2.1.1.

PM SFE

General astrophysics and exoplanet science drive the PM surface specification as a function of spatial frequency. General astrophysics is most sensitive to the shape and stability of the PSF driven by low-spatial frequency errors. Exoplanet science high-contrast imaging is most sensitive to mid- and high-spatial frequency errors. Mid-spatial frequency errors blur or spread the core. High-spatial frequency errors and surface roughness scatter light out of the core and over the entire PSF. Thus, per Table 2, the total PM surface figure specification is further divided into low-, mid-, and high-spatial frequency bands.

Table 2

HabEx PM SFE specification.

The specification of Table 2 assumes computer-controlled polishing for spatial frequencies of up to 30 cycles (50 mm minimum tool size) to correct quilting error with a power spectral density (PSD) slope of $-2.5$ for high-spatial frequencies. The 100-cycle boundary between mid- and high-spatial errors assumes that the coronagraph uses a $64\times 64$ deformable mirror (DM). A $64\times 64$ DM can theoretically correct spatial frequencies up to 32 cycles (or half the number of DM elements). This could create a dark hole with an IWA of $\lambda /D$ and an outer working angle (OWA) of $32\lambda /D$. The systems-engineering consideration is that PM spatial frequency errors up to $3\times$ beyond what can be corrected by the DM can scatter energy back into the dark hole. Therefore, the PM needs to have the smoothest surface figure possible for spatial frequency errors from 30 to 100 cycles.^11,12

Finally, note that if the PM is segmented, then the low-spatial SFE must be subdivided between segment SFE ($N-30\text{cycles}/\text{diameter}$) and segment-to-segment phasing ($<N\text{cycles}/\text{diameter}$ where $N=\text{number of segments}$). A $50/50$ division would be 2.5 nm rms for each.

2.1.2.

PM CTE error

CTE errors can introduce low- and mid-spatial frequency SFE. CTE errors have two components. First is knowledge of the exact temperature at which the mirror’s CTE is zero. Any operational departure from this temperature will introduce a WFE—primarily power. Fortunately, this error can be mitigated by adjusting the on-orbit operating temperature for a minimum WFE. The second, and more important, error is mid-spatial frequency residual cryo-null-polishing error. Because of CTE inhomogeneity in the mirror substrate, the mirror will have a different shape at its operating temperature than at its manufacture temperature. This is a problem for cryogenic telescopes, which can be mitigated via cryo-null-figuring (CNF). For example, the JWST mirror segments, which operate at 30 K (260 K below their manufacture temperature), had to undergo CNF for many tens of nanometers of cryodeformation.¹⁰ HabEx requires CNF because it has an extremely smooth SFE specification and it plans to operate at a nominal temperature of 270 K (i.e., 30 K below its likely fabrication temperature). The exact amount of WFE that must undergo CNF depends on the mirror substrate’s CTE homogeneity. For example, Fig. 6 shows a 1.2-m extreme-lightweight ZERODUR^® mirror (ELZM) with a 9.8-nm rms cryodeformation over a 62-K temperature range. Analysis indicates that this mirror has a CTE homogeneity of $\pm 5\mathrm{ppb}/\mathrm{K}$ (parts per billion/K), and its surface changes by $0.18\mathrm{nm}/\mathrm{K}$.¹³ Thus, the maximum CTE error shift from its 300-K manufacture temperature to its 270-K operating temperature is the error budget allocation of 3.6 nm rms. Obviously, this error can be mitigated by CNF or DM correction.

Fig. 6

1.2-m Schott ELZM 62 K thermal deformation.

2.1.3.

PM gravity sag figure error

A gravity sag error is primarily low-spatial frequency. It occurs because, during fabrication, mirrors experience self-weight deflection when attached to their metrology mount. However, in the 0-G environment of space, there is no self-weight deflection. The change in the mirror’s shape from 1 to 0 (zero) G is called G-release. To produce a diffraction limited space telescope, the PM must be fabricated to its required on-orbit 0-G figure. This is accomplished by quantifying and removing gravity sag from the mirror fabrication process. Any uncertainty in this removal is called G-release error. For HabEx, the desired G-release error is $<4\mathrm{nm}$ rms. Of course, this error could be corrected via a DM, but it is better for the PM to have a good 0-G figure.

Gravity sag consists of two components: self-weight mount deflection and facesheet deflection. Self-weight mount deflection occurs due to the reaction or bending of the mirror gravity against its optomechanical support mount in a gravitational field. Facesheet deflection occurs because space mirrors are typically light-weight—leaving parts of the facesheet without support. These unsupported parts deflect due to the acceleration of gravity against the vertical core ribs. For the purposes of this paper, we will ignore facesheet gravity deflection because it is typically a smaller effect than fabrication quilting. The reason is that force exerted onto the facesheet by grinding and polishing is typically greater than that of gravity. Also, methods to mitigate facesheet deflection and quilting at the scale of HabEx’s 4-m mirror are Technology Readiness Level 9 (TRL-9). One proven approach for mitigating this issue is to optimize the mirror’s facesheet thickness and core pocket dimensions. Another approach is to pressurize the mirror core cells to balance gravity during the fabrication process.

3.3.1.

Monolithic versus segmented

The single most important design decision is whether to make the PM monolithic or segmented. The current largest space mirror is the 2.4-m monolith on Hubble; however, once Webb flies, it will carry a 6.5-m segmented mirror. It is much easier to achieve the ultrastable wavefront required for coronagraphy with a monolithic mirror than with a segmented mirror. As discussed in Sec. 2.3 and shown in Fig. 10, habitable-zone coronagraphy requires a wavefront with picometer-level stability in the mid-spatial frequency regime (i.e., 2 to 10 cycles per aperture). To meet this specification, a segmented aperture mirror must align and phase its segments to each other and maintain that phasing with a stability of a few pm.^5,25,29–32 A monolith has no such requirement.

With its $2.0\text{-}\mu \mathrm{m}$ diffraction limit, JWST was obviously not designed to meet the requirements of a UVOIR exoplanet science mission. But JWST defines the current state of the art. As shown in Fig. 14, JWST’s primary mirror segment assemblies (PMSA) are predicted to have $\sim 20\text{-}\mathrm{nm}$-rms rocking modes in the 40- to 50-Hz range, which impact WFE. In addition, the JWST PMA is predicted to have an $\sim 40\text{-}\mathrm{nm}$-rms “wing flap” mode at $\sim 17\mathrm{Hz}$ (Fig. 14), driven entirely by back-plane stiffness.²¹ While JWST was not designed for coronagraphy, its design can be improved. Increasing the backplane stiffness from 17 to 170 Hz could potentially reduce the wing flap amplitude to $\sim 400\mathrm{pm}$ rms. And an $\sim 10\times$ improvement in WFE might be achieved by operating JWST while warm (because of dampening). But to get into the 1 to 2 pm stability requires increasing its vibration isolation from 70 to 140 dB.²² All of this complexity and engineering difficulty is avoided with a monolithic mirror. Two other examples of segmented aperture telescopes are the Large Aperture Mirror Program (LAMP) and Segmented Mirror Telescope (SMT). LAMP (Fig. 16) was a ground demonstration of a 4-m, actively controlled, segmented mirror. Each of its seven 2-m segments were $\sim 17\text{-}\mathrm{mm}$ thick and attached to surface control actuators mounted in a stiff, lightweight, graphite-epoxy, and back-up structure. A total of 312 surface control and 42 segment position actuators enabled the LAMP mirror to be controlled to a wavefront accuracy of $<100\mathrm{nm}$ rms (i.e., diffraction limit of $1.3\mu \mathrm{m}$).³³ SMT (Fig. 17) is a 3-m telescope with six hexagonal petals. Each segment has more than 100 actuators for surface control, as well as three fine and six coarse actuators to bring the segments into alignment after deployment.³⁴ SMT was designed to be diffraction limited at $0.5\mu \mathrm{m}$ but only achieved $5\mu \mathrm{m}$ because of structural and thermal instability.

Fig. 16

4-m LAMP telescope.

Fig. 17

3-m SMT telescope.

Another important design trade is aperture diameter, because size drives science yield. While the SLS 8.4-m fairing can accommodate an aperture larger than 6 m, 4 m was selected for programmatic reasons. A 4-m PM can be affordably manufactured using existing infrastructure available at multiple potential suppliers. Schott has existing infrastructure to melt, cast, and lightweight-machine 4.2-m ZERODUR^® mirror substrates (Fig. 18).³⁵ Similarly, Corning has infrastructure to assemble 4-m “class” ULE^® mirror substrates via either frit bond or low temperature-fusion (LTF). And multiple organizations have existing infrastructure to grind and polish 4-m class substrates into space mirrors, including: Collins Aerospace in Danbury, Connecticut; L3/Brashears in Pittsburgh, Pennsylvania; Harris Corporation in Rochester, New York; Arizona Optical Systems and the University of Arizona in Tucson, Arizona; and RESOC outside of Paris, France. Figure 19 shows the 4.2-m Southern Astrophysical Research Telescope (SOAR) PM being polished via computer control at Collins Aerospace.³⁶ In addition, several organizations are considering, planning, or implementing the ability to coat 4- to 6-m class mirrors by scaling up the proven process demonstrated on 2.5-m class mirrors, including: Collins Aerospace,³⁶ ZeCoat Corporation,³⁷ and Harris Corporation. Because 4-m mirrors can be manufactured by multiple organizations using existing infrastructure and processes, they are relatively affordable and low risk.

Fig. 18

Schott Glass 5-m, 5-axis CNC machine center-loaded with a 4.5-m glassy ZERODUR^® blank.

Fig. 19

Collins Aerospace computer-controlled manufacture of 4.2-m SOAR PM.

3.3.2.

PM material

Selecting the mirror material involves thermal, mechanical, and programmatic considerations. For HabEx, a key metric is CTE. CTE and CTE homogeneity are important because they determine how the mirror’s shape deforms as a function of bulk temperature or thermal gradient changes. Such deformations impact the telescope’s ability to meet its diffraction-limited performance and wavefront-stability specifications. The typical approach for mitigating this risk is to select a material with zero CTE and extreme homogeneity. A second approach is to measure the mirror’s shape change between its manufacture and operational temperatures and then “cryo-null figure” the mirror to have the desired surface shape at the operational temperature. This is the approach taken for the JWST beryllium mirrors. A final approach is to actively control the mirror’s on-orbit shape.

The HabEx telescope design team evaluated ZERODUR^® and ULE^®. Both materials are TRL-9 with multiple mirrors currently flying in space. Both Schott and Corning can tailor their respective material’s zero CTE temperature (Figs. 20 and 21) and both claim similar CTE homogeneity (i.e., $\pm 5\mathrm{ppb}$). Therefore, a mirror manufactured from either material should have similar thermal performance. But the real impact of this design decision is architectural—whether the mirror is open-backed or closed-back. Because ZERODUR^® is a ceramic, it must be machined from a single boule, resulting in an open-back architecture. By comparison, ULE^® is a glass and can be assembled via frit bonding or LTF processes into a closed-back architecture. The advantage is that closed-back mirrors have significantly higher stiffness. Yet, at the same time, because a ZERODUR^® mirror is machined from a single boule, their CTE distribution can be smoother and more homogeneous.

Fig. 20

Schott can tailor its Zerodur^® ceramic to have (a) a zero slope $dL/L$ or (b) zero CTE at temperatures ranging from $\sim -20°\mathrm{C}$ to $+40°\mathrm{C}$ with an uncertainty of $\pm 4°\mathrm{C}$.³⁸

Fig. 21

Corning can tailor its ULE^® glass to have a zero CTE at temperatures ranging from $\sim 5°\mathrm{C}$ to 40°C based on the amount of titanium dopant with an uncertainty of $\pm 4°\mathrm{C}$. Less titanium shifts the zero-CTE to lower temperatures. More titanium shifts zero-CTE to higher temperatures.³⁹

3.3.3.

Primary mirror substrate design

The most important design criterion is mirror stiffness. Stiffness enables performance and simplifies manufacture. When the mirror’s stiffness is higher, it is easier to produce the smooth surface needed to achieve 400-nm diffraction limited performance. The higher the mirror’s stiffness, the smaller its gravity sag or self-weight deflection, which results in a smaller inertial error and lowers the potential of a significant G-release error. Also, a stiffer mirror is easier to handle (i.e., mounting to machinery or simply turning over), which reduces fabrication risk.

Inertial WFE is the deformation that a mirror experiences when it is accelerated against its mount. To first order, it is a scaled vector-superposition of the mirror’s ($X,Y,Z$) 1 G gravity sag. For example, if a mirror has $25\mu \mathrm{m}$ of 1 G gravity sag, it will have 25 pm of deformation when accelerated by 1 micro-G. Note that inertial WFE is not the same as resonant WFE. Resonant WFE typically has a large amplitude and occurs only at specific temporal frequencies. Inertial WFE, on the other hand, occurs at all nonresonant temporal frequencies. Again, JWST provides a definitive example of inertial error. JWST’s 220-Hz open-back beryllium PM segments on a three-point mount have a horizontal gravity sag of $\sim 200\mathrm{nm}$. When driven (in a horizontal configuration) at 87.3 Hz, they have an inertial astigmatic WFE of $\sim 220\mathrm{nm}$ per G of driving force.⁴⁰ There are two approaches for minimizing inertial WFE: make the mirror as stiff as possible to minimize its gravity sag and/or minimize the mirror’s exposure to acceleration by minimizing thruster noise or isolating the mirror from such noise.

G-release error, as discussed in Sec. 2.1.3, is the error between a mirror’s 1-G shape and its on-orbit 0-G shape. The risk that a mirror will have significant G-release error is correlated with the mirror’s stiffness. The lower the mirror’s stiffness, the greater will be its 1-G gravity sag—and the greater will be the uncertainty in the fabrication process’s ability to characterize and compensate for or back-out that gravity sag. Thus, it is best to make the mirror as stiff as possible.

For a 400-nm diffraction-limited telescope, G-release error should be $<4\mathrm{nm}$ rms. While TRL-9 techniques exist to characterize self-weight deflection to $<10\mathrm{nm}$ rms, it is also possible to mitigate the risk of G-release error via an active mirror (e.g., PM, SM, or deformable flat mirrors). An active PM can easily correct 50 to 200 nm rms of error using TRL-9 technology. While they were never used, the Hubble PM had 24 actuators to mitigate the risk of astigmatic G-release error.⁴¹ The Webb PM has 18 actuators to mitigate the risk of segment-radius matching error and segment-level astigmatism.

Two TRL-9 methods for characterizing and removing self-weight deflection to $<10\mathrm{nm}$ rms are a metrology mount that compensates for or off-loads the effect of gravity during testing and testing the mirror under multiorientation to mathematically solve for gravity deformation separate from surface shape. Regarding gravity off-loading, two implementations are multipoint mount and air bag. For mid-quality mirrors, air bags are adequate. For high-precision mirrors, multipoint mounts are preferred. Multipoint metrology mount technology was developed with NASA funding in the 1970s for the Large Space Telescope Program (Hubble).^41–43 The Hubble PM’s $7.6\text{-}\mu \mathrm{m}$ self-weight deflection was characterized to an accuracy of 1.4 nm rms using a 135-point metrology mount.^41,44,45 Regarding gravity orientation testing, two methods are the vertical face-up/face-down test⁴⁶ and the horizontal rotation testing.^36,47,48 Typically, a multipoint mount or air bag is used for in-process metrology, and a multiorientation test is used for final metrology. The Kepler PM was tested using an air bag, a 108-point metrology mount, and a face-up/face-down gravity orientation test.⁴⁶ The air bag was estimated to off-load gravity sag with an uncertainty of 5.6 nm rms. The difference between the air bag test and multipoint mount test was 16.4 nm rms. The difference between the air bag and face-up/face-down test was 18.4 nm rms. In both cases, the largest component of this difference was spherical aberration. By inference, the difference between the multipoint mount and the up/down test was 8.3 nm rms.

Whether the PM is active or passive, it is important to make it as stiff as possible to minimize its gravity sag (because when the amplitude is smaller, it is easier to characterize and remove the amplitude from the 1 G metrology data). For any given mirror material, mirror gravity sag is proportional to geometry, mass density, and mount interface:

To minimize gravity sag for a 4-m mirror, one must make the mirror as thick as possible and with the lowest areal density possible. For ULE^®, the state-of-art thickness is $\sim 30\mathrm{cm}$. Specialized abrasive waterjet machines can cut core elements as thick as 28 cm.⁴⁹ These elements can be frit bonded or low-temperature fused to form a mirror substrate. To make thicker mirrors, the AMTD project successfully demonstrated a stack and seal process by manufacturing a 40-cm-thick test mirror.^27,28,49 For ZERODUR^®, Schott has demonstrated 42-cm-thick substrates and they are working to produce 45-cm-thick mirrors.³⁵ Other design elements that impact stiffness include: face-sheet thickness; whether not the back of the mirror is open, closed, or partially closed and thickness of the back-sheet; and geometry of the core structure (i.e., iso-grid, rectilinear-grid, or hex-grid), pocket size, core wall thickness, etc. For example, hex-grid is more mass efficient but less stiff than iso-grid. Thus, hex-grid is typically used for closed-back mirrors (because the back sheet adds stiffness), and iso-grid is typically used for open-back mirrors. Also, the mirror mount geometry is important. A complete discussion of mirror design methodology can be found in Yoder and Vukobratovich.⁴¹

The HabEx design team conducted multiple trade studies with literally hundreds of variations to optimize the stiffness, mass, and gravity sag of candidate open-back ZERODUR^® and closed-back ULE^® PMs. Design parameters traded included facesheet thickness, mirror thickness, core wall thickness, core pocket size, core geometry, and mount geometry (i.e., edge mount $R=100\%$ versus $R=80\%$ or 67%).^50–54 Early in the design process, the goal was to achieve the highest first-mode frequency possible. The reason was to minimize inertial WFE stability caused by reaction wheels. However, once microthrusters with their significantly lower noise properties were baselined, the emphasis shifted to optimizing thermal WFE stability, inertial WFE stability, and demonstrated manufacturability. ZERODUR^® was selected as the baseline material because Schott had demonstrated a routine ability to fabricate 4.2-m diameter ZERODUR^® substrates and turn them into lightweight structures via their ELZM machining process. Furthermore, a 1.2-m ELZM owned by Schott and tested at NASA Marshall Space Flight Center (MSFC) showed better thermal stability than a 1.5-m ULE^® mirror manufactured as part of the AMTD project.⁵⁵ Part of ZERODUR^® CTE advantage over ULE is that the mirror is fabricated from a single boule of material. Thus, a ZERODUR^® mirror has a more homogeneous CTE distribution than a mirror assembled from multiple ULE^® components.

The baseline ZERODUR^® mirror assembly provides an excellent balance between mass and stiffness. The substrate has a flat-back geometry with a 42-cm edge thickness and mass of $\sim 1400\mathrm{kg}$ (Fig. 22). The mirror’s free first mode frequency is 88 Hz. Its mounted first-mode frequency is 70 Hz. Mass is important because it provides thermal capacity for a thermally stable mirror. In addition, mass allows for local stiffening of the substrate to minimize gravity sag.⁵⁰ The mirror substrate geometry and hexapod mount designs were optimized to produce the most uniform $X,Y,Z$ gravity sag deformation possible. The mirror is attached at three edge locations to a hexapod-mount system. This geometry was selected to allow defocus and minimize spherical gravity sag based on VVC aberration sensitivity (Fig. 8). Figure 23 shows the baseline mirror’s predicted 1-G surface gravity sag in global telescope $X,Y,Z$ coordinate system.

Fig. 22

PMA.

Fig. 23

PM gravity sag (surface).

As a matter of completeness, other substrate designs considered include an MSFC 45-cm-thick closed-back ULE^® mirror with total mass of 1388 kg and first mode frequency of 180 Hz,⁵² a Harris Corporation 40-cm-thick closed-back ULE^® mirror with total mass of 440 kg and first mode frequency of 137 Hz, developed under the AMTD project,⁵⁶ a Schott AG ELZM on-axis 34-cm-thick design with 718 kg and $\sim 80\mathrm{Hz}$ first mode,⁵⁷ and a Collins Corp ZERODUR^® shaped-back mirror with mass of 1200 kg and first mode frequency of 120 Hz.³⁶

3.3.4.

PM support structure

The PM support structure is a simple truss. It is specified to be manufactured from TRL-9 M46J with quasi-isotopic laminate properties of 25% 0 deg, 50% 45 deg, and 25% 90 deg, as well as a density of $1.58{\mathrm{g}/\mathrm{cm}}^3$ with a total mass of $\sim 865\mathrm{kg}$ (Fig. 22). As discussed earlier in this section, JWST’s PMA is predicted to have a flapping mode at 17 Hz, which, as discussed in Sec. 2.2, is above the anticipated ZWFS 10-Hz control frequency. Therefore, to minimize WFE stability, the HabEx PM hexapod supports and truss structure are designed for its rigid body and bending modes to be above 40 Hz. For example, Figs. 24 and 25 show a 43.5-Hz rocking mode and a 50-Hz bouncing mode.

Fig. 24

43.5-Hz rocking mode.

Fig. 25

50-Hz bouncing mode.

Finally, the PM truss structure is designed to accommodate a launch constraint system consisting of 18 axial and 12 radial launch locks (Fig. 26). While ZERODUR^® is suitable for applications with mechanical loads from 4.3 to 14.5 psi and when manufactured using Schott’s standard surface etch protocol can withstand up to 17.4 K psi for short durations,⁵⁸ standard engineering practice is to limit the maximum launch load to 600 psi. The HabEx design team modeled stress experienced by the baseline PM when exposed to different launch acceleration loads with and without a launch lock system (Table 6). Without a constraint system, launch stress of as much as 1000 psi is concentrated at the three hexapod attachment locations (Fig. 27). And, while according to Schott a Zerodur^® mirror can tolerate this level of stress, we designed a launch constraint system that is predicted to expose no point on the mirror to greater than 300 psi (Fig. 27). Also, if necessary, the launch constraint support structure could be used as a reaction structure for an active figure control system to mitigate G-release error.

Fig. 26

PM launch constraint system has 18 axial and 12 radial launch locks.

Table 6

Baseline HabEx 4-m PM launch stress.

Fig. 27

Launch locks redistribute launch stress from the three hexapod attachment locations to the entire mirror.

3.3.5.

PM actuators

The PM is attached to six hexapod actuators and 30 launch lock mechanisms (18 axial and 12 radial). The SM is also attached to six hexapod actuators and launch lock mechanisms. These actuators and mechanisms are assumed TRL-9 components given that similar actuators and mechanisms are currently flying on Hubble and Kepler and will fly on JWST and WFIRST. For example, JWST uses a two-stage stepper-motor actuator for both launch restraint and alignment. Its coarse stage has a range of 20 mm (12.5 mm of which is used to deploy from the launch restraints). The fine stage uses a mechanical gear to drive an eccentric cam shaft with a step size of 7.7 nm.⁵⁹ Given that the fine stage is mechanical, smaller step sizes required for HabEx or Large UV/Optical/IR Surveyor (LUVOIR) can be achieved with a different gear ratio. In addition, the AMTD project designed, built, and characterized a fine-stage actuator with a range of $15\mu \mathrm{m}$, step size of 0.8 nm, mass of 0.313 kg, and axial-stiffness of $41\mathrm{N}/\mu \mathrm{m}$ (Fig. 28).²⁷ Figure 2 (0.8 nm step 8: AMTD fine stage) actuator.²⁷

Fig. 28

AMTD fine stage (0.8-nm step) actuator.

3.3.6.

PM reflective coating

The baseline reflectance coating for the PM, SM, and TM is a Hubble-like aluminum coating with a magnesium-fluoride protective overcoat. The coating and its deposition processes are TRL-9 and have been used in flight programs since the 1970s. HabEx has the same spectral range science requirement as Hubble.

3.3.7.

PM thermal control system

The PM thermal control system is critical to the HabEx telescope’s ability to achieve science-required diffraction-limited performance and wavefront stability. The function of the thermal control system is to uniformly set the PM’s front surface to the desired operating temperature and keep it at that temperature regardless of where the telescope points on the sky relative to the Sun. Control system accuracy impacts diffraction limit performance, signal-to-noise ratio, and spectral throughput. The precision with which the system can maintain temperature determines wavefront stability. Any gradient or bias error in the mirror’s bulk temperature will introduce a static “cryo-deformation” WFE. And any temporal variation in the mirror’s temperature will introduce instability.

Similar to Hubble, HabEx is cold biasing the PM and SM and heating them to the desired operating temperature. The desired operating temperature is constrained by two competing requirements. Near-IR science needs the mirrors to be cold to minimize in-field thermal noise. However, UV science requires the mirrors to be free of any contamination, such as a monolayer of water ice or other out-gassed molecules, to maximize spectral throughput. HabEx has selected an operating temperature of 260 to 270 K for its mirrors because it is above the sublimation temperature for water ice. The amount of cold bias is also constrained by competing requirements. When the bias is larger, it is easier to control the mirror temperature; however, the system will require more electrical power to achieve that control. The minimum amount of bias is one where the mirrors are cold biased for all potential Sun orientation angles. Cold bias is produced by thermal isolation around the telescope. Preliminary analysis required the removal of some spacecraft panels on the anti-Sun side to increase the cold bias.

Making the temperature of the PM’s front surface uniform anywhere it points on the sky relative to the Sun is a bit more difficult. Because of the radiative transfer solid-angle view factor of the PM through the straylight tube to deep space, the middle of the mirror will be colder than the edge. Also, depending on the pointing angle to the Sun, the thermal load into the Sun-side of the telescope will change as a function of the angle cosine—thus changing the lateral gradient experienced by the PM. To compensate, the HabEx plans to use a thermal control system with radial and azimuthal heater zones behind and around the perimeter of the PM. The radial heater zones compensate for the view factor radial gradient to create a uniform front surface temperature. The azimuthal heater zones compensate for changes in the lateral thermal gradient as a function of pointing angle.

The baseline HabEx active radial thermal control concept is an engineering scale-up of systems built by the Harris Corporation. Zonal active thermal control of PMs is currently TRL-9 with systems currently flying on the Harris Corporation commercial 0.7- and 1.1-m Spaceview™ telescopes. These systems enable on-orbit focus adjustment for optimal image quality.^60,61 In addition, under the NASA Astrophysics Division-funded PTCS, Harris Corporation has built and delivered to NASA for testing with candidate mirror assemblies a 1.5-m system with 37 thermal control zones (Fig. 29).⁶² This system has six azimuthal heater zones in each of five radial and circumferential zones. In addition, there are heater zones on each mirror strut and the support structure. The test article is fully instrumented to quantify radiative and conductive heat flow.

Fig. 29

Predictive thermal control study zonal thermal control system technology demonstrator.

The ability to achieve the wavefront stability required for coronagraphy depends on the PM’s thermal sensitivity and the thermal control system’s controllability. Regarding thermal sensitivity, the rate at which the PM’s SFE changes (i.e., stability) depends on its CTE, mass, and thermal capacity⁶³

The key to achieving an ultrastable thermal wavefront is to sense and correct fluctuations in the mirror’s thermal environment faster than the mirror’s response time to those changes. For any given mirror design, stability is achieved by a combination of two engineering specifications: control period and thermal fluctuation (sensor noise). For a generic mirror design, shown in Fig. 30, if the thermal sensors are uncertain to 50 mK, the control period needs to be about 50 s. But, if the sensors are uncertain to 5 mK, the control period can be 500 s. And a factor of $10\times$ improvement is accomplished if the thermal system is controlled at 50 s with a 5-mK noise sensor. The relationship is linear. As long as the thermal control system ($\text{period}\times \text{fluctuation}$) product is below the mirror architecture tolerance line, the thermal control system will provide the specified wavefront stability.⁶⁴

Fig. 30

Thermal wavefront stability is achieved by balancing thermal sensing noise and control period.

Given that thermal sensing system control periods can be on the order of a few seconds, the technical design parameter that determines wavefront stability performance is sensing noise. The WFE will be more stable when the sensing noise is lower. The current TRL-9 capability is defined by the Harris Corporation Spaceview™ telescope’s thermal control systems sensors, which have a noise of $\sim 50\mathrm{mK}$ and control their 1.1-m telescope to a temperature of 100 to 200 mK.⁶⁵ It is important to note that these are Earth-viewing systems in low-Earth orbit. These same systems would be much more thermally stable if they were doing astrophysics science at SE-L2. STOP analysis presented in Sec. 4.3.2 shows that the more massive HabEx PM can be controlled to a temperature of $\sim 1\mathrm{mK}$ with a system of 50 mK noise and 30-s control period. Performance margin can be obtained using sensors with lower noise. The PTCS thermal enclosure system is using commercial platinum resistance thermometers with $\pm 5\mathrm{mK}$ reproducibility and $\pm 10\mathrm{mK}$ long-term stability.⁶⁶ Currently, for WFIRST, Harris Corporation is designing a thermal control system that uses 4-wire bridge-circuit thermistors with $<4\mathrm{mK}$ noise.⁶⁵

Finally, it is necessary to “tune” the thermal enclosure to “set” temperatures. Schott Corporation has the ability to “tailor” ZERODUR^® to have 0 CTE at a specific temperature. However, the exact CTE at a specific temperature is uncertain by $\pm 10\mathrm{ppb}/\mathrm{K}$.⁶⁷ As shown in Fig. 20, the CTE/K slope at $\sim 270\mathrm{K}$ is $\sim 2.5\mathrm{ppb}/\mathrm{K}$; thus, a $\pm 10\mathrm{ppb}/\mathrm{K}$ uncertainty translates into the actual 0 CTE temperature being uncertain by $\pm 4\mathrm{K}$. Therefore, it may be necessary to adjust the on-orbit temperature of the PM and SM thermal enclosures a few K to optimize wavefront stability. In addition, zonal temperature tuning can be accomplished to compensate for CTE inhomogeneity.

4.1.1.

Mechanical model

The integrated observatory FEM was created using the MSC Patran, preprocessor, and geometry created in ProEngineer computer aided design (CAD). The PM and SM FEMs were created independently using the NASA MSFC-developed Arnold Mirror Modeler. Using the integrated NASTRAN model, analyses were performed to ensure strength/stability and stiffness requirements were satisfied in accordance with NASA-STD5001B and the launch vehicle payload users guide (United Launch Alliance, ULA-Delta IV Heavy). In addition, the integrated FEM was used to perform dynamic response and thermal analysis. Table 7 summarizes the models.

Table 7

Integrated model details.

Structural elements utilize composite construction where possible to provide a rigid and lightweight design. Where possible, M55J carbon composite material is used due to its excellent strength/stiffness and low mass-density ($1.58{\mathrm{g}/\mathrm{cm}}^3$) specifications. Telescope structure skins, circumferential ribs, axial webs, and the forward contamination door utilize Honeycomb Sandwich Construction with M55J facesheets with Hexcel honeycomb core. Mirror support truss members assume M55J circular tube construction with titanium end fittings. The M55J unidirectional composite layup orientations were tailored for maximum performance and minimum mass. Structural damping is specified to be 0.0005 (0.05%).

4.1.2.

Thermal model

The integrated observatory thermal model was created in Thermal Desktop using the geometry created in Pro-Engineer CAD. The Thermal Desktop model has 20 K elements and calculates the telescope’s structure and mirror temperature distribution at 10 K nodes. The temperature distribution for each node is mapped onto the NASTRAN FEM, and the deflections created by each node’s CTE are calculated using NASTRAN solution 101. RBM and mirror surface deformations are calculated from the NASTRAN deflections using SigFit. During the analysis stage, the calculated RBMs were replaced by a residual error to simulate correction via the laser metrology system. The PM and SM’s mesh grids were sized to enable SigFit to fit thermally induced SFE to higher-order Zernike polynomials.

The model assumes multilayer insulation (MLI) to control heat loss and to isolate thermal disturbances (i.e., the Sun). Radiators pull heat from the science instruments and spacecraft electronics. Between the MLI and radiators, the payload is passively cold biased. Active thermal enclosures are used to maintain the operating temperature of the PM and SM at $\sim 270\mathrm{K}$ (see Sec. 3.3.7). Without these heaters, the model predicts a PM temperature of 206 K and an SM temperature of 196 K. The model assumes TRL-9 capabilities for the enclosure specifications: thermal sensors with 50-mK measurement uncertainty and proportional controller systems (proportional integral derivative, PID) operating with 30-s periods. The model has a total of 133 control zones. Of these, 36 are bangbang survival heaters set at 212 K and 97 are PID control zones (Table 8). They are set to keep the PM and SM front face temperatures at $\sim 270\mathrm{K}$. The model predicts that the PM front surface will have $\sim 200\text{-}\mathrm{mK}$ “trefoil” gradient (Fig. 33). The source of this gradient is thermal conduction into the hexapod struts. The mirror will have $\sim 3\mathrm{K}$ front to back gradient caused by radiative coupling to deep space as well.

Table 8

Thermal model details.

Fig. 33

Predicted 200-mK trefoil thermal distribution of PM front surface.

The PM and SM CTE are modeled as consisting of a uniform “bulk” CTE and a CTE homogeneity distribution. The uniform CTE value determines the mirror’s low-order shape response to bulk temperature changes and/or gradient temperature changes (i.e., axial, radial, or lateral). Such temperature changes can produce low-order errors such as power and astigmatism. The homogeneity distribution determines the mirror’s mid-spatial response. The model calculates mirror shape changes from two effects: response of mirror with uniform CTE to changes in temperature at each of the 10-K nodes and response of a mirror with a CTE inhomogeneity distribution to a uniform bulk temperature change.

Per Fig. 20, Schott Corp. has the ability to tailor ZERODUR^® to have a 0 CTE at 275 K ($-18°\mathrm{C}$). The uncertainty of Schott’s ability to achieve 0 CTE at 275 K is $\pm 10\mathrm{ppb}/\mathrm{K}$.⁶⁷ Therefore, the model assumes a uniform bulk CTE of $20\mathrm{ppb}/\mathrm{K}$. One method to estimate CTE inhomogeneity is to measure the thermal deformation of the mirror and assume that CTE is linear with temperature. As part of the AMTD project, a 1.2-m ELZM was measured to have an $\sim 11\text{-}\mathrm{nm}$-rms deformation over a 62-K thermal range (from 292 to 230 K). Figure 6 shows the measured error and its decomposition into Zernike polynomials.¹³ The model assumes this measured thermal signature for its CTE inhomogeneity distribution. Obviously, the final flight mirror substrate must be selected based on measured CTE homogeneity. To first order, CTE homogeneity data can provide an initial estimate of the required mirror thermal stability. For example, if a 60 K thermal change produces 5 nm rms of trefoil wavefront and the error budget tolerance is 0.5 pm rms, then the mirror needs to be kept thermally stable to 6 mK.

The primary weakness of the current thermal design is the structure, which is very simple. It assumes that the PM support truss structure, telescope tower, and tube are uniformly composed of TRL-9 composite material M46J with $-334.2\mathrm{ppb}/\mathrm{K}$. There are no elements to represent adhesive joints, and there has been no effort to athermalize the structure by application of positive/negative CTE materials. As a result, the telescope model does not accurately model structural thermoelastic effects. In addition, the current design does not actively control the structure temperature. Consequently, the model is highly sensitive to thermal changes. The justification for this design/analysis decision is that the telescope will have a laser truss system to track slow moving thermal structural changes. As shown in Fig. 34, the thermal model predicts that, without any heaters, the tube will have a gradient of over 100 K, and the PM truss will have an $\sim 20\text{-}\mathrm{K}$ gradient. As discussed in Sec. 4.2.1, these gradients will produce $<0.5\mathrm{mm}$ of mechanical displacement—which can be easily corrected by the actuators, discussed above in Sec. 3.3.5.

Fig. 34

On-orbit temperature of (a) baseline telescope structure baffle tube and (b) PMA.

4.1.3.

Mechanical analysis input

Dynamic mechanical errors (LOS jitter, LOS WFE stability, inertial WFE stability, and impulse ring-down) are caused by structural response to mechanical stimuli. To minimize the source of such stimuli, the baseline HabEx observatory architecture does not use reaction wheels for slewing or pointing. Instead, it uses thrusters and microthrusters. Thrusters, located at the base of the telescope, slew and point the telescope. They are then turned off and microthrusters maintain pointing for the duration of a science exposure. The microthruster noise is the only source of mechanical stimuli. Both thrusters are attached to the spacecraft, and their transmission path to the telescope is via the interface ring. If necessary, this ring can be isolated from the telescope. As shown in Fig. 35, there are four microthruster modules located at the base of the observatory (aft) and another four modules located above the center of solar pressure (forward). Each forward module consists of four heads, each with nine emitters, covering a 90-deg cone. Each aft module consists of four heads, each with 18 emitters.

Fig. 35

Microthruster locations.

Microthrusters run continuously with thrust proportional to applied current. Figure 36 shows a measured noise PSD for a single colloidal microthruster emitter. The data indicate that microthruster emitters have a maximum noise of $0.03\mu \mathrm{N}/\sqrt{\mathrm{Hz}}$ with a roll-off after about 2 Hz.⁶⁸ To be conservative, because the data are noisy and have not been measured beyond 5 Hz, the HabEx STOP analysis assumed that each microthruster emitter has a flat or “white” noise spectrum of $0.1\mu \mathrm{N}$—providing at least a $3\times$ specification margin. Note that this noise profile is identical to the specification used by LISA and ST-7.⁶⁸ Finally, because the aft modules have twice as many emitter heads as the forward modules, each forward module is specified to have a flat noise spectrum of $0.4\mu \mathrm{N}/\sqrt{\mathrm{Hz}}$ and each aft module is specified to have $0.8\mu \mathrm{N}/\sqrt{\mathrm{Hz}}$.

Fig. 36

PSD noise plot for colloidal micro-thrusters.

Finally, because the noise “forcing-function” amplitudes are extremely small, values reported by the analysis could have significant uncertainty. While the FEM’s predicted performance is linear as a function of input, the physical system being modeled may not be linear. To mitigate this risk, a model uncertainty factor (MUF) is used in some analyses, while in others an MUF is not applied. For each analysis, the MUF status is made explicit.

4.1.4.

Thermal analysis input

Dynamic thermal errors (LOS drift, LOS WFE, and thermal WFE stability) are caused by structural and optical component response to thermal stimuli. To predict the telescope’s thermal performance at SE-L2, DRMs were analyzed in Thermal Desktop. The calculated temperature distribution is mapped onto the NASTRAN FEM and the deflections of each node calculated. RBMs and SFE of the PM and SM were calculated from the NASTRAN deflections using SigFit.

The DRM starts by pointing the telescope at a reference star to dig the dark hole. The analysis assumes that the telescope reaches a steady-state thermal condition at this Sun orientation. Next, the telescope is pointed at the science star. To make the analysis worst-case, it is assumed that this requires a $+\theta \text{-}\mathrm{deg}$ pitch away from the Sun (Fig. 37). To facilitate speckle subtraction, the telescope is rolled $\pm \mathrm{\Phi }\mathrm{deg}$ about the science star vector (Fig. 38).

Fig. 37

Nominal observing scenario slews for thermal analysis.

Fig. 38

Telescope motions as viewed from the Sun.

The specified DRM analyzed is (Fig. 39):

Fig. 39

Analyzed DRM.

The STOP Analysis calculates as a function of DRM (time):

Figure 40 shows how well the modeled active zonal thermal enclosure controls the temperature of the PM. Once the science observation begins, the PM temperature only changes by $\pm 0.15\mathrm{mK}$. The main reason for this stability is the PM’s thermal mass. The thermal time constant of the PM is $\sim 20\mathrm{h}$. Note that roll maneuvers produce negligible thermal error. Also note that, if the science star and reference stars were located such that the Sun angle remained unchanged during the slew (i.e., rotate around the sun vector), then there would be no change to the telescope’s thermal environment.

Fig. 40

250-hour DRM average PM temperature with PID control for 270 K.

4.2.1.

On-orbit optical alignment

Space telescopes are not launched in their final optical alignment. To survive launch, mirrors are retracted against launch restraints. They are then deployed and aligned on-orbit. For example, the JWST PMSAs are deployed 12.5 mm from their launch restraints to their operational position.⁵⁹ In addition, space telescopes are not launched in their thermal state. Space telescopes are launched “warm,” nominally 300 K, and then cooled to their operational temperature. This thermal shift can change the shape of both the mirrors and the structure. For example, because the baseline HabEx structure design has not been athermalized, the integrated thermal model predicts that structure changes will result in rigid body displacements for the PM of 250 to $500\mu \mathrm{m}$, 25 to $50\mu \mathrm{m}$ for the SM, and rotations on the order of 20 to $50\mu$-radian for both.

Thus, an on-orbit system is required to establish and maintain alignment between the PM, SM, and TM. To sense these thermodeformation errors, the baseline telescope architecture has a laser metrology system. And, to correct these errors, the baseline telescope architecture has hexapod actuator systems for the PM and SM with a 20-mm range and subnanometer resolution (Sec. 3.3.5). Assuming that the individual laser gauges have a 0.1-nm rms measurement uncertainty, the alignment system is predicted to have a positional uncertainty (Table 9) sufficient to align the telescope to an LOS accuracy of $<0.2\mathrm{mas}$ (on-sky) with a wavefront residual error of $<10\mathrm{pm}$ rms.

Table 9

Laser metrology system positional uncertainty relative to the TM.69

4.2.2.

PM G-release error

As discussed in Sec. 2.1.3, G-release error is a significant risk to achieving a 400-nm diffraction-limited telescope. Section 3.3.3 discusses how the HabEx PM is specifically designed to minimize the risk of G-release error by making the mirror stiff. Figure 23 shows that the baseline ZERODUR^® mirror has a predicted 1 G $X,Y,Z$ gravity sag of (18.6, 18.4, and $12.6\mu \mathrm{m}$) RMS. And, as also discussed in Sec. 3.3.3, while methods exist to characterize self-weight deflection and produce 0-G space mirrors, it is also possible to mitigate G-release error risk via an active mirror (PM, SM, or DM). For example, analysis of the alternative ULE^® HabEx PM indicates that 15 actuators can reduce G-release error by $20\times$, 25 can reduce G-release error by $40\times$, and 50 actuators can reduce G-release error by $100\times$ (Fig. 41).⁷⁰

Fig. 41

Correctability of alternative ULE^® 4-m PM design as a function of actuators.

4.3.1.

Predicted mechanical LOS stability performance

To predict mechanical LOS jitter, the rigid-body motion of each mirror was calculated as a result of the structure’s response from 0 to 350 Hz to the microthruster noise specification applied to the structure from 0 to 10 Hz. Per Fig. 36, the microthruster specification provides at least a $3\times$ margin at low frequencies and more margin at higher frequencies—because the flat specification ignores mass damping—relative to the anticipated microthruster performance. In addition, an MUF of 2 was applied for a total margin of $\sim 4\times$. Figure 43 shows the predicted displacement for the PM relative to the SM as a function of frequency. This presentation format is typical for an observatory that uses reaction wheels, because the wheels excite each mode as their speed changes. But microthrusters excite all modes simultaneously. Therefore, it is necessary to rss the individual components into a running sum. Figures 44 and 45 show the cumulative rigid body displacement and rotations for the PM and SM.

Fig. 43

PM-to-SM rigid body amplitudes for microthruster noise specification.

Fig. 44

Cumulative PM and SM rigid body displacement for microthruster noise specification.

Fig. 45

Cumulative PM and SM rigid body rotations for microthruster noise specification.

Table 10 lists the maximum predicted amplitude for each RBM. These motions predict an on-sky LOS jitter of 0.03 mas, which provides a $\sim 10\times$ margin against the 0.3-mas specification. This stability implies that the baseline telescope does not need a low-order, ZWFS-controlled, fast-steering mirror.

Table 10

Predicted maximum “jitter” motion of PM and SM from microthruster noise.

Figures 44 and 45 show information regarding frequencies of various structural modes and the coupling of those modes with optical components. For example, the displacements that occur at 11 Hz are associated with a twisting mode between the spacecraft and telescope (Fig. 46). There is a tube-bending mode at 12.8 Hz and a flexing mode at 24.7 Hz. The 30-Hz mode is bending between the SM tower and PM truss structure. Moreover, the displacements in the 45- to 50-Hz range are caused by motions of the PM on its mount (Figs. 24 and 25).

Fig. 46

Tube modes.

4.3.2.

Predicted thermal LOS stability

To predict thermal LOS drift, the rigid-body motions of each mirror were calculated as a result of the 250-h DRM discussed in Sec. 4.1.4. Figures 47 and 48 show the $X,Y,Z$ rigid body residual displacements of the PM and SM (i.e., the amount of thermal RBM that is not corrected by the laser metrology system that senses and controls optical alignment of the PM and SM).

Fig. 47

Displacements of PM due to thermal DRM.

Fig. 48

Displacements of SM due to thermal DRM.

Table 11 lists the maximum amplitude of the uncorrectable motion for each rigid-body DOF of the PM and SM. These motions predict an on-sky LOS jitter of $<0.2\mathrm{mas}$ (i.e., $\sim 1.5\times$ margin). In this case, no MUF is applied.

Table 11

Predicted maximum uncorrectable “drift” motion of PM and SM for DRM.

4.3.3.

Predicted LOS stability performance

rss’ing the rigid-body motions of the PM and SM from the mechanical and thermal model STOP analysis (Tables 11 and 12) predicts their maximum rigid-body motions (Table 12). These motions predict an on-sky LOS stability of $<0.2\mathrm{mas}$ for a $1.5\times$ margin.

Table 12

Predicted maximum RBM of PM and SM for a DRM.

4.4.1.

Predicted LOS WFE stability

LOS WFE instability is produced by LOS drift and jitter. Thermal LOS drift and mechanical LOS jitter causes beam shear on the SM and TM. Also, because these mirrors are conics, beam shear manifests itself as low-order astigmatism and coma. Conic mirrors correct spherical aberration, and the shear of full aperture spherical aberration is coma. When viewed over an off-axis subaperture, coma appears as astigmatism. As shown in Fig. 49, each mirror’s RBM produces its own WFE “signature” that can decompose into a set of Zernike polynomial coefficients. As expected, there are no significant higher-order errors. The input for the analysis in Fig. 49 is the rss’d rigid body displacements in Table 12.

Fig. 49

WFE decomposed into Zernike polynomial coefficients for PM and SM rigid-body motions.

Figure 50 compares the predicted LOS WFE calculated in Fig. 49 to the LOS WFE allocation in the simple VVC-6 error budget of Fig. 10. As expected, the largest LOS errors are tilt, power, and astigmatism. However, these errors are not significant to the VVC-6. The most significant are trefoil and secondary astigmatism. But, because the LOS jitter is so small, the predicted amplitudes for these errors have $>100\times$ margin relative to their error budget allocation.

Fig. 50

LOS WFE stability predicted to have $>100\times$ margin relative to error budget allocation.

4.4.2.

Predicted inertial WFE stability

Inertial WFE instability occurs when the PM is accelerated by mechanical disturbances causing it to react (i.e., bend) against its mounts. Figures 24 and 25 show two such mechanical disturbances—43.5-Hz rocking and 50-Hz bouncing. Figures 51 and 52 show how the mirror bends as it reacts against the hexapod mount for the rocking and bouncing modes. Note that inertial WFE is not the same as modal resonant motion. Two methods were used to predict inertial WFE instability: linear scaling of gravity sag deformation and predicted deformation using NASTRAN decomposed into Zernike polynomials using SigFit.

Fig. 51

Mirror bending from 49.5-Hz rocking mode.

Fig. 52

Mirror bending from 50-Hz bouncing mode.

Linear scaling of gravity sag deformation

To first order, inertial WFE should be similar to the mirror’s static ($X,Y,Z$) gravity sag (i.e., bending of the mirror when it is exposed to a 1-G acceleration in the $X,Y,Z$ directions) (Fig. 23), scaled by the on-orbit acceleration to which it is exposed. To estimate on-orbit acceleration, the observatory FEM was used to calculate $X,Y,Z$-accelerate from the microthrusters at each PM hexapod interface. Figure 53 shows the maximum calculated $X,Y,Z$-acceleration into the mirror at each frequency. Nevertheless, because microthrusters are always on, it is necessary to rss over all frequencies to determine the maximum acceleration into the mirror (Fig. 54). The absolute maximum $X,Y,Z$ acceleration imparted into the mirror is (170, 240, and 390 nG). Scaling the RMS gravity sag WFE of (37.3, 36.8, and $25.2\mu \mathrm{m}$) yields an $X,Y,Z$ RMS inertial WFE of (6.3, 8.8, and 9.8 pm). Assuming that $X$ and $Y$ gravity sags are entirely astigmatism and the $Z$-sag is trefoil, this first-order linear analysis indicates that the PM will have $\sim 11\mathrm{pm}$ of astigmatism and $\sim 10\mathrm{pm}$ of trefoil. Obviously, the trefoil is about $10\times$ higher than allowed by the simple error budget.

Fig. 53

Maximum PM ($X,Y,Z$) acceleration at each frequency.

Fig. 54

Maximum PM ($X,Y,Z$) acceleration at each frequency.

Fortunately, this analysis is conservative. The maximum acceleration amplitudes at each frequency are $\sim 10\times$ larger than the RMS acceleration at each frequency. In addition, the analysis does not assume mass dampening. The analysis assumes that the PM responds to all acceleration frequencies equally. In practice, this is not real. In a real system, the PM’s inertia prevents responding to high frequency acceleration.

Predicted inertial surface deformation

To obtain a more accurate estimate of inertial WFE, NASTRAN calculated the displacement of PM surface nodes from 0 to 350 Hz as a function of the microthruster noise specification applied to the structure from 0 to 10 HZ (multiplied times 2 for wavefront). To eliminate rigid body displacements, the node displacements were calculated relative to a fixed surface reference point. The WFE produced by these displacements were fit to Zernike polynomials using SigFig. Because the microthrusters excite all modes simultaneously, the individual frequencies were rss’d to yield a total accumulated WFE for each Zernike term (Fig. 55). Consistent with mass dampening (i.e., where the mirror has a smaller response to higher frequency accelerations), the nonlinear analysis predicts an astigmatism WFE that is $3\times$ smaller, a focus error that is $4\times$ smaller, and a trefoil WFE that is $5\times$ smaller than the simple linear gravity sag scaling approach. Figure 56 shows how the calculated inertial WFE of the PM compares with its simple error budget allocation of Fig. 10. As expected, trefoil is the dominant term, but still has $1.6\times$ margin.

Fig. 55

Total PM inertial deformation of select Zernikes: $\mathrm{astig}=3.6\mathrm{pm}$ rms; $\text{focus}=1.5\mathrm{pm}$ rms; $\text{trefoil}=1.1\mathrm{pm}$ rms.

Fig. 56

Estimated inertial WFE.

4.4.3.

Predicted thermal WFE stability

Thermal WFE instability occurs when the PM’s bulk temperature or temperature gradient changes. If the mirror’s CTE is completely homogeneous and constant, then a bulk temperature should only result in a defocus error. But any inhomogeneity in the mirror’s CTE will result in a temperature-dependent WFE. In addition, because CTE is itself temperature dependent, any change in the mirror’s thermal gradient will also result in a WFE. Unlike mechanical WFE, which is mostly low-spatial-frequency, thermal WFE can have significant high-spatial-frequency content. Therefore, the best mitigation strategy is to minimize thermal disturbances. This can be done either by passive isolation or active sense and control.

Thermal WFE instability was predicted by applying the DRM discussed in Sec. 4.1.4 to the integrated model. Thermal Desktop calculated the temperature distribution as a function of time. With this distribution, NASTRAN calculated surface deformations, which were then used by SigFit to decompose the temporal WFE into Zernike polynomials as a function of time. Figure 57 shows the predicted change in PM SFE decomposed into Zernike polynomials. As symmetric errors, power and trefoil’s changes are caused by the DRM pitch angle, which changes the total solar load on the telescope. Moreover, as an asymmetric error, astigmatism’s change is caused by the DRM roll that shifts thermal load from one side to the other. Figure 58 shows predicted change for the SM. Because of its location, it is relatively insensitive to roll.

Fig. 57

DRM PM SFE changes.

Fig. 58

DRM SM SFE changes.

Total DRM WFE was calculated by rss’ing the PM and SM Zernike terms as a function of time and selecting the maximum amplitude for each. This analysis has a $3\times \mathrm{X}$ MUF. Figure 59 shows each Zernike compared to the simple error budget of Fig. 10. Obviously, trefoil is a problem. Fortunately, there is a solution: reallocate the error budget.

Fig. 59

Estimated PM/SM thermal WFE.

The thermal STOP analysis pipeline does not evaluate as many of the higher-order Zernike terms as the optomechanical STOP analysis pipeline. Therefore, there are some “holes” in Fig. 59 data analysis. However, it is reasonable to assume that these terms will be similar in amplitude to those reported and the $3\times$ MUF is an analysis artifact. The thermal STOP model used PV CTE homogeneity measured WFE instead of RMS. But, because there is an interaction between bulk CTE and CTE homogeneity, the results were not “scaled.”