Abstract

Misalignment induced third-order coma with respect to misaligned parameters in TMA optical systems is derived by using Nodal Aberration Theory, which yields the compensation factors that can be used to accomplish coma compensation in both coaxial and off-axis misaligned TMA telescopes. By using the compensation factors, coma free point for the tertiary mirror in TMA telescopes is derived and proved to be the negative form of the one for the secondary mirror in the Cassegrain telescope. The compensation factors can also be used to design the off-axis TMAs due to their capability of eliminating the coma over the field of view.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
    [Crossref]
  2. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).
  3. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
    [Crossref] [PubMed]
  4. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
    [Crossref] [PubMed]
  5. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
    [Crossref] [PubMed]
  6. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
    [Crossref] [PubMed]
  7. K. P. Thompson, T. Schmid, and J. P. Rolland, “Recent Discoveries from Nodal Aberration Theory,” Proc. SPIE 7652, 76522Q (2010).
    [Crossref]
  8. H. Shi, H. Jiang, X. Zhang, C. Wang, and T. Liu, “Analysis of nodal aberration properties in off-axis freeform system design,” Appl. Opt. 55(24), 6782–6790 (2016).
    [Crossref] [PubMed]
  9. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
    [Crossref] [PubMed]
  10. A. Bauer, J. P. Rolland, and K. P. Thompson, “Ray-based optical design tool for freeform optics: coma full-field display,” Opt. Express 24(1), 459–472 (2016).
    [Crossref] [PubMed]
  11. X. Zhang, S. Xu, H. Ma, and N. Liu, “Optical compensation for the perturbed three mirror anastigmatic telescope based on nodal aberration theory,” Opt. Express 25(11), 12867–12883 (2017).
    [Crossref] [PubMed]
  12. 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]
  13. R. N. Wilson, Reflecting Telescope Optics I (Springer, 1996).
  14. K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using Nodal Aberration Theory to understand the aberrations of multiple unobscured Three Mirror Anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
    [Crossref]

2017 (1)

2016 (2)

2012 (1)

2011 (2)

2010 (2)

2009 (2)

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
[Crossref] [PubMed]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using Nodal Aberration Theory to understand the aberrations of multiple unobscured Three Mirror Anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

2005 (1)

1980 (1)

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

Bauer, A.

Fuerschbach, K.

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
[Crossref] [PubMed]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using Nodal Aberration Theory to understand the aberrations of multiple unobscured Three Mirror Anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

Jiang, H.

Jin, G.

Liu, N.

Liu, T.

Ma, H.

Ren, B.

Rolland, J. P.

A. Bauer, J. P. Rolland, and K. P. Thompson, “Ray-based optical design tool for freeform optics: coma full-field display,” Opt. Express 24(1), 459–472 (2016).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, and J. P. Rolland, “Recent Discoveries from Nodal Aberration Theory,” Proc. SPIE 7652, 76522Q (2010).
[Crossref]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using Nodal Aberration Theory to understand the aberrations of multiple unobscured Three Mirror Anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

Schmid, T.

K. P. Thompson, T. Schmid, and J. P. Rolland, “Recent Discoveries from Nodal Aberration Theory,” Proc. SPIE 7652, 76522Q (2010).
[Crossref]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using Nodal Aberration Theory to understand the aberrations of multiple unobscured Three Mirror Anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

Shack, R. V.

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

Shi, H.

Thompson, K.

Thompson, K. P.

A. Bauer, J. P. Rolland, and K. P. Thompson, “Ray-based optical design tool for freeform optics: coma full-field display,” Opt. Express 24(1), 459–472 (2016).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, and J. P. Rolland, “Recent Discoveries from Nodal Aberration Theory,” Proc. SPIE 7652, 76522Q (2010).
[Crossref]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
[Crossref] [PubMed]

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using Nodal Aberration Theory to understand the aberrations of multiple unobscured Three Mirror Anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

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

Wang, C.

Xu, S.

Zhang, X.

Zhong, X.

Appl. Opt. (2)

J. Opt. Soc. Am. A (4)

Opt. Express (3)

Proc. SPIE (3)

K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using Nodal Aberration Theory to understand the aberrations of multiple unobscured Three Mirror Anastigmatic (TMA) telescopes,” Proc. SPIE 7433, 74330B (2009).
[Crossref]

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

K. P. Thompson, T. Schmid, and J. P. Rolland, “Recent Discoveries from Nodal Aberration Theory,” Proc. SPIE 7652, 76522Q (2010).
[Crossref]

Other (2)

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

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

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Figures (11)

Fig. 1
Fig. 1 Gaussian optics of a coaxial TMA telescope
Fig. 2
Fig. 2 The five lines in the object space of the TM in a misaligned TMA system
Fig. 3
Fig. 3 LV2 in the object space of the TM
Fig. 4
Fig. 4 Contribution centers of the SM and the TM
Fig. 5
Fig. 5 αx(βx) of a surface
Fig. 6
Fig. 6 (a) Plot of the coaxial TMA in Table 1. (b) FFD in terms of coma at the image plane in the aligned system. (c) FFD when αy = 0.02° is imposed on the SM.
Fig. 7
Fig. 7 (a) By = 2mm, αy = 0.01°, βy = 0.02° and Ay = –0.0271mm are imposed on the surfaces. (b) On the basis of (a), Bx = 0.5mm, βx = –0.03°, Ax = 0.02mm and αx = 0.006663° are imposed on the surfaces at the same time.
Fig. 8
Fig. 8 An off-axis TMA system
Fig. 9
Fig. 9 (a) Plot of the off-axis TMA in Table 2. (b) FFD when system is aligned. (c) FFD when Δαy = 0.01° is imposed on the SM
Fig. 10
Fig. 10 FFDs in terms of coma at the image field when use (a) ΔAy, (b) ΔBy and (c) Δβy to accomplish the compensation while Δαy = 0.01° are imposed on SM.
Fig. 11
Fig. 11 Residual coma when use the compensation factors to redesign the TMA in Table 2.

Tables (2)

Tables Icon

Table 1 Optical parameters of a coaxial TMA telescope. Units are mm and degrees.

Tables Icon

Table 2 Optical parameters for an off-axial TMA telescope. Units are mm and degrees.

Equations (29)

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W 131 = j W 131j ( H · ρ )( ρ · ρ ) = j W 131j o ( H o j · ρ )( ρ · ρ ) + j W 131j * ( H * j · ρ )( ρ · ρ )
W 131 = j W 131j o (( H σ o j )· ρ )( ρ · ρ ) + j W 131j * (( H σ * j )· ρ )( ρ · ρ ) = j W 131j ( H · ρ )( ρ · ρ ) j W 131j o ( σ o j · ρ )( ρ · ρ ) j W 131j * ( σ * j · ρ )( ρ · ρ )
Δ W 131 = 1 2 ( j S II,j o ( σ o j · ρ )( ρ · ρ ) + j S II,j * ( σ * j · ρ )( ρ · ρ ) )
{ S II,j o = A j A prj y j ( u j n j u j n j ) S II,j * =( y prj y j ) c j 3 ( n j n j ) b sj y j 4
{ S II,2 o = 1 4 ( y 1 f ) 3 m 3 3 ( m 2 +1) ( m 2 1) 2 [ d 1 2 f m 3 ( m 2 1) ] u pr1 S II,2 * = 1 4 ( y 1 f ) 3 m 3 3 ( m 2 +1) 3 d 1 b s2 u pr1
A 3 = n 3 i 3 = u 2 u 3 2 = m 3 1 2 u 3 = 1 m 3 2 y 1 f
u 3 n 3 u 3 n 3 = u 3 u 3 =( m 3 +1) u 3 =( m 3 +1) y 1 f
y 3 = s 3 u 3 = s 3 m 3 m 3 u 3 = y 1 f s 3
c 3 = 1 2 f 3 = m 3 +1 2 s 3
y pr3 = y pr2 + d 2 u pr2 = d 1 u pr1 d 2 ( d 1 f 2 +1) u pr2 =( d 1 d 2 f 2 + d 2 d 1 ) u pr1
{ s pr2 = d 1 f 2 d 1 + f 2 , u pr3 = u pr2 =( d 1 f 2 +1) u pr1 u pr3 =( s pr2 d 2 f 3 1) u pr3 , i pr3 = u pr3 u pr3 2
A pr3 = n 3 i pr3 = ( d 2 +2 f 3 )( d 1 + f 2 ) d 1 f 2 2 f 2 f 3 u pr1
{ S II,3 o = 1 4 ( y 1 f ) 3 ( m 3 +1) 2 (1 m 3 )[ d 1 d 2 f 2 + d 2 d 1 +2 f 3 ( d 1 f 2 +1) ] u pr1 S II,3 * = 1 4 ( y 1 f ) 3 ( m 3 +1) 3 ( d 1 d 2 f 2 + d 2 d 1 ) b s3 u pr1
s E2 = f 2 d 1 d 1 + f 2
P= 2 d 1 (2 f 2 α+A) d 1 + f 2
{ U V2 = PA s E2 , U O2 = A+2 f 2 αP 2 f 2 s E2 U V3 = BP d 2 s E2 , U O3 = B+2 f 3 βP 2 f 3 + d 2 s E2 , U Lq =P/ s E2
U x,3 = U x β
s V2,3 = s E2 d 2 + s V = s E2 d 2 PB+( d 2 s E2 )β U V2,3
{ U V2,3 = (AP)( d 2 + 3 f 3 ) s E2 f 3 + BA+β f 3 f 3 U O2,3 = B 2 f 3 +β AP+2α f 2 2 f 2 s E2 (A+2α f 2 )( d 2 s E2 )+P(2 f 3 d 2 ) f 3 (2 f 2 s E2 ) U V3,3 =β BP d 2 s E2 U O3,3 = BP( d 2 s E2 )β 2 f 3 + d 2 s E2 U Lq,3 = B+β f 3 f 3 P( d 2 + f 3 ) s E2 f 3
σ j = s E ( U xj,3 U Lq,3 ) e
{ σ 2 * =( m f 2 f 3 + m+ f 2 d 1 f 3 )A s E e = k A2 * A s E e σ 2 o =( m f 2 f 3 m f 2 f 3 ( d 1 +2 f 2 ) )(A+2 f 2 α) s E e =( k A2 o A+ k α2 o α) s E e σ 3 * = n+t n f 2 f 3 ( d 2 A f 2 B+2 d 2 f 2 α) s E e =( k A3 * A+ k B3 * B+ k α3 * α) s E e σ 3 o = n+t f 2 f 3 (n+2t) [ ( d 2 +2 f 3 )A f 2 B+2( d 2 f 2 +2 f 2 f 3 )α2 f 2 f 3 β ] s E e =( k A3 o A+ k B3 o B+ k α3 o α+ k β3 o β) s E e
{ m= d 2 f 2 + f 3 n= d 1 d 2 d 1 f 2 + d 2 f 2 t= d 1 f 3 + f 2 f 3
Δ W 131 = 1 2 [ ( S II,2 o σ 2 o + S II,2 * σ 2 * + S II,3 o σ 3 o + S II,3 * σ 3 * )· ρ ]( ρ · ρ ) = s E 2 [ S II,2 o ( k A2 o A+ k α2 o α)+ S II,2 * k A2 * A+ S II,3 o ( k A3 o A+ k B3 o B+ k α3 o α+ k β3 o β)+ S II,3 * ( k A3 * A+ k B3 * B+ k α3 * α)]( e · ρ )( ρ · ρ ) = s E 2 ( k A A+ k B B+ k α α+ k β β)( e · ρ )( ρ · ρ )
{ k A = S II,2 o k A2 o + S II,2 * k A2 * + S II,3 o k A3 o + S II,3 * k A3 * k B = S II,3 o k B3 o + S II,3 * k B3 * k α = S II,2 o k α2 o + S II,3 o k α3 o + S II,3 * k α3 * k β = S II,3 o k β3 o
{ f mx = k A A x + k B B x + k α α x + k β β x f my = k A A y + k B B y + k α α y + k β β y
{ k β =( m 3 1) ( m 3 +1) 2 [ 2 d 2 2 d 1 + r 3 + 2 d 1 (2 d 2 + r 3 ) r 2 ] B β = k β k B =T, B α = k α k B =G, A α = k α k A =1/( 1 r 2 + H GM )
{ M= b s3 ( m 3 +1)+(1 m 3 ),T= r 3 (1 m 3 )/M,H= ( m 2 +1) 3 m 3 3 b s2 / ( m 3 +1 ) 2 N= m 3 3 (1 m 2 2 ) ( m 3 +1 ) 2 M ( d 1 d 1 m 2 + m 2 r 1 d 1 + r 2 ),G=N r 2 +2 d 2 +2T
Z CFP3 = B β = k β k B = s 3 /[ ( m 3 +1 2 ){ 1( m 3 +1 m 3 1 ) } b s3 ]
Δ W 131 = s E 2 ( k A ΔA+ k B ΔB+ k α Δα+ k β Δβ)( e · ρ )( ρ · ρ )

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