Abstract

This paper introduces the path forward for the integration of freeform optical surfaces, particularly those related to φ-polynomial surfaces, including Zernike polynomial surfaces, with nodal aberration theory. With this formalism, the performance of an optical system throughout the field of view can be anticipated analytically accounting for figure error, mount-induced errors, and misalignment. Previously, only misalignments had been described by nodal aberration theory, with the exception of one special case for figure error. As an example of these new results, three point mounting error that results in a Zernike trefoil deformation is studied for the secondary mirror of a two mirror and three mirror telescope. It is demonstrated that for the case of trefoil deformation applied to a surface not at the stop, there is the anticipated field constant contribution to elliptical coma (also called trefoil) as well as a newly identified field dependent contribution to astigmatism: field linear, field conjugate astigmatism. The magnitude of this astigmatic contribution varies linearly with the field of view; however, it has a unique variation in orientation with field that is described mathematically by a concept that is unique to nodal aberration theory known as the field conjugate vector.

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  1. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE251, 146–153 (1980).
  2. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A22(7), 1389–1401 (2005).
    [CrossRef] [PubMed]
  3. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express18(16), 17433–17447 (2010).
    [CrossRef] [PubMed]
  4. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing phi-polynomial surfaces,” Opt. Express19(22), 21919–21928 (2011).
    [CrossRef] [PubMed]
  5. K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am.103, 159–165 (1980).
  6. J. Wang, B. Guo, Q. Sun, and Z. Lu, “Third-order aberration fields of pupil decentered optical systems,” Opt. Express20(11), 11652–11658 (2012).
    [CrossRef] [PubMed]
  7. K. P. Thompson, “Reinterpreting Coddington: correcting 150 years of confusion,” in Robert Shannon and Roland Shack, Legends in Applied Optics, J. E. Harvey, and R. B. Hooker, eds. (SPIE Press, 2005), 41–49.
  8. C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc.103, 159–165 (1942).
  9. A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE7652, 765230, 765230-11 (2010).
    [CrossRef]
  10. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A26(5), 1090–1100 (2009).
    [CrossRef] [PubMed]
  11. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A27(6), 1490–1504 (2010).
    [CrossRef] [PubMed]
  12. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A28(5), 821–836 (2011).
    [CrossRef] [PubMed]
  13. Synopsys Inc, “Zernike Polynomials,” in CODE V Reference Manual (2012), Volume IV, Appendix C.
  14. R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express20(15), 16436–16449 (2012).
    [CrossRef]
  15. J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE679, 21–24 (1986).
  16. J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
    [CrossRef]
  17. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A26(6), 1503–1517 (2009).
    [CrossRef] [PubMed]

2012 (2)

2011 (2)

2010 (3)

2009 (2)

2005 (1)

2004 (1)

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

1986 (1)

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE679, 21–24 (1986).

1980 (2)

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE251, 146–153 (1980).

K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am.103, 159–165 (1980).

1942 (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc.103, 159–165 (1942).

Atcheson, P. D.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Atkinson, C. B.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Au, D.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Bronowicki, A. J.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Bujanda, E.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Burch, C. R.

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc.103, 159–165 (1942).

Cakmakci, O.

Cohen, A.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Davies, D.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Dunn, C.

Fuerschbach, K.

Gray, R. W.

Guo, B.

Lightsey, P. A.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Lu, Z.

Lundquist, R.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Lynch, R.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Macenka, S. A.

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE679, 21–24 (1986).

Menzel, M. T.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Mohan, M.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Nella, J.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Pohner, J.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Rakich, A.

Reynolds, P.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Rivera, H.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Rolland, J. P.

Schmid, T.

Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE251, 146–153 (1980).

Shuckstes, D. V.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Simmons, D. D. F.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Smith, R. C.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Stacy, J. E.

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE679, 21–24 (1986).

Sullivan, P. C.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Sun, Q.

Texter, S. C.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Thompson, K.

Thompson, K. P.

R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express20(15), 16436–16449 (2012).
[CrossRef]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A28(5), 821–836 (2011).
[CrossRef] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing phi-polynomial surfaces,” Opt. Express19(22), 21919–21928 (2011).
[CrossRef] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A27(6), 1490–1504 (2010).
[CrossRef] [PubMed]

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express18(16), 17433–17447 (2010).
[CrossRef] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A26(5), 1090–1100 (2009).
[CrossRef] [PubMed]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A26(6), 1503–1517 (2009).
[CrossRef] [PubMed]

K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am.103, 159–165 (1980).

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE251, 146–153 (1980).

Waldie, D. D.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

Wang, J.

Woods, R.

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am.103, 159–165 (1980).

J. Opt. Soc. Am. A (5)

Mon. Not. R. Astron. Soc. (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc.103, 159–165 (1942).

Opt. Express (4)

Proc. SPIE (4)

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE679, 21–24 (1986).

J. Nella, P. D. Atcheson, C. B. Atkinson, D. Au, A. J. Bronowicki, E. Bujanda, A. Cohen, D. Davies, P. A. Lightsey, R. Lynch, R. Lundquist, M. T. Menzel, M. Mohan, J. Pohner, P. Reynolds, H. Rivera, S. C. Texter, D. V. Shuckstes, D. D. F. Simmons, R. C. Smith, P. C. Sullivan, D. D. Waldie, and R. Woods, “James Webb Space Telescope (JWST) Observatory architecture and performance,” Proc. SPIE5487, 576–587 (2004).
[CrossRef]

A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE7652, 765230, 765230-11 (2010).
[CrossRef]

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE251, 146–153 (1980).

Other (2)

K. P. Thompson, “Reinterpreting Coddington: correcting 150 years of confusion,” in Robert Shannon and Roland Shack, Legends in Applied Optics, J. E. Harvey, and R. B. Hooker, eds. (SPIE Press, 2005), 41–49.

Synopsys Inc, “Zernike Polynomials,” in CODE V Reference Manual (2012), Volume IV, Appendix C.

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Figures (11)

Fig. 1
Fig. 1

When the aspheric corrector plate of a Schmidt telescope is displaced longitudinally from the aperture stop along the optical axis, the beam for an off-axis field point will displace along the corrector plate. The amount of relative beam displacement defined by Eq. (2) depends on the paraxial quantities for the marginal ray height,y, the chief ray height, y ¯ , the chief ray angle, u ¯ , and the distance between the stop and the corrector plate, t .

Fig. 2
Fig. 2

Generation of coma and astigmatism as the aspheric corrector plate in a Schmidt telescope is moved longitudinally (along the optical axis) from the physical aperture stop located at the center of curvature of the spherical primary mirror for various positions (a-d). For each field point in the FFD, the plot symbol conveys the magnitude and orientation of the aberration. (e) Plots of the magnitude of coma and astigmatism generated as the aspheric plate is moved longitudinally for two field points, (0°, 2°) (blue square) and (0°, 4°) (red triangle).

Fig. 3
Fig. 3

Surface map describing the characteristic error induced in some cases by a kinematic three point mount on an optical surface as measured interferometrically, over the full aperture of the part, on-axis. The error is quantified by its magnitude | z 10/11 | MNTERR and its orientation ξ MNTERR 10/11 that is measured clockwise with respect to the y ^ axis. The coordinate system assumes that the part is evaluated looking from the interferometer towards the part. P and V denote where the surface error is a peak rather than a valley.

Fig. 4
Fig. 4

The characteristic field dependence of field linear, field conjugate astigmatism that is generated by a mount-induced trefoil error on an optical surface placed away from the aperture stop of an optical system.

Fig. 5
Fig. 5

(a) In the presence of conventional third order field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a two mirror telescope with three point mount-induced error on the secondary mirror, the astigmatic field dependence displays four nodes, i.e., quadranodal behavior. The nodal behavior is displayed in a reduced field coordinate, Π . The node located by 2( x ¯ MNTERR 222 ) has an orientation angle of ξ MNTERR 10/11 and a magnitude that is proportional to | C MNTERR 333,SM | . The two related nodes on the equilateral triangle are then advanced by 120° and 240° for this special case. (b) A measurement or simulation of the mount-induced error on the secondary mirror yields the magnitude and orientation of C MNTERR 333,SM .

Fig. 6
Fig. 6

(a) Layout for a F/8, 300 mm Ritchey-Chrétien telescope and (b) a Full Field Display (FFD) of the RMS wavefront error (RMS WFE) of the optical system at 0.633µm over a ± 0.2° FOV. Each circle represents the magnitude of the RMS WFE at a particular location in the FOV.

Fig. 7
Fig. 7

Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for (a) a Ritchey-Chrétien telescope in its nominal state and (b) the telescope when 0.5λ of three point mount-induced error oriented at 0° has been added to the secondary mirror. It is important to recognize that these displays of data are full field displays that are based on a Zernike polynomial fit to real ray trace optical path difference data evaluated on a grid of points in the FOV. For each field point, the plot symbol conveys the magnitude and orientation of the Zernike coefficients pairs, Z5/6 on the left and Z10/11 on the right.

Fig. 8
Fig. 8

(a) Layout for a JWST-like telescope geometry and (b) a Full Field Display (FFD) of the RMS WFE of the optical system at 1.00 µm over a ± 0.2° FOV. The system utilizes a field bias (outlined in red) to create an accessible focal plane.

Fig. 9
Fig. 9

Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for (a) a JWST-like telescope in its nominal state and (b) the telescope when 0.5λ of three point mount-induced error oriented at 0° has been added to the secondary mirror.

Fig. 10
Fig. 10

Displays of the magnitude and orientation of FRINGE Zernike astigmatism (Z5/6) and FRINGE Zernike trefoil, elliptical coma, (Z10/11) throughout the FOV for a JWST-like telescope with 0.5λ of three point mount-induced error oriented at 0° on the off-axis tertiary mirror.

Fig. 11
Fig. 11

In the presence of conventional third order field quadratic astigmatism and Zernike trefoil at a surface away from the stop, e.g., a two mirror telescope with a three point mount-induced error on the secondary mirror, the astigmatic field dependence displays four nodes, i.e., quadranodal behavior. (a) The nodal behavior is displayed in a reduced field coordinate, Π , where the node located by 2( x ¯ MNTERR 222 ) has an orientation angle of ξ MNTERR 10/11 and a magnitude that is proportional to | C MNTERR 333,SM | . The two related nodes on the equilateral triangle are then advanced by 120° and 240° for this special case. (b) When the nodal solutions are re-mapped to the conventional field coordinate, H , the node located by 2( x ¯ MNTERR 222 ) has an orientation angle of ξ MNTERR 10/11 and a magnitude that is proportional to | C MNTERR 333,SM 3 | . The two related nodes on the equilateral triangle are then advanced by 120° and 240°.

Tables (1)

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Table 1 Field aberration terms that are generated from the longitudinal shift of an aspheric plate from the stop surface in a Schmidt telescope.

Equations (31)

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W Corrector,Stop = W 040 ( ASPH ) ( ρ · ρ ) 2 ,
Δ h ( y ¯ y ) H =( u ¯ t y ) H ,
W Corrector,NotStop = W 040 ( ASPH ) [ ( ρ +Δ h )·( ρ +Δ h ) ] 2 = W 040 ( ASPH ) [ ( ρ · ρ ) 2 +4( Δ h · ρ )( ρ · ρ )+4( Δ h ·Δ h )( ρ · ρ ) +2( Δ h 2 · ρ 2 )+4( Δ h ·Δ h )( Δ h · ρ )+ ( Δ h ·Δ h ) 2 ].
( Z 10 Z 11 )=( z 10 ρ 3 cos( 3ϕ ) z 11 ρ 3 sin( 3ϕ ) ),
| z 10/11 | MNTERR = z 10 2 + z 11 2
ξ MNTERR 10/11 Test = 1 3 tan 1 ( z 11 z 10 ),
if ρ =ρ( sin( ϕ ) cos( ϕ ) ),then ρ 3 = ρ 3 ( sin( 3ϕ ) cos( 3ϕ ) ),
ξ MNTERR 10/11 = 1 3 tan 1 ( z 10 z 11 ).
W 333,Stop = 1 4 ( C MNTERR 333 3 · ρ 3 ),
C MNTERR 333 3 4 | z 10/11 | MNTERR exp( i3 ξ MNTERR 10/11 ).
W 333,NotStop = 1 4 [ C MNTERR 333 3 · ( ρ +Δ h ) 3 ] = 1 4 [ C MNTERR 333 3 · ρ 3 +3 C MNTERR 333 3 ·Δ h ρ 2 +3 C MNTERR 333 3 ·Δ h 2 ρ + C MNTERR 333 3 ·Δ h 3 ].
A · B C = A B * · C ,
B * =| B |exp( iβ )= B x x ^ + B y y ^ .
W 333,NotStop = 1 4 [ C MNTERR 333 3 · ρ 3 +3 C MNTERR 333 3 Δ h * · ρ 2 +3 C MNTERR 333 3 Δ h *2 · ρ + C MNTERR 333 3 ·Δ h 3 ].
3 4 C MNTERR 333 3 Δ h * · ρ 2 = 3 4 ( y ¯ j y j ) C MNTERR 333,j 3 H * · ρ 2 ,
W AST =[ 1 2 W 222 H 2 3 4 ( y ¯ SM y SM ) C MNTERR 333,SM 3 H * ]· ρ 2 ,
1 2 W 222 H 2 3 4 ( y ¯ SM y SM ) C MNTERR 333,SM 3 H * = 0 .
1= H H * | H | 2 = H * H | H | 2 ,
1 | H | 2 [ 1 2 W 222 H 3 3 4 ( y ¯ SM y SM ) C MNTERR 333,SM 3 H * H ] H * = 0 .
[ 1 2 W 222 H 3 | H | 2 3 4 ( y ¯ SM y SM ) C MNTERR 333,SM 3 ] H * = 0 .
2( x ¯ MNTERR 222 ),( x ¯ MNTERR 222 )+i 3 ( x ˜ MNTERR 222 ),( x ¯ MNTERR 222 )i 3 ( x ˜ MNTERR 222 ),
W AST =[ 3 4 ( y ¯ SM y SM ) C MNTERR 333,SM 3 H * ]· ρ 2 .
( σ FF ) j = ( δ v FF * ) j y ¯ j ,
( H FF ) j = H ( σ FF ) j .
W AST =[ 1 2 W 222 H 2 3 4 ( y ¯ j y j ) C MNTERR 333,j 3 ( H FF * ) j ]· ρ 2 .
W AST =[ 3 4 ( y ¯ j y j ) C MNTERR 333,j 3 ( H FF * ) j ]· ρ 2 ,
H 3 | H | 2 3 2 W 222 ( y ¯ SM y SM ) C MNTERR 333,SM 3 = 0 .
Π 3 H 3 | H | 2 = | H | 3 e i3θ | H | 2 =| H | e i3θ = ( | H | 1 3 ) 3 e i3θ ,
Π 3 3 2 W 222 ( y ¯ SM y SM ) C MNTERR 333,SM 3 = 0 .
x ¯ MNTERR 222 = x ˜ MNTERR 222 = [ 3 2 W 222 ( y ¯ SM y SM ) ] 1 3 ( C MNTERR 333,SM 3 ) 1 3 .
2( x ¯ MNTERR 222 ),( x ¯ MNTERR 222 )+i 3 ( x ˜ MNTERR 222 ),( x ¯ MNTERR 222 )i 3 ( x ˜ MNTERR 222 ).

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