Abstract

Extremely large optical telescopes are being designed with primary mirrors composed of hundreds of segments. The “out-of-plane” piston, tip, and tilt degrees of freedom of each segment are actively controlled using feedback from relative height measurements between neighboring segments. The “in-plane” segment translations and clocking (rotation) are not actively controlled; however, in-plane motions affect the active control problem in several important ways, and thus need to be considered. We extend earlier analyses by constructing the “full” interaction matrix that relates the height, gap, and shear motion at sensor locations to all six degrees of freedom of segment motion, and use this to consider three effects. First, in-plane segment clocking results in height discontinuities between neighboring segments that can lead to a global control system response. Second, knowledge of the in-plane motion is required both to compensate for this effect and to compensate for sensor installation errors, and thus, we next consider the estimation of in-plane motion and the associated noise propagation characteristics. In-plane motion can be accurately estimated using measurements of the gap between segments, but with one unobservable mode in which every segment clocks by an equal amount. Finally, we examine whether in-plane measurements (gap and/or shear) can be used to estimate out-of-plane segment motion; these measurements can improve the noise multiplier for the “focus-mode” of the segmented-mirror array, which involves pure dihedral angle changes between segments and is not observable with only height measurements.

© 2012 Optical Society of America

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References

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  1. J. Nelson, G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
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    [CrossRef]
  5. T. Sebring, “The Cornell Caltech Atacama Telescope: progress and plans,” Proc. SPIE 7733, 77331X (2010).
    [CrossRef]
  6. G. Chanan, D. G. MacMartin, J. Nelson, T. Mast, “Control and alignment of segmented-mirror telescopes: Matrices, modes, and error propagation,” Appl. Opt. 43, 1223–1232(2004).
    [CrossRef]
  7. H. Bonnet, ESO Internal Report (2008).
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    [CrossRef]
  9. D. G. MacMartin, G. Chanan, “Measurement accuracy in control of segmented-mirror telescopes,” Appl. Opt. 43, 608–615 (2004).
    [CrossRef]

2010 (1)

T. Sebring, “The Cornell Caltech Atacama Telescope: progress and plans,” Proc. SPIE 7733, 77331X (2010).
[CrossRef]

2008 (3)

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

J. Nelson, G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
[CrossRef]

R. Gilmozzi, J. Spyromilio, “The 42m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

2004 (2)

2003 (1)

1990 (1)

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Andreae, S.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Arthur, A. A.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Biocca, A.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Bonnet, H.

H. Bonnet, ESO Internal Report (2008).

Chanan, G.

Cohen, R. W.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Franck, J.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Fuertes, J. M.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Gabor, G.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Gilmozzi, R.

R. Gilmozzi, J. Spyromilio, “The 42m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Jared, R. C.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Llacer, J.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

MacDonald, D. R.

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

MacMartin, D. G.

Mast, T.

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

G. Chanan, D. G. MacMartin, J. Nelson, T. Mast, “Control and alignment of segmented-mirror telescopes: Matrices, modes, and error propagation,” Appl. Opt. 43, 1223–1232(2004).
[CrossRef]

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Meng, J.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Merrick, T.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Minor, R.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Nelson, J.

J. Nelson, G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
[CrossRef]

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

G. Chanan, D. G. MacMartin, J. Nelson, T. Mast, “Control and alignment of segmented-mirror telescopes: Matrices, modes, and error propagation,” Appl. Opt. 43, 1223–1232(2004).
[CrossRef]

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Orayani, M.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Padin, S.

Roberts, L. C.

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

Salz, P.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Sanders, G. H.

J. Nelson, G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
[CrossRef]

Schaefer, B.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Sebring, T.

T. Sebring, “The Cornell Caltech Atacama Telescope: progress and plans,” Proc. SPIE 7733, 77331X (2010).
[CrossRef]

Seo, B.-J.

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

Shelton, C.

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

Sirota, M. J.

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

Spyromilio, J.

R. Gilmozzi, J. Spyromilio, “The 42m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Troy, M.

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

Witebsky, C.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

Appl. Opt. (3)

Proc. SPIE (5)

C. Shelton, T. Mast, G. Chanan, J. Nelson, L. C. Roberts, M. Troy, M. J. Sirota, B.-J. Seo, D. R. MacDonald, “Advances in edge sensors for the Thirty Meter Telescope primary mirror,” Proc. SPIE 7012, 701210 (2008).
[CrossRef]

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996–1008 (1990).
[CrossRef]

T. Sebring, “The Cornell Caltech Atacama Telescope: progress and plans,” Proc. SPIE 7733, 77331X (2010).
[CrossRef]

J. Nelson, G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
[CrossRef]

R. Gilmozzi, J. Spyromilio, “The 42m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Other (1)

H. Bonnet, ESO Internal Report (2008).

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Figures (10)

Fig. 1.
Fig. 1.

Hexagonal segmentation geometry used in examples. Each segment has three actuators, and two sensors on each intersegment edge; for the 492-segment geometry shown (that of TMT) this leads to a total of 1476 actuators and 2772 sensors.

Fig. 2.
Fig. 2.

Sensor geometry of an individual segment, and definition of gap and shear at sensor location. Sensors on a segment are numbered clockwise, and the segment radius a and distance b from edge center to sensor are indicated.

Fig. 3.
Fig. 3.

Constructing the single-segment 12 × 3 matrices g and s that relate in-plane x and y translation and clocking to the gap (top row) and shear motion (bottom row) at the sensor locations; these correspond to Eq. (6).

Fig. 4.
Fig. 4.

Each subplot shows, for a particular deflection pattern of the mirror, the resulting response at each of the n s sensor locations: the height (mixed with dihedral; first set of n s = 2772 points), gap (second set of 2772 points), and shear motion (third set). In each case, the response is normalized by the maximum response. The pattern of motion within a given sensor type depends on the sensor numbering convention; it is the relative response between height, gap, and shear for each deflection pattern that is of interest. The three deflection patterns shown are focus-mode (left), breathing-mode (center), corresponding to pure translation of every segment radially along the optical prescription, and torsion-mode (right), corresponding to uniform clocking of every segment. Focus-mode also produces a nearly-uniform dihedral signature that is small compared to the gap signature.

Fig. 5.
Fig. 5.

Torsion mode pattern (illustrated for Keck segmentation geometry); equal clocking of every segment produces only shear at sensor locations, and no change in the gaps between segments.

Fig. 6.
Fig. 6.

Response pattern that results from the control system canceling the sensor height readings that result from in-plane clocking of a single segment (left; the rms surface error is 40 nm / mrad ) and due to torsion mode (right; the rms surface error is 6.8 μm / mrad ).

Fig. 7.
Fig. 7.

In-plane noise multiplier scaling as a function of number of segments, for estimating gap and shear motion at sensor locations from either gap information alone, or from both gap and shear. The estimation errors using only shear information are significantly larger, as shown in Table 1.

Fig. 8.
Fig. 8.

Modal noise multiplier for estimating in-plane motion from measurements of gap alone, shear alone, or both gap and shear. The unobservable torsion mode using only gap sensors is not shown (the noise multiplier is infinite for this mode). Units are nm / nm with clocking scaled by length a .

Fig. 9.
Fig. 9.

Least-observable pattern of in-plane motion with shear sensors (for a 54-segment array); there is also some very small segment clocking not shown.

Fig. 10.
Fig. 10.

Noise multiplier for estimating focus-mode, per nm of height/dihedral noise, for 492-segment example with L eff = 24 mm . Both the gap and shear sensor noise standard deviations are assumed to be higher than the height/dihedral noise by the factor on the abscissa.

Tables (1)

Tables Icon

Table 1. Noise Multipliers for the 492-Segment Example for Estimating In-Plane Motion from Measurements of Gap Only, Shear Only, or Gap and Shear Combined a

Equations (16)

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

[ y h y g y s ] = [ A h z A h w A g z A g w A s z A s w ] [ z w ] + [ η h η g η s ] .
g = [ 3 / 2 1 / 2 b / a 3 / 2 1 / 2 b / a 0 1 b / a 0 1 b / a 3 / 2 1 / 2 b / a 3 / 2 1 / 2 b / a 3 / 2 1 / 2 b / a 3 / 2 1 / 2 b / a 0 1 b / a 0 1 b / a 3 / 2 1 / 2 b / a 3 / 2 1 / 2 b / a ] and s = [ 1 / 2 3 / 2 3 / 2 1 / 2 3 / 2 3 / 2 1 0 3 / 2 1 0 3 / 2 1 / 2 3 / 2 3 / 2 1 / 2 3 / 2 3 / 2 1 / 2 3 / 2 3 / 2 1 / 2 3 / 2 3 / 2 1 0 3 / 2 1 0 3 / 2 1 / 2 3 / 2 3 / 2 1 / 2 3 / 2 3 / 2 ] .
z ^ = A h z # A h w w ,
w ^ = A g w # y g ,
y ^ s = A s w w ^ = A s w A g w # y g ,
y ^ g = A g w w ^ = A g w A g w # y g .
x ^ = lim ϵ 0 ( A T R 1 A + ϵ I ) 1 A T R 1 y ,
A = U Σ V T ,
x ^ = V Σ 1 U T y .
x ~ = V Σ 1 U T η
1 n a E { x ~ T x ~ } = 1 n a tr [ E { x ~ x ~ T } ] = 1 n a tr [ V Σ 2 V T ] = 1 n a tr [ V T V Σ 2 ] = 1 n a i σ i 2
E { s ~ T s ~ } = tr [ H V Σ 2 V T H T ] .
E { ( f f ^ ) 2 } = i ( v f T v i σ i ) 2 .
E { ( f f ^ ) 2 } = ( v f T V Σ 1 U T q ) 2 ,
E { ( f f ^ ) 2 } = 1 σ j 2 ( u j T q ) 2 .
E { ( f f ^ ) 2 } = Q / σ j 2 .

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