Abstract

We present a differential wavefront sampling method for the efficient alignment of centred optical systems. Using the inter-element effects reported in our previous study, this method generates a linear symmetric matrix that relates the optical wavefront to misalignments within the system. The solution vector of this matrix equation provides a unique description of decentre and tilt misalignments of the system. We give a comparison of this approach to the existing method in the first case study and then illustrate characteristics of the new approach using the subsequent four case studies and Monte-Carlo alignment simulations. The results reveal superiority of the method over the existing one in misalignment estimation accuracy and demonstrate the practical feasibility and robustness.

©2007 Optical Society of America

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References

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  1. R. V. Shack and K. 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. SPIE251, 146–153 (1980)
  2. B. McLeod, “Collimation of Fast Wide-Field Telescopes,” PASP 108, 217–219 (1996).
    [Crossref]
  3. 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]
  4. 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).
  5. W. Sutherland, Alignment and Number of Wavefront Sensors for VISTA, VIS-TRE-ATC-00112-0012 (Technical report, Astronomy Technology Center, UK, 2001).
  6. S. Kim, H.-S. Yang, Y.-W. Lee, and S.-W. Kim, “Merit function regression method for efficient alignment control of two-mirror optical systems,” Opt. Express 15, 5059–5068 (2007)
    [Crossref] [PubMed]
  7. H. J. Jeong, G. N. Lawrence, and K. B. Nahm, “Auto-alignment of a three mirror off-axis telescope by reverse optimization and end-to-end aberration measurements,” in Current Developments in Optical Engineering II, R. E. Fischer and W. J. Smith, eds., Proc. SPIE818, 419–430 (1987)
  8. M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991)
    [Crossref]
  9. H. Chapman and D. 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]
  10. H. Lee, “Amon-Ra system alignment,” in Novel space optical instrument for deep space earth albedo monitoring, 250–289 (Master thesis, Yonsei University, 2005).
  11. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W Kim, “Computer-guided alignment I : Phase and amplitude modulation of the alignment-influenced wavefront,” Opt. Express 15, 3127–3139 (2007)
    [Crossref] [PubMed]
  12. G. B. Arfken and H. Weber, Mathematical Methods for Physicist 4th ed., (Academic press, London, 1996).
  13. A. Malvick and E. Pearson, “Theoretical elastic deformations of a 4-m diameter optical mirror using dynamic relaxation,” Appl. Opt. 7, 1207–1212 (1968).
    [Crossref] [PubMed]
  14. R. Tyson and B. W. Frazier, Field Guide to Adaptive Optics, (SPIE, Washington, 2004).
    [Crossref]
  15. Private communication (Kevin Middleton, Oxford, 2007).
  16. E. P. Goodwin and J. C. Wyant, Field Guide to Interferometric Optical Testing, (SPIE, Washington, 2004).
  17. A. E. Bryson, Applied Linear Optimal Control, (Cambridge University Press, Cambridge, 2002)
  18. A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Statistics and computing  10, 197–208 (2000)

2007 (2)

2004 (1)

E. P. Goodwin and J. C. Wyant, Field Guide to Interferometric Optical Testing, (SPIE, Washington, 2004).

2000 (2)

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Statistics and computing  10, 197–208 (2000)

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]

1991 (1)

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991)
[Crossref]

1968 (1)

Andrieu, C.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Statistics and computing  10, 197–208 (2000)

Arfken, G. B.

G. B. Arfken and H. Weber, Mathematical Methods for Physicist 4th ed., (Academic press, London, 1996).

Bryson, A. E.

A. E. Bryson, Applied Linear Optimal Control, (Cambridge University Press, Cambridge, 2002)

Chapman, H.

H. Chapman and D. 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.

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]

Doucet, A.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Statistics and computing  10, 197–208 (2000)

Frazier, B. W.

R. Tyson and B. W. Frazier, Field Guide to Adaptive Optics, (SPIE, Washington, 2004).
[Crossref]

Godsill, S.

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Statistics and computing  10, 197–208 (2000)

Goodwin, E. P.

E. P. Goodwin and J. C. Wyant, Field Guide to Interferometric Optical Testing, (SPIE, Washington, 2004).

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).

Jeong, H. J.

H. J. Jeong, G. N. Lawrence, and K. B. Nahm, “Auto-alignment of a three mirror off-axis telescope by reverse optimization and end-to-end aberration measurements,” in Current Developments in Optical Engineering II, R. E. Fischer and W. J. Smith, eds., Proc. SPIE818, 419–430 (1987)

Kim, S.

Kim, S.-W

Kim, S.-W.

Lawrence, G. N.

H. J. Jeong, G. N. Lawrence, and K. B. Nahm, “Auto-alignment of a three mirror off-axis telescope by reverse optimization and end-to-end aberration measurements,” in Current Developments in Optical Engineering II, R. E. Fischer and W. J. Smith, eds., Proc. SPIE818, 419–430 (1987)

Lee, H.

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W Kim, “Computer-guided alignment I : Phase and amplitude modulation of the alignment-influenced wavefront,” Opt. Express 15, 3127–3139 (2007)
[Crossref] [PubMed]

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

Lee, Y.-W.

Lundgren, M. A.

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991)
[Crossref]

Malvick, A.

McLeod, B.

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

Nahm, K. B.

H. J. Jeong, G. N. Lawrence, and K. B. Nahm, “Auto-alignment of a three mirror off-axis telescope by reverse optimization and end-to-end aberration measurements,” in Current Developments in Optical Engineering II, R. E. Fischer and W. J. Smith, eds., Proc. SPIE818, 419–430 (1987)

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).

Pearson, E.

Shack, R. V.

R. V. Shack and K. 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. SPIE251, 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.

H. Chapman and D. 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]

Thompson, K.

R. V. Shack and K. 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. SPIE251, 146–153 (1980)

Tosh, I. A. J.

Tyson, R.

R. Tyson and B. W. Frazier, Field Guide to Adaptive Optics, (SPIE, Washington, 2004).
[Crossref]

Weber, H.

G. B. Arfken and H. Weber, Mathematical Methods for Physicist 4th ed., (Academic press, London, 1996).

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]

Wolfe, W. L.

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991)
[Crossref]

Wyant, J. C.

E. P. Goodwin and J. C. Wyant, Field Guide to Interferometric Optical Testing, (SPIE, Washington, 2004).

Yang, H.-S.

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).

Appl. Opt. (1)

Opt. Eng. (1)

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30, 307–311 (1991)
[Crossref]

Opt. Express (2)

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]

Other (11)

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

H. J. Jeong, G. N. Lawrence, and K. B. Nahm, “Auto-alignment of a three mirror off-axis telescope by reverse optimization and end-to-end aberration measurements,” in Current Developments in Optical Engineering II, R. E. Fischer and W. J. Smith, eds., Proc. SPIE818, 419–430 (1987)

R. V. Shack and K. 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. SPIE251, 146–153 (1980)

H. Chapman and D. 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, 250–289 (Master thesis, Yonsei University, 2005).

G. B. Arfken and H. Weber, Mathematical Methods for Physicist 4th ed., (Academic press, London, 1996).

R. Tyson and B. W. Frazier, Field Guide to Adaptive Optics, (SPIE, Washington, 2004).
[Crossref]

Private communication (Kevin Middleton, Oxford, 2007).

E. P. Goodwin and J. C. Wyant, Field Guide to Interferometric Optical Testing, (SPIE, Washington, 2004).

A. E. Bryson, Applied Linear Optimal Control, (Cambridge University Press, Cambridge, 2002)

A. Doucet, S. Godsill, and C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Statistics and computing  10, 197–208 (2000)

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Figures (8)

Fig. 1.
Fig. 1. Alignment state evolutions by MFWS (A-C) and DWS (D-F). Normalised field locations for MFWS (Hx,Hy) = {(0,0),(±1,0),(0,±1),(±0.5,0),(0,±0.5), (±0.5,0.5), (±0.5,-0.5), (+0.25,+0.25), (-0.25,-0.25)}.
Fig. 2.
Fig. 2. RMS aberration after corrections by MFWS and DWS.
Fig. 3.
Fig. 3. Case 2 : Nm = 10, σ m = 1.0nm, δuL = 250μm, δuA = 180arcsec, σ c,L = 1.0μm, σ c,A = 1 .0arcsec
Fig. 4.
Fig. 4. Case 3 : Nm = 10, σm = 1.0nm, σd = D1(see the note in Table 2), δuL=250μm, δuA = 180arcsec, σc,L = 1.0μm, σc,A = 1.0arcsec
Fig. 5.
Fig. 5. Case 4 : Nm = 10, σ m = 1.0nm, σ d = D2(see the note in Table 2 ), δuL=250μm, δuA = 180arcsec, σ c,L = 1.0μm, σ c,A = 1.0arcsec
Fig. 6.
Fig. 6. Case 5 : Nm = 10, σ m = 1.0nm, σ d = D3(see the note in Table 2), δuL=250μm, δ uA = 180arcsec, σ c,L = 1.0μm, σ c,A = 1.0arcsec
Fig. 7.
Fig. 7. Monte Carlo alignment simulations : Each point derived from 101 realisations; M2’s residual angular misalignment against σ m with (A) varying k in δuL = k ∙ 500μm and δ uA = k ∙ 360arcsec and (B) varying Nm from its default value of 100. (C,D) are The RMS system aberrations corresponding to (A) and (B) respectively. (Note : σ c,L = 1.0μm, σ c,A = 1.0arcsec were used. Nm=10 in (A) and k=0.5 in (B). Error bars represent the standard deviations.)
Fig. 8.
Fig. 8. Monte Carlo alignment simulations : Each point derived from 101 realisations; M2’s residual angular misalignment against (A) σ c, (C) the amount of surface astigmatism, and (E) the amount of surface trefoil. (B,D,F) are the RMS system aberrations corresponding to (A), (C), and (E), respectively. (Note : σ m = 10nm, Nm = 10 were used in (A). σ c,L = 1.0μm, σ c,A = 1.0arcsec were used in (C,E). Error bars represent the standard deviations.)

Tables (3)

Tables Icon

Table 1. Classification of surface deformation elements

Tables Icon

Table 2. Alignment simulation input parameters

Tables Icon

Table 3. Comparison 1 : Misaligned M2 and M3

Equations (16)

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Φ = Φ 0 + Φ A + Φ D + Φ C + Φ R + Φ S
Φ A u i = L i ( 1 ) + j = 0 M ( 1 + δ ij ) L ij ( 2 ) u j +
Z 5 , L = M 5 , M 2 u 2 = ( F 5 L 5 , M 2 ) u 2
[ Z 5 , L Z 6 , L ] = ( [ F 5 , 0 0 , F 6 ] [ L 5 , M 2 L 6 , M 2 ] ) u 2
[ Z ' 5 , L Z ' 6 , L ] = ( [ F ' 5 , 0 0 , F ' 6 ] [ L 5 , M 2 , 0 0 , L 5 , M 3 L 6 , M 2 , 0 0 , L 6 , M 3 ] )     [ u 2 u 3 ]
[ x 2 Z 5 y 2 Z 5 θ 2 Z 5 ϕ 2 Z 5 ] = [ 2 L x 2 x 2 , 0 , 0 , L x 2 ϕ 2 0 , 2 L y 2 y 2 , L y 2 θ 2 , 0 0 , L θ 2 y 2 , 2 L θ 2 θ 2 , 0 L ϕ 2 x 2 , 0 , 0 , 2 L ϕ 2 ϕ 2 ] [ dx 2 dy 2 2 2 ]
u j Z 5 = k ( 1 + δ jk ) L u j , u k ( 2 ) u k , v j Z 5 = k ( 1 + δ jk ) L v j , v k ( 2 ) v k
Φ ( u i ± δu i ) = Φ 0 + L u i ( 1 ) { ( u i ± δu i ) + q ( 0 , σ c ) } + L u i , u i ( 2 ) { ( u i ± δu i ) + q ( 0 , σ c ) } 2
+ j ( i ) L u i , u j ( 2 ) { ( u i ± δu i ) + q ( 0 , σ c ) } u j + q ( 0 , σ m N m )
Φ u i = L u i ( 1 ) { 1 + q ( 0 , σ c 2 ) 2 δ u i } + L u i , u i ( 2 ) { 2 ( 1 + q ( 0 , σ c 2 ) 2 δ u i ) u i + q ( 0 , σ c 2 ) + q 2 ( 0 , σ c 2 ) 2 δ u i }
+ j ( i ) { L u i , u j ( 2 ) ( 1 + q ( 0 , σ c 2 ) 2 δ u i ) u j } + q ( 0 , σ m 2 N m ) 2 δ u i
σ ( u i ) 2 ~ 1 2 [ ( 2 L u i , u i ( 2 ) ) 2 { 1 + 2 u i δu i + ( u i δu i ) 2 } + j ( i ) { L u i , u j ( 2 ) u j δu i } 2 ] σ c 2 + σ m 2 2 N m δu i 2
lim u 0 σ ( u i ) 2 ~ 1 2 { ( 2 L u i , u i ( 2 ) ) 2 σ c 2 + σ m 2 N m δu i 2 }
lim u 0 σ ( u i ) 2 ~ 1 2 { ( 2 L u i , u i ( 2 ) ) 2 σ c 2 + C u i 2 σ d 2 + σ m 2 N m δu i 2 } + σ n 2
V = P T diag { σ ( u i ) 2 } P
dx + dy i = dL e i a ( Linear ) , + i = dR e i b ( Angular )

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