Abstract

In manufacturing an aspheric lens or mirror, it is often convenient to generate a spherical surface as a starting point. The radius of the sphere should be such that the volume of glass which must be removed to achieve the final aspheric is a minimum. Equations for this best-fit sphere are developed and arranged in a form well suited for computer programming.

© 1966 Optical Society of America

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Figures (1)

Fig. 1
Fig. 1

Parameters used in best-fit sphere program.

Equations (27)

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x = C y 2 1 + ( 1 - C 2 b y 2 ) ½ + 1 5 a 2 k y 2 k ,
z = y 2 / h 2 ,
x 1 5 A k z k
A k = a 2 k h 2 k + ( r 0 b ) M k ( b h 2 r 0 2 ) k ,
X = R + ω - ( R 2 - y 2 ) ½ ω + R 1 5 M k ( y / R ) 2 k .
Δ x = X - x ω - B k z k ,
B k = A k - R M k ( h 2 / R 2 ) k .
k B k z k - 1 = 0.
V = 2 π y 1 y 2 Δ x y d y .
ω = B k z T k .
Δ x = B k ( z T k - z k ) .
V = π h 2 z 1 z 2 Δ x d z = π h 2 B k 1 2 ( z T k - z k ) d z = π h 2 B k [ Δ z z T k - z 2 k + 1 - z 1 k + 1 k + 1 ] ,
V k , R = π h 2 [ B k , R ( Δ z z T k - z 2 k + 1 - z 1 k + 1 k + 1 ) + k B k Δ z z T k - 1 z T , R ] = 0.
B k , R = ( 2 k - 1 ) M k ( h 2 / R 2 ) k .
v = h / R , C k = ( z 2 k + 1 - z 1 k + 1 ) / [ ( k + 1 ) Δ z ] , E k = A k / h ,
f ( v , z ) ( 2 k - 1 ) M k v 2 ( k - 1 ) ( z k - C k ) = 0 , g ( v , z ) k z k - 1 ( M k v 2 k - 1 - E k ) = 0 ,
x = y 2 / 2 r 0 .
Δ x = R + ω - ( R 2 - y 2 ) ½ - y 2 / 2 r 0 .
y T = ( R 2 - r 0 2 ) ½ .
ω = ( 1 / 2 r 0 ) ( R 2 + r 0 2 ) - R , ω / R = ( R - r 0 ) / r 0 .
V = 2 π [ 1 2 ( R + ω ) y ^ 2 + 1 3 ( R 2 - y ^ 2 ) ³ / - 1 3 R 3 - y ^ 4 / 8 r 0 ] .
V R = 0 = 2 π [ y ^ 2 2 r 0 R + R ( R 2 - y ^ 2 ) ½ - R 2 ] ,
R = r 0 + y ^ 2 / 4 r 0 ,
y T = y ^ 2 ( 1 + y ^ 2 8 r 0 2 ) ½ .
y T y ^ ,
y ^ 2 ( 2 ) ½ r 0 .
d f = f , z d z + f , v d v , d g = g , z d z + g , v d v ,

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