Abstract

An elegant and accurate way to determine the zero-gravity surface figure of an optic from ground-based interferometric metrology is to average the figures found in two or more configurations that are rotated with respect to the direction of gravity, so gravity forces in the frame of the optic cancel in the average. In a recent elucidation of this technique, we emphasized that care must be taken to ensure that mount forces at each attachment point similarly cancel, and we presented some specific mounting schemes that gave accurate zero-gravity surface determinations during fabrication and acceptance testing of the Space Interferometry Mission PT-M1 mirror. Here we show that multiconfiguration averaging techniques work well for the most important special case of a mirror in a flightlike hexapod mount clocked into either two or three symmetrically placed positions. We explicitly compute mount forces (axial forces in the six struts of the hexapod) and show that at any attachment point their average over multiple clocked configurations vanishes in the frame of the optic, ensuring the success of zero-gravity surface figure extraction.

© 2009 Optical Society of America

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References

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  1. J. W. Zinn and G. W. Jones, “Kepler primary mirror assembly: FEA surface figure analyses and comparison to metrology,” Proc. SPIE 6671, 667105 (2007).
    [CrossRef]
  2. P. Glenn and G. Ruthven, “Off-loading of a cylindrical optic to simulate zero-gravity,” Proc. SPIE 518, 172-180(1984).
  3. J. C. Marr IV, M. Shao, and R. Goullioud, “SIM-Lite: progress report,” Proc. SPIE 7013, 70132M (2008).
    [CrossRef]
  4. R. A. Laskin, “Successful completion of SIM-PlanetQuest technology,” Proc. SPIE 6268, 626823 (2006).
    [CrossRef]
  5. R. Goullioud, C. A. Lindensmith, and I. Hahn, “Results from the TOM3 testbed: thermal deformation of optics at the picometer level,” in IEEE 2006 Aerospace Conference (IEEE, 2006).
    [CrossRef]
  6. D. Stewart, “A platform with six degrees of freedom,” Proc. Inst. Mech. Eng. 180, 371-378 (1965).
    [CrossRef]
  7. E. E. Bloemhof, J. C. Lam, V. A. Feria, and Z. Chang, “Precise determination of the zero-gravity surface figure of a mirror without gravity-sag modeling,” Appl. Opt. 46, 7670-7678(2007).
    [CrossRef] [PubMed]
  8. Z. Chen, Finite Element Methods and Their Applications (Springer-Verlag, 2005).
  9. D. W. Nicholson, Finite Element Analysis: Thermomechanics of Solids (CRC Press, 2003).
    [CrossRef]
  10. R. Richards, Jr., Principles of Solid Mechanics (CRC Press, 2001).
  11. 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]

2008 (1)

J. C. Marr IV, M. Shao, and R. Goullioud, “SIM-Lite: progress report,” Proc. SPIE 7013, 70132M (2008).
[CrossRef]

2007 (2)

J. W. Zinn and G. W. Jones, “Kepler primary mirror assembly: FEA surface figure analyses and comparison to metrology,” Proc. SPIE 6671, 667105 (2007).
[CrossRef]

E. E. Bloemhof, J. C. Lam, V. A. Feria, and Z. Chang, “Precise determination of the zero-gravity surface figure of a mirror without gravity-sag modeling,” Appl. Opt. 46, 7670-7678(2007).
[CrossRef] [PubMed]

2006 (2)

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]

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

1984 (1)

P. Glenn and G. Ruthven, “Off-loading of a cylindrical optic to simulate zero-gravity,” Proc. SPIE 518, 172-180(1984).

1965 (1)

D. Stewart, “A platform with six degrees of freedom,” Proc. Inst. Mech. Eng. 180, 371-378 (1965).
[CrossRef]

Bloemhof, E. E.

Chang, Z.

Chen, Z.

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

Feria, V. A.

E. E. Bloemhof, J. C. Lam, V. A. Feria, and Z. Chang, “Precise determination of the zero-gravity surface figure of a mirror without gravity-sag modeling,” Appl. Opt. 46, 7670-7678(2007).
[CrossRef] [PubMed]

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]

Glenn, P.

P. Glenn and G. Ruthven, “Off-loading of a cylindrical optic to simulate zero-gravity,” Proc. SPIE 518, 172-180(1984).

Goullioud, R.

J. C. Marr IV, M. Shao, and R. Goullioud, “SIM-Lite: progress report,” Proc. SPIE 7013, 70132M (2008).
[CrossRef]

R. Goullioud, C. A. Lindensmith, and I. Hahn, “Results from the TOM3 testbed: thermal deformation of optics at the picometer level,” in IEEE 2006 Aerospace Conference (IEEE, 2006).
[CrossRef]

Hahn, I.

R. Goullioud, C. A. Lindensmith, and I. Hahn, “Results from the TOM3 testbed: thermal deformation of optics at the picometer level,” in IEEE 2006 Aerospace Conference (IEEE, 2006).
[CrossRef]

Jones, G. W.

J. W. Zinn and G. W. Jones, “Kepler primary mirror assembly: FEA surface figure analyses and comparison to metrology,” Proc. SPIE 6671, 667105 (2007).
[CrossRef]

Lam, J. C.

E. E. Bloemhof, J. C. Lam, V. A. Feria, and Z. Chang, “Precise determination of the zero-gravity surface figure of a mirror without gravity-sag modeling,” Appl. Opt. 46, 7670-7678(2007).
[CrossRef] [PubMed]

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]

Laskin, R. A.

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

Lindensmith, C. A.

R. Goullioud, C. A. Lindensmith, and I. Hahn, “Results from the TOM3 testbed: thermal deformation of optics at the picometer level,” in IEEE 2006 Aerospace Conference (IEEE, 2006).
[CrossRef]

Marr IV, J. C.

J. C. Marr IV, M. Shao, and R. Goullioud, “SIM-Lite: progress report,” Proc. SPIE 7013, 70132M (2008).
[CrossRef]

Nicholson, D. W.

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

Richards, R.

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

Ruthven, G.

P. Glenn and G. Ruthven, “Off-loading of a cylindrical optic to simulate zero-gravity,” Proc. SPIE 518, 172-180(1984).

Shao, M.

J. C. Marr IV, M. Shao, and R. Goullioud, “SIM-Lite: progress report,” Proc. SPIE 7013, 70132M (2008).
[CrossRef]

Stewart, D.

D. Stewart, “A platform with six degrees of freedom,” Proc. Inst. Mech. Eng. 180, 371-378 (1965).
[CrossRef]

Van Buren, D.

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]

Zinn, J. W.

J. W. Zinn and G. W. Jones, “Kepler primary mirror assembly: FEA surface figure analyses and comparison to metrology,” Proc. SPIE 6671, 667105 (2007).
[CrossRef]

Appl. Opt. (1)

Proc. Inst. Mech. Eng. (1)

D. Stewart, “A platform with six degrees of freedom,” Proc. Inst. Mech. Eng. 180, 371-378 (1965).
[CrossRef]

Proc. SPIE (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]

J. W. Zinn and G. W. Jones, “Kepler primary mirror assembly: FEA surface figure analyses and comparison to metrology,” Proc. SPIE 6671, 667105 (2007).
[CrossRef]

P. Glenn and G. Ruthven, “Off-loading of a cylindrical optic to simulate zero-gravity,” Proc. SPIE 518, 172-180(1984).

J. C. Marr IV, M. Shao, and R. Goullioud, “SIM-Lite: progress report,” Proc. SPIE 7013, 70132M (2008).
[CrossRef]

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

Other (4)

R. Goullioud, C. A. Lindensmith, and I. Hahn, “Results from the TOM3 testbed: thermal deformation of optics at the picometer level,” in IEEE 2006 Aerospace Conference (IEEE, 2006).
[CrossRef]

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Views of front, optical surface (top), and back structure (middle) of PT-M1, a spherical prototype of the primary mirror for the SIM three-mirror compressor. The mirror face is built on a lightweighted hub, to which a flightlike hexapod mount is attached (bottom). The mirror is 343 mm in diameter and has an 2.2 m radius of curvature. The top two drawings show the nodes used for FEA modeling.

Fig. 2
Fig. 2

Interferometric measurements of the surface figure of PT-M1 in a two-configuration gravity-reversing scheme incorporating the principles of Ref.  [7]. The mount consists of three point contacts bonded to the hub at the back of the mirror (cartoons at top, but with threefold azimuthal symmetry, not depicted); mount forces thus reverse and are applied at nearly the same positions in the two orientations, far from the mirror surface. As a result, deformations in the surface figures due to gravity and mount forces (top two data panels) are nearly exactly complementary, resulting in an average map (lower data panel) that accurately captures the zero-gravity surface. The rms departure from a spherical surface is 5.7 nm , including surface errors of a fold flat in the optical train and errors in the transmission flat, with a formal error from the gravity-reversal scheme itself of only 0.0003 nm rms . [Metrology data courtesy of S. Spanjian, T. Roff, L. Dettmann, and J. Daniel, Tinsley Laboratories.]

Fig. 3
Fig. 3

Interferometric measurements of the surface figure of PT-M1 in a six-configuration gravity-reversing scheme incorporating the principles of Ref. [7]. The mount consists of two horizontal pegs under lightweighting ribs, near the vertical center; mount forces thus reverse and are applied at nearly the same positions in the two orientations (differing only by the thickness of the center ribs). As a result, deformations in the surface figures due to gravity and mount forces (top six data panels, all fixed in the frame of the optic) are nearly exactly complementary, resulting in an average map (lower data panel) that accurately captures the zero-gravity surface. The rms departure from a spherical surface is 3.8 nm , including errors in the transmission flat, with a formal error from this extremely simple and convenient gravity-reversal scheme of only 0.87 nm rms . [Metrology data courtesy S. Spanjian, T. Roff, L. Dettmann, and J. Daniel, Tinsley Laboratories.]

Fig. 4
Fig. 4

Surface metrology of PT-M1 measured at the Jet Propulsion Laboratory with the mirror mounted in the flightlike hexapod mount of Fig. 1. (Top) Surface maps in three orientations, clocked at 0°, 120°, and 240°, rotated back to show the mirror in a fixed position; gravity vectors are sequentially rotated in these plots. Substantial astigmatism is caused by gravity sag; surfaces are 14.5, 13.9, and 12.7 nm rms. (Bottom) Average of these three surface maps, 5.6 nm rms . Astigmatism is largely removed, confirming theoretical expectations that this average should give a zero-gravity surface map. (As a practical issue, static patterns that are due to the transmission sphere and fold flat are also somewhat suppressed.) Color bars are in nanometers. The main zero-gravity surface features are common with Figs. 2, 3.

Fig. 5
Fig. 5

Superposition of two on-edge hexapod mount configurations clocked by 180 ° (top) and three clocked by 120 ° (bottom) showing cancellation of gravity and mount forces: the average surface figures will thus accurately give the zero-gravity figure. Hexapod struts are axially stiff but transversely compliant and in PT-M1 are attached at right angles to the plane of the figure, implying that the forces they exert are primarily tangential to the mirror hub. By symmetry, the mount forces must be as shown. Although not immediately apparent, both schemes work for arbitrary initial orientation or clocking angle, as will be seen from more detailed calculations in Section 5.

Fig. 6
Fig. 6

Computation of static forces at each of the three mount points as a function of clocking angle of the optic for PT-M1 in a hexapod mount. (Top) Geometric layout: f 1 , f 2 , and f 3 are the magnitudes of the three mount forces that must act tangentially (see discussion in text). The clocking angle (angle that the first mount point makes with the vertical y axis) is θ 1 . (Middle and bottom) Mount force solutions obtained by balancing horizontal force and net torque and requiring the net vertical force to balance the weight of the mirror, Mg, and requiring mount forces to act tangentially (see discussion in text). These results can be combined to show that mount forces, superimposed in the frame of the optic, cancel at each attachment point when averaged over clocking configurations.

Equations (11)

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A i j δ x j = f i = f i g + f i m ,
A i j δ x j = f i = f i g + f i m .
( R A R 1 ) ( R δ x ) = R ( f g + f m ) .
A ( R δ x ) = f g f m .
1 2 [ ( x + δ x ) + ( R x + R δ x ) ] = x ,
[ 1 1 1 0 0 0 0 0 0 1 1 1 cos θ 1 cos θ 2 cos θ 3 sin θ 1 sin θ 2 sin θ 3 sin θ 1 0 0 cos θ 1 0 0 0 sin θ 2 0 0 cos θ 2 0 0 0 sin θ 3 0 0 cos θ 3 ] [ f 1 x f 2 x f 3 x f 1 y f 2 y f 3 y ] = [ 0 1 0 0 0 0 ] .
f 1 ( θ 1 ) + R ( 180 ) f 1 ( θ 1 + 180 ° ) = 0 ;
f 1 ( θ 1 ) + R ( 120 ) f 2 ( θ 2 ) + R ( 240 ) f 3 ( θ 3 ) = 0 .
R ( θ ) = [ cos θ - sin θ 0 sin θ cos θ 0 0 0 1 ] .
f ^ 1 = [ sin ψ , cos ψ sin ϕ , cos ψ cos ϕ ] , f ^ 2 = [ sin ψ , cos ψ sin ϕ , cos ψ cos ϕ ] , r ^ = [ 0 , 1 , 0 ] .
[ [ f ^ 1 ] x [ f ^ 2 ] x [ R 1 f ^ 1 ] x [ R 1 f ^ 2 ] x [ R 2 f ^ 1 ] x [ R 2 f ^ 2 ] x [ f ^ 1 ] y [ f ^ 2 ] y [ R 1 f ^ 1 ] y [ R 1 f ^ 2 ] y [ R 2 f ^ 1 ] y [ R 2 f ^ 2 ] y [ f ^ 1 ] z [ f 2 ^ ] z [ R 1 f ^ 1 ] z [ R 1 f ^ 2 ] z [ R 2 f ^ 1 ] z [ R 2 f ^ 2 ] z [ r ^ × f ^ 1 ] x [ r ^ × f ^ 2 ] x [ ( R 1 r ^ ) × ( R 1 f ^ 1 ) ] x [ ( R 1 r ^ ) × ( R 1 f ^ 2 ) ] x [ ( R 2 r ^ ) × ( R 2 f ^ 1 ) ] x [ ( R 2 r ^ ) × ( R 2 f ^ 2 ) ] x [ r ^ × f ^ 1 ] y [ r ^ × f ^ 2 ] y [ ( R 1 r ^ ) × ( R 1 f ^ 1 ) ] y [ ( R 1 r ^ ) × ( R 1 f ^ 2 ) ] y [ ( R 2 r ^ ) × ( R 2 f ^ 1 ) ] y [ ( R 2 r ^ ) × ( R 2 f ^ 2 ) ] y [ r ^ × f ^ 1 ] z [ r ^ × f ^ 2 ] z [ ( R 1 r ^ ) × ( R 1 f ^ 1 ) ] z [ ( R 1 r ^ ) × ( R 1 f ^ 2 ) ] z [ ( R 2 r ^ ) × ( R 2 f ^ 1 ) ] z [ ( R 2 r ^ ) × ( R 2 f ^ 2 ) ] z ] [ f 1 f 2 f 3 f 4 f 5 f 6 ] = [ 0 1 0 0 0 0 ] .

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