Abstract

By applying a new method for aberration derivation to anamorphic systems made from cylindrical surfaces, we obtain a complete set of primary aberration coefficients. This set is in a form similar to the well-known Seidel aberrations for rotationally symmetrical optical systems (RSOS). The aberration coefficients include first-order quantities only. By tracing four nonskew paraxial marginal and chief rays in the two associated RSOS of the anamorphic system, the aberration coefficients can be derived.

© 2009 Optical Society of America

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  1. S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems. I. First-order foundation and method for deriving the anamorphic primary aberration coefficients,” Appl. Opt. 48, xxx-xxx (2009).
    [CrossRef]
  2. H. Chretien, “Anamorphotic lens system and method of making the same,” U.S. patent 1,962,892 (12 June 1934).
  3. G. H. Cook, “Anamorphotic systems,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, 1965), Vol. 3, pp. 127-132.
  4. R. Wartmann and U. Schauss, “Anastigmatic anamorphic lens,” U.S. patent 6,310,731 B1 (30 October 2001).
  5. C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. London 67b, 529-537 (1954).
  6. S. Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (University of Arizona, 2008).
  7. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).
  8. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).

2009

S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems. I. First-order foundation and method for deriving the anamorphic primary aberration coefficients,” Appl. Opt. 48, xxx-xxx (2009).
[CrossRef]

1954

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. London 67b, 529-537 (1954).

Chretien, H.

H. Chretien, “Anamorphotic lens system and method of making the same,” U.S. patent 1,962,892 (12 June 1934).

Cook, G. H.

G. H. Cook, “Anamorphotic systems,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, 1965), Vol. 3, pp. 127-132.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).

Sasian, J.

S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems. I. First-order foundation and method for deriving the anamorphic primary aberration coefficients,” Appl. Opt. 48, xxx-xxx (2009).
[CrossRef]

Schauss, U.

R. Wartmann and U. Schauss, “Anastigmatic anamorphic lens,” U.S. patent 6,310,731 B1 (30 October 2001).

Wartmann, R.

R. Wartmann and U. Schauss, “Anastigmatic anamorphic lens,” U.S. patent 6,310,731 B1 (30 October 2001).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

Wynne, C. G.

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. London 67b, 529-537 (1954).

Yuan, S.

S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems. I. First-order foundation and method for deriving the anamorphic primary aberration coefficients,” Appl. Opt. 48, xxx-xxx (2009).
[CrossRef]

S. Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (University of Arizona, 2008).

Appl. Opt.

S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems. I. First-order foundation and method for deriving the anamorphic primary aberration coefficients,” Appl. Opt. 48, xxx-xxx (2009).
[CrossRef]

Proc. Phys. Soc. London

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. London 67b, 529-537 (1954).

Other

S. Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (University of Arizona, 2008).

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).

H. Chretien, “Anamorphotic lens system and method of making the same,” U.S. patent 1,962,892 (12 June 1934).

G. H. Cook, “Anamorphotic systems,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, 1965), Vol. 3, pp. 127-132.

R. Wartmann and U. Schauss, “Anastigmatic anamorphic lens,” U.S. patent 6,310,731 B1 (30 October 2001).

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Figures (2)

Fig. 1
Fig. 1

Example of parallel-cylindrical anamorphic attachment system.

Fig. 2
Fig. 2

Example of cross-cylindrical anamorphic system.

Tables (1)

Tables Icon

Table 1 Primary Aberration Coefficients for Cylindrical Anamorphic Systems

Equations (30)

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z = 1 2 x 2 r x + 1 8 x 4 r x 3 .
z = 1 2 y 2 r y + 1 8 y 4 r y 3 .
z = 1 2 ( c x x 2 + c y y 2 ) + 1 8 ( c x 3 x 4 + c y 3 y 4 ) ,
δ ξ k 3 = Δ [ j = 1 k A x , j ( x j 1 δ N j 2 + z j 2 L j 1 ) + H x ψ x j = 1 k δ N j 2 j = 1 k h x , j ( c x , j 2 z j 2 x j 1 + α j 3 γ j 3 ) n j ] / n k u x , k , δ η k 3 = Δ [ j = 1 k A y , j ( y j 1 δ N j 2 + z j 2 M j 1 ) + H y ψ y j = 1 k δ N j 2 j = 1 k h y , j ( c y , j 2 z j 2 y j 1 + β j 3 γ j 3 ) n j ] / n k u y , k ,
δ ξ k 3 = Δ [ j = 1 k A x , j ( x j 1 δ N j 2 + z j 2 L j 1 ) + H x ψ x j = 1 k δ N j 2 ] / n k u x , k , δ η k 3 = Δ [ j = 1 k A y , j ( y j 1 δ N j 2 + z j 2 M j 1 ) + H y ψ y j = 1 k δ N j 2 ] / n k u y , k .
x j 1 = ρ x h x , j + H x h ¯ x , j , y j 1 = ρ y h y , j + H y h ¯ y , j , L j 1 = ρ x u x , j + H x u ¯ x , j , M j 1 = ρ y u y , j + H y u ¯ y , j , δ N j 2 = L j 1 2 + M j 1 2 2 = 1 2 [ ( ρ x u x , j + H x u ¯ x , j ) 2 + ( ρ y u y , j + H y u ¯ y , j ) 2 ] z j 2 = 1 2 ( c x , j x j 1 2 + c y , j y j 1 2 ) = 1 2 [ 1 r x , j ( ρ x h x , j + H x h ¯ x , j ) 2 + 1 r y , j ( ρ y h y , j + H y h ¯ y , j ) 2 ] .
δ ξ k 3 = Δ [ j = 1 k A x , j ( x j 1 δ N j 2 + z j 2 L j 1 ) + H x ψ x j = 1 k δ N j 2 ] / n k u x , k = { 1 2 j = 1 k A x , j [ ( ρ x h x , j + H x h ¯ x , j ) ( ρ 2 x Δ u 2 x , j + H 2 x Δ u ¯ x , j 2 + 2 ρ x H x Δ u x , j u ¯ x , j ) 1 2 j = 1 k A x , j [ ( ρ x h x , j + H x h ¯ x , j ) ( ρ 2 y Δ u 2 y , j + H 2 y Δ u ¯ y , j 2 + 2 ρ y H y Δ u y , j u ¯ y , j ) 1 2 j = 1 k A x , j [ c x , j ( h 2 x , j ρ x 2 + 2 h x , j h ¯ x , j H x ρ x + h ¯ x , j 2 H 2 x ) ( ρ x Δ u x , j + H x Δ u ¯ x , j ) ] 1 2 j = 1 k A x , j [ c y , j ( h 2 y , j ρ y 2 + 2 h y , j h ¯ y , j H y ρ y + h ¯ y , j 2 H 2 y ) ( ρ x Δ u x , j + H x Δ u ¯ x , j ) ] + 1 2 ψ x H x j = 1 k ( ρ 2 x Δ u 2 x , j + H 2 x Δ u ¯ x , j 2 + 2 ρ x H x Δ u x , j u ¯ x , j ) + 1 2 ψ x H x j = 1 k ( ρ 2 y Δ u 2 y , j + H 2 y Δ u ¯ y , j 2 + 2 ρ y H y Δ u y , j u ¯ y , j ) } / n k u x , k ,
δ η k 3 = Δ [ j = 1 k A y , j ( y j 1 δ N j 2 + z j 2 M j 1 ) + H y ψ y j = 1 k δ N j 2 ] / n k u y , k = { 1 2 j = 1 k A y , j [ ( ρ y h y , j + H y h ¯ y , j ) ( ρ 2 x Δ u 2 x , j + H 2 x Δ u ¯ x , j 2 + 2 ρ x H x Δ u x , j u ¯ x , j ) 1 2 j = 1 k A y , j [ ( ρ y h y , j + H y h ¯ y , j ) ( ρ 2 y Δ u 2 y , j + H 2 y Δ u ¯ y , j 2 + 2 ρ y H y Δ u y , j u ¯ y , j ) 1 2 j = 1 k A y , j [ c x , j ( h 2 x , j ρ x 2 + 2 h x , j h ¯ x , j H x ρ x + h ¯ x , j 2 H 2 x ) ( ρ y Δ u y , j + H y Δ u ¯ y , j ) ] 1 2 j = 1 k A y , j [ c y , j ( h 2 y , j ρ y 2 + 2 h y , j h ¯ y , j H y ρ y + h ¯ y , j 2 H 2 y ) ( ρ y Δ u y , j + H y Δ u ¯ y , j ) ] + 1 2 ψ y H y j = 1 k ( ρ 2 x Δ u 2 x , j + H 2 x Δ u ¯ x , j 2 + 2 ρ x H x Δ u x , j u ¯ x , j ) + 1 2 ψ y H y j = 1 k ( ρ 2 y Δ u 2 y , j + H 2 y Δ u ¯ y , j 2 + 2 ρ y H y Δ u y , j u ¯ y , j ) } / n k u y , k .
δ ξ k 3 = ( 4 D 1 ρ x 3 + 2 D 3 ρ x ρ y 2 + 3 D 4 H x ρ x 2 + 2 D 5 H y ρ x ρ y + D 6 H x ρ y 2 + 2 D 8 H x 2 ρ x + 2 D 10 H y 2 ρ x + D 12 H x H y ρ y + D 13 H x 3 + D 15 H x H y 2 ) / n k u x , k , δ η k 3 = ( 4 D 2 ρ y 3 + 2 D 3 ρ x 2 ρ y + D 5 H y ρ x 2 + 2 D 6 H x ρ x ρ y + 3 D 7 H y ρ y 2 + 2 D 9 H y 2 ρ y + 2 D 11 H x 2 ρ y + D 12 H x H y ρ x + D 14 H y 3 + D 16 H x 2 H y ) / n k u y , k ,
D 1 = 1 8 j = 1 k A x , j ( h x , j Δ u 2 x , j + c x , j h x , j 2 Δ u x , j ) ,
D 2 = 1 8 j = 1 k A y , j ( h y , j Δ u 2 y , j + c y , j h y , j 2 Δ u y , j ) ,
D 3 = 1 4 j = 1 k A y , j ( h y , j Δ u 2 x , j + c x , j h x , j 2 Δ u y , j ) ,
D 4 = 1 6 j = 1 k [ A x , j ( h ¯ x , j Δ u 2 x , j + c x , j h 2 x , j Δ u ¯ x , j + 2 h x , j Δ u x , j u ¯ x , j + 2 c x , j h x , j h ¯ x , j Δ u x , j ) ψ x Δ u 2 x , j ] ,
D 5 = 1 2 j = 1 k [ A y , j ( h ¯ y , j Δ u 2 x , j + c x , j h 2 x , j Δ u ¯ y , j ) ψ y Δ u 2 x , j ] ,
D 6 = 1 2 j = 1 k A y , j ( h y , j Δ u x , j u ¯ x , j + c x , j h x , j h ¯ x , j Δ u y , j ) ,
D 7 = 1 6 j = 1 k [ A y , j ( h ¯ y , j Δ u 2 y , j + c y , j h 2 y , j Δ u ¯ y , j + 2 h y , j Δ u y , j u ¯ y , j + 2 c y , j h y , j h ¯ y , j Δ u y , j ) ψ y Δ u 2 y , j ] ,
D 8 = 1 4 j = 1 k [ A x , j ( h x , j Δ u ¯ x , j 2 + c x , j h ¯ x , j 2 Δ u x , j + 2 h ¯ x , j Δ u x , j u ¯ x , j + 2 c x , j h x , j h ¯ x , j Δ u ¯ x , j ) 2 ψ x Δ u x , j u ¯ x , j ] ,
D 9 = 1 4 j = 1 k [ A y , j ( h y , j Δ u ¯ y , j 2 + c y , j h ¯ y , j 2 Δ u y , j + 2 h ¯ y , j Δ u y , j u ¯ y , j + 2 c y , j h y , j h ¯ y , j Δ u ¯ y , j ) 2 ψ y Δ u y , j u ¯ y , j ] ,
D 10 = 1 4 j = 1 k A x , j ( h x , j Δ u ¯ y , j 2 + c y , j h ¯ y , j 2 Δ u x , j ) ,
D 11 = 1 4 j = 1 k A y , j ( h y , j Δ u ¯ x , j 2 + c x , j h ¯ x , j 2 Δ u y , j ) ,
D 12 = j = 1 k [ A y , j ( h ¯ y , j Δ u x , j u ¯ x , j + c x , j h x , j h ¯ x , j Δ u ¯ y , j ) ψ y Δ u x , j u ¯ x , j ] ,
D 13 = 1 2 j = 1 k [ A x , j ( h ¯ x , j Δ u ¯ x , j 2 + c x , j h ¯ x , j 2 Δ u ¯ x , j ) ψ x Δ u ¯ x , j 2 ] ,
D 14 = 1 2 j = 1 k [ A y , j ( h ¯ y , j Δ u ¯ y , j 2 + c y , j h ¯ y , j 2 Δ u ¯ y , j ) ψ y Δ u ¯ y , j 2 ] ,
D 15 = 1 2 j = 1 k [ A x , j ( h ¯ x , j Δ u ¯ y , j 2 + c y , j h ¯ y , j 2 Δ u ¯ x , j ) ψ x Δ u ¯ y , j 2 ] ,
D 16 = 1 2 j = 1 k [ A y , j ( h ¯ y , j Δ u ¯ x , j 2 + c x , j h ¯ x , j 2 Δ u ¯ y , j ) ψ y Δ u ¯ x , j 2 ] .
A x , j = n j u x , j + n j h x , j c x , j , A ¯ x , j = n j u ¯ x , j + n j h ¯ x , j c x , j , A y , j = n j u y , j + n j h y , j c y , j , A ¯ y , j = n j u ¯ y , j + n j h ¯ y , j c y , j , Ψ x = n j ( h ¯ x , j u x , j h x , j u ¯ x , j ) = A x , j h ¯ x , j A ¯ x , j h x , j , Ψ y = n j ( h ¯ y , j u y , j h y , j u ¯ y , j ) = A y , j h ¯ y , j A ¯ y , j h y , j .
P x , j = Δ c x , j n j , P y , j = Δ c y , j n j .
D 1 = 1 8 j = 1 k A x , j 2 h x , j Δ u x , j n j = 1 8 S I x , D 4 = 1 2 j = 1 k A x , j A ¯ x , j h x , j Δ u x , j n j = 1 2 S I I x , D 8 = 1 4 j = 1 k ( 3 A ¯ x , j 2 h x , j Δ u x , j n j + Ψ 2 x P x , j ) = 1 4 ( 3 S I I I x + S I V x ) , D 13 = 1 2 j = 1 k [ A ¯ x , j 3 h x , j Δ 1 n j 2 + A ¯ x , j h ¯ x , j ( Ψ x A ¯ x , j h x , j ) P x , j ] = 1 2 S V x .
D 2 = 1 8 j = 1 k A y , j 2 h y , j Δ u y , j n j = 1 8 S I y , D 7 = 1 2 j = 1 k A y , j A ¯ y , j h y , j Δ u y , j n j = 1 2 S I I y , D 9 = 1 4 j = 1 k ( 3 A ¯ y , j 2 h y , j Δ u y , j n j + Ψ y 2 P y , j ) = 1 4 ( 3 S I I I y + S I V y ) , D 14 = 1 2 j = 1 k [ A ¯ y , j 3 h y , j Δ 1 n j 2 + A ¯ y , j h ¯ y , j ( Ψ y A ¯ y , j h y , j ) P y , j ] = 1 2 S V y .
D 3 = 1 4 j = 1 k A y , j ( h y , j Δ u x , j 2 + h x , j 2 c x , j Δ u y , j ) , D 5 = 1 2 j = 1 k ( A ¯ y , j h y , j Δ u x , j 2 + A y , j h x , j 2 c x , j Δ u ¯ y , j ) , D 6 = 1 2 j = 1 k A y , j ( h y , j Δ u x , j u ¯ x , j + h x , j h ¯ x , j c x , j Δ u y , j ) , D 10 = 1 4 j = 1 k A x , j ( h x , j Δ u ¯ y , j 2 + h ¯ y , j 2 c y , j Δ u x , j ) , D 11 = 1 4 j = 1 k A y , j ( h y , j Δ u ¯ x , j 2 + h ¯ x , j 2 c x , j Δ u y , j ) , D 12 = j = 1 k ( A ¯ y , j h y , j Δ u x , j u ¯ x , j + A y , j h x , j h ¯ x , j c x , j Δ u ¯ y , j ) , D 15 = 1 2 j = 1 k ( A ¯ x , j h x , j Δ u ¯ y , j 2 + A x , j h ¯ y , j 2 c y , j Δ u ¯ x , j ) , D 16 = 1 2 j = 1 k ( A ¯ y , j h y , j Δ u ¯ x , j 2 + A y , j h ¯ x , j 2 c x , j Δ u ¯ y , j ) .

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