Abstract

Misalignments always occur in real optical systems. These misalignments do not generate new aberration forms, but they change the aberration field dependence. Two-mirror telescopes have been used in several applications. We analyze a two-mirror telescope configuration that has negligible sensitivity to decenter misalignments. By applying the wave aberration theory for plane-symmetric optical systems it is shown that the asphericity in the secondary mirror, if properly chosen, can compensate for any decenter perturbation allowing third-order coma unchanged across the field of view. For any two-mirror system it is possible to find a configuration in which decenter misalignments do not generate field-uniform coma.

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References

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  1. W. J. Smith, Modern Optical Engineering 4th ed. (McGraw Hill, 2008), Chap.5 74, Chap.18 508–514.
  2. J. M. Sasian, “The world of unobstructed reflecting telescopes” in ATM Journal Issue 1 10–15 (1992).
  3. T. Schmid, K. P. Thompson, and J. P. Rolland, “Misalignment-induced nodal aberration fields in two-mirror astronomical telescopes,” Appl. Opt.49(16), D131–D144 (2010).
    [CrossRef] [PubMed]
  4. J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng.33(6), 2045–2061 (1994).
    [CrossRef]
  5. R. N. Wilson, Reflecting Telescope OpticsVol. II 2nd ed. (Springer, 2001), Chap.2 105–119.
  6. 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]
  7. T. Schmid, K. P. Thompson, and J. P. Rolland, “A unique astigmatic nodal property in misaligned Ritchey-Chrétien telescopes with misalignment coma removed,” Opt. Express18(5), 5282–5288 (2010).
    [CrossRef] [PubMed]
  8. A. M. Manuel, “Field-dependent aberrations for misaligned reflective optical systems” PhD. Dissertation, Optical Sciences Center, University of Arizona (2009).
  9. B. A. McLeod, “Collimation of fast wide-field telescopes,” Publ. Astron. Soc. Pac.108, 217–219 (1996).
    [CrossRef]
  10. G. Catalan, “Design method of an astronomical telescope with reduced sensitivity to misalignment,” Appl. Opt.33(10), 1907–1915 (1994).
    [CrossRef] [PubMed]
  11. R. N. Wilson, Reflecting Telescope OpticsVol.I 2nd ed. (Springer, 2000), Chap. 3 61–62, 260–267.
  12. K. P. 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]
  13. M. Bottema and R. A. Woodruff, “Third order aberrations in Cassegrain-type telescopes and coma correction in servo-stabilized images,” Appl. Opt.10(2), 300–303 (1971).
    [CrossRef] [PubMed]
  14. B. Ren, G. Jin, and X. Zhong, “Third-order coma-free point in two-mirror telescopes by a vector approach,” Appl. Opt.50(21), 3918–3923 (2011).
    [CrossRef] [PubMed]
  15. O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003).
    [CrossRef]

2011 (1)

2010 (2)

2009 (1)

2005 (1)

2003 (1)

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003).
[CrossRef]

1996 (1)

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

1994 (2)

G. Catalan, “Design method of an astronomical telescope with reduced sensitivity to misalignment,” Appl. Opt.33(10), 1907–1915 (1994).
[CrossRef] [PubMed]

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng.33(6), 2045–2061 (1994).
[CrossRef]

1992 (1)

J. M. Sasian, “The world of unobstructed reflecting telescopes” in ATM Journal Issue 1 10–15 (1992).

1971 (1)

Bottema, M.

Cakmakci, O.

Catalan, G.

Guyon, O.

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003).
[CrossRef]

Jin, G.

McLeod, B. A.

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

Ren, B.

Rolland, J. P.

Sasian, J. M.

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng.33(6), 2045–2061 (1994).
[CrossRef]

J. M. Sasian, “The world of unobstructed reflecting telescopes” in ATM Journal Issue 1 10–15 (1992).

Schmid, T.

Thompson, K. P.

Woodruff, R. A.

Zhong, X.

Appl. Opt. (4)

Astron. Astrophys. (1)

O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys.404(1), 379–387 (2003).
[CrossRef]

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

Opt. Eng. (1)

J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng.33(6), 2045–2061 (1994).
[CrossRef]

Opt. Express (1)

Publ. Astron. Soc. Pac. (1)

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

Other (5)

R. N. Wilson, Reflecting Telescope OpticsVol.I 2nd ed. (Springer, 2000), Chap. 3 61–62, 260–267.

R. N. Wilson, Reflecting Telescope OpticsVol. II 2nd ed. (Springer, 2001), Chap.2 105–119.

W. J. Smith, Modern Optical Engineering 4th ed. (McGraw Hill, 2008), Chap.5 74, Chap.18 508–514.

J. M. Sasian, “The world of unobstructed reflecting telescopes” in ATM Journal Issue 1 10–15 (1992).

A. M. Manuel, “Field-dependent aberrations for misaligned reflective optical systems” PhD. Dissertation, Optical Sciences Center, University of Arizona (2009).

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Figures (7)

Fig. 1
Fig. 1

Two-mirror telescope configuration.

Fig. 2
Fig. 2

The left hand graphic shows the on-axis sensitivity for systems with different combinations of conic constants of the mirrors as a function of the x decenter of the secondary. The right hand side graphic exhibits the on-axis sensitivity due to x tilts. The conic constants of the primary mirror for the different systems are shown in the legend. RMS in waves (λ = 0.55µm).

Fig. 3
Fig. 3

Graphics of the Zernike Fringe Polynomial coefficients: A) Z 5 ; B) Z 6 ; C) Z 7 ; D) Z 8 and E) RMS referenced to the centroid (waves, λ = 0.55µm) as a function of tilt perturbation for the classical Cassegrain, the RC and the LS configurations.

Fig. 4
Fig. 4

Graphics of the Zernike Fringe Polynomial coefficients: A) Z 5 ; B) Z 6 ; C) Z 7 ; D) Z 8 and E) RMS referenced to the centroid (waves, λ = 0.55µm) as a function of decenter perturbation for the classical Cassegrain, the RC and the LS configurations.

Fig. 5
Fig. 5

Decenter of a spherical surface.

Fig. 6
Fig. 6

Graphic of ZCEP as a function of the conic constant of the secondary mirror for two systems with different BFL and mirrors separation distance.

Fig. 7
Fig. 7

Graphic of coefficient Z 7 (Zernike fringe polynomial), on-axis as a function of decenter perturbation for the LS configuration considering the nominal value of k p (−0.4) and k s (1.934) and three other values for k s with Δ k s of 4E-3, 3E-2 and 1E-1. The conic constant of the secondary mirror for the different error conditions is shown in the legend.

Tables (2)

Tables Icon

Table 1 Conic constants of the primary and secondary mirrors for the different configurations.

Tables Icon

Table 2 Conic constants, M p aspheric terms, residual linear coma and quadratic astigmatism for the classical Cassegrain, RC and LS configurations for systems with 3750mm EFL, 1200mm BFL and distance between mirrors of 800mm.

Equations (40)

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R p = 2df Bf ,
R s = 2dB B+df   .
k p =( B f ). ( B+df ) ( Bf ) 3 .[ ( B+df ) 2 k s + ( f+dB ) 2 ]1  .
k p = R p 3 ( fB ) 3 8 d 3 f 3 ,
k s = R s 3 ( fdB ) ( f+dB ) 2 8 d 3 B 3 .
k s = R s 3 [ 2f (Bf) 2 +(fdB)( f+dB )( dfB ) ] 8 d 3 B 3 .
W 03001 o ={ J II p + J II s }= 1 2 [ n p sin( I p ) A p Δ ( u n ) p x p + n s sin( I s ) A s Δ ( u n ) s x s ].
A p = m 2f x p cos( I p ),
A s = m f [ 1 ( m+1 ) 2m cos( I s ) ] x p ,
Δ ( u n ) p = m f x p ,
Δ ( u n ) s = m f ( 1+ 1 m ) x p .
W 03001 0 = 1 2 [ 1 2 ( m f ) 2 sin( I p )cos( I p )+ Bm f 3 sin( I s )(m+1)[ 1 (m+1) 2m cos( I s ) ] ] x p 3 ,
W 03001 o = Bm 2 f 3 sin( I s )( m+1 )[ 1 ( m+1 ) 2m cos( I s ) ] x p. 3
sin( I s )= δ sphere R s .
W 03001 o = 1 8 ( x p f ) 3 ( m+1 ) 2 ( m1 ) δ sphere .
W 03001 * =βΔ[ ncos(I) ] x 3 .
Z= ( x 2 + y 2 ) 2R +( k+1 ) ( x 2 + y 2 ) 2 8 R 3  ,
Z asphere = k s ( x s 2 + y s 2 ) 2 8 R s 3 .
W * = k s ( x s 2 + y s 2 ) 2 8 R s 3 Δ ( n ) s .
W decenter * = k s ( ( x s + δ asphere ) 2 + y s 2 ) 2 8 R s 3 Δ ( n ) s .
W 03001 * = k s 4 x s 3 δ asphere 8 R s 3 Δ ( n ) s .
W 03001 * = 1 8 ( x p f ) 3 ( m+1 ) 3 k s δ asphere .
W 03001 = 1 8 ( x p f ) 3 ( m+1 ) 2 [ ( m1 ) δ sphere ( m+1 ) k s δ asphere ].
W 03001 = 1 8 ( x p f ) 3 ( m+1 ) 2 [ ( m1 )( m+1 ) k s ]δ.
[ ( m1 )( m+1 ) k s ]=0,
k s = ( m1 ) ( m+1 ) .
k s = Bfd Bf+d  .
Z CFP = 2f(m1) R A (m+1) 2 [ (m1) (m+1) k s ]  ,
W 03001 residual = 1 8 ( x p f ) 3 ( m+1 ) 2 [ ( m+1 )Δ k s ] δ asphere .
W(H,ρ)= k,m,n,p,q W 2k+n+p,2m+n+q,n,p,q (HH) k (ρρ) m (Hρ) n (iH) p (iρ) q ,
W 13100 (Hρ)(ρρ),
W 03001 (iρ)(ρρ).
W 03001 = W 03001 o + W 03001 * .
W 03001 0 = i=1 i=j { J II } i ,
J II = 1 2 nsin( I )AΔ( u n )x.
A=ni=n( u+ xcos(I) R ),
Δ( u n )= u' n' u n   .
Z= Z sphere + Z asphere ,
Z asphere =α ( iρ ) 2 +β( iρ )( ρρ )+γ ( ρ.ρ ) 2 .
W 03001 * =βΔ[ ncos(I) ] x 3 .

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