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Journal of the Korean Physical Society, Vol. 57, No. 3, September 2010, pp. 440∼445
Optimization and Tolerance Scheme for a Mirror Mount Design Based on
Optomechanical Performance
Hagyong Kihm∗and Yun-Woo Lee
Center for Space Optics, Korea Research Institute of Standards and Science, Daejeon 305-340
(Received 28 May 2010, in final form 13 August 2010)
We present the design optimization and tolerance scheme for the fold mirror system in our satellite
telescope. A new type of mirror and mount flexure is proposed. The mirror is light-weight and is
designed with an asymmetric off-axis configuration. The mount is a combination of monolithic axial
and lateral supports. The design parameters of the flexure mount are optimized for optomechanical
performances by using a simulated annealing method. The sensitivities of the optomechanical
performances to the design parameters are tabulated to obtain tolerance for the critical dimensions
by using finite element analysis(FEA). Optical distortions are examined with Zernike polynomials
for qualitative analysis and design feedback. This unprecedented mirror and mount design would
be a promising candidate for an asymmetric or off-axis optical system.
PACS numbers: 42.15.Eq, 07.87.+v, 95.55.Fw
Keywords: Mirror mount, Optomechanics, Flexure, Tolerance, Space optics
DOI: 10.3938/jkps.57.440
I. INTRODUCTION
Optical systems require support structures that isolate
the optical parts from mechanical and thermal loads.
Mechanical loads are gravity, vibration, assembly or
mounting errors, and fabrication residual stress. Ther-
mal effects might distort optical surfaces and induce dis-
placements due to a mismatch of CTE(coefficients of
thermal expansion) or to a temperature gradient within
the optical system. Therefore, the performance of optical
mirrors, such as those employed in satellite telescopes,
may be severely degraded by an inappropriate mounting
configuration. The general design objective and philoso-
phy of an optical mirror mount are introduced by Chin
[1]. The concepts for a successful mounting should min-
imize optical distortions and provide a simple means of
alignment. Also, the mount must be athermal. Achiev-
ing good thermal stability performance might be critical
even in laboratory environment [2].
A kinematic mount is an ideal support constraining
three orthogonal axes without redundancy. However, the
point contact desirable for a kinematic support is not fea-
sible in environmentally challenged systems. Instead, a
semi-kinematic mount with a finite contact area is usu-
ally adopted to disperse local stresses. Flexure mounting
may be regarded as a semi-kinematic design. A flexure
is a monolithic structure providing elastic motions in a
predefined way. The benefits of using flexures include
∗E-mail: hkihm@kriss.re.kr
lack of the hysteresis and the friction effects inherent in
semi-kinematic mounts. Also, maintenance is unneces-
sary and fabrication has become common practice with
electrical discharge machining. A mirror mount flexure is
not intended for linear or precise motions. Different from
the flexure hinges used in actuator mechanism, a mirror
mount flexure minimizes optical surface distortions and
maintains optical alignment under operation or trans-
port. Kinematic principles determine the location and
the direction of a mounting flexure. The line of action
of the flexure should pass through the mirror’s center of
gravity. Compliance should be provided to athermalize
the mirror and mounting flexures. For example, radial
compliance should be added in an axisymmetric mirror
element. Tangential compliance is also required to pre-
vent assembly stress from propagating toward the mirror
surface [3]. Lateral mirror supports are usually respon-
sible for radial and tangential compliance. Axial mirror
supports, in the case of massive large aperture telescopes,
reduce mirror surface distortion.
Flexure mounts can be categorized according to the
type of flexure element. Simple blade flexures are usu-
ally used as tangential edge supports for relatively small
axisymmetric mirrors [3]. A bipod flexure, which is the
most common support type, generally gives better re-
sults in terms of optical performances [4, 5]. Flexure
hinges can also be categorized according to the cross-
sectional shape. There are corner-filleted flexure hinges
and conic-section (circular, elliptical, parabolic, and hy-
perbolic) flexure hinges [6–8]. Corner-filleted flexures
are more bending-compliant and induce lower stresses.
-440-
Optimization and Tolerance Scheme for a Mirror Mount Design ··· – Hagyong Kihm and Yun-Woo Lee -441-
Table 1. Design loads and requirements for the mirror assembly
Design loads
vibration 59 G (gravitation)
assembly error 10 µm
isothermal 20 ◦C±12
Optomechanical
performances
fundamental frequency >210 Hz
mirror distortion <15 nm
mirror decenter <2µm
mirror tip/tilt <0.01◦
Safety factors for
survival
flexure yield >1.25
flexure buckling >10
mirror fracture >2
adhesive breakage >3
Material strength
mirror fracture (Schott Zerodur) 10 MPa
flexure yield (Invar36) 210 MPa
adhesive (3M EC2216 B/A Gray) 6 MPa
Less precision in rotational motion does not matter in
mirror mount flexures. Most mirror mount flexures, be
they blade or bipod, have corner-filleted cross sections
for these reasons [3,9].
Optimal design of flexure hinges is possible for pre-
cision mechanisms based on theoretical closed-form so-
lutions [10, 11] or finite element analysis (FEA) [12].
Empirical formulation and dimensionless graph analy-
sis have also been reported [7,13]. Those research-efforts
have provided initial design steps and even optimized
simple cases for precision mechanisms. Mirror mount
flexures can also benefit from those results being used
as design guidelines. Optical performance, however, is
a major criterion in mirror mounting flexures. For ex-
ample, a theoretical derivation of optimum mount solu-
tion was made for a simple mirror disk [14]. The FEA
has gained popularity in optical mirror mount design as
the mirrors are light-weight and more complicated [15–
17]. A comparison between a FEA and an interferomet-
ric measurement showed the two results proved to be in
good agreement [18].
This paper presents a flexure design procedure with
a simulated annealing (SA) algorithm [19] for optimal
optomechanical performance. Also, we propose a new
methodology to obtain tolerances for the critical dimen-
sions by using a FEA. Section II. shows the overall con-
figuration of a folding mirror system and design opti-
mization. Section III. explains the performance results
and tolerances based on a sensitivity table. Conclusions
follow in Section IV.
II. PARAMETRIC OPTIMIZATION USING
SIMULATED ANNEALING
The mirror and flexure assembly presented in this pa-
per has a unique mounting scheme. The mount has three
sets of bipod flexures on a single frame. The main bi-
pod supports at the center of gravity, and the others
prevent bending moments at the rear. Traditional mir-
ror mounts have axial and/or lateral supports fabricated
and assembled separately, but this mount is monolithic
due to the mechanical interference with other mirror as-
semblies. With this concept, two different mirror sets
are made in a space telescope system. The design and
the tolerance processes developed in this paper are ap-
plied directly to those two cases. For optical systems for
military or space missions, the design and tolerance pro-
cesses have been veiled in the optomechanical engineer’s
expertise or have been determined by their experiences.
Although this research seems to focus on a specific appli-
cation, we tried to generalize the optomechanical design
process with our example.
Design loads and optomechanical requirements for the
mirror system are summarized in Table 1. The vibration
load comes from the launch environment and has three
orthogonal directions. The assembly load is related to
the fixture base flatness. Flexure mount is fixed with
fasteners on flat bezel inserts, and their irregularities dis-
tort the flexure mount, resulting in optical distortion.
The flatness tolerance of the bezel inserts is 10 µm. The
satellite telescope suffers temperature variations in or-
bit, and a mismatch of CTE between the mirror, flexure,
and adhesive should not affect the optical performance
or the system’s survival. The mirror surface distortion
should be less than 15 nm for diffraction-limited system
performance. Safety factors for system’s survival differ
depending on the parts’ material and the failure modes.
Material strengths are taken from the vendors’ reports.
The adhesive shear strength, however, was obtained by
using an in-house coupon tests to reflect the real appli-
cation.
Figure 1 shows a schematic of the mirror assembly.
The mirror’s center of mass, which determines the angle
and the position of bipod flexures, is marked for refer-
ence. Blade flexures provide radial compliance for ther-
mal expansion mismatch, and bipod flexures have tan-
gential compliance for the assembly load and the bending
-442- Journal of the Korean Physical Society, Vol. 57, No. 3, September 2010
moments. The front flexure is positioned in the vicinity
of mirror’s center of mass while the other flexures sup-
port the rear. Critical dimensions of concern are denoted
for parametric optimization. Their names and optimized
dimensions are given in Table 2. The rotation centers of
the bipod flexures coincide with the mirror’s center of
mass. The base frame of the flexure mount is triangle-
shaped and has tree holes for joining bolts with bezel
inserts.
We implemented a simulated annealing (SA) method
for structural optimization [19]. SA is proven to be useful
in global optimization and avoids being trapped at local
minima [20]. The algorithm employs a random search,
which accepts some increasing changes in the objective
function f, as well as decreasing changes. Values increas-
ing fare accepted with a probability
p= exp −
δf
T,(1)
where δf is the increase in fand Tis referred to as the
temperature. Tplays a role similar to the temperature
in a physical annealing process. To avoid getting trapped
at local minima, the reduction rate should be slow.
The objective of this study is to find a design meet-
ing all the requirements shown in Table 1. We used
MATLABrfor SA routine and optical analysis with
Zernike polynomials [21]. MATLABrgets FEA results
from CATIArand updates dimensional parameters af-
ter optomechanical analysis. We used four constraints
for mechanical safety in each gravity direction. They are
listed in Table 1 as safety factors for survival. In total,
13 constraints, including a frequency requirement, were
used in the SA. Also, we tried to minimize optical dis-
tortions and stresses in the adhesive. However, we could
not observe any significant improvement in optical dis-
tortions, which proved to be insensitive to dimensional
variations of the flexures. We found from this result
that optical performance was determined mainly by the
mounting configuration rather than by the detailed di-
mensions of the flexures. This manifests the importance
of proven design examples and optomechanical experi-
ences. On the other hand, the stresses in the adhesive
under vibrational loads were sensitive to the flexure di-
mensions. Figure 2 shows the parametric variations to
minimize stresses in the adhesive pads. Over 90 itera-
tions were conducted for convergence. Figure 3 shows
the history of shear stress at the adhesive from Fig. 2.
About a 16% improvement was observed.
III. PARAMETRIC OPTOMECHANICAL
PERFORMANCE EVALUATION
Aside from design and optimization of the mirror and
its mounting structures for the given requirement, tol-
erances for the critical dimensions are important. Tol-
Fig. 1. (a) Schematic of the mirror assembly. (b) Section
view showing the interface between the mirror and the flexure
mount. (c) Independent design parameters for the flexure
mount.
Fig. 2. (Color online) Parametric variations from simu-
lated annealing optimization.
erance determines manufacturing methods and assem-
bly procedures. Fabrication cost, time, and difficulty all
depend on dimensional tolerance. Performance repro-
ducibility also rely on tolerances. Mechanical designers
usually decide the tolerance of each part from the view-
point of manufacturability, assembly, and cost. In op-
tomechanical systems, however, tolerance cannot be de-
termined solely from mechanical viewpoints. We should
consider optical surface distortion, tip-tilt, and a decen-
ter induced by gravity, assembly load, and thermal load.
Adhesive bonding, which is generally adopted for cou-
pling an optical component with a metallic mount, might
be a critical factor. Several authors have reported on op-
tomechanical sensitivity and tolerances [22,23], but the
results were confined to the lens assembly and the fabri-
Optimization and Tolerance Scheme for a Mirror Mount Design ··· – Hagyong Kihm and Yun-Woo Lee -443-
Table 2. Flexure design parameters and their values (mm).
D1
Front radial flexure
width 31.68
D2height 28.35
D3thickness 2.04
D4Front tangential
flexure
width 6.96
D5thickness 4.17
D6Rear radial flexure height 16.15
D7thickness 2.56
D8Rear tangential flexure width 10.04
D9thickness 3.85
Fig. 3. Shear stress minimization with the parametric vari-
ations in Fig. 2.
cation. In this paper, we propose to derive dimensional
tolerances for a mirror mount from the sensitivities of
the optomechanical performances, which, to our knowl-
edge, is the time such an approach has been suggested.
The tolerance is assumed to be inversely proportional to
the sensitivity, which is defined as
Sij =∂Pi
∂Dj
,(2)
where Sij is the sensitivity of performance Piwith re-
spect to dimension Dj. The tolerance of dimension Dj
is proportional to the allowable tolerance ratio Tjas fol-
lows.
Tj∝min
i
Pi
Sij
δi
,where δi=
Qi−Pi
Qi
.(3)
Qiis the performance requirement, and δiis the safety
distance indicating how critical the dimension is. The
tolerance is tightened when the safety distance has a
small value.
Table 3 shows the performance parameters and their
values under environmental loads. The mirror’s rigid
body motion due to the assembly load is ignored as it can
be adjusted during integration. Qiwas calculated from
the design load, safety factor, and material strength of
each part. For example, Q1was obtained by dividing the
material strength (210 MPa) by the gravity load(59 G)
Fig. 4. (Color online) Mirror surface distortion under 1 G
gravity in y-direction.
and the safety factor (1.25). There are three orthogonal
directions in gravity and three positions for the assembly
load. In this paper, however, only maximum values of
concern are presented. In the case of a 1-G gravity load,
most performances give small safety distances, except
for the mirror’s tip/tilt. This means that the mechanical
safety largely depends on the flexure design. The mir-
ror’s decenter under a 1-G gravity shows a small safety
distance δ12, which will be released in a non-gravity space
environment. The CTE mismatch between the mirror,
flexure, and adhesive results in a high shear stress at the
adhesive, which is evident in the small distance δ9.
The Optical distortion of the mirror surface was evalu-
ated by fitting the displacement with Zernike polynomi-
als up to 49 terms [21]. Piston and tilt terms were used
to calculate the decenter and the tip/tilt errors. The rest
terms relate to the wavefront errors degrading the optical
performances. Figure 4 shows the distortion map under
1-G gravity in the y-direction. Astigmatism is dominant
with 23 nm, and the overall root-mean-square (rms) value
is 4.1 nm, showing satisfactory performance. Figure 5
shows the distortion map under an assembly load(10 µm)
at the rear insert position. The rms value is 5.2nm,
which also gives enough safety distance δ17.
Table 4 is the sensitivity Sij and derived tolerance of
each dimension Dj. The tolerance ratios Tjwere divided
by the minimum value T4and are plotted in Figure 6
-444- Journal of the Korean Physical Society, Vol. 57, No. 3, September 2010
Table 3. Performance parameters and their values.
i Performance parameter Design load QiPiδi
1
σVon Mises @ flexure (MPa)
1 G gravity 2.85 2.23 0.217
2 Assembly (10 µm) 210 10.3 0.951
3 Isothermal (1 ◦C) 5.25 0.73 0.861
4
σVon Mises @ mirror (kPa)
1 G gravity 84.7 58.5 0.309
5 Assembly (10 µm) 5000 480 0.904
6 Isothermal (1 ◦C) 330 120 0.636
7
σShear @ adhesive (kPa)
1 G gravity 33.8 18.3 0.459
8 Assembly (10 µm) 1500 150 0.9
9 Isothermal (1 ◦C) 150 110 0.266
10 Fundamental frequency 210 315 0.5
11 Buckling factor 10 43 3.3
12
Mirror decenter (µm)
1 G gravity 2 1.46 0.27
13 Isothermal (1 ◦C) 2 0.197 0.902
14
Mirror tip/tilt (deg)
1 G gravity 0.01 0.00039 0.961
15 Isothermal (1 ◦C) 0.01 0.00011 0.989
16
Mirror distortion (nm)
1 G gravity 15 10.6 0.293
17 Assembly (10 µm) 15 7.66 0.489
18 Isothermal (1 ◦C) 15 0.4 0.973
Fig. 5. (Color online) Mirror surface distortion under an
assembly load of 10 µm displacement at the rear insert.
for qualitative comparison. One can then distribute the
tolerances based on these parameters. Table 4 can also
be used as a lookup table to interrogate performance
degradation due to dimensional errors. The front radial
flexure (D1−3) has relatively large tolerances compared
with other dimensions. Tangential flexures are proven
to be critical for performance stability, so care must be
taken in the manufacturing process. The last column
shows the tolerance example for our case.
IV. CONCLUSIONS
We presented a new type of mirror/mount flexure de-
sign and proposed a new tolerance scheme for mirror
mounting flexures based on FEA. The mirror is a pris-
Table 4. Sensitivities and derived tolerances.
j S1jS2jS3j··· Tjtol. (mm)
1 000.0E+0 10.1E+0 -100.0E+0 69.97 0.50
2 -10.0E+3 -8.9E+0 -105.4E+0 39.95 0.20
3 80.0E+3 228.7E+0 645.1E+0 4.99 0.04
4 -400.0E+3 55.6E+0 321.9E+0 1.00 0.01
5 250.0E+3 -83.8E+0 49.5E+0 1.60 0.02
6 40.0E+3 -1.0E+3 95.0E+0 9.99 0.10
7 -130.0E+3 1.4E+3 3.1E+3 2.24 0.02
8 160.0E+3 -312.8E+0 -326.9E+0 2.50 0.02
9 -320.0E+3 -5.0E+3 -634.7E+0 1.25 0.01
Fig. 6. Tolerance ratio Tjfor each dimension is plotted for
comparison.
matic off-axis folding mirror. The mount has three sets
of monolithic flexures on a single frame, and each flex-
ure is a combination of blade and bipod flexures. The
optomechanical performance and the sensitivity method
Optimization and Tolerance Scheme for a Mirror Mount Design ··· – Hagyong Kihm and Yun-Woo Lee -445-
were proposed as new tolerance guidelines rather than
relying on the arbitrary tolerances made by experienced
engineers. We applied this procedure to two other sets of
mirror assemblies for an infrared space telescope. Mis-
sion critical systems, like space and military optics, will
benefit from our results.
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