Performance Prediction of the LSST Secondary Mirror Myung K. Cho*, Ming Liang2, and Douglas R. Neill2 *
GSMT Program Office, National Optical Astronomy Observatory 2 LSST Project, National Optical Astronomy Observatory 950 N. Cherry Ave., Tucson, AZ, USA 85719 ABSTRACT
The Large Synoptic Survey Telescope (LSST) is an 8.4 meter telescope with a field of view of 10 square degrees. This telescope will be capable of mapping the entire visible sky every few nights via sequential 15second exposures, opening new windows on the universe from dark energy to time variable objects. The LSST optics calls for an annular 3.5 m diameter Secondary Mirror (M2), which is a large meniscus convex asphere (ellipse). The M2 converts the beam reflected from the f/1.2 primary mirror into a beam for the f/0.83 Tertiary Mirror (M3). The M2 has a mass of approximately 1.5 metric tons and the mirror support system will need to maintain the mirror figure at different gravity orientations. The optical performance evaluations were made based on the optimized support systems consisting of 72 axial supports, mounted at the mirror back surface, and 6 tangent link lateral supports mounted around the outer edge. The predicted print-though errors of the M2 supports are 8nm RMS surface for axial gravity and 10nm RMS surface for lateral gravity. The natural frequencies were calculated for the M2 dynamic performance. In addition, thermo-elastic analyses of M2 for thermal gradient cases were conducted. The LSST M2 support system has an active optics capability to maintain optical figure and its performance to correct low-order aberrations has been demonstrated. The optical image qualities and structure functions for the axial and lateral gravity print-through cases, and thermal gradient effects were calculated. Keywords: wide field survey telescope, large secondary mirror performance, support system optimization, active optics system, image quality and structure function
1. INTRODUCTION The optical systems of LSST have stringent requirements to achieve scientific goals. The M2 support system was optimized to meet the requirements defined in “LSST Image Size and Wavefront Error Budgets”[4]. The requirements are defined in terms of structure functions. Preliminary wavefront error budget allocation as a goal for the M2 figure are as follows: 20nm RMS at Zenith and 20nm RMS at 75 degrees elevation for gravity induced fabrication error. To fulfill the optical and mechanical performance requirements, extensive finite element analyses using I-DEAS and optical analyses with PCFRINGE[8] have been conducted. Mechanical and optical analyses performed include static gravity induced deformations, natural frequency calculations, and support system sensitivity evaluations. The influence matrix of M2 was established to compensate potential errors using an M2 active optics system. Performances of the M2 support system were evaluated for the sample sensitivity cases before and after active optics corrections. As a baseline LSST optical configuration in this study, the M2 was modeled based on the optical prescription in Table 1. In the table, higher order aspheric terms, A4 and A6 in Zemax, are listed for each of the optical mirrors. The physical and ULE™ glass material properties used in the M2 FE mirror models are summarized in Table 2. The LSST telescope CAD model and the optical system configurations are shown in Figure 1.
*
[email protected]; phone 1 520 318-8544; fax 1 520 318-8424; www.noao.edu
Advances in Optomechanics, edited by Alson E. Hatheway, Proc. of SPIE Vol. 7424, 742407 · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.823716
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Table 1. Optical Prescription of LSST (length in meters)
Configuration Primary Diameter Primary F ratio System F ratio FOV
ROC 8.4 m Primary 1.18 Secondary 1.23 Tertiary 3.5 degrees
-19.835 -6.788 -8.345
K -1.2150 -0.2220 0.1500
A4
A6
Distance between optics
1.381E-09 M1 - M2 -1.274E-05 -9.680E-07 M2 - M3 -4.500E-07 -8.150E-09 M3 - Image
-6.156 (m) 6.390 -4.666
Table 2. ULE Material properties used in FE mirror models. Coefficient of thermal expansion Thermal conductivity Specific hea t Density Modulus of elasticity Poisson’s ratio
15 x 10 -9 m/mºC 1.3 W/mºC 766 J/kgºC 2205 kg/m 3 9.2 x 10 10 Pa 0.17
Fig. 1. The LSST telescope CAD model and its optical system configurations
2. SECONDARY MIRROR CONFIGURATION The Secondary mirror is a large annular meniscus convex ellipse. The M2 converts the beam reflected from the f/2.4 Primary Mirror (M1) into an f/28 beam through the Tertiary mirror for the wide field camera. The mirror substrate is made out of ULE™ manufactured by Corning at their Canton facility in New York. The substrate was fabricated via Corning’s high temperature fusion process out of 8 boules, machined and arranged in a petal-shaped geometry. The oversized fused substrate was machined plano/plano to include critical zone placement. Finally, the substrate was slumped to near net shape over a sag form and is currently undergoing final grinding and acid etching, with final delivery in December 2009. A schematic manufacturing process of the mirror substrate is shown in Figure 2. The design concept of the M2 Cell Assembly (M2CA) developed by the LSST is shown in Figure 3. The M2CA consists of M2, support system, and the cell. Several different FE models were created to serve various calculations. A typical FE mirror model of M2 is composed with four layers of elements with a total of 10,080 solid elements and 12,245 nodes. This model assumes a solid convex meniscus mirror with a diameter of 3.47m, 100mm thick, a central obscuration of 1.78m, and a radius of curvature of -6.8m (best fit sphere). The M2 mass was estimated as 1469Kg from a M2 FE solid model. A local coordinate system in the FE model was assumed as follows: (1) the positive Z-axis corresponds to the line which connects the vertex of the primary mirror to the vertex of the secondary mirror in the telescope; (2) the positive X-axis corresponds to the telescope’s mechanical elevation axis; (3) the positive Y-axis is defined by the right hand rule.
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For the thermal time constant, a convection analysis was performed with a M2 thermal FE mirror model. An air convection was applied on the front and back surfaces having heat transfer coefficients of 3 W/m2 ºC and 1 W/m2 ºC, respectively. Air convection along the edge of the mirror was incorporated. The thermal time constant (τ) of 12 hours was calculated by obtaining the required time for the mirror to reach 3.7 ºC as 1/e of the initial temperature at 10ºC.
Fig. 2. Secondary mirror substrate manufacturing process (substrate status in Corning Canton facility as of July 2, 2009). Top left shows a schematic layout of M2 with the 8 boule petals; top center shows quality inspection of a boule petal, and top right shows the fused substrate in Corning’s 8-meter furnace (center hole plugged during fusing). The bottom right shows the fused substrate before plano/plano machining and the bottom left photo shows the recently slumped M2 substrate ready for final grinding and acid etching.
Fig. 3. Secondary mirror Cell Assembly (M2CA) consisting of M2, Support system, and mirror Cell (Earlier axial actuator configuration shown)
3. SECONDARY MIRROR SUPPORT SYSTEM A parametric study for the M2 support design was made with several support patterns. The Optimum mirror support system was determined, and it contains a total of 72 axial supports arranged in three equally spaced rings with an active optics capability and 6 active lateral supports tangentially mounted around the outer edge of the mirror. This M2 support system was optimized for minimum gravity induced errors and minimum variation in axial support forces. The axial support was optimized for the telescope at Zenith pointing, and the lateral was optimized at horizon pointing. The optical performance was evaluated for gravity variations between Zenith and Horizon. The active forces were evaluated based on the influence matrix calculated from 72 axial support points. In addition, random support force errors and sample support failure modes were estimated as a part of the sensitivity and tolerance analyses. To predict the mirror stiffness, fundamental mirror frequencies were calculated with a free-free boundary condition. Active optics correction was attempted to low order Zernike modes to demonstrate the M2 figure corrections. Detailed mechanical and optical performance analyses were conducted using I-DEAS finite element analysis program and the PCFRINGE optical program.
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3.1 M2 AXIAL SUPPORT Parametric modeling iterations were conducted for the support system optimization. These iterative calculations utilize an optimization scheme for a minimum global surface deformation over the optical surface. The key metric, during the optimization process, was the optical surface RMS error. This optimization process was initially developed for the Gemini 8m telescopes [1][3] and applied to the Gemini primary mirrors. In order to achieve the optical performance goal of 20nm RMS surface, extensive parametric calculations were made for an optimum axial support system. The axial support optimization yields an optical surface RMS error of 8nm with a PV of 39nm. This optimized axial support features a 72 axial support system arranged in a three (3) concentric ring pattern. This support optimization involves two main optimization processes to achieve the goal. First, the support locations were determined for a minimum global RMS surface error without constraining support force magnitudes. Next, a further optimization was processed to select support forces into groups without sacrificing the RMS surface errors. This optimization process is being adopted in the Primary Mirror of the Advanced Technology Solar Telescope (ATST) Project[2] and the Secondary Mirror and Tertiary Mirror of the Thirty Mirror Telescope (TMT) Project[5][6] . Grouping supports will simplify the manufacturing and design processes. The axial support forces were optimized in two groups, nominally 191N on the innermost ring and 218N on the rest. The optical surface contour map and the support locations are shown in Figures 4. The M2 optimized axial support system descriptions and the support configuration are summarized in Table 3. Table 3. The axial support system configuration.
Ring 1 2 3
Radial position (m) 1.002 1.284 1.601
No. of Supports 18 24 30
Support Force (N) 191 218 218
Fig. 4. Axial support print through and optimized support radial locations. Axial support print through (RMS=8nm, surface) with two groups of axial support forces (191N and 218N).
The LSST plans to have the support print through polished out for telescope zenith pointing; therefore, the negative of the axial support print through will appear as the telescope moves away from the zenith in proportion to (1-cos(Z)), where Z is a zenith angle. Since manufacturing limitations may preclude this approach, a more conservative assumption is utilized in this analysis that the mirror is figured convex up without accommodation for axial support print through. The axial support print though will then be manifest itself according to (2-cos(Z)). 3.2 M2 LATERAL SUPPORT
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The M2 lateral support configuration has 6 lateral supports equally spaced around the periphery of the mirror mounted at the M2 mid-plane of the edge. The lateral support is actively controlled and provides tangential forces to compensate the lateral gravity of the mirror when the telescope is away from Zenith. This lateral support concept was employed in the SOAR 4m telescope and its performance is well proven[7]. A schematic layout of the M2 cell is shown in Figure 5.
Fig. 5. M2 cell with M2 positioner and M2 baffle. The 6 Lateral supports were mounted at the M2 mid-plane of the edge.
(a) (b) Fig. 6. Lateral support layout and lateral gravity print through error during horizon pointing. (a) Lateral support print through (RMS= 10nm surface, (b) Axial support force distribution (Fmax = 133 N).).
As the telescope is pointed away from Zenith, the lateral support will take the lateral gravity of M2. Since the line of action of the lateral force is not on the plane of the M2 center of gravity, the mirror is exerted additionally by a set of axial support forces to balance the coupling moment. At the extreme telescope elevation (horizon pointing position), optimum axial support force set was determined for a minimum surface RMS errors. The optimized lateral support was obtained with the optical surface error of 10nm, RMS and 152nm, P-V with the maximum axial support force of 133N. At horizon pointing, the optical surface map and optimized axial support forces from the lateral support are shown in Figure 6. Since each lateral support carries a high resultant force as a point load, a local effect due to this lateral force was examined. This local effect (called Poisson’s effect) was calculated at the 6 lateral support locations. This is an important parameter in defining the clear aperture of Secondary mirror. As a first order approximation, the Poisson’s effect was estimated by examining the relative displacements between the top and bottom of the mirror surface under the lateral gravity carried by the lateral supports. A maximum local deformation of 65nm was calculated at the lateral support locations. The Poisson’s effect was drastically reduced at radial distances of 30mm away from the edge of the mirror. Since the clear aperture starts 20mm from the OD the Poisson's effect produces negligible influence on the image quality. The corresponding maximum surface displacement is 20nm. The maximum stress level at the lateral support locations, support pad physical dimension, and its material choice will be investigated further by advanced opto-mechanical and thermal analyses.
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If the secondary mirror was polished, figured, and tested at its face up position, then no gravity support error would exist at the face up position. After the M2 is installed in the telescope, the M2 would be in a -2g axial gravity (reversed gravity impact from null figuring at faced up position) at the telescope in Zenith position. At a 90 degrees Zenith angle (horizon position), the support gravity print-through would be a combination of the axial gravity and lateral gravity errors. Therefore, the resulting surface error becomes 12nm RMS with a quadratic sum of errors from -1g axial (reversed gravity impact) and 1g lateral gravity cases. 3.3
IMAGE QUALITY
Image quality calculations were made to quantify gravity print-though effects from the M2 support. For this printthrough analysis, three configurations were considers as: nominal case (ideal system, no errors), case with axial gravity print-through of -2g, and 1g lateral gravity case. The wavefront maps on-axis images at 632nm were calculated and the corresponding spot diagrams for the three cases in R band on axis and 1.75 degree off-axis positions were obtained. In Figure 7, the wavefront maps are shown for two gravity print through cases, axial –2g and lateral 1g, respectively . Diffraction energy concentrations were calculated at 50% and 80% for the three cases, respectively. The diffraction encircled energy is listed in Table 4, where D50 stands for a 50% diffraction encircle energy diameter in arcseconds and D80 is a 80% diffraction encircle energy diameter in arcseconds.
Fig 7. Wavefront maps for axial –2g case, and lateral 1g case, respectively. Table 4. Diffraction Encircled Energy calculations in arc-seconds for three cases. On Axis D50
3.4
1.75 degree off axis D80
D50
D80
No error
0.0795
0.1910
0.1117
0.1961
-2 dz, print through
0.0887
0.2513
0.1235
0.2421
1 dy, print through
0.0818
0.1975
0.1141
0.2021
SUPPORT SENSITIVITY
Sensitivity and tolerance analyses were performed to quantify the optical surface deformations affected by uncertainties in the design and potential errors involved in manufacturing and system integration. The sensitivity cases are commonly as: support position error, support failure mode, single actuator force error, lateral support force errors, and random active force errors. For a case in which a single axial support has a force error of 1 N, the net change in the optical surface error is nominally 6nm RMS surface errors after piston, tilts, and focus removed. Since the displacements are small, we utilize a linear FE analysis and the optical effect can be scaled linearly.
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Axial support force sensitivity calculations were made assuming the axial forces in a random distribution. This can be a case to simulate actuator manufacturing errors, force repeatability limit, or force accuracy. In order to demonstrate the effect on random force errors, a sample FE model was created with a maximum deviation of 0.5 N (+/- 0.25 N) axial force errors. The calculation further assumed that the axial force set was in a random three-sigma Gaussian distribution. A histogram of the random force distribution is shown in Figure 8(a). A random axial force distribution at 72 supports shown in Figure 8(b). The resulting optical surface map is shown in Figure 8(c) after removing piston, tilts, and focus. The result indicates an RMS surface of 7.1 nm with predominantly astigmatic shape. Monte Carlo scheme can be adopted to simulate the random effect for further investigation.
(a) (b) (c) Fig 8. Axial support force sensitivity due to random axial forces (in Gaussian distribution). (a) Histogram of random
axial forces (Fmax=+/- 0.25N) (b) random axial forces in Gaussian distribution. (c) Optical surface response to a random axial forces (RMS=7.1nm surface).
4. MIRROR FREQUENCY MODES Natural frequencies of the mirror were calculated by using a solid full FE mirror model with a free-free boundary condition. These frequency modes are characteristic mirror bending shapes and were obtained after removing rigid body motions (piston and tilts). The natural frequencies, up to 24 modes, were calculated and the first 10 characteristic mode shapes are listed in Table 5. Each of these first 10 characteristic shapes is shown in Figure 9. The lowest mode was found at 37 Hz, as an astigmatic shape. These low frequency modes are similar to low order Zernike polynomials, but not in the same order. Table 5. The first 10 natural frequency mode shapes. mode ID
frequency (hz)
mode shape
1 2 3 4 5 6 7 8 9 10
37 37 99 99 152 181 181 194 194 222
45 astigmatism 0 astigmatism 0 trefoil 30 trefoil focus 0 quadfoil 45 quadfoil 90 coma 0 coma 2nd order 0 coma
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Fig. 9. First 10 natural mirror mode shapes (free-free)
5. ACTIVE OPTICS PERFORMANCE The M2 active optics (aO) system has 72 axial support actuators in the M2 support system. An influence matrix was constructed by calculating the influence function of each of the 72 axial support actuators. A 1 N force load was applied to one axial support actuator at a time. The M2 aO system will correct the Secondary mirror optical surface errors, closed loop, using a look-up-table (LUT) as the initial values. The wave front error analysis system will provide off sets to these values to correct measured errors. The M2 figure errors are mostly low order aberrations which can be decomposed into a set of low order orthogonal functions. Examples of orthogonal functions (sets) are Zernike polynomials, natural mirror bending modes, and modes from singular value decompositions (SVD). The first 10 natural bending modes were modeled to evaluate the M2 aO performance. These are the same natural modes shown in Figure 12. Each of the first 10 modes was scaled to a RMS surface error of 1000nm as a reference. Active optics corrections were applied to each individual mode to minimum surface RMS error. Active optics correction capabilities for the mirror bending modes are summarized in Table 6. The maximum aO correction forces required to compensate each surface are listed along with the residual surface errors and the “gain”, defined as the ratio of the RMS input amplitude (1000 nm in each case) to the residual RMS surface error. For example, the lowest mirror bending mode (Mode ID 1 in Table 6), is an astigmatic optical surface and can be corrected almost entirely. In this case, the astigmatism of 1000nm RMS surface error can be reduced to a residual RMS surface error of 0.2 nm with a maximum active force of 2.2 N. The mode shape and its active optics correction forces are shown in Figure 10. Table 6. Active optics performance with First 10 Natural bending modes.
mode ID 1 2 3 4 5 6 7 8 9 10
Reference P-V rms (nm) (nm) 3835 3806 4156 4150 3541 4357 4442 5794 5797 5738
1000 1000 1000 1000 1000 1000 1000 1000 1000 1000
Active Optics Correction P-V rms Fmax Gain (nm) (nm) (N) 2.3 1.9 3.7 3.8 8.6 13.7 15.0 19.4 19.4 30.6
0.2 0.2 0.4 0.4 1.5 1.5 1.5 2.2 2.3 4.0
2.18 2.18 16.57 16.16 28.85 59.80 59.42 70.01 71.15 130.57
6417 6111 2424 2434 685 687 674 450 444 251
For this astigmatic mirror mode, the result shows an active optics error (aO error) of 0.2%, or 0.2nm/1000nm. This is equivalent to a “gain” of 6417, where the “gain” (reciprocal of “aO error”) is defined as the ratio of the surface RMS input amplitude to the surface RMS residual error. The surface map of this lowest mode, residual surface map, and the aO force distribution are shown in Figure 6. The least correctable among these natural modes is 10th mode, a higher order coma mode. The residual RMS surface error of 4nm was calculated and its aO error was 4%, or a gain of 251. The
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performance summary indicates that the M2 aO system works adequately with the low order aberration modes but does not perform as well for the higher order modes. The higher order modes require higher active forces with higher residual errors.
(a) (b) Fig. 10. Active optics correction to astigmatic mirror mode. (a) Astigmatism (Reference): RMS = 1000 nm. (b) Active optics force distribution (Fmax = 2.2 N).
It is desirable to apply the active support system to correct potential errors from external or internal loads after the M2 is installed in the telescope. As a sample case, a thermal M2 model was considered to simulate the loading or environmental conditions which can be exerted onto the M2. Note, there are no plans to provide active thermal control of M2 on telescope. For the thermal distortions due to temperature variations on the M2, FE models were created with a unit thermal gradient of 1oC each of the local coordinate directions. Modeled, as a most dominant gradient case, was a linear gradient of 1oC through the thickness (1oC/0.1m), indicating the top surface 1oC warmer than the back surface. For this particular case, a P-V surface error of 185 nm and RMS surface error of 53 nm were calculated, after removing piston and tilt. After applying aO correction, the optical surface RMS error was reduced to 0.7 nm with a maximum aO correction force of 4.7 N. The surface error maps before and after aO correction and the correction force distribution are shown in Figure 11. Note that insignificant RMS surface error (less than 1nm) was calculated for thermal gradient cases along the local X or Y-axis after removing piston and tilt before applying aO corrections.
(a) (b) (c) Fig. 11. Active optics correction to thermal gradient through the thickness. (a) Optical surface map from delta T of 1oC over a thickness of 0.1m , RMS surface error = 53 nm. (b) Residual RMS surface error = 0.7 nm. (c) aO force distribution (Fmax = 4.7N)
6. STRUCTURE FUNCTIONS In order to control the amplitude of surface figure errors as a function of their spatial frequency, the LSST Telescope Error Budgets Document [4] specifies the requirement for surface figure accuracy in terms of a Structure Function (SF). The value of the SF for each separation distance is calculated in terms of the optical path difference (OPD) for each pair of points on the OPD map. SF is defined as: D(r) = < [Φ(x + r) - Φ(x)]2 >, where Φ is the OPD at a position x with a separation of r. A SF was calculated for the axial gravity support print-though at all spatial scales and is shown in Figure 12. A scale factor of 4 was used to convert the surface error to wavefront error and from facing up during fabrication to
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facing down during operation. Additionally, a SF of the lateral gravity support print-through was calculated though at all spatial scales and is shown in Figure 13. A scale factor of 2 was used to convert the surface error to wavefront error.
Fig. 12. OPD map of the axial gravity support print-through (8nm RMS surface, or 17nm RMS OPD) and the square root of the structure function, D(r), calculated from the OPD map, where: D(r) = < [Φ(x + r) - Φ(x)]2 >.
Fig. 13. OPD map of the lateral gravity support print-through (10nm RMS surface, or 20nm RMS OPD) and the square root of the structure function, D(r), calculated from the OPD map, where: D(r) = < [Φ(x + r) - Φ(x)]2 >.
7. SUMMARY Extensive finite element analyses and optical calculations were performed to optimize a secondary mirror support system for the LSST. In the optimization process, iterative parametric analyses were utilized to achieve a minimum global surface deformation (surface RMS error). The optimized axial support system achieved an optical surface error RMS of 10nm. The axial support system used 72 active axial support actuators, arranged in a three concentric ring pattern. The lateral support system was optimized to achieve an optical surface RMS error of 8nm. The lateral support system used six actively control lateral supports, equally spaced along the periphery of the mirror mounted tangentially at the midplane. The optical surface deformations for various Zenith angles were evaluated by combining cases of the effects from axial and lateral gravities. The results showed that the current LSST secondary mirror support system adequately meets the optical performance goal of 20nm surface RMS and satisfies the M2 surface figure accuracy requirement defined in terms of a Structure Function. Sensitivity analyses were conducted with several sample cases to quantify the optical surface deformations affected by uncertainties in design and potential errors involved in polishing, assembly and system integrations. Performances of the M2 aO system were demonstrated using the first 10 natural mirror bending modes. A few sensitivity cases, corrected by the M2 aO, are also implemented. The results demonstrated that the M2 aO system is capable of adequately correcting the optical figure errors. Integrated FE models with the mirror, supports, and mirror cell structure need to be established
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for further optimizations to refine design parameters of the mirror cell and support systems. A high fidelity finite element model will be required to evaluate more extensive sensitivity cases, structural interaction effects, thermal mismatches, or other opto-mechanical effects. This FE model may include features of support pads, mounting blocks, linkage, and other detail hardware parts which may contribute to mechanical and optical performance degradation.
ACKNOWLEDGMENTS This research was carried out at the National Optical Astronomy Observatory, and was sponsored in part by the LSST. The authors gratefully acknowledge the support of the LSST partner institutions. LSST is a public-private partnership. Funding for the design and development activity comes from the National Science Foundation, private donations, grants to universities, and in-kind support at Department of Energy laboratories and other LSSTC Institutional Members. Work described in this paper is supported by the National Science Foundation under Scientific Program Order No. 9 (AST0551161) and Scientific Program Order No. 1 (AST-0244680) through Cooperative Agreement AST-0132798. The authors would like to acknowledge Jacques Sebag and William Gressler of the LSST for their review and support.
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[6] [7] [8]
Cho, M. K. and Price, R. S., “Optimization of Support Point Locations and Force Levels of the Primary Mirror Support System,” RPT-O-G0017, Gemini Technical Report, November, (1993). Cho, M. K. Price, R. S., Moon, I. K, Optimization of the ATST Primary Mirror Support System, Proc. SPIE 6263, May (2006). Cho, M. K. and Roberts, J. L., “Response of the Primary Mirror to Support System Errors,” RPT-O-G0021, Gemini Technical Report, November, (1993) . Sebag, J., “LSST Telescope Error Budgets,” LSST ARCHIVE DOCUMENT #3535, (2/2007). Blanco, D., Cho, M., Daggert, L., Daly, P., DeVries, J., Elias, J., Fitz-Patrick, B., Hileman, E., Hunten, M., Liang, M., Nickerson, M., Pearson, E., Rosin, D., Sirota, M. and Stepp, L., “Control and support of 4-meter class secondary and tertiary mirrors for the Thirty Meter Telescope,” Opt mechanical Technologies for Astronomy, ed. E. AtadEttedgui, J. Antebi and D. Lemke, SPIE Proc. 6273, TMT.OPT.JOU.06.002, (2006). Cho, M. K, “Performance Prediction of TMT Secondary Mirror Support System,” Ritchey-Chrétien design, SPIE 7018-65, Marseille, France, (2008). Neill, D. R. and Krabbendam, V. “Active tangent link system for transverse support of large thin meniscus mirrors,” Proc. SPIE 6665, (2007). Cho, M. K. and Richard, R. M., “PCFRINGE Program – Optical Performance Analysis using Structural Deflections and Optical Test Data,” Version 3.5, the Optical Sciences Center, University of Arizona, (1990).
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