High Precision CTE-Measurement of SiC-100 for Cryogenic
Space-Telescopes
K. Enya1, N. Yamada2, T. Onaka3, T. Nakagawa1, H. Kaneda1, M. Hirabayashi4,
Y. Toulemont5, D. Castel5, Y. Kanai6, and N. Fujishiro6
enya@ir.isas.jaxa.jp
Received ; accepted
1 Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency,
3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan
2 National Metrology Institute of Japan, Advanced Industrial Science and Technology, 3
Tsukuba Central, Tsukuba, Ibaraki 305-8563, Japan
3 Department of Astronomy, Graduate School of Science, University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
4 Sumitomo Heavy Industries, Ltd., 5-2 Niihama Works, Soubiraki-cho, Niihama, Ehime
792-9599, Japan
5 Astrium Satellites, Earth Observation, Navigation & Science (F), 31 av des Cosmo-
nautes, 31042 Toulouse Cedex 4, France
6 Genesia Corporation, Mitaka Sangyo Plaza 601, 3-38-4 Shimorenjyaku, Mitaka, Tokyo
181-0013, Japan
http://arxiv.org/abs/0704.1515v1
– 2 –
ABSTRACT
We present the results of high precision measurements of the thermal expan-
sion of the sintered SiC, SiC-100, intended for use in cryogenic space-telescopes,
in which minimization of thermal deformation of the mirror is critical and pre-
cise information of the thermal expansion is needed for the telescope design. The
temperature range of the measurements extends from room temperature down to
∼ 10K. Three samples, #1, #2, and #3 were manufactured from blocks of SiC
produced in different lots. The thermal expansion of the samples was measured
with a cryogenic dilatometer, consisting of a laser interferometer, a cryostat, and
a mechanical cooler. The typical thermal expansion curve is presented using the
8th order polynomial of the temperature. For the three samples, the coefficients
of thermal expansion (CTE), α#1, α#2, and α#3 were derived for temperatures
between 293K and 10K. The average and the dispersion (1σ rms) of these three
CTEs are 0.816 and 0.002 (× 10−6/K), respectively. No significant difference was
detected in the CTE of the three samples from the different lots. Neither in-
homogeneity nor anisotropy of the CTE was observed. Based on the obtained
CTE dispersion, we performed an finite-element-method (FEM) analysis of the
thermal deformation of a 3.5m diameter cryogenic mirror made of six SiC-100
segments. It was shown that the present CTE measurement has a sufficient accu-
racy well enough for the design of the 3.5m cryogenic infrared telescope mission,
the Space Infrared telescope for Cosmology and Astrophysics (SPICA).
Subject headings: instrumentation: miscellaneous — methods: laboratory —
techniques: miscellaneous
– 3 –
1. Introduction
Development of cryogenic light-weight mirrors is a key technology for infrared
astronomical space-telescope missions, which have large advantages owing to being free
from the turbulence, thermal background, and absorption caused by atmosphere. The
light-weight mirror technology is essential to bring large mirrors into space that enable a
high sensitivity and a high spatial resolution. The infrared sensitivity of the space-telescope
is vastly improved by reduction of the thermal background by cooling the telescope to
cryogenic temperatures. Thus infrared space telescopes badly need cooled light-weight
mirrors with a sufficient optical quality.
Silicon-carbide (SiC) is one of the most promising materials for space telescopes because
of its high ratio of stiffness to density. Japanese space mission for infrared astronomy,
AKARI, carries a 68.5 cm aperture telescope whose mirrors are made of sandwich type SiC
(Murakami 2004; Kaneda et al. 2005). The entire AKARI telescope system is cooled down
to 6K by helium gas vaporizing from liquid helium. AKARI was launched in February 2006
and the telescope system performance has been confirmed to be as expected from pre-launch
ground tests (Kaneda et al. 2007). The Herschel Space Observatory, a submillimeter
satellite mission by the European Space Agency (Pilbratt 2004), employs mirrors of sintered
SiC (SiC-100) provided by Boostec Industries and EADS-Astrium (Breysse et al. 2004).
The 3.5m diameter primary mirror of the Herschel telescope is made of 12 segments brazed
together. The Herschel Space Observatory will be launched in 2008 to make observations
in 60–670µm, whilst the telescope will be kept to ∼ 80K by passive cooling. SiC-100 is
one of the most frequently used SiCs for space optics. It has been used for the telescopes
of ALADIN, GAIA, ROCSAT and other missions (Breysse et al. 2004 and the references
therein).
The Space Infrared telescope for Cosmology and Astrophysics (SPICA) is the next
– 4 –
generation mission for infrared astronomy, planned by the Japan Aerospace Exploration
Agency (Nakagawa 2004; Onaka et al. 2005). The SPICA telescope is required to have a
3.5m diameter aperture and will be cooled down to 4.5K by the combination of radiative
cooling and mechanical coolers (Sugita et al. 2006). SPICA is planned to be launched in
the middle of the 2010s and execute infrared observations in 5–200µm. SiC-100 is one of
the promising candidate materials for the mirrors and structures of the SPICA telescope,
whilst carbon-fiber reinforced silicon-carbide is another candidate now being investigated
(Ozaki et al. 2004; Enya et al. 2006; Enya et al. 2004). Monolithic primary mirrors can
be manufactured by the joint segment technology, similar to the primary mirror for the
Herschel telescope. The requirement for the surface figure accuracy of the SPICA primary
mirror is, however, better than 0.06µm rms. This requirement is ∼ 20 times more severe
than the Herschel Space Observatory because of the difference in the targeted wavelength
range.
In the development of cryogenic space-telescopes, it is important to suppress the
thermal deformation of the mirror caused by cooling to satisfy the requirement for the
surface figure accuracy (Kaneda et al. 2003; Kaneda et al. 2005). Therefore, the study
of the coefficients of thermal expansion (CTE) of the material used for the mirror is
important. The CTE data of the mirror material are indispensable for the design of
cryogenic space-telescopes because the actual telescope mirror needs to accommodate
complicated support structures consisting of materials different from the mirror. If the CTE
measurement of test pieces of the mirror material has a sufficient accuracy, it will enable us
to predict the thermal deformation of a segmented mirror caused by the dispersion in the
CTE of each segment. The most direct and highly sensitive test of the thermal deformation
is an interferometer measurement of the actual mirror at cold temperatures. The CTE data
are useful to interpret the result of direct measurements of the mirror and investigate the
origin of the observed deformation.
– 5 –
However, measurements of the CTE of SiC and its dispersion have not so far been
performed with a sufficient accuracy and for a wide temperature range. Especially, little
data are available at temperatures lower than 77K, which can be realized by only liquid
nitrogen cooling. Prior to this work, available CTE data of SiC-100 were limited at
temperatures higher than ∼ 77K (Toulemont 2005). Pepi & Altshuler (1995) presented the
CTE data of reaction bonded optical grade (RBO) SiC down to 4K based on measurements
with samples made from one block of the RBO SiC. The CTE of the new-technology SiC
(NT-SiC), developed high-strength reaction-sintered SiC, has been reported down to 20K
for one sample by Suyama et al. (2005).
In this work, we present the results of high precision CTE measurements of SiC-100
down to cryogenic temperatures for samples from three different lots. We set two major
goals for the present work: one is to provide the typical CTE of SiC-100 for the development
of the SPICA telescope, and the other is to estimate the thermal deformation of segmented
mirrors from the measured CTEs.
2. Experiment
2.1. Sample
Figure 1 shows three samples (#1, #2, and #3) of SiC-100 measured in this
work. All of the samples were manufactured by Boostec Industries and EADS-
Astrium (Breysse et al. 2004). Each of the samples was extracted from blocks of
SiC produced in different lots. The locations of each sample in the SiC blocks are
arbitrary. The samples are of a rectangular parallelepiped shape with the dimension of
20.00+0.05
−0.00mm×20.00
+0.05
−0.00mm×6.0
−0.0 mm. Flatness, parallelism, and roughness of the
20.00mm×6.00mm surfaces are important for the measurement of this work: The flatness
– 6 –
of these surfaces is ≤ λ/10 rms, where λ is the wavelength of the He-Ne laser, 632.8 nm.
The parallelism of the opposing 20.00mm×6.00mm surfaces is less than 2′′, whilst the
parallelism of the opposing 20.00mm×20.00mm surfaces is less than 1.0◦. All of the
20.00mm ×6.00mm surfaces are polished to an optical grade and the surface roughness
finally achieved is less than 3 nm rms. Owing to the polished surface, the directly reflected
laser light by the sample was used for the measurement. As the result, the measurement
was free from the uncertainty due to any additional mirrors or coating, which would be
needed if the direct reflected light could not be used.
2.2. Measurement
The measurement of the thermal expansion was carried out with the laser interferometric
dilatometer system for low temperature developed in the National Metrology Institute of
Japan, Advance Industrial Science and Technology (Yamada & Okaji 2000, hereafter the
paper–I; Okaji & Yamada 1997; Okaji et al. 1997). The system consists of the cryostat,
the cryogenic mechanical refrigerator of the GM cycle (V204SC) by Daikin Industries Ltd.,
and the interferometer utilizing acousto-optical modulators and the stabilized He-Ne laser
system (05STP905) by Melles Griot. The cooling is performed with the refrigerator alone
and no cryogen is needed. The minimum temperature achieved in this work is about 10K.
The space around the cold stage of the cryostat, containing the installed sample, is filled
with 130Pa helium gas and sealed off at room temperature before cooling to ensure thermal
uniformity. The configuration of the whole cryostat and the sample installation into the cold
stage of the cryostat are shown in figures 1 and 2 of the paper–I, respectively. The change
of length of the sample, ∆L, is measured with the double path type laser interferometer
of the optical heterodyne method with the digital lock-in amplifier (Model SR-850) from
Stanford Research System. Details of this system are given in the paper–I.
– 7 –
We made totally 6 measurements of the CTEs of SiC-100. In each measurement, the
sample was initially cooled down to ∼ 10K and then the proportional integral derivative
(PID) temperature control was applied. After the the temperature had been stabilized, the
temperature and the change of the sample length ∆L were measured. The temperature
stability during the measurement was less than 0.02K per hour. This process was repeated
at approximately 16 temperatures of roughly equal intervals up to room temperature. One
dataset of the temperature vs. ∆L was obtained for one cooling cycle. To compensate the
systematic uncertainty, each measurement was repeated by rotating the sample by 90◦ as
described in figure 2 of the paper–I. We measured CTEs of two orthogonal directions (A and
B directions as shown in figure 1) of each sample. The directions A and B were arbitrary
chosen.
3. Result and discussion
3.1. Typical thermal expansion
The results of the measurement of the thermal expansion are presented in figure 2. A
fit with the 8th order polynomial is applied to each of the six datasets and the thermal
expansion (contraction) ∆L/L is set as 0 at 293K for each of the six curves. The six
datasets are plotted in figure 2 (a). We present the curve derived from the fit with all of the
six datasets as the typical thermal contraction of SiC-100. It is shown by the solid line in
figure 2 (a). The coefficients of the 8th order polynomial, ∆L/L =
i, are presented
in table 1. Figure 2 (b) shows the residual dispersion of the data after subtracting the fit
curve. The shape of the curve in figure 2 (a) is roughly compatible with those of the RBO
SiC (Pepi & Altshuler 1995) and NT-SiC (Suyama et al. 2005), though slightly negative
CTEs observed in the RBO SiC and NT-SiC observed at temperatures less than 50K are
not seen in the present measurements for SiC-100. Because of the uncertainties in the other
– 8 –
measurements, it is difficult to further investigate the origins of the differences at present.
For each of the six curves, the average ∆L/L per temperature between 293K and 10K
is derived and summarized in table 2. The average of the six values and their dispersion (1σ)
are 0.816 and 0.005 (× 10−6/K), respectively. The dispersion is smaller than the previous
upper-limit obtained with high-purity single crystal silicon in paper–I, 0.01 (× 10−6/K),
indicating that the present measurements have reached the limit set by the instrument.
Pepi & Altshuler (1995) have shown the 1σ dispersion of 0.04 (× 10−6/K) for their
measurements. Karlmann et al. (2006) presented the repeatability of the CTE measurement
to be 0.004 (× 10−6/K) from 35K to 305K of single crystal silicon by the interferometer
based cryogenic dilatometer. Comparing with Karlmann et al. (2006), our measurement
reached lower temperatures. Thus the present dispersion is concluded to be well below
the measurement uncertainty and we do not detect any significant variations in the CTEs
of the present 6 measurements. We do not detect either any differences in the CTEs in
different directions of the same sample or any differences in the samples extracted from
different lots. Thus SiC-100 we have measured is homogeneous and isotropic within the
present measurement uncertainty. Finally, we average two α in directions A and B of each
of three samples to obtain α#1 = 0.8145, α#2 = 0.8160, and α#3 = 0.8185 (× 10
−6/K). The
average and the dispersion (1σ rms) of these three CTEs are 0.816 and 0.002 (× 10−6/K),
respectively.
3.2. Alternative measurements of the dispersion of CTE
Differential tests of the CTE dispersion is another strong tool to investigate the
variations in the CTEs. The quite small dispersion in the CTEs of SiC-100 has been
confirmed by the systematic check made by Astrium in the manufacturing process of the
– 9 –
mirror segments of the Herschel Space Observatory. All the 12 segments of the Herschel
primary mirror came from different SiC powder batches. The measurement of the bending
deflection was made for a brazed couple of two thin SiC samples coming from different
batches of SiC. The couples were placed in a vacuum chamber and the thermal deformation
of the surface figure of the samples was measured by a interferometer between room
temperature and 150K. The deformation data is directly linked to the curvature of the
bending deflection and to the difference of the CTE between the two samples.
In order to validate the sufficient homogeneity of the SiC material, two kinds of
verification on the material have been performed: One of the tests was of homogeneity inside
one spare segment. For this test, samples were cut out from one spare sintered segment
at several locations (along radial, tangent, and thickness directions). The other tests was
of homogeneity between samples belonging to different flight segments. For this test, as it
is was not possible to take samples from the flight segments after sintering, the samples
were cut out from different segments at their green body stage, before the sintering of the
segments. Those samples were taken from arbitrary locations of the segments (different
orientations and locations were therefore present in the test samples). Those samples were
made similar to the segments themselves by taking care of sintering them in the same run
as the associated segments. By this way, it was possible to reproduce the differential CTE
characteristics. In the telescope manufacturing process, 12 samples of the 12 segments were
brazed on the reference samples and tested at 150K. The results indicate the dispersion (1σ
rms) to be smaller than 0.0025 (× 10−6/K).
The agreement of the dispersion of the CTE derived by two different methods signifies
that both of the measurements are reliable and the uniformity of the CTE of the SiC-100
is well confirmed. The direct measurement of the CTE of this work and the differential
measurement are complementary: the direct measurement provides absolute CTE data
– 10 –
down to 10K with high accuracy, which is indispensable to design the space telescope
including the surrounding structures, whilst the differential test checks the uniformity of
the CTE for a large number of the samples.
3.3. Simulation of the mirror deformation
It is fruitful to relate the accuracy of the CTE measurement with corresponding thermal
deformation of the mirror. In this section the thermal deformation of the segmented mirror
is estimated on the basis of the measured dispersion of the CTE values. To examine this
issue, we perform a case study by using a simple model of the finite-element-method (FEM)
analysis. All of the simulated thermal deformation is derived for the case of cooling down
from 293K to 10K. Figure 3 shows the model used in the FEM analysis. One mirror with a
center hole is constructed from six segments with the rib structure as shown in figures 3 (a)
and b as a light-weight mirror design. In this model, it was assumed that the mirror
surface is flat for simplicity. Figures 3 (c) and d show a three-dimensional view and the
geometry of one segment, respectively. The diameter of the whole mirror is 3.5m, equating
to the designed diameter of the primary mirror of the SPICA telescope and the Herschel
Space Observatory. The thicknesses of the rib structure and mirror surface are 3 mm.
The Young’s modulus and Poisson’s ratio of SiC-100 at room temperature are reported to
be 420GPa and 0.17, respectively (Breysse et al. 2004; Toulemont 2005). We use these
values in the simulation of the thermal deformation since the temperature dependence of
these quantities is usually small and not available at present. In the FEM analysis, the
rotationally symmetric axis of the mirror is set along the z axis of the Cartesian coordinates.
The constraints of the model are shown in figure 3 (b): Three points on the rib of the
mirror, indicated by triangles, are constrained on the lines in the x-y plane as shown by the
arrows in figure 3 (b). These points are free along the radial direction within the constraint
– 11 –
lines. Therefore, this constraints cause no inner stress in the mirror in the simulation of the
cooling and thermal deformation.
The results of the simulation are presented in figure 4. Figure 4 (a) shows the thermal
deformation toward the z-direction of the mirror surface obtained from the FEM analysis,
in which α#1, α#2, and α#3 are given for the segment s5 and s6, s1 and s2, s3 and
s4, respectively (case–I). Figure 4 (b) shows the simulated z-direction deformation, in
which α#1, α#2, and α#3 were given for the segment s3 and s6, s2 and s5, s1 and s4,
respectively (case–II). The case–I and case–II are the configuration, in which the two
segments having the same CTE are allocated to be adjoined and confronted position,
respectively. As the result, the configuration of the case–I corresponds to a mirror, which
consists of three segments having α#1, α#2, and α#3, whilst the configuration of the case–II
corresponds to a mirror having a CTE distribution of 180◦ rotationally symmetric. Both of
figures 4 (a) and (b) show the surface after the tilt correction. For the case–I and case–II,
0.032µm and 0.040µm(1σ rms) are obtained as the surface deformation after the tilt
correction, respectively. Since the requirement for the surface figure accuracy of SPICA is
0.06µm rms, the FEM analysis indicates that the thermal deformation estimated based
on the measured CTE dispersion among the segments is sufficiently small for the SPICA
telescope.
The temperature expected for the SPICA is 4.5K, whilst the lowest temperature of
the CTE measurement in this work is ∼ 10K. However, the thermal contraction between
10K and 4.5K is negligible and does not affect the mirror deformation at all. It is shown
that the accuracy of the CTE measurement achieved in the present study is sufficient to
investigate the thermal deformation for the wave front error of less than 0.06µm for a
segmented SiC mirror of 3.5m size.
– 12 –
4. Conclusion
In this work, we performed high precision measurement of the thermal expansion of
the sintered SiC, SiC-100 for use in cryogenic space-telescopes. Three samples of SiC-100
from different lots are measured. The temperature the measurement ranges from room
temperature to ∼ 10K. The following results are obtained.
1. The typical thermal expansion of SiC-100 is given in the form of ∆L/L =
The coefficients are shown in table 1.
2. The CTEs were measured for three samples of two orthogonal directions., The
average and dispersion (1σ) of these six values between 293K and 10K are 0.816 and
0.005 (10−6/K), respectively. The dispersion is well below the present measurement
uncertainty.
3. The homogeneity and the anisotropy of the CTE of SiC-100 has been confirmed
within the present measurement accuracy.
4. The small dispersion of the absolute CTE obtained is compatible with the results of
the differential CTE measurements using brazed samples made at Astrium.
5. For the three samples, nominal CTEs, α#1, α#2, and α#3 were derived for temperatures
between 293K and 10K by averaging two CTE data of tow directions for each sample.
6. The thermal deformation of a segmented mirror is estimated by a FEM analysis,
using α#1, α#2, and α#3. The result indicates that the present measured dispersion is
sufficiently small and well below the SPICA requirement on thermal deformation of
the mirror, 0.06µm rms.
– 13 –
We are grateful for Dr. Masahiro Okaji in National Metrology Institute of Japan,
Advance Industrial Science and Technology for kind and large support for the measurement.
This work was supported in part by a grant from the Japan Science and Technology Agency.
– 14 –
REFERENCES
Breysse, J., Castel, D., Laviron, B., Logut, D., & Bougoin, M., 2004, Proc. of the 5th
International Conference on Space Optics, 659
Enya, K., Nakagawa, T., Kataza, H., Kaneda, H., Y.Y.Yamashita, T. Onaka, T. Oshima, &
T. Ozaki, 2004, Proc. of SPIE, 5487, 1092
Enya, K., Nakagawa, T., Kaneda, H., Onaka, T., Ozaki, T., & Kume, M., 2007, Applied
Optics, in press
Kaneda, H., Onaka, T., Yamashiro, R., & Nakagawa, T., 2003, Proc. of SPIE, 4850, 230
Kaneda, H., Onaka, T., Nakagawa, T., Enya, K., Murakami, H., Yamashiro, R., Ezaki, T.,
Numao, Y., & Sugiyama, Y., 2005, Applied Optics, 44, 6823
Kaneda, H., Kim, W., Onaka, T., Wada, T., Ita, Y., Sakon, I., & Takagi, T., 2007, Publ.
Astron. Soc. Japan, submitted
Karlmann, P. B., Klein, K. J., Halverson, P. G., Peters, R. D., Levine, M. B., van Buren, D.,
& Dudik, M. J., 2006, Proc. of AIP Conferenc, Advances in Cryogenic Engineering,
824, 35
Murakami, H., 2004, Proc. of SPIE, 5487, 330
Nakagawa, T., & SPICA working group, 2004, Adv. Sp. Res., 34, 645
Okaji, M., & Yamada, N., 1997, High Temp. & High Press., 29, 89
Okaji, M., Yamada, N., Kato, H., & Nara, H., 1997, Cryogenics, 37, 251
Onaka, T., Kaneda, H., Enya, K., Nakawaga, T., Murakami, H., Matsuhara, H., & Kataza,
H., 2005, Proc. of SPIE, 5494, 448
– 15 –
Ozaki, T., Kume, M., Oshima, T., Nakagawa, T., Matsumoto, T., Kaneda, H., Murakami,
H., Kataza, H., Enya, K., T. Onaka, & M. Krodel, 2004, Proc. of SPIE, 5494, 366
Sugita, H., Nakagawa, T., Murakami, H., Okamoto, A., Nagai, H., Murakami, H., Narasaki,
K., & Hirabayashi, M., 2006, Cryogenics, 46, 149
Toulemont, Y., private communication, 2005
Pepi, J. W., & Altshuler, T. L., 1995, Proc. of SPIE, 2543, 201
Pilbratt, G. T., 2004, Proc. of SPIE, 5487, 401
Suyama, S., Itoh, Y., Tsuno, K., & Ohno, K., 2005, Proc. of SPIE, 5868, 96
Yamada, N., & Okaji, M., 2000, High Temp. & High Press., 32, 199
This manuscript was prepared with the AAS LATEX macros v5.2.
– 16 –
Fig. 1.— Samples measured in this work. The geometry of the samples was
20.00mm× 20.00mm× 6.0mm. The surfaces of ×20.00mm× 6.0mm were polished as de-
scribed in the text.
– 17 –
Fig. 2.— Result of the measurements. (a): Thermal expansion data for the direction A
and B of the sample #1, #2, and #3. The solid line represents the fitting curve with 8th
order polynomial which is expressed by the coefficients presented in table 1. (b): Residual
dispersion of the thermal expansion data around the fitting curve.
– 18 –
Fig. 3.— The model for the FEM analysis. (a) 3D view of the model of the whole mirror seen
from the reflective side. (b) The same model but seen from the back side of the mirror. Three
triangles on the rib indicate points for constraints. All of the three points were constrained
in the three lines shown by the arrows. s1∼ s6 are for ID of the segments (see also figure 4).
(c) and (d) shows a 3D view and the geometry of one segment, respectively.
– 19 –
Fig. 4.— The simulated surface deformation by the FEM analysis. z-direction deforma-
tion (i.e., perpendicular to the surface) after tilt correction is shown by color map. (a): The
result of the case–I simulation. α#1, α#2, and α#3 were given for the segment s5 and s6, s1
and s2, s3 and s4, respectively. (b): The result of the case–II simulation. α#1, α#2, and α#3
were given for the segment s3 and s6, s2 and s5, s1 and s4, respectively.
– 20 –
Table 1: Coefficients of the 8th order polynomial ∆L/L =
i to represent the typical
thermal expansion of SiC-100 below 300 K.
coefficient value
a0 +2.43165× 10
a1 −9.25541× 10
a2 +7.38688× 10
a3 +2.44225× 10
a4 +5.68470× 10
a5 +5.94436× 10
a6 −4.04320× 10
a7 +8.70017× 10
a8 −6.64445× 10
– 21 –
Table 2: Summary of the measured CTE.
Samplea Directionb α (10−6/K) c
#1 A 0.820
#1 B 0.809
#2 A 0.821
#2 B 0.811
#3 A 0.815
#3 B 0.822
a, b: #1, #2, and #3 correspond to the sample number and A and B correspond to
the direction of the sample for measurement shown in figure 1. xc: ∆L/L between 293K
and 10K.
	Introduction
	Experiment
	Sample
	Measurement
	Result and discussion
	Typical thermal expansion
	Alternative measurements of the dispersion of CTE
	Simulation of the mirror deformation
	Conclusion

  We present the results of high precision measurements of the thermal
expansion of the sintered SiC, SiC-100, intended for use in cryogenic
space-telescopes, in which minimization of thermal deformation of the mirror is
critical and precise information of the thermal expansion is needed for the
telescope design. The temperature range of the measurements extends from room
temperature down to $\sim$ 10 K. Three samples, #1, #2, and #3 were
manufactured from blocks of SiC produced in different lots. The thermal
expansion of the samples was measured with a cryogenic dilatometer, consisting
of a laser interferometer, a cryostat, and a mechanical cooler. The typical
thermal expansion curve is presented using the 8th order polynomial of the
temperature. For the three samples, the coefficients of thermal expansion
(CTE), $\bar{\alpha}_{#1}$, $\bar{\alpha}_{#2}$, and $\bar{\alpha}_{#3}$ were
derived for temperatures between 293 K and 10 K. The average and the dispersion
(1 $\sigma$ rms) of these three CTEs are 0.816 and 0.002 ($\times 10^{-6}$/K),
respectively. No significant difference was detected in the CTE of the three
samples from the different lots. Neither inhomogeneity nor anisotropy of the
CTE was observed. Based on the obtained CTE dispersion, we performed an
finite-element-method (FEM) analysis of the thermal deformation of a 3.5 m
diameter cryogenic mirror made of six SiC-100 segments. It was shown that the
present CTE measurement has a sufficient accuracy well enough for the design of
the 3.5 m cryogenic infrared telescope mission, the Space Infrared telescope
for Cosmology and Astrophysics (SPICA).

Introduction
Development of cryogenic light-weight mirrors is a key technology for infrared
astronomical space-telescope missions, which have large advantages owing to being free
from the turbulence, thermal background, and absorption caused by atmosphere. The
light-weight mirror technology is essential to bring large mirrors into space that enable a
high sensitivity and a high spatial resolution. The infrared sensitivity of the space-telescope
is vastly improved by reduction of the thermal background by cooling the telescope to
cryogenic temperatures. Thus infrared space telescopes badly need cooled light-weight
mirrors with a sufficient optical quality.
Silicon-carbide (SiC) is one of the most promising materials for space telescopes because
of its high ratio of stiffness to density. Japanese space mission for infrared astronomy,
AKARI, carries a 68.5 cm aperture telescope whose mirrors are made of sandwich type SiC
(Murakami 2004; Kaneda et al. 2005). The entire AKARI telescope system is cooled down
to 6K by helium gas vaporizing from liquid helium. AKARI was launched in February 2006
and the telescope system performance has been confirmed to be as expected from pre-launch
ground tests (Kaneda et al. 2007). The Herschel Space Observatory, a submillimeter
satellite mission by the European Space Agency (Pilbratt 2004), employs mirrors of sintered
SiC (SiC-100) provided by Boostec Industries and EADS-Astrium (Breysse et al. 2004).
The 3.5m diameter primary mirror of the Herschel telescope is made of 12 segments brazed
together. The Herschel Space Observatory will be launched in 2008 to make observations
in 60–670µm, whilst the telescope will be kept to ∼ 80K by passive cooling. SiC-100 is
one of the most frequently used SiCs for space optics. It has been used for the telescopes
of ALADIN, GAIA, ROCSAT and other missions (Breysse et al. 2004 and the references
therein).
The Space Infrared telescope for Cosmology and Astrophysics (SPICA) is the next
– 4 –
generation mission for infrared astronomy, planned by the Japan Aerospace Exploration
Agency (Nakagawa 2004; Onaka et al. 2005). The SPICA telescope is required to have a
3.5m diameter aperture and will be cooled down to 4.5K by the combination of radiative
cooling and mechanical coolers (Sugita et al. 2006). SPICA is planned to be launched in
the middle of the 2010s and execute infrared observations in 5–200µm. SiC-100 is one of
the promising candidate materials for the mirrors and structures of the SPICA telescope,
whilst carbon-fiber reinforced silicon-carbide is another candidate now being investigated
(Ozaki et al. 2004; Enya et al. 2006; Enya et al. 2004). Monolithic primary mirrors can
be manufactured by the joint segment technology, similar to the primary mirror for the
Herschel telescope. The requirement for the surface figure accuracy of the SPICA primary
mirror is, however, better than 0.06µm rms. This requirement is ∼ 20 times more severe
than the Herschel Space Observatory because of the difference in the targeted wavelength
range.
In the development of cryogenic space-telescopes, it is important to suppress the
thermal deformation of the mirror caused by cooling to satisfy the requirement for the
surface figure accuracy (Kaneda et al. 2003; Kaneda et al. 2005). Therefore, the study
of the coefficients of thermal expansion (CTE) of the material used for the mirror is
important. The CTE data of the mirror material are indispensable for the design of
cryogenic space-telescopes because the actual telescope mirror needs to accommodate
complicated support structures consisting of materials different from the mirror. If the CTE
measurement of test pieces of the mirror material has a sufficient accuracy, it will enable us
to predict the thermal deformation of a segmented mirror caused by the dispersion in the
CTE of each segment. The most direct and highly sensitive test of the thermal deformation
is an interferometer measurement of the actual mirror at cold temperatures. The CTE data
are useful to interpret the result of direct measurements of the mirror and investigate the
origin of the observed deformation.
– 5 –
However, measurements of the CTE of SiC and its dispersion have not so far been
performed with a sufficient accuracy and for a wide temperature range. Especially, little
data are available at temperatures lower than 77K, which can be realized by only liquid
nitrogen cooling. Prior to this work, available CTE data of SiC-100 were limited at
temperatures higher than ∼ 77K (Toulemont 2005). Pepi & Altshuler (1995) presented the
CTE data of reaction bonded optical grade (RBO) SiC down to 4K based on measurements
with samples made from one block of the RBO SiC. The CTE of the new-technology SiC
(NT-SiC), developed high-strength reaction-sintered SiC, has been reported down to 20K
for one sample by Suyama et al. (2005).
In this work, we present the results of high precision CTE measurements of SiC-100
down to cryogenic temperatures for samples from three different lots. We set two major
goals for the present work: one is to provide the typical CTE of SiC-100 for the development
of the SPICA telescope, and the other is to estimate the thermal deformation of segmented
mirrors from the measured CTEs.
2. Experiment
2.1. Sample
Figure 1 shows three samples (#1, #2, and #3) of SiC-100 measured in this
work. All of the samples were manufactured by Boostec Industries and EADS-
Astrium (Breysse et al. 2004). Each of the samples was extracted from blocks of
SiC produced in different lots. The locations of each sample in the SiC blocks are
arbitrary. The samples are of a rectangular parallelepiped shape with the dimension of
20.00+0.05
−0.00mm×20.00
+0.05
−0.00mm×6.0
−0.0 mm. Flatness, parallelism, and roughness of the
20.00mm×6.00mm surfaces are important for the measurement of this work: The flatness
– 6 –
of these surfaces is ≤ λ/10 rms, where λ is the wavelength of the He-Ne laser, 632.8 nm.
The parallelism of the opposing 20.00mm×6.00mm surfaces is less than 2′′, whilst the
parallelism of the opposing 20.00mm×20.00mm surfaces is less than 1.0◦. All of the
20.00mm ×6.00mm surfaces are polished to an optical grade and the surface roughness
finally achieved is less than 3 nm rms. Owing to the polished surface, the directly reflected
laser light by the sample was used for the measurement. As the result, the measurement
was free from the uncertainty due to any additional mirrors or coating, which would be
needed if the direct reflected light could not be used.
2.2. Measurement
The measurement of the thermal expansion was carried out with the laser interferometric
dilatometer system for low temperature developed in the National Metrology Institute of
Japan, Advance Industrial Science and Technology (Yamada & Okaji 2000, hereafter the
paper–I; Okaji & Yamada 1997; Okaji et al. 1997). The system consists of the cryostat,
the cryogenic mechanical refrigerator of the GM cycle (V204SC) by Daikin Industries Ltd.,
and the interferometer utilizing acousto-optical modulators and the stabilized He-Ne laser
system (05STP905) by Melles Griot. The cooling is performed with the refrigerator alone
and no cryogen is needed. The minimum temperature achieved in this work is about 10K.
The space around the cold stage of the cryostat, containing the installed sample, is filled
with 130Pa helium gas and sealed off at room temperature before cooling to ensure thermal
uniformity. The configuration of the whole cryostat and the sample installation into the cold
stage of the cryostat are shown in figures 1 and 2 of the paper–I, respectively. The change
of length of the sample, ∆L, is measured with the double path type laser interferometer
of the optical heterodyne method with the digital lock-in amplifier (Model SR-850) from
Stanford Research System. Details of this system are given in the paper–I.
– 7 –
We made totally 6 measurements of the CTEs of SiC-100. In each measurement, the
sample was initially cooled down to ∼ 10K and then the proportional integral derivative
(PID) temperature control was applied. After the the temperature had been stabilized, the
temperature and the change of the sample length ∆L were measured. The temperature
stability during the measurement was less than 0.02K per hour. This process was repeated
at approximately 16 temperatures of roughly equal intervals up to room temperature. One
dataset of the temperature vs. ∆L was obtained for one cooling cycle. To compensate the
systematic uncertainty, each measurement was repeated by rotating the sample by 90◦ as
described in figure 2 of the paper–I. We measured CTEs of two orthogonal directions (A and
B directions as shown in figure 1) of each sample. The directions A and B were arbitrary
chosen.
3. Result and discussion
3.1. Typical thermal expansion
The results of the measurement of the thermal expansion are presented in figure 2. A
fit with the 8th order polynomial is applied to each of the six datasets and the thermal
expansion (contraction) ∆L/L is set as 0 at 293K for each of the six curves. The six
datasets are plotted in figure 2 (a). We present the curve derived from the fit with all of the
six datasets as the typical thermal contraction of SiC-100. It is shown by the solid line in
figure 2 (a). The coefficients of the 8th order polynomial, ∆L/L =
i, are presented
in table 1. Figure 2 (b) shows the residual dispersion of the data after subtracting the fit
curve. The shape of the curve in figure 2 (a) is roughly compatible with those of the RBO
SiC (Pepi & Altshuler 1995) and NT-SiC (Suyama et al. 2005), though slightly negative
CTEs observed in the RBO SiC and NT-SiC observed at temperatures less than 50K are
not seen in the present measurements for SiC-100. Because of the uncertainties in the other
– 8 –
measurements, it is difficult to further investigate the origins of the differences at present.
For each of the six curves, the average ∆L/L per temperature between 293K and 10K
is derived and summarized in table 2. The average of the six values and their dispersion (1σ)
are 0.816 and 0.005 (× 10−6/K), respectively. The dispersion is smaller than the previous
upper-limit obtained with high-purity single crystal silicon in paper–I, 0.01 (× 10−6/K),
indicating that the present measurements have reached the limit set by the instrument.
Pepi & Altshuler (1995) have shown the 1σ dispersion of 0.04 (× 10−6/K) for their
measurements. Karlmann et al. (2006) presented the repeatability of the CTE measurement
to be 0.004 (× 10−6/K) from 35K to 305K of single crystal silicon by the interferometer
based cryogenic dilatometer. Comparing with Karlmann et al. (2006), our measurement
reached lower temperatures. Thus the present dispersion is concluded to be well below
the measurement uncertainty and we do not detect any significant variations in the CTEs
of the present 6 measurements. We do not detect either any differences in the CTEs in
different directions of the same sample or any differences in the samples extracted from
different lots. Thus SiC-100 we have measured is homogeneous and isotropic within the
present measurement uncertainty. Finally, we average two α in directions A and B of each
of three samples to obtain α#1 = 0.8145, α#2 = 0.8160, and α#3 = 0.8185 (× 10
−6/K). The
average and the dispersion (1σ rms) of these three CTEs are 0.816 and 0.002 (× 10−6/K),
respectively.
3.2. Alternative measurements of the dispersion of CTE
Differential tests of the CTE dispersion is another strong tool to investigate the
variations in the CTEs. The quite small dispersion in the CTEs of SiC-100 has been
confirmed by the systematic check made by Astrium in the manufacturing process of the
– 9 –
mirror segments of the Herschel Space Observatory. All the 12 segments of the Herschel
primary mirror came from different SiC powder batches. The measurement of the bending
deflection was made for a brazed couple of two thin SiC samples coming from different
batches of SiC. The couples were placed in a vacuum chamber and the thermal deformation
of the surface figure of the samples was measured by a interferometer between room
temperature and 150K. The deformation data is directly linked to the curvature of the
bending deflection and to the difference of the CTE between the two samples.
In order to validate the sufficient homogeneity of the SiC material, two kinds of
verification on the material have been performed: One of the tests was of homogeneity inside
one spare segment. For this test, samples were cut out from one spare sintered segment
at several locations (along radial, tangent, and thickness directions). The other tests was
of homogeneity between samples belonging to different flight segments. For this test, as it
is was not possible to take samples from the flight segments after sintering, the samples
were cut out from different segments at their green body stage, before the sintering of the
segments. Those samples were taken from arbitrary locations of the segments (different
orientations and locations were therefore present in the test samples). Those samples were
made similar to the segments themselves by taking care of sintering them in the same run
as the associated segments. By this way, it was possible to reproduce the differential CTE
characteristics. In the telescope manufacturing process, 12 samples of the 12 segments were
brazed on the reference samples and tested at 150K. The results indicate the dispersion (1σ
rms) to be smaller than 0.0025 (× 10−6/K).
The agreement of the dispersion of the CTE derived by two different methods signifies
that both of the measurements are reliable and the uniformity of the CTE of the SiC-100
is well confirmed. The direct measurement of the CTE of this work and the differential
measurement are complementary: the direct measurement provides absolute CTE data
– 10 –
down to 10K with high accuracy, which is indispensable to design the space telescope
including the surrounding structures, whilst the differential test checks the uniformity of
the CTE for a large number of the samples.
3.3. Simulation of the mirror deformation
It is fruitful to relate the accuracy of the CTE measurement with corresponding thermal
deformation of the mirror. In this section the thermal deformation of the segmented mirror
is estimated on the basis of the measured dispersion of the CTE values. To examine this
issue, we perform a case study by using a simple model of the finite-element-method (FEM)
analysis. All of the simulated thermal deformation is derived for the case of cooling down
from 293K to 10K. Figure 3 shows the model used in the FEM analysis. One mirror with a
center hole is constructed from six segments with the rib structure as shown in figures 3 (a)
and b as a light-weight mirror design. In this model, it was assumed that the mirror
surface is flat for simplicity. Figures 3 (c) and d show a three-dimensional view and the
geometry of one segment, respectively. The diameter of the whole mirror is 3.5m, equating
to the designed diameter of the primary mirror of the SPICA telescope and the Herschel
Space Observatory. The thicknesses of the rib structure and mirror surface are 3 mm.
The Young’s modulus and Poisson’s ratio of SiC-100 at room temperature are reported to
be 420GPa and 0.17, respectively (Breysse et al. 2004; Toulemont 2005). We use these
values in the simulation of the thermal deformation since the temperature dependence of
these quantities is usually small and not available at present. In the FEM analysis, the
rotationally symmetric axis of the mirror is set along the z axis of the Cartesian coordinates.
The constraints of the model are shown in figure 3 (b): Three points on the rib of the
mirror, indicated by triangles, are constrained on the lines in the x-y plane as shown by the
arrows in figure 3 (b). These points are free along the radial direction within the constraint
– 11 –
lines. Therefore, this constraints cause no inner stress in the mirror in the simulation of the
cooling and thermal deformation.
The results of the simulation are presented in figure 4. Figure 4 (a) shows the thermal
deformation toward the z-direction of the mirror surface obtained from the FEM analysis,
in which α#1, α#2, and α#3 are given for the segment s5 and s6, s1 and s2, s3 and
s4, respectively (case–I). Figure 4 (b) shows the simulated z-direction deformation, in
which α#1, α#2, and α#3 were given for the segment s3 and s6, s2 and s5, s1 and s4,
respectively (case–II). The case–I and case–II are the configuration, in which the two
segments having the same CTE are allocated to be adjoined and confronted position,
respectively. As the result, the configuration of the case–I corresponds to a mirror, which
consists of three segments having α#1, α#2, and α#3, whilst the configuration of the case–II
corresponds to a mirror having a CTE distribution of 180◦ rotationally symmetric. Both of
figures 4 (a) and (b) show the surface after the tilt correction. For the case–I and case–II,
0.032µm and 0.040µm(1σ rms) are obtained as the surface deformation after the tilt
correction, respectively. Since the requirement for the surface figure accuracy of SPICA is
0.06µm rms, the FEM analysis indicates that the thermal deformation estimated based
on the measured CTE dispersion among the segments is sufficiently small for the SPICA
telescope.
The temperature expected for the SPICA is 4.5K, whilst the lowest temperature of
the CTE measurement in this work is ∼ 10K. However, the thermal contraction between
10K and 4.5K is negligible and does not affect the mirror deformation at all. It is shown
that the accuracy of the CTE measurement achieved in the present study is sufficient to
investigate the thermal deformation for the wave front error of less than 0.06µm for a
segmented SiC mirror of 3.5m size.
– 12 –
4. Conclusion
In this work, we performed high precision measurement of the thermal expansion of
the sintered SiC, SiC-100 for use in cryogenic space-telescopes. Three samples of SiC-100
from different lots are measured. The temperature the measurement ranges from room
temperature to ∼ 10K. The following results are obtained.
1. The typical thermal expansion of SiC-100 is given in the form of ∆L/L =
The coefficients are shown in table 1.
2. The CTEs were measured for three samples of two orthogonal directions., The
average and dispersion (1σ) of these six values between 293K and 10K are 0.816 and
0.005 (10−6/K), respectively. The dispersion is well below the present measurement
uncertainty.
3. The homogeneity and the anisotropy of the CTE of SiC-100 has been confirmed
within the present measurement accuracy.
4. The small dispersion of the absolute CTE obtained is compatible with the results of
the differential CTE measurements using brazed samples made at Astrium.
5. For the three samples, nominal CTEs, α#1, α#2, and α#3 were derived for temperatures
between 293K and 10K by averaging two CTE data of tow directions for each sample.
6. The thermal deformation of a segmented mirror is estimated by a FEM analysis,
using α#1, α#2, and α#3. The result indicates that the present measured dispersion is
sufficiently small and well below the SPICA requirement on thermal deformation of
the mirror, 0.06µm rms.
– 13 –
We are grateful for Dr. Masahiro Okaji in National Metrology Institute of Japan,
Advance Industrial Science and Technology for kind and large support for the measurement.
This work was supported in part by a grant from the Japan Science and Technology Agency.
– 14 –
REFERENCES
Breysse, J., Castel, D., Laviron, B., Logut, D., & Bougoin, M., 2004, Proc. of the 5th
International Conference on Space Optics, 659
Enya, K., Nakagawa, T., Kataza, H., Kaneda, H., Y.Y.Yamashita, T. Onaka, T. Oshima, &
T. Ozaki, 2004, Proc. of SPIE, 5487, 1092
Enya, K., Nakagawa, T., Kaneda, H., Onaka, T., Ozaki, T., & Kume, M., 2007, Applied
Optics, in press
Kaneda, H., Onaka, T., Yamashiro, R., & Nakagawa, T., 2003, Proc. of SPIE, 4850, 230
Kaneda, H., Onaka, T., Nakagawa, T., Enya, K., Murakami, H., Yamashiro, R., Ezaki, T.,
Numao, Y., & Sugiyama, Y., 2005, Applied Optics, 44, 6823
Kaneda, H., Kim, W., Onaka, T., Wada, T., Ita, Y., Sakon, I., & Takagi, T., 2007, Publ.
Astron. Soc. Japan, submitted
Karlmann, P. B., Klein, K. J., Halverson, P. G., Peters, R. D., Levine, M. B., van Buren, D.,
& Dudik, M. J., 2006, Proc. of AIP Conferenc, Advances in Cryogenic Engineering,
824, 35
Murakami, H., 2004, Proc. of SPIE, 5487, 330
Nakagawa, T., & SPICA working group, 2004, Adv. Sp. Res., 34, 645
Okaji, M., & Yamada, N., 1997, High Temp. & High Press., 29, 89
Okaji, M., Yamada, N., Kato, H., & Nara, H., 1997, Cryogenics, 37, 251
Onaka, T., Kaneda, H., Enya, K., Nakawaga, T., Murakami, H., Matsuhara, H., & Kataza,
H., 2005, Proc. of SPIE, 5494, 448
– 15 –
Ozaki, T., Kume, M., Oshima, T., Nakagawa, T., Matsumoto, T., Kaneda, H., Murakami,
H., Kataza, H., Enya, K., T. Onaka, & M. Krodel, 2004, Proc. of SPIE, 5494, 366
Sugita, H., Nakagawa, T., Murakami, H., Okamoto, A., Nagai, H., Murakami, H., Narasaki,
K., & Hirabayashi, M., 2006, Cryogenics, 46, 149
Toulemont, Y., private communication, 2005
Pepi, J. W., & Altshuler, T. L., 1995, Proc. of SPIE, 2543, 201
Pilbratt, G. T., 2004, Proc. of SPIE, 5487, 401
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Yamada, N., & Okaji, M., 2000, High Temp. & High Press., 32, 199
This manuscript was prepared with the AAS LATEX macros v5.2.
– 16 –
Fig. 1.— Samples measured in this work. The geometry of the samples was
20.00mm× 20.00mm× 6.0mm. The surfaces of ×20.00mm× 6.0mm were polished as de-
scribed in the text.
– 17 –
Fig. 2.— Result of the measurements. (a): Thermal expansion data for the direction A
and B of the sample #1, #2, and #3. The solid line represents the fitting curve with 8th
order polynomial which is expressed by the coefficients presented in table 1. (b): Residual
dispersion of the thermal expansion data around the fitting curve.
– 18 –
Fig. 3.— The model for the FEM analysis. (a) 3D view of the model of the whole mirror seen
from the reflective side. (b) The same model but seen from the back side of the mirror. Three
triangles on the rib indicate points for constraints. All of the three points were constrained
in the three lines shown by the arrows. s1∼ s6 are for ID of the segments (see also figure 4).
(c) and (d) shows a 3D view and the geometry of one segment, respectively.
– 19 –
Fig. 4.— The simulated surface deformation by the FEM analysis. z-direction deforma-
tion (i.e., perpendicular to the surface) after tilt correction is shown by color map. (a): The
result of the case–I simulation. α#1, α#2, and α#3 were given for the segment s5 and s6, s1
and s2, s3 and s4, respectively. (b): The result of the case–II simulation. α#1, α#2, and α#3
were given for the segment s3 and s6, s2 and s5, s1 and s4, respectively.
– 20 –
Table 1: Coefficients of the 8th order polynomial ∆L/L =
i to represent the typical
thermal expansion of SiC-100 below 300 K.
coefficient value
a0 +2.43165× 10
a1 −9.25541× 10
a2 +7.38688× 10
a3 +2.44225× 10
a4 +5.68470× 10
a5 +5.94436× 10
a6 −4.04320× 10
a7 +8.70017× 10
a8 −6.64445× 10
– 21 –
Table 2: Summary of the measured CTE.
Samplea Directionb α (10−6/K) c
#1 A 0.820
#1 B 0.809
#2 A 0.821
#2 B 0.811
#3 A 0.815
#3 B 0.822
a, b: #1, #2, and #3 correspond to the sample number and A and B correspond to
the direction of the sample for measurement shown in figure 1. xc: ∆L/L between 293K
and 10K.
	Introduction
	Experiment
	Sample
	Measurement
	Result and discussion
	Typical thermal expansion
	Alternative measurements of the dispersion of CTE
	Simulation of the mirror deformation
	Conclusion
