Abstract

Zernike polynomials are generally used to predict the optical performance of a mirror. However, it can also be done by a numerical iterative method. As piston, tip, tilt, and defocus (P.T.T.F) aberrations can be easily removed by optical alignment, we iteratively used a rotation transformation and a paraboloid graph subtraction for removal of the aberrations from a raw deformation of the optical surface through a Finite Element Method (FEM). The results of a 30 cm concave circular mirror corrected by the iterative method were almost the same as those yielded by Zernike polynomial fitting, and the computational time was fast. In addition, a concave square mirror whose surface area is π was analyzed in order to visualize the deformation maps of a general mirror aperture shape. The iterative method can be applicable efficiently because it does not depend on the mirror aperture shape.

© 2014 Optical Society of America

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References

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    [Crossref]
  5. M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
    [Crossref]
  6. M. Cho, M. Liang, and D. R. Neill, “Performance prediction of the LSST secondary mirror,” Proc. SPIE 7424, 742407 (2009).
    [Crossref]
  7. K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).
  8. P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
    [Crossref]
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    [Crossref]
  13. A. H. Al-Hamdani and S. Y. Hasan, “Zernike polynomiales for optical systems with rectangular and square apertures of area equal to π,” IJPAP 51(12), 837–843 (2013).

2013 (1)

A. H. Al-Hamdani and S. Y. Hasan, “Zernike polynomiales for optical systems with rectangular and square apertures of area equal to π,” IJPAP 51(12), 837–843 (2013).

2012 (1)

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

2011 (1)

M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
[Crossref]

2010 (1)

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

2009 (1)

M. Cho, M. Liang, and D. R. Neill, “Performance prediction of the LSST secondary mirror,” Proc. SPIE 7424, 742407 (2009).
[Crossref]

2008 (1)

M. Cho, “Performance prediction of the TMT secondary mirror support system,” Proc. SPIE 7018, 70181S (2008).
[Crossref]

2005 (1)

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 586701 (2005).
[Crossref]

1994 (1)

1976 (1)

Al-Hamdani, A. H.

A. H. Al-Hamdani and S. Y. Hasan, “Zernike polynomiales for optical systems with rectangular and square apertures of area equal to π,” IJPAP 51(12), 837–843 (2013).

Bisson, G. R.

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 586701 (2005).
[Crossref]

Bouchez, A.

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

Cho, M.

M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
[Crossref]

M. Cho, M. Liang, and D. R. Neill, “Performance prediction of the LSST secondary mirror,” Proc. SPIE 7424, 742407 (2009).
[Crossref]

M. Cho, “Performance prediction of the TMT secondary mirror support system,” Proc. SPIE 7018, 70181S (2008).
[Crossref]

Chow, W. W.

Corredor, A.

M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
[Crossref]

Doyle, K. B.

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 586701 (2005).
[Crossref]

Dribusch, C.

M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
[Crossref]

Genberg, V. L.

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 586701 (2005).
[Crossref]

Han, W.

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

Hart, M.

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

Hasan, S. Y.

A. H. Al-Hamdani and S. Y. Hasan, “Zernike polynomiales for optical systems with rectangular and square apertures of area equal to π,” IJPAP 51(12), 837–843 (2013).

Hinz, P. M.

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

Jeong, W. S.

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

Johns, M.

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

Kim, Y. S.

M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
[Crossref]

Lee, D. H.

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

Liang, M.

M. Cho, M. Liang, and D. R. Neill, “Performance prediction of the LSST secondary mirror,” Proc. SPIE 7424, 742407 (2009).
[Crossref]

McGregor, P.

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

McLeod, B.

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

Michels, G. J.

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 586701 (2005).
[Crossref]

Moon, B.

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

Moon, I. K.

M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
[Crossref]

Nam, U. W.

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

Neill, D. R.

M. Cho, M. Liang, and D. R. Neill, “Performance prediction of the LSST secondary mirror,” Proc. SPIE 7424, 742407 (2009).
[Crossref]

Noll, R. J.

Park, K.

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
[Crossref]

Park, Y.

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

Pyo, J.

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

Shectman, S.

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

Swantner, W.

Appl. Opt. (1)

IJPAP (1)

A. H. Al-Hamdani and S. Y. Hasan, “Zernike polynomiales for optical systems with rectangular and square apertures of area equal to π,” IJPAP 51(12), 837–843 (2013).

J. Opt. Soc. Am. (1)

JASS (1)

K. Park, B. Moon, D. H. Lee, W. S. Jeong, U. W. Nam, Y. Park, J. Pyo, and W. Han, “Performance Analysis for mirrors of 30 cm cryogenic space infrared telescope,” JASS 29(3), 321–328 (2012).

Proc. SPIE (5)

P. M. Hinz, A. Bouchez, M. Johns, S. Shectman, M. Hart, B. McLeod, and P. McGregor, “The GMT adaptive optics system,” Proc. SPIE 7736, 77360C(2010).
[Crossref]

M. Cho, “Performance prediction of the TMT secondary mirror support system,” Proc. SPIE 7018, 70181S (2008).
[Crossref]

M. Cho, A. Corredor, C. Dribusch, K. Park, Y. S. Kim, and I. K. Moon, “Design and Development of a Fast Steering Secondary Mirror for the Giant Magellan Telescope,” Proc. SPIE 8125, 812505 (2011).
[Crossref]

M. Cho, M. Liang, and D. R. Neill, “Performance prediction of the LSST secondary mirror,” Proc. SPIE 7424, 742407 (2009).
[Crossref]

K. B. Doyle, V. L. Genberg, G. J. Michels, and G. R. Bisson, “Optical modeling of finite element surface displacements using commercial software,” Proc. SPIE 5867, 586701 (2005).
[Crossref]

Other (4)

E. B. Becker, G. F. Carey, and J. T. Oden, Finite Elements: An Introduction. Volume 1 (PRENTICE-Hall, 1981).

D. S. Watkins, Fundamentals of Matrix Computations (WILEY, 2010).

P. R. Yoder, Opto-mechanical System Design (Marcel Dekker, 1993).

P. Y. Bely, The Design and Construction of Large Optical Telescope (Springer, 2003).

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Figures (9)

Fig. 1
Fig. 1 P.T.T. correction process steps by rotation transformation.
Fig. 2
Fig. 2 Paraboloid graph according to coefficient increment (Δc1).
Fig. 3
Fig. 3 Flow chart of iterative method.
Fig. 4
Fig. 4 Optical ray path of mirror.
Fig. 5
Fig. 5 A 30 cm concave circular mirror and three flexures: (a) Solid model, (b) Finite element model.
Fig. 6
Fig. 6 The analysis results for Y-Gravity: (a) FEM result, (b) Raw deformation map, (c) Correction maps by the iterative method, (d) Correction maps by Zernike polynomial fitting, (e) Difference map.
Fig. 7
Fig. 7 Optical deformation map of the mirror for each loading; (a) Raw deformation map: from left, X-Gravity, Z-Gravity, and a 10 K temperature change, (b) Correction map by iterative method: from left, X-Gravity, Z-Gravity, and a 10 K temperature change, (c) Difference map: from left, X-Gravity, Z-Gravity, and a 10 K temperature change.
Fig. 8
Fig. 8 A concave square mirror and four flexures: (a) Solid model, (b) FE model.
Fig. 9
Fig. 9 Optical deformation maps of a square mirror before and after the iterative method and Zernike polynomial fitting, for each load condition: (a) X-Gravity, (b) Y-Gravity, (c) Z-Gravity, (d) 10 K temperature change.

Tables (2)

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Table 1 Surface P-V and RMS by the Iterative Method and Zernike Polynomial Fitting for Each Load Condition

Tables Icon

Table 2 Time Taken To Calculate P.T.T.F. Aberration by the Iterative Method and Zernike Polynomial Fitting for Each Load Condition

Equations (3)

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

( x i ' y i ' Δ z n,i ' )=[ 1 0 0 0 cosθ sinθ 0 sinθ cosθ ]( x i y i Δ z n,i ).
( x i * y i * z n,i * )=[ cosθ 0 sinθ 0 1 0 sinθ 0 cosθ ]( x i ' y i ' z n,i ' ).
z f,i =c1 r i 2 +c0,

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