TMT Metrology study for M2 and M3

Transcription

1 Consultants for Optical Metrology Tucson, Arizona 1 Introduction Requirements for optical test hardware Subaperture interferometry with stitching software Fizeau interferometry Subaperture stitching Fabrication tests for secondary mirror Fizeau test Stitching of interferometry data Profilometry Fabrication tests tertiary mirror Fizeau test of M Stitching of interferometry data Scanning pentaprism measurement Test of secondary mirror in its support cell Fizeau test Profilometry Test of tertiary mirror in its support cell Fizeau test Scanning pentaprism measurement Ritchey-Common Conclusion References

2 1 Introduction A preliminary study was commissioned by TMT to investigate possibilities for metrology for the 3.1-m diameter convex aspheric secondary mirror and for the 2.5 x 3.5-m elliptically shaped flat tertiary mirror. The measurement systems take into consideration the various requirements for the metrology: Need to provide data to guide grinding and polishing in the optics shop Evaluate performance of optics in their as-used orientation Evaluate performance of the active support cells For both the secondary (M2) and tertiary (M3) mirrors, subaperture Fizeau interferometry was studied with1-m nominal subaperture diameter. The large scale surface errors can be measured by stitching the data from the subaperture measurements. This method provides decreasing sensitivity for the larger scale errors, but the requirements for these low order shapes will be relaxed because they will be adjusted in the telescope using the active supports based on wavefront measurement. The performance of these tests in the presence of alignment errors and noise was simulated directly to provide estimates of performance. The subaperture size of 1 meter was chosen based on estimates of cost and performance, but no trade study was performed. Larger subapertures are always better for the optical test. Fewer measurements are needed to cover the optic and fewer degrees of freedom need to be determined by the stitching. However, difficulty manufacturing, handling, and illuminating the Fizeau reference optic increases sharply as it becomes large than about 1 meter. The performance for measuring with 1-m subapertures is very good. There are two competing effects that nearly cancel. The measurement errors from the subapertures tend to accumulate as the overall surface map is determined. And the oversampling provided tends to average out some of the errors. These two effects nearly balance for the 3-m optics measured by 1-m subapertures as is proposed. It is always important to corroborate measurements for large optics. The shape error for both mirrors could be measured with a custom coordinate measuring machine called a swing arm profilometer. The geometry of this machine allows measurement of low order errors in the 3-m class optics to < 0.1 µm rms. In addition, the flat mirror can be measured using a scanning autocollimation test, where slope variations are measured by scanning the measurement beam from an autocollimator with a moving pentaprism. This measurement can provide measurement of astigmatism and power with ~30 nm rms accuracy. These measurements are most easily performed in the shop with the TMT mirrors face up. Each measurement can also be performed at other orientations, but with difficulty. In order to validate by the polished surface and the telescope support, the following methodology for mirror metrology is assumed: 1. High resolution subaperture Fizeau measurement of the entire surface in the polishing shop, with the optic face up. These measurements can be made on a suitably design support that needs only to work in this orientation 2

3 2. Corroboration of the surface in the polishing shop with a scanning test, such as profilometer or scanning pentaprism. 3. Evaluation of the surface in the telescope cell in the as-used orientation. A high resolution full aperture map is not required here. There are two concerns that can each be addressed separately: a. Local deformation due to high stress. This can be measured using a subaperture test or a scanning test without the requirement to stitch the information into a full surface b. Low order shape errors. These can be adjusted in the telescope using the support actuators. It is useful to measure the low order modes for the mirrors in their as-used orientation to verify that the actuator forces are within acceptable limits. The surface measurement accuracy is relaxed for these controllable modes because the final adjustments will be made using wavefront data from the telescope. 2 Requirements for optical test hardware The details of the flowdown of telescope performance requirements to the optical test hardware have not yet been defined. However, a few simple facts will guide this: The lowest order shapes will be controlled by actuators based on wavefront data. Such shape errors, whether they come from the measurement or any other source, use some of the actuator force budget, but do not cause any error in the optical aspects of the telescope. The telescope will be limited by the atmosphere. As long as the surfaces are significantly better than the atmosphere at all spatial scales, they do not affect the optical performance. This is true whether the system uses adaptive optics or not. The budgeting and specification of such errors for the different surfaces is not addressed here. Optical shop tests must measure all of the necessary spatial scales to allow finishing of the surface It is highly advantageous to validate the performance of the mirror in its telescope cell, in the as-used orientation, before the hardware is shipped to the telescope. The secondary and tertiary mirrors are both expected to use thin, solid glass substrates and to be supported by an array of force actuators. The forces will be set to give the desired shape, as measured using wavefront data from the telescope. The control of the mirrors is optimally performed using the control modes, which include the mirror rigidity and the actuator layout. The control modes can be determined numerically from the influence functions of the individual actuators. Typically a finite element model is perturbed by applying a unit force at the each of the actuator locations. This set of influence functions is processed using singular value decomposition to give a different set of control modes which are orthogonal and which are listed in order of increasing force/displacement. The lowest of these modes will be controlled in the telescope without any coupling into other modes. As such, any error that can be represented by a combination of these control modes can be fully corrected. 3

4 The higher order modes cannot be corrected as well because they are stiffer, requiring more force, and because they do not follow the finite element predictions perfectly. The increased force and the imperfect modeling allow parasitic effects to occur which limit performance. The adjustment of such a high order mode will always introduce some other shape errors which can affect optical performance. The analysis of the test optics was performed here to estimate how the measurement errors will affect uncertainty in the surface as represented by a linear combination of the lowest modes and higher order residual after fitting to the modes. Zernike polynomials were assumed for the secondary mirror modes. Actual bending modes were used for the tertiary mirror. 3 Subaperture interferometry with stitching software 3.1 Fizeau interferometry Both the secondary and tertiary mirrors can be measured with subaperture Fizeau interferometry. The data from the subaperture measurements can be stitched together to give full surface maps. There are two key technical issues with this method: efficiency and accuracy. A nominal subaperture size of 1-m was chosen for both measurements to achieve good technical performance at acceptable cost, although no trade study was explicitly performed. The 1 meter size is feasible, sensible, and involves low risk. Preliminary designs for both test configurations are given below, but they include some common elements, shown in Figure 1 Illumination Optic Polarization Instantaneous interferometer LHC (B) RHC (A) Mirror Under Test 1 m reference flat Spots from the reference surface Alignment mode A B A B Spots from the test surface Software screen Figure 1. Layout of Fizeau test for subaperture interferometric measurement 4

5 The Fizeau interferometer uses reflection from a reference surface that is held only a few mm from the surface being measured. The reflection from the reference surface is interfered with light reflected from the surface under test, and the difference between the two reflected wavefronts is used for the measurement. Phase shifting interferometry can be performed by either pushing the reference plate with PZTs or using a simultaneous phase shifting interferometer that uses polarization to separate the reflections. The simultaneous phase shifting interferometer uses two beams with orthogonal polarization with some tilt difference. These two are each reflected from the reference surface and from the surface under test. By setting the relative tilts for the optics, one can select the reference and test beam with appropriate orthogonal polarizations. Then the phase shifting is performed using polarization optics to create 3 or 4 interferograms that have 90 relative phase shift and can be read out simultaneously. The wavefront phase is then calculated from the set of interferograms. This freezes out the effect of vibration. Commercially available systems that can be used in this mode are produced by ESDI and 4D Technologies. As long as the physical separation is small, then the effects of imperfect wavefronts due to limitations in the illumination optics and the test plate glass do not significantly affect the measurement. The analysis for such errors is given by Burge 1. Also, the effect of birefringence in the test plate has been shown to have negligible effect for the simultaneous phase shifting interferometers as long as circular polarization is used. 2 The use of a standard vibration insensitive Fizeau interferometer with the remote test plate was demonstrated for 1-m aperture measurements. 3 This system used the H-1000 instantaneous Fizeau system from ESDI and included an off-axis parabola for illumination. Measurements achieve a noise level of 3 nm rms over the 1-m aperture. 3.2 Subaperture stitching Software is used to combine, or stitch, the data from the subapertures to create a full aperture map. Stitching is typically a least-squares type of process where the relative alignment degrees of freedom are determined based on the consistency of the different measurements. There are two different types of stitching algorithms those that concentrate on the overlap regions 4,5 and those that solve for more global consistency. 6 They have both been proven to work to the noise limit. For this project, we have written software that solves for the tip, tilt, and piston of each map in order to optimize the data consistency in the overlap regions. For the case of the secondary mirror, power for each subaperture was determined as well. The software was tested with some examples and shown to be quite robust. The noise sensitivity for the individual measurements is presented separately for the cases of the secondary mirror and tertiary mirror. This software was never carefully optimized for this application. It would not be difficult to develop software that has slightly better noise rejection, but the gains will be marginal. 5

6 The basic algorithm was taken from a paper by Otsubo et al. 4 The software was written in MatLab whose powerful matrix manipulation functions are taken advantage of. Little effort was devoted to better memory management or improving calculation speed. Test cases were used to verify the accuracy of the software. The algorithm as applied to the TMT mirrors is briefly described here: A full-aperture map and valid data mask was created using synthetic data. Each sub-aperture map has the same matrix size and valid data was masked by its own mask and the full-aperture mask. Sub-aperture masks determined the overlapped area. In the overlapped area, average was taken from the fitted sub-aperture data, which was done by adding up the masks to determine how many sub-apertures cover a specific pixel. The least squares algorithm described above was used to solve for the alignment degrees of freedom to minimize the difference between the measurements in the overlap region. The stitched full-aperture map can be decomposed into Zernike polynomials and the residual is calculated after Zernike decomposition. Extensive checks have been done to make sure the stitching works properly: Full-aperture maps defined by combinations of Zernikes divided into subapertures without adding tilts and piston to each sub-aperture. Same as above but with random tilts and piston added to each sub-aperture. The simulations for the tests included noise in the subaperture measurements. Random noise was modeled with as 3 nm rms surface error with correlation length equal to about1/4 the diameter of the 1-m subaperture. A typical plot of the noise is shown in Figure 2. Figure 2. Typical noise for the subaperture measurements of 3 nm rms with correlation length about 0.25 meter 6

7 4 Fabrication tests for secondary mirror 4.1 Fizeau test Optical figure Clear Aperture 3.046m (FOV=15 arcmin, EL=3.5m) Vertex Radius of Curvature m (convex) Optical Surface Shape Hyperbola, K = The preliminary design for a subaperture Fizeau test of the secondary mirror was developed. Since one meter subapertures were used, two different test plates are required, with their centers 45 and 115 cm from the center of the secondary mirror. Figure 3. Distribution of sub-aperture measurements for SM. There are total 18 measurements in two rings. 7

8 The convex secondary mirror is highly aspheric, so the Fizeau reference plates must be made with matching concave aspheric surfaces. These can be individually manufactured as off-axis aspheres and can be measured from center of curvature using an interferometer with computer generated hologram. The hologram itself can be used for the alignment of the test. 7 The aspheric departure for the concave test plates are shown in Figure 4. Test plate 1. 1-m diameter 45 cm off center TMT secondary mirror 191 µm P-V aspheric departure (34 µm rms) Test plate 2. 1-m diameter 115 cm off center TMT secondary mirror 766 µm P-V aspheric departure (155 µm rms) Figure 4. Aspheric departure for the two 1-m Fizeau test plates needed for measuring the TMT secondary mirror. The design of the illumination system was not developed here. This system could use refractive or reflective optics, and the details would be driven by a combination of the space of the optics shop and available optics. The quality of the illumination optics does not need to be high, as these errors do not significantly affect the secondary measurement. 4.2 Stitching of interferometry data A complete simulation was performed for the measurement of the secondary mirror using 18 subapertures. The tip, tilt, piston, and power for the subapertures were fit using methods described above. In addition, there are two alignment modes per measurement that must be fit radial motion of the test plate and clocking of the test plate about its local axis. The effects of these degrees of freedom are shown below in Table 1. In this analysis, the alignment errors for radial position and clocking are assumed to be constant for the test i.e. they have the same value for all measurements. The tip, tilt, piston, and power are assumed to be independent for all 18 subaperture measurements. 8

9 Table 1. The apparent wavefront errors when the reference mirrors have alignment errors relative to SM. Radial shift 1mm Clocking 0.05 degree Test plate 1 Test plate 2 Zernike standard coefficients (rms nm): Z4 (power): -173 Z6 (0 astigmatism): 122 Z7 (90 coma): 38 Zernike standard coefficients (rms nm): Z5 (45 astigmatism): -48 Z8 (0 coma): 15 Zernike standard coefficients (rms nm): Z4 (power): -407 Z6 (0 astigmatism): 278 Z7 (90 coma): 28 Zernike standard coefficients (rms nm): Z5 (45 astigmatism): -298 Z8 (0 coma): 36 As an example, synthetic data were created for the secondary mirror and random alignment errors were included for the subaperture measurements. For this case, in the absence of noise, the stitching is perfect. This is shown in Table 2. 9

10 Table 2. Simulation for stitching 18 maps together, including alignment errors, to recreate the secondary mirror surface. Given SM full Input full aperture map: 0.747μm rms (synthetic data) aperture map: SA measurements: Ref radial shift and clocking: Ref1: 0.25mm, Ref2: -0.25mm, Random power of 1μm sigma added to each SA measurements Stitching result: reference alignment error REMOVED SA measurement 1: 0.460μm rms Stitched map: 0.747μm rms SA measurement 2: 0.737μm rms Fitted map: 0.747μm rms SA measurement 3: 0.604μm rms Residual: 0.001μm rms Noise was added to this simulation with as 3 nm rms random errors with the form as shown in Figure 2. The noise for each subaperture measurement was independent. The surface was then reconstructed from the subaperture data and evaluated. An example from one of the Monte Carlo trials is shown in Figure 5. In the simulations, we used 100x100 pixels for each sub-aperture and 300x300 pixels for the full M2 aperture. The actual matrix for the full-aperture map is 400x400 pixels. We ran 50 Monte-Carlo simulations of stitching sub-aperture measurements with 3nm correlated noise. We fitted the stitched maps to the 11 modes (equivalent to 2 Zernike astigmatisms, 2 trefoils, 1 focus, 2 quadfoils, 2 comas and 2 pentafoils, see Figure 6) and obtained statistics for the fitted coefficients of the modes as listed in Table 3. Because the axial alignment of the reference mirrors affects the power measurement of subapertures, there is uncertainty in determining the power of the stitched surface map such 10

11 that the power is indeterminate. The real power in the surface would be measured by controlling the spacing from the test plate to the mirror for one of the measurements. Stitched map: 8.2nm rms Fitted: rms 6.4nm Residual: rms 3nm Figure 5. Example for one of the Monte Carlo runs which includes the effect of 3 nm rms noise per measurement. Figure 6. List of the modes of SM which are similar to lower order Zernikes. Table 3. The uncertainties of the fitted coefficients of the modes and the residual of the stitched map obtained from 50 M-C simulations given 3 nm rms noise for each of the 18 measurements. Mirror Mode Uncertainty (nm) (power) Indeterminate RSS 3.7 Residual mean

12 To check each M2 mode s fitablity in presence of noise, we generate sub-aperture measurements with given full-aperture map of a certain mode, plus 3nm correlated noise in each sub-aperture, and then stitch SA measurements to recover the given FA map. The result is listed in Table 4. These simulations show that each mode can be recovered from stitching of SA measurements with high accuracy. Table 4. Fitablity of each mode of M2 in presence of noise (3nm rms in sub-aperture measurements). Mirror Mode Input Fitted (nm rms) (nm rms) 1 (astigmatism) (astigmatism) (trefoil) (trefoil) (power) NA NA NA 6 (quadfoil) (quadfoil) (coma) (coma) (pentafoil) (pentafoil) Residual (nm rms) Summary: Our analysis and simulation show that noises in the sub-aperture Fizeau measurements will cause a small amount of lower order errors which are well within the correctable range. Other errors, such as registration errors between sub-apertures, can be estimated. 4.3 Profilometry A second test for the TMT secondary mirror uses a coordinate measuring machine called the swing arm profilometer 8, which utilizes advantageous geometry for measuring the optical surface. The swing-arm profilometer uses a displacement probe at the end of an arm to make mechanical measurements of the optical surface. The geometry for this test is shown in Figure 7. This geometry can be used to make scans across the diameter of the surface. Multiple scans can be combined by using data from circumferential scans, made by rotating the substrate under the probe. 12

13 Surface figure in um EXHIBIT D - TMT.OPT.TEC REL02 rotary stage arm probe and alignment stages secondary mirror probe trajectory axis of rotation optical axis center of curvature Figure 7. The swing arm profilometer measures along arcs on the surface by pivoting an arm about an axis that goes through the center of curvature of the mirror being measured. A probe at the end stays normal to the surface and measures only the aspheric departure. This geometry works equally well for concave and convex measurements, such as the measurement of the 1.7- m convex mirror shown here. For the swingarm profilometer, the probe is mounted at the end of an arm that swings across the test optic such that the axis of rotation goes through the center of curvature of the optic. The arc defined by the probe tip trajectory lies on a spherical surface defined by this center. (This geometry is used for generating spherical surfaces using cup wheels.) For measuring the aspheric optics, the probe, which is aligned so its travel is in the direction normal to the optical surface, reads only the surface departure from spherical. The University of Arizona has used the swingarm profilometer to provide surface measurements with accuracy of ~0.03 µm rms for meter sized convex aspheres without calibrating the bearing. Figure 7 shows a comparison of a profilometer measurement with results from a full aperture Fizeau test with CGH. Swing Arm Profilometer Data Full scan data Normalized radial position 0: Magellan f a aa scan rms: figure rms: Noise in a single scan of 18 nm rms. This is measured by subtracting the average of many scans from a single scan Figure 8. The swing arm profilometer achieves excellent accuracy without even calibrating the air bearing. The comparison between profilometer measurements and interferometry is excellent. Noise of 18 nm rms for a single scan of a 1.7-m convex asphere is shown. 13

14 The accuracy of making full surface measurements for the TMT secondary was modeled for the swing arm profilometer assuming measurement noise of 0.1 µm rms per point. A Matlab program was written to process the measurement data. The data is fit to low order Zernikes which are close to the M2 s natural modes. The software was extensively checked to make sure it processes data correctly (see Table 5). Table 5. An example of SAP data processing. Surface map of lower order Zernike combinations is correctly fitted from scan data. Fitted map and scans (8 radial scans and 2 A typical scan (measured and fitted plot are circumferential scans): perfectly overlapped): Zernikes Input coefficient Fitted coefficient Z4-1.6 µm µm Z5 1.2 µm µm Z8-1 µm µm Z µm µm A Monte-Carlo simulation of the swing arm profilometer was performed assuming 0.1 µm uncertainty at each measurement point. With 6 scans (4 radial scans and 2 circumferential scans) and 200 points per scan, the total uncertainty in the lowest 11 modes is less than 15nm. Table 6 shows the different scan configurations that were simulated. The larger number of scans improves performance in two ways: The data density is higher, so the sampling is better More points are measured, so the noise is reduced by averaging The performance is plotted as functions of the number of modes fit and the number of scans made in Figure 9. For the case of measuring all 11 modes, ten scans (8 diameters and 2 circumferential) can determine the surface to 13 nm rms. These numbers are achievable for a system with performance similar to that of the University of Arizona machine. This was a custom built machine which could be duplicated. Clearly the ultimate performance for any machine will depend on the quality of the engineering and the components. 14

15 Table 6. Plots of scan configurations: 6 scans (4 radial scans + circumferential scans): 8 scans (6 radial scans + circumferential scans): 10 scans (8 radial scans + circumferential scans): 12 scans (10 radial scans + circumferential scans): Figure 9. Monte Carlo analysis of M2 measurements with a swing arm profilometer with 0.1 µm noise per measurement point. Different scan configurations are included here. 15

16 It is difficult to measure the radius of curvature using the swing arm profilometer. A misalignment of the arm appears as power in the scan. The radius of curvature corroboration would have to be performed using a different test. 5 Fabrication tests tertiary mirror The tertiary mirror is a 2.5 x 3.5 meter elliptically shaped turning flat. The beam footprint from a single field point is smaller, but the errors are evaluated here over the full aperture. Two different measurement techniques are proposed here: Subaperture Fizeau interferometry + stitching Scanning pentaprism slope measurements The performance of each of these tests is excellent. The elliptical shape of the tertiary mirror introduces some complication. The secondary tests were evaluated in terms of the Zernikes, which nearly match the bending modes. The tertiary modes are much different. These modes are giving in Table 7. 16

17 Table 7. List of M3 s modes and the corresponding surface figure errors and slope errors. RMS surface: 1μm RMS slope: 2.05μrad RMS surface: 1μm RMS slope: 2.25μrad RMS surface: 1μm RMS slope: 3.18μrad RMS surface: 1μm RMS slope: 2.77μrad RMS surface: 1μm RMS slope: 2.86μrad RMS surface: 1μm RMS slope: 3.84μrad 17

18 Table 7 (cont.) List of M3 s modes and the corresponding surface figure errors and slope errors. RMS surface: 1μm RMS slope: 3.43μrad RMS surface: 1μm RMS slope: 3.46μrad RMS surface: 1μm RMS slope: 4.45μrad RMS surface: 1μm RMS slope: 4.53μrad 5.1 Fizeau test of M3 Like the secondary mirror, the flat can be measured using a 1-m Fizeau test, processed with stitching software. The measurement of the flat is significantly easier than that of the secondary for the following reasons: Only one test plate is required. The M2 test required two aspheric test plates The two Fizeau test plates for the M2 test are not only different, they must be supported at different tilt angles, following the curved surface. The reference for the flat is always horizontal. The illumination light is collimated. The reference surface is flat, and easier to make There is no sensitivity to radial position or clocking. In fact the reference flat can be measured by changing these alignment parameters and separating features that stay with the mirror from those that move with the test plate. 9 There is no sensitivity to axial position. The test for M2 accommodated an uncertainty in power for each of the subapertures. 18

19 5.2 Stitching of interferometry data Like M2, M3 can be measured in sub-aperture and the full-aperture surface map can be obtained from stitching sub-aperture measurements together. We studied the case of using 1m diameter reference mirror for sub-aperture measurements. A total of 19 measurements are needed to cover the whole M2 surface. Figure 10 shows the subaperture distribution assumed for this analysis. This geometry seems sensible, but it is probably not optimal. No trade studies were performed to optimize this. Figure 10. Sub-aperture distributions for the M3 test. Reference mirror is 1m in diameter. Like the stitching simulations for the secondary mirror, we assumed 3 nm rms noise per measurement with 25 cm nominal correlation length. An example of the stitching simulation is shown in Figure 11. An additional 50 Monte- Carlo simulations were run and the full-aperture map was stitched from the 19 subaperture measurements each of which has correlated noise of 3nm. In the simulations, we used 100x100 pixels for each sub-aperture and 250x350 pixels for the full M3 aperture. The actual matrix for the full-aperture map is 350x350 pixels. The stitched maps were fitted to the 10 modes of M3 shown in Table 7. The uncertainties of the fitted coefficients of the 10 modes are listed in Table 8. 19

20 M3 SA example: 3nm M3 stitched: 6.4nm rms M3 fitted: 3.8nm rms M3 residual: 5.2nm rms Figure 11. An example of stitched map of sub-aperture measurements with 3nm rms noise. After removing tilt, the residual noise is about 3 nm rms. Table 8. The uncertainties of the fitted coefficients of the modes and the residual of the stitched map obtained from 50 M-C simulations. Mirror Mode Uncertainty (nm) RSS 4.6 Residual 3 20

21 Summary: TM can be measured in 1m sub-apertures of 1m diameter. Our simulation shows the measurement noise causes little low order aberrations easily corrected by active optics. High order residual errors are reasonable. 5.3 Scanning pentaprism measurement The shape of the flat mirror can be sampled with a scanning pentaprism 10 system, such as that shown in Figure 12. This technique scans the beam from an autocollimator across the pupil using a pentaprism. Slope variations are measured and integrated to give the profile along the line of the measurement. Multiple scans can be made to build up a full surface. This test has been proven at the University of Arizona where a 1.6-m diameter flat mirror was measured to 12 nm rms. 11 This simple system achieves such high accuracy by taking advantage of a special property of the pentaprism the light is deviated by an angle, nominally 90, which is completely insensitive to tilt of the prism itself. Thus no additional reference is required. Scanning Pentaprism Test Two pentaprisms are co-aligned to a high resolution autocollimator The beam is deviated by 90 to the test surface Any additional deflection in the return beam is a direct measure of surface slope changes Electronically controlled shutters are used to select the reference path or the test path One prism remains fixed (reference) while the other scans across the mirror Feedback mirror UDT (Alignment AC) Fixed prism (reference ) Coupling wedge Scanning prism ELCOMAT (Measuring AC) Autocollimator system Shutters Mechanical A second autocollimator (UDT) supports maintains angular alignment of the scanning prism through an active feedback control Figure 11. Scanning pentaprism measurement for large flat mirrors. 1 21

22 Figure 12. Data from University of Arizona scanning pentaprism measurement of 1.6-m flat. 9 It is difficult to determine the absolute power with the Fizeau test. (We rely on knowledge of flatness for the reference surface.) We evaluate the performance of the scanning pentaprism system for determining the lowest order shape errors in the flat, power and astigmatism, shown in Tables 9 and 10. Measurement of M3 with the scanning pentaprism system was simulated, including the data reduction. Results from a test case are shown in Tables 9 and 10. Table 9. Test results for the PPT data processing software. Test Case First 3 Input coefficient Fitted modes (RMS μm) Output coefficients (RMS μm) 1 Mode Mode Mode Mode Mode Mode Residual (RMS μrad)

23 Table 10. List of fitted surface map and scan plots. Fitted surface map + scans Scan 1, 0 degree scan Scan 2, 45 degree scan Scan 3, 90 degree scan Scan 4, 135 degree scan A Monte-Carlo simulation was run to see the uncertainty in determining the first three modes of the M3: mostly astigmatism and power. The pentaprism test was modeled with a random Gaussian distribution with standard deviation of 0.2 µrad. The data was then fit with the MatLab program to the first three modes. 100 Monte-Carlo runs were performed and statistics of the fitted coefficients of the three modes and the residual were calculated. Table 11 lists the results. 23

24 Table 11. Results of 100 Monte-Carlo simulations of the penta-prism measurements of M3. 0.2μrad measurement uncertainty is assumed. Sigma of mode coefficients (nm) Mode 1 20 Mode 2 24 Mode 3 9 Mean of the residual slope error (μrad) 0.19 Table 12. Plots of a typical Monte-Carlo run output. Fitted surface map + scans Scan 1, 0 degree scan Scan 2, 45 degree scan Scan 3, 90 degree scan Scan 4, 135 degree scan 24

25 The other modes have increased slope variation per µm of surface height, so they are measured more accurately than power and astigmatism, but only if enough scans are made to fully sample them. With 0.2 µrad rms noise and multiple scans, each mode can be determined to < 20 nm rms. 6 Test of secondary mirror in its support cell 6.1 Fizeau test There is no reason why the secondary test cannot be performed wit the optical surface face down. This will require mechanical modifications and the shape of the test plate must be measured in this orientation. The main difficulties with this test configuration will be: Getting access to the Fizeau test. Either the secondary mirror will need to be mounted on a platform or a large fold mirror will be needed. The test plate support must work and must be calibrated in this orientation To measure the full mirror, the mirror must be rotated. This may be difficult face down However, if the goal of the test is to sample some regions on the surface to look for small scale errors due to mounting problems, many of the difficulties go away. 6.2 Profilometry The swing arm profilometer can measure the convex surface as it hangs from its support. The principal difficulty will be mounting the secondary so that it is stable. Again, it is desirable to rotate the mirror, which may be difficult in this orientation. 7 Test of tertiary mirror in its support cell 7.1 Fizeau test The tertiary mirror is used at nominally 45 from zenith pointing. It would be difficult to support the Fizeau reference in this orientation. 7.2 Scanning pentaprism measurement The scanning pentaprism system can easily accommodate this geometry. One would need only to support the ends of the rails at appropriate locations to give the desired scans. It would be difficult to measure the full surface this way, but it would test the low order modes to ~20 nm rms and could measure for local bumps of a few nm rms. 7.3 Ritchey-Common It may be easier to perform the Ritchey-Common test for this configuration. A spherical mirror could be mounted above the tertiary and measured using the geometry shown in 25

26 Figure 13. Like the Fizeau test, this can be performed as a sub-aperture measurement and the data can be stitched if desired. Reference mirror (spherical) θ Flat surface under test Figure 13. Layout for the Ritchie Common test of M3 in its support. This test can be performed as a subaperture test. 8 Conclusion The tests for TMT M2 and M3 are challenging because of the large size, but they do not require new technology. The methods already proven at the University of Arizona will work for measuring these mirrors 26

27 References 1 J. H. Burge, Fizeau interferometry for large convex surfaces, in Optical Manufacturing and Testing, V. J. Doherty and H. P. Stahl, Editors, Proc. SPIE 2536, (1995). 2 C. Zhao, D. Kang, and J. Burge, Effects of birefringence on Fizeau interferometry that uses a polarization phase-shifting technique, Applied Optics 44, (2005). 3 C. Zhao, R.A. Sprowl, M. Bray, J.H. Burge, Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique, in Interferometry XIII, edited by E. Novak, W. Osten, C. Gorecki, Proc. of SPIE 6293, 62930K, (2006). 4 M. Otsubo, K. Okada, J Tsujiuchi, Measurement of large plane surface shapes by connecting small aperture interferograms, Optical Engineering, 33 (2), , (1994). 5 M. Bray, Stitching interferometer for large plano optics using a standard interferometer, in Optical Manufacturing and Testing II, ed. H. P. Stahl, Proc. SPIE 3134, (1997). 6 P. Su, J.H. Burge, R. Sprowl, J. Sasian, Maximum Likelihood Estimation as a General Method of Combining Sub-Aperture Data for Interferometric Testing, in International Optical Design Conference 2006, Proc. SPIE 6342 (2006). 7 J. H. Burge, R. Zehnder, C. Zhao, Optical alignment with computer-generated holograms, in Optical System Alignment and Tolerancing, ed. by J. Sasian and M. Ruda, Proc. SPIE 6676, (2007). 8 D. S. Anderson and J. H. Burge, Swing arm profilometry of aspherics, in Optical Manufacturing and Testing, V. J. Doherty and H. P. Stahl, Editors, Proc. SPIE 2536, (1995). 9 J. Yellowhair, P. Su, M. Novak, J. Burge, Fabrication and testing of large flats, in Optical Manufacturing and Testing VII, ed. by J. Burge, O. Faehnle, and R. Williamson, Proc. SPIE 6671, (2007). 10 P. C. V. Mallik, C. Zhao, J. H. Burge, Measurement of a 2-meter flat using a pentaprism scanning system, Optical Engineering 46 (2), (2007). 11 J. Yellowhair, J. H. Burge, Analysis of a scanning pentaprism system for measurements of large flat mirrors, accepted for publication in App. Opt. (2007). 27