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Control and Alignment of Segmented-Mirror Telescopes: Matrices, Modes, and Error Propagation

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Control and Alignment of Segmented-Mirror Telescopes: Matrices, Modes, and Error Propagation Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation Gary Chanan, Douglas G. MacMartin, Jerry Nelson, and Terry Mast Starting from the successful Keck telescope design, we construct and analyze the control matrix for the active control system of the primary mirror of a generalized segmented-mirror telescope, with up to 1000 segments and including an alternative sensor geometry to the one used at Keck. In particular we examine the noise propagation of the matrix and its consequences for both seeing-limited and diffraction- limited observations. The associated problem of optical alignment of such a primary mirror is also analyzed in terms of the distinct but related matrices that govern this latter problem. © 2004 Optical Society of America OCIS codes: 010.7350, 110.6770, 120.5050, 220.1140. 1. Introduction get for diffraction-limited or for seeing-limited obser- The next generation of ground-based optical and in- vations and to help resolve the question of the frared telescopes is currently in the planning stages, optimal number of segments for a given aperture. with a variety of projects in the 20–100-m class cur- The analysis presented here is a generalization of rently under consideration.1–7 The majority of these that used for the successful control and alignment8,9 telescopes involve a highly segmented primary mir- of the 36-segment primary mirrors of the two tele- ror, with several hundred up to even a few thousand scopes of the W. M. Keck Observatory.10 The Keck segments. The optimal number of segments used to experience has led to a number of practical consider- fill a particular aperture size is the result of a key ations that inform the analysis presented here. design trade-off: Larger segments are in general In this paper we take the proposed 30-m California more difficult to fabricate whereas smaller segments Extremely Large Telescope ͑CELT͒ as the prototype are more difficult to align and control, principally as of the next generation of highly segmented tele- a result of their larger numbers. scopes. ͑Its current design calls for a total of 1080 In this paper we discuss alignment and control of segments.͒ An introduction to the CELT segment highly segmented telescopes in terms of the control control matrices, position actuators, edge sensors, matrices that define these tasks. A modal analysis and error propagation can be found elsewhere.11–13 provides physical insight into the control and align- We also consider intermediate designs between Keck ment problems, and we pay particular attention to and CELT. the question of error propagation. When supple- This paper is organized as follows. In Section 2 mented with estimates of sensor noise and measure- we describe the construction of and error propagation ment uncertainty, this analysis can be used to by the matrices responsible for the active control of estimate the relevant terms in the optical error bud- segmented-mirror telescopes. Implications for both diffraction-limited and seeing-limited observations are discussed. In Section 3 we describe the con- G. Chanan ͑[email protected]͒ is with the Department struction of and error propagation by the matrices of Physics and Astronomy, University of California, Irvine, Irvine, used in the phasing of these mirrors. Our conclu- California 92697-4575. D. G. MacMartin is with Control and Dy- sions are summarized in Section 4. namical Systems, California Institute of Technology, Pasadena, California 91125. J. Nelson and T. Mast are with University of 2. Active Control California Observatories͞Lick Observatory, University of Califor- nia, Santa Cruz, Santa Cruz, California 95064. A. General Considerations Received 12 August 2003; revised manuscript received 17 No- vember 2003; accepted 20 November 2003. At Keck and at other segmented-mirror tele- 0003-6935͞04͞061223-10$15.00͞0 scopes,14,15 the primary mirror segments are actively © 2004 Optical Society of America positioned in their 3 out-of-plane degrees of freedom 20 February 2004 ͞ Vol. 43, No. 6 ͞ APPLIED OPTICS 1223 Fig. 1. Segment geometry of ͑a͒ an 11-ring telescope and ͑b͒ CELT. The 36-segment Keck geometry corresponds to the central three rings of ͑a͒. by three mechanical actuators. ͑Because the optical tolerances on the in-plane degrees of freedom are considerably less restrictive, these 3 degrees of free- dom are positioned passively.͒ The relative dis- placements of adjacent mirror segments are sensed Fig. 2. Geometry of the three actuator positions and 12 sensing by precision capacitive edge sensors, of which there points on a segment. are two per intersegment edge. The segments are actively controlled by means of a two-step process: B. Construction of the Active Control Matrix ͑ ͒ 1 Initially, the desired readings of the edge sensors for Horizontal Sensors are determined by external optical means16; ͑2͒ sub- The Keck sensors are horizontal, that is, the plates of sequently, the mirror is stabilized against perturba- the differential capacitors that make up the sensors tions due to gravity and thermal effects by the are parallel to the segment surface.10 ͑The bodies of moving the actuators so as to maintain the sensor the sensors are below the lower surface of the seg- readings at their desired values. At Keck the actua- ment.͒ The geometric relationships between the 12 tors are updated every 0.5 s; for the large segmented half-sensors and the segment that they monitor are telescopes of the future, the update rate is likely to be defined by Fig. 2; the placement of the three segment higher. actuators is also indicated. The Keck parameters In this paper we consider a variety of N-ring tele- are given by a ϭ 900 mm, f ϭ 173 mm, g ϭ 55 mm, scopes as well as the nominal CELT design. An and h ϭ 706 mm. The sensors sense the relative N-ring telescope consists of N hexagonal rings of seg- edge height, that is, the height of a segment relative ments, with the central segment missing, for a total to its neighbor, at the points indicated by the num- 2 ϩ of 3N 3N segments. The Keck telescopes are bered squares in Fig. 2. For the Keck geometry, and three-ring telescopes with 36 segments each. CELT for all geometries considered in this paper, the sens- has a circularized design; that is, it starts with 20 ing points for sensors 7 and 12 are both above the line rings, but is then circularized by exclusion of those connecting actuators 2 and 3; the simple sign conven- segments whose centers lie at a distance of greater tion below is affected if this is not the case. The than 30.1a from the optic axis, where a is the hexagon values of the ratios f͞a, g͞a, and h͞a for Keck are side length. In addition, the central 19 segments close to optimal in the sense of minimal noise multi- are removed in the CELT design. The total number plication ͑see below͒, but they also reflect various of CELT segments is then 1080. Figure 1 shows an practical considerations; that is, the precise values do 11-ring telescope as well as CELT. not have a fundamental significance. Nevertheless, In the following subsections we discuss the matri- for simplicity and directness of comparison, we as- ces that govern such active control systems. We dis- sume that these same ratios obtain for all cases con- cuss separately the matrices corresponding to the sidered in this paper, except where we explicitly take horizontal sensors used for the Keck telescopes and g ϭ 0. Other practical concerns ͑specifically, the the mechanically simpler vertical sensors that have interchangeability of segments͒ dictate that the ori- been proposed for CELT.11 As we describe, both sen- entation of the actuator triangle will vary from seg- sor designs rely on changes in differential capaci- ment to segment, but for simplicity we take all tance resulting from the relative motion of actuator triangles to have the same orientation. neighboring segments. Both designs are sensitive to This simplification will not change the basic proper- out-of-plane displacement as well as to changes in the ties of the associated control matrix; in particular it dihedral angle between the segments. These two will have no effect at all on the error multipliers or designs are representative of a variety of possible other similar quantities derived below. sensors that can be used to make measurements of Now, if actuator 1 is pistoned by an amount ⌬s, the the relative segment motion. segment will rotate about a line through actuators 2 1224 APPLIED OPTICS ͞ Vol. 43, No. 6 ͞ 20 February 2004 and 3, so that the reading of each sensor on the seg- ment ͑in height units͒ will change by r⌬z ⌬s ϭ , (1) h where r is the perpendicular distance from the sensor to the rotation axis, and where the sign of r is positive if the sensor and the actuator are on the same side of the rotation axis and negative if they are on opposite sides. Suppose we move actuator 1 by an amount ⌬z. From Eq. ͑1͒ the corresponding edge height incre- ment at sensor positions 1 through 6 will then be 1 ͑ ͒ ⌬s ϭ ⌬zͩ h ϩ f cos 30° Ϫ g sin 30°ͪ͞h, Fig. 3. Geometry of the vertical CELT sensors. 1,1 3 1 ⌬s ϭ ⌬zͫ h ϩ ͑a Ϫ f ͒cos 30° ϩ g sin 30°ͬ͞h, where the subscripts 1 and 2 denote top and bottom, 2,1 3 respectively; the minus sign holds for those sensors with the U-groove on the segment of interest, and the 1 ⌬s ϭ ⌬zͩ h ϩ a cos 30° Ϫ gͪ͞h, plus sign holds for those sensors with the paddle on 3,1 3 the segment of interest.
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