Abstract

A modification introduced in the damped least-squares method automatically assigns a damping factor to each parameter in a manner that compensates for the relative sensitivities of the variables. The convergence rate of the lens-design process is thereby considerably improved. The operation of the program is described, in particular the use of Lagrangian undetermined multipliers for the control of boundary conditions.

© 1965 Optical Society of America

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References

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  1. A. Girard, Rev. Opt. 37, 225, 397 (1958).
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  11. K. Levenberg, Quart. Appl. Math. 2, 164 (1944).
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  14. M. Herzberger, Quart. Appl. Math. 7, 217 (1949).

1964 (1)

1963 (5)

1960 (1)

1959 (2)

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

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

1958 (1)

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

1957 (2)

1949 (1)

M. Herzberger, Quart. Appl. Math. 7, 217 (1949).

1944 (1)

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

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Figures (5)

F. 1
F. 1

Initial configuration of cathode-ray-tube transfer lens (see Table I).

F. 2
F. 2

Gradual improvement of transfer lens after 5 and 9 cycles, respectively. Aberration curves describe the intersection heights y of rays on image surface relative to principal ray vs relative incidence height of rays on entrance pupil. The three curves correspond to image heights y=0, y=8.8, and y=12.5 mm. Tangential—, sagittal – – –.

F. 3
F. 3

Lens configuration and aberration curves of intermediate design. The power of the second element had been reduced to zero by the computer and was consequently removed from the design.

F. 4
F. 4

Final design and aberration curves of transfer lens (see Table II) for an image position of −0.055 mm from Gaussian-image plane.

F. 5
F. 5

Modulation transfer function of final design (Fig. 4 and Table II) in tangential and sagittal directions.

Tables (2)

Tables Icon

Table I Initial design data of cathode-ray-tube transfer lens (f/2.0, E.F.L. 70.6, magnification ×0.5, object diam. 50.0. All dimensions in mm).

Tables Icon

Table II Final design data of cathode-ray-tube transfer lens (f/2.0, E.F.L. 70.6, magnification ×0.5, object diam. 50.0. All dimensions in mm).

Equations (16)

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f i = j = 1 n a i j x j + f i 0 i = 1 , 2 , , m ,
a i j = f i / x j ,
j = 1 n a i j x j + f i 0 = 0 i = 1 , 2 , , m .
ϕ = i = 1 m ( j = 1 n a i j x j + f i 0 ) 2 = i = 1 m f i 2 .
à A x = à f 0 ,
ψ = i = 1 m f i 2 + p 2 j = 1 n q j 2 x j 2 ,
q j = ( i = 1 m a i j 2 ) 1 2 .
à A x + p 2 Q x = à f 0 ,
x j ( Σ i a i j f i 0 ) / ( p 2 Σ i a i j 2 ) = ( ϕ 0 / x j ) / ( 2 p 2 Σ i a i j 2 ) .
x j = x j / p ( Σ a i j 2 ) 1 2 ,
x = T x ,
d 2 = x M x
φ k ( x 1 , x 2 , , x n ) = 0 k = 1 , 2 , , l .
C x = φ 0 ,
ψ ¯ ( x 1 , x 2 , , x n , λ 1 , λ 1 , , λ l ) = ψ ( x 1 , x 2 , , x n ) + k = 1 l λ k φ k ( x 1 , x 2 , , x n )
à A x + p Q x + C λ = à f 0 ,