Abstract

A collimation method of misaligned optical systems is proposed. The method is based on selectively nullifying main alignment-driven aberration components. This selective compensation is achieved by the optimal adjustment of chosen alignment parameters. It is shown that this optimal adjustment can be obtained by solving a linear matrix equation of the low-order alignment-driven terms of primary field aberrations. A significant result from the adjustment is to place the centers of the primary field aberrations, initially scattered over the field due to misalignment, to a desired common field location. This aberration concentering naturally results in recovery of image quality across the field of view. Error analyses and robustness tests show the method’s feasibility in efficient removal of alignment-driven aberrations in the face of measurement and model uncertainties. The extension of the method to the collimation of a misaligned system with higher-order alignment-driven aberrations is also shown.

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References

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  1. 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,” Proc. SPIE 818, 419–430 (1987).
  2. 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]
  3. W. Sutherland, Alignment and Number of Wavefront Sensors for VISTA, VIS-TRE-ATC-00112-0012 (Technical report, Astronomy Technology Center, UK, 2001).
  4. 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]
  5. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II : Optical system alignment using differential wavefront sampling,” Opt. Express 15, 15424–15437 (2007).
    [CrossRef] [PubMed]
  6. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Practical implementation of the complex wavefront modulation model for optical alignment,” Proc. SPIE 6617, 66170N (2007).
    [CrossRef]
  7. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Implementation of differential wavefront sampling in optical alignment of pupil-segmented telescope systems,” Proc. SPIE 7017, 70171T (2008).
    [CrossRef]
  8. H. N. Chapman and D. W. Sweeney, “Rigorous method for compensation selection and alignment of microlitho-graphic optical systems,” Proc. SPIE 3331, 102–113 (1998).
    [CrossRef]
  9. A. M. Hvisc and J. H. Burge, “Alignment analysis of four-mirror spherical aberration correctors,” Proc. SPIE 7018, 701819 (2008).
    [CrossRef]
  10. D. O’Donoghue, South African Large Telscope, Observatory, 7935, South Africa (Personal communication, 2009).
  11. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1975).
    [CrossRef]
  12. 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).
  13. B. McLeod, “Collimation of Fast Wide-Field Telescopes,” Publ. Astron. Soc. Pac. 108, 217–219 (1996).
    [CrossRef]
  14. 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]
  15. L. Noethe and S. Guisard, “Analytic expressions for field astigmatism in decentered two mirror telescopes and application to the collimation of the ESO VLT,” Acta Anat. Suppl. 144, 157–167 (2000).
  16. H. Lee, G. B. Dalton, I. A. J. Tosh, and S. Kim, “Computer-guided alignment I : Phase and amplitude modulation of the alignment-influenced wavefront,” Opt. Express 15, 3127–3139 (2007).
    [CrossRef] [PubMed]
  17. H. Lee, G. Dalton, I. Tosh, and S. Kim, “Computer-guided alignment III: Description of inter-element alignment effect in circular-pupil optical systems,” Opt. Express 16, 10992–11006 (2008).
    [CrossRef] [PubMed]
  18. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C 2nd ed. (Cambridge, 2002).
  19. G. D’Agostini, Bayesian Reasoning in Data Analysis (World Scientific, 2005).

2008 (3)

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Implementation of differential wavefront sampling in optical alignment of pupil-segmented telescope systems,” Proc. SPIE 7017, 70171T (2008).
[CrossRef]

A. M. Hvisc and J. H. Burge, “Alignment analysis of four-mirror spherical aberration correctors,” Proc. SPIE 7018, 701819 (2008).
[CrossRef]

H. Lee, G. Dalton, I. Tosh, and S. Kim, “Computer-guided alignment III: Description of inter-element alignment effect in circular-pupil optical systems,” Opt. Express 16, 10992–11006 (2008).
[CrossRef] [PubMed]

2007 (4)

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,” Acta Anat. Suppl. 144, 157–167 (2000).

1998 (1)

H. N. Chapman and D. W. Sweeney, “Rigorous method for compensation selection and alignment of microlitho-graphic 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]

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]

1987 (1)

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,” Proc. SPIE 818, 419–430 (1987).

1975 (1)

Burge, J. H.

A. M. Hvisc and J. H. Burge, “Alignment analysis of four-mirror spherical aberration correctors,” Proc. SPIE 7018, 701819 (2008).
[CrossRef]

Chapman, H. N.

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

D’Agostini, G.

G. D’Agostini, Bayesian Reasoning in Data Analysis (World Scientific, 2005).

Dalton, G.

Dalton, G. B.

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Implementation of differential wavefront sampling in optical alignment of pupil-segmented telescope systems,” Proc. SPIE 7017, 70171T (2008).
[CrossRef]

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II : Optical system alignment using differential wavefront sampling,” Opt. Express 15, 15424–15437 (2007).
[CrossRef] [PubMed]

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

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Practical implementation of the complex wavefront modulation model for optical alignment,” Proc. SPIE 6617, 66170N (2007).
[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]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C 2nd ed. (Cambridge, 2002).

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,” Acta Anat. Suppl. 144, 157–167 (2000).

Hvisc, A. M.

A. M. Hvisc and J. H. Burge, “Alignment analysis of four-mirror spherical aberration correctors,” Proc. SPIE 7018, 701819 (2008).
[CrossRef]

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,” Proc. SPIE 818, 419–430 (1987).

Kim, S.

Kim, S.-W.

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Implementation of differential wavefront sampling in optical alignment of pupil-segmented telescope systems,” Proc. SPIE 7017, 70171T (2008).
[CrossRef]

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II : Optical system alignment using differential wavefront sampling,” Opt. Express 15, 15424–15437 (2007).
[CrossRef] [PubMed]

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]

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Practical implementation of the complex wavefront modulation model for optical alignment,” Proc. SPIE 6617, 66170N (2007).
[CrossRef]

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,” Proc. SPIE 818, 419–430 (1987).

Lee, H.

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]

McLeod, B.

B. McLeod, “Collimation of Fast Wide-Field Telescopes,” Publ. Astron. Soc. Pac. 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,” Proc. SPIE 818, 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,” Acta Anat. Suppl. 144, 157–167 (2000).

Noll, R. J.

O’Donoghue, D.

D. O’Donoghue, South African Large Telscope, Observatory, 7935, South Africa (Personal communication, 2009).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C 2nd ed. (Cambridge, 2002).

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. W.

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

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C 2nd ed. (Cambridge, 2002).

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.

Tosh, I. A. J.

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Implementation of differential wavefront sampling in optical alignment of pupil-segmented telescope systems,” Proc. SPIE 7017, 70171T (2008).
[CrossRef]

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II : Optical system alignment using differential wavefront sampling,” Opt. Express 15, 15424–15437 (2007).
[CrossRef] [PubMed]

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

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Practical implementation of the complex wavefront modulation model for optical alignment,” Proc. SPIE 6617, 66170N (2007).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C 2nd ed. (Cambridge, 2002).

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]

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]

Yang, H.-S.

Acta Anat. Suppl. (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,” Acta Anat. Suppl. 144, 157–167 (2000).

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

Proc. SPIE (5)

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,” Proc. SPIE 818, 419–430 (1987).

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Practical implementation of the complex wavefront modulation model for optical alignment,” Proc. SPIE 6617, 66170N (2007).
[CrossRef]

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Implementation of differential wavefront sampling in optical alignment of pupil-segmented telescope systems,” Proc. SPIE 7017, 70171T (2008).
[CrossRef]

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

A. M. Hvisc and J. H. Burge, “Alignment analysis of four-mirror spherical aberration correctors,” Proc. SPIE 7018, 701819 (2008).
[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 (5)

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

D. O’Donoghue, South African Large Telscope, Observatory, 7935, South Africa (Personal communication, 2009).

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).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C 2nd ed. (Cambridge, 2002).

G. D’Agostini, Bayesian Reasoning in Data Analysis (World Scientific, 2005).

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Figures (10)

Fig. 1.
Fig. 1.

Initial field aberration scans of the two-mirror system: (A) Coma field scans, (B) Astigmatism field scans, (C) Curvature field scans, (D) Field scans of RMS wavefront error (wv=632.8 nanometers).

Fig. 2.
Fig. 2.

Field scans of the two-mirror system after correction: (A) Coma field scans, (B) Astigmatism field scans, (C) Curvature field scans, (D) Field scans of RMS wavefront error (wv=632.8 nanometers).

Fig. 3.
Fig. 3.

Expected probability distributions of the residuals of the quadractic terms: (A) Normalized Probability Distribution Function (PDF) and Cumulative Distribution Function (CDF) of the quadratic terms of Curv, (B) PDF and CDF of the quadratic terms of Astg 1, (C) PDF and CDF of the quadratic terms of Astg 2 (wv=632.8 nanometers).

Fig. 4.
Fig. 4.

Initial field scans the three-mirror camera system: (A) Coma field scans, (B) Astigmatism field scans, (C) Curvature field scans, (D) Field scans of RMS wavefront error (wv=632.8 nanometers).

Fig. 5.
Fig. 5.

Field scans of the three-mirror camera system after correction: (A) Coma field scans, (B) Astigmatism field scans, (C) Curvature field scans, (D) Field scans of RMS wavefront error (wv=632.8 nanometers).

Fig. 6.
Fig. 6.

Robustness test against measurement error (top row), model error (middle row), and sample size for averaging (bottom row). (Column A) error in y-decenter of M2, (Column B) error in θ of M2, (Column C) error in θ of M3, (Column D) rms wavefront error (wv=632.8 nanometers).

Fig. 7.
Fig. 7.

Initial field scans of the example telescope system: (A) Coma field scans, (B) Astigmatism field scans, (C) Curvature field scans, (D) Field scans of RMS wavefront error (wv=632.8 nanometers).

Fig. 8.
Fig. 8.

Field scans of Coma, Astg 1, and Curv after correction by Method I: (A) Coma field scans, (B) Astigmatism field scans, (C) Curvature field scans, (D) Field scans of RMS wavefront error (wv=632.8 nanometers).

Fig. 9.
Fig. 9.

Field scans of Coma, Astg 1, and Curv after correction by Method II: (A) Coma field scans, (B) Astigmatism field scans, (C) Curvature field scans, (D) Field scans of RMS wavefront error (wv=632.8 nanometers).

Fig. 10.
Fig. 10.

Full field ray-spot diagrams of the example telescope system: (A) initial, (B) After correction by Method I, (C) After correction by Method II, (D) Nominal.

Tables (2)

Tables Icon

Table 1. Curve fit coefficients of Comax , Astg 1, and Curv along Hx axis (unit in wv)

Tables Icon

Table 2. The three-mirror system used in Section 3.2

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

Coma x = C H x H x + ( C x x + C ϕ ϕ ) + O ( 2 ) Coma y = C H y H y + ( C y y + C θ θ ) + O ( 2 ) Astg 1 = A H x H x H x 2 + ( A H x x x + A H x ϕ ϕ ) H x + [ A x x x 2 + A ϕ ϕ ϕ 2 + A x ϕ x ϕ ] + A H y H y H y 2 + ( A H y y y + A H y θ θ ) H y + [ A y y y 2 + A θ θ θ 2 + A y θ y θ ] + O ( 3 ) Astg 2 = B H x H y H x H y + ( B H x y y + B H x θ θ ) H x + ( B H y x x + B H y θ ϕ ) H y + [ B x y x y + B x θ x θ + B y ϕ y ϕ + B θ ϕ ϕ θ ] + O ( 3 ) Curv = F 0 + F H x H x H x 2 + ( F H x x x + F H x ϕ ϕ ) H x + [ F x x x 2 + F ϕ ϕ ϕ 2 + F x ϕ x ϕ ] + F H y H y H y 2 + ( F H y y y + F H y θ θ ) H y + [ F y y y 2 + A θ θ θ 2 + F y θ y θ ] + O ( 3 )
Coma : ( X 1 C x , Y 1 C y ) , Astg 1 : ( X 2 2 A H x H x , Y 2 2 A H y H y ) , Curv : ( X 3 2 F H x H x , Y 3 2 F H y H y )
C x x + C ϕ ϕ = X 1 C y y + C θ θ = Y 1 A H x x x + A H x ϕ ϕ = X 2 A H y y y + A H y θ θ = Y 2 F H x x x + F H x ϕ ϕ = X 3 F H y y y + F H y θ θ = Y 3 B H y x x + B H y ϕ ϕ = X 4 B H x y y + B H x θ θ = Y 4
[ C x C ϕ A H x x A H x ϕ ] [ x ϕ ] = [ C x C ϕ A H x x A H x ϕ ] [ x ϕ ] = [ X 1 X 2 ]
[ C x , A x , F x ] T x = M x x = X , [ C y , A y , F y ] T y = M y y = Y
x = m x 1 M x x = n x M x x = M ˜ x x and y = m y 1 M y y = n y M x x = M ˜ y y
x = ( 1 [ M ˜ x 0 ] ) x = M ̂ x x and y = ( 1 [ M ˜ y 0 ] ) y = M ̂ y y
Curv = x T ( M ̂ x T F x x M ̂ x ) x + y T ( M ̂ y T F y y M ̂ y ) y = x T F ̂ x x x + y T F ̂ y y y
Astg 1 = x T ( M ̂ x T A x x M ̂ x ) x + y T ( M ̂ y T A y y M ̂ y ) y = x T A ̂ x x x + y T A ̂ y y y
  Astg 2 = x T ( M ̂ x T B x y M ̂ y ) y = x T B ̂ x y y
E [ Curv ] = Σ i N ( F ̂ x i x i σ x i 2 + F ̂ y i y i σ y i 2 ) Var [ Curv ] = Σ i N ( F ̂ x i x i 2 σ x i 4 + F ̂ y i y i 2 σ y i 4 ) E [ Astg 1 ] = Σ i N ( A ̂ x i x i σ x i 2 + A ̂ y i y i σ y i 2 ) Var [ Astg 1 ] = Σ i N ( A x i x i 2 σ x i 4 + A y i y i 2 σ y i 4 ) E [ Astg 2 ] = Σ i N B ̂ x i y i σ x i σ y i Var [ Astg 2 ] = Σ i N B ̂ x i y i 2 σ x i 2 σ y i 2
E [ Curv ] + 3 var [ Curv ] Curv req
p = [ R T R ] 1 R T q where q 1 = w i δ w i and R ij = H i j δ w i
x = [ m x T m x ] 1 m x T X = D x m x T X with m ij , x = m ij , x σ X i and X i = X i σ X i
y = [ m y T m y ] 1 m y T Y = D y m y T Y with m ij , y = m ij , y σ Y i and Y i = Y i σ Y i
σ x i 2 = D ii , x M and σ y i 2 = D ii , y M
x = m x 1 ( M x + δ M x ) x = n x ( X + δ X ) , y = m y 1 ( M y + δ M y ) y = n y ( Y + δ Y )
σ x i 2 j N n ij , x 2 σ M ij , x 2 σ x j 2 and σ y i 2 j N n ij , y 2 σ M ij , y 2 σ y j 2
Coma x = ( C dh x dh x + C d h x 3 dh x 3 + C dh x dh y 2 dh x dh y 2 )
+ ( C H x + C dh x 2 H x dh x 2 + C dh y 2 H x dh y 2 + C dh x dh y H x dh x dh y ) H x + C dh x H x 2 dh x H x 2 + O ( 4 )

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