Abstract

The actuator influence functions of a thin meniscus mirror can be expanded in Zernike polynomials. And the correctness of influence functions has a great effect on solving the correction forces. The Zernike coefficients are applied as parameters in the all-floating support system. We analyze the main interferential factors when different deformation modes are corrected. The influence caused by fitting errors is studied in this paper. A preferable result can be obtained after eliminating the interferential factors. The method can obtain useful correction forces. Comparing the peak-to-valley and root mean square values among the results calculated by different accuracy influence functions, we find that there is a limited convergence property when the accuracy increases.

© 2013 Optical Society of America

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  1. P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
    [CrossRef]
  2. J. DeVries, D. Neill, and E. Hileman, “LSST telescope primary/tertiary mirror hardpoints,” Proc. SPIE 7739, 77391J (2010).
    [CrossRef]
  3. J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors—optimization of aperture size,” Appl. Opt. 32, 2431–2438 (1993).
    [CrossRef]
  4. K. E. Moore and G. N. Lawrence, “Zonal model of an adaptive mirror,” Appl. Opt. 29, 4622–4628 (1990).
    [CrossRef]
  5. Z. G. Shi, Y. X. Sui, Z. Y. Liu, J. Peng, and H. J. Yang, “Mathematical construction and perturbation analysis of Zernike discrete orthogonal points,” Appl. Opt. 51, 4210–4214 (2012).
    [CrossRef]
  6. D. R. Iskander, M. J. Collins, B. Davis, and L. G. Carney, “Monochromatic aberrations and characteristics of retinal image quality,” Clin. Exp. Optom. 83, 315–322 (2000).
    [CrossRef]

2012 (1)

2010 (2)

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

J. DeVries, D. Neill, and E. Hileman, “LSST telescope primary/tertiary mirror hardpoints,” Proc. SPIE 7739, 77391J (2010).
[CrossRef]

2000 (1)

D. R. Iskander, M. J. Collins, B. Davis, and L. G. Carney, “Monochromatic aberrations and characteristics of retinal image quality,” Clin. Exp. Optom. 83, 315–322 (2000).
[CrossRef]

1993 (1)

1990 (1)

Alda, J.

Boreman, G. D.

Capaccioli, M.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Carney, L. G.

D. R. Iskander, M. J. Collins, B. Davis, and L. G. Carney, “Monochromatic aberrations and characteristics of retinal image quality,” Clin. Exp. Optom. 83, 315–322 (2000).
[CrossRef]

Collins, M. J.

D. R. Iskander, M. J. Collins, B. Davis, and L. G. Carney, “Monochromatic aberrations and characteristics of retinal image quality,” Clin. Exp. Optom. 83, 315–322 (2000).
[CrossRef]

D’Orsi, S.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Davis, B.

D. R. Iskander, M. J. Collins, B. Davis, and L. G. Carney, “Monochromatic aberrations and characteristics of retinal image quality,” Clin. Exp. Optom. 83, 315–322 (2000).
[CrossRef]

De Paris, G.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

DeVries, J.

J. DeVries, D. Neill, and E. Hileman, “LSST telescope primary/tertiary mirror hardpoints,” Proc. SPIE 7739, 77391J (2010).
[CrossRef]

Ferragina, L.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Fierro, D.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Hileman, E.

J. DeVries, D. Neill, and E. Hileman, “LSST telescope primary/tertiary mirror hardpoints,” Proc. SPIE 7739, 77391J (2010).
[CrossRef]

Iskander, D. R.

D. R. Iskander, M. J. Collins, B. Davis, and L. G. Carney, “Monochromatic aberrations and characteristics of retinal image quality,” Clin. Exp. Optom. 83, 315–322 (2000).
[CrossRef]

Lawrence, G. N.

Liu, Z. Y.

Magrin, D.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Marty, L.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Molfese, C.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Moore, K. E.

Neill, D.

J. DeVries, D. Neill, and E. Hileman, “LSST telescope primary/tertiary mirror hardpoints,” Proc. SPIE 7739, 77391J (2010).
[CrossRef]

Peng, J.

Perina, F.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Perrotta, F.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Recchia, S.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Rossettini, P.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Schipani, P.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Shi, Z. G.

Sui, Y. X.

Tomelleri, R.

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

Yang, H. J.

Appl. Opt. (3)

Clin. Exp. Optom. (1)

D. R. Iskander, M. J. Collins, B. Davis, and L. G. Carney, “Monochromatic aberrations and characteristics of retinal image quality,” Clin. Exp. Optom. 83, 315–322 (2000).
[CrossRef]

Proc. SPIE (2)

P. Schipani, M. Capaccioli, S. D’Orsi, L. Ferragina, L. Marty, C. Molfese, F. Perrotta, G. De Paris, D. Fierro, R. Tomelleri, P. Rossettini, F. Perina, S. Recchia, and D. Magrin, “The VST active primary mirror support system,” Proc. SPIE 7739, 773931 (2010).
[CrossRef]

J. DeVries, D. Neill, and E. Hileman, “LSST telescope primary/tertiary mirror hardpoints,” Proc. SPIE 7739, 77391J (2010).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Mirror’s deformation calculated by different accuracy of influence functions. (a), (b) Restoration of the 5th Zernike mode by using 10 and 20 Zernike polynomials as influence functions, respectively.

Fig. 2.
Fig. 2.

Force distribution caused by different Zernike modes in influence functions. When calculating the force vector a⃗, individually add the 17th, 21st, 22nd, 31st, and 32nd Zernike term of influence functions. Compared with (a), calculated by the influence function without correlation terms, (b)–(f) obviously show the bad results.

Fig. 3.
Fig. 3.

Correlative Zernike mode terms when correcting the 5th Zernike mode. (a) The 17th Zernike mode, (b) the 21st Zernike mode, (c) the 22nd Zernike mode, (d) the 31st Zernike mode, and (e) the 32nd Zernike mode.

Fig. 4.
Fig. 4.

Influence Zernike modes when correcting the 7th and 9th Zernike modes. (a) The target deformation, the 7th Zernike polynomial surface, (b) the 23th Zernike mode, (c) the 34th Zernike mode, (d) the 35th Zernike mode. (e) The target deformation, the 9th Zernike polynomial surface, (f) the 16th Zernike mode, and (g) the 36th Zernike mode.

Fig. 5.
Fig. 5.

(a) Surface deformation supported by the central actuator. (b) Difference in Zernike coefficients when describing the deformation (a) by different precisions. The D-value obtained by (b) represents that the deformation (a) was separately decomposed to 36, 64, 100, and 144 Zernike polynomials. The coefficients, which calculated by 36 terms Zernike polynomials in reconstruction, are subtrahend. And the mutual 36 coefficients, which calculated 64, 100, and 144 terms Zernike polynomials in reconstruction, are minuend. Line 1, line 2, and line 3 are the difference between the coefficients obtained by 64, 100, and 144 Zernike polynomials and the coefficients obtained by 36 Zernike polynomials in reconstruction.

Fig. 6.
Fig. 6.

PV and RMS values of the residual error by different accuracy influence function in calculating 5th, 7th, and 9th Zernike deformation. The PV value of the idea deformation which we want to get is 2 μm.

Equations (20)

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

m(x,y)=i=1Nwi(x,y)ai,
m(2D)=W(3D)·a⃗(1D),
a⃗=(WtW)1Wt.
m=W·(WtW)1Wt.
m=W·Wt.
wi(x,y)=k=1KcikZk(x,y).
W=ZC,
m=ZC·a⃗.
=Zb⃗,
m=ZC(CtZtZC)1CtZtZb⃗=ZC(CtC)1Ctb⃗.
a⃗=(CtC)1Ctb⃗,
W=ZC+Δw,
(WtW)1=((ZC+Δw)t(ZC+Δw))1.
m=(ZC+Δw)·((ZC+Δw)t(ZC+Δw))1(ZC+Δw)t,
m=(ZC+Δm)a⃗,
=Zb⃗+Δ,
m=(ZC+Δw)·((ZC+Δw)t(ZC+Δw))1(ZC+Δw)t(Zb⃗+Δ).
m=(ZC+Δm)a⃗=(ZC+Δw+ΔmΔw)a⃗=(ZC+Δw)a⃗+(ΔmΔw)a⃗(ZC+Δw)a⃗.
a⃗=((ZC+Δw)t(ZC+Δw))1(ZC+Δw)t(Zb⃗+Δ).
a⃗=((ZC+Δw)t(ZC+Δw))1(ZC+Δw)tZb⃗.

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