Opt. Soc. Am. 50, 494 (1960).
    [CrossRef]
  32. J. C. Holladay, Computer Design of Optical Lens Systems (IBM 704), Proc. of the 1960 Comp. Applications Symp. sponsored by Armour Research Foundation (Macmillan Co., New York, 1961).
  33. S. Kato, Science of Light (Tokyo) 10, 1 (1961).
  34. R. Hooke, T. A. Jeeves, J. Assoc. Computing Mach. 8, 212 (1961).
    [CrossRef]
  35. H. Slevogt, Optik 18, 571 (1961).
  36. E. Glatzel, Optik 18, 577 (1961).
  37. Proceedings of the Conference on Optical Instruments and Techniques, London 1961 (Chapman and Hall, London, 1962). (This contains a section on optical design techniques by various authors.)
  38. R. E. Hopkins, G. H. Spencer, J. Opt. Soc. Am. 52, 172 (1962).
    [CrossRef]
  39. D. P. Feder, J. Opt. Soc. Am. 52, 177 (1962).
    [CrossRef]
  40. R. R. Willey, Appl. Opt., 1, 368 (1962).
    [CrossRef]
  41. H. A. Spang, SIAM Rev. 4, (1962).
  42. R. E. Hopkins, J. Opt. Soc. Am. 52, 1218 (1962).
    [CrossRef]
  43. D. S. Grey, J. Opt. Soc. Am. 53, 672 (1963).
    [CrossRef]

1963 (1)

1962 (5)

1961 (4)

S. Kato, Science of Light (Tokyo) 10, 1 (1961).

R. Hooke, T. A. Jeeves, J. Assoc. Computing Mach. 8, 212 (1961).
[CrossRef]

H. Slevogt, Optik 18, 571 (1961).

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

1960 (2)

1959 (4)

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

M. Nunn, C. G. Wynne, Proc. Phys. Soc. (London) 74, 316 (1959).
[CrossRef]

J. Meiron, J. Opt. Soc. Am. 49, 293 (1959).
[CrossRef]

H. D. Korones, R. E. Hopkins, J. Opt. Soc. Am. 49, 869 (1959).
[CrossRef]

1958 (1)

A. Girard, Rev. opt. 37, 225, 397 (1958).

1957 (3)

1955 (5)

G. Black, Nature 175, 164 (1955); also, an expansion of this theme in Proc. Phys. Soc. (London) B68, 729 (1955).
[CrossRef]

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

R. E. Hopkins, C. A. McCarthy, R. Walters, J. Opt. Soc. Am. 45, 363 (1955).
[CrossRef]

F. Wachendorf, Optik 12, 329 (1955).

C. A. McCarthy, J. Opt. Soc. Am. 45, 1087 (1955).
[CrossRef]

1954 (2)

1952 (2)

M. L. Stein, J. Research Natl. Bur. Standards 48, 407 (1952).
[CrossRef]

M. R. Hestenes, E. Stiefel, J. Research Natl. Bur. Standards 49, 409 (1952).
[CrossRef]

1951 (1)

1949 (1)

H. R. J. Grosch, J. Opt. Soc. Am. 39, 1059 (1949).

1945 (1)

H. R. J. Grosch, J. Opt. Soc. Am. 35, 803 (1945).

1944 (2)

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

H. B. Curry, Quart. Appl. Math. 2, 258 (1944).

1847 (1)

A. Cauchy, Compt. rend. 25, 536 (1847).

Baker, J. G.

J. G. Baker, Design and Development of an Automatically Focusing 40-inch f/5.0 Distortionless Telephoto and Related Lenses for High-altitude Aerial Reconnaissance, N.D.R.C., Section 16.1, Optical Instruments (1944).

J. G. Baker, Technical Reports 1–12, and a Supplement, prepared by the Perkin-Elmer Corp., under Air Force Contract AF 33(038)-10836 Declassified 1959 (issued at intervals from 1951–1955).

Black, G.

G. Black, Nature 175, 164 (1955); also, an expansion of this theme in Proc. Phys. Soc. (London) B68, 729 (1955).
[CrossRef]

Cauchy, A.

A. Cauchy, Compt. rend. 25, 536 (1847).

Chernoff, H.

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

Crockett, J. B.

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

Curry, H. B.

H. B. Curry, Quart. Appl. Math. 2, 258 (1944).

Davidon, W. C.

W. C. Davidon, Variable Metric Method for Minimization, Argonne National Laboratory, Rept. ANL-5990 (May1959).
[CrossRef]

Eldert, C.

Feder, D. P.

Girard, A.

A. Girard, Rev. opt. 37, 225, 397 (1958).

Glatzel, E.

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

Grey, D. S.

Grosch, H. R. J.

H. R. J. Grosch, J. Opt. Soc. Am. 39, 1059 (1949).

H. R. J. Grosch, J. Opt. Soc. Am. 35, 803 (1945).

Handy, B. F.

D. P. Feder, B. F. Handy, Optical Ray Tracing Problems and The Card Programed Calculator, Paper presented to the Association for Computing Machinery, Rutgers Univ. (March 1950).

Hennessy, W. P.

Hestenes, M. R.

M. R. Hestenes, E. Stiefel, J. Research Natl. Bur. Standards 49, 409 (1952).
[CrossRef]

Holladay, J. C.

J. C. Holladay, Computer Design of Optical Lens Systems (IBM 704), Proc. of the 1960 Comp. Applications Symp. sponsored by Armour Research Foundation (Macmillan Co., New York, 1961).

Hooke, R.

R. Hooke, T. A. Jeeves, J. Assoc. Computing Mach. 8, 212 (1961).
[CrossRef]

Hopkins, R. E.

Jeeves, T. A.

R. Hooke, T. A. Jeeves, J. Assoc. Computing Mach. 8, 212 (1961).
[CrossRef]

Kato, S.

S. Kato, Science of Light (Tokyo) 10, 1 (1961).

Korones, H. D.

Levenberg, K.

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

Loebenstein, H. M.

McCarthy, C. A.

Meiron, J.

Nunn, M.

M. Nunn, C. G. Wynne, Proc. Phys. Soc. (London) 74, 316 (1959).
[CrossRef]

Robinson, G.

E. T. Whittaker, G. Robinson, The Calculus of Observations (Blackie and SonsLondon, 1932), p. 214.

Rosen, S.

Slevogt, H.

H. Slevogt, Optik 18, 571 (1961).

Smith, D. E.

D. E. Smith, Source Book in Mathematics (Macmillan, New York, 1929), pp. 576–579.

Spang, H. A.

H. A. Spang, SIAM Rev. 4, (1962).

Spencer, G. H.

Stavroudis, O. N.

Stein, M. L.

M. L. Stein, J. Research Natl. Bur. Standards 48, 407 (1952).
[CrossRef]

Stiefel, E.

M. R. Hestenes, E. Stiefel, J. Research Natl. Bur. Standards 49, 409 (1952).
[CrossRef]

Volinez, G.

Wachendorf, F.

F. Wachendorf, Optik 12, 329 (1955).

Walters, R.

Whittaker, E. T.

E. T. Whittaker, G. Robinson, The Calculus of Observations (Blackie and SonsLondon, 1932), p. 214.

Willey, R. R.

Wynne, C. G.

M. Nunn, C. G. Wynne, Proc. Phys. Soc. (London) 74, 316 (1959).
[CrossRef]

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

Appl. Opt. (1)

Compt. rend. (1)

A. Cauchy, Compt. rend. 25, 536 (1847).

J. Assoc. Computing Mach. (1)

R. Hooke, T. A. Jeeves, J. Assoc. Computing Mach. 8, 212 (1961).
[CrossRef]

J. Opt. Soc. Am. (18)

J. Research Natl. Bur. Standards (2)

M. L. Stein, J. Research Natl. Bur. Standards 48, 407 (1952).
[CrossRef]

M. R. Hestenes, E. Stiefel, J. Research Natl. Bur. Standards 49, 409 (1952).
[CrossRef]

Nature (1)

G. Black, Nature 175, 164 (1955); also, an expansion of this theme in Proc. Phys. Soc. (London) B68, 729 (1955).
[CrossRef]

Optik (3)

F. Wachendorf, Optik 12, 329 (1955).

H. Slevogt, Optik 18, 571 (1961).

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

Pacific J. Math. (1)

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

Proc. Phys. Soc. (London) (2)

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

M. Nunn, C. G. Wynne, Proc. Phys. Soc. (London) 74, 316 (1959).
[CrossRef]

Quart. Appl. Math. (2)

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

H. B. Curry, Quart. Appl. Math. 2, 258 (1944).

Rev. opt. (1)

A. Girard, Rev. opt. 37, 225, 397 (1958).

Science of Light (Tokyo) (1)

S. Kato, Science of Light (Tokyo) 10, 1 (1961).

SIAM Rev. (1)

H. A. Spang, SIAM Rev. 4, (1962).

Other (8)

W. C. Davidon, Variable Metric Method for Minimization, Argonne National Laboratory, Rept. ANL-5990 (May1959).
[CrossRef]

Proceedings of the Conference on Optical Instruments and Techniques, London 1961 (Chapman and Hall, London, 1962). (This contains a section on optical design techniques by various authors.)

J. C. Holladay, Computer Design of Optical Lens Systems (IBM 704), Proc. of the 1960 Comp. Applications Symp. sponsored by Armour Research Foundation (Macmillan Co., New York, 1961).

J. G. Baker, Design and Development of an Automatically Focusing 40-inch f/5.0 Distortionless Telephoto and Related Lenses for High-altitude Aerial Reconnaissance, N.D.R.C., Section 16.1, Optical Instruments (1944).

D. E. Smith, Source Book in Mathematics (Macmillan, New York, 1929), pp. 576–579.

E. T. Whittaker, G. Robinson, The Calculus of Observations (Blackie and SonsLondon, 1932), p. 214.

J. G. Baker, Technical Reports 1–12, and a Supplement, prepared by the Perkin-Elmer Corp., under Air Force Contract AF 33(038)-10836 Declassified 1959 (issued at intervals from 1951–1955).

D. P. Feder, B. F. Handy, Optical Ray Tracing Problems and The Card Programed Calculator, Paper presented to the Association for Computing Machinery, Rutgers Univ. (March 1950).

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Figures (2)

Fig. 1
Fig. 1

The entrance pupil plane showing a typical ray pattern. The coordinates of each ray are y = Vaη and z = where a is the semiaperture and V is the ratio by which the aperture is reduced in the meridian direction. The normalized ray coordinates are η and ζ.

Fig. 2
Fig. 2

The program guarantees that the upper and lower meridian rays are transmitted. It calculates the length L along the ray intercepted in each medium and establishes the implicit boundary conditions (b = Lt ≥ 0) where t is the minimum permitted for a particular medium.

Equations (66)

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Design A is preferred to design B . ( A B ) Neither design A nor design B is preferred . ( A = B ) Design A is less preferred than design B . ( A B ) }
A B B A ,             A = B B = A .
A B and B C A C , A = B and B = C A = C .
φ = μ 1 f 1 2 + μ 2 f 2 2 + + μ M f M 2 .
φ = f 1 2 + f 2 2 + + f M 2 .
f 1 = f 2 ( x 1 , x 2 , , x N ) f 2 = f 2 ( x 1 , x 2 , , x N ) f M = f M ( x 1 , x 2 , , x N ) } .
b 1 ( x 1 , x 2 , , x N ) 0 b 2 ( x 1 , x 2 , , x N ) 0 b T ( x 1 , x 2 , , x N ) 0 } .
A j k = f j / x k
A ( A 11 A 12 . A 1 N A 21 A 22 . A 2 N A M 1 A M 2 .. A M N )
φ x k = 2 ( f 1 f 1 x k f 2 f 2 x k + f 3 f 3 x k + + f M f M x k ) .
G k ½ ( φ / x k ) .
G k = f 1 A 1 k + f 2 A 2 k + + f M A M k .
x ( x 1 x 2 x N )             f ( f 1 f 2 f M )             G ( G 1 G 2 G N )
G = A T f ,
f = A ( x - x 0 ) + f 0 .
D 0 = - G 0 x 1 = x 0 + h 0 D 0 D 1 = - G 1 x 2 = x 1 + h 1 D 1 D j = - G j x j + 1 = x j + h j D j } .
D 0 T G 1 = D 1 T G 2 = D 2 T G 3 = = D j T G j + 1 = = 0
h j = G j 2 / H j 2 , where H j = A D j .
x j + 1 = x j + h j D j
A ( x j + 1 - x j ) = h j A D j .
A ( x j + 1 - x j ) = f j + 1 - f j .
f j + 1 - f j = h j A D j .
G j + 1 - G j = h j A T A D j .
D j T G j + 1 - D j T G j = h j D j T A T A D j = h j ( A D j ) T ( A D j ) .
D j T G j + 1 = 0 from ( 17 ) and D j T G j = - G j 2 from ( 16 ) } .
h j ( A D j ) T ( A D j ) = G j 2
D 0 = - G 0 x 1 = x 0 + h 0 D 0 D 1 = - G 1 + ( G 1 2 / G 0 2 ) D 0 x 2 = x 1 + h 1 D 1 D j = - G j + ( G j 2 / G j - 1 2 ) D j - 1 x j + 1 = x j + h j D j } .
D 0 T G 1 = D 1 T G 2 = D 2 T G 3 = = D j T G j + 1 = = 0.
f = Ax + f 0
G = A T Ax + G 0 .
x = - ( A T A ) - 1 G 0 .
ψ φ + p x T x φ + p ( x 1 2 + x 2 2 + + x N 2 ) .
grad ψ = grad φ + p grad ( x T x ) .
grad ( x T x ) = 2 x .
½ grad ψ = A T f + p x .
½ grad ψ = A T ( Ax + f 0 ) + p x = A T Ax + G 0 + p x .
A T Ax + p x = - G 0 ,
( A T A + p I ) x = - G 0 .
x = - ( A T A + p I ) - 1 G 0 ,
x = - ( p I ) - 1 G 0 = - p - 1 I G 0 = - p - 1 G 0 .
d 2 = x T x x 1 2 + x 2 2 + + x N 2 .
d 2 = m 11 x 1 2 + m 22 x 2 2 + + m N N x N 2 ,
d 2 = m 11 x 1 2 + 2 m 12 x 1 x 2 + + 2 m 1 N x 1 x N + m 22 x 2 2 + + 2 m 2 N x 2 x N + + + m N N x N 2 .
d 2 = x T Mx ,
x = - M - 1 G ,
M = A T A
M = A T A + p I
x 1 = ( c 1 - c 2 ) ( N - 1 ) x 2 = ( c 1 + c 2 ) } .
x = ( N - 1 1 - N 1 1 ) c , where x ( x 1 x 2 ) and c ( c 1 c 2 )
d 2 = x T x
d 2 = c T ( N - 1 1 - N 1 1 ) T ( N - 1 1 - N 1 1 ) c .
M = ( 1 + ( N - 1 ) 2 1 - ( N - 1 ) 2 1 - ( N - 1 ) 2 1 + ( N - 1 ) 2 ) .
b ( b 1 b 2 b T ) and the matrix B = ( B 11 B 12 B 1 N B 21 B 22 B 2 N B T 1 B T 2 B T N ) ,
Q u i e s c e n t those for which b j > , E f f e c t i v e those for which b j , F o r b i d d e n those for which b j < - .
Bx = b - b 0 = Δ b .
( A B ) x = ( Δ f Δ b ) .
( A B r B s ) x = ( Δ f Δ b r Δ b s ) .
B r x = Δ b r = 0.
( A r A q B r r B r q B s r B s q ) ( x r x q ) = ( Δ f 0 r Δ b s ) .
( I - A r B r r - 1 0 0 B r r - 1 0 0 - B s r B r r - 1 I )
( 0 A * I B r r - 1 B r q 0 B * ) ( x r x q ) = ( Δ f 0 r Δ b s ) ,
A * A q - A r B r r - 1 B r q and B * B s q - B s r B r r - 1 B r q .
A * x q = Δ f , B r r - 1 B r q x q = - x r , B * x q = Δ b s . } .
Bx = - b .
x = B T λ ,
B B T λ = - b .

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