Abstract

This paper offers an alternative to current practice in the specification of aspheric optical surfaces. Spline functions, a numerical analog of a flexible draftsman’s curve, are used to represent a rotationally symmetric aspheric surface. We develop the interpolation formulas pertinent to ray tracing and apply an optimization procedure to design a simple Schmidt system as an illustration. We conclude by showing that the new formulation seems to be more compatible with the optimization process than the usual polynomial aspheric.

© 1971 Optical Society of America

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References

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  1. J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  3. D. S. Grey, J. Opt. Soc. Amer. 53, 672, 677 (1963).
    [CrossRef]
  4. E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

1963

D. S. Grey, J. Opt. Soc. Amer. 53, 672, 677 (1963).
[CrossRef]

Ahlberg, J. H.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Grey, D. S.

D. S. Grey, J. Opt. Soc. Amer. 53, 672, 677 (1963).
[CrossRef]

Isaacson, E.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Keller, H. B.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Nilson, E. N.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Walsh, J. L.

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

J. Opt. Soc. Amer.

D. S. Grey, J. Opt. Soc. Amer. 53, 672, 677 (1963).
[CrossRef]

Other

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications (Academic, New York, 1967).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

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Figures (4)

Fig. 1
Fig. 1

Sample points x0 to xn and their spline interpolation.

Fig. 2
Fig. 2

Two steps in Newton’s method of convergence on the intersection of a skew ray with the surface.

Fig. 3
Fig. 3

A pathologic case of convergence failure.

Fig. 4
Fig. 4

Illustrating the localized effect of parameter perturbation.

Equations (10)

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S ( ρ ) = M i - 1 ( ρ i - ρ ) h i + M i ( ρ - ρ i - 1 ) h i ,             h i = ρ i - ρ i - 1 .
S ( ρ ) = M i - 1 ( ρ i - ρ ) 3 6 h i + M i ( ρ - ρ i - 1 ) 3 6 h i + ( x i - 1 - M i - 1 h i 2 6 ) ( ρ i - ρ ) h i + ( x i - M i h i 2 6 ) ( ρ - ρ i - 1 ) h i
S ( ρ ) = - M i - 1 ( ρ i - ρ ) 2 2 h i + M i ( ρ - ρ i - 1 ) 2 h i + x i - x i - 1 h i - M i - M i - 1 6 h i .
S ( ρ i ) = h i M i - 1 6 + h i M i 3 + x i - x i - 1 h i
S ( ρ i ) = - h i + 1 M i 3 - h i + 1 M i + 1 6 + x i + 1 - x i h i + 1 .
h i 6 M i - 1 + h i + h i + 1 3 M i + h i + 1 M i + 1 6 = x i + 1 - x i h i + 1 - x i - x i - 1 h i ,             i = 1 , 2 , , n - 1.
[ 2 λ 0 μ 1 2 λ 1 μ 2 2 λ 2 · · · · μ n - 1 2 λ n - 1 μ n 2 ] [ M 0 M 1 M 2 · · · · M n - 1 M n ] = [ d 0 d 1 d 2 · · · · d n - 1 d n ] ,
λ i = h i + 1 h i + h i + 1 , μ i = 1 - λ i , d i = 6 [ ( x i - 1 - x i h i + 1 - x i - x i - 1 h i ) / ( h i + h i + 1 ) ] ,             i = 1 , 2 , , n - 1 , λ 0 = 1 , μ n = 1 , d 0 = 6 h 1 [ x 1 - x 0 h 1 ] , d n = 6 h n [ x n - x n - x n - 1 h n ] .
P = R 0 + α d ,
( P - Q j ) · N j = 0

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