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,” in Optical alignment, R. M. Shagam and W. C. Sweatt, eds., 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,” PASP 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,” PASP 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,” in Emerging Lithographic Technology, Y. Vladimirsky, eds., Proc. SPIE3331,102–113 (1998).
    [Crossref]
  9. H. Lee, “Amon-Ra system alignment,” in Novel space optical instrument for deep space earth albedo monitoring, pp250–289 (Master thesis, Yonsei University, 2005).
  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)

2000 (1)

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).

1997 (1)

R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109,53–60 (1997).
[Crossref]

1996 (1)

B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 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,” in Optical alignment, R. M. Shagam and W. C. Sweatt, eds., Proc. SPIE 251,146–153 (1980)

Born, M.

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

Chapman, H. N.

H. N. Chapman and D.W. Sweeney, “Rigorous method for compensation selection and alignment of microlithographic optical systems,” in Emerging Lithographic Technology, Y. Vladimirsky, eds., Proc. SPIE3331,102–113 (1998).
[Crossref]

Dalton, G. B.

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”.

Delabre, B.

R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109,53–60 (1997).
[Crossref]

Doyle, K. B.

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

Flannery, B.

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

Genberg, V. L.

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

Guisard, S.

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).

Kim, S.

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”.

Lee, H.

H. Lee, “Amon-Ra system alignment,” in Novel space optical instrument for deep space earth albedo monitoring, pp250–289 (Master thesis, Yonsei University, 2005).

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”.

McLeod, B.

B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 108,217–219 (1996).
[Crossref]

Michels, G. J.

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

Noethe, L.

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).

Press, W.

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

Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” in Optical alignment, R. M. Shagam and W. C. Sweatt, eds., Proc. SPIE 251,146–153 (1980)

Sutherland, W.

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

Sweeney, D.W.

H. N. Chapman and D.W. Sweeney, “Rigorous method for compensation selection and alignment of microlithographic optical systems,” in Emerging Lithographic Technology, Y. Vladimirsky, eds., Proc. SPIE3331,102–113 (1998).
[Crossref]

Teukolsky, S.

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

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,” in Optical alignment, R. M. Shagam and W. C. Sweatt, eds., Proc. SPIE 251,146–153 (1980)

Tosh, I. A. J.

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”.

Vetterling, W.

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

Wilson, R. N.

R. N. Wilson and B. Delabre, “Concerning the Alignment of Modern Telescopes: Theory, Practice, and Tolerance Illustrated by the ESO NTT,” PASP 109,53–60 (1997).
[Crossref]

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

Wolf, E.

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

A&A Supp. (1)

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).

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

PASP (2)

B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 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,” PASP 109,53–60 (1997).
[Crossref]

Proc. SPIE (1)

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” in Optical alignment, R. M. Shagam and W. C. Sweatt, eds., Proc. SPIE 251,146–153 (1980)

Other (8)

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

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

H. N. Chapman and D.W. Sweeney, “Rigorous method for compensation selection and alignment of microlithographic optical systems,” in Emerging Lithographic Technology, Y. Vladimirsky, eds., Proc. SPIE3331,102–113 (1998).
[Crossref]

H. Lee, “Amon-Ra system alignment,” in Novel space optical instrument for deep space earth albedo monitoring, pp250–289 (Master thesis, Yonsei University, 2005).

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

Tables Icon

Table 3. The modeling accuracy for three cases

Tables Icon

Table 1. The optical prescription of the example system

Tables Icon

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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