Abstract

Formulas are presented for the computation of derivatives of the optical aberration functions S, T, V, W, and K with respect to the surface parameters for symmetrical optical systems. The important case of conic-section surfaces of revolution is considered separately. The formulas are suitable for use in computer programs for automatic computation of optical aberration coefficients.

© 1985 Optical Society of America

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References

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  1. T. B. Andersen, “Automatic Computation of Optical Aberration Coefficients,” Appl. Opt. 19, 3800 (1980).
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  3. T. B. Andersen, “Automatic Computation of Optical Focal Surfaces,” Appl. Opt. 20, 2754 (1981).
    [CrossRef] [PubMed]
  4. T. B. Andersen, “Optical Aberration Functions: Computation of Caustic Surfaces and Illuminance in Symmetrical Systems,” Appl. Opt. 20, 3723 (1981).
    [CrossRef] [PubMed]
  5. S. C. Tam, G. D. W. Lewis, S. Doric, D. Heshmaty-Manesh, “Diffraction Analysis of Rotationally Symmetric Optical Systems Using Computer-Generated Aberration Polynomials,” Appl. Opt. 22, 1181 (1983).
    [CrossRef] [PubMed]
  6. J. Barcala, M. Alvarez-Claro, F. Garralón, “Sensitivity of Optical Systems to Changes in Curvatures. I: Finite Sensitivity Coefficients,” Atti Fond. Giorgio Ronchi 36, 348 (1981).
  7. T. B. Andersen, “Optical Aberration Functions: Derivatives with Respect to Axial Distances for Symmetrical Systems,” Appl. Opt. 21, 1817 (1982).
    [CrossRef] [PubMed]
  8. T. B. Andersen, “Optical Aberration Functions: Chromatic Aberrations and Derivatives with Respect to Refractive Indices for Symmetrical Systems,” Appl. Opt. 21, 4040 (1982).
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  9. T. B. Andersen, A. Reiz, “Positions of Stars in Regions of 14 Southern Galactic Clusters,” Astron. Astrophys. Suppl. Ser. 53, 181 (1983).
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  11. G. W. Forbes, “Order Doubling in the Computation of Aberration Coefficients,” J. Opt. Soc. Am. 73, 782 (1983).
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  12. G. W. Forbes, “Chromatic Coordinates in Aberration Theory,” J. Opt. Soc. Am. A 1, 344 (1984).
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  13. G. W. Forbes, “Weighted Truncation of Power Series and the Computation of Chromatic Aberration Coefficients,” J. Opt. Soc. Am. A 1, 350 (1984).
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  14. G. W. Forbes, “Weighted Order Doubling in the Computation of Chromatic Aberration Coefficients,” J. Opt. Soc. Am. A 1, 974 (1984).
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  15. G. W. Forbes, “Order Doubling in the Determination of Characteristic Functions,” J. Opt. Soc. Am. 72, 1097 (1982).
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  16. G. W. Forbes, “New Class of Characteristic Functions in Hamiltonian Optics,” J. Opt. Soc. Am. 72, 1698 (1982).
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  17. G. W. Forbes, “Concatenation of Restricted Characteristic Functions,” J. Opt. Soc. Am. 72, 1702 (1982).
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1984 (3)

1983 (4)

1982 (5)

1981 (4)

1980 (1)

Alvarez-Claro, M.

J. Barcala, M. Alvarez-Claro, F. Garralón, “Sensitivity of Optical Systems to Changes in Curvatures. I: Finite Sensitivity Coefficients,” Atti Fond. Giorgio Ronchi 36, 348 (1981).

Andersen, T. B.

Andrews, M.

Barcala, J.

J. Barcala, M. Alvarez-Claro, F. Garralón, “Sensitivity of Optical Systems to Changes in Curvatures. I: Finite Sensitivity Coefficients,” Atti Fond. Giorgio Ronchi 36, 348 (1981).

Doric, S.

Forbes, G. W.

Garralón, F.

J. Barcala, M. Alvarez-Claro, F. Garralón, “Sensitivity of Optical Systems to Changes in Curvatures. I: Finite Sensitivity Coefficients,” Atti Fond. Giorgio Ronchi 36, 348 (1981).

Heshmaty-Manesh, D.

Lewis, G. D. W.

Reiz, A.

T. B. Andersen, A. Reiz, “Positions of Stars in Regions of 14 Southern Galactic Clusters,” Astron. Astrophys. Suppl. Ser. 53, 181 (1983).

Tam, S. C.

Appl. Opt. (7)

Astron. Astrophys. Suppl. Ser. (1)

T. B. Andersen, A. Reiz, “Positions of Stars in Regions of 14 Southern Galactic Clusters,” Astron. Astrophys. Suppl. Ser. 53, 181 (1983).

Atti Fond. Giorgio Ronchi (1)

J. Barcala, M. Alvarez-Claro, F. Garralón, “Sensitivity of Optical Systems to Changes in Curvatures. I: Finite Sensitivity Coefficients,” Atti Fond. Giorgio Ronchi 36, 348 (1981).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

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Tables (3)

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Table I Optical Specifications for the Danish 1.5-m Ritchey-Chrétien Telescope

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Table II Computed Aberration Coefficients for the 1.5-m Telescope

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Table III Surface-Derivatives of the Aberration Coefficients for the 1.5-m Ritehey-Chrétien Telescope

Equations (69)

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( x i y i ξ i η i ) = [ S i ( ρ 0 , ψ 0 , κ 0 ) T i ( ρ 0 , ψ 0 , κ 0 ) V i ( ρ 0 , ψ 0 , κ 0 ) W i ( ρ 0 , ψ 0 , κ 0 ) ] ( x 0 y 0 ξ 0 η 0 ) .
ρ i = x i 2 + y i 2 ; ψ i = ξ i 2 + η i 2 ; κ i = x i ξ i + y i η i .
Q i = ( S i T i V i W i ) ,
Q i + 1 = J i B i Q i , Q 1 = J 0 ,
J i = ( 1 d i 0 1 ) ,
B i = ( 1 0 0 1 ) + ( μ i N i N i + 1 1 ) [ 2 f i df i d ρ f i ( 1 + 2 f i df i d ρ ) 2 df i d ρ 1 + 2 f i df i d ρ ] E + ( μ i N i N i + 1 1 ) C i .
x i 2 + y i 2 ρ i = ρ i + ψ i f i ( ρ i ) 2 + 2 κ i f i ( ρ i ) ,
cos θ i = γ i N i { 1 2 [ κ i + ψ i f i ( ρ i ) ] df i d ρ } ,
cos θ i = ( 1 μ i 2 + μ i 2 cos θ i ) 1 / 2 ,
γ i = [ 1 + 4 ρ i ( df i d ρ ) 2 ] 1 / 2
N i = ( 1 + ξ i 2 + η i 2 ) 1 / 2 = ( 1 + ψ i ) 1 / 2 .
N i + 1 = μ i N i + γ i ( cos θ i μ i cos θ i ) .
H i = J Γ B Γ J Γ 1 B Γ 1 J i + 1 B i + 1 J i B i ,
H i = H i + 1 J i B i , H Γ + 1 = E .
Q Γ + 1 = H i + 1 Q i + 1 .
Q i + 1 p ij = J i B i p ij Q i = p ij ( μ i N i N i + 1 ) J i C i Q i + ( μ i N i N i + 1 1 ) J i C i p ij Q i .
ρ i p ij = 2 [ κ i + ψ i f i ( ρ i ) ] f i p ij ,
γ i p ij = 4 γ i 3 [ ( κ i + ψ i f i ) ( df i d ρ ) 2 f i p ij + ρ i d f i d ρ p ij d f i d ρ ] ,
cos θ i p ij = cos θ i γ i γ i p ij 2 N i γ i ψ i df i d ρ f i p ij 2 N i γ i ( κ i + ψ i f i ) p ij df i d ρ = 2 N i γ i 3 { 2 ( κ i + ψ i f i ) ( df i d ρ ) 2 [ 1 2 ( κ i + ψ i f i ) df i d ρ ] + ψ i df i d ρ [ 1 + 4 ρ i ( df i d ρ ) 2 ] } f i p ij 2 N i γ i 3 ( 2 ρ i df i d ρ + κ i + ψ i f i ) p ij d f i d ρ ,
cos θ i p ij = μ i 2 cos θ i cos θ i cos θ i p ij ,
μ i N i N i + 1 p ij = μ i N i N i + 1 2 N i + 1 p ij = μ i N i N i + 1 2 ( cos θ i μ i cos θ i ) ( γ i p ij μ i γ i cos θ i cos θ i p ij ) .
C i p ij = [ 2 df i d ρ ( 1 + 4 f i df i d ρ ) 0 2 df i d ρ ] f i p ij + ( 2 f i 2 f i 2 2 2 f i ) p ij df i d ρ .
Q Γ + 1 p ij = p ij ( H i + 1 Q i + 1 ) = H i + 1 p ij Q i + 1 + H i + 1 Q i + 1 p ij = ( H i + 1 ρ i + 1 ρ i + 1 p ij + H i + 1 ψ i + 1 ψ i + 1 p ij + H i + 1 κ i + 1 κ i + 1 p ij ) Q i + 1 + H i + 1 Q i + 1 p ij .
[ φ 1 , φ 2 , φ 3 ] = det ( φ 1 , φ 2 , φ 3 ) ( ρ 0 , ψ 0 , κ 0 ) .
Q Γ + 1 p ij = [ H i + 1 , ψ i + 1 , κ i + 1 ] ρ i + 1 p ij + [ ρ i + 1 , H i + 1 , κ i + 1 ] ψ i + 1 p ij + [ ρ i + 1 , ψ i + 1 , H i + 1 ] κ i + 1 p ij [ ρ i + 1 , ψ i + 1 , κ i + 1 ] Q i + 1 + H i + 1 Q i + 1 p ij .
( ρ i + 1 κ i + 1 κ i + 1 ψ i + 1 ) = Q i + 1 ( ρ 0 κ 0 κ 0 ψ 0 ) Q i + 1 T ,
p ij ( ρ i + 1 κ i + 1 κ i + 1 ψ i + 1 ) = Q i + 1 p ij ( ρ 0 κ 0 κ 0 ψ 0 ) Q i + 1 T + Q i + 1 ( ρ 0 κ 0 κ 0 ψ 0 ) Q i + 1 T p ij ,
f i ( ρ i ) = j = 1 a ij ρ i j ,
df i d ρ = j = 1 j a ij ρ i j 1 .
f i a ij = df i d ρ ρ i a ij + ρ i j ,
f i a ij = ρ i j 1 2 ( κ i + ψ i f i ) df i ρ = N i γ i cos θ i ρ i j .
a ij ( df i d ρ ) = d 2 f i d ρ 2 ρ i a ij + j ρ i j 1 = 2 N i γ i cos θ i ( κ i + ψ i f i ) d 2 f i d ρ 2 ρ i j + j ρ i j 1 .
f i ( ρ ) = R i 1 + b i [ 1 ( 1 1 + b i R i 2 ρ ) 1 / 2 ] ;
f i ( ρ ) = R i j = 1 ( 2 j 2 ) ! 2 2 j 1 j ! ( j 1 ) ! ( 1 + b i ) j 1 R i 2 j ρ j .
f ( ρ ) = a 1 ρ + a 2 ρ 2 + a 3 ρ 3 +
f ( ρ ) = 1 2 R ρ + 1 + b 8 R 3 ρ 2 + ( 1 + b ) 2 16 R 5 ρ 3 + ,
R = 1 2 a 1 , b = a 2 a 1 3 1 .
f i R i = 1 κ i R i [ ( 1 κ i R i ) 2 ( 1 + b i + ψ i ) ρ i R i 2 ] 1 / 2 1 + b i + ψ i ,
df i d ρ = 1 2 [ R i ( 1 + b i ) f i ] .
ρ i R i 2 = f i R i [ 2 ( 1 + b i ) f i R i ] ,
γ i = 1 ( 1 + b i ) f i R i ( 1 b i ρ i R i 2 ) 1 / 2 ,
cos θ i N i = ( ( 1 κ i R i ) 2 ( 1 + b i + ψ i ) ρ i R i 2 1 b i ρ i R i 2 ) 1 / 2 .
f i R i = f i R i ( 1 + b i + ψ i ) f i R i ( 1 κ i R i ) ,
f i b i = ½ f i f i R i .
R i df i d ρ = 2 ( df i d ρ ) 2 ψ i f i R i ( 1 κ i R i ) ( 1 + b i + ψ i ) f i R i ( 1 κ i R i ) ,
b i df i d ρ = ½ f i R i df i d ρ + ( df i d ρ ) 2 f i .
f i c i = R i 2 f i R i ,
c i d f i d ρ = R i 2 R i d f i d ρ .
K ( ρ 0 , ψ 0 , κ 0 ) = i = 0 Γ K i = i = 0 Γ d i + ( μ i N i + 1 N i 1 ) f i μ 0 μ i N i + 1 .
K i p ij = ( μ i N i + 1 N i 1 ) f i p ij ( d i f i ) N i + 1 N i + 1 p ij μ 0 μ i N i + 1 .
K p ij = K i p ij + n = i + 1 Γ K n p ij = K i p ij + n = i + 1 Γ ( K n ρ i + 1 + ρ i + 1 p ij + K n ψ i + 1 + ψ i + 1 p ij + K n κ i + 1 + κ i + 1 p ij ) = K i p ij + [ K i + 1 , ψ i + 1 , κ i + 1 ] ρ i + 1 p ij + [ ρ i + 1 , K i + 1 , κ i + 1 ] ψ i + 1 p ij + [ ρ i + 1 , ψ i + 1 , K i + 1 ] κ i + 1 p ij [ ρ i + 1 , ψ i + 1 , κ i + 1 ] ,
K i + 1 = n = i + 1 Γ K n ,
K 0000 = k = 0 Γ d i μ 0 μ i
S 3 c 1
S 3 c 2
S 3 b 1
S 3 b 2
T 3 c 1
T 3 c 2
T 3 b 1
T 3 b 2
V 3 c 1
V 3 c 2
V 3 b 1
V 3 b 2
W 3 c 1
W 3 c 2
W 3 b 1
W 3 b 2

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