Abstract

The zero-gravity surface figure of optics used in spaceborne astronomical instruments must be known to high accuracy, but earthbound metrology is typically corrupted by gravity sag. Generally, inference of the zero-gravity surface figure from a measurement made under normal gravity requires finite-element analysis (FEA), and for accurate results the mount forces must be well characterized. We describe how to infer the zero-gravity surface figure very precisely using the alternative classical technique of averaging pairs of measurements made with the direction of gravity reversed. We show that mount forces as well as gravity must be reversed between the two measurements and discuss how the St. Venant principle determines when a reversed mount force may be considered to be applied at the same place in the two orientations. Our approach requires no finite-element modeling and no detailed knowledge of mount forces other than the fact that they reverse and are applied at the same point in each orientation. If mount schemes are suitably chosen, zero-gravity optical surfaces may be inferred much more simply and more accurately than with FEA.

© 2007 Optical Society of America

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References

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  1. M. Shao, "Science overview and status of the SIM project," Proc. SPIE 5491, 328-333 (2004).
    [CrossRef]
  2. R. A. Laskin, "Successful completion of SIM-PlanetQuest technology," Proc. SPIE 6268, 626823 (2006).
    [CrossRef]
  3. D. Korsch, Reflective Optics (Academic, 1991).
  4. R. Goullioud, "Results from the TOM3 testbed: thermal deformation of optics at the picometer level," presented at the IEEE Aerospace Conference, Big Sky, Montana (IEEE, 2006).
  5. V. A. Feria, J. C. Lam, and D. Van Buren, "M1 mirror print-thru investigation and performance on the thermo-opto-mechanical testbed for the Space Interferometry Mission," Proc. SPIE 6273, 62731C (2006).
    [CrossRef]
  6. Z. Chen, Finite Element Methods and Their Applications (Springer-Verlag, 2005).
  7. D. W. Nicholson, Finite Element Analysis: Thermomechanics of Solids (CRC, 2003).
    [CrossRef]
  8. R. Richards, Jr., Principles of Solid Mechanics (CRC, 2001).

2006 (2)

R. A. Laskin, "Successful completion of SIM-PlanetQuest technology," Proc. SPIE 6268, 626823 (2006).
[CrossRef]

V. A. Feria, J. C. Lam, and D. Van Buren, "M1 mirror print-thru investigation and performance on the thermo-opto-mechanical testbed for the Space Interferometry Mission," Proc. SPIE 6273, 62731C (2006).
[CrossRef]

2004 (1)

M. Shao, "Science overview and status of the SIM project," Proc. SPIE 5491, 328-333 (2004).
[CrossRef]

Proc. SPIE (3)

M. Shao, "Science overview and status of the SIM project," Proc. SPIE 5491, 328-333 (2004).
[CrossRef]

R. A. Laskin, "Successful completion of SIM-PlanetQuest technology," Proc. SPIE 6268, 626823 (2006).
[CrossRef]

V. A. Feria, J. C. Lam, and D. Van Buren, "M1 mirror print-thru investigation and performance on the thermo-opto-mechanical testbed for the Space Interferometry Mission," Proc. SPIE 6273, 62731C (2006).
[CrossRef]

Other (5)

Z. Chen, Finite Element Methods and Their Applications (Springer-Verlag, 2005).

D. W. Nicholson, Finite Element Analysis: Thermomechanics of Solids (CRC, 2003).
[CrossRef]

R. Richards, Jr., Principles of Solid Mechanics (CRC, 2001).

D. Korsch, Reflective Optics (Academic, 1991).

R. Goullioud, "Results from the TOM3 testbed: thermal deformation of optics at the picometer level," presented at the IEEE Aerospace Conference, Big Sky, Montana (IEEE, 2006).

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Figures (7)

Fig. 1
Fig. 1

(Color online) Views of front (top) and back (middle) of PT-M1, a prototype of the primary mirror for SIM's three-mirror compressor. The mirror is 343 mm in diameter and has a spherical surface with radius of curvature 2.2   m . Nodes for FEA modeling are shown, and bonding pads for attaching hexapod mounts are visible. At the bottom is a photograph of the mirror during fabrication, showing details of backside lightweighting. [Photograph courtesy S. Spanjian and J. Daniel, Tinsley.]

Fig. 2
Fig. 2

(Color online) Mount-induced print-through (dimples) on the TOM3 brassboard model M1, as measured (top) and as simulated with a detailed FEA model (bottom). (From [5]; color-bar units in this and all later figures are meters.) The detailed origin of the dimples is discussed in the text, Section 2. The surface map here describes a normal-gravity on-edge configuration, and includes gravity sag in brassboard M1's flightlike mount in addition to the deformation caused by the fabrication mount. Surface errors on the measured map amount to 20 nm rms , with peaks exceeding 100 nm . The FEA model has 98,000 nodes and 49,000 elements and provides an excellent qualitative match but predicts a surface error of 14 nm rms . Since flight tolerance for this optic is 6.3 nm rms, dimpling of this magnitude would not be permissible in the ultimate flight mirror.

Fig. 3
Fig. 3

(Color online) FEA modeling of the surface figure of PT-M1 during gravity reversal in a simple mirror mount in which the mirror rests on three points of contact near the rim (the threefold azimuthal symmetry is not depicted in the cartoons at left). The mirror model is specified to have a spherical surface in the absence of applied forces. The face-up and face-down orientations experience gravity forces that are reversed. However, mount forces in the two cases are applied at positions separated by the thickness of the mirror rim, so are only imperfectly reversed. Deformations in the two configurations are shown in the top two panels; their average is shown in the bottom panel, which recovers the ideal spherical surface (which would look flat in this display of departure from sphericity) marred by some dimple artifacts near the rim. The rms error in the average is 11.5 nm .

Fig. 4
Fig. 4

(Color online) FEA modeling of the surface figure of PT-M1 during gravity reversal in an improved scheme incorporating the principles of Section 4. The mount now consists of three point contacts bonded to the hub at the back of the mirror (cartoons at left, but with threefold azimuthal symmetry, not depicted); mount forces thus reverse and are applied at very nearly the same positions in the two orientations, far from the mirror surface. As a result, the surface figures (top two panels at right) are nearly exactly complementary, giving an average map (lower right panel) with a formal rms error of only 0.0003 nm .

Fig. 5
Fig. 5

Nanometer-scale optical surface metrology of the actual PT-M1 mirror for the face-up (upper left) and face-down (upper right) configurations; as with the modeling of Fig. 4, these are very nearly complementary to each other. The lower panel shows the average of the two, which gives the zero-gravity mirror figure; it departs from a spherical surface by only 5.7 nm rms . Included in this value are the surface errors of a fold flat in the optical train and some known imperfections in the mounts. The central spot is high by perhaps 35 nm but covers too small an area to have significant impact on the rms. [Metrology data courtesy S. Spanjian, T. Roff, L. Dettmann, and J. Daniel, Tinsley.]

Fig. 6
Fig. 6

FEA modeling of the surface figure of PT-M1 for a simple, near-optimal, on-edge gravity-reversal mounting scheme that can conveniently be implemented during the fabrication cycle. Two horizontal pegs under lightweighting ribs, near the vertical center, provide support. The average map (lower panel) has an rms error of only 1.5 nm . All figure panels at right are fixed in the frame of the optic; arrows indicate the direction of gravity.

Fig. 7
Fig. 7

Nanometer-scale optical surface metrology (upper panels) of the actual PT-M1 mirror for six different clockings of the near-optimal on-edge mount modeled in Fig. 6, showing a strong resemblance to the FEA modeling of Fig. 6. Each successive clocking is rotated by an angle of 60°. The lower map is the average of all six clockings, representing three gravity-reversed pairs; it departs from a sphere by only 3.8 nm rms . All figure panels are fixed in the frame of the optic; arrows indicate the direction of gravity. [Metrology data courtesy S. Spanjian, T. Roff, L. Dettmann, and J. Daniel, Tinsley.]

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