Abstract

The Advanced Technology Solar Telescope (ATST) is an off-axis Gregorian astronomical telescope design. The ATST is expected to be subject to thermal and gravitational effects that result in misalignments of its mirrors and warping of its primary mirror. These effects require active, closed-loop correction to maintain its as-designed diffraction-limited optical performance. The simulation and modeling of the ATST with a closed-loop correction strategy are presented. The correction strategy is derived from the linear mathematical properties of two Jacobian, or influence, matrices that map the ATST rigid-body (RB) misalignments and primary mirror figure errors to wavefront sensor (WFS) measurements. The two Jacobian matrices also quantify the sensitivities of the ATST to RB and primary mirror figure perturbations. The modeled active correction strategy results in a decrease of the rms wavefront error averaged over the field of view (FOV) from 500 to 19  nm, subject to 10   nm rms WFS noise. This result is obtained utilizing nine WFSs distributed in the FOV with a 300   nm rms astigmatism figure error on the primary mirror. Correction of the ATST RB perturbations is demonstrated for an optimum subset of three WFSs with corrections improving the ATST rms wavefront error from 340 to 17 .8   nm. In addition to the active correction of the ATST, an analytically robust sensitivity analysis that can be generally extended to a wider class of optical systems is presented.

© 2006 Optical Society of America

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2004

1994

1985

1983

Barrett, H. H.

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

Chanan, G.

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

Johnston, D. C.

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

Maeda, J.

McMartin, D.

Milster, T.

Myers, K.

H. H. Barrett and K. Myers, Foundations of Image Science (Wiley, 2004).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

Paxman, R.

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

Ruda, M.

M. Ruda, "Methods for null testing sections of aspheric surfaces," Ph.D. dissertation (Optical Sciences Center, University of Arizona, 1979).

Smith, W.

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Brooks/Cole, 1988).

Sweeney, D.

D. Sweeney, LSST Project Manager, Large Synoptic Survey Telescope, Tucson, Ariz. 85712 (personal communication, 2005).

Tyson, R.

R. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1997), Chap. 7.

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

Wallner, E. P.

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

Welsh, B. M.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

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, M. Warner, and the rest of the ATST team, "Technical challenges of the Advanced Technology Solar Telescope," in Proc. SPIE 4837,94-109 (2003).

MATLAB is a licensed product of The MathWorks, www.mathworks.com.

CODE V is a licensed product of Optical Research Associates (ORA), www.opticalres.com.

H. H. Barrett and K. Myers, Foundations of Image Science (Wiley, 2004).

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Brooks/Cole, 1988).

R. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, 1997), Chap. 7.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, 1996).

D. Sweeney, LSST Project Manager, Large Synoptic Survey Telescope, Tucson, Ariz. 85712 (personal communication, 2005).

M. Ruda, "Methods for null testing sections of aspheric surfaces," Ph.D. dissertation (Optical Sciences Center, University of Arizona, 1979).

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Figures (17)

Fig. 1
Fig. 1

Schematic of the ATST shown with nine fields. The mirrors and optical surfaces are numbered.

Fig. 2
Fig. 2

Nine WFSs are arranged in a 3 × 3 distribution in the FOV. The numbers in parentheses label the locations of the WFSs in units of degrees.

Fig. 3
Fig. 3

Logical flow between the ATST optical model in CODE V and the reconstructor in MATLAB. The model represented in the figure corrects the RB misalignments of the ATST; then the model corrects for the M1 LBMs. Both correction strategies are contained in separate boxes.

Fig. 4
Fig. 4

Schematic of the pupil coordinate system that is used to define the polar angle ϕ.

Fig. 5
Fig. 5

Singular values of H (crosses) and H ˜ TRUN ( triangles ) are plotted for every iteration of the mode truncation algorithm. The plot is a graphical representation of the truncation algorithm, which shows the condition number of H ˜ TRUN decreasing with each iteration of the algorithm. The units of the y axis are consistent with the singular values of the system, which are waves per perturbation type.

Fig. 6
Fig. 6

Variation of the rms 45° astigmatism averaged over the ATST FOV as a function of the number of DLSR iterations.

Fig. 7
Fig. 7

rms Zernike coefficients averaged over the ATST FOV and averaged over the 25 Monte Carlo realizations of the ATST as a function of the number of reconstructor iterations.

Fig. 8
Fig. 8

WFS modes before the DLSR is applied and the WFS modes after the DLSR is applied for a single Monte Carlo realization of the ATST. The WFS modes are explicitly indicated in the plot. That is, the WFS modes are defined as the rms Zernike polynomials distributed in each ATST WFS.

Fig. 9
Fig. 9

Total rms aberrations averaged over the ATST FOV and the Monte Carlo realizations of the ATST for 100, 200, and 300   nm of X astigmatism ( Z 4 ) M1 figure errors. The pseudoinverse reconstructor is applied after three iterations of the DLSR.

Fig. 10
Fig. 10

WFS modes before and after the DLSR and pseudoinverse reconstructor are applied. The WFS modes shown correspond to a single Monte Carlo realization of the ATST with 100   nm Z 4 LBMs introduced into M1.

Fig. 11
Fig. 11

Total rms aberrations averaged over the ATST FOV and the Monte Carlo realizations of the ATST for 100, 200, and 300   nm of 45° astigmatism ( Z 6 ) M1 figure errors. The pseudoinverse reconstructor is applied after three iterations of the DLSR.

Fig. 12
Fig. 12

Total rms aberrations averaged over the ATST FOV and the Monte Carlo realizations of the ATST for 100, 200, and 300   nm of third-order spherical aberration ( Z 13 ) M1 figure errors. The pseudoinverse reconstructor is applied after three iterations of the DLSR.

Fig. 13
Fig. 13

WFS modes before and after the DLSR and pseudoinverse reconstructor are applied. The WFS modes shown correspond to a single Monte Carlo realization of the ATST with 100   nm Z 6 LBMs introduced into M1.

Fig. 14
Fig. 14

WFS modes before and after the DLSR and pseudoinverse reconstructor are applied. The WFS modes shown correspond to a single Monte Carlo realization of the ATST with 100   nm Z 13 LBMs introduced into M1.

Fig. 15
Fig. 15

rms modulation of g for increasing left singular vectors included in the sum shown in Eq. (9). The rms variation quantifies the modulation of the wavefront signal sensed by the WFS in the ATST FOV.

Fig. 16
Fig. 16

WFS modes subject to increasing amounts of truncation according to Eq. (9). The WFS modes shown correspond to the average Monte Carlo realization of the ATST optical model.

Fig. 17
Fig. 17

Residual rms aberration across the ATST FOV as a function of reconstructor iteration for three WFS cases.

Tables (9)

Tables Icon

Table 1 Optical Prescription of the ATST Given to Two Significant Figures a

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Table 2 RBDOFs for M1 and M2

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Table 3 Zernike Polynomials Used to Define the Optical Performance of the ATST a

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Table 4 Order of ATST RBDOF Sensitivities a

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Table 5 Operational Sensitivities of the ATST RBDOFs a

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Table 6 Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithm a

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Table 7 Torsional Motions of M1 and M2 that Result in the Dominant Aberration 45° Astigmatism Z6 a

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Table 8 Singular Values of H SEC and Eigenvalues for H SEC T H SEC a

Tables Icon

Table 9 Order of the ATST M1 LBMs Truncated Using the Mode Truncation Algorithm a

Equations (48)

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g = h 0 + H f + B a .
h i j = δ g i δ f j ,
b i j = δ g i δ a j .
B # = ( B T B ) 1 B T ,
H # = ( H T H + ρ 2 I ) 1 H T .
g = [ Z 4 , WFS 1 Z 4 , WFS 9 Z 10 , WFS 1 Z 10 , WFS 9 ] .
H = V Σ U T .
f = n = 1 R ( u n     T f ) u n ,
g = n = 1 R ( v n     T g ) v n .
( v n     T g ) = μ n ( u n     T f ) .
f n = 1 P ( u n     T f ) u n ,
g n = 1 P ( v n     T g ) v n ,
H = U T .
H ˜ = Σ U T .
H ˜ = V ˜ Σ ˜ U ˜ T .
H ˜ TRUN = V ˜ Σ ˜ U ˜ T .
Σ U T = Σ ˜ U ˜ T .
H ˜ = [ μ 1 0 0 0 0 μ 2 0 0 0 0 0 0 0 0 μ P ] × [ u 1 ( 1 ) u 1 ( 2 ) ... u 1 ( P ) u 2 ( 1 ) u 2 ( 2 ) ... u 2 ( P ) ... u P ( 1 ) u P ( 2 ) ... u P ( P ) ] .
H ˜ = [ μ 1 u 1 ( 1 ) μ 1 u 1 ( 2 ) μ 1 u 1 ( P ) μ 2 u 2 ( 1 ) μ 2 u 2 ( 2 ) μ 2 u 2 ( P ) μ P u P ( 1 ) μ P u P ( 2 ) μ P u P ( P ) ] .
H ˜ TRUN = [ μ 1 u 1 ( 1 ) t μ 1 u 1 ( P ) μ 2 u 2 ( 1 ) t μ 2 u 2 ( P ) μ P u P ( 1 ) t μ P u P ( P ) ] .
1 cond ( H ) = μ P μ 1 .
σ n = δ g n δ s ,
σ n , O = σ n × δ s .
f ^ = U [ Σ 2 + R 2 ] 1 U T H T [ g h 0 ] .
H T H = U Σ 2 U T .
f ^ = U [ Σ 2 + σ R       2 I ] 1 U T H T [ g h 0 ] .
v n     T g = μ n u n     T f .
σ g , rms = n P ( v n     T g ) 2 ,
σ g , rms = n P μ n ( u n     T f ) 2 .
H # = [ H T H + ρ 2 I ] 1 H T .
B # = [ B T B ] 1 B T .
f ^ = [ H T H + Σ n Σ f     1 ] 1 H T [ g h 0 ] .
g = h 0 + [ H B ] [ f a ] .
g = H f + n .
ln [ P ( n ) ] n T Σ n     1 n .
Σ n = n n T .
ln [ P ( f ) ] f T Σ f     1 f .
Σ f = f f T .
n ^ = g H f ^ .
P ( f , n ) exp ( f T Σ f     1 f [ g H f ^ ] T Σ n     1 [ g H f ^ ] ) .
f ^ = [ H T H + Σ n Σ f     1 ] 1 H T g .
Σ n     1 = σ n     2 I ,
Σ f = σ f     2 I .
f ^ = [ H T H + σ n     2 σ f     2 I ] 1 H T g .
H = V Σ U T .
f ^ = [ U Σ 2 U T + σ n     2 σ f     2 U U T ] 1 U Σ V T .
f ^ = U [ Σ 2 + σ n     2 σ f     2 ] 1 U T U Σ V T ,
f ^ = U [ Σ 2 + σ n     2 σ f     2 ] 1 Σ V T .

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