SNAP Telescope Mirror Structural Analysis

# EXECUTIVE SUMMARY

The four mirrors comprising the SuperNova/Acceleration Probe (SNAP)
space telescope optics have been studied to verify feasibility, predict
mechanical performance, and create baseline designs. Figure 1
illustrates the telescope and the general optical layout. The major
performance parameters examined are surface distortions in optical
testing on the ground and due to mounting errors, stresses from launch
loads, and natural frequencies. Additionally, other factors drive the
designs, including thermal stability requirements, packaging
limitations, and mass considerations. For each of the mirrors, baseline
designs have been developed that will meet optical, mechanical, thermal,
and mass constraints. These designs are briefly summarized in Table 1.
These baselines will be useful for further developing the optical
testing plan, for evolving the basic design of the telescope structure
itself, and for initiating dialogue with potential optics vendors. The
designs presented here are intended to be starting points for working
with industry.

  -------------------- ---------------- -------------- ------------ --------------
                       Primary          Secondary      Folding Flat Tertiary

  Mechanical outside/  2050/686         440/N/A        700 x 495/   700/N/A
  inside diameter (mm)                                 196 x 139    

  Optical outside/     2010/750         430/N/A        680 x 480/   690/N/A
  inside diameter (mm)                                 200 x 141    

  Radius of curvature  4854.5           1022.8         N/A          1282.3
  (mm)                                                              

  Asphericity (conic   -0.9830          -1.786         N/A          -0.6131
  const.)                                                           

  Shape of front/back  Concave/Convex   Convex/Flat    Flat/Flat    Concave/Flat

  Thickness (mm)       200, uniform     65 at center   100          90 at edge

  Construction         Closed-back,     Closed-back,   Open-back    Closed-back,
                       frit-bonded ULE  frit-bonded    Silicon      frit-bonded
                       glass            ULE glass      Carbide      ULE glass

  Mass of mirror (kg)  204              4.1            14.4         13.8

  Areal density        70               28             54           36
  (kg/m\^2)                                                         

  Percent              85%              76%            74%          76%
  lightweighted                                                     

  Bipod supports\*     3 'normal'       3 'inverted'   3 'inverted' 3 'normal'

  Surface              6.8/10 (57       7.5/9          9.1/10       8.8/10
  distortion/budget in offloaders)                                  
  one-g test (nm rms)                                               

  First natural        111/283          137/2020       188/923      202/760
  frequency: on                                                     
  mounts/free-free                                                  
  (Hz)                                                              

  Minimum mirror/bipod 16/2.7           35/14          14/11        23/7.5
  material safety                                                   
  factor                                                            

  \*'Normal' bipods                                                 
  converge at the                                                   
  mirror; 'inverted'                                                
  bipods converge at                                                
  the other end.                                                    
  -------------------- ---------------- -------------- ------------ --------------

**Table 1-Baseline mirror design summary**

![](media/image1.jpeg){width="5.230555555555555in"
height="8.781944444444445in"}

## Figure 1-Optical Telescope Assembly (representative of general configuration only) 

**BACKGROUND**

# SNAP Telescope

![](media/image2.png){width="6.375in"
height="2.4451388888888888in"}Figure 1 is a rendered view of the SNAP
telescope, and Figure 2 shows a schematic view illustrating the light
paths. As currently envisioned, the SNAP telescope employs a
three-mirror anastigmat (TMA) configuration^1,2,3^. The concave primary
mirror and convex secondary mirror are on a common axis, and the folding
flat mirror folds the image 90 degrees toward the concave tertiary
mirror, which projects the image through a hole in the folding flat
mirror onto the flat focal plane.

## Figure 2-Schematic of SNAP Telescope

Information on the SNAP mission and science are available at the SNAP
home page, http://snap.lbl.gov.

**Materials**

Due to the mass constraints inherent in a space mission, it is expected
the mirrors will be lightweighted designs. To minimize thermal
distortions, the mirrors and the telescope structure will be actively
thermally controlled, so that the optics are isothermal, around room
temperature, within about one degree Kelvin. In general, it is also
necessary to use materials with extremely low coefficient of thermal
expansion (CTE). Additionally, distortions due to thermal gradients will
be mitigated by selecting materials with high ratios of thermal
conductivity to CTE.

Materials included in the analyses in this study are listed with some of
their properties in Table 2.

+-------------+--------+-----------+-----------+-----------+-----------+
|             | C      | Schott    | Silicon   | Invar 36  | M55J      |
|             | orning | Zerodur   | Carbide   |           | carbon    |
|             | ULE    | glass     |           |           | fibe      |
|             | glass  |           |           |           | r/cyanate |
|             |        |           |           |           | ester     |
+-------------+--------+-----------+-----------+-----------+-----------+
| Young's     | 67.6   | 91.0      | 310       | 148       | 96.5      |
| modulus     |        |           |           |           |           |
| (Gpa)       |        |           |           |           |           |
+-------------+--------+-----------+-----------+-----------+-----------+
| Density     | 2203   | 2530      | 2950      | 8047      | 1924      |
| (kg/m\^3)   |        |           |           |           |           |
+-------------+--------+-----------+-----------+-----------+-----------+
| Strength    | 50     | 70        | 81        | 483       | 690       |
| (MPa)       |        |           |           |           |           |
+-------------+--------+-----------+-----------+-----------+-----------+
| Failure     | M      | Maximum   | Worst of  | Von Mises | Maximum   |
| criterion   | aximum | Principal | Maximum   |           | P         |
| used        | Pri    |           | P         |           | rincipal, |
|             | ncipal |           | rincipal, |           | Minimum   |
|             |        |           | Von Mises |           | Principal |
+-------------+--------+-----------+-----------+-----------+-----------+
| CTE,        | -0.03  | -0.02 to  | 2.4       | 1.3       | -0.07     |
| Coefficient | to     | 0.02      |           |           | (         |
| of thermal  | 0.03   |           |           |           | in-plane) |
| expansion   |        | -0.05 to  |           |           |           |
| (10^-6^/K)  |        | 0.05      |           |           |           |
|             |        |           |           |           |           |
|             |        | -0.10 to  |           |           |           |
|             |        | 0.10      |           |           |           |
|             |        |           |           |           |           |
|             |        | (various  |           |           |           |
|             |        | grades)   |           |           |           |
+-------------+--------+-----------+-----------+-----------+-----------+
| k, Thermal  | 1.31   | 1.46      | 156       | 10.2      | 35        |
| c           |        |           |           |           |           |
| onductivity |        |           |           |           |           |
| (W/m-K)     |        |           |           |           |           |
+-------------+--------+-----------+-----------+-----------+-----------+
| Steady      | 23     | 14        | 15        | 130       | 2         |
| state       |        |           |           |           |           |
| thermal     |        | 34        |           |           |           |
| distortion  |        |           |           |           |           |
| parameter   |        | 68        |           |           |           |
| \|CTE/k\|   |        |           |           |           |           |
| (nm/W)      |        | (various  |           |           |           |
|             |        | grades)   |           |           |           |
+-------------+--------+-----------+-----------+-----------+-----------+

**Table 2-Materials and properties used in this study**

Carbon fiber/cyanate ester is employed in the structure of the telescope
(not addressed in this report) and in a proposed strongback for the
secondary mirror. It has low density and very high stiffness and
strength, making it very suitable for load-bearing structures. Its low
in-plane thermal expansion and fairly high thermal conductivity make for
thermally stability. It has comparatively high through-the-thickness
thermal expansion, which must be accounted for in design details. Note
that the cyanate ester matrix material exhibits some dimensional change
after launch due to moisture loss. This puts a practical limit on
material thickness of around 2mm to limit dimensional changes beyond a
dryout period of a month or so.

The two glass materials exhibit extremely low thermal expansion
coefficients, making them suitable for very stable mirrors. They have
comparable stiffness-to-density and strength-to-density values, with a
moderate advantage going to Zerodur. It is assumed that different
processes would be used on the two materials, resulting in different
forms of mirrors.

Lightweighted Zerodur mirrors are ground from monolithic shapes, meaning
they are either open back or partially closed back, to allow tool
access. Tooling practicalities put some lower limitations on wall
thickness and corner fillet radius. The connection between the face
sheet and the backing ribs has the strength of the bulk material.

ULE mirrors may be made using face sheets bonded to a lightweighted core
by frit bonding or low temperature fusion (LTF). Because the core is be
fabricated independent of the face sheets, abrasive water jetting may be
employed, allowing for thinner walls and tighter inside corner radii, as
compared to Zerodur. In frit bonding, the frit forms the bond between
the ULE parts, while LTF involves direct cohesion of the ULE parts. LTF
has less heritage than frit bonding, and is believed to be inferior
structurally. The LTF process does not introduce a foreign material
(frit) with a slightly mismatched CTE relative to that of ULE. Material
homogeneity inherent to LTF is expected to produce a mirror with lower
mid and high spatial frequency surface figure errors---important in
planet-finder missions, but not crucial to SNAP science. Either joining
technology requires inspection and possible testing to ensure quality of
joints.

Silicon carbide (SiC) is a candidate mirror material that exhibits
superior mechanical properties as compared with ULE and Zerodur. It has
significantly higher CTE than the glasses, but this higher CTE is partly
offset by SiC's proportionally higher thermal conductivity. The higher
thermal conductivity means localized distortions should be similar for
SiC and glass mirrors. Bulk thermal expansion or contraction due to
uniform temperature changes cause changes in radii of curvature for
mirrors with curved optical surfaces, affecting the prescription of the
telescope. Therefore, the curved mirrors require materials with
extremely low CTE, a restriction that does not apply to flat-surfaced
mirrors. SNAP's folding flat mirror is a good candidate for SiC, both
because of the mirror's comparative insensitivity to thermal changes,
and because of structural needs due to its central hole, overall size,
and packaging constraints.

Invar 36 is used as a low-expansion baseline for struts and flexures.
The focus of the studies is on the mirrors and general support schemes,
and only cursory development work has been done on the supports
themselves.

# PERFORMANCE GOALS AND ANALYSIS METHODS

The major performance parameters examined in this study are: surface
distortions in optical testing on the ground and due to mounting errors;
stresses from launch loads; and natural frequencies. Other factors
driving the design are thermal stability requirements, packaging
limitations, and mass considerations.

The main tool used for predicting mirror performance was Algor finite
element analysis (FEA) software, along with Microsoft Excel and
QuickBasic for data reduction of FEA output.

Numerous variants and iterations of each mirror and its supports were
modeled in FEA. The mirror models were comprised of 3- and 4-node plate
elements, and the supports were comprised of a combination of 6- and
8-node brick elements and 3- and 4-node plate elements. Except in the
cases of free-free modal analysis and mount misalignment sensitivity
analysis, the base of each support was fully constrained using
six-degree of freedom boundary conditions. Free-free modal analysis had
no constraints, while mount misalignments were modeled by releasing one
degree of freedom and replacing it with a displacement boundary
condition.

# Optical Ground Testing

Pre-launch verification of the optics is planned using interferometric
methods along with actual imaging of clean, focused images. The clean,
focused imaging requires that in the ground test, the telescope and each
of the mirrors adhere to their zero-g shape within some tolerance. The
tolerance budget for the difference between ground-test and on-orbit
mirror shapes is included in Table 1. Note that these values are optical
surface distortions, which for mirrors is one half the wave front error
(WFE).

Wavefront error budgets for the mirrors are a subset of the overall
wavefront error budget for the telescope. This budget was generated by
starting with science requirements for a Strehl (ratio of actual to
theoretical telescope point-source brighness) of 0.71 at 633nm. Effects
of multiple mirrors may be combined by vector addition of the RMS
optical path difference (OPD) produced by imperfect mirrors. Such vector
addition of the OPD errors occurs in the pupil plane, and vector
addition is called root sum square (RSS). Alternately, effects of mirror
errors may be combined at the focal plane by multiplying Strehl
contributions of each mirror (values ≤1). Strehl and RMS OPD (σ,
measured in same units as wavelength λ) are related by the Marechal
approximation^4^, under the assumption of small errors σ, which are
incoherent between mirrors.

$$\text{Strehl}\text{\textasciitilde}1 - \left( \frac{2\text{πσ}}{\lambda} \right)^{2}$$

Preliminary studies on the primary mirror indicated that meeting the
surface distortion budget with reasonably light designs would be highly
impractical without offloading the mirror. Furthermore, with the primary
mirror axis aligned horizontal to gravity, it is essentially impossible
to meet the surface distortion budget, even employing supplemental axial
force applications.

Therefore, clean-image testing of the primary mirror requires that its
optical axis be aligned with gravity. This implies that clean-image
testing of the entire telescope occurs in the 'vertical beam'
configuration, as illustrated in Figure 1. It is assumed that primary
mirror offloaders push on the mirror's back surface, implying that the
primary mirror normal is aligned with, but of opposite sense to the
gravity vector. (Note that looking down would not alter the magnitudes
of any of the results of this study.) The orientations of the secondary,
folding flat, and tertiary mirrors relative to gravity for ground-test
imaging are dictated by the primary mirror's orientation. Namely, the
secondary mirror axis is aligned with gravity, the tertiary mirror axis
is horizontal, and the surface normal to the folding flat mirror aims 45
degrees above horizontal.

Optical surface distortions were predicted by applying a one-g
acceleration in the appropriate direction to the mirror/support FEA
model. Offloaders, where used, were modeled using a combination of
boundary conditions and nodal forces. In each mirror model, the mirror's
axis or surface normal was along the Z direction. The Z components of
the front face sheet node displacements were data-reduced to calculate
rms surface distortions, correcting for piston, tilt, and radius of
curvature change where appropriate.

Graphical post-processing of the FEA results allows for visualization of
the optical surface distortions. Hiding all but the front face sheet and
setting the output parameter to Z displacement quickly indicates
shortcomings of various mirror designs and support schemes. Some of
these graphical outputs are included in this report.

# Mounting Errors

Due to imperfections in the mounting interface, when each mirror is
integrated with the telescope structure, there will be some displacement
from its completely stress-free condition. It is expected that during
the integration flow, the mirrors may need to be uninstalled from the
telescope. Whenever a mirror is removed then reinstalled for maintenance
or service, there will be some mount position repeatability uncertainty.
While the mirror mounts would ideally be kinematic, in reality they will
have some elastic stiffness. Therefore, they will impose loads and
moments on the mirrors if they are displaced from their unstressed
position.

Surface distortions caused by displacing a support strut were studied
for each mirror. Displacement boundary elements and guiding boundary
conditions were applied along one axis at a time to the base of a single
strut, without gravity. The magnitude of displacement in each case was
0.025mm, implying a 0.050mm target zone centered on nominal, which is an
achievable degree of repeatability. Resulting surface distortions were
analyzed to find rms net piston and tilt components and rms distortions,
corrected for piston and tilts. The budgets for surface distortions due
to mount errors are 3nm rms for the folding flat mirror and 4nm rms for
the three other mirrors.

# Launch Stresses 

A preliminary, conservative analysis indicates launch limit loads
(design loads) are 12.5g along the launch vehicle thrust direction
(axial) and 8g lateral to the thrust direction. The limit loads will be
refined using coupled loads analysis with the launch vehicle as the
satellite design develops. The current limit load accelerations were
applied to the FEA models of the mirrors and their supports. Maximum
stresses were recorded using post-processing visualizations. Safety
factors against material failure are the quotient of material strength
to the maximum stress. Table 2 summarizes the failure strengths and
failure criteria, along with other properties, used for the materials
modeled in this study.

# Modal Analysis

Modal analyses were done on each mirror both with its mounts fixed at
the base and with the mirror in the free-free configuration. Free-free
analysis includes only the mirror, without its mounts, with no
constraints, free in space. Free-free analysis gives an indication of
the relative stiffness of a design, while analysis on mounts is more
relevant to performance during launch.

For a Delta IV launch vehicle, minimum natural frequencies for primary
structures are 27 Hz axial and 10 Hz lateral, and the minimum natural
frequency for secondary structures is 35 Hz, regardless of
orientation^5^. Each of the mirror/mount assemblies will attach to the
telescope structure, which in turn will attach to the spacecraft. Due to
structural compliance, the mirror natural frequencies will be lower than
those predicted with the mounts constrained at their bases. It is
desirable to have higher natural frequencies, and a general goal in this
study was to have a fundamental frequency on mounts around 100 Hz or
greater.

## PRIMARY MIRROR

# Preferred Design

Figures 3 and 4 illustrate the preferred configuration for the primary
mirror. It is a closed-back ULE meniscus with front-to-back thickness of
200mm. It employs a lightweighted core with hexagonal cells measuring
158mm flat-to-flat. The walls of the core material are 2.5mm thick,
except at the inside and outside diameters (8mm thick). While this
design employs three 'normal' bipods, it provides six mounting
positions, for redundancy. The front face sheet is 12mm thick, and the
back face sheet is 8mm thick. The front face thickness helps minimize
quilting effects from polishing and gravity sag.

Normal bipods are preferred over inverted bipods for this application
because they induce less distortion when subjected to lateral loads.
This means distortions are minimized in horizontal beam testing and in
vertical beam testing with slight misalignments causing lateral loads.
The geometry of the bipods is optimized to minimize distortions under
lateral loads.

![](media/image3.jpeg){width="5.103472222222222in"
height="5.022222222222222in"}

**Figure 3-Preferred design of primary mirror**

![](media/image4.jpeg){width="5.90625in" height="1.2652777777777777in"}

**Figure 4-Side view of preferred design of primary mirror**

# 

# Optical Testing-Summary

The surface distortion budget for the primary mirror for clean-image
testing is 10nm rms. It is necessary to find at least one configuration
that meets that budget. However, verification and repeatability testing
using interferometry may have looser requirements on surface
distortions.

Optical surface distortions have been studied for several possible one-g
test configurations. Configurations include horizontal and vertical beam
orientations and bipod, sling, and offloader support. The results are
summarized in Table 3, and each configuration is described in the
following subsections. In each configuration, material failure safety
factors are 50 or greater for the glass and 20 or greater for the
bipods.

  ------------------------------------------ ----------------------------
  Configuration                              Optical surface distortion
                                             (nm rms)

  Vertical beam, normal bipods at 70% radius 757

  Vertical beam, inverted bipods at 70%      241
  radius                                     

  Vertical beam, normal bipods, 57           6.8
  offloaders                                 

  Horizontal beam, normal bipods at 70%      81
  radius                                     

  Horizontal beam, inverted bipods at 70%    3600
  radius                                     

  Horizontal beam, normal bipods at mirror   49
  periphery                                  

  Horizontal beam, no bipods, sling support  179
  ------------------------------------------ ----------------------------

**Table 3-Surface distortions for various one-g test configurations**

# Optical Testing-Vertical Beam, Normal Bipods at 70% Radius

![](media/image5.jpeg){width="5.602777777777778in"
height="4.022222222222222in"}

**Figure 5-Primary mirror surface distortions (in meters, along mirror
axis) in one-g on normal bipods, 757nm rms, corrected for piston**

# Optical Testing-Vertical Beam, Inverted Bipods at 70% Radius

![](media/image6.jpeg){width="5.394444444444445in" height="3.90625in"}

**Figure 6-Primary mirror surface distortions (in meters, along optical
axis) in one-g on inverted bipods, 241nm rms, corrected for piston**

**Optical Testing-Vertical Beam, Multiple Offloaders**

Clean-image optical testing of the primary mirror in one-g requires
offloading the mirror to meet the surface distortion budget of 10nm rms.
The left side of Figure 7 shows a pattern of 57 individually-force-tuned
offloaders that reduce gravity-induced surface distortions to 6.8nm rms,
as illustrated on the right side of the same figure. A concept for
mechanical implementation of these offloaders is illustrated in Figure
8. In this concept, each offloader is a pneumatic plunger employing a
flexible diaphragm rather than sliding seals, to minimize stiction. Each
plunger uses a load cell to provide feedback for tuning its force to the
optimal predicted value. The array of offloaders is installed on a
fixture that is installed in the space between the back of the primary
mirror and the optics bench for optical testing. As currently
configured, the distance from the back of the primary mirror to the
optics bench ranges from 150mm at the center hole to 245mm at the
periphery. The offloader/fixture assembly is removed once the optics are
verified.

See also Tech Note
<http://snap.lbl.gov/pub/bscw.cgi/d120747/SNAP-TECH-05004.ppt> for more
detailed study of offloading schemes.

![](media/image7.jpeg){width="2.6020833333333333in"
height="2.604861111111111in"}

Figure 7-Primary mirror offloader positions and one-g offloaded
distortions (in meters, along mirror axis), 6.8nm rms, corrected for
piston

![](media/image9.jpeg){width="6.175694444444445in"
height="4.513888888888889in"}

Figure 8-Concept for primary mirror offloading scheme, one sector of
open-back mirror shown in this example

**Optical Testing-Horizontal Beam, General**

During its manufacture and for verification at the component level, the
primary mirror may be tested interferometrically in the horizontal beam
orientation. When testing with the optical axis horizontal, rather than
offloading, the effects of gravity sag may by minimized by rotating the
mirror 180 degrees about its axis and averaging measured distortions in
the two orientations. In the horizontal beam orientation, it is
envisioned that the mirror may be supported on its flight bipods, on
temporary peripheral bipods, or from a strap-like sling cradling the
lower half of the periphery.

# Optical Testing-Horizontal Beam, Normal Bipods at 70% Radius

![](media/image10.jpeg){width="5.190972222222222in"
height="3.7534722222222223in"}

**Figure 9-Primary mirror one-g horizontal beam distortions (in meters,
along mirror axis) on normal bipods at 70% radius, 81nm rms, corrected
for tilt and piston. Gravity vector to the right in this figure.**

# 

# Optical Testing-Horizontal Beam, Inverted Bipods at 70% Radius

![](media/image11.jpeg){width="5.457638888888889in"
height="3.954861111111111in"}

Figure 10-Primary mirror one-g horizontal beam distortions (in meters,
along mirror axis) on inverted bipods at 70% radius, 3600nm rms,
corrected for tilt and piston

# Horizontal beam, normal bipods at mirror periphery

Figure 11 illustrates distortions along the optical axis the mirror
supported on peripheral bipods whose axial positions are optimized for
the horizontal beam orientation. Optical surface distortions are 49nm
rms for this configuration.

![](media/image12.jpeg){width="6.302777777777778in"
height="4.929166666666666in"}

**Figure 11-Primary mirror one-g horizontal beam distortions (in meters,
along mirror axis) on temporary peripheral bipods, 49nm rms, corrected
for tilt and piston**

# Horizontal beam, no bipods, sling support

Figure 12 shows surface distortions predicted for the mirror supported
on a sling in the horizontal beam orientation. The sling is approximated
by applying constraints on the lower half of the mirror periphery that
may only react radially, behaving effectively like rollers. Surface
distortions for this configuration are 179nm rms.

![](media/image13.jpeg){width="6.298611111111111in" height="4.525in"}

Figure 12- Primary mirror one-g horizontal beam distortions (in meters,
along mirror axis) on a sling, 179nm rms corrected for tilt and piston

# Mounting Errors

Table 4 shows rms surface distortion (corrected for tilts and piston)
along with tilts about lateral axes resulting from the displacement of a
single strut by 0.025mm in each of three axes. These displacements are
applied to the strut indicated in Figure 3. The surface distortion
budget for mount errors for the primary mirror is 4nm rms.

  --------------------- ----------------- ----------------- -----------------
  Displacement          Surface           Tilt magnitude    Tilt magnitude
  direction             distortion (nm    about X-axis      about Y-axis
                        rms)              (μrad)            (μrad)

  X (\~radial)          0.33              0                 0

  Y (\~circumferential) 0.52              0                 11

  Z (axial)             0.47              0                 11
  --------------------- ----------------- ----------------- -----------------

**Table 4-Distortions and tilts due to displacement of a primary mirror
bipod strut**

**Comparison of Material/Construction Configurations**

A comparative study was undertaken to evaluate the suitability of
various primary mirror constructions and materials. The mirrors compared
in the study had outside diameters of 2.00 meters, rather than the 2.05
meters of the current design, and they were supported peripherally,
rather than at 2/3 of the outside diameter. The make-up and performance
of the representative mirror designs for each configuration is listed in
Table 5. The preferred design (as described in a previous subsection,
highlighted) is included for reference and should not be directly
compared with the other designs because its diameter and mounting scheme
are distinct.

  ------------------------------------ ---------- ---------- --------- -------- --------- --------
  Material                             ULE        Zerodur    Zerodur   ULE      Zerodur   ULE

  Open/closed/partially closed back    closed     open       partial   closed   open      closed

  Cell shape                           triangle   triangle   hex       hex      hex       hex

  Cell size (hex=flat-flat,            294        294        155       155      155       158
  triangle=side length)                                                                   

  Outside diameter (OD, mm)            2000       2000       2000      2000     2000      2050

  Bipod location                       OD         OD         OD        OD       OD        2/3 OD

  Meniscus thickness (mm)              200        200        200       200      200       200

  Front face thickness (mm)            10         11.5       10        10       10        12

  Back face thickness (mm,             10         N/A        8/15      8        N/A       8
  nominal/reinforced)                                                                     

  Core rib thickness (mm,              2.5/5      6/9        5/10      2.5/10   5/10      2.5
  nominal/reinforced)                                                                     

  Periphery thickness (mm,             10/15      10/17      10/15     10/15    10/15     8
  nominal/reinforced)                                                                     

  Mass of glass (kg)                   205        231        234       197      202       204

  rms surf. distortion from pure       10.8       6.9        5.4       6.4      5.4       5.0
  gravity quilting (nm)                                                                   

  P-V deflection on kinematic edge     7.5        14.3       10.9      7        47        7.6
  mounts (μm)                                                                             

  rms surf. distortion on kinematic    1.7        3.3        2.4       1.6      10.2      1.7
  edge mounts (μm)                                                                        

  First free-free natural frequency    300        189        212       279      99        284
  (Hz)                                                                                    

  First natural frequency on bipods    121        111        114       123      63        111
  (Hz)                                                                                    

  Material safety factors at launch                                                        
  loads:                                                                                  

  Bipods, lateral loads                2.8        2.5        2.5       2.9      2.8       2.7

  Glass, lateral loads                 12         12         16        15       13        17

  Bipods, axial loads                  3.1        2.8        2.8       3.2      3.1       3.1

  Glass, axial loads                   15         7.7        7.9       7.5      3.2       16
  ------------------------------------ ---------- ---------- --------- -------- --------- --------

**Table 5-Comparison of materials and construction for primary mirror**

Distortion due to pure gravity quilting is predicted by constraining
translation of the front face sheet where it intersects ribs and
peripheral elements. The only distortions are due to gravity acting on
unsupported portions of the face sheet within each cell. Distortions on
kinematic edge mounts are included for direct comparison of self-sagging
characteristics. Three equally spaced constraints are used in the
reinforced area of the periphery, and distortions are expressed both in
peak-valley and rms terms. Note that the units for these measurements
are microns, not nanometers.

This comparative study demonstrated that open, closed, and partially
closed back designs can all deliver acceptable performance, if
configured correctly. Predictably, fully closed-back designs can be
significantly stiffer while being lighter than other configurations. As
would be expected, employing hexagonal cells in an open back design
makes for a dramatically less stiff mirror, due to the lack of
continuous, collinear stiffeners. The results of this study helped drive
the preferred design in the direction of the stiff and structurally
efficient closed-back configuration.

**SECONDARY MIRROR**

**Preferred Design**

Preliminary design efforts illustrated some trends with respect to
mirror construction, mass, and performance (see Appendix A). These
efforts have lead to the selection of the material selection and
construction employed in the comparatively more mass-efficient design
presented here.

Figure 13 shows the preferred configuration for the secondary mirror,
along with a proposed support scheme. Because the secondary mirror
position and orientation will be adjustable in flight, it is modeled
with linear actuators. The closed-back ULE, plano-convex mirror is 65mm
thick at its center, thinning to 43mm at the perimeter. The 5mm thick
front face sheet and 3mm thick back face sheet are frit-bonded to the
core. The core has 1.5mm thick ribs and a 2mm thick periphery, with
triangular cells, 72mm long on a side. The mirror mounts on inverted
Invar bipods to the carbon fiber/cyanate ester strongback, which is
supported on the actuators. The modular design allows for testing and
qualification of the mirror separate from the actuators and adds layout
flexibility. It also allows for replacement or maintenance of one or
more actuators during the integration flow without affecting the figure
of the mirror.

![](media/image14.wmf)

**Figure 13-Preferred design of secondary mirror**

Employing six actuators when five degrees of freedom are required (due
to axial symmetry) provides functional redundancy. The axial compliance
of the actuators is modeled by constructing one end of the actuator body
(in purple) as a membrane, tuning the stiffness in a separate model. In
this model, the axial stiffness of the actuators is 20kN/mm.

In the one-g vertical beam optical test, the secondary mirror axis is
aligned with gravity. Employing inverted bipods and evenly spacing the
support points on a circle approximately two thirds the mirror outside
diameter minimizes optical surface distortions, as shown in Figure 14.
Surface distortions of 7.5nm rms meet the budget of 9nm rms.

![](media/image15.jpeg){width="6.286111111111111in"
height="3.6145833333333335in"}

**Figure 14-Secondary mirror surface distortions (in meters, along
mirror axis) in one-g, 7.5nm rms, corrected for piston**

Secondary mirror surface rms surface distortions (corrected for tilts
and piston) and tilts about lateral axes due to displacements of an
actuator's foot are shown in Table 6. The actuator displaced is
indicated in Figure 13. The magnitude of displacement is 0.025mm along
each axis. The surface distortion budget for mount errors for the
secondary mirror is 4nm rms.

  --------------------- ----------------- ----------------- -----------------
  Displacement          Surface           Tilt magnitude    Tilt magnitude
  direction             distortion (nm    about X-axis      about Y-axis
                        rms)              (μrad)            (μrad)

  X (\~radial)          1.30              54                36

  Y (\~circumferential) 0.86              36                24

  Z (axial)             1.20              50                33
  --------------------- ----------------- ----------------- -----------------

**Table 6-Distortions and tilts due to displacement of a secondary
mirror actuator base**

## FOLDING FLAT MIRROR

# Preferred Design

The design of the folding flat mirror is shown in Figures 15 and 16. The
divergent and tapered ribs near the central hole are necessary to avoid
obscuring the diverging beam from the tertiary mirror to the focal
plane. The absence of material in this area, and other packaging
constraints, like the presence of the shutter, provide challenges to
maintain structural integrity against sag in one-g and launch loading.
Fortunately, the flat optical surface makes this mirror largely
insensitive to bulk thermal expansion effects, while the curved mirrors'
radii of curvature change with uniform expansions or contractions.
Therefore, candidate materials for the folding flat mirror have looser
coefficient of thermal expansion requirements. The greater elastic
modulus silicon carbide exhibits compared to the glass options make it
suitable for this application.

# 

![](media/image16.wmf)

**Figure 15-Preferred design of folding flat mirror**

![](media/image17.jpeg){width="5.485416666666667in"
height="1.6006944444444444in"}

**Figure 16-Side view of preferred design of folding flat mirror**

The mirror is 100mm thick overall, with 5mm thick face and walls except
the 10mm zone near where the upper struts mount.

In the vertical beam optics test, this design has 9.1nm rms surface
distortion, corrected for tilt and piston. Comparable, or even much
bulkier and more massive glass designs could not meet the surface
distortion budget of 10nm rms without offloading. Avoiding offloading
this mirror greatly simplifies testing. The surface distortion is
illustrated in Figure 17. This shows the predominance of the tilt of
approximately 0.1 arcsecond. The rms surface distortion value is very
sensitive to the positioning of the uppermost struts.

![](media/image18.jpeg){width="6.295833333333333in"
height="3.5493055555555557in"}

Figure 17-Folding flat mirror surface distortions (in meters, along
mirror axis) in one-g, 9.1nm rms, corrected for tilt and piston

The mirror on its supports has natural frequencies all above 180 hertz
and material safety factors greater than ten subject to launch loads.

Surface distortions (corrected for tilts and piston) and tilts about
lateral axes due to displacements of bipod feet are in Table 7. Figure
15 illustrates the nomenclature to indicate which strut is displaced.
The magnitude of displacement is 0.025mm along each axis. The surface
distortion budget for mount errors for the folding flat mirror is 3nm
rms.

+---------+-----------------+-------------+------------+------------+
| Di      | Displacement    | Surface     | Tilt       | Tilt       |
| splaced | direction       | distortion  | magnitude  | magnitude  |
| strut   |                 |             | about      | about      |
|         |                 | (nm rms)    | X-axis     | Y-axis     |
|         |                 |             |            |            |
|         |                 |             | (μrad)     | (μrad)     |
+---------+-----------------+-------------+------------+------------+
| Lower   | X               | 0.91        | 25         | 21         |
|         | (\~c            |             |            |            |
|         | ircumferential) |             |            |            |
+---------+-----------------+-------------+------------+------------+
|         | Y (\~radial)    | 1.06        | 30         | 26         |
+---------+-----------------+-------------+------------+------------+
|         | Z (axial)       | 1.19        | 30         | 26         |
+---------+-----------------+-------------+------------+------------+
| Middle  | X (\~radial)    | 0.43        | 0.9        | 13         |
+---------+-----------------+-------------+------------+------------+
|         | Y               | 0.77        | 2.3        | 32         |
|         | (\~c            |             |            |            |
|         | ircumferential) |             |            |            |
+---------+-----------------+-------------+------------+------------+
|         | Z (axial)       | 0.76        | 2.3        | 32         |
+---------+-----------------+-------------+------------+------------+
| Upper   | X (\~radial)    | 0.76        | 27         | 7.7        |
+---------+-----------------+-------------+------------+------------+
|         | Y               | 0.76        | 28         | 7.9        |
|         | (\~c            |             |            |            |
|         | ircumferential) |             |            |            |
+---------+-----------------+-------------+------------+------------+
|         | Z (axial)       | 0.93        | 33         | 9.2        |
+---------+-----------------+-------------+------------+------------+

**Table 7-Distortions and tilts due to displacement of folding flat
mirror struts**

Employing 'cathedral' ribs behind the face sheet, within the existing
rib cells might allow for a thinner face sheet or mitigate effects of
polishing quilting or gravity quilting. This feature is one that may be
implemented based on experience and preferences of potential vendors.

## TERTIARY MIRROR

# Preferred Design

Figure 18 shows the design of the tertiary mirror and its bipods. The
plano-concave, closed back ULE mirror is 90mm thick at the perimeter and
42mm thick in the center. Five mm thick front and 3mm back face sheets
bond to a triangular-celled core with 101mm sides to each cell. The ribs
of the core are 2mm thick, except those that form a triangle whose
vertices are at the bipod mounting positions, which are 4mm thick. The
periphery is 4mm thick generally, 15mm thick where the bipods mount,
tapering to 9mm thick at the face sheets. The thick periphery tapers
near the face sheets because the practical upper limit on frit bond
width is understood to be 9.5mm.

![](media/image19.wmf)

**Figure 18- Preferred design of tertiary mirror**

The bipods attach to the end of the optics bay. In the vertical beam
optical test, the mirror's axis is horizontal. In this orientation, the
rms surface distortion is very sensitive to the position of the bipods
along the optical axis. FEA modeling predicts approximately where the
ideal mounting position is, but the actual hardware will require
adjustment to optimize the mounting positions. Figure 19 shows surface
distortions in the direction parallel to the optical axis with the
mirror loaded as in the one-g vertical beam optical test. The bipods are
in the optimal position to minimize distortions, 8.8nm rms, meeting the
budget of 10nm rms. The highly localized distortions near where the
upper bipods illustrate the need for the thickened periphery in those
areas.

![](media/image20.jpeg){width="6.299305555555556in"
height="4.634027777777778in"}

Figure 19-Tertiary mirror surface distortions (in meters, along mirror
axis) in one-g, 8.8nm rms, corrected for tilt and piston. Bipods omitted
for clarity.

Table 8 shows rms surface distortion (corrected for tilts and piston)
and tilts about lateral axes resulting from the displacement of a single
strut by 0.025mm in each of three axes. These displacements are applied
to the strut indicated in Figure 18. The surface distortion budget for
mount errors for the tertiary mirror is 4nm rms.

  --------------------- ----------------- ----------------- -----------------
  Displacement          Surface           Tilt magnitude    Tilt magnitude
  direction             distortion (nm    about X-axis      about Y-axis
                        rms)              (μrad)            (μrad)

  X (\~circumferential) 0.57              23                0

  Y (\~radial)          0.05              0                 0

  Z (axial)             0.56              23                0
  --------------------- ----------------- ----------------- -----------------

**Table 8-Distortions and tilts due to displacement of a tertiary bipod
strut**

**CONCLUSION**

Baseline designs of the four mirrors comprising the SNAP telescope
optics are complete. They are all feasible, manufacturable
configurations that meet the performance requirements for the mission,
including surface distortions in optical testing, mass and volume
constraints, and survival of the launch" environment. They may be
removed for service and reinstalled without inducing unacceptable
optical distortions. They demonstrate the viability of the optical
layout. The designs are at a state appropriate to engage possible
vendors to develop detailed designs.

## REFERENCES

1.  M. Lampton, et al, "SNAP Telescope" SPIE **4849** pp.215-226, 2002.

2.  M. Lampton, et al, "SNAP Telescope: an Update" SPIE **5166** pp.
    113-123, 2004.

3.  M. Sholl, et al, "SNAP Telescope" SPIE **5487-73**, in press 2004.

4.  Born & Wolf, Principles of Optics, Sixth Edition, PP. 459-464,
    Pergamon Press, 1980.

5.  Boeing Delta IV Payload Planners Guide, October 2000.

**Appendix A, Preliminary Trades on Secondary Mirror Design**

Various construction configurations for the secondary mirror were
studied to compare the merits of each design type. This comparative
study was made assuming kinematic support at three points, somewhat
representative of 'normal' bipod mounting for the mirror. In addition,
in this study, material wall thicknesses were assumed to be 5mm or
greater. Therefore, these results may not be directly comparable with
the present preferred design, with much thinner wall construction and
inverted bipods. However, the results are illustrative of the
performance of the various configurations and the trends within each
configuration.

Figures A1 and A2 show rms surface distortions subject to one g in the
vertical beam test for different secondary mirror designs. Figure A1
illustrates trends against mirror mass, while Figure A2 shows trends
related to overall thickness for reasonable designs. Each data point
represents a best realistic effort given the configuration and mass
constraint. The most efficient designs in both charts are toward the
origin, where surface distortions and size or mass are minimized.

![](media/image21.jpeg){width="6.298611111111111in"
height="4.721527777777778in"}

Figure A1-Secondary mirror surface distortions in one g vertical beam
test vs mirror mass (450mm mirror OD, kinematic support)

![](media/image22.jpeg){width="6.299305555555556in" height="4.84375in"}

Figure A2-Secondary mirror surface distortions in one g vertical beam
test vs mirror thickness (450mm mirror OD, kinematic support)

These charts illustrate some informative trends. As might be expected, a
solid glass mirror is the most volume-efficient design, but is the least
mass-efficient. Supporting the mirror near 2/3 the outside diameter
produces roughly one half to one third the rms surface distortion.
Closed back ULE is more efficient than open back Zerodur for the range
of this study. For greater mirror thicknesses, the benefits of the back
face sheet diminish as shear deflections become more significant
compared with bending deflections. For realistic designs fitting the
constraints of the SNAP telescope, closed back ULE is the best choice.