Abstract

We apply a new method for optical aberration derivation to anamorphic systems made from toroidal surfaces and obtain a complete set of primary aberration coefficients. This set is written in a form similar to the well-known Seidel aberrations for rotationally symmetrical optical systems and includes first- order quantities only, thus it can be easily applied to anamorphic lens design practice. By tracing four nonskew paraxial marginal and chief rays, the 16 anamorphic primary aberration coefficients can be easily calculated.

© 2010 Optical Society of America

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  1. J. C. Burfoot, “Third-order aberrations of doubly symmetric systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
    [CrossRef]
  2. H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970), Chap. 8, pp. 199–202.
  3. R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
    [CrossRef]
  4. J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183–188 (1997).
    [CrossRef]
  5. T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” J. Appl. Phys. 17, 131–136 (1978).
    [CrossRef]
  6. J. H. Jung and J. W. Lee, “Anamorphic lens for a CCD camera apparatus,” U.S. patent 5,671,093 (3 September 1997).
  7. I. A. Neil, “Anamorphic imaging system,” U.S. patent 7,085,066 (1 August 2006).
  8. J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1–12(2005).
    [CrossRef]
  9. S Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (The University of Arizona, 2008), free online accessible, http://gradworks.umi.com/33/31/3331455.html.
  10. H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).
  11. W. T. Welford, Aberrations of Optical System (Hilger, 1986), Chap. 11.
  12. V. N. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics (SPIE Press, 1998).
  13. A. Cox, A System of Optical Design (Focal Press, 1964).
  14. R. R. Shannon, The Art and Science of Optical Design(Cambridge University, 1997).
  15. S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems. I: the first-order foundation and method for deriving the anamorphic primary aberration coefficients,” Appl. Opt. 48, 2574–2584 (2009).
    [CrossRef] [PubMed]
  16. C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
    [CrossRef]
  17. G. G. Slyusarev, Aberration and Optical Design Theory(Hilger, 1984).
  18. L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik (Jena) 94, 167–172 (1993).
  19. P. J. Sands, “Aberration coefficients of double-plane-symmetric systems,” J. Opt. Soc. Am. 63, 425–430 (1973).
    [CrossRef]
  20. E. Delano, “Primary aberrations of a system of toroidal Fresnel surfaces,” J. Opt. Soc. Am. A 10, 1529–1534 (1993).
    [CrossRef]
  21. S. Yuan and J. Sasian, “Aberrations of anamorphic optical systems II: the primary aberration theory for cylindrical anamorphic systems,” Appl. Opt. 48, 2836–2841 (2009).
    [CrossRef] [PubMed]

2009 (2)

2005 (1)

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1–12(2005).
[CrossRef]

1997 (1)

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183–188 (1997).
[CrossRef]

1993 (2)

L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik (Jena) 94, 167–172 (1993).

E. Delano, “Primary aberrations of a system of toroidal Fresnel surfaces,” J. Opt. Soc. Am. A 10, 1529–1534 (1993).
[CrossRef]

1978 (1)

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” J. Appl. Phys. 17, 131–136 (1978).
[CrossRef]

1973 (1)

1966 (1)

R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

1954 (2)

J. C. Burfoot, “Third-order aberrations of doubly symmetric systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
[CrossRef]

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
[CrossRef]

Barakat, R.

R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Blanc, B.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1–12(2005).
[CrossRef]

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970), Chap. 8, pp. 199–202.

Burfoot, J. C.

J. C. Burfoot, “Third-order aberrations of doubly symmetric systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
[CrossRef]

Chen, C.

L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik (Jena) 94, 167–172 (1993).

Cox, A.

A. Cox, A System of Optical Design (Focal Press, 1964).

Delano, E.

E. Delano, “Primary aberrations of a system of toroidal Fresnel surfaces,” J. Opt. Soc. Am. A 10, 1529–1534 (1993).
[CrossRef]

Doucet, M.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1–12(2005).
[CrossRef]

Gauvin, J.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1–12(2005).
[CrossRef]

He, L.

L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik (Jena) 94, 167–172 (1993).

Hopkins, H. H.

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

Houston, A.

R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Jung, J. H.

J. H. Jung and J. W. Lee, “Anamorphic lens for a CCD camera apparatus,” U.S. patent 5,671,093 (3 September 1997).

Kasuya, T.

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” J. Appl. Phys. 17, 131–136 (1978).
[CrossRef]

Lee, J. W.

J. H. Jung and J. W. Lee, “Anamorphic lens for a CCD camera apparatus,” U.S. patent 5,671,093 (3 September 1997).

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics (SPIE Press, 1998).

Neil, I. A.

I. A. Neil, “Anamorphic imaging system,” U.S. patent 7,085,066 (1 August 2006).

Sands, P. J.

Sasian, J.

Sasian, J. M.

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183–188 (1997).
[CrossRef]

Shannon, R. R.

R. R. Shannon, The Art and Science of Optical Design(Cambridge University, 1997).

Shimoda, K.

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” J. Appl. Phys. 17, 131–136 (1978).
[CrossRef]

Slyusarev, G. G.

G. G. Slyusarev, Aberration and Optical Design Theory(Hilger, 1984).

Suzuki, T.

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” J. Appl. Phys. 17, 131–136 (1978).
[CrossRef]

Thibault, S.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1–12(2005).
[CrossRef]

Wang, M.

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1–12(2005).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical System (Hilger, 1986), Chap. 11.

Wynne, C. G.

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
[CrossRef]

Yuan, S

S Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (The University of Arizona, 2008), free online accessible, http://gradworks.umi.com/33/31/3331455.html.

Yuan, S.

Appl. Opt. (2)

J. Appl. Phys. (1)

T. Kasuya, T. Suzuki, and K. Shimoda, “A prism anamorphic system for Gaussian beam expander,” J. Appl. Phys. 17, 131–136 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

R. Barakat and A. Houston, “The aberrations on non-rotationally symmetric systems and their diffraction effects,” Opt. Acta 13, 1–30 (1966).
[CrossRef]

Opt. Eng. (1)

J. M. Sasian, “Double-curvature surfaces in mirror system design,” Opt. Eng. 36, 183–188 (1997).
[CrossRef]

Optik (Jena) (1)

L. He and C. Chen, “The primary aberration coefficients of a torus,” Optik (Jena) 94, 167–172 (1993).

Proc. Phys. Soc. B (2)

C. G. Wynne, “The primary aberrations of anamorphic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954).
[CrossRef]

J. C. Burfoot, “Third-order aberrations of doubly symmetric systems,” Proc. Phys. Soc. B 67, 523–528 (1954).
[CrossRef]

Proc. SPIE (1)

J. Gauvin, M. Doucet, M. Wang, S. Thibault, and B. Blanc, “Development of new family of wide-angle anamorphic lens with controlled distortion profile,” Proc. SPIE 5874, 1–12(2005).
[CrossRef]

Other (11)

S Yuan, “Aberrations of anamorphic optical systems,” Ph.D. dissertation (The University of Arizona, 2008), free online accessible, http://gradworks.umi.com/33/31/3331455.html.

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

W. T. Welford, Aberrations of Optical System (Hilger, 1986), Chap. 11.

V. N. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics (SPIE Press, 1998).

A. Cox, A System of Optical Design (Focal Press, 1964).

R. R. Shannon, The Art and Science of Optical Design(Cambridge University, 1997).

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge University, 1970), Chap. 8, pp. 199–202.

G. G. Slyusarev, Aberration and Optical Design Theory(Hilger, 1984).

J. H. Jung and J. W. Lee, “Anamorphic lens for a CCD camera apparatus,” U.S. patent 5,671,093 (3 September 1997).

I. A. Neil, “Anamorphic imaging system,” U.S. patent 7,085,066 (1 August 2006).

E. Delano, “Primary aberrations of a system of toroidal Fresnel surfaces,” J. Opt. Soc. Am. A 10, 1529–1534 (1993).
[CrossRef]

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Tables (1)

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Table 1 Primary Aberration Coefficients for Toroidal Anamorphic Systems

Equations (34)

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

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