Abstract

We present the effects of misalignments on the field dependence of the third-order aberration fields of traditional, two-mirror astronomical telescopes in the context of nodal aberration theory, which we believe is the most general and extensible framework for describing and improving on-station performance. While many of the advantages of nodal aberration theory, compared to other, often power series expansion-based descriptions of misalignment effects on aberrations, become particularly important when analyzing telescopes with more than two mirrors, or in the presence of figure errors; this paper aims to provide and demonstrate the fundamental concepts needed to fully describe the state of correction of misaligned two-mirror telescopes. Importantly, it is shown that the assumption that perfect performance on axis ensures a fully aligned telescope is false, and we demonstrate that if Ritchey–Chrétien telescopes are aligned for zero coma on axis as the sole criterion, formidable misalignments will likely remain, leading to image quality degradation, particularly beyond midfield caused by astigmatism with binodal field dependence (i.e., astigmatism goes to zero at two points in the field).

© 2010 Optical Society of America

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References

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  1. R. N. Wilson, Reflecting Telescope Optics, 2nd ed. (Springer-Verlag, 2004), Vols. 1 and 2.
  2. R. K. Bhatia, “Telescope alignment: is the zero-coma condition sufficient?,” Proc. SPIE 2479, 354–363 (1995).
    [CrossRef]
  3. B. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).
    [CrossRef]
  4. 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]
  5. R. V. Shack and K. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
  6. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).
  7. 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. A 26, 1503–1517 (2009).
    [CrossRef]
  8. H. H. Hopkins, in The Wave Theory of Aberrations (Oxford on Clarendon, 1950).
  9. D. Hestenes, “Oersted Medal lecture 2002: reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121(2003).
    [CrossRef]
  10. J. R. Fienup and C. C. Wackermann, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907(1986).
    [CrossRef]
  11. C. Roddier and F. Roddier, “Wave-front reconstruction from defocused images and the testing of ground-based optical telescopes,” J. Opt. Soc. Am. A 10, 2277–2287 (1993).
    [CrossRef]
  12. D. J. Schroeder, Astronomical Optics (Academic, 1987).
  13. K. P. Thompson, “Aberrations fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).
  14. 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]
  15. 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]
  16. K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16, 20345–20353 (2008).
    [CrossRef] [PubMed]

2010 (1)

2009 (2)

2008 (1)

2005 (1)

2003 (1)

D. Hestenes, “Oersted Medal lecture 2002: reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121(2003).
[CrossRef]

1996 (1)

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

1995 (1)

R. K. Bhatia, “Telescope alignment: is the zero-coma condition sufficient?,” Proc. SPIE 2479, 354–363 (1995).
[CrossRef]

1993 (1)

1986 (1)

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).

Bhatia, R. K.

R. K. Bhatia, “Telescope alignment: is the zero-coma condition sufficient?,” Proc. SPIE 2479, 354–363 (1995).
[CrossRef]

Buchroeder, R. A.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).

Cakmakci, O.

Fienup, J. R.

Hestenes, D.

D. Hestenes, “Oersted Medal lecture 2002: reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121(2003).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, in The Wave Theory of Aberrations (Oxford on Clarendon, 1950).

McLeod, B.

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

Roddier, C.

Roddier, F.

Rolland, J. P.

Schmid, T.

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.

Wackermann, C. C.

Wilson, R. N.

R. N. Wilson, Reflecting Telescope Optics, 2nd ed. (Springer-Verlag, 2004), Vols. 1 and 2.

Am. J. Phys. (1)

D. Hestenes, “Oersted Medal lecture 2002: reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121(2003).
[CrossRef]

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

Opt. Express (2)

Proc. SPIE (2)

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

R. K. Bhatia, “Telescope alignment: is the zero-coma condition sufficient?,” Proc. SPIE 2479, 354–363 (1995).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

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

Other (5)

H. H. Hopkins, in The Wave Theory of Aberrations (Oxford on Clarendon, 1950).

R. N. Wilson, Reflecting Telescope Optics, 2nd ed. (Springer-Verlag, 2004), Vols. 1 and 2.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, 1976).

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

K. P. Thompson, “Aberrations fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, 1980).

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Figures (11)

Fig. 1
Fig. 1

FFD for FRINGE Zernike coefficients (a) Z 9 (spherical aberration), (b) Z 7 and Z 8 (coma), (c) Z 5 and Z 6 (astigmatism), (d) Z 4 (medial focal surface), and (e) distortion.

Fig. 2
Fig. 2

Concept of the shifted aberration field center and the effective field height.

Fig. 3
Fig. 3

FFDs for FRINGE Zernike coefficients Z 7 and Z 8 for an (a) aligned and (b) misaligned Cassegrain or Gregorian telescope.

Fig. 4
Fig. 4

(a) Wave aberration coefficients for coma for the Mount Hopkins telescope, (b) aberration field vector for coma for a misaligned secondary mirror configuration of the Mount Hopkins telescope (secondary mirror perturbations: x decenter, 932 μm ; y decenter, 0 μm ; x tilt, 6 arcmin ; and y tilt, 6 arcmin ), and (c) FFD for coupled FRINGE Zernike coefficients Z 7 and Z 8 , computed using only real ray tracing combined with a FRINGE Zernike polynomial wavefront fitting algorithm.

Fig. 5
Fig. 5

(a) Wave aberration contributions for astigmatism for the Mount Hopkins telescope, (b) sigma vectors (spherical and aspheric) after collimating the initially misaligned telescope (secondary mirror perturbations: x decenter, 932 μm ; y decenter, 0 μm ; x tilt, 6 arcmin ; and y tilt, 6 arcmin ) to zero coma (secondary mirror perturbations: x decenter, 929 μm ; y decenter, 917 μm ; x tilt, 6 arcmin ; and y tilt, 6 arcmin ), and squared sigma vectors, (c) construction of the a 222 vector denoting the midpoint between the two astigmatic nodes, and the squared vector a 222 2 , (d) construction of the b 222 2 vector, (e) construction of the b 222 vector, and (f) construction of the astigmatic node locations ( a 222 ± i b 222 ).

Fig. 6
Fig. 6

FFD for coupled FRINGE Zernike coefficients Z 5 and Z 6 .

Fig. 7
Fig. 7

Sensitivity of the eccentricity of the obscuration shadow in defocused images on alignment-induced (field-constant) coma ( Z 7 / Z 8 ).

Fig. 8
Fig. 8

Concept of the coma-free pivot point in two-mirror telescopes.

Fig. 9
Fig. 9

Through-focus photographic plate taken with the 90 in. telescope of the Steward Observatory, located on Kitt Peak. The boxes to the left and right are highly magnified regions of the star plate.

Fig. 10
Fig. 10

(a) FFD for FRINGE Zernike coefficients Z 5 and Z 6 for the aligned telescope, (b) FFD for FRINGE Zernike coefficients Z 5 and Z 6 for the misaligned telescope, (c) point spread functions computed for three field points through focus for the aligned telescope, and (d) point spread functions computed for three field points through focus for the misaligned telescope.

Fig. 11
Fig. 11

Geometric illustration of the concept of vector multiplication.

Equations (31)

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W = j p n m W ( k l m ) j ( H A j · H A j ) p ( ρ · ρ ) n ( H A j · ρ ) m ,
H A j = H σ j ,
W COMA = j W 131 j [ ( H σ j ) · ρ ] ( ρ · ρ ) ,
a 131 = 1 W 131 j ( W 131 j σ j ) ,
a 131 = 1 W 131 j ( W 131 j ( sph ) σ j ( sph ) + W 131 j ( asph ) σ j ( asph ) ) ,
W COMA = [ A 131 · ρ ] ( ρ · ρ ) ,
A 131 = j W 131 j σ j ,
A 131 = j ( W 131 j ( sph ) σ j ( sph ) + W 131 j ( asph ) σ j ( asph ) ) ,
W AST = 1 2 j W 222 j [ ( H σ j ) 2 ρ 2 ] .
W AST = 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] ρ 2 ,
a 222 A 222 W 222 = 1 W 222 j ( W 222 j σ j ) ,
b 222 2 B 222 2 W 222 a 222 2 = 1 W 222 j ( W 222 j σ j 2 ) a 222 2 ,
a 222 = 1 W 222 j ( W 222 j ( sph ) σ j ( sph ) + W 222 j ( asph ) σ j ( asph ) ) ,
b 222 2 = 1 W 222 j ( W 222 j ( sph ) σ j ( sph ) 2 + W 222 j ( asph ) σ j ( asph ) 2 ) a 222 2 ,
a = | δ z δ z 0 δ z + δ z 0 | ,
Z 56 = δ z 8 F # 2 ,
ϕ = 0.5 tan 1 ( Z 5 Z 6 ) ,
c = a b = [ a x a y ] [ b x b y ] = [ a y b x + a x b y a y b y a x b x ] ,
c = A B = | A | | B | exp ( i ( α + β ) ) .
W AST = 1 2 j [ W 222 j ( H σ j ) 2 ρ 2 ] .
W = 1 2 [ j W 222 j H 2 2 H ( j W 222 j σ j ) + j W 222 j σ j 2 ] ρ 2 .
A 222 j W 222 j σ j ,
B 2 222 j W 222 j σ j 2 ,
W AST = 1 2 [ W 222 H 2 2 H A 222 + B 222 2 ] ρ 2 ,
H 2 = [ 2 H x H y H y 2 H x 2 ] ,
A 222 H = [ A 222 , y H x + A 222 , x H y A 222 , y H y A 222 , x H x ] ,
B 222 2 = [ 2 B 222 , x B 222 , y B 222 , y 2 B 222 , x 2 ] .
W AST , x = W 222 H x H y ( A 222 y H x + A 222 x H y ) + B 222 x B 222 y ,
W AST , y = 1 2 W 222 ( H y 2 H x 2 ) + ( A 222 x H x A 222 y H y ) + 1 2 ( B 222 y 2 B 222 x 2 ) .
Z 6 = 2 B 0 H x H y + B 1 ( α y H x + α x H y ) + 2 B 2 α x α y ,
Z 5 = B 0 ( H y 2 H x 2 ) + B 1 ( α x H x α y H y ) + B 2 ( α x 2 α y 2 ) .

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