Abstract

Algebraic ray-trace equations for axially symmetric optical systems are expanded in terms of system parameters and paraxial variables. The transfer parameter used is optical path, and equations are given for second-, fourth-, sixth-, and eighth-order differences in optical path of a ray from the axial value. Selected rays are traced to a tilted reference sphere in the exit pupil, optical path differences of given order are equated to the wave aberration polynomial of corresponding order and with proper coordinates, and the resulting linear equations are solved for wave aberration coefficients.

© 1976 Optical Society of America

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References

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  1. G. W. Hopkins, “Proximate ray tracing and optical aberration coefficients,” J. Opt. Soc. Am. 66, 405–410 (1976).
    [Crossref]
  2. A. Cox, A System of Optical Design (Focal, London, 1964), pp. 175–181.
  3. H. H. Hopkins, “The development of image evaluation methods,” Proceedings of the Society of Photo-Optical Instrumentation Engineers, Image Assessment and Specification 64, 2–18 (1974).
    [Crossref]
  4. G. W. Hopkins, “Abberrational Analysis of Optical Systems: A Proximate Ray Trace Approach,” Dissertation (The University of Arizona) (University Microfilms, Ann Arbor, 1976).
  5. H. H. Hopkins, The Wave Theory of Aberrations (Clarendon, Oxford, 1950), p. 49.

1976 (1)

1974 (1)

H. H. Hopkins, “The development of image evaluation methods,” Proceedings of the Society of Photo-Optical Instrumentation Engineers, Image Assessment and Specification 64, 2–18 (1974).
[Crossref]

Cox, A.

A. Cox, A System of Optical Design (Focal, London, 1964), pp. 175–181.

Hopkins, G. W.

G. W. Hopkins, “Proximate ray tracing and optical aberration coefficients,” J. Opt. Soc. Am. 66, 405–410 (1976).
[Crossref]

G. W. Hopkins, “Abberrational Analysis of Optical Systems: A Proximate Ray Trace Approach,” Dissertation (The University of Arizona) (University Microfilms, Ann Arbor, 1976).

Hopkins, H. H.

H. H. Hopkins, “The development of image evaluation methods,” Proceedings of the Society of Photo-Optical Instrumentation Engineers, Image Assessment and Specification 64, 2–18 (1974).
[Crossref]

H. H. Hopkins, The Wave Theory of Aberrations (Clarendon, Oxford, 1950), p. 49.

J. Opt. Soc. Am. (1)

Proceedings of the Society of Photo-Optical Instrumentation Engineers, Image Assessment and Specification (1)

H. H. Hopkins, “The development of image evaluation methods,” Proceedings of the Society of Photo-Optical Instrumentation Engineers, Image Assessment and Specification 64, 2–18 (1974).
[Crossref]

Other (3)

G. W. Hopkins, “Abberrational Analysis of Optical Systems: A Proximate Ray Trace Approach,” Dissertation (The University of Arizona) (University Microfilms, Ann Arbor, 1976).

H. H. Hopkins, The Wave Theory of Aberrations (Clarendon, Oxford, 1950), p. 49.

A. Cox, A System of Optical Design (Focal, London, 1964), pp. 175–181.

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Tables (2)

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TABLE I Rays used in determining aberration coefficients. H is normalized image height; ρx and ρy are normalized pupil coordinates.

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TABLE II Ray errors required for calculating aberration coefficients are marked with an ×.

Equations (9)

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c x = c x ( 1 ) + δ c x ( 3 ) + δ c x ( 5 ) + δ c x ( 7 ) + , c y = c y ( 1 ) + δ c y ( 3 ) + δ c y ( 5 ) + δ c y ( 7 ) + , c z = 1 + δ c z ( 2 ) + δ c z ( 4 ) + δ c z ( 6 ) + δ c z ( 8 ) + .
d c c + c x ( 2 ) + c y 2 = d c c ( 2 ) + δ d c c ( 4 ) + δ d c c ( 6 ) + δ d c c ( 8 ) + ,
w = w ( 0 ) + δ w ( 2 ) + δ w ( 4 ) + δ w ( 6 ) + δ w ( 8 ) + .
x o = x o ( 1 ) ,             y o = y o ( 1 ) , z o = z o ( 2 ) + δ z o ( 4 ) + δ z o ( 6 ) + δ z o ( 8 ) .
k = k ( 0 ) + δ k ( 2 ) + δ k ( 4 ) + δ k ( 6 ) + .
W = ( W 020 ρ 2 + W 111 H ρ y + W 200 H 2 ) + ( W 040 ρ 4 + W 131 H ρ 2 ρ y + W 220 H 2 ρ 2 + W 222 H 2 ρ y 2 + W 311 H 3 ρ y + W 400 H 4 ) + ( W 060 ρ 6 + W 151 H ρ 4 ρ y + W 240 H 2 ρ 4 + W 242 H 2 ρ 2 ρ y 2 + W 331 H 3 ρ 2 ρ y + W 333 H 3 ρ y 3 + W 420 H 4 ρ 2 + W 422 H 4 ρ y 2 + W 511 H 5 ρ y + W 600 H 6 ) + ( W 080 ρ 8 + W 171 H ρ 6 ρ y + W 260 H 2 ρ 6 + W 262 H 2 ρ 4 ρ y 2 + W 351 H 3 ρ 4 ρ y + W 353 H 3 ρ 2 ρ y 3 + W 440 H 4 ρ 4 + W 442 H 4 ρ 2 ρ y 2 + W 444 H 4 ρ y 4 + W 531 H 5 ρ 2 ρ y + W 533 H 5 ρ y 3 + W 620 H 6 ρ 2 + W 622 H 6 ρ y 2 + W 711 H 7 ρ y + W 800 H 8 ) .
y = ( G 1 ρ y + G 2 H ) + ( P 1 ρ 2 ρ y + P 2 H ρ 2 + P 3 H ρ y 2 + P 4 H 2 ρ y + P 5 H 2 ρ y + P 6 H 3 ) + ( S 1 ρ 4 ρ y + S 2 H ρ 4 + S 3 H ρ 2 ρ y 2 + S 4 H 2 ρ 2 ρ y + S 5 H 2 ρ 2 ρ y + S 6 H 2 ρ y 2 + S 7 H 3 ρ 2 + S 8 H 3 ρ y 2 + S 9 H 3 ρ y 2 + S 10 H 4 ρ y + S 11 H 4 ρ y + S 12 H 5 ) + ( T 1 ρ 6 ρ y + T 2 H ρ 6 + T 3 H ρ 4 ρ y 2 + T 4 H 2 ρ 4 ρ y + T 5 H 2 ρ 4 ρ y + T 6 H 2 ρ 2 ρ y 3 + T 7 H 3 ρ 4 + T 8 H 3 ρ 2 ρ y 2 + T 9 H 3 ρ 2 ρ y 2 + T 10 H 3 ρ y 4 + T 11 H 4 ρ 2 ρ y + T 12 H 4 ρ y 3 + T 13 H 4 ρ 2 ρ y + T 14 H 4 ρ y 3 + T 15 H 5 ρ 2 + T 16 H 5 ρ y 2 + T 17 H 5 ρ y 2 + T 18 H 6 ρ y + T 19 H 6 ρ y + T 20 H 7 )
x = ( G 1 ρ x ) + ( P 1 ρ 2 ρ x + P 3 H ρ x ρ y + P 5 H 2 ρ x ) + ( S 1 ρ 4 ρ x + S 3 H ρ 2 ρ x ρ y + S 5 H 2 ρ 2 ρ x + S 6 H 2 ρ x ρ y 2 + S 9 H 3 ρ x ρ y + S 11 H 4 ρ x ) + ( T 1 ρ 6 ρ x + T 3 H ρ 4 ρ x ρ y + T 5 H 2 ρ 4 ρ x + T 6 H 2 ρ 2 ρ x ρ y 2 + T 9 H 3 ρ 2 ρ x ρ y + T 10 H 3 ρ x ρ y 3 + T 13 H 4 ρ 2 ρ x + T 14 H 4 ρ x ρ y 2 + T 17 H 5 ρ x ρ y + T 19 H 6 ρ x ) .
δ y = ( G ¯ 1 H + G ¯ 2 ρ y ) + ( P ¯ 1 H 3 + P ¯ 2 H 2 ρ y + P ¯ 3 H 2 ρ y + P ¯ 4 H ρ y 2 + P ¯ 5 H ρ 2 + P ¯ 6 ρ 2 ρ y ) + ( S ¯ 1 H 5 + S 2 H 4 ρ y + S ¯ 3 H 4 ρ y + S ¯ 4 H 3 ρ y 2 + S ¯ 5 H 3 ρ y 2 + S ¯ 6 H 3 ρ 2 + S ¯ 7 H 2 ρ y 3 + S ¯ 8 H 2 ρ 2 ρ y + S ¯ 9 H 2 ρ 2 ρ y + S ¯ 10 H ρ 2 ρ y 2 + S ¯ 11 H ρ 4 + S ¯ 12 ρ 4 ρ y ) , δ x = ( G ¯ 2 ρ x ) + ( P ¯ 2 H 2 ρ x + P ¯ 4 H ρ x ρ y + P ¯ 6 ρ 2 ρ x ) + ( S ¯ 2 H 4 ρ x + S ¯ 4 H 3 ρ x ρ y + S ¯ 7 H 2 ρ x ρ y 2 + S ¯ 8 H 2 ρ 2 ρ x + S ¯ 10 H ρ 2 ρ x ρ y + S ¯ 12 ρ 4 ρ x ) .