Control and alignment of segmented- : matrices, modes, and error propagation

Gary Chanan, Douglas G. MacMartin, Jerry Nelson, and Terry Mast

Starting from the successful Keck design, we construct and analyze the control matrix for the active control system of the 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 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. We define the sensor gain G as the absolute value 1 ⌬ ͑͞⌬ Ϫ⌬ ͒ ⌬s ϭ ⌬zͩ h ϩ a cos 30° ϩ gͪ͞h, of the ratio s C1 C2 , or in terms of fixed 4,1 3 sensor parameters, 1 ⌬s ϭ ⌬zͫ h ϩ ͑a Ϫ f ͒cos 30° Ϫ g sin 30°ͬ͞h, 5,1 1 d 3 G ϭ . (5) 2 C 1 ⌬s ϭ ⌬zͩ h ϩ f cos 30° ϩ g sin 30°ͪ͞h. (2) 6,1 3 The control matrix element Aij is then defined to be the increase in G ͑⌬C Ϫ⌬C ͒ for the ith capacitor ⌬ 1 2 For sensor positions 7 through 12, sj,1 can be ob- associated with an incremental motion of 1 ␮mofthe ⌬ tained from sjϪ6,1, but with sin 30° and cos 30° jth actuator. For actuator 1 and the sensors in Fig. Ϫ Ϫ replaced by sin 30° and cos 30°, respectively. A 2 we have useful check on the signs and normalizations of the above relations is provided by the closure relations, A ϭ Ϯ⌬s ͞⌬z, (6) which follow from the symmetries of the system: j,1 j,1 ϭ ⌬s ϩ ⌬s ϩ ⌬s ϭ ⌬z, where the sign is positive for j 2, 4, 6, 7, 9, 11 and 1,1 5,1 9,1 negative for the remaining values of j. The entire ⌬s2,1 ϩ ⌬s6,1 ϩ ⌬s10,1 ϭ ⌬z, control matrix can readily be filled out in this way. ͑ ⌬s ϩ ⌬s ϩ ⌬s ϭ ⌬z, Each actuator affects 12 sensors or fewer for periph- 3,1 7,1 11,1 eral segments because these do not have six nearest ⌬s4,1 ϩ ⌬s8,1 ϩ ⌬s12,1 ϭ ⌬z. (3) neighbors͒, and each sensor is affected by six actua- tors ͑in all cases͒. Thus each row of the control ma- We now need to relate the edge height increments trix has up to 12 nonzero elements, and each column to the readings on the differential capacitors that has exactly six nonzero elements. constitute the edge sensors. The Keck edge sen- sors10 consist of a paddle on one segment that fits into C. Construction of the Active Control Matrix a U-groove on its neighboring segment across the for Vertical Sensors intersegment gap to form two parallel-plate capaci- The sensors proposed for CELT are vertical, that is, tors, one above the other, each having a capacitance the capacitor plates are perpendicular to the segment ϭ⑀ ͞ C 0A d. So that all segments are similar, the surface.11 In particular, one half of the sensor con- U-groove is attached to the segment of interest at the sists of a single vertical sense plate bonded or plated odd-sensor locations in Fig. 2 and to the paddle at the directly onto the side of one segment, and the other even-sensor locations. As one segment moves up by half consists of two vertical drive plates on the side of ⌬ an amount s relative to its neighbor, the differential its neighbor segment directly across the interseg- capacitance will change according to ment gap. The CELT sensor geometry is shown in Fig. 3; here w is the width of the capacitor drive ⌬ Ϫ ⌬ ⌬ C1 C2 2 s ϭ ͞ ϭ Ϯ , (4) plates, l is the height of the single plate, and A wl 2 C d is the area of each drive plate. We assume that the

20 February 2004 ͞ Vol. 43, No. 6 ͞ APPLIED OPTICS 1225 vertical gap between plates is small. Figure 2 still and the primes distinguish these results for vertical applies but with g ϭ 0. sensors from the above unprimed results for horizon- In this case, unlike the horizontal sensors, there tal sensors. Note that ␩ quantifies the relative sen- are two contributions to the differential capacitance: sitivity to rotation ͑dihedral angle͒ versus shear. Ј ϭ The matrix elements A1, j with j 7 through 12 can Ј Ј Ϫ ⌬C1 Ϫ ⌬C2 ⌬A1 Ϫ ⌬A2 ⌬d1 Ϫ ⌬d2 be obtained from A1,jϪ6 with cos 30 replaced by cos ϭ Ϫ . (7) ␩ Ϫ␩ ͑ C A d 30° and replaced by but sin 30° is not replaced by Ϫsin 30°͒. Note that the signs and trigonometric The first ͑area͒ term on the right-hand side of Eq. ͑7͒ factors associated with the second terms on the right- is due to the shear of the segment of interest relative hand side of Eqs. ͑9͒ are determined by the orienta- to its nearest neighbor across the gap; the second tion of the segment edges; in particular, the signs of ͑rotation͒ term is due to the rotation of the segment these latter terms do not alternate as one goes around relative to its neighbor, i.e., to the change in the the segment. As before, from this point it is dihedral angle. Again, to make all segments simi- straightforward to fill out the entire control matrix. lar, the construction of the sensors should alternate The nominal CELT segments have a circumscribed as one proceeds around the segment; for the sake of radius of a ϭ 500 mm and a thickness of 50 mm; for definiteness, and so that the signs are similar to the such segments typical sensor geometries would be l ϭ above, we take the even sensors in Fig. 2 to have the 20 mm, h ϭ 400 mm, and ␦ϭ2 mm, for a nominal double plates and the odd sensors to have the single value of ␩ϭ0.06. In this paper we mainly consider plates. a slightly more conservative sensor geometry defined It is easiest to consider first what happens to sensor by ␩ϭ0.05. 3 in Fig. 2 when actuator 1 is pistoned out of the page D. Error Propagation by the Control Matrix by an amount ⌬z. ͑This case is relatively simple because the associated segment edge is then parallel With the matrix that governs the active control sys- to the rotation axis.͒ Under these circumstances, tem now specified, we can proceed to discuss the in- Eq. ͑7͒ reduces to version of this system ͑how to obtain the desired actuator changes from the observed sensor readings͒ ⌬C Ϫ ⌬C 4r⌬z l⌬z and its associated error propagation. The relation 1 2 ϭ Ϫ Ϫ , (8) between sensors and actuators is given by C lh 2hd ⌬ ϭ ⌬ ͑ ͒ ͑ ͒ si ͚ Aij zj, (11) where all symbols are as defined in Eqs. 1 and 4 . j The first term on the right-hand side of Eq. ͑8͒ comes from the change in the overlap area. The second where i runs over all sensors and j over all actuators, comes from the change in the effective separation of and where the symbol ⌬ refers to the difference be- the plates. tween the actual sensor or actuator readings and Putting in the appropriate signs, trigonometric fac- their desired ͑absolute͒ readings. The desired sen- tors, and normalizations for all terms, and proceeding sor readings are defined when the alignment of the as before, we obtain the typical matrix elements: telescope is correct as determined by external optical means; actuator lengths are changed further only to 1 maintain these desired sensor readings in the face of A Ј ϭ Ϫ ͩ h ϩ f cos 30°ͪ͞h Ϫ ␩ sin 30°, deformations due to gravity and temperature 1,1 3 changes. The actuator changes that will maintain 1 the desired sensor readings are calculated with the A Ј ϭ ͫ h ϩ ͑a Ϫ f ͒cos 30°ͬ͞h Ϫ ␩ sin 30°, aid of the pseudo-inverse matrix whose elements are 1,2 3 Bji: 1 A Ј ϭ Ϫ ͩ h ϩ a cos 30°ͪ͞h Ϫ ␩, ⌬ ϭ ⌬ 1,3 3 zj ͚ Bji si. (12) i 1 A Ј ϭ ͩ h ϩ a cos 30°ͪ͞h Ϫ ␩, 1. Singular-Value Decomposition 1,4 3 A powerful technique used to calculate the pseudoin- 1 verse matrix B is singular-value decomposition A Ј ϭ Ϫ ͫ h ϩ ͑a Ϫ f ͒cos 30°ͬ͞h Ϫ ␩ sin 30°, 1,5 ͑ ͒ 17,18 3 SVD of the original matrix. We briefly review this method here. In SVD the m ϫ n matrix A 1 ͑where m Ն n͒ can be written as the product of three A Ј ϭ ͩ h ϩ f cos 30°ͪ͞h Ϫ ␩ sin 30°, (9) 1,6 3 matrices:

T where A ϭ UWV , (13) ϫ 2 where U is an m n column orthogonal matrix, W is l ϫ ␩ ϭ (10) an n n diagonal matrix whose diagonal elements wi 8h␦ are positive or zero and are referred to as the singular

1226 APPLIED OPTICS ͞ Vol. 43, No. 6 ͞ 20 February 2004 values of the matrix A, V is an n ϫ n orthogonal matrix, and the symbol T denotes transpose. The matrix B is then obtained as

B ϭ VWϪ1UT, (14) ͞ Ϫ1 where the jth diagonal element 1 wj of W is re- ϭ placed by 0 in the event that wj 0. The matrix V defines an essentially unique orthonormal basis set of modes of the system, such that any arbitrary config- uration of the system can be expressed as a unique ͑ ͒ linear combination of these modes. In particular, Vij Fig. 4. Two lowest-spatial-frequency highest error multiplier gives the value of the ith actuator in the jth mode. modes for CELT, assuming vertical sensors with ␩ϭ0.05: ͑a͒ For the control matrices considered here, three of mode 1, ͑b͒ mode 2. the singular values are equal to zero; the correspond- ing singular modes are the three actuator vectors corresponding to rigid body motion ͑global piston, tip, high-spatial-frequency modes have large edge discon- ͒ ͑ tinuities that are easily detected by the edge sensors. and tilt of the primary mirror as a whole because ␣ such motion has no effect on the sensor readings͒.If The error multiplier j associated with the jth there are n segments, there are 3n actuators and mode can be shown to be 3n Ϫ 3 modes of interest in the basis set. The nor- 2 malization of the Vij is defined by V ␣ 2 ϭ ͩ ijͪ . (19) j ͚ 2 i wj ͚ Vij ϭ 1 (15) i ␣ The individual error multipliers j and the overall for all j. error multiplier ␣ are independent of the hexagon 2. Error Multipliers from Singular-Value side length a. Decomposition This analysis is also applicable to the limiting case of vertical sensors with ␩ϭ0, except that here there If we were to put random uncorrelated noise equally is an additional singular mode resulting from the into all sensors, then the actuators would respond fact that these sensors cannot sense a change in the proportionally as determined by the A matrix: intersegment dihedral angle. Thus this mode corre- ␦ ϭ ␣␦ sponds to a constant dihedral angle, with no segment- a s, (16) to-segment shear. Because the resulting surface where ␦s and ␦a are the rms values of the sensors and resembles the error surface corresponding to an over- ͑ ͒ actuators, and we refer to the dimensionless param- all focus error except that the former is faceted ,itis eter ␣ as the ͑overall͒ noise multiplier. Alterna- referred to as focus mode. For sensors with nonzero ␩ tively, we could put random noise into the sensors , the corresponding mode is nonsingular, but is gen- ␦␣ erally the mode with the largest error multiplier. and determine the rms amplitude k for each of the above 3n Ϫ 3 modes. Figure 4 shows the two CELT modes with the low- ͑ ͒ By the orthogonality of the modes, we have est spatial frequencies largest error multipliers for the case of vertical sensors with ␩ϭ0.05; Figure 5 ␦ 2 ϭ ␦ 2 ϭ ␣ 2␦ 2 shows the two highest-spatial-frequency modes a ͚ ak ͚ k s . (17) ͑ ͒ k k smallest error multipliers . These results are typi- cal of all sensor geometries ͑except for the extra sin- It is convenient to order the modes according to the gular mode associated with ␩ϭ0͒. Inspection of the size of their error multipliers, from largest to small- modes shows that there is a close correspondence est. With this ordering, we define a residual error multiplier rk, which includes the error multiplier of the kth mode and all higher modes:

2 2 rk ϭ ͚ ␣j . (18) jՆk ͑ Note that r1 is then the same as the global error multiplier ␣.͒ When the modes are ordered in this way, they are also more or less ordered in spatial frequency from lowest to highest. The reason for this correspondence is not hard to understand: Low-spatial-frequency modes have small edge discon- tinuities that are difficult for the sensors to detect Fig. 5. Same as Fig. 4 but for the two highest-spatial-frequency and therefore for the active control system to control; ͑lowest error multiplier͒ modes: ͑a͒ mode 3236, ͑b͒ mode 3237.

20 February 2004 ͞ Vol. 43, No. 6 ͞ APPLIED OPTICS 1227 the electromechanical sensor readings with wave- front information. We considered this question in detail elsewhere19,20 and concluded that a supple- mentary wave-front sensor is not likely to be needed for stabilization purposes. Here we just summarize the argument. For diffraction-limited ͓adaptive op- tics ͑AO͔͒ observing, a wave-front sensor will already be present as a part of the AO system. If we assume that the Keck sensor noise levels of a few nanometers or less are representative of extremely large tele- scopes as well, the wave-front errors associated with the active control system—even allowing for the rel- atively large error multipliers calculated above—will be a small perturbation on the aberrations associated Fig. 6. Individual error multipliers for primary mirror active con- with atmospheric turbulence. It follows that a rea- trol systems of telescopes with 3, 7, and 11 rings and for CELT. sonable AO system will necessarily have the dynamic All cases correspond to vertical sensors with ␩ϭ0.05. The range and bandwidth to correct automatically any straight lines ͑plotted only for the three-ring telescope and for residual low-spatial-frequency errors left over from ͒ ͞ CELT show that the error multipliers for the ns 3 lowest modes the active control system; supplementary alignment ͌ ͞ can be approximated by the function 0.5 ns i, where ns is the wave-front sensors should not be necessary. For number of segments and i is the mode number. seeing-limited observations, we will not have an AO wave-front sensor to correct the misalignments auto- matically, but the fact remains that the atmospheric between the lowest-order modes and the Zernike aberrations will dominate by a large factor. If the polynomials. image quality is tolerable in the presence of atmo- Figure 6 shows the full range of error multipliers spheric aberrations, it is likely to be tolerable still in for 3-ring, 7-ring, and 11-ring telescopes and CELT, the presence of the additional relatively small aber- all with CELT-style sensors with ␩ϭ0.05. Figure 7 rations resulting from the active control system. shows a similar plot but for the residual error multi- The above considerations should not be taken as an plier ͓see Eq. ͑18͔͒. For CELT with ␩ϭ0.05, 84% of argument against closed-loop focus control for large the overall control system noise comes from focus segmented telescopes. A low-bandwidth focus cor- mode alone; 98% of the noise comes from the ten rection loop is motivated by considerations of control lowest modes of the system. The error multiplier of the secondary mirror, whereas the above argument depends sensitively on ␩ for focus mode ͑mode 1͒: was in the context of the segmented primary. Im- ␣ ϭ ␩ϭ ␣ ϭ For CELT we have 1 24.5 for 0.05, but 1 plementation of closed-loop focus control would be a 12.6 for ␩ϭ0.10. However, for modes 2 and higher, good addition to the Keck telescopes, which may still the error multipliers are only weakly dependent on ␩ be retrofitted for this control in the future. ͑ ␣ ϭ ␩ϭ ␣ ϭ ␩ϭ ͒ e.g., 2 8.8 for 0.05 and 2 8.2 for 0.10 . The error multiplier curves scale as the square root of 3. Surface Errors from Singular-Value the total number of segments. Decomposition for Diffraction-Limited Observing The sharp decline in error multiplier with increas- For diffraction-limited observing, a useful optical fig- ing mode number raises the possibility of our being ure of merit for the telescope is the rms wave-front able to improve the mirror control by supplementing error, which is equal to twice the rms surface error. Here we show that the rms surface error is nearly equal to the rms actuator error. As discussed above, the latter quantity is readily obtained as the overall error multiplier times the rms sensor error. Let zij represent the displacement of the ith actua- tor ͑i ϭ 1, 2, 3͒ on the jth segment,

pj ϭ ͑ z1j ϩ z2j ϩ z3j͒͞3 (20) ␤ represent the piston error on that segment, and x ␤ and y represent the angles through which the seg- ment is rotated about the x and y axes. Let single brackets denote the appropriate average over the ͗ ͘ ϭ telescope; thus z1 is the average of all the i 1 ͗ ͘ actuators over the ns segments, but z is the average over all 3ns actuators. Double brackets denote the Fig. 7. Same as Fig. 6 but for residual error multipliers, defined average over both the telescope and the ensemble. to include the multipliers for the indicated mode and for all higher- We take the ensemble to be defined by a Gaussian order modes. distribution of sensor ͑not actuator͒ errors, or equiv-

1228 APPLIED OPTICS ͞ Vol. 43, No. 6 ͞ 20 February 2004 alently, by the collection of SVD modes weighted by Table 1. Rms Raya Tilt ͑One Dimension͒ per Nanometer of Sensor b the error multipliers calculated above. Because ro- Noise for Various Telescopes tating the primary mirror by 120° and 240° rotates rms Tilt ͑mas͞nm͒ the i ϭ 1 actuator to the i ϭ 2 and i ϭ 3 positions, various double-bracketed actuator averages cannot Sensor ␩ 3-ring 7-ring 11-ring CELT depend on i. Thus, for example, we have Vertical 0 1.60 1.67 1.74 1.87 ͗͗ ͘͘ ϭ ͗͗ ͘͘ ϭ ͗͗ ͘͘ Vertical 0.05 3.08 2.91 2.90 2.98 z1 p z3 p z3 p , (21) Vertical 0.10 1.99 1.96 1.99 2.09 from which it follows that Horizontal — 2.29 2.22 2.24 2.33 aThe rms segment tilt is half as large. b ͗͗z1 z2͘͘ ϭ ͗͗z2 z3͘͘ ϭ ͗͗z3 z1͘͘. (22) This assumes a segment side length of 0.500 m. It follows from Eqs. ͑21͒ and ͑22͒ that the mean squared values of the various quantities will be re- latter ͑as indicated by the prime͒ that for a given i we lated by include only those j values corresponding to an ac- tuator on the same segment. This calculation shows 9 2 2 2 2 that the exact ratio z˜ ͞z is indeed close to unity ͗͗␤x ͘͘ ϭ ͗͗␤y ͘͘ ϭ ͑͗͗z ͘͘ Ϫ ͗͗p ͒͘͘. (23) rms rms 4h2 for most practical telescope geometries. However, the ratio is weakly dependent on a even for constant We let z˜ represent the continuous height variable a͞h; for example, for the Keck geometry, the ratio is over the segment surface, where z˜ ϭ z at the actuator i 1.028 for 0.5-m segments and 0.971 for 0.9-m seg- locations. Again, averaging as appropriate, we have ments. For 0.5-m segments and a telescope of seven ͑ ͒ ͗͗z˜ 2͘͘ ϭ ͗͗␤ 2͗͘͘x2͘ ϩ ͗͗␤ 2͗͘͘y2͘ ϩ ͗͗p2͘͘, (24) rings 168 segments or larger, the rms surface and x y actuator errors differ by less than 1% and there is where x and y are the coordinates in the nominal little point in maintaining a distinction between plane of the segment relative to an origin at the seg- these two quantities. ment center. For a regular hexagon of side a,we have 4. Tip–Tilt Errors from Singular-Value Decomposition for Seeing-Limited Observing 5 For seeing-limited observations, we are interested ͗x2͘ ϭ ͗y2͘ ϭ a2, (25) 24 not in the rms surface error ͑as discussed above͒, but rather in the rms surface ͑that is, segment͒ tip and and thus tilt, or the rms ray tip and tilt, which includes the ͗͗ 2͘͘ ϭ ␬͗͗ 2͘͘ ϩ ͑ Ϫ ␬͒͗͗ 2͘͘ factor of 2 for doubling on reflection. Here we z˜ z 1 p , (26) present a prescription for obtaining the rms ray tip where and tilt directly from the SVD formalism. A straightforward extension of the preceeding 15 a2 analysis gives the mean-squared one-dimensional tilt ␬ ϭ . (27) 16 h2 in the x direction as

As noted above we take the ensemble of actuator 1 cixcjxVikVjk t 2 ϭ Ј ͩ ͪ , (29) x 2 ͚ 2 errors to be defined by a Gaussian distribution of ns h ijk wk sensor errors; we can perform the ensemble averag- ing efficiently and exactly by summing over the mir- where the cix are dimensionless and defined below. ror modes with the above error multipliers as The units of tx are radians per meter when h is ex- weighting factors. Because the modes are strongly pressed in meters. The values of cix are given by weighted toward low spatial frequencies, we expect 2 2 2 2 ϭ ϭ ͱ ϭ Ϫͱ that ͗͗z ͘͘Ϸ͗͗p ͘͘ and thus ͗͗z˜ ͘͘Ϸ͗͗z ͘͘, i.e., that the c1x 0, c2x 3, c3x 3 (30) actuator errors provide an excellent estimate of the for the three actuators on segment 1 and similarly for overall surface error. In terms of V and W and the actuators on all other segments. The calculation of above formalism, we have the mean-squared y tilt is the same except for the 1 V V numerical values: ͗p2͘ ϭ Ј ͩ ik jkͪ , (28) ͚ 2 9ns ijk wk c1y ϭ Ϫ2, c2y ϭ 1, c3y ϭ 1. (31) where Vik refers to the value of the ith actuator in the A useful computational check is provided by the fact kth mode, wk is the singular value in the kth mode that the mean-squared tilts in x and y should be ͑unless it is equal to zero, in which case the corre- identical. ͒ sponding term is omitted , ns is the total number of In Table 1 we present the one-dimensional rms ray segments, and h is the height of the actuator triangle tip and tilt per nanometer of sensor noise for various as defined above. The sum is over all modes k and telescope and sensor geometries. Although the er- over all actuators i and j, with the restriction on the ror multipliers discussed above were independent of

20 February 2004 ͞ Vol. 43, No. 6 ͞ APPLIED OPTICS 1229 the segment size and scaled as the square root of the by means of diffraction measurements made at the total number of segments, the tip–tilt noise scales centers of the intersegment edges. Relevant math- inversely as the segment size, can vary by a factor of ematical details are presented in Subsections 3.B and ϳ2 with sensor geometry, and is virtually indepen- 3.C. dent of the number of segments. Thus the typical Now we want to determine whether, in a given rms one-dimensional image blur is 2–3 mas per nm of circumstance, we should use the single-step align- sensor noise for a segment with a ϭ 0.500 m or 1–1.5 ment approach ͑analogous to the control approach͒ or mas͞nm for a segment with a ϭ 1m. the two-step approach. Let the angular uncertainty ͑in one dimension and as measured in arcseconds on 3. Alignment the sky͒ in the Shack–Hartmann-type measurements be ␦␪, the height uncertainty ͑as measured in nano- A. General Considerations meters at the surface͒ in the diffraction-type edge The problem of segment alignment is closely related measurements be ␦e, and the ratio ͑from Table 1͒ of to, but different from, the above problem of segment rms ray tilt to sensor noise ͑in arcseconds per nano- active control. The task of the active control system meter͒ be ␮. The one-step versus two-step decision is to freeze the relative positions of the primary mir- hinges formally on whether ␦e is less than ␦␪͞␮,in ror segments in the face of perturbations due to grav- which case the single-step approach is favored, or ity and temperature once the mirror is in the desired greater than ␦␪͞␮, in which case the two-step ap- configuration. Conversely, it is the task of the mir- proach is preferred. In practice, however, the situ- ror alignment system to determine the desired con- ation is more complicated because of the additional figuration ͑desired sensor readings͒ in the first place. singular mode associated with the single-step ap- The active control loop runs continuously ͑at2Hzfor proach and because of the additional optical difficul- Keck, probably somewhat higher for CELT͒; the mir- ties associated with making two measurements per ror alignment is done only infrequently—perhaps edge in the single-step approach, as opposed to one once a month, depending on the drift rate of the sen- measurement per edge in the two-step approach. sor electronics. Both of these practical considerations favor the two- We have shown elsewhere9,21 that, by exploiting step method. At Keck, we have ␦e ϭ 10 nm, ␦␪ ϭ diffraction effects from misaligned segment edges, we 0.030 arc sec,21,22 and ␮ϭ0.001 arc sec͞nm ͑from can make optical edge measurements that are in Table 1, scaling to 0.9-m segments͒, but the two-step some sense analogous to the electromechanical edge method is used, even though ␦␪͞␮ϭ30 nm. This measurements provided by the capacitive edge sen- decision should be reevaluated for extremely large sors. One could thus approach the initial alignment telescopes, but we note that in general smaller seg- problem in a way that is almost identical mathemat- ments favor the two-step approach because the corre- ically to the control problem analyzed above. A sec- sponding values of ␮ are larger and the two-per-edge ond approach is also possible ͑this is the one actually measurements are more difficult to make. used at Keck͒, in which one proceeds in two steps: One could also imagine a hybrid alignment ap- First, the segments are aligned in tip and tilt and proach in which the low-spatial-frequency tip–tilt then, in a separate procedure, in the piston degree of modes are measured with a Shack–Hartmann sensor freedom. In this subsection we briefly describe and the high-spatial-frequency modes are measured these two approaches and their relative advantages. by two-per-edge phase-type measurements ͑which First, suppose that there are two optical measure- would also determine the segment piston errors͒, but ments per intersegment edge, made in essentially the a detailed treatment of this possibility is beyond our same physical locations as the electromechanical present scope. edge sensors. The optical measurements are made exactly at the intersegment edge ͑that is, g ϭ 0͒ so B. Construction of the Phasing Matrix that the geometry is essentially that of the vertical In this subsection we describe the control aspects of sensors considered above, except that in this case the mathematics associated with the phasing proce- there is no sensitivity to rotation, only to shear. dure, i.e., minimizing the intersegment edge steps ͑as Thus the analysis summarized in Section 2 applies, measured at the edge centers͒. A detailed discus- with ␩ϭ0. In this case there is an extra singular sion of this so-called narrowband phasing algorithm mode—the focus mode ͑in addition to the usual sin- can be found elsewhere.21 gular modes of global piston, tip, and tilt͒—that can- The control matrix for phasing is easily con- not be extracted from the optical measurements, but structed. The jth edge is determined by the values we neglect this complication for now. of the pistons of the adjacent segments: Alternatively, suppose that the segment tips and tilts are measured with a Shack–Hartmann-type q ϭ p ͑ ͒ Ϫ p ͑ ͒, (32) scheme,16 similar to that commonly employed in AO j iϩ j iϪ j systems. After these measurements are made and the segments are tipped and tilted appropriately, the where iϩ͑ j͒ and iϪ͑ j͒ are the indices of the segments segment pistons must still be adjusted or phased. on the positive and negative sides, respectively, of the The phasing procedure is defined mathematically as jth edge. This matrix is independent of the details of minimizing the steps between segments; this is done the sensor geometry. In the case of the phasing ma-

1230 APPLIED OPTICS ͞ Vol. 43, No. 6 ͞ 20 February 2004 phasing matrix, the error multiplier is almost inde- pendent of the number of segments, increasing only from 0.620 for the 36-segment Keck geometry to 0.725 for the 1080-segment CELT geometry. For the phasing modes ͑as was the case for control modes͒, the error multipliers fall off rapidly with in- creasing spatial frequency, so that we expect the spatial-frequency content of the residual piston er- rors after piston alignment to be significantly lower than that of uncorrelated piston errors. For one con- sequence of this, note from Fig. 8 that modes 1 and 2 Fig. 8. Two lowest-spatial-frequency ͑highest error multiplier͒ resemble global coma, for which the best-fit plane has phasing modes for CELT: ͑a͒ mode 1, ͑b͒ mode 2. Note the large a nonzero slope. Because these modes are the larg- tilt components. est contributors to the overall error multiplier, this means that the best-fit plane to the overall residual phase errors will have a larger slope than that for trix, there is only one singular mode—that corre- random phase errors. Thus a fraction of the resid- sponding to global piston. ual wave-front error associated with phasing is indis- C. Properties of the Phasing Matrix tinguishable from global tilt. For the Keck geometry, subtracting off the best-fit plane will re- The phasing modes with the lowest and highest spa- duce the rms piston error by approximately 20%; for tial frequencies are shown for the nominal CELT the CELT geometry, the corresponding reduction is geometry in Figs. 8 and 9, respectively. Plots of the ͑ ͒ approximately 12%. In quoting phasing errors at error multipliers individual and residual for all Keck, we have generally subtracted off this tip–tilt phasing modes are shown in Fig. 10 for the CELT component. For similar reasons, the deformable geometry. The scaling law for the phasing matrix is mirror in an AO system may help to mitigate phasing different from that of the control matrix described errors, depending on how the wave-front sensor deals above. In the case of the control matrix, the overall with the wave-front discontinuities. error multiplier scaled as the square root of the total Experience at Keck has shown that the segment tip number of segments or actuators; in the case of the and tilts need to be adjusted considerably more often than the segment pistons.22 The question then aris- es: What set of actuator commands will correct the segment tip and tilts while having the smallest del- eterious effects on the segment pistons ͑as the seg- ments will not always be phased immediately following tip–tilt adjustment͒? This question is an important one because the least-stable, lowest-order modes are in fact dominated by segment pistons. Conceptually, it is simplest to break up the solution into two steps. ͑1͒ Determine the actuator com- mands that will zero out the segment tip and tilts, subject to the constraint that the mean actuator ͑ Fig. 9. Same as Fig. 8, but for the two highest-spatial-frequency length change for each segment is zero. This can ͑lowest error multiplier͒ modes: ͑a͒ mode 1078, ͑b͒ mode 1079. always be done exactly.͒ Also determine the change in all the intersegment edge heights implicit in these actuator commands. The actuator commands are then sent. ͑2͒ Determine the ͑pure͒ piston com- mands that will remove, or at least minimize, the edge height changes predicted in the previous step. This can be done with the phasing matrix determined above. Sending these latter commands will produce the optimal configuration of the mirror for these given circumstances.

4. Summary and Conclusions We have given a general prescription for the con- struction of the mirror control matrices and also for the phasing matrices for essentially arbitrary highly segmented mirrors. We have described the modes Fig. 10. Error multipliers for phasing modes of CELT: upper and error propagation for both types of matrix. In curve, residual multipliers ͑defined as in Fig. 7͒; lower curve, in- the mirror control matrix, the overall error multiplier dividual mode multipliers. scales as the square root of the number of segments,

20 February 2004 ͞ Vol. 43, No. 6 ͞ APPLIED OPTICS 1231 which may present a problem for telescopes with sev- Future Giant Telescopes, J. R. P. Angel and R. Gilmozzi, eds., eral thousand segments. The only singular modes Proc. SPIE 4840, 116–128 ͑2002͒. associated with the control matrices are the three 8. R. Cohen, T. Mast, and J. Nelson, “Performance of the W. M. global rigid body degrees of freedom, plus one addi- Keck Telescope active mirror control system,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE tional singular mode in the limiting case of very short ͑ ͒ ͑␩ ϭ ͒ 2199, 105–116 1994 . vertical sensors 0 , which are not sensitive to 9. G. A. Chanan, M. Troy, F. G. Dekens, S. Michaels, J. Nelson, changes in the intersegment dihedral angle. In the T. Mast, and D. Kirkman, “Phasing the mirror segments of the phasing matrix, the overall error multiplier is virtu- Keck telescopes: the broadband phasing algorithm,” Appl. ally independent of the number of segments, which Opt. 37, 140 ͑1998͒. suggests that straightforward extensions of current 10. J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the phasing techniques should be adequate even for the Keck Observatory and Telescope,” Keck Observatory Rep. No. largest of the extremely large telescopes. Although 90 ͑W. M. Keck Observatory, Kamuela, Hawaii, 1985͒. further analysis is needed, it is possible, if not likely, 11. J. Nelson, ed., “The California Extremely Large Telescope,” ͑ that segment tip–tilt alignment and phasing of ex- CELT Rep. No. 34 University of California and California Institute of Technology, Santa Cruz, Calif., 2002͒. tremely large telescopes will best be carried out in 12. T. Mast and J. Nelson, “Segmented mirror control system two separate procedures, as is currently done for the hardware for CELT,” in Optical Design, Materials, Fabrica- Keck telescopes. We have argued elsewhere that tion, and Maintenance, P. Dierickx, ed., Proc. SPIE 4003, 226– supplemental wave-front control, in terms of an aux- 240 ͑2000͒. iliary wave-front sensor, should not be needed for the 13. J. Nelson, “CELT segment positioning actuators— primary mirror, although a low-order system— requirements,” CELT Tech. Note No. 5 ͑University of Califor- essentially an enhanced guider—is probably neces- nia, Santa Cruz, Santa Cruz, Calif. 2001͒. sary to keep the secondary mirror properly aligned. 14. T. G. Barnes, M. T. Adams, J. A. Booth, M. E. Cornell, N. I. Gaffney, J. R. Fowler, G. J. Hill, G. M. Hill, C. E. Nance, F. We thank Mitchell Troy and Lothar Noethe for Piche, L. W. Ramsey, R. L. Ricklets, W. J. Spiesman, and P. T. many useful discussions. Worthington, “Commissioning experience with the 9.2-m Hobby-Eberly Telescope,” in Telescope Structures, Enclosures, References Controls, Assembly͞Integration͞Validation, and Commission- 1. J. E. Nelson, “Progress on the California Extremely Large ing, T. A. Sebring and T. Andersen, eds., Proc. SPIE 4004, Telescope ͑CELT͒,” in Future Giant Telescopes, J. R. P. Angel 14–25 ͑2000͒. and R. Gilmozzi, eds., Proc. SPIE 4840, 47–59 ͑2002͒. 15. P. Alvarez, J. M. Rodriguez Espinoza, and F. R. Kabana, “GTC 2. T. Anderson, A. L. Ardeberg, J. Beckers, A. Goncharov, M. project: present and future,” in Telescope Structures, Enclo- Owner-Petersen, H. Riewaldt, R. Snel, and D. Walker, “The sures, Controls, Assembly͞Integration͞Validation, and Com- Euro-50 extremely large telescope,” in Future Giant Tele- missioning, Proc. SPIE 4004, 26–35 ͑2000͒. scopes, J. R. P. Angel and R. Gilmozzi, eds., Proc. SPIE 4840, 16. G. A. 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