Abstract

Optical design may be thought of as an exercise in the solution of sets of simultaneous, nonlinear, algebraic equations with prescribed boundary conditions. Several methods for solving such sets of equations are discussed, and numerical examples are given for some of them. These methods are believed suitable for use with automatic computing machinery.

In the optical problem there are usually many more equations than variables, so that an exact solution is seldom possible. If, however, the quality of a system is specified by a merit function φ, then a set of parameters which make φ a minimum can be interpreted as a solution of the equations.

© 1957 Optical Society of America

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References

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  1. Augustin Cauchy, Compt. rend. 25, 536–538 (1847).
  2. H. B. Curry, Quart. Appl. Math. 2, 258–261 (1944).
  3. J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33–50 (1950).
    [CrossRef]
  4. M. L. Stein, J. Research Natl. Bur. Standards 48, 407–413 (1952).
    [CrossRef]
  5. A. I. Forsythe and G. E. Forsythe, Natl. Bur. Standards, Appl. Math. Ser. 39, 55–69 (1954).
  6. G. Black, Proc. Phys. Soc. (London) B68, 729–736 (1955).
  7. R. E. Hopkins and C. A. McCarthy, J. Opt. Soc. Am. 45, 363–365 (1955). Also a note on this paper by C. A. McCarthy, J. Opt. Soc. Am. 45, 1087 (1955).
    [CrossRef]
  8. S. Rosen and C. Eldert, J. Opt. Soc. Am. 44, 250–252 (1954).
    [CrossRef]
  9. K. Levenberg, Quart. Appl. Math. 2, 164–168 (1944).
  10. F. Wachendorf, Optik 12, 329–359 (1955).

1955 (3)

1954 (2)

A. I. Forsythe and G. E. Forsythe, Natl. Bur. Standards, Appl. Math. Ser. 39, 55–69 (1954).

S. Rosen and C. Eldert, J. Opt. Soc. Am. 44, 250–252 (1954).
[CrossRef]

1952 (1)

M. L. Stein, J. Research Natl. Bur. Standards 48, 407–413 (1952).
[CrossRef]

1950 (1)

J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33–50 (1950).
[CrossRef]

1944 (2)

H. B. Curry, Quart. Appl. Math. 2, 258–261 (1944).

K. Levenberg, Quart. Appl. Math. 2, 164–168 (1944).

1847 (1)

Augustin Cauchy, Compt. rend. 25, 536–538 (1847).

Augustin Cauchy,

Augustin Cauchy, Compt. rend. 25, 536–538 (1847).

Black, G.

G. Black, Proc. Phys. Soc. (London) B68, 729–736 (1955).

Chernoff, H.

J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33–50 (1950).
[CrossRef]

Crockett, J. B.

J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33–50 (1950).
[CrossRef]

Curry, H. B.

H. B. Curry, Quart. Appl. Math. 2, 258–261 (1944).

Eldert, C.

Forsythe, A. I.

A. I. Forsythe and G. E. Forsythe, Natl. Bur. Standards, Appl. Math. Ser. 39, 55–69 (1954).

Forsythe, G. E.

A. I. Forsythe and G. E. Forsythe, Natl. Bur. Standards, Appl. Math. Ser. 39, 55–69 (1954).

Hopkins, R. E.

Levenberg, K.

K. Levenberg, Quart. Appl. Math. 2, 164–168 (1944).

McCarthy, C. A.

Rosen, S.

Stein, M. L.

M. L. Stein, J. Research Natl. Bur. Standards 48, 407–413 (1952).
[CrossRef]

Wachendorf, F.

F. Wachendorf, Optik 12, 329–359 (1955).

Compt. rend. (1)

Augustin Cauchy, Compt. rend. 25, 536–538 (1847).

J. Opt. Soc. Am. (2)

J. Research Natl. Bur. Standards (1)

M. L. Stein, J. Research Natl. Bur. Standards 48, 407–413 (1952).
[CrossRef]

Natl. Bur. Standards, Appl. Math. Ser. (1)

A. I. Forsythe and G. E. Forsythe, Natl. Bur. Standards, Appl. Math. Ser. 39, 55–69 (1954).

Optik (1)

F. Wachendorf, Optik 12, 329–359 (1955).

Pacific J. Math. (1)

J. B. Crockett and H. Chernoff, Pacific J. Math. 5, 33–50 (1950).
[CrossRef]

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

G. Black, Proc. Phys. Soc. (London) B68, 729–736 (1955).

Quart. Appl. Math. (2)

H. B. Curry, Quart. Appl. Math. 2, 258–261 (1944).

K. Levenberg, Quart. Appl. Math. 2, 164–168 (1944).

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Figures (3)

Fig. 1
Fig. 1

This indicates the operation of the optimum gradient method for a case of two simple linear equations. An arbitrary initial guess is taken to be at the point P0. The successive approximations to the solution are at the points P1, P2, P3, ⋯. The solution for this case is located at the origin. The ellipses are the level lines of the function φ.

Fig. 2
Fig. 2

This illustrates a metric determined by a matrix B. The distance d from the center of the ellipse O to any point P on the ellipse is defined to be a constant.

Fig. 3
Fig. 3

This shows the complex level lines of a function φ arising from two extremely simple equations. The function has two minima, which lie on the dotted hyperbola |A|=0. It might be expected that a corresponding figure arising from an optical problem, and having, say, 50 variables, would be very intricate.

Tables (7)

Tables Icon

Table I This table lists the merit function φ and the third-order aberrations of a set of Cooke triplets. The first column refers to the design and the second to the merit function. The first row is a Cooke Triplet Patent No. 155640; the second row is an arbitrary perturbation. The remaining rows are successive improvements to reduce φ using the optimum gradient procedure. There are many more intermediate stages not shown in this table. The final design No. 3 appears to be as good or better than the original, and at this point the process was discontinued.

Tables Icon

Table II This table shows the lens parameters of three of the lenses listed in Table I. The perturbed lens is an arbitrary variation of the original patent No. 155640. The 3rd improvement was the best design reached starting with this perturbed lens. The original lens is shown for comparison. The headings c, t, N, D denote, respectively, curvature, separation, index, and dispersion. The first separation is the distance from the entrance pupil plane to the first surface.

Tables Icon

Table III(a) This shows a run of 9 steps using the optimum gradient method starting at x=1.00, y=−4.00. A large gain is made at step 1, but from then on the gains are rapidly decreasing. Even after 9 steps φ is not very close to φmin(0.022034).

Tables Icon

Table III(b) A second run starting with step 5 shows the effect of an acceleration. The * shows the acceleration step, which has clearly made a big improvement. At the end of 9 steps φ is practically at the minimum value of 0.022034.

Tables Icon

Table III(c) This shows the generalization of the Newton-Raphson procedure using the same initial point as in III(a). The convergence is much slower, and the last column shows that the iteration is approaching the curve |A|=0, which effectively halts the process.

Tables Icon

Table III(d) Using the Newton-Raphson method starting at step 5 of III(a). This is a much more favorable place to begin this method, and a considerable improvement is seen in φ.

Tables Icon

Table IV The optimum gradient method was applied to a doublet having 7 variables. The values of φ, G, and h are given at each step. Steps 12, 17, and 24 are acceleration steps and are indicated by asterisks.

Equations (54)

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f 1 = f 2 = = f M = 0
φ μ 1 f 1 2 + μ 2 f 2 2 + + μ M f M 2
φ = j = 1 M f j 2 .
G 1 2 grad φ 1 2 ( φ / x 1 , φ / x 2 , , φ / x N ) .
φ / x k = 2 j = 1 M f j ( f j / x k ) = 2 j = 1 M A j k f j
A j k f j / x k .
G = A T f .
L 2 φ x j x k G j x k .
G = 0
d φ = k = 1 N ( φ / x k ) d x k = 2 G · d s
d s ( d x 1 , d x 2 , , d x N ) .
Δ s = - h G
Δ φ - 2 h G 2 .
φ ( h ) < φ ( 0 ) .
f 1 = x - 10 y f 2 = x + 10 y } .
φ f 1 2 + f 2 2 = 2 x 2 + 200 y 2 .
( 2 0 0 200 )
y = 0.1 y x = x }
d φ = k = 1 N ( φ / x k ) d x k
k = 1 N ( d x k ) 2 = C .
k = 1 N g k d x k 2 = C ,
i = 1 N j = 1 N B i j d x i d x j = C
D = B - 1 G
B j k = 0             if             j k .
f j = f j ( 0 ) + k = 1 N A j k Δ x k + ,
f f ( 0 ) + A Δ x ,
Δ x = - A - 1 f ( 0 ) .
P 1 = P 0 + Δ x
G = A T f = 0.
P 1 = P 0 + h Δ x ,
G T Δ x = - ( A T f ) T ( A - 1 f ) = - f T f = - φ .
G = A T f A T f ( 0 ) + A T A Δ x .
Δ x = - ( A T A ) - 1 A T f ( 0 )
G j G j ( 0 ) + k = 1 N ( G j / x ) k Δ x k
Δ x = - L - 1 G ( 0 ) .
[ φ x k ( x j + Δ x j ) - φ x k ( x j ) ] Δ x j - 1 [ φ x j ( x k + Δ x k ) - φ x j ( x k ) ] Δ x k - 1 .
f 1 = ( x - 1 ) 2 + ( y - 1 ) 2 - 1 f 2 = y - ( x - 3 4 ) 2 + 1 φ = f 1 2 + f 2 2 .
A 11 = f 1 / x = 2 ( x - 1 ) , A 12 = f 1 / y = 2 ( y - 1 ) A 21 = f 2 / x = - 2 ( x - 3 4 ) , A 22 = f 2 / y = 1 } .
G x = A 11 f 1 + A 21 f 2 ,             G y = A 12 f 1 + A 22 f 2 ,
D = h 0 G 0 + h 1 G 1
Δ f j k = 1 N A j k y k ,
k = 1 N A j k y k = 0 ,
k = 1 N y k 2 = C
Δ φ = k = 1 N G k y k .
G ( G 1 , G 2 , , G N ) y ( y 1 , y 2 , , y N ) A j ( A j 1 , A j 2 , , A j N ) }
G · y ψ
A j · y = 0
d ψ = G · d y = 0
y · d y = 0
A 1 · d y = 0 A 2 · d y = 0 A M · d y = 0 } .
( G + λ 0 y + λ 1 A 1 + λ 2 A 2 + + λ M A M ) · d y = 0.
- λ 0 y = G + λ 1 A 1 + λ 2 A 2 + + λ M A M .
A 1 · A 1 λ 1 + A 1 · A 2 λ 2 + + A 1 · A M λ M = - A 1 · G A 2 · A 1 λ 1 + A 2 · A 2 λ 2 + + A 2 · A M λ M = - A 2 · G A M · A 1 λ 1 + A M · A 2 λ 2 + + A M · A M λ M = - A M · G } .
P 1 = P 0 + h 0 λ 0 y