Abstract

Simplified formulas are presented for the computation of derivatives of the optical aberration functions S,T,V,W, and K with respect to the axial surface distances for symmetrical optical systems. The formulas are well-suited for implementation in computer programs for automatic computation of optical aberration coefficients.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. P. Feder, Appl. Opt. 2, 1209 (1963).
    [CrossRef]
  2. C. G. Wynne, P. M. J. H. Wormell, Appl. Opt. 2, 1233 (1963).
    [CrossRef]
  3. G. H. Spencer, Appl. Opt. 2, 1257 (1963).
    [CrossRef]
  4. B. Brixner, Appl. Opt. 2, 1281 (1963).
    [CrossRef]
  5. D-Q. Su, Y-N. Wang, Acta Astronomica Sinica 15, 51 (1974);English translation in Chinese Astronomy and Astrophysics 2, 171 (1978).
  6. D. P. Feder, J. Opt. Soc. Am. 58, 1494 (1968).
    [CrossRef]
  7. F. D. Cruickshank, G. A. Hills, J. Opt. Soc. Am. 50, 379 (1960).
    [CrossRef]
  8. C. J. Woodruff, Opt. Acta 22, 933 (1975).
    [CrossRef]
  9. P. N. Robb, J. Opt. Soc. Am. 66, 1037 (1976).
    [CrossRef]
  10. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  11. P. J. Sands, Appl. Opt. 9, 828 (1970).
    [CrossRef] [PubMed]
  12. Ref. 10, Part 3.
  13. T. B. Andersen, Appl. Opt. 19, 3800 (1980).
    [CrossRef] [PubMed]
  14. T. B. Andersen, Appl. Opt. 20, 3263 (1981).
    [CrossRef] [PubMed]
  15. T. B. Andersen, Appl. Opt. 20, 2754 (1981).
    [CrossRef] [PubMed]
  16. T. B. Andersen, Appl. Opt. 20, 3723 (1981).
    [CrossRef] [PubMed]
  17. H. W. Epps, P. J. Peters, in Astronomical Observations with Television-Type Sensors, J. W. Glaspey, G. A. H. Walker, Eds. (U. British Columbia, Vancouver, 1973), pp. 415–432.

1981 (3)

1980 (1)

1976 (1)

1975 (1)

C. J. Woodruff, Opt. Acta 22, 933 (1975).
[CrossRef]

1974 (1)

D-Q. Su, Y-N. Wang, Acta Astronomica Sinica 15, 51 (1974);English translation in Chinese Astronomy and Astrophysics 2, 171 (1978).

1970 (1)

1968 (1)

1963 (4)

1960 (1)

Andersen, T. B.

Brixner, B.

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Cruickshank, F. D.

Epps, H. W.

H. W. Epps, P. J. Peters, in Astronomical Observations with Television-Type Sensors, J. W. Glaspey, G. A. H. Walker, Eds. (U. British Columbia, Vancouver, 1973), pp. 415–432.

Feder, D. P.

Hills, G. A.

Peters, P. J.

H. W. Epps, P. J. Peters, in Astronomical Observations with Television-Type Sensors, J. W. Glaspey, G. A. H. Walker, Eds. (U. British Columbia, Vancouver, 1973), pp. 415–432.

Robb, P. N.

Sands, P. J.

Spencer, G. H.

Su, D-Q.

D-Q. Su, Y-N. Wang, Acta Astronomica Sinica 15, 51 (1974);English translation in Chinese Astronomy and Astrophysics 2, 171 (1978).

Wang, Y-N.

D-Q. Su, Y-N. Wang, Acta Astronomica Sinica 15, 51 (1974);English translation in Chinese Astronomy and Astrophysics 2, 171 (1978).

Woodruff, C. J.

C. J. Woodruff, Opt. Acta 22, 933 (1975).
[CrossRef]

Wormell, P. M. J. H.

Wynne, C. G.

Acta Astronomica Sinica (1)

D-Q. Su, Y-N. Wang, Acta Astronomica Sinica 15, 51 (1974);English translation in Chinese Astronomy and Astrophysics 2, 171 (1978).

Appl. Opt. (9)

J. Opt. Soc. Am. (3)

Opt. Acta (1)

C. J. Woodruff, Opt. Acta 22, 933 (1975).
[CrossRef]

Other (3)

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Ref. 10, Part 3.

H. W. Epps, P. J. Peters, in Astronomical Observations with Television-Type Sensors, J. W. Glaspey, G. A. H. Walker, Eds. (U. British Columbia, Vancouver, 1973), pp. 415–432.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Notation introduced when changing the position of the entrance pupil from z = z0 to z = z 0 *.

Tables (1)

Tables Icon

Table I Distance-Derivatives of the Aberration Coefficients for the UV Camera

Equations (79)

Equations on this page are rendered with MathJax. Learn more.

d i = z i + 1 z i , i = 0 , 1 , Γ ,
z z i = f i ( x , y ) = f i ( x 2 + y 2 ) , i = 1 , Γ .
σ i = ( L i , M i , N i )
ξ i = L i / N i , η i = M i / N i
ρ i = x i 2 + y i 2 , ψ i = ξ i 2 + η i 2 , κ i = x i ξ 1 + y i η i .
[ x i y i ] = S i ( ρ 0 , ψ 0 , κ 0 ) [ x 0 y 0 ] + T i ( ρ 0 , ψ 0 , κ 0 ) [ ξ 0 η 0 ] ,
[ ξ i η i ] = V i ( ρ 0 , ψ 0 , κ 0 ) [ x 0 y 0 ] + W i ( ρ 0 , ψ 0 , κ 0 ) [ ξ 0 η 0 ] .
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 ) ] d f i d ρ } ,
cos θ i = ( 1 μ i 2 + μ i 2 cos 2 θ i ) 1 / 2 ,
γ i = [ 1 + 4 ρ i ( d f 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 ) ,
β i = 2 γ i N i + 1 ( cos θ i μ i cos θ i ) d f i d ρ ,
[ S i + 1 T i + 1 V i + 1 W i + 1 ] = [ 1 β i ( d i f i ) f i + ( d i f i ) ( μ i N i N i + 1 β i f i ) β i μ i N i N i + 1 β i f i ] [ S i T i V i W i ]
Q i = [ S i T i V i W i ] , J i = [ 1 d i 0 1 ] , B i = [ 1 + β i f i f i f i ( μ i N i N i + 1 β i f i ) β i μ i N i N i + 1 β i f i ] ,
Q i + 1 = J i B i J i 1 B i 1 J 2 B 2 J 1 B 1 J 0 ,
[ x i + 1 y i + 1 ξ i + 1 η i + 1 ] = Q i + 1 [ x 0 y 0 ξ 0 η 0 ] .
d 0 * = d 0 + Δ
x 0 = x 0 * + ξ 0 * Δ , y 0 = y 0 * + η 0 * Δ , ξ 0 = ξ 0 * , η 0 = η 0 * , ρ 0 = ρ 0 * + ψ 0 * Δ 2 + 2 κ 0 * Δ , ψ 0 = ψ 0 * , κ 0 = ψ 0 * Δ + κ 0 * .
[ x Γ + 1 * y Γ + 1 * ξ Γ + 1 * η Γ + 1 * ] ( x 0 * , y 0 * , ξ 0 * , η 0 * ) = [ x Γ + 1 y Γ + 1 ξ Γ + 1 η Γ + 1 ] ( x 0 , y 0 , ξ 0 , η 0 ) = Q Γ + 1 ( ρ 0 * + ψ 0 * Δ 2 + 2 κ 0 * Δ , ψ 0 * , ψ 0 * Δ + κ 0 * ) [ x 0 * + ξ 0 * Δ y 0 * + η 0 * Δ ξ 0 * η 0 * ]
Δ [ x Γ + 1 * y Γ + 1 * ξ Γ + 1 * η Γ + 1 * ] Δ = 0 = d 0 [ x Γ + 1 y Γ + 1 ξ Γ + 1 η Γ + 1 ] = Q Γ + 1 [ ξ 0 η 0 0 0 ] + ( 2 κ 0 Q Γ + 1 ρ 0 + ψ 0 Q Γ + 1 κ 0 ) [ x 0 y 0 ξ 0 η 0 ] = { Q Γ + 1 [ 0 1 0 0 ] + 2 κ 0 Q Γ + 1 ρ 0 + ψ 0 Q Γ + 1 κ 0 } [ x 0 y 0 ξ 0 η 0 ] ,
Q Γ + 1 d 0 = Q Γ + 1 [ 0 1 0 0 ] + 2 κ 0 Q Γ + 1 ρ 0 + ψ 0 Q Γ + 1 κ 0 .
S Γ + 1 d 0 = 2 κ 0 S Γ + 1 ρ 0 + ψ 0 S Γ + 1 κ 0 , T Γ + 1 d 0 = S Γ + 1 + 2 κ 0 T Γ + 1 ρ 0 + ψ 0 T Γ + 1 κ 0 , V Γ + 1 d 0 = 2 κ 0 V Γ + 1 ρ 0 + ψ 0 V Γ + 1 κ 0 , W Γ + 1 d 0 = V Γ + 1 + 2 κ 0 W Γ + 1 ρ 0 + ψ 0 W Γ + 1 κ 0 .
H i = J Γ B Γ J Γ 1 B Γ 1 J i + 1 B i + 1 J i B i .
Q Γ + 1 d i = d i ( J Γ B Γ J i + 1 B i + 1 J i ) B i J i 1 B i 1 J 1 B 1 J 0 = d i ( H i + 1 J i ) B i Q i .
x i * = x i + 1 ξ i + 1 d i , y i * = y i + 1 η i + 1 d i , ξ i * = ξ i + 1 , η i * = η i + 1 , ρ i * = ρ i + 1 + ψ i + 1 d i 2 2 κ i + 1 d i , ψ i * = ψ i + 1 , κ i * = ψ i + 1 d i + κ i + 1 .
d i ( H i + 1 J i ) ( ρ i * , ψ i * , κ i * ) = H i + 1 J i [ 0 1 0 0 ] + 2 κ i * ρ i * ( H i + 1 J i ) + ψ i * κ i * ( H i + 1 J i ) .
[ φ 1 , φ 2 , φ 3 ] = det ( φ 1 , φ 2 , φ 3 ) ( ρ 0 , ψ 0 , κ 0 ) = φ 1 ρ 0 ( φ 2 ψ 0 φ 3 κ 0 φ 2 κ 0 φ 3 ψ 0 ) + φ 1 ψ 0 ( φ 2 κ 0 φ 3 ρ 0 φ 2 ρ 0 φ 3 κ 0 ) + φ 1 κ 0 ( φ 2 ρ 0 φ 3 ψ 0 φ 2 ψ 0 φ 3 ρ 0 ) .
[ φ 1 , φ 2 , φ 3 ] = [ φ 2 , φ 3 , φ 1 ] = [ φ 3 , φ 1 , φ 2 ] = [ φ 1 , φ 3 , φ 2 ] = [ φ 3 , φ 2 , φ 1 ] = [ φ 2 , φ 1 , φ 3 ] ; [ φ 1 , φ 2 , φ 3 ] = 0 ; [ φ 1 , φ 2 , φ 3 φ 4 ] = φ 3 [ φ 1 , φ 2 , φ 4 ] + φ 4 [ φ 1 , φ 2 , φ 3 ] .
[ φ ρ 0 φ ψ 0 φ κ 0 ] = [ ρ i * ρ 0 ψ i * ρ 0 κ i * ρ 0 ρ i * ψ 0 ψ i * ψ 0 κ i * ψ 0 ρ i * κ 0 ψ i * κ 0 κ i * κ 0 ] [ φ ρ i * φ ψ i * φ κ i * ] ,
φ ρ i * = [ φ , ψ i * , κ i * ] [ ρ i * , ψ i * , κ i * ] , φ ψ i * = [ ρ i * , φ , κ i * ] [ ρ i * , ψ i * , κ i * ] , φ κ i * = [ ρ i * , ψ i * , φ ] [ ρ i * , ψ i * , κ i * ] .
[ ρ i * , ψ i * , κ i * ] = [ ρ i + 1 + ψ i + 1 d i 2 2 κ i + 1 d i , ψ i + 1 ψ i + 1 d i + κ i + 1 ] = [ ρ i + 1 , ψ i + 1 , κ i + 1 ] ,
2 κ i * [ φ , ψ i * , κ i * ] + ψ i * [ ρ i * , ψ i * , φ ] = 2 ( ψ i + 1 d i + κ i ) [ φ , ψ i + 1 ψ i + 1 d i + κ i + 1 ] + ψ i + 1 [ ρ i + 1 + ψ i + 1 d i 2 2 κ i + 1 d i , ψ i + 1 , φ ] = 2 ( ψ i + 1 d i + κ i ) [ φ , ψ i + 1 , κ i + 1 ] + ψ i + 1 [ ρ i + 1 , ψ i + 1 , φ ] 2 ψ i + 1 d i [ κ i + 1 , ψ i + 1 , φ ] = 2 κ i + 1 [ φ , ψ i + 1 , κ i + 1 ] + ψ i + 1 [ ρ i + 1 , ψ i + 1 , φ ] = [ φ , ψ i + 1 , κ i + 1 2 ] [ φ , ψ i + 1 , ρ i + 1 ψ i + 1 ] = [ φ , ψ i + 1 , κ i + 1 2 ρ i + 1 ψ i + 1 ] .
d i ( H i + 1 J i ) = H i + 1 J i [ 0 1 0 0 ] + [ H i + 1 J i , ψ i + 1 , κ i + 1 2 ρ i + 1 ψ i + 1 ] [ ρ i + 1 , ψ i + 1 , κ i + 1 ] .
Q Γ + 1 d i = { H i + 1 [ 0 1 0 0 ] + [ H i + 1 , ψ i + 1 , κ i + 1 2 ρ i + 1 ψ i + 1 ] [ ρ i + 1 , ψ i + 1 , κ i + 1 ] } Q i + 1 ,
Q Γ + 1 d Γ = [ 0 1 0 0 ] Q Γ + 1 = [ V Γ + 1 W Γ + 1 0 0 ] ,
J i B i = Q i + 1 Q i 1 ,
H i + 1 = Q Γ + 1 Q Γ 1 Q Γ Q Γ 1 1 Q i + 3 Q i + 2 1 Q i + 2 Q i + 1 1 = Q Γ + 1 Q i + 1 1 .
( 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 ,
Q Γ + 1 d i = Q Γ + 1 Q i + 1 1 [ 0 1 0 0 ] Q i + 1 + Q Γ + 1 [ Q i + 1 1 , ψ i + 1 , κ i + 1 2 ρ i + 1 ψ i + 1 ] Q i + 1 + [ Q Γ + 1 , ψ i + 1 , κ i + 1 2 ρ i + 1 ψ i + 1 ] [ ρ i + 1 , ψ i + 1 , κ i + 1 ] ,
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 ( ρ i ) μ 0 μ i N i + 1 .
K d 0 = 1 μ 0 1 + ψ 0 + 2 κ 0 K ρ 0 + ψ 0 K κ 0 ,
K d i = 1 + ψ i + 1 μ 0 μ i + [ j = 1 Γ K j , ψ i + 1 , κ i + 1 2 ρ i + 1 ψ i + 1 ] [ ρ i + 1 , ψ i + 1 , κ i + 1 ] ,
K d Γ = 1 + ψ Γ + 1 μ 0 μ Γ = 1 + V Γ + 1 2 ρ 0 + W Γ + 1 2 ψ 0 + 2 V Γ + 1 W Γ + 1 κ 0 μ 0 μ Γ .
G ( ρ 0 , ψ 0 , κ 0 ) = n = 1 j = 0 n k = 0 j G n , n j , j k , k ρ 0 n j ψ 0 j k κ 0 k ,
G ρ 0 = n = 0 j = 0 n k = 0 j ( n + 1 j ) G n + 1 , n + 1 j , j k , k ρ 0 n j ψ 0 j k κ 0 k ,
G ψ 0 = n = 0 j = 0 n k = 0 j ( j + 1 k ) G n + 1 , j + 1 k , k ρ 0 n j ψ 0 j k κ 0 k ,
G κ 0 = n = 0 j = 0 n k = 0 j ( k + 1 ) G n + 1 , n j , j k , k + 1 ρ 0 n j ψ 0 j k κ 0 k ,
H i = H i + 1 J i B i ,
j = i 1 Γ K j = K i 1 + j = 1 Γ K j ,
[ j = i 1 Γ K j , ψ i , κ i 2 ρ i ψ i ] ,
2 κ i + 1 [ φ , ψ i + 1 , κ i + 1 ] + ψ i + 1 [ ρ i + 1 , ψ i + 1 , φ ]
S 6 d 0
S 6 d 1
S 6 d 2
S 6 d 3
S 6 d 4
T 6 d 0
T 6 d 1
T 6 d 2
T 6 d 3
T 6 d 4
V 6 d 0
V 6 d 1
V 6 d 2
V 6 d 3
V 6 d 4
W 6 d 0
W 6 d 1
W 6 d 2
W 6 d 3
W 6 d 4
S ( ρ 0 , ψ 0 , κ 0 ) = S 0000 + S 1100 ρ 0 + S 1010 ψ 0 + S 1001 κ 0 + S 2200 ρ 0 2 + S 2110 ρ 0 ψ 0 + S 2101 ρ 0 κ 0 + S 2020 ψ 0 2 + S 2011 ψ 0 κ 0 + S 2002 κ 0 2 + ,
T ( ρ 0 , ψ 0 , κ 0 ) = T 0000 + T 1100 ρ 0 + T 1010 ψ 0 + T 1001 κ 0 + T 2200 ρ 0 2 + T 2110 ρ 0 ψ 0 + T 2101 ρ 0 κ 0 + T 2020 ψ 0 2 + T 2011 ψ 0 κ 0 + T 2002 κ 0 2 + ,
S nijk ( Δ = 0 ) = S nijk , T nijk ( Δ = 0 ) = T nijk .
S 0000 = S 0000 , T 0000 = T 0000 + S 0000 Δ ;
S 1100 = S 1100 , S 1010 = S 1010 + S 1001 Δ + S 1100 Δ 2 , S 1001 = S 1001 + 2 S 1100 Δ , T 1100 = T 1100 + S 1100 Δ , T 1010 = T 1010 + ( S 1010 + T 1001 ) Δ + ( S 1001 + T 1100 ) Δ 2 + S 1100 Δ 3 , T 1001 = T 1001 + ( S 1001 + 2 T 1100 ) Δ + 2 S 1100 Δ 2 ;
S 2200 = S 2200 , S 2110 = S 2110 + S 2101 Δ + 2 S 2200 Δ 2 , S 2101 = S 2101 + 4 S 2200 Δ , S 2020 = S 2020 + S 2011 Δ + ( S 2110 + S 2002 ) , Δ 3 101 + S 2200 Δ 4 , S 2011 = S 2011 + 2 ( S 2110 + S 2002 ) Δ + 3 S 21 4 S 2200 Δ 3 , S 2002 = S 2002 + 2 S 2101 Δ + 4 S 2200 Δ 2 , T 2200 = T 2200 + S 2200 Δ , T 2110 = T 2110 + ( S 2110 + T 2101 ) Δ + ( S 2101 + 2 T 2200 ) Δ 2 + 2 S 2200 Δ 3 , T 2101 = T 2101 + ( S 2101 + 4 T 2200 ) Δ + 4 S 2200 Δ 2 , T 2020 = T 2020 + ( S 2020 + T 2011 ) Δ + ( S 2011 + T 2110 + T 2002 ) Δ 2 + ( S 2110 + S 2002 + T 2101 ) Δ 3 + ( S 2101 + T 2200 ) Δ 4 + S 2200 Δ 5 , T 2011 = T 2011 + ( S 2011 + 2 T 2110 + 2 T 2002 ) Δ + ( 2 S 2110 + 2 S 2002 + 3 T 2101 ) Δ 2 + ( 3 T 2101 + 4 T 2200 ) Δ 3 + 4 S 2200 Δ 4 , T 2002 = T 2002 + ( S 2002 + 2 T 2101 ) Δ + 2 ( S 2101 + 2 T 2200 ) Δ 2 + 4 S 2200 Δ 3 .

Metrics