Abstract

In this paper, two-mirror telescopes having the secondary mirror decentered and/or tilted are considered. Equations for third-order coma are derived by a vector approach. Coma-free condition to remove misalignment-induced coma was obtained. The coma-free point in two-mirror telescopes is found as a conclusion of our coma-free condition, which is in better agreement with the result solved by Wilson using Schiefspiegler theory.

© 2011 Optical Society of America

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References

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  1. B. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).
    [CrossRef]
  2. R. N. Wilson and B. Delabre, “Concerning the alignment of modern telescopes: theory, practice, and tolerances illustrated by the ESO NTT,” Publ. Astron. Soc. Pac. 109, 53–60 (1997).
    [CrossRef]
  3. W. B. Wetherell and M. P. Rimmer, “General analysis of aplanatic Cassegrain, Gregorian, and Schwarzschild telescopes,” Appl. Opt. 11, 2817–2832 (1972).
    [CrossRef] [PubMed]
  4. R. N. Wilson, Reflecting Telescope Optics I (Springer, 1996).
  5. D. J. Schroeder, Astronomical Optics (Academic, 1987).
  6. R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
  7. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389–1401 (2005).
    [CrossRef]
  8. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry; spherical aberration,” J. Opt. Soc. Am. A 26, 1090–1100 (2009).
    [CrossRef]
  9. T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
    [CrossRef]
  10. T. Schmid, K. P. Thompson, and J. P. Rolland, “A unique astigmatic nodal property in misaligned Ritchey-Chrétien telescopes with misalignment coma removed,” Opt. Express 18, 5282–5288 (2010).
    [CrossRef] [PubMed]
  11. L. Noethe and S. Guisard, “Analytical expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” Astron. Astrophys. Suppl. Ser. 144, 157–167 (2000).
    [CrossRef]
  12. A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IThB7P.
  13. H. H. Hopkins, Wave Theory of Aberrations (Oxford, 1950).
  14. W. Welford, Aberration of the Symmetrical Optical System (Academic, 1974).
  15. J. M. Sasian, “How to approach the design of bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
    [CrossRef]

2010 (2)

A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IThB7P.

T. Schmid, K. P. Thompson, and J. P. Rolland, “A unique astigmatic nodal property in misaligned Ritchey-Chrétien telescopes with misalignment coma removed,” Opt. Express 18, 5282–5288 (2010).
[CrossRef] [PubMed]

2009 (1)

2008 (1)

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[CrossRef]

2005 (1)

2000 (1)

L. Noethe and S. Guisard, “Analytical expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” Astron. Astrophys. Suppl. Ser. 144, 157–167 (2000).
[CrossRef]

1997 (1)

R. N. Wilson and B. Delabre, “Concerning the alignment of modern telescopes: theory, practice, and tolerances illustrated by the ESO NTT,” Publ. Astron. Soc. Pac. 109, 53–60 (1997).
[CrossRef]

1996 (2)

B. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).
[CrossRef]

R. N. Wilson, Reflecting Telescope Optics I (Springer, 1996).

1994 (1)

J. M. Sasian, “How to approach the design of bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

1987 (1)

D. J. Schroeder, Astronomical Optics (Academic, 1987).

1980 (1)

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

1974 (1)

W. Welford, Aberration of the Symmetrical Optical System (Academic, 1974).

1972 (1)

1950 (1)

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

Delabre, B.

R. N. Wilson and B. Delabre, “Concerning the alignment of modern telescopes: theory, practice, and tolerances illustrated by the ESO NTT,” Publ. Astron. Soc. Pac. 109, 53–60 (1997).
[CrossRef]

Guisard, S.

L. Noethe and S. Guisard, “Analytical expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” Astron. Astrophys. Suppl. Ser. 144, 157–167 (2000).
[CrossRef]

Hopkins, H. H.

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

McLeod, B.

B. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).
[CrossRef]

Noethe, L.

L. Noethe and S. Guisard, “Analytical expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” Astron. Astrophys. Suppl. Ser. 144, 157–167 (2000).
[CrossRef]

Rakich, A.

A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IThB7P.

Rimmer, M. P.

Rolland, J. P.

T. Schmid, K. P. Thompson, and J. P. Rolland, “A unique astigmatic nodal property in misaligned Ritchey-Chrétien telescopes with misalignment coma removed,” Opt. Express 18, 5282–5288 (2010).
[CrossRef] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[CrossRef]

Sasian, J. M.

J. M. Sasian, “How to approach the design of bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Schmid, T.

T. Schmid, K. P. Thompson, and J. P. Rolland, “A unique astigmatic nodal property in misaligned Ritchey-Chrétien telescopes with misalignment coma removed,” Opt. Express 18, 5282–5288 (2010).
[CrossRef] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[CrossRef]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, 1987).

Shack, R. V.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Thompson, K.

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Thompson, K. P.

Welford, W.

W. Welford, Aberration of the Symmetrical Optical System (Academic, 1974).

Wetherell, W. B.

Wilson, R. N.

R. N. Wilson and B. Delabre, “Concerning the alignment of modern telescopes: theory, practice, and tolerances illustrated by the ESO NTT,” Publ. Astron. Soc. Pac. 109, 53–60 (1997).
[CrossRef]

R. N. Wilson, Reflecting Telescope Optics I (Springer, 1996).

Appl. Opt. (1)

Astron. Astrophys. Suppl. Ser. (1)

L. Noethe and S. Guisard, “Analytical expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” Astron. Astrophys. Suppl. Ser. 144, 157–167 (2000).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

J. M. Sasian, “How to approach the design of bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[CrossRef]

Opt. Express (1)

Proc. SPIE (2)

T. Schmid, K. P. Thompson, and J. P. Rolland, “Alignment induced aberration fields of next generation telescopes,” Proc. SPIE 7068, 70680E (2008).
[CrossRef]

R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Publ. Astron. Soc. Pac. (2)

B. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).
[CrossRef]

R. N. Wilson and B. Delabre, “Concerning the alignment of modern telescopes: theory, practice, and tolerances illustrated by the ESO NTT,” Publ. Astron. Soc. Pac. 109, 53–60 (1997).
[CrossRef]

Other (5)

R. N. Wilson, Reflecting Telescope Optics I (Springer, 1996).

D. J. Schroeder, Astronomical Optics (Academic, 1987).

A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IThB7P.

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

W. Welford, Aberration of the Symmetrical Optical System (Academic, 1974).

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Figures (5)

Fig. 1
Fig. 1

Field displacement vectors for surface contribution. σ j o is for the center of symmetry for the base sphere, and σ j * is for the center of symmetry for the aspheric departure of the jth surface.

Fig. 2
Fig. 2

Tilted and decentered two-mirror telescope: the stop is at the primary mirror; tilt and decenter are both in the meridian plane.

Fig. 3
Fig. 3

Gaussian optics of a Cassegrain two-mirror telescope.

Fig. 4
Fig. 4

Pure decenter of the center of curvature of the secondary mirror C 2 .

Fig. 5
Fig. 5

Pure decenter of the aspheric vertex V 2 of the secondary mirror.

Equations (35)

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W coma , j = W 131 , j H ρ 3 cos ϕ ,
W coma , j = W 131 , j ( H · ρ ) ( ρ · ρ ) .
W coma = j W coma , j .
H j o = H σ j o ,
H j * = H σ j * ,
W coma , j o = W 131 , j o ( H j o · ρ ) ( ρ · ρ ) = W 131 , j o [ ( H σ j o ) · ρ ] ( ρ · ρ ) ,
W coma , j * = W 131 , j * ( H j * · ρ ) ( ρ · ρ ) = W 131 , j * [ ( H σ j * ) · ρ ] ( ρ · ρ ) ,
W coma , j = W coma , j o + W coma , j * = W 131 , j o [ ( H σ j o ) · ρ ] ( ρ · ρ ) + W 131 , j * [ ( H σ j * ) · ρ ] ( ρ · ρ ) = ( W 131 , j o + W 131 , j * ) ( H · ρ ) ( ρ · ρ ) [ ( W 131 , j o σ j o + W 131 , j * σ j * ) · ρ ] ( ρ · ρ ) = W 131 , j ( H · ρ ) ( ρ · ρ ) [ ( W 131 , j o σ j o + W 131 , j * σ j * ) · ρ ] ( ρ · ρ ) .
W coma , 1 = ( W 131 , 1 o + W 131 , 1 * ) ( H · ρ ) ( ρ · ρ ) = W 131 , 1 ( H · ρ ) ( ρ · ρ ) .
W coma , 2 = W 131 , 2 ( H · ρ ) ( ρ · ρ ) [ ( W 131 , 2 o σ 2 o + W 131 , 2 * σ 2 * ) · ρ ] ( ρ · ρ ) .
W coma = W coma , 1 + W coma , 2 = ( W 131 , 1 + W 131 , 2 ) ( H · ρ ) ( ρ · ρ ) [ ( W 131 , 2 o σ 2 o + W 131 , 2 * σ 2 * ) · ρ ] ( ρ · ρ ) .
o c 1 = ( L s 2 ) ( δ C 2 ) s 2 r 2 ,
o c 2 = ( L + d 1 ) ( δ C 2 ) d 1 + r 2 ,
c 1 c 2 = o c 2 o c 1 = ( L s 2 s 2 r 2 + L + d 1 d 1 + r 2 ) δ C 2 = [ ( L r 2 ) ( d 1 + s 2 ) ( s 2 r 2 ) ( d 1 + r 2 ) ] δ C 2 .
σ 2 o = c 1 c 2 u pr 1 f .
W 131 = 1 2 ( y 1 y m 1 ) 3 ( η η m ) S II cos ϕ ,
W 131 , j = 1 2 ( S II ) j .
W 131 , 2 o = 1 2 ( S II ) 2 0 ,
W 131 , 2 * = 1 2 ( S II ) 2 * .
W 131 , 2 o σ 2 o = 1 2 ( S II ) 2 0 σ 2 o ,
W 131 , 2 * σ 2 * = 1 2 ( S II ) 2 * σ 2 * ,
( S II ) 2 = ( y 1 f ) 3 [ d 1 ξ + f 2 ( m 2 2 1 ) s pr 1 L f ξ ] u pr 1 ,
ξ = ξ 0 + ξ * = ( m 2 + 1 ) 3 4 ( m 2 1 m 2 + 1 ) 2 + ( m 2 + 1 ) 3 4 b s 2 .
( S II ) 2 0 = ( y 1 f ) 3 ξ 0 [ d 1 + 2 f m 2 1 ] u pr 1 = 1 4 ( y 1 f ) 3 ( 1 m 2 2 ) ( d 1 + L + f ) u pr 1 ,
( S II ) 2 * = ( y 1 f ) 3 ξ * d 1 u pr 1 = 1 4 ( y 1 f ) 3 ( m 2 + 1 ) 3 b s 2 d 1 u pr 1 .
W 131 , 2 o σ 2 o = 1 8 ( y 1 f ) 3 ( 1 m 2 2 ) ( d 1 + L + f ) u pr 1 c 1 c 2 u pr 1 f = 1 8 ( y 1 f ) 3 ( 1 m 2 2 ) ( m 2 + 1 ) δ C 2 .
δ V 2 c 1 c 3 = ( s 2 ) E L ( s 2 ) E ,
( s 2 ) E = d 1 f 2 d 1 + f 2 .
c 1 c 3 = L ( s 2 ) E ( s 2 ) E δ V 2 = [ 1 L ( d 1 + f 2 ) d 1 f 2 ] δ V 2 .
σ 2 * = c 1 c 3 u pr 1 f .
W 131 , 2 * σ 2 * = 1 8 ( y 1 f ) 3 ( m 2 + 1 ) 3 b s 2 d 1 u pr 1 c 1 c 3 u pr 1 f = 1 8 ( y 1 f ) 3 ( m 2 + 1 ) 3 b s 2 δ V 2 .
( W 131 , 2 o σ 2 o + W 131 , 2 * σ 2 * ) = 1 8 ( y 1 f ) 3 ( 1 m 2 2 ) ( m 2 + 1 ) δ C 2 1 8 ( y 1 f ) 3 ( m 2 + 1 ) 3 b s 2 δ V 2 = 1 8 ( y 1 f ) 3 ( m 2 + 1 ) 2 [ ( 1 m 2 ) δ C 2 + ( m 2 + 1 ) b s 2 δ V 2 ] .
( 1 m 2 ) δ C 2 + ( m 2 + 1 ) b s 2 δ V 2 = 0.
δ C 2 δ V 2 = m 2 + 1 m 2 1 b s 2 = r 2 z CFP z CFP .
z CFP = r 2 / [ 1 ( m 2 + 1 m 2 1 ) b s 2 ] = s 2 / [ ( m 2 + 1 2 m 2 ) { 1 ( m 2 + 1 m 2 1 ) b s 2 } ] .

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