Abstract

The James Webb Space Telescope (JWST) Optical Telescope Element is a three mirror anastigmat consisting of a 6.5 m segmented primary mirror (PM), a secondary mirror, and a tertiary mirror. The primary mirror comprises 18 individual hexagonal segments. The telescope and instruments will be assembled at Goddard Space Flight Center (GSFC) to build the Optical Telescope Element-Integrated Science Instrument Module (OTIS). While at GSFC, the OTIS will go through a series of environmental tests. In these tests the OTIS will be exposed to launch level acoustics and vibrations. To assure that OTIS’s performance has not changed due to these environmental tests, the assembly will be tested optically at the center of curvature of the PM. A high-speed interferometer has been designed and built to characterize both static and dynamic changes due to environmental exposure. This paper describes the details of these measurement techniques. To validate and develop the techniques that will be used on OTIS assembly two spare JWST PM segments were measured and the results presented here.

© 2015 Optical Society of America

1. Introduction

The James Webb Space Telescope (JWST) primary mirror segments will be tested statically and dynamically by measuring the surface of each segment using a computer-generated hologram (CGH) at the center of curvature (CoC) of the primary mirror. The main instrument used to perform both static and dynamic measurements is a custom-made high-speed interferometer (HSI) designed and fabricated by 4D Technology to the specifications of the Goddard Space Flight Center [1]. To assure integration times are kept short, this interferometer is equipped with a high-power 25 mW He–Ne laser with path matching capability. The interferometer uses instantaneous phase-shifting techniques to capture all interferogram information in a single camera frame. The heart of the system is a pixel-wise phase-shifting element which allows each pixel to have a unique phase-shift. The HSI camera is a high-speed CMOS. The frame rate is limited by the size of the camera detector region of interest selected for data transfer. As an example, the interferometer can take data at 1 KHz for a 720×720 pixel format. A higher speed of 2.25 KHz can be achieved but the detector size must be reduced to 400×400 pixels.

The static surface measurements will measure changes to the segment surfaces before and after environmental vibration and acoustic tests. The static surface measurements are fit to Zernike polynomials to assess changes to specific aberrations like astigmatism. As has been modeled and demonstrated during the mirror manufacturing process, astigmatism is the dominant aberration expected from adverse changes. Unfortunately, it is also the dominant aberration from optical test misalignment. Therefore, great care must be taken to repeat the alignment of the test setup. The noise level for the static measurements is set by the environmental repeatability of the clean room. This includes temperature, both absolute and induced mirror gradients, air turbulence, and vibration. Testing on JWST flight spares has been performed to measure the noise level in the clean room for the static measurements. The rms repeatability noise floor for the surface figure is approximately 10 nm and for astigmatism is 20 nm. This is achieved under tight temperature and air flow control of the clean room environment.

The optical dynamics tests are similar to the static one in that the surface figure of the mirror is measured using the same interferometer and CGH test setup. However, rather than a single snapshot of the surface a series of figure measurements are taken at a high rate of speed exceeding 1 KHz. This type of measurement is typically performed for a duration of 10 s such that at 1 KHz, 10,000 individual surface figure measurements are taken. The movement of the mirror is compared to the input vibration levels by stimulating the telescope assembly at several places and measuring the dynamic behavior of the primary mirror segments at the CoC. The stimuli are applied at the hard points of the telescope assembly. Vibration levels of the mirror segments are in micrometers and the frequencies are spread over 3–250 Hz. Both random and sine vibration stimuli are measured. The input forces are synced to the figure measurements by using a trigger signal to initiate the test. Power spectral densities (PSDs), spatial modes, and phase and modulation transfer functions are computed from the measurements. PSDs per pixel are measured by temporal phase unwrapping individual pixels and then performing a fast Fourier transform (FFT) of the displacement in time. This is also normalized to PSDs of the forcing function of the stimuli to create transfer functions. Spatial mirror modes are computed by creating surfaces at each point in time of the temporally phase unwrapped pixels. These surfaces are then fitted to Zernike polynomials. An FFT of each Zernike coefficient is then computed and normalized at each frequency to the amplitude and phase of the force function. These normalized amplitudes and phases will be compared before and after environmental tests. Individual modes, both rigid body and mirror deformation, are observed using these same techniques. These modes include absolute piston and also deformation about the points of contact between the back structure and the mirror due to mirror inertia as they move as a rigid body [2].

2. Metrology Validation

The primary mirror segments will be tested statically and dynamically by measuring the surface of each segment using a CGH which acts as a diffractive null lens. The primary mirror segments are each off-axis sections of the parent conic. There are three prescription types defined as A, B, and C segment types. For the JWST, one spare primary mirror segment was manufactured for each of the three prescription types. The A segment spare is also called the engineering development unit (EDU), as it was the first JWST primary mirror segment built. This mirror has been through all processing including gold coating and cryo-null figuring, a process where the mirror is polished for optimized cryogenic performance. The C segment spare has been polished but not cryo-null figured or coated. Finally, the B spare has not been polished and is in a smooth ground state [3].

In order to develop and validate the metrology described in this paper the A segment spare, EDU, and the C spare mirror were used. This paper will focus on the results of the EDU mirror. The EDU was mounted on a test stand within the Spacecraft Systems Development and Integration Facility (SSDIF) cleanroom at NASA/Goddard Space Flight Center. This test stand was originally used at L-3’s Integrated Optical Systems Tinsley for ambient optical metrology during the polishing process. However, it has been modified to eliminate most motorized motion capability to allow for easier operation. Figure 1 shows a picture of the front of the EDU and back of the C7 mirrors as mounted on the test stand.

 figure: Fig. 1.

Fig. 1. EDU and C7 mirrors on the test stand.

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The optical test layout used for primary mirror testing is that of a classical interferometric CoC test. This test involves the placement of an interferometer with a diverging lens such that its focus is coincident with the CoC of the mirror under test. If the test mirror were a sphere then all optical test rays would be normal to the mirror surface. However, as previously described the JWST primary mirror segments are not spheres but rather off-axis aspheres. Therefore, a diffractive null lens is required to convert the interferometer’s spherical test beam to an aspheric one. When the mirror is placed at the correct location relative to the CGH and interferometer then all optical test rays are normal to the mirror surface and retrace their path through the test layout. Deviations in the mirror figure create optical path differences that show up as phase differences in the interferometer. These differences are then converted to a surface figure map through software.

In order to distinguish figure error from alignment error, the mirror under test is precisely aligned to the CGH in six degrees of freedom. This alignment is broken up into three steps. The tip/tilt of the mirror is aligned using interferometer tilt fringes as feedback. The decenter and clocking of the mirror relative to the CGH is achieved using an alignment camera system. This system is mounted on the same breadboard as the interferometer and consists of a camera attached to a Meade telescope. The system relies on additional features built into the CGH diffractive pattern as well as fiducial targets mounted to the sides of the mirror. The final alignment degree of freedom is the axial spacing between the mirror and the CGH. This distance is measured using a Leica ADM (absolute distance meter) and set accordingly. An alternate method to set the axial mirror distance is to reduce the amount of power in the interferometric measurements by altering the spacing. While this second method is not adequate for absolute determination of the mirror radius of curvature (RoC), it is sufficient for performing delta measurements where the RoC is not of primary interest [4].

The RoC of the primary mirror is approximately 16 m. This is the distance that the mirror must be positioned away from the interferometer in a CoC test. A long optical path test such as this leads to a number of challenges, the most notable of which is turbulence and stratification within the air path. This was especially challenging for the EDU test as the optical setup was positioned right next to the cleanroom air outlet. However, we were able to work closely with the SSDIF facilities personnel to optimize the thermal environment. This was also very important as the primary mirror segments are designed for cryogenic use. The beryllium substrate material has a low coefficient of thermal expansion at cryogenic temperatures but is relatively high at room temperature. This means that at room temperature the mirrors are much more sensitive to thermal gradients. These gradients within the mirror assembly cause very large and measureable changes to the figure, RoC, and astigmatism contributions. In order to minimize these effects the temperature within SSDIF needed to be very stable. An air temperature rate of change requirement of less than 0.05°C per 10 min was established as the limit. This kept mirror gradients, and thus mirror deformations, within acceptable values. Figure 2 shows the test layout and location within the cleanroom.

 figure: Fig. 2.

Fig. 2. Center of curvature test layout for EDU testing.

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3. Static Measurement Results

Static surface figure measurements of each of the primary mirror segments will be used to determine if any detrimental changes occurred as a result of the Optical Telescope Element-Integrated Science Instrument Module (OTIS) level vibro-acoustic testing. This is accomplished by performing a measurement both before and after vibration tests and then simply subtracting them from one another. This delta measurement philosophy is of key importance when establishing the metrology setup and requirements. Absolute errors, provided they are repeatable, will not affect the measurement capability. A discussion of all errors associated with the static measurement test is beyond the scope of this paper. However, a quick summary of the errors that will potentially degrade the measurement results include: alignment of the interferometer to the CGH, alignment of the mirror to the CGH, proper registration of the figure data, interferometer and diverger wavefront error, vibration effects, optical air path turbulence and stratification, primary mirror segment thermal gradients, primary mirror segment absolute soak temperature, and orientation changes relative to gravity. For each of the effects listed here the absolute error is not of concern but rather the difference between the before and after measurements. Therefore, the sum of all metrology errors will be called the measurement reproducibility.

Figure 3 shows typical measurement results for the surface figure and astigmatism. This example is the comparison of the SSDIF measurement to that taken previously at Marshall Space Flight Center’s X-Ray Cryogenic Facility (XRCF) during final verification testing [5].

 figure: Fig. 3.

Fig. 3. Surface figure example showing comparison measurement.

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To get a sense of the level of reproducibility error a series of measurements were taken with the EDU mirror. The testing took place over the course of four days and included four distinctive sets of measurements (A, B, C, and D). Each measurement being an average of 250 fringe captures from the interferometer. To properly assess reproducibility the system was completely realigned for each measurement and the time of day and operator were varied. The SSDIF thermal environment had not been optimized at the time of these measurements so monitoring of the air temperature was used to pick reasonable times to perform the measurements. Figure 4 shows the difference in each SSDIF measurement from the other three.

 figure: Fig. 4.

Fig. 4. Static reproducibility measurement results.

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Reproducibility testing showed that metrology was good to 8.1 nm rms figure and 27.5 nm rms astigmatism worst case. The goals were 10 nm rms and 20 nm rms, respectively. While these results look very promising, the test results shown are preliminary as further work is being done to optimize the facility and move the test setup to a location that better approximates the OTIS test layout. At that point, we fully expect to meet the metrology goals for static reproducibility.

4. Dynamic Measurement Method and Results

The objective of the dynamics test is to examine the OTIS and measure its characteristic structural responses to applied stimuli of known and controlled parameters. The OTIS will be characterized prior to the vibration and acoustic testing, and the measurements will be repeated after the environmental tests are completed. Changes in the observed characteristic responses of the primary mirror (PM) may be indications of a change in the structural arrangement of the OTIS. The complexion of the changes across the field could provide some specificity on the elements of the structural changes experienced by the OTIS.

The OTIS characteristic responses are defined through the dynamic response of the structure to singular, low-energy disturbances applied at various locations around the main structure. An input disturbance on the structure will produce a vibration pattern throughout the structure, including the primary mirror segment assembly (PMSA) surfaces. The response at the PMSA surfaces is the target measurement of the test using the HSI. The observed response at a particular frequency will have an amplitude that, when normalized to the input stimulus level, will be called the gain and have units of m/N. Additionally, the peak displacement response will be time phased with a lag relative to the peak of the input displacement. This lag will be called the phase lag, or just phase, and will be reported in degrees.

The combination of gain and phase at a particular frequency for any measured or calculated parameter defines a transfer function. This transfer function is the characteristic structural response that is sought, with changes in the gain and phase parameters the expected indicators of an altered structural arrangement.

The characteristic functions can be developed at many interesting levels. Gain and phase for the response measured at each pixel can be developed by treating each pixel as a channel of data unwrapped in the time domain and then reduced to spectral response characteristics. Alternatively, the field of pixels can be considered as a whole for each individual image. In this case, the measured displacements are unwrapped spatially and reduced into Zernike terms. The Zernike terms are then assembled in time-order and reduced into spectral response characteristics. This shows the strength of the HSI in the large amount of data available and the flexibility in the data reduction options.

The HSI setup is currently limited to measuring a single PMSA at a time. The language of the transfer function will allow the individual PMSA measurements to be assembled into a complete PM response field without any extrapolations or assumptions for stitching.

JWST EDU and C7 PMSA segments were used as test objects to validate the capability of the HSI to accomplish the dynamic target objectives described previously. Each segment was mounted in the Tinsley mirror support stand. A stinger was used as the stimulus source and the HSI was used to measure the response of the mirror surface. Figure 5 shows two positions for the applied stinger force, one on the mirror assembly and the other on the mirror support stand.

 figure: Fig. 5.

Fig. 5. Vibration stinger attached to the mirror and stand assemblies.

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The test was ordered in two steps. The first step measured the nominal PMSA and test stand configuration at multiple times over a relatively long period of time. The data from this first step defined the initial reference baseline response characteristics and provided a measure of the noise floor in the test setup. The second step was accomplished by first modifying the test stand stiffness by coupling together various stand components and then repeating the multiple measurements over a long period of time. The mean values for each of these two sets of results are compared by applying a standard statistical hypothesis known as a T-test, with means and variances unknown. The statistic of the measurement, referred to as the Tstat, is shown in the following equation:

Tstat=μ1μ2s12N1+s22N2.

The parameters in this equation are defined as follows:

  • N1 and N2 are the number of repetitions of the test in each set (for this effort we collected 10 datasets for each configuration).
  • μ1 and μ2 are the calculated averages from the N repetitions.
  • s1 and s2 are the calculated standard deviations for each set of data.

As the two average mean values deviate from one another the Tstat value increases. Therefore, larger values of the Tstat indicate a higher level of confidence that a real change has been detected. This statistical test is applied to each pixel individually but is shown as a composite mirror image for easier visualization.

Various structural modifications were made to the test stand to simulate both subtle and not so subtle alterations in the structural load path. Figure 6 shows two aluminum brackets used to effect structural changes to the test stand without altering the structural characteristics of the PMSA itself.

 figure: Fig. 6.

Fig. 6. Test stand with brackets applied to modify stiffness.

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The structural short circuit channel (SCCH) bracket affected all parameters of the test results, including resonant frequency shifts of 2% to 5%, and changes to the PMSA characteristic phase and gain responses. The structural short circuit L (SCL) bracket had a more subtle effect, often only discernable by a shift in phase angle in the transfer function. The brackets were generally used individually and not simultaneously as shown in the figure.

The test control system and data flow are represented in Fig. 7. A function generator sends a string of pulses that serves to trigger the data acquisition system (DAQ) and the HSI. The pulses in the signal also control the HSI frame rate. The DAQ sends an excitation signal to the stinger to provide the excitation to the system under test. The test database includes results for excitations that were random, sine dwell, and sine sweep. The DAQ, which collected the input force versus time, and the HSI, which collected the PMSA surface images versus time, were under the control of separate workstations. The datasets are ultimately brought together by synchronizing the clocks of the two workstations. The common trigger pulse used to initiate both the DAQ and the HSI synchronizes the two datasets with a high degree of precision. The synchronization of the two datasets was determined by measurement to be within ±10μs of each other.

 figure: Fig. 7.

Fig. 7. Test control system and data flow.

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For initial testing, each test dataset consists of measuring the PMSA surface with the HSI using frequency sampling of 900 Hz for 10 s and a detector spatial sampling of 720×720. This generates 9000 images that are then reduced and analyzed by developing spectrum transfer functions, spatial transfer functions, and spatial modes. The resulting transfer function parameters are compared by the hypothesis Tstat test. The following sections discuss results for each of the types of data modes evaluated.

A. Spectrum Transfer Function Evaluation

Each pixel in the image field can be treated as a singular and independent interferometer. The 9000 sequential data points for a single pixel are phase unwrapped in time and then used to generate a power spectral density (PSD) for that particular pixel. The pixel PSD can be normalized by the PSD of the input forcing function (stimulus) and the resulting product is what is called the spectrum transfer function (STF). In actuality, the resulting product contains a measure of the gain versus frequency and does not necessarily include the phase lag. Therefore, it is not a complete transfer function since it contains only half the story. Figure 8 shows incoherent and coherent STFs for several pixels.

 figure: Fig. 8.

Fig. 8. PSDs for several pixels under varying input forces.

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Phase unwrapping pixel data in the time domain (i.e., temporal phase unwrapping) constrains the dynamic range of the response of the PMSA surface to a quarter wavelength of the laser between sample intervals. This is because pixel displacements greater than this leads to phase ambiguities. A sampling at 900 Hz limits the maximum velocity of the mirror under test to ±142.5μm/s. This limitation constrains the magnitude of the forcing function or stimulus and precludes unlimited range of that parameter to create a more favorable signal-to-noise ratio (SNR). Figure 9 shows histograms of the velocity at one pixel due to random stimuli at different levels. The background velocity without any stimulus is also shown. The velocity is within the dynamic range when the forcing function is set at 110 mV. However, when the forcing function is increased to 400 mv the velocity limit is exceeded.

 figure: Fig. 9.

Fig. 9. Velocities for several pixels showing the effect of exceeding limit.

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With this velocity constraint in mind, the maximum SNR for a random input signal case is approximately 7, while the SNR for sine testing can be as high as 400. This observation suggests a strategy in testing that includes using random testing to identify frequency ranges of interest, followed by specific sine tests to get well above the noise floor.

B. Spatially Mapped Transfer Functions

Spectrum transfer functions become cumbersome with the large quantities of pixels and make comparing differences challenging. Spatially mapping STF results for all pixels at a common frequency can be more insightful and easier to interpret. Mapping the reduced gain and phase lag at each pixel for a specific frequency is referred to as the spatially mapped transfer function (SMTF). Figure 10 shows the SMTF of four test cases at 87.3 Hz. This frequency was chosen as it is approximately at a rigid body tip/tilt mode of the mirror.

 figure: Fig. 10.

Fig. 10. Spatially mapped transfer functions for 87.3 Hz sine input (frequency is the tilt mode of the mirror).

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In Fig. 10, the first or leftmost column of the two images labeled as Set A is a baseline measurement with no brackets. Set B is a similar measurement that is repeated to set a noise floor for the Tstat evaluation. The third and fourth columns are tests that included the SCCH bracket and SCL bracket, respectively. The results in each dataset are derived from the averaging of 10 replicates each. The images show that Sets A and B are qualitatively identical and that adding the SCCH or the SCL bracket causes an obvious difference. The Tstat calculation is used as a quantitative measure of how much change can be detected and the level of confidence that one can ascribe to the likelihood that a change actually occurred. Figure 11 shows the Tstat map that includes all the pixels when comparing Set B to Set A. The histogram charts count the number of pixels that have a Tstat within the sampling range. The Tstat range for the gain values is between 2 and +5. Similarly for the phase lag, the Tstat range is between 3 and zero. This calculation suggests that the Tstat noise floor associated with this test is of the order of ±5 for gain and ±3 for the phase lag. If future observations have a large number of pixels with Tstat values greater than 15 or 20, then one can conclude with confidence that a change occurred in either the mirror or the structure behind mirror.

 figure: Fig. 11.

Fig. 11. Tstat values for 87.3 Hz sine input.

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Next, we develop the Tstat values for the comparison between Set A, the baseline, and the SCL bracket case. The results are also shown in Fig. 11. The Tstat values range from 120 to +500 for gain and -140 to +120 for phase lag. The ranges for both phase and gain, along with a visual inspection of the histograms indicate that a large number of nodes are outside the noise floor developed previously. This states with confidence what we already know, i.e., the system was altered. The same evaluation was completed that compared Set A and the SCCH bracket case. The results were similar with even larger Tstat ranges.

The results presented thus far were all generated using a stimulus at 87.3 Hz. This was chosen because it is between 86.5 and 88.2 Hz, which are the modeled tip and tilt modes of the mirror segment. The stimulus was changed to 65 Hz and the test cycle was repeated. The 65 Hz frequency was selected because it will resonate a tilt mode of the support stand holding the mirror segment. The test results were reduced to gain and phase as before, and then comparisons are developed using the Tstat evaluation. The results are presented in Figs. 1214.

 figure: Fig. 12.

Fig. 12. Spatially mapped transfer functions for 65 Hz sine input (mode of the test stand).

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 figure: Fig. 13.

Fig. 13. Tstat values for 65 Hz sine input.

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 figure: Fig. 14.

Fig. 14. Tstat values for 65 Hz sine input.

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Comparing Set A and Set B puts the noise floor at ±7 for gain and ±3 for phase lag. The observed ranges in the comparison between Set A and the SCL cases has Tstat ranges of 240 to +50 for gain and 20 to +120 for phase lag. The ranges for Set A compared to SCCH results has ranges of 450 to +10 for gain and 240 to +60 for lag. Again, the results show convincingly the measurement tool and the data reduction methods are capable of detecting system level changes by observing the characteristic response of the mirror surface.

C. Modal Content Evaluation

The surface density of data in each image collected by the HSI allows for mode shape extraction and evaluation of the mirror surface. The total motion of the PMSA segment can be categorized as rigid body or deformation. The deformations in the surface can be expressed as the superposition of a set of orthogonal functions and the coefficient of each of those functions or terms is called the content of that particular term. The set of orthogonal Zernike functions was chosen for this evaluation. Zernike functions are orthonormal polynomials that are used in optics routinely to describe surfaces. At each measurement time the mirror figure is measured and fit to Zernike polynomials, relative to the time zero initial static shape. Zernike coefficients of interest are piston (z1), tip (z2), tilt (z3), 0 deg astigmatism (z5), and 45 deg astigmatism (z6). The resultant time varying Zernike coefficients are Fourier transformed into the frequency domain. Measuring the specific Zernike coefficient for each surface and Fourier transforming this time variable function results in an amplitude spectral density (ASD) of this specific Zernike. This ASD is normalized to the input force ASD at each specific frequency to express the result in language of transfer functions. In this case, there are two components to the ASD; gain and phase. To reduce the random noise in the measurements, averaging of the modulus and phase are done in the frequency domain by averaging multiple measurement sets at each frequency of interest. Measurements are done before and after modifications to the test stand (i.e., changing brackets) and the measured transfer functions are compared through T-statistics. Any Zernike coefficient, rigid body, and/or deformation at each frequency can be measured and analyzed. Table 1 summarizes Zernike transfer functions for 5 coefficients at 87.3 Hz stimulus for Set A, Set B, SCCH, and SCL data that were previously analyzed using gain and lag transfer functions.

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Table 1. Sine 87 Hz Gain Zernike Terms

The first column is the sample mean for gain Zernike transfer function in units of μm/N for 10 measurements averaged for Set A. The second column is the sample standard deviation for the 10 measurements averaged for Set A. The third and the fourth columns are the same but for Set B. Fifth and sixth columns are for SCCH mean and standard deviation for gain transfer functions. The last two columns are for SCL Zernike transfer functions. We use T-statistics to compare every set to Set A. The result is summarized in Table 2. Note that the information in this table condenses all the spatial information previously presented into clear and concise scalar values.

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Table 2. Sine 87 Hz Gain Zernike Tstat Results

The first column, no bracket Set B, is the repeatability (noise) and the second and third columns show large changes in the Zernike transfer functions which measure changes to the state of the mirror system through modification of the stand. This approach has a noise floor lower than direct comparison of gain and lag transfer functions because the measurements are averaged spatially as well as temporally (or ensemble averaging). In Table 3, phase transfer functions of the Zernike’s are tabulated. The units are in degrees for this table. In Table 4, T-statistics are summarized.

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Table 3. Sine 87 Hz Phase Lag Zernike Terms

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Table 4. Sine 87 Hz Phase Tstat Results

The following data show the same evaluation for the data generated at 65 Hz. The first column compares the phase Zernike transfer functions of Set B to Set A. This sets the noise floor for the phase Zernike transfer functions. The second column compares the phase Zernike transfer function of the mirror system after adding the SCCH bracket to Set A (no bracket). Similarly, comparison is done for SCL bracket to Set A. Large changes in the phase Zernike transfer functions of the system after modification to the no bracket system are shown. Similar calculations have been done on the 65 Hz data. The results are tabulated in Tables 58.

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Table 5. Sine 65 Hz Gain Zernike Terms

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Table 6. Sine 65 Hz Gain Zernike Tstat Results

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Table 7. Sine 65 Hz Phase Lag Zernike Terms

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Table 8. Sine 65 Hz Phase Lag Zernike Tstat Results

D. Modeling

To see if the real changes to OTIS structure can be detected using gain/phase transfer functions, a finite element model (FEM) of OTIS was used. This model consists of two JWST PM mirrors, C6 and B5 on the primary mirror backplane support structure (PMBSS), see Fig. 15. Each mirror is attached at three fittings on the support structure. The mirror models were replaced with finite element direct input matrix at a grid (DMIG) representations to reduce the degrees of freedom in the model. This approximation is acceptable for this initial feasibility assessment. The load application point was selected to be near the top of the structure, see Fig. 16.

 figure: Fig. 15.

Fig. 15. Model used for backplane and mirror analysis.

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 figure: Fig. 16.

Fig. 16. Model load and boundary conditions.

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This load application point was selected because it was an accessible and pre-existing interface location on the OTIS structure that was near the mirror surface and far away from the test boundary conditions. The load is a 1 N unit load applied in the Y direction to define the force vector. That direction is perpendicular to both gravity (Z direction) and the mirror surface focal point vector (+X direction) to reduce the likelihood that any changes in the structure response during trade analyses were lost in the global response of the structure. Transfer functions of the structure were developed at two sine vibrations, 40 and 60 Hz, through the measurement of the mirror motions. These transfer functions are the baseline for our measurements. Next, a modification to backplane structure was performed, removing a coupon as shown in Fig. 17, and then transfer functions were developed.

 figure: Fig. 17.

Fig. 17. Location of the removed bond for trade analysis.

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Direct subtraction of the gain and phase transfer functions before and after modification shows that the amplitude change is detectable by our proposed methodology, see Figs. 1821.

 figure: Fig. 18.

Fig. 18. Difference in gain results (trade–nominal, 40 Hz input).

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 figure: Fig. 19.

Fig. 19. Difference in phase results (trade–nominal, 40 Hz input).

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 figure: Fig. 20.

Fig. 20. Difference in gain results (trade–nominal, 60 Hz input).

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 figure: Fig. 21.

Fig. 21. Difference in phase results (trade–nominal, 40 Hz input).

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E. Spatial Mode Development

The HSI data can be used to study spatial modes of the mirror surface in the test assembly. The data is of such high resolution that rigid body and deformation displacements can be resolved and distinguished. The ability to separate rigid body from deformation is of interest because further study of the deformation field may yield insight to the stresses associated with inertial pressures on the mirror substrate.

As mentioned previously, testing of the EDU and C7 mirrors on the support stand was performed at 87.3 Hz as that particular frequency splits the difference between two mirror rigid body tip/tilt modes. Given a sine wave stimulus of 87.3 Hz, it is expected that the resulting mirror displacements will be a combination of these two tilt modes.

The surface analysis is accomplished by applying two mathematical filters to the data. The first filter is a frequency band filter applied to the data associated with each pixel, which is followed by applying a Zernike decomposition of the surface at each point in time to filter true rigid body content from deformation content in the displacement field.

The approach starts by applying the conventional Fourier transform from time domain to frequency domain. A frequency band filter is applied in the frequency domain and the resulting signal is then transformed back to the time domain through the inverse Fourier transform. The result of this process is a time varying signal associated with frequency filtering of interest for each individual pixel.

Surfaces are then generated by synchronizing the filtered data of all pixels at each time measurement. The change in this surface in time is the spatial mode of interest. The surface at each time step is decomposed into Zernike functions to extract rigid body (tip/tilt) and deformation (two astigmatism modes) content. Astigmatism is chosen since this is the first deformation mode of the mirror. The analysis is applied to all pixels and at each time slice. In effect, the resulting images are still frames of the animation sequence for the motions.

The results presented in Fig. 22 are the effect of applying a delta function at 87.3 Hz for the frequency filter. The left mosaic (3×5 snapshots) of the figure shows the rigid body motions and the right mosaic shows the deformation motions over a cycle due to stimulus at 87.3 Hz. It is noted that one cycle of 87.3 Hz stimulus at 900 Hz sampling corresponds to 11 frames. The two mosaics for rigid body and deformation motions are going through full cycle by the 11th frame. Figure 23 shows the amplitude of the rigid body motion and associated deformation versus frame (i.e., time).

 figure: Fig. 22.

Fig. 22. Rigid body and deformation spatial modes at 87.3 Hz with a delta function bandwidth.

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 figure: Fig. 23.

Fig. 23. Amplitude of rigid body and associated deformation versus time for 87.3 Hz with a delta function bandwidth.

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This demonstrates that the so-called rigid body mode of the mirror is not completely a rigid body. The changing acceleration vector is the generator of the deformation of the mirror at the rigid body frequency. This mirror deformation is an imprint of the reaction at the supporting points holding the mirror due the relative inertial changes caused by the mirror rigid body motion. For each rigid body motion at any frequency there is a corresponding deformation motion mostly in the form of astigmatism at that specific frequency. This is independent of the first resonant deformation mode of the mirror at 240 Hz and its spectral distribution.

Coupling between the tip and tilt modes of the mirror is examined by widening the frequency filter to include both frequency modes with a band width of 7 Hz centered about the 87.3 Hz frequency of interest. No rotation of the tip/tilt mode was observed; hence no coupling between the modes. The mosaic for this 7 Hz filter was the same as the motion with delta function bandwidth and is therefore not shown here.

Increasing the bandwidth to 20 Hz we see coupling of energy between the tip/tilt modes along with other modes. This coupling shows up as nutation of the tip/tilt mode about the mirror axis, see Fig. 24. Figure 25 shows the relative amplitudes of the rigid body and deformation modes, respectively, for each figure map. This mode coupling can be seen in Figure 26, which also demonstrates the occurrence of additional frequency modes when applying a stimulus at 87.3 Hz.

 figure: Fig. 24.

Fig. 24. Rigid body and deformation spatial modes at 87.3 Hz with a 20 Hz bandwidth.

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 figure: Fig. 25.

Fig. 25. Amplitude of rigid body and associated deformation versus time for 87.3 Hz with a 20 Hz bandwidth.

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 figure: Fig. 26.

Fig. 26. Zernike PSD for the first three Zernike terms.

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F. Temporal Phase Unwrapping in Extreme Environment

The dynamic range of the HSI scales linearly with sampling frequency. The highest sampling frequency tested to date is 2.25 KHz. This translates to a velocity limit of 450 μm/s for each pixel which is sufficient in most experimental conditions. However, to enhance the dynamic range for more extreme conditions, a displacement measuring interferometer (DMI) with a corner cube mounted to the side of the mirror is used in conjunction with the HSI. A DMI sampling frequency of 5 KHz is chosen with a trigger signal that initiates the HSI, the DMI, and the vibration stinger at the same time. The amplitude of the stinger is set beyond the dynamic range of the HSI at 900 Hz sampling. The HSI phase at each pixel is wrapped in time domain and since each pixel moved from frame to frame with velocities beyond the dynamic range, the temporal phase unwrapping cannot be done utilizing only the HSI data. Spatial phase unwrapping can still be done because it is not subject to the temporal dynamic range. Rather, it is limited by the amount of mirror motion relative to the integration time of the camera that results in acceptable fringe contrast. The spatial phase unwrapping limitation has not presented itself as an issue and will not be discussed here.

The temporal limitation can be overcome by using the DMI data to correct for the phase of a single mirror image pixel which is near the location of the DMI retro reflector. This correction is applied to all of the HSI frames. The wrapped phase of the HSI gets corrected using DMI data by utilizing the equation below. This equation determines the number of additional 2π phase wraps observed by the DMI versus the HSI over the time period of two sequential measurements [6]:

(φHSI)Corrected=MOD[(DMI(φHSI)λ4π)+λ4,λ2].

Following the DMI correction, the temporal phase unwrapping between frames can then be done for this specific pixel. Using this corrected pixel, spatial phase unwrapping can be accomplished for each frame. The last step is to add the DMI amplitude back to all pixels per frame per time. This would be a constant that gets added to all pixels. As a result, although the velocity of the mirror is beyond the dynamic range of the HSI, a DMI enables us to do temporal phase unwrapping beyond this limitation.

Figure 27 displays temporal phase unwrapping of the HSI in which the mirror is shaken to a random input vibration with an excitation level of 400 mv and the DMI is used to correct for the wrapped phase of the HSI. Next, the subtracted value of DMI is added back on to each frame. Notice that the DMI and HSI pixels are not identical. This is to be expected since the image pixel and retro reflector are spatially separated.

 figure: Fig. 27.

Fig. 27. Reference pixel corrected for DMI.

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5. Summary

We have established the capability to take accurate and repeatable JWST primary mirror surface figure measurements in the Goddard Space Flight Center’s SSDIF cleanroom. This static-type workmanship testing at the PM segment level is possible and enabled by improved environmental conditions, most notably the temperature stability, within the SSDIF. Additionally, an HSI was developed to allow for dynamic-type testing of PMSAs for workmanship purposes. HSI dynamic surface measurements were successfully obtained, demonstrating the capability to observe structural changes in either the mirrors themselves or the structure holding the mirrors. Dynamics testing will be used for diagnostics of the structural conditions of the OTIS and the measurements may help build confidence in telescope and OTIS dynamics models. Data processing has been challenging due to the large volumes of data and processing speed required. However, these challenges have been overcome by the optimization of the required software and improvements to the storage and computing power utilized. The optimization of all software and hardware which will be used for OTIS testing is continuing. In addition, we are developing diagnostics criteria to determine the level of significance of any noted changes due to the planned OTIS vibro-acoustics testing.

NASA’s James Webb Space Telescope project.

References

1. 4D Technology, in Tucson, AZ, http://www.4dtechnology.com.

2. B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014). [CrossRef]  

3. L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).

4. J. B. Hadaway, D. M. Chaney, and L. M. Carey, “The optical metrology system for cryogenic testing of the JWST primary mirror segments,” Proc. SPIE 8126, 81260P (2011). [CrossRef]  

5. D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011). [CrossRef]  

6. B. Saif, M. Bluth, P. Greenfield, W. Hack, B. Eegholm, P. Blake, R. Keski-Kuha, L. Feinberg, and J. W. Arenberg, “Measurement of large cryogenic structures using a spatially phase-shifted digital speckle pattern interferometer,” Appl. Opt. 47, 737–745 (2008). [CrossRef]  

References

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  1. 4D Technology, in Tucson, AZ, http://www.4dtechnology.com .
  2. B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
    [Crossref]
  3. L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).
  4. J. B. Hadaway, D. M. Chaney, and L. M. Carey, “The optical metrology system for cryogenic testing of the JWST primary mirror segments,” Proc. SPIE 8126, 81260P (2011).
    [Crossref]
  5. D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011).
    [Crossref]
  6. B. Saif, M. Bluth, P. Greenfield, W. Hack, B. Eegholm, P. Blake, R. Keski-Kuha, L. Feinberg, and J. W. Arenberg, “Measurement of large cryogenic structures using a spatially phase-shifted digital speckle pattern interferometer,” Appl. Opt. 47, 737–745 (2008).
    [Crossref]

2014 (1)

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

2012 (1)

L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).

2011 (2)

J. B. Hadaway, D. M. Chaney, and L. M. Carey, “The optical metrology system for cryogenic testing of the JWST primary mirror segments,” Proc. SPIE 8126, 81260P (2011).
[Crossref]

D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011).
[Crossref]

2008 (1)

Arenberg, J. W.

Atkinson, C.

L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).

Blake, P.

Bluth, M.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

B. Saif, M. Bluth, P. Greenfield, W. Hack, B. Eegholm, P. Blake, R. Keski-Kuha, L. Feinberg, and J. W. Arenberg, “Measurement of large cryogenic structures using a spatially phase-shifted digital speckle pattern interferometer,” Appl. Opt. 47, 737–745 (2008).
[Crossref]

Brown, B.

D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011).
[Crossref]

Carey, L. M.

J. B. Hadaway, D. M. Chaney, and L. M. Carey, “The optical metrology system for cryogenic testing of the JWST primary mirror segments,” Proc. SPIE 8126, 81260P (2011).
[Crossref]

Chaney, D.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

Chaney, D. M.

J. B. Hadaway, D. M. Chaney, and L. M. Carey, “The optical metrology system for cryogenic testing of the JWST primary mirror segments,” Proc. SPIE 8126, 81260P (2011).
[Crossref]

D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011).
[Crossref]

Clampin, M.

L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).

Eegholm, B.

Feinberg, L.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

B. Saif, M. Bluth, P. Greenfield, W. Hack, B. Eegholm, P. Blake, R. Keski-Kuha, L. Feinberg, and J. W. Arenberg, “Measurement of large cryogenic structures using a spatially phase-shifted digital speckle pattern interferometer,” Appl. Opt. 47, 737–745 (2008).
[Crossref]

Feinberg, L. D.

L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).

Gallagher, B.

D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011).
[Crossref]

Greenfield, P.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

B. Saif, M. Bluth, P. Greenfield, W. Hack, B. Eegholm, P. Blake, R. Keski-Kuha, L. Feinberg, and J. W. Arenberg, “Measurement of large cryogenic structures using a spatially phase-shifted digital speckle pattern interferometer,” Appl. Opt. 47, 737–745 (2008).
[Crossref]

Hack, W.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

B. Saif, M. Bluth, P. Greenfield, W. Hack, B. Eegholm, P. Blake, R. Keski-Kuha, L. Feinberg, and J. W. Arenberg, “Measurement of large cryogenic structures using a spatially phase-shifted digital speckle pattern interferometer,” Appl. Opt. 47, 737–745 (2008).
[Crossref]

Hadaway, J. B.

D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011).
[Crossref]

J. B. Hadaway, D. M. Chaney, and L. M. Carey, “The optical metrology system for cryogenic testing of the JWST primary mirror segments,” Proc. SPIE 8126, 81260P (2011).
[Crossref]

Keski-Kuha, R.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).

B. Saif, M. Bluth, P. Greenfield, W. Hack, B. Eegholm, P. Blake, R. Keski-Kuha, L. Feinberg, and J. W. Arenberg, “Measurement of large cryogenic structures using a spatially phase-shifted digital speckle pattern interferometer,” Appl. Opt. 47, 737–745 (2008).
[Crossref]

Lewis, J.

D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011).
[Crossref]

Saif, B.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

B. Saif, M. Bluth, P. Greenfield, W. Hack, B. Eegholm, P. Blake, R. Keski-Kuha, L. Feinberg, and J. W. Arenberg, “Measurement of large cryogenic structures using a spatially phase-shifted digital speckle pattern interferometer,” Appl. Opt. 47, 737–745 (2008).
[Crossref]

Sanders, J.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

Smith, S.

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

Texter, S.

L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).

Appl. Opt. (1)

Proc. SPIE (4)

B. Saif, R. Keski-Kuha, L. Feinberg, D. Chaney, M. Bluth, P. Greenfield, W. Hack, S. Smith, and J. Sanders, “New method for characterizing the state of optical and opto-mechanical systems,” Proc. SPIE 9143, 91430C (2014).
[Crossref]

L. D. Feinberg, M. Clampin, R. Keski-Kuha, C. Atkinson, and S. Texter, “James Webb Space Telescope optical element development history and results,” Proc. SPIE 8442, 84422B (2012).

J. B. Hadaway, D. M. Chaney, and L. M. Carey, “The optical metrology system for cryogenic testing of the JWST primary mirror segments,” Proc. SPIE 8126, 81260P (2011).
[Crossref]

D. M. Chaney, J. B. Hadaway, J. Lewis, B. Gallagher, and B. Brown, “Cryogenic performance of the JWST primary mirror segment engineering development unit,” Proc. SPIE 8150, 815008 (2011).
[Crossref]

Other (1)

4D Technology, in Tucson, AZ, http://www.4dtechnology.com .

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Figures (27)

Fig. 1.
Fig. 1. EDU and C7 mirrors on the test stand.
Fig. 2.
Fig. 2. Center of curvature test layout for EDU testing.
Fig. 3.
Fig. 3. Surface figure example showing comparison measurement.
Fig. 4.
Fig. 4. Static reproducibility measurement results.
Fig. 5.
Fig. 5. Vibration stinger attached to the mirror and stand assemblies.
Fig. 6.
Fig. 6. Test stand with brackets applied to modify stiffness.
Fig. 7.
Fig. 7. Test control system and data flow.
Fig. 8.
Fig. 8. PSDs for several pixels under varying input forces.
Fig. 9.
Fig. 9. Velocities for several pixels showing the effect of exceeding limit.
Fig. 10.
Fig. 10. Spatially mapped transfer functions for 87.3 Hz sine input (frequency is the tilt mode of the mirror).
Fig. 11.
Fig. 11. Tstat values for 87.3 Hz sine input.
Fig. 12.
Fig. 12. Spatially mapped transfer functions for 65 Hz sine input (mode of the test stand).
Fig. 13.
Fig. 13. Tstat values for 65 Hz sine input.
Fig. 14.
Fig. 14. Tstat values for 65 Hz sine input.
Fig. 15.
Fig. 15. Model used for backplane and mirror analysis.
Fig. 16.
Fig. 16. Model load and boundary conditions.
Fig. 17.
Fig. 17. Location of the removed bond for trade analysis.
Fig. 18.
Fig. 18. Difference in gain results (trade–nominal, 40 Hz input).
Fig. 19.
Fig. 19. Difference in phase results (trade–nominal, 40 Hz input).
Fig. 20.
Fig. 20. Difference in gain results (trade–nominal, 60 Hz input).
Fig. 21.
Fig. 21. Difference in phase results (trade–nominal, 40 Hz input).
Fig. 22.
Fig. 22. Rigid body and deformation spatial modes at 87.3 Hz with a delta function bandwidth.
Fig. 23.
Fig. 23. Amplitude of rigid body and associated deformation versus time for 87.3 Hz with a delta function bandwidth.
Fig. 24.
Fig. 24. Rigid body and deformation spatial modes at 87.3 Hz with a 20 Hz bandwidth.
Fig. 25.
Fig. 25. Amplitude of rigid body and associated deformation versus time for 87.3 Hz with a 20 Hz bandwidth.
Fig. 26.
Fig. 26. Zernike PSD for the first three Zernike terms.
Fig. 27.
Fig. 27. Reference pixel corrected for DMI.

Tables (8)

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Table 1. Sine 87 Hz Gain Zernike Terms

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Table 2. Sine 87 Hz Gain Zernike Tstat Results

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Table 3. Sine 87 Hz Phase Lag Zernike Terms

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Table 4. Sine 87 Hz Phase Tstat Results

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Table 5. Sine 65 Hz Gain Zernike Terms

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Table 6. Sine 65 Hz Gain Zernike Tstat Results

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Table 7. Sine 65 Hz Phase Lag Zernike Terms

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Table 8. Sine 65 Hz Phase Lag Zernike Tstat Results

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

T stat = μ 1 μ 2 s 1 2 N 1 + s 2 2 N 2 .
( φ HSI ) Corrected = MOD [ ( DMI ( φ HSI ) λ 4 π ) + λ 4 , λ 2 ] .

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