Design Considerations for a Highly Segmented Mirror

Design considerations for a highly segmented mirror

Stephen Padin

Design issues for a 30-m highly segmented mirror are explored, with emphasis on parametric models of simple, inexpensive segments. A mirror with many small segments offers cost savings through quantity production and permits high-order active and adaptive wave-front corrections. For a 30-m f͞1.5 paraboloidal mirror made of spherical, hexagonal glass segments, with simple warping harnesses and three- point supports, the maximum segment diameter is ϳ100 mm, and the minimum segment thickness is ϳ5 mm. Large-amplitude, low-order gravitational deformations in the mirror cell can be compensated if the segments are mounted on a plate ﬂoating on astatic supports. Because gravitational deformations in the plate are small, the segment actuators require a stroke of only a few tens of micrometers, and the segment positions can be measured by a wave-front sensor. © 2003 Optical Society of America OCIS codes: 350.1260, 010.1080, 220.4880, 120.4640.

1. Introduction design considerations are explored here for a 30-m Multiple mirrors seem to be the only viable approach f͞1.5 visible and infrared telescope with AO. for optical telescopes with apertures much larger For a large mirror with small segments, the cost than ϳ8 m. Many designs have been proposed, and and complexity of sensors and actuators are key is- these are broadly distinguished by the size of the sues. If gradients in the telescope structural defor- subapertures. The 20–20 telescope elements, which mations are repeatable at the level of ϳ1 ␮m across have seven 8.4-m-diameter segments,1 and the a segment diameter, changes in the segment posi- Phased Array Mirror Extendible Large Aperture tions can be measured by a wave-front sensor instead ͑PAMELA͒, with 8-cm-diameter segments,2–4 repre- of edge sensors. The wave-front sensor could also be sent the extremes of the design space. The Keck shared by a high-order AO system. If segment-to- telescopes, with 1.8-m-diameter segments,5 and the segment piston errors are not repeatable at the level Hobby-Eberly Telescope, with 1.15-m-diameter seg- of approximately half of the wave-front sensor wave- ments,6,7 are the only working multiple-mirror tele- length, an iterative piston-stepping algorithm,9 or scopes. measurements at multiple wavelengths, must be Small segments have potential for a lightweight, used to resolve 2␲ phase ambiguities. The deforma- low-cost telescope mirror and can support adaptive tion gradient constraint is severe, but feasible: Ax- optics ͑AO͒ if the segment diameter is approximately ial gravitational deformation of the primary in the half of the Fried parameter.8 PAMELA addressed 30-m California Extremely Large Telescope ͑CELT͒ some of these issues, using a scaled version of the design is ϳ10-mm p.-p., i.e., ϳ700-␮m p.-p. on 1-m Keck approach with inductive edge sensors and voice scales10; gravitational deformations in the Keck tele- coil actuators, but the large number of edge sensors scopes are repeatable at the level of 1 part per 1000, and high actuator power consumption make the de- so the nonrepeatable deformation in the CELT pri- sign unattractive for a very large mirror. Simpler mary should be Ͻ1-␮m p.-p. on 1-m scales. A thin small-segment schemes are possible, and some of the highly segmented mirror will have low mass, which will help to achieve small, repeatable structural deformations. Small gravitational deformations on the scale of a segment diameter also allow the mirror to be constructed with small gaps between segments. The author is with the California Institute of Technology, Mail This is important because it gives a low infrared Stop 105-24, 1200 East California Boulevard, Pasadena, California 91125. His e-mail address is [email protected]. background and low diffraction pattern sidelobes. Received 26 July 2002; revised manuscript received 16 Decem- The stroke of the segment actuators is set by grav- ber 2002. itational deformations in the telescope structure, but 0003-6935͞03͞163305-08$15.00͞0 these are mainly on large spatial scales, so it is not © 2003 Optical Society of America efﬁcient for the segments to have expensive, long-

1 June 2003 ͞ Vol. 42, No. 16 ͞ APPLIED OPTICS 3305 stroke actuators. An obvious development is a hier- and the more sparse array of mirror plate cells. Be- archy of actuators, with a few long-stroke actuators cause the mirror plate has a simple force-based sup- compensating most of the structural deformation and port, the number of support points can be large, and short-stroke actuators on each segment providing the plate can then be a fairly thin, lightweight struc- fast and small spatial scale wave-front control. This ture. scheme is particularly appropriate if the mirror is The floating mirror plate averages segment actua- made of several rafts of segments because the low- tor reaction forces over large spatial scales, and the order deformations can be compensated by use of force-based support decouples the mirror plate from actuators on each raft.3 Compensation of low-order the telescope structure. These effects minimize deformations could also be moved to a fairly small control–structure interaction for adaptive correc- active mirror at an image of the primary, leaving just tions, so the segments do not require reaction masses, high-order corrections in the primary. Several dif- and this helps to reduce the cost and mass of the ferent optical designs are possible,11–14 but the mirror. The mirror plate is tightly coupled to the schemes with good image quality have two or three segment actuators, and keeping the natural frequen- additional reflections that cause a significant sensi- cies of the plate modes above the frequencies of adap- tivity degradation for long-wavelength infrared ob- tive corrections is an important constraint on the servations.15 A third option, explored in more detail plate design. The fundamental frequency of the ϳ͑ ͞ ␲͒͑ ͞ ͒1͞2 ϰ 3͞ 2 in this paper, is to mount the small segments on a plate is f1 1 2 K1 m1 , where K1 t1 R1 is 18 ϭ␲ 2 ␳ plate floating in a force field, as in a monolithic mirror the stiffness of the plate ; and m1 R1 t1 , t1, R1, mount ͑Ref. 16 and references therein͒. The plate and ␳ are the mass, thickness, radius, and density of ϰ ͞ 2 supports provide axial and radial forces to compen- the plate, respectively. Thus f1 t1 R1 . Modes sate gravity, so the plate is essentially free of gravi- with Zernike radial degree n have features with a ϳ ͞ tational deformations. This is analogous to a scale size of R1 n, so the natural frequency for ͑ ͒ϰ 2 scheme with raft actuators, but the control require- modes in the nth radial degree is f1 n n . Adap- ments are simpler because the plate supports can all tive corrections must be made on the time scale of exert the same elevation-dependent force. The seg- the wind-crossing time for a mode feature, so the ment actuators can be fairly inexpensive, short- frequency for corrections in the nth radial degree is 17 ͑ ͒ϳ ͞ stroke electrostrictive devices. These are fast fAO n un R1, where u is the wind speed. Be- ͑ ͒͞ ͑ ͒ϰ enough for AO and have low hysteresis so they can be cause f1 n fAO n n, a mirror plate that is not operated without local feedback from sensors on the excited by the lowest-order adaptive corrections will segments. not be excited by higher-order corrections if the The following sections contain an overview of de- coupling between different mirror plate modes is sign issues for a mirror with many small segments small. For a 30-m telescope, and a wind speed of mounted on a floating plate. The material is in 30 msϪ1 in the turbulent atmospheric layers, the three parts: design of the floating plate; parametric frequency of the lowest-order adaptive corrections modeling of a simple, inexpensive segment ͑which is is ϳ2 Hz, so the floating mirror plate should have a the emphasis of this paper͒; and optical alignment fundamental frequency of at least a few hertz. This and AO issues. requires an ϳ1-m-thick steel box-style mirror plate ͑see Fig. 1͒. 2. Floating Mirror Plate The force-based support of the floating mirror plate The classical flotation support for a monolithic mirror offers no resistance to wind buffeting, but wind- can be applied to compensate structural deformations induced deformations in the mirror can be corrected in a segmented mirror telescope. In this scheme, if they are smaller than the stroke of the segment the segments are mounted on a plate floating in an actuators. Position-based support can be applied on astatic, gravity-compensating force field. The shape short time scales to provide wind resistance, but at of the plate is determined only by the support forces the expense of a lower fundamental frequency for the and is independent of deformations in the telescope plate and increased coupling to the telescope struc- structure. This is an important advantage because ture. If the mirror plate is supported by hydraulic the stiffness requirements for the structure can be actuators, the cylinders can be connected to an accu- relaxed. The plate is stiff on small spatial scales, mulator with small orifices to restrict the hydraulic and this helps to minimize piston errors between seg- fluid flow. This couples the mirror plate to the tele- ments. Segment motions that are due to gravity are scope structure on short time scales, and the struc- also small, so the gaps between segments are set ture provides stiffness. On long time scales, fluid mainly by manufacturing tolerances. This may give can flow freely in and out of the cylinders, and the a lower infrared background than a mirror with much system behaves like a constant force support. The larger segments mounted in an uncompensated cell. mirror plate control is simple with force actuators, The mirror plate could be a space-frame structure, but position actuators can be used if they are fitted or a box with cells made of steel or fiber-reinforced with load cells to measure the forces applied to the composite sheets. A space frame or box with a stiff plate. Hydraulic and pneumatic actuators are par- face sheet is attractive because it reduces deforma- ticularly appropriate, because many actuators can be tions on small scales and provides a convenient in- connected to a single reservoir to provide constant terface between the dense array of segment actuators pressure over the entire mirror plate.

3306 APPLIED OPTICS ͞ Vol. 42, No. 16 ͞ 1 June 2003 of the mirror.19 For a spherical segment, the surface profile is R2 R4 R6 z ͑r, ␪͒ ϭ ͑2r2 Ϫ 1͒ͩ ϩ ϩ ϩ ...ͪ s 4k 16k3 32k5 ϩ r␶ cos ␪, (2) where k is the radius of curvature of the segment and ␶ is a tilt in the ͑z, ␪ϭ0͒ plane. In a highly segmented mirror, R ϽϽ k, and only the lowest-order terms are significant. A spherical mirror ͑b ϭ 0͒ is easily synthesized from spherical segments, but for a paraboloidal mirror ͑b ϭϪ1͒ we must warp the seg- ␶ Ϫ ments and adjust k and to minimize z zs.In fairly slow systems with small segments, zcoma is neg- Fig. 1. Fundamental frequencies for several 30-m-diameter ligible ͑e.g., a 100-mm-diameter segment at the edge box-style floating mirror plates made of steel. The results are ͞ ϭ ͒ of a 30-m f 1.5 paraboloid has zcoma 0.3-nm p.-p. . from a finite-element model with rectangular plates for the cell High-order segment figure errors Ͻ25-nm p.-p. walls and triangular plates for the front and back face sheets. ͑␭͞20 at ␭ϭ0.5 ␮m͒ are reasonable for small, fairly ϰ 3͞ ϰ For a mirror plate without face sheets, K1 wt1 l and m1 ͞ inexpensive optics. Low-order segment figure er- wt1 l, where K1 and m1 are the stiffness and mass of the mirror plate; hence the fundamental frequency f ϳ͑1͞2␲͒͑K ͞m ͒1͞2 ϰ rors are dominated by the error in the segment radius 1 1 1 of curvature, but this can be at least partially cor- t1 is independent of w and l. The small increase in f1 with l in 5,20 the finite-element model is the result of use of single ideal plates rected by a simple warping harness. If the p.-p. for the cell surfaces. manufacturing error in the radius of curvature is 1%, and the error in warping the segments is 10% of the maximum warp, i.e., 0.1% of the radius of curvature, the corresponding figure error is 10Ϫ3R2͞2k p.-p. 3. Segment Design For a small, slow segment, a 1% radius of curvature In a highly segmented mirror, the segments must error represents a fairly tight manufacturing toler- achieve adequate optical performance with a simple, ance; e.g., with R ϭ 50 mm and k ϭ 90 m, a 1% error inexpensive design. This suggests use of spherical in k corresponds to a segment sagitta error of just 139 surfaces, warping harnesses, and simple three-point nm. Astigmatism can also be compensated with a supports. The key performance metric for the mir- warping harness ͑see Fig. 2͒, and in this case errors ror is the surface error, which has contributions from in both the orientation and the depth of the warp are figure errors, gravitational deformation, warping important. An error ⌬␪ in the orientation of the ͉ ͉ 2␨2⌬␪͞ 3 ⌬␪ ϽϽ harness print-through, actuator position errors, and warp gives a p.-p. figure error b R k1 for support-point torques. The latter are caused by 1, and a fractional error ⌬F͞F in the warping force ͑⌬ ͞ ͉͒ ͉ 2␨2͞ 3 bending of the segment supports because of gravity, gives a p.-p. figure error F F b R 2k1 . For 2 2 1͞2 temperature changes, and actuator operation. Seg- ͓͑⌬F͞F͒ ϩ͑2⌬␪͒ ͔ ϭ 0.1, i.e., 10% amplitude or 3° ment design constraints are explored in this section orientation error in the warp, the total p.-p. segment with simple parametric and finite-element models. figure error is The initial design of a segment mirror assumes an Ϫ d Ϸ ͓͑0.1bR2␨2͞2k 3͒2 ϩ ͑10 3R2͞2k͒2 ideal support; surface errors that are due to support- fig 1 point torques are included at the end of the section. ϩ ͑25 nm͒2͔1͞2. (3) The required segment profile, in cylindrical polar coordinates ͑r, ␪, z͒ centered on the segment, with z The p.-p. gravitational deformation for a segment normal to the segment surface, is on a three-point support is mg␲R2 R4 ␲2g␳͑1 Ϫ ␯2͒ Ϸ ϭ dgrav 0.003 0.036 2 , (4) z͑r, ␪͒ Ϸ zfocus ϩ zastig ϩ zcoma D t E R2 bR2␨2 where m is the mass of the segment; t is the segment Ϸ ͑2r2 Ϫ 1͒ͩ ϩ ͪ ϩ ͑r2 cos 2␪͒ 4k 4k 3 thickness; g is the acceleration that is due to gravity; 1 1 D ϭ Et3͞12͑1 Ϫ␯2͒ is the flexural rigidity of the bR2␨2 bR3␨ segment; and ␳, ␯, and E are the density, Poisson’s ϫ ͩ ͪ ϩ ͓͑3r2 Ϫ 2r͒cos ␪͔ͩ ͪ , (1) 3 3 ratio, and Young’s modulus for the segment material, 4k1 6k1 respectively.21 For a glass segment, with mechani- Ϸ ϫ cal properties given in Table 1, dgrav 9.12 Ϫ8 4͞ 2 ͑ ͒ where r is the segment radial coordinate, normalized 10 R t p.-p. dgrav, R, and t are in meters . to the segment radius R; ␨ is the distance of the The warping harness support print-through can be segment center from the axis of the mirror, and k1 estimated from the pressure required to deflect a and b are the radius of curvature and conic constant plate through a distance equal to the depth of the

1 June 2003 ͞ Vol. 42, No. 16 ͞ APPLIED OPTICS 3307 Fig. 2. Nine-point warping harness. This provides a radius of curvature adjustment and a variable-orientation astigmatic warp. The warping spring is clamped to the center of the back of the segment, and the spring arms engage slotted nuts on studs attached to the segment. The slotted nuts are adjusted to set the Fig. 3. Finite-element analysis of a 100-mm-diameter, 5-mm- required figure. If the warping spring is made of steel and the thick hexagonal glass segment in a four-point astigmatic warping segment is made of glass, a change in temperature will cause a harness. The mechanical properties of the segment material are ϳ␰ ͑ ͒ surface error temp see Section 3 . This effect could be reduced given in Table 1. The small black dots show the deflection of 2400 if the warping spring is made from a fiber-reinforced composite points across the segment surface, and the large black dots show with a low thermal-expansion coefficient. the residuals ͑51.8-nm p.-p.͒ after we fit z ϭϪ0.4 Ϫ 8.0͑2r2 Ϫ 1͒ϩ 283.0r2 cos͑2␪͒ nm. For a 100-mm-diameter segment at the edge ͞ ϭ 2 ͑ ␪͒ ͓ of a 30-m f 1.5 paraboloid, zastig 190r cos 2 nm approxima- warp. For a change in radius of curvature, the tion ͑1͔͒. This requires 0.67 N at each point in the warping har- warping harness must support the segment at three ness, and the residual surface profile error is 34.8-nm p.-p., i.e., Ϸ ϭ ϫ Ϫ5 2 ͑ ͒ points, and the support print-through is dfocus dastig 1.39 10 R dastig and R are in meters . 0.003q͑␲R2͒2͞D, where q is the applied pressure ͓see approximation ͑4͔͒. The deflection of a simply supported plate is ␦ϭqR4͑5 ϩ␯͒͞64͑1 ϩ␯͒D,18 and a change in radius of curvature ⌬k changes the seg- For a glass segment at the edge of a 30-m f͞1.5 pa- 2 Ϸ ϫ Ϫ5 2 ͑ ment depth by ␦ϭ͑R ͞2k͒͑⌬k͞k͒, so the p.-p. support raboloid, dastig 1.15 10 R dastig and R are in ͒ ϭ print-through is meters .Afinite-element analysis gives dastig 1.39 ϫ 10Ϫ5R2 ͑see Fig. 3͒, which is close to the value R2 ⌬k 1 ϩ ␯ d Ϸ 0.003 ͩ ͪ64␲2. (5) from the plate model. The total support print- focus 2k k 5 ϩ ␯ through for a glass segment with 1% p.-p. manufacturing error in the radius of curvature, at the edge of a For a glass segment in a 30-m f͞1.5 mirror, and ⌬k͞ 2 2 1͞2 Ϫ5 2 30-m f͞1.5 paraboloid, is d ϭ͑d ϩ d ͒ Ϸ k ϭ 0.005, d Ϸ 1.24 ϫ 10 R ͑d and R are in Ϫ warp focus astig focus focus 1.69 ϫ 10 5R2 p.-p. ͑d and R are in meters͒. meters͒. An astigmatic warp requires forces at four warp The segment actuator stroke s must be large points, and the print-through is a factor 3 smaller enough to accommodate uncompensated deforma- than for a three-point support. With a change in tions of a few micrometers in the floating mirror segment depth equal to the amplitude of z , the astig plate, segment mounting errors of a few micrometers p.-p. print-through is ͑see Section 4͒, and atmospheric path-length varia- ͉b͉R2␨2 1 ϩ ␯ tions if the mirror is used for AO. For Kolmogorov d Ϸ 0.001 ͩ ͪ64␲2. (6) astig 3 ϩ ␯ turbulence, the wave-front variance, excluding the 4k1 5 ͗␸2͘ϭ ͑ ͞ ͒5͞3 2 piston mode, is 1.03 D1 r0 rad , where D1 is the diameter of the mirror and r0 is the Fried Table 1. Material Properties parameter.8,22 A correction range of five times the rms path-length variation requires s Ͼ͑D ͞r ͒5͞65␭͞ Glass Stainless 1 0͞ ␲ ͑ ͒ϳ ͓ ␭͑␮ ͔͒6 5 23 4 ; and at a good site, r0 m 0.2 2 m , so Property Ceramic Steel Units ͑␮ ͒Ͼ ͓ ͑ ͒͞ ͔5͞6 s m 13 D1 m 30 . The tip–tilt part of the Young’s modulus 9.0 ϫ 1010 2.0 ϫ 1011 NmϪ2 wave-front error can be corrected by a separate fast- Poisson’s ratio 0.24 0.30 steering mirror, so s ϳ 20 ␮m should be adequate for Ϫ Density 2.53 ϫ 103 7.85 ϫ 103 kg m 3 both AO and segment position error corrections for a Ϫ7 Ϫ5 Ϫ1 Thermal-expansion 10 10 K 30-m telescope. Electrostrictive actuators with po- coefficient sition feedback only from a wave-front sensor make

3308 APPLIED OPTICS ͞ Vol. 42, No. 16 ͞ 1 June 2003 Fig. 4. Surface error contributions for a 5-mm-thick glass seg- ͞ ͓ Fig. 5. Segment radius as a function of thickness for a 75-nm p.-p. ment at the edge of a 30-m f 1.5 paraboloid. dfig approximation ͑3͔͒ is the segment figure error, when we assume 0.1% error in the total surface error. radius of curvature, 10% amplitude or 3° orientation error in astig- ͓ matism, and 25-nm p.-p. high-order errors; dgrav approximation ͑4͔͒ is the gravitational deformation of the segment on a three- is also a good choice for extrasolar planet observa- ͓ ͑ ͒ ϳ ϫ ␭͑␮ ͒ point support; dwarp quadrature sum of approximations 5 and tions because scattering within 2 arc sec m of ͑6͔͒ is the support print-through when the segment is warped to the star is controlled by the segment actuators rather change the radius of curvature by 0.50% and compensate astigma- than by polishing errors. tism; dact is the segment position error that is due to hysteresis in A segment must be attached to its actuators with ͑ ␮ ͒ the segment actuators 1% of a 2- m step ; and dtot is the quadra- flexures that accommodate changes in temperature ture sum of all the error contributions. The horizontal line at and actuator strain. As the flexures bend they apply 75-nm p.-p. surface error corresponds to a 30-nm rms wave-front torques at the segment support points, and the seg- error. ment deforms. The following deformation estimates are for simple rod flexures of length h and diameter d. the segment simple, but at the expense of position Each flexure supports one third of the weight of the segment, so the torque at a segment support point errors that are due to actuator hysteresis. This is a ␶ ϳ ͞ when the telescope is at the horizon is grav mgh 3. small effect for closed-loop operation because the ac- ϳ␶ ͑͞ Ϫ ͒ tuators in a high-order AO system do not move much This corresponds to a force Fgrav grav R Ract at the segment edge, where Ract is the radius of the in the wave-front sensor integration time. A posi- ϭ ͌͞ ␮ actuator pattern. Ract R 2 for minimum tion error of 2 m times the fractional hysteresis is 24 used here as a rough estimate. For electrostrictive dgrav. The torques at the three support points are materials, the hysteresis is ϳ0.01s,17 so the segment aligned, so the axial force at the center of the segment ϳ is small. The p.-p. segment deformation is position error is dact 20-nm p.-p. For an ideal torque-free support, the p.-p. segment 2͑ ϩ ␯͒ ͱ 3␳ ͑ ϩ ␯͒͑ Ϫ ␯͒ ϭ͑ 2 ϩ 2 ϩ 2 ϩ 2FgravR 3 2R gh 3 1 surface error is dtot dfig dgrav dwarp ␰ ϳ ϭ , 2͒1͞2 grav ␲͑ ϩ ␯͒ 2͑ ͱ Ϫ ͒ dact . This is shown in Fig. 4 for a 5-mm-thick 16 1 D 2Et 2 1 ͞ glass segment at the edge of a 30-m f 1.5 paraboloid. (7) For a wave-front error Ͻ30-nm rms, the surface error must be Ͻ75-nm p.-p., which requires R Յ 52 mm. which is a rough estimate based on the deflection of a This is actually an underestimate of the maximum simply supported segment with force 2Fgrav at the 18 segment radius because dfig and dwarp are for a seg- center. ment at the edge of the mirror ͑and the worst-case If just one actuator is fully extended, the segment ͒ ␥ϭ ͞ manufacturing error and dgrav is for the full zenith tilts through an angle 2s 3Ract. The flexures angle range, but most observations are made at ze- bend a distance ϳh␥, and the torque at each support Ͻ ϳ 2␥ ϭ ͞ 3 ϭ␲ 4͞ nith angles 70°. Figure 5 shows the segment ra- point is h Kf, where Kf 3EfIf h and If d 64 ϭ dius as a function of thickness for dtot 75-nm p.-p. are the bending stiffness and moment of inertia of a 25 For thin segments, gravitational deformations dom- flexure and Ef is Young’s modulus for the flexure inate, and the segments must be made smaller; but material. The corresponding p.-p. segment defor- for thick segments, figure errors and warping har- mation is ness print-through limit the segment radius. The ϳ 3sE d4͑3 ϩ ␯͒͑1 Ϫ ␯͒ maximum segment radius is 50 mm, and the cor- ␰ ϳ f ϳ act . (8) responding minimum thickness of 5 mm gives an 32Et3h͑ ͱ2 Ϫ 1͒ exceptionally lightweight mirror with a short thermal time constant. A segment diameter of ϳ100 A temperature variation ⌬T changes the actuator mm is a good choice for AO at visible wavelengths pattern radius and bends the flexures a distance ϳ ␭ϭ ␮ ␣ ⌬ ␣ because r0 200 mm at 0.5 m at a good site. It Ract T, where is the difference between the ex-

1 June 2003 ͞ Vol. 42, No. 16 ͞ APPLIED OPTICS 3309 crease away from the axis, so each batch would also have a slightly different segment diameter. For the simple segments in this paper, the fundamental frequency is set by the bending stiffness of the support flexures, so there is a trade-off between the AO bandwidth and the segment surface error that is due to support-point torques. The fundamental ϳ͑͞ ␲͒͑ ͞ ͒1͞2 frequency is f 1 2 3Kf m ; and for a 100- mm-diameter, 5-mm-thick glass segment on 0.5- mm-diameter, 2-mm-long steel flexures, f ϳ 420 Hz. This model tends to underestimate Kf and f because it constrains the flexures at only one end. A finite- element analysis yields f ϭ 628 Hz. For a typical wind speed of 30 msϪ1 in a high turbulent layer, the wind-crossing frequency for a segment radius of 50 Fig. 6. Surface error that is due to support-point torques for a mm is 600 Hz, so the simple segment design is just 100-mm-diameter, 5-mm-thick glass segment on 0.5-mm-diameter able to support the highest-order AO modes. The steel rod flexures attached to a steel mount. Mechanical proper- response of a segment to wind buffeting is also set by ties of the segment and flexure materials are given in Table 1. the stiffness of the support flexures, but in this case ␰ ͓approximation ͑7͔͒ is the segment deformation that is due to grav the axial stiffness is relevant. A 100-mm-diameter gravitational deflection of the flexures when the telescope is at the horizon; ␰ ͓approximation ͑8͔͒ is the deformation that is due to segment on 0.5-mm-diameter, 2-mm-long steel flex- act ures has a stiffness of ϳ7 ϫ 109 Pa mϪ1; compare this bending of the flexures when just one actuator is fully extended; 7 Ϫ1 ␰ ͓ ͑ ͔͒ with ϳ10 Pa m for the 1-m-diameter segments in temp approximation 9 is the deformation that is due to the ␰ 10 flexures bending because ofa5Ktemperature change; and tot is the CELT. On small spatial scales, a highly seg- the quadrature sum of the surface error contributions. The cross mented mirror can have high stiffness, and hence ␰ ␰ and circle are finite-element analysis results for grav and act. good wind resistance, because the support-point den- ␰ temp could be reduced substantially if the segment mount was sity is high; on large spatial scales, the stiffness is made from a composite material with a low thermal-expansion limited by the floating mirror plate and its support coefficient. structure ͑see Section 2͒.

4. Optical Alignment pansion coefficients of the segment and its mount. An important feature of the mirror design described The torque at each support point is ␶ ϭ temp in this paper is use of only a wave-front sensor to hK ␣R ⌬T, which corresponds to a force ␶ ͑͞R Ϫ f act temp measure the segment positions. This is attractive R ͒ at the segment edge and Ϫ␶ ͞R at the cen- act temp act because the segment control depends directly on the ter. The three support-point torques are roughly quality of the wave front delivered by the telescope, equivalent to a single force 3␶ ͓1͞R ϩ 1͑͞R Ϫ temp act but the approach is not simple because a large wave- R ͔͒ at the center of a simply supported segment. act front sensor is required. A mirror with N segments, The p.-p. deformation is and three actuators per segment, must have at least 3N͞2 wave-front sensor subapertures to measure all 27R2E d4␣⌬T͑3 ϩ ␯͒͑1 Ϫ ␯͒ the degrees of freedom. A hexagonal subaperture ␰ ϳ f . (9) ͑ temp 3 2͑ Ϫ ͱ ͒ array with one subaperture per segment edge i.e., 128Et h 2 2 three subapertures per segment͒ gives the average tilt of the mirror surface at each segment edge and ␰ ␰ 4 Both act and temp are proportional to d ,soitis should be enough to calculate all the segment tilts important to make the flexures thin. Figure 6 shows and relative piston errors. Adding a subaperture at the segment surface error that is due to support-point the center of each segment fills the array and pro- torques for a 100-mm-diameter, 5-mm-thick glass vides an independent measurement of the tilt of each segment on 0.5-mm-diameter steel flexures attached segment ͑see Fig. 7͒. This scheme requires a hexag- to a steel mount. In this case, the optimum flexure onal lenslet array with 4N elements and a rectangu- length is 3 mm, and the support contributes only lar detector array with an aspect ratio of cos͑␲͞6͒ and ϳ20-nm p.-p. to the segment surface error. 16N pixels ͑i.e., ϳ4 ϫ 105 lenslets and 1.6 million Figures 4 and 6 suggest that a 30-m f͞1.5 mirror, pixels for a 30-m telescope with 100-mm-diameter with good optical performance, can be made from segments͒. The wave-front reconstruction for 4N ϳ100-mm-diameter spherical segments with warp- subapertures and 3N actuators requires 8N ϫ 3N ing harnesses and simple three-point supports. multiply-and-add operations, which is possible for a This requires ϳ105 segments, with radius of curva- 1-Hz update rate for active corrections with 105 seg- ture varying from 90 m on axis to 92.3 m at the edge. ments, but requires some development for AO. For 1% p.-p. manufacturing tolerance on the radius of The wave-front sensor subapertures are small for a curvature, three or four different batches, each of a highly segmented mirror, so a laser guide star is few ϫ104 identical segments, would be required. In required. For a Shack–Hartmann sensor measur- a concave mirror, the segment diameter must de- ing all 3N modes of the mirror, the wave-front vari-

3310 APPLIED OPTICS ͞ Vol. 42, No. 16 ͞ 1 June 2003 mounted near the secondary. Prior to segment in- stallation, each mounting node must be adjusted to match its segment, based on the segment height and node height measurements. Installing 105 segments is a substantial task that would require par- allel operations and automated tracking of all the measurements.

5. Discussion The maintenance of a highly segmented mirror is an important consideration because mirror coatings have a finite lifetime. Mounting the segments on rafts allows them to be replaced in batches, but this approach breaks up the mirror support structure and encourages piston errors at raft boundaries. If the segments have a simple release mechanism, they could be handled one at a time, perhaps by a robot. An important advantage of small segments is that Fig. 7. Wave-front sensor subaperture and detector configura- multilayer coatings can be used to improve the re- tion. The bold hexagons are mirror segments, the circles are flectivity and lifetime of the surface.29–31 This sim- wave-front sensor subapertures ͑1 ϩ 6͞2 per segment͒, and the plifies the logistics of recoating and increases the rectangles represent the assignment of detector pixels to subaper- throughput of the telescope. Small segments also ͑ ͒ tures four pixels per subaperture if there are no guard bands . reduce the cost of the recoating facility. The segment design details given in this paper suggest that a large mirror, with good optical perfor- ance in each sensed mode is ͗␴2͘Ϸ␲2͞3NpA rad2, mance, can be made from small spherical segments where p is the number of photons per unit area inci- with warping harnesses. A simple three-point seg- dent on the telescope and A is the area of a wave-front ment support is adequate for both axial and radial sensor subaperture in the entrance pupil.26 For a loads. Mounting the segments on a floating plate hexagonal array of circular subapertures, A ϭ␲͓R that compensates gravitational deformations allows cos͑␲͞6͔͒2͞4 Ϸ R2͞2. The total wave-front variance use of inexpensive, short-stroke segment actuators is then 3N ͗␴2͘Ϸ2␲2͞pR2 rad2, and the total path- and can significantly relax the stiffness requirements length variance is ͗⑀2͘ϭ3N͗␴2͑͘␭͞2␲͒2 Ϸ␭2͞2pR2. for the telescope structure. This scheme is also ap- At ␭ϭ0.63 ␮m, an mth magnitude spectral-type AO propriate for AO because it decouples the segment Ϫ Ϫ star provides p ϳ 2.4 ϫ 1010 0.4m photons m 2 in a actuators from the structure. AO is an attractive 1-s integration time and 0.3-␮m bandwidth; so for feature of a highly segmented mirror, but the ap- ϭ ϳ ϩ ͑␶ 1͞2 ͗⑀2͘1͞2͒ ␶ R 50 mm, m 51.2 5 log int , where int proach also offers real potential for cost savings is the wave-front sensor integration time in seconds through quantity production of simple optics. ⑀ ͗⑀2͘1͞2 ϭ ␶ ϭ and is in meters. For 5 nm and int 1s, m ϳ 9.7, which requires a laser guide star for most This research was supported by the Caltech Dis- observations.27 Gravitational and thermal deforma- covery Fund. ␶ tions in the telescope structure vary slowly, so int could probably be increased to 10 s for active correc- References tions, which would allow a 12th magnitude guide 1. R. Angel, M. Lloyd-Hart, K. Hedge, R. Sarlot, and C. Peng, ͞ star. “The 20 20 telescope: MCAO imaging at the individual and Because the segment actuators have a short stroke, combined foci,” in Proceedings of ESO Conference on Beyond Conventional Adaptive Optics, R. Ragazzoni and S. Esposito, initial alignment of the segments is an important eds. ͑to be published͒. consideration. Of course the corollary is that, once 2. A. D. Gleckler, D. J. Markason, and G. H. Ames, “PAMELA: this problem is solved, the segments can be phased control of a segmented mirror via wavefront tilt and segment quickly,9,28 and recovery from a segment change is piston sensing,” in Active and Adaptive Optical Components, rapid. During assembly, the radius of curvature M. A. Ealey, ed., Proc. SPIE 1543, 176–189 ͑1991͒. and astigmatism of each segment must be set within 3. S. C. Fawcett, “Development of adaptive optical segments with ϳ0.1% and ϳ10%, respectively, and the height of the integrated wave front sensing,” in Active and Adaptive Optical segment and its mount must be measured with an Components and Systems II, M. A. Ealey, ed., Proc. SPIE 1920, ͑ ͒ accuracy of a few micrometers. These measure- 193–199 1993 . ments and adjustments could be automated. The 4. J. M. Rakoczy, E. E. Montgomery, and J. L. Lindner, “Recent enhancements of the Phased Array Mirror Extendible Large relative heights of the segment mounting nodes on Aperture ͑PAMELA͒ telescope testbed at MSFC,” in Telescope the telescope must also be measured with an accu- Structures, Enclosures, Controls, Assembly͞Integration͞Vali- racy of a few micrometers. This could be done with dation and Commissioning, T. A. Sebring and T. Anderson, a laser interferometer mounted on a plate that re- eds., Proc. SPIE 4004, 352–362 ͑2000͒. places a segment assembly, with a fixed retroreflector 5. J. E. Nelson, T. S. Mast, and S. M. Faber, eds., “The design of

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3312 APPLIED OPTICS ͞ Vol. 42, No. 16 ͞ 1 June 2003