Abstract

We present formalism and analysis of three active alignment reconstruction techniques applied to the Advanced Technology Solar Telescope. The three reconstructors generate optical control signals that are a matrix product of a wavefront-sensing signal and the reconstructors themselves. The optical control signals are fed to the six rigid body degrees of freedom of the telescope secondary mirror and the eight bending modes of the primary mirror. The resulting aligned state is a least-squares alignment of the telescope subject to perturbations that result from thermal and gravitational flexures. Two of the reconstructors utilize a single guide object in the telescope field of view and the third reconstructor utilizes three guide objects. One of the single-guide-object reconstructors is developed with an explicit minimum force constraint that minimizes the actuator forces exerted on the telescope primary mirror during active alignment. The force optimized reconstructor also achieves close to the minimum residual wavefront error. Simulation results, optical control analysis, and a discussion of the reconstructor methods and properties are presented.

© 2010 Optical Society of America

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References

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  1. T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
    [CrossRef]
  2. ATST project website, http://atst.nso.edu.
  3. F. Woeger and T. Rimmele, “Effect of anisoplanatism on the measurement accuracy of an extended-source Hartmann–Shack wavefront sensor,” Appl. Opt. 48, A35–A46 (2009).
    [CrossRef]
  4. Giant Magellan Telescope project website, http://www.gmto.org.
  5. J. H. Burge, L. B. Kot, H. M. Martin, R. Zehnder, and C. Zhao, “Design and analysis for interferometric measurements of the GMT primary mirror segments,” Proc. SPIE 6273, 62730M(2006).
    [CrossRef]
  6. R. Upton, “Optical control of the Advanced Technology Solar Telescope,” Appl. Opt. 45, 5881–5896 (2006).
    [CrossRef] [PubMed]
  7. C. R. Vogel, Computational Methods for Inverse Problems (SIAM2002).
    [CrossRef]
  8. R J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  9. The condition number is the ratio of the greatest to the least singular values of a positive semidefinite matrix, which are common in active optical control problems where the matrix is overdetermined. The condition number can also be used to quantify the degree of linear independence between the compensating DOFs that are encapsulated in the columns of Jacobian matrices such as HM2
  10. MATLAB is a licensed product of The Mathworks, http://www.mathworks.com.
  11. ZEMAX is a licensed product of the ZEMAX Development Corporation, http://www.zemax.com.

2009 (1)

2006 (2)

J. H. Burge, L. B. Kot, H. M. Martin, R. Zehnder, and C. Zhao, “Design and analysis for interferometric measurements of the GMT primary mirror segments,” Proc. SPIE 6273, 62730M(2006).
[CrossRef]

R. Upton, “Optical control of the Advanced Technology Solar Telescope,” Appl. Opt. 45, 5881–5896 (2006).
[CrossRef] [PubMed]

2003 (1)

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

2002 (1)

C. R. Vogel, Computational Methods for Inverse Problems (SIAM2002).
[CrossRef]

1976 (1)

Briggs, J.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Burge, J. H.

J. H. Burge, L. B. Kot, H. M. Martin, R. Zehnder, and C. Zhao, “Design and analysis for interferometric measurements of the GMT primary mirror segments,” Proc. SPIE 6273, 62730M(2006).
[CrossRef]

Dalrymple, N. E.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Goodrich, B. D.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Hegwer, S. L.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Hill, F.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Hubbard, R.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Keil, S. L.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Keller, C. U.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Kot, L. B.

J. H. Burge, L. B. Kot, H. M. Martin, R. Zehnder, and C. Zhao, “Design and analysis for interferometric measurements of the GMT primary mirror segments,” Proc. SPIE 6273, 62730M(2006).
[CrossRef]

Martin, H. M.

J. H. Burge, L. B. Kot, H. M. Martin, R. Zehnder, and C. Zhao, “Design and analysis for interferometric measurements of the GMT primary mirror segments,” Proc. SPIE 6273, 62730M(2006).
[CrossRef]

Noll, R J.

Oschmann, J. M.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Radick, R. R.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Ren, D.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Rimmele, T.

Rimmele, T. R.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Upton, R.

Vogel, C. R.

C. R. Vogel, Computational Methods for Inverse Problems (SIAM2002).
[CrossRef]

Wagner, J.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Wampler, S.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Warner, M.

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

Woeger, F.

Zehnder, R.

J. H. Burge, L. B. Kot, H. M. Martin, R. Zehnder, and C. Zhao, “Design and analysis for interferometric measurements of the GMT primary mirror segments,” Proc. SPIE 6273, 62730M(2006).
[CrossRef]

Zhao, C.

J. H. Burge, L. B. Kot, H. M. Martin, R. Zehnder, and C. Zhao, “Design and analysis for interferometric measurements of the GMT primary mirror segments,” Proc. SPIE 6273, 62730M(2006).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Proc. SPIE (2)

T. R. Rimmele, S. L. Keil, C. U. Keller, F. Hill, J. Briggs, N. E. Dalrymple, B. D. Goodrich, S. L. Hegwer, R. Hubbard, J. M. Oschmann, R. R. Radick, D. Ren, J. Wagner, S. Wampler, and M. Warner, “Technical challenges of the Advanced Technology Solar Telescope,” Proc. SPIE 4837, 94–109 (2003).
[CrossRef]

J. H. Burge, L. B. Kot, H. M. Martin, R. Zehnder, and C. Zhao, “Design and analysis for interferometric measurements of the GMT primary mirror segments,” Proc. SPIE 6273, 62730M(2006).
[CrossRef]

Other (6)

C. R. Vogel, Computational Methods for Inverse Problems (SIAM2002).
[CrossRef]

ATST project website, http://atst.nso.edu.

Giant Magellan Telescope project website, http://www.gmto.org.

The condition number is the ratio of the greatest to the least singular values of a positive semidefinite matrix, which are common in active optical control problems where the matrix is overdetermined. The condition number can also be used to quantify the degree of linear independence between the compensating DOFs that are encapsulated in the columns of Jacobian matrices such as HM2

MATLAB is a licensed product of The Mathworks, http://www.mathworks.com.

ZEMAX is a licensed product of the ZEMAX Development Corporation, http://www.zemax.com.

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Figures (11)

Fig. 1
Fig. 1

Profile view of the ATST with the primary and secondary mirrors, which control the low-order aberrations, and the M3 and M6 mirrors, which control pupil and image boresight.

Fig. 2
Fig. 2

Distribution of the three guide objects in the ATST FOV for R3GO. The radius of the circle is 1 arcmin .

Fig. 3
Fig. 3

Flow chart describing the correction strategy used to introduce the force optimized corrections to the secondary mirror, and bending mode corrections to the primary mirror. The correction strategy begins with boresight corrections to the pupil and image.

Fig. 4
Fig. 4

Screen capture of the active optical modeling tool developed by the NSO for active optical control modeling of the ATST.

Fig. 5
Fig. 5

Flow of control between MATLAB and ZEMAX that is accomplished with QATST. The dynamic flow of control is enabled using the dynamic data exchange protocol available in Windows operating systems. The model presented is used to form the Jacobian matrices and to perform optical modeling simulations.

Fig. 6
Fig. 6

RMS wavefront error maps at the center of the science field corresponding to the initial postcommissioning perturbation, the correction using R3GO, the correction using R1GO, and the correction using RFO in units of nanometers.

Fig. 7
Fig. 7

RMS wavefront error contour maps in units of nanometers over the ATST FOV corresponding to the initial postcommissioning perturbation, the correction using R3GO, the correction using R1GO, and the correction using RFO.

Fig. 8
Fig. 8

Actuator force maps in units of Newtons mapped onto the primary mirror surface resulting from applying the R3GO, the R1GO, and the RFO.

Fig. 9
Fig. 9

Singular values of the Jacobian used to form the R3GO reconstructor. The singular values are plotted with no damping ρ = 0 and damping ρ = 100 . The M1 figure modes and the M2 rigid body DOFs have similar singular values. The effect of this is that the M1 figure modes are engaged when the wavefront error is large, resulting in large actuator forces applied to M1. The main contributors of the M1 and M2 DOFs to each right singular vector are represented by the groupings of the M1 and M2 DOFs shown. Strictly speaking, all the M1 and M2 DOFs contribute to each singular vector and singular value, but the dominant contributors in each case are shown.

Fig. 10
Fig. 10

Plots of the square force eigenvalues corresponding to RFO and R1GO. The force eigenvalues are identifiable by their relative magnitudes. Spherical aberration requires the greatest amount of actuator force to introduce into the mirror, and astigmatism the least. The forces for defocus are not shown because focus is introduced by a uniform piston of the primary mirror along the telescope axis, which requires zero net actuator force.

Fig. 11
Fig. 11

Plots of the first two wavefront-sensing singular vectors v 1 and v 2 of the M2 Jacobian H M 2 . These two singular vectors have the two greatest singular values, which are almost identical. The correction of the astigmatism components is most likely to occur.

Tables (2)

Tables Icon

Table 1 Perturbation Parameters for Simulation of the ATST Commissioning Case a

Tables Icon

Table 2 Mean RMS Residual Wavefront Errors, the Mean RMS Actuator Forces, and the PV Actuator Forces Required to Engage the Primary Mirror for All Three Reconstructor Cases

Equations (17)

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

g H M 2 f M 2 + 2 S a .
a K [ g H M 2 f M 2 ] ,
d a T a d f M 2 = 0.
f M 2 [ ( K H M 2 ) T ( K H M 2 ) ] 1 ( K H M 2 ) T g .
f M 2 [ ( K H M 2 ) T ( K H M 2 ) + ρ I ] 1 ( K H M 2 ) T g .
f M 2 [ H M 2 T H M 2 + ρ I ] 1 H M 2 T g .
[ f M 1 f M 2 ] = ( [ H M 1 , 3 G O H M 2 , 3 G O ] T [ H M 1 , 3 G O H M 2 , 3 G O ] + ρ I ) 1 [ H M 1 , 3 G O H M 2 , 3 G O ] T [ g 1 g 2 g 3 ] .
f M 1 = S a .
Δ a K [ g H M 2 f M 2 ] .
[ θ x , M 3 θ y , M 3 θ x , M 6 θ y , M 6 ] = [ H bore T H bore ] 1 H bore T [ x S H y S H x C X T y C X T ] .
H bore = [ δ θ x , M 3 δ x S H δ θ x , M 3 δ y S H δ θ x , M 3 δ x C X T δ θ x , M 3 δ y C X T δ θ y , M 3 δ x S H δ θ y , M 3 δ y S H δ θ y , M 3 δ x C X T δ θ y , M 3 δ y C X T δ θ x , M 6 δ x S H δ θ x , M 6 δ y S H δ θ x , M 6 δ x C X T δ θ x , M 6 δ y C X T δ θ y , M 6 δ x S H δ θ y , M 6 δ y S H δ θ y , M 6 δ x C X T δ θ y , M 6 δ y C X T ] .
θ M 3 , M 6 = [ H bore T H bore ] 1 H bore T x S H , C X T , f M 2 H # g , Δ a K [ g H M 2 f M 2 ] , f M 1 = S Δ a , θ M 3 , M 6 = [ H bore T H bore ] 1 H bore T x S H , C X T .
Δ a K [ I H M 2 H # ] g ,
Δ a T Δ a = g T [ I H M 2 H # ] T K T K [ I H M 2 H # ] g .
M = [ I H M 2 H # ] T K T K [ I H M 2 H # ] .
M = Q Λ Q T .
Δ a T Δ a = k λ k g T q k q k T g .

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