Abstract

A set of algorithms is developed to compute partial derivatives of transverse-ray errors with respect to the optical parameters. These are based on a generalized version of the Aldis equations. Results are given for a test case that demonstrates feasibility of application of these derivatives to lens optimization.

© 1976 Optical Society of America

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References

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  1. A. Cox, A System of Optical Design (Focal, London, New York, 1964).
  2. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).

Cox, A.

A. Cox, A System of Optical Design (Focal, London, New York, 1964).

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).

Other (2)

A. Cox, A System of Optical Design (Focal, London, New York, 1964).

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974).

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Figures (1)

FIG. 1
FIG. 1

Ray-trace parameters.

Tables (4)

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TABLE I Tessar—initial configuration. Focal length is 10.00 cm, relative aperture is 3.57, and coverage angle is ±19.8°.

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TABLE II Tessar ray—trace data.

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TABLE III Tessar—optimized. Focal length is 10.3297 cm, relative aperture is 3.69, coverage angle is 19.21°.

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TABLE IV Transverse image errors—optimized Tessar. (X0, Y0) ≡ entrance pupil coordinates, full field = 19.21°.

Equations (46)

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X s - X s - 1 l s - 1 = Y s - Y s - 1 m s - 1 = Z s - Z s - 1 + T s - 1 n s - 1 ,
l s - l s - 1 X s = m s - m s - 1 Y s = ρ s ( n s - 1 - n s ) .
b n n n X n μ n = s = 1 n [ X s ( b s n s μ s - b s - 1 n s - 1 μ s - 1 ) - Z s ( b s l s μ s - b s - 1 l s - 1 μ s - 1 ) ] - s = 1 n β s ( l s - l s - 1 ) ,
b n n n μ n ( Y n - Y n ) = s = 1 n [ Y s ( b s n s μ s - b s - 1 n s - 1 μ s - 1 ) - Z s ( b s m s μ s - b s - 1 m s - 1 μ s - 1 ) ] - s - 1 n β s ( m s - m s - 1 ) - b n Y n s = 1 n ( n s μ 2 - n s - 1 μ s - 1 ) ,
U s = μ n X s b n n n ( b s n s μ s - b s - 1 n s - 1 μ s - 1 ) - μ n Z s b n n n ( b s l s μ s - b s - 1 l s - 1 μ s - 1 ) - μ n β s b n n n ( l s - l s - 1 ) ,
V s = μ n Y s b n n n ( b s n s μ s - b s - 1 n s - 1 μ s - 1 ) - μ n Z s b n n n ( b s m s μ s - b s - 1 m s - 1 μ s - 1 ) - μ n β s b n n n ( m s - m s - 1 ) - μ n Y n n n ( n s μ s - n s - 1 μ s - 1 ) ,
X n = s = 1 N U s ,
Δ Y = Y n - Y n = s = 1 n V s .
U s P = μ n b n n n [ ( l s - 1 - l s ) β s P + X s n s - l s Z s μ s b s P - ( β s + b s Z s μ s ) l s P + b s X s μ s n s P + ( b s n s μ s - b s - 1 n s - 1 μ s - 1 ) X s P - ( b s l s μ s - b s - 1 l s - 1 μ s - 1 ) Z s P ] ,
V s P = μ n b b n n [ ( m s - 1 - m s ) β s P + Y s n s - m s Z s μ s b s P - ( β s + b s Z s μ s ) m s P + ( b s Y s μ s - b n Y n μ s ) n s P + ( b s n s μ s - b s - 1 n s - 1 μ s ) Y P - ( b s m s μ s - b s - 1 m s - 1 μ s - 1 ) Z s P ] .
X s P = l s - 1 n s - 1 Z s P
Y s P = m s - 1 n s - 1 Z s P .
β s ρ s = 0 ,             b s ρ s = β s ( μ s - 1 - μ s ) .
Z s ρ s = n s - 1 Z s ρ s μ s - 1 cos i s .
l s ρ s = Ψ s ( ρ s X s ρ s + X s [ 1 + ρ s 2 ( Z s / ρ s ) ] 1 - ρ s Z s ) - ρ s X s ( 1 - ρ s Z s ) n s σ ρ s ,
m s ρ s = Ψ s ( ρ s Y s ρ s + Y s [ 1 + ρ s 2 ( Z s / ρ s ) ] 1 - ρ s Z s ) - ρ s Y s ( 1 - ρ s Z s ) n s p s ,
n s ρ s = - Ψ s μ s cos r s [ ( l s X s + m s Y s ) ( 1 + ρ s 2 Z s ρ s ) + ρ s ( 1 - ρ s Z s ) ( l s X s ρ s + m s Y s ρ s ) ] ,
Ψ s = ( n s - 1 - n s ) / ( 1 - ρ s Z s ) .
β s T s - 1 = b s - 1 μ s - 1 ,
b s T s - 1 = μ s - 1 - μ s μ s - 1 ρ s b s - 1 ,
X s T s - 1 = l s - 1 n s - 1 ( 1 + Z s T s - 1 ) ,
Y s T s - 1 = m s - 1 n s - 1 ( 1 + Z s T s - 1 ) ,
Z s T s - 1 = ρ s ( X s l s - 1 + Y s m s - 1 ) μ s - 1 cos i s ,
I s T s - 1 = ρ s Ψ s ( X s T s - 1 + ρ s X s 1 - ρ s Z s Z s T s - 1 ) - ρ s X s 1 - ρ s Z s n s T s - 1 ,
m s T s - 1 = ρ s Ψ s ( Y s T s - 1 + ρ s Y s 1 - ρ s Z s Z s T s - 1 ) - ρ s Y s 1 - ρ s Z s n s T s - 1 ,
n s T s - 1 = - ρ s Ψ s μ s cos r s [ ρ s ( l s X s + m s Y s ) Z s T s - 1 + ( 1 - ρ s Z s ) ( l s X s T s - 1 + m s Y s T s - 1 ) ] .
l s P = X s ( n s - 1 - n s ) ρ s P + ρ s ( n s - 1 - n s ) X s P - ρ s X s n s P ,
m s P = Y s ( n s - 1 - n s ) p s P + ρ s ( n s - 1 - n s ) Y s P - ρ s Y s n s P ,
n s P = n s - n s - 1 n s - ρ s ( l s X s + m s Y s ) ( ( l s X s + m s Y s ) p s P s + ρ s n s - 1 ( l s l s - 1 + m s m s - 1 ) Z s P s ) .
Z s ρ s = Q s 2 n s - 1 H s ( 1 - κ ρ 2 Q 2 ) 1 / 2 [ n s - 1 - ρ s ( X s l s - 1 + Y s m s - 1 ) ] ,
H s = 1 + ( 1 - κ ρ 2 Q 2 ) 1 / 2 ,
ρ s ρ s = 1 ( 1 - κ ρ s 2 Q s 2 ) 3 / 2 + [ ρ s 3 κ s ( 1 - κ ρ 2 Q 2 ) 3 / 2 + 8 A + 24 B Q s 2 + 48 C Q s 4 + 80 D Q s 6 + 120 E Q s 8 ] ( X S X s ρ s + Y s Y s ρ s ) .
Z s T s - 1 = ρ s ( X s l s - 1 + Y s m s - 1 ) n s - 1 - ρ s ( X s l s - 1 + Y s m s - 1 ) ,
ρ s T s - 1 = ( ρ s 3 κ s ( 1 + κ s ρ s 2 Q s 2 ) 3 / 2 + 8 A + 24 B Q 2 + 48 C Q 4 + 80 D Q 6 + 120 E Q 8 ) ( X s X s T s - 1 + Y s Y s T s - 1 ) .
β s κ s = 0 ,             b s κ s = 0 , Z s κ s = ρ s 3 Q s 4 n s - 1 2 H 2 ( 1 - κ s ρ s 2 Q s 2 ) 1 / 2 [ n s - 1 - ρ s ( X s l s - 1 + Y s m s - 1 ) ] ,
ρ s κ s = ρ s 3 Q s 2 2 ( 1 - κ s ρ s 2 Q s 2 ) 3 / 2 .
β s A s = 0 ,             b s A s = 0 , Z s A s = Q s 4 n s - 1 n s - 1 - 4 A s ( X s 3 l s - 1 + Y s 3 m s - 1 ) ,
ρ s A s = 4 Q s 2 + 8 A s n s - 1 ( X s l s - 1 + Y s m s - 1 ) Z s A s .
β s B s = 0 ,             b s B s = 0 , Z s B s = Q s 6 n s - 1 n s - 1 - 6 B s ( X s 5 l s - 1 + Y s 5 m s - 1 ) ,
ρ s B s = 6 Q s 4 + 24 B Q s 2 n s - 1 ( X s l s - 1 + Y s m s - 1 ) Z s B s .
β s C s = 0 ,             b s C s = 0 , Z s C s = Q s 8 n s - 1 n s - 1 - 8 C s ( X s 7 l s - 1 + Y s 7 m s - 1 ) ,
ρ s C s = 8 Q s 6 + 48 C s Q s 4 n s - 1 ( X s l s - 1 + Y s m s - 1 ) Z s C s .
β s D s = 0 ,             b s D s = 0 , Z s D s = Q s 10 n s - 1 n s - 1 - 10 D s ( X s 9 l s - 1 + Y s 9 m s - 1 ) ,
ρ s D s = 10 Q s 8 + 80 D s Q s 6 n s - 1 ( X s l s - 1 + Y s m s - 1 ) Z s D s .
β s E s = 0 ,             b s E s = 0 , Z s E s = Q s 12 n s - 1 n s - 1 - 12 E s ( X s 11 l s - 1 + Y s 11 m s - 1 ) ,
ρ s E s = 12 Q s 10 + 120 E s Q s 8 n s - 1 ( X s l s - 1 + Y s m s - 1 ) Z s E s .