Abstract

As the first part of a development programme on computer-guided alignment(CGA), we model the alignment influence on the optical wavefront in terms of the phase and amplitude modulation. This modulation is derived from the interaction between alignment parameters and influence functions, both expressed in complex form. The alignment influence model is used to approximate the ray-traced target wavefront of a randomly mis-aligned multi-element system. The approximated wavefront shows a factor of ~ 100 improvement in predicting the target, when coupled non-linear influences among elements are included. This demonstrates the significance of the inter-element effect. We discuss the possibility of adopting the model for rectifying mis-alignment of multi-element systems.

© 2007 Optical Society of America

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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)
  2. 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]
  3. B. McLeod, "Collimation of Fast Wide-Field Telescopes," Publ. Astron. Soc. Pac. 108, 217-219 (1996).
    [CrossRef]
  4. R. N. Wilson and B. Delabre, "Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT," Publ. Astron. Soc. Pac. 109, 53-60 (1997).
    [CrossRef]
  5. R. N. Wilson, Reflecting Telescope Optics Vol. I, 2nd ed. (Springer-Verlag, Berlin, 2004).
  6. L. Noethe and S. Guisard, "Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT," A&A Supp. 144, 157-167 (2000).
  7. W. Sutherland, Alignment and Number of Wavefront Sensors for VISTA, VIS-TRE-ATC-00112-0012 (Technical report, Astronomy Technology Center, UK, 2001).
  8. H. N. Chapman and D. W. Sweeney, "Rigorous method for compensation selection and alignment of microlithographic optical systems," Proc. SPIE 3331, 102-113 (1998).
    [CrossRef]
  9. H. Lee, "Amon-Ra system alignment," in Novel space optical instrument for deep space earth albedo monitoring, (Master thesis, Yonsei University, 2005) pp. 250-289.
  10. H. Lee, G. B. Dalton, I. A. J. Tosh, and S. Kim are preparing a manuscript to be called "Computer-guided alignment II: Optical alignment via deliberate mis-alignment."
  11. W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in C 2nd ed., (Cambridge University Press, Cambridge, 1999).
  12. M. Born and E. Wolf, Principles of Optics, 7th ed, (Cambridge University Press, Cambridge, 2004).
  13. K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis, (SPIE Press, Washington, 2002).
    [CrossRef]

2005 (1)

1998 (1)

H. N. Chapman and D. W. Sweeney, "Rigorous method for compensation selection and alignment of microlithographic optical systems," Proc. SPIE 3331, 102-113 (1998).
[CrossRef]

1997 (1)

R. N. Wilson and B. Delabre, "Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT," Publ. Astron. Soc. Pac. 109, 53-60 (1997).
[CrossRef]

1996 (1)

B. McLeod, "Collimation of Fast Wide-Field Telescopes," Publ. Astron. Soc. Pac. 108, 217-219 (1996).
[CrossRef]

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)

Chapman, H. N.

H. N. Chapman and D. W. Sweeney, "Rigorous method for compensation selection and alignment of microlithographic optical systems," Proc. SPIE 3331, 102-113 (1998).
[CrossRef]

Delabre, B.

R. N. Wilson and B. Delabre, "Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT," Publ. Astron. Soc. Pac. 109, 53-60 (1997).
[CrossRef]

McLeod, B.

B. McLeod, "Collimation of Fast Wide-Field Telescopes," Publ. Astron. Soc. Pac. 108, 217-219 (1996).
[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)

Sweeney, D. W.

H. N. Chapman and D. W. Sweeney, "Rigorous method for compensation selection and alignment of microlithographic optical systems," Proc. SPIE 3331, 102-113 (1998).
[CrossRef]

Thompson, K. P.

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]

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)

Wilson, R. N.

R. N. Wilson and B. Delabre, "Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT," Publ. Astron. Soc. Pac. 109, 53-60 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

Proc. SPIE (2)

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)

H. N. Chapman and D. W. Sweeney, "Rigorous method for compensation selection and alignment of microlithographic optical systems," Proc. SPIE 3331, 102-113 (1998).
[CrossRef]

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 Tolerance Illustrated by the ESO NTT," Publ. Astron. Soc. Pac. 109, 53-60 (1997).
[CrossRef]

Other (8)

R. N. Wilson, Reflecting Telescope Optics Vol. I, 2nd ed. (Springer-Verlag, Berlin, 2004).

L. Noethe and S. Guisard, "Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT," A&A Supp. 144, 157-167 (2000).

W. Sutherland, Alignment and Number of Wavefront Sensors for VISTA, VIS-TRE-ATC-00112-0012 (Technical report, Astronomy Technology Center, UK, 2001).

H. Lee, "Amon-Ra system alignment," in Novel space optical instrument for deep space earth albedo monitoring, (Master thesis, Yonsei University, 2005) pp. 250-289.

H. Lee, G. B. Dalton, I. A. J. Tosh, and S. Kim are preparing a manuscript to be called "Computer-guided alignment II: Optical alignment via deliberate mis-alignment."

W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in C 2nd ed., (Cambridge University Press, Cambridge, 1999).

M. Born and E. Wolf, Principles of Optics, 7th ed, (Cambridge University Press, Cambridge, 2004).

K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis, (SPIE Press, Washington, 2002).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

λ(1) as made by ray-tracing and Chebyshev decomposition.

Fig. 2.
Fig. 2.

(A) Amplitude at (ξ,η) = (-1,0) as a function of t, (B) Top-view of the wavefront.

Fig. 3.
Fig. 3.

A selection of λ (2) for couplings between two parameters associated with M1.

Fig. 4.
Fig. 4.

A selection of λ (2) for couplings between one parameter of M1 and another of M2.

Fig. 5.
Fig. 5.

(A) Amplitude at (ξ,η) = (-1,0) with π period, (B) Top-view of the wavefront.

Fig. 6.
Fig. 6.

λ (2) of the source (unit : μm/(0.8°)2)

Fig. 7.
Fig. 7.

(A), (B) λ (3) of each term associated with cos 3φ in Eq. 13 (unit : μm/(0.8°)3), (C) Amplitude at (ξ,η) = (-1,0) with 2π/3 period, (D) Top-view of the wavefront.

Fig. 8.
Fig. 8.

λ (1) of δϕ 1 sampled by fixing (A) δθ 0 = 0.8° and (B) δθ 0 = 0.0°, (C) Difference of (A) from (B), (D) Difference computed from summing up the coupled influences of δϕ 1 and δθ j 0 from j = 1 up to 6, (E) Difference of (D) from (C) (unit : μm/0.1°)

Fig. 9.
Fig. 9.

Modeling accuracy estimation

Tables (4)

Tables Icon

Table 2. The perturbation grid

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Table 3. The modeling accuracy for three cases

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Table 1. The optical prescription of the example system

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Table 4. Zernike coefficients of some of the presented alignment influence functions

Equations (21)

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u i = x i y i z i θ i ϕ i ω i
δ Φ = i = 0 M ( i Φ ) δ u i + i = 0 M j = 0 M δ u i ( i T j Φ ) δ u j T +
z a = δ x i δ y = a e i f , z b = δ z = b e i g , z c = δ ϕ i δ θ = c e i h
δ Φ ( 2 ) = δ u i λ ii ( 2 ) δ u i T + 2 δ u i λ ij ( 2 ) δ u j T + δ u j λ jj ( 2 ) δ u j T
( where λ ii ( 2 ) = i T i Φ , λ ij ( 2 ) = i T j Φ , λ jj ( 2 ) = i T j Φ )
λ ( m ) = Real { k l 2 ( k + 1 ) 1 + δ l 0 m A k , l R k l ( ρ ) Θ l ( φ ) }
Δ m l = i = 0 M ( z a ) p 1 ( i ) ( z b ) p 2 ( i ) ( z c ) p 3 ( i )
δ Φ ( m ) = Real { k l m A k , l R k l ( ρ ) Θ l ( φ ) Δ m l }
with k l 0 ( even ) and n l 0 ( even )
δ Φ a ( 1 ) = k { 1 A k , 1 R k 1 ( ρ ) ( δ x cos φ + δ y sin φ ) }
δ Φ c , c ( 2 ) = k { 2 A k , 0 R k 0 ( ρ ) ( δ ϕ 2 + δ θ 2 ) }
+ k { 2 A k , 2 R k 2 ( ρ ) ( ( δ ϕ 2 δ θ 2 ) cos 2 φ + 2 δ ϕ δθ sin 2 φ ) }
Δ 2 0 = z a , 1 z c , 2 * + z a , 1 * z c , 2
= a 1 c 2 ( e i f 1 e i h 2 + e i f 1 e i h 2 )
Δ 2 2 = z a , 1 z c , 2 = a 1 c 2 e i f 1 e i h 2
δ Φ a 1 , c 2 ( 2 ) = k ( 2 A k , 0 + 2 A ˜ k , 0 ) R k 0 ( ρ ) ( δ x 1 δ ϕ 2 + δ y 1 δ θ 2 )
+ k 2 A k , 2 R k 2 ( ρ ) { ( δ x 1 δ ϕ 2 δ y 1 δ θ 2 ) cos 2 φ
+ ( δ x 1 δ θ 2 + δ y 1 δ ϕ 2 ) sin 2 φ }
δ Φ c 0 ( 3 ) = k 3 A k , 1 R k 1 ( ρ ) { ( δ ϕ 0 2 + δ θ 0 2 ) ( δ ϕ 0 cos φ + δ θ 0 sin φ ) }
+ k 3 A k , 3 R k 3 ( ρ ) { ( δ ϕ 0 3 3 δ θ 0 2 δ ϕ 0 ) cos 3 φ + ( 3 δ ϕ 0 2 δ θ 0 δ θ 0 3 ) sin 3 φ }
λ ( 1 ) = λ 1 ( 1 ) + λ 1,0 ( 2 ) δ u 0 T + δ u 0 λ 1,0 ( 3 ) δ u 0 T

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