Abstract

Giant segmented telescopes in general require in situ control of the figures of their hexagonal segments. The necessary inputs may be either direct, through surface measurements, or indirect, through measurements of the intersegment edge heights. We investigate the limitations on the indirect (edge height) control that follow from the geometrical properties of hexagonal surfaces and find that these may be substantial. In particular, there are many unmeasurable modes, which we construct and count systematically, and whose potential effect on the segment figure problem we discuss.

© 2009 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  13. “APS conceptual design report--Volume 1: Technical plan,” TMT report TMT.CTR.CDD.07.002.REL01 (TMT, 2007).

2009 (1)

L. Noethe, European Southern Observatory, Karl-Schwartzschild-Strasse 2, Garching bei Munchen, Germany (private communication, 2009).

2008 (1)

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219-701219-10 (2008).

2007 (2)

“APS conceptual design report--Volume 1: Technical plan,” TMT report TMT.CTR.CDD.07.002.REL01 (TMT, 2007).

E. Williams, “Warping harness analysis--1.44 m segments,” TMT report TMT.OPT.PRE.07.011.REL01 (TMT, 2007).

2006 (1)

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[CrossRef]

2002 (1)

2000 (2)

G. A. Chanan, C. Ohara, and M. Troy, “Phasing the mirror segments of the Keck telescopes: the narrowband phasing algorithm,” Appl. Opt. 39, 4706-4714 (2000).
[CrossRef]

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

1998 (2)

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

G. A. Chanan, M. Troy, F. G. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37, 140-155 (1998).
[CrossRef]

1989 (1)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ. Press, 1989).

1985 (1)

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, 1985).

1976 (1)

Alvarez, P.

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

Andersen, T.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Ardeberg, A.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Beckers, J.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Chanan, G. A.

Dekens, F. G.

Devaney, N.

Faber, S. M.

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, 1985).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ. Press, 1989).

Flicker, R.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Gilmozzi, R.

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219-701219-10 (2008).

Gontcharov, A.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Jessen, N. C.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Kirkman, D.

Mannery, E.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Mast, T.

Mast, T. S.

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, 1985).

Michaels, S.

Montoya, L.

Nelson, J.

Nelson, J. E.

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, 1985).

Noethe, L.

L. Noethe, European Southern Observatory, Karl-Schwartzschild-Strasse 2, Garching bei Munchen, Germany (private communication, 2009).

Noll, R. J.

Ohara, C.

Owner-Petersen, M.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ. Press, 1989).

Riewaldt, H.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

Rodriguez-Espinoza, J. M.

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

Sanchez, F.

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

Sanders, G. H.

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[CrossRef]

Schumacher, A.

Spyromilio, J.

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219-701219-10 (2008).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ. Press, 1989).

Troy, M.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ. Press, 1989).

Williams, E.

E. Williams, “Warping harness analysis--1.44 m segments,” TMT report TMT.OPT.PRE.07.011.REL01 (TMT, 2007).

Appl. Opt. (3)

Astrophys. Space Sci. (1)

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. SPIE (2)

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[CrossRef]

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219-701219-10 (2008).

Other (6)

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, 1985).

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., ESO Conference and Workshop Proceedings (ESO, 2000), Vol. 57, pp. 72-82.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge Univ. Press, 1989).

E. Williams, “Warping harness analysis--1.44 m segments,” TMT report TMT.OPT.PRE.07.011.REL01 (TMT, 2007).

L. Noethe, European Southern Observatory, Karl-Schwartzschild-Strasse 2, Garching bei Munchen, Germany (private communication, 2009).

“APS conceptual design report--Volume 1: Technical plan,” TMT report TMT.CTR.CDD.07.002.REL01 (TMT, 2007).

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Figures (9)

Fig. 1
Fig. 1

Unobservable mode (surface A) for a hexagon constrained at three places along each of the two right-hand edges with piston, tip, tilt, focus, and two astigmatism degrees of freedom controlled.

Fig. 2
Fig. 2

Edge profiles for the unobservable mode (surface A) shown in Fig. 1. Note that the edges opposite the zero edges are straight lines and can thus be perfectly aligned to flat neighboring segments having piston and tilt errors only.

Fig. 3
Fig. 3

One of the unobservable folding modes for the three-ring telescope, with two columns of segments folded. See text for details of the construction. There are 18 such folding modes for the three-ring telescope and 6 n such modes for the n-ring telescope.

Fig. 4
Fig. 4

Unobservable pseudo-trefoil mode for the three-ring telescope. To see that this mode is linearly independent of the other unobservable modes in the six degree of freedom problem, note that it disappears if the missing central segment is replaced, but none of the other modes is affected.

Fig. 5
Fig. 5

One of the 91 unobservable modes determined numerically by singular value decomposition for the 492-segment Thirty Meter Telescope. Here we have assumed that six segment degrees of freedom—piston, tip, tilt, focus, and two astigmatisms—are controlled so as to minimize the intersegment edge heights at three points along each intersegment edge. Such modes are linear combinations of the more intuitive (mostly folding) modes described in the text.

Fig. 6
Fig. 6

Unobservable mode (surface B) for a hexagon constrained at three places along each of the three lower edges with piston, tip, tilt, focus, two astigmatisms, and two trefoils controlled.

Fig. 7
Fig. 7

Edge profiles for the unobservable mode (surface B) shown in Fig. 6. The edge opposite the middle zero edge has a parabolic profile and can thus be perfectly aligned to a neighboring segment with a pure focus (plus piston) error.

Fig. 8
Fig. 8

Typical segment-centered mode for the eight degree of freedom problem for a two-ring telescope. This mode is constructed by placing surface B directly above the missing central segment and surrounding it with surface B segments of appropriate orientation. All other segments are flat. Here there are five surface B segments, but for most locations in larger telescopes there are six.

Fig. 9
Fig. 9

Unobservable “outer ring” mode for the eight degree of freedom problem for a two-ring telescope, in which the outer ring of segments is nonflat, with a 6 θ symmetry, but the inner ring is flat.

Tables (5)

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Table 1 Zernike Polynomials Used in This Work

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Table 2 Zernike Expansion Coefficients for Single Hexagon Modes

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Table 3 Number of Unobservable Modes for Various Telescope Geometries

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Table 4 Prealignment and Postalignment Segment Figures for the 6 DoF Problem for TMT a

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Table 5 Prealignment and Postalignment Segment Figures for the 8 DoF Problem for TMT a

Equations (2)

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Aa = b ,
u 6 n + 8 = v j = 1 6 n + 7 v · u j

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