Maximum heat transfer along a sapphire suspension fiber for a cryogenic interferometric gravitational wave detector

Transcription

1 Maximum heat transfer along a sapphire suspension fiber for a cryogenic interferometric gravitational wave detector T. Tomaru 1, T. Suzuki, T. Uchiyama, A. Yamamoto, T. Shintomi High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki, , Japan C. T. Taylor 2, K. Yamamoto, S. Miyoki, M. Ohashi, K. Kuroda Institute for Cosmic Ray Research (ICRR), The University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba, , Japan We have shown experimentally that the thermal conductivity of a sapphire fiber, which is used as a suspension and thermal conductor of mirrors in a cryogenic interferometric gravitational wave detector, is limited by boundary scattering of phonons in the temperature range below 40 K, and that maximum heat transfer along the fiber is proportional not to the square of the diameter of the fiber, but to the cubic. For the design of a sapphire suspension fiber for the LCGT project (1 mm in diameter, 25 cm in length and two loops), the maximum heat transfer along the fiber from the mirror at 20 K to the medium mass at 10 K is 340 mw. Key words: Cryogenic mirror; Cryogenic interferometer; Sapphire; Thermal conductivity; Boundary scattering; Gravitational wave; LCGT 1 Introduction To detect several gravitational wave (GW) events from the coalescence of neutron star binaries every year, we have to improve the sensitivity by three 1 Corresponding author. address: tomaru@post.kek.jp 2 Present address: Rio Tinto Australian Science Olympiads BOX 7251, Canberra MC, ACT 2610, Australia Preprint submitted to Elsevier Preprint 5 July 2002

2 orders of magnitude higher than that of the currently most sensitive GW interferometer, TAMA300[1,2]. A clear method for improving the sensitivity is to reduce both the thermal noise[3] and the thermoelastic noise[4] (the sensitivity limitations of GW interferometers) by cooling the sapphire mirrors and suspensions[5,6]. For the Large-scale Cryogenic Gravitational wave Telescope (LCGT) project[7], which is a Japanese future project, a mirror temperature of 20 K is required. A technical issue concerning a cryogenic GW interferometer is to cool the mirrors against heating up by absorption of the laser power, with keeping the performance of the vibrational isolation of the mirrors. Because seismic vibration is also one of the sensitivity limitations of GW interferometers, high performance vibrational isolators are necessary[8,9]. Fig. 1. Schematic diagram of the mirror suspension and cooling system. In the LCGT project, we plan to cool the mirrors and suspensions by using thermal conduction of the suspension fibers. Figure 1 shows a schematic diagram of the mirror suspension and cooling system. Mono-crystalline sapphire fibers are used as the suspension fibers since they have both large thermal conductivity[10] and a high mechanical quality factor (Q) at cryogenic temperature[11]. A fundamental study of cooling against a few dozen milliwatts of heating up has been done[12]. However, since we showed that the present optical absorption coefficient of sapphire mirror substrates is approximately 100 ppm/cm[13] 3, a few watts of heat generation in the mirror is estimated in the LCGT. Therefore, we studied a larger heat transfer along the sapphire fiber than in our previous study. 3 For small samples, 25 ppm/cm of optical absorption in sapphire substrate has been reported[14]. 2

3 The thermal conductivity of materials is described as κ = 1 3 C V vl, (1) where C V is the volumetric specific heat of the material, v is the sound velocity in the material and l is the mean free path of a phonon. Since the mean free path of a phonon in a thin sapphire fiber can be limited by boundary scattering at cryogenic temperature[15], the thermal conductivity of a sapphire suspension fiber can be limited by the fiber diameter. Therefore, since the maximum heat transfer along the fiber can be proportional to the cube of the diameter of the fiber, the heat transfer along the fiber would largely depend on the fiber diameter. In this paper, we report on a measurement of the thermal conductivity of sapphire fibers, and an estimation of the maximum heat transfer along them. 2 Thermal conductivity measurement We used a longitudinal heat flow method[10] to measure the thermal conductivity of sapphire fibers. Figure 2 shows the experimental setup. A temperature gradient in the sample was produced by Heater-1 and the temperature difference between two points of the sample was measured by Carbon-Grass Resistance thermometers. This measurement corresponds to the four-wire method in electrical resistance measurements. The total temperature of the sample was controlled by Heater-2. The measured temperature range was between 4.2 K and 100 K. Three sapphire fibers with different diameters (φ160 µm, φ250 µm and φ390 µm) were used in this measurement in order to investigate the effect of the boundary scattering of phonons. These sapphire fibers were manufactured by Photran LLC using the Edge defined Film-fed Growth (EFG) method[16]. The c-axis of these samples was along with the fiber axis. Table 1 Comparison of the ratio of the thermal conductivity at 5 K with that of the sample diameter. φ160 µm φ250 µm φ390 µm Thermal conductivity Diameter of the sample Figure 3 shows the results of the thermal conductivity measurements of the sapphire fibers. The measurement errors came from the calibration of the thermometers and the linear model of the thermal conductivity. Table 1 gives 3

4 Fig. 2. Experimental setup of a thermal conductivity measurement of sapphire fibers (longitudinal heat flow method). k f390mm f250mm f160mm Maximum value in a data book Fig. 3. Measurement results of the thermal conductivity of mono-crystalline sapphire fibers. The closed circles, triangles and squares are the measured thermal conductivity of fibers of φ390 µm, φ250 µm and φ160 µm, respectively. The dashed line shows the maximum thermal conductivity of sapphire reported in a data book[10] (φ2.5 mm sample). 4

5 a comparison of the ratio of the thermal conductivity at 5 K as well as the ratio of the sample diameter. Since the ratio of the thermal conductivity of these fibers agrees well with the ratio of the sample diameter, the mean free path of a phonon is limited by boundary scattering in the temperature range below 40 K. k Fig. 4. Effect of phonon reflection at the sample surface. The closed triangles and rhombuses indicate the thermal conductivity of a φ250 µm sample with the original surface and with a scratched surface, respectively. Only for a φ250 µm sample, we measured the thermal conductivity again after scratching the surface by a diamond paste of #8000 to investigate the effect of phonon reflection at the sample surface. The original surface quality of the sample was commercial level as an optical fiber, and we could not find a scratch larger than on the micrometer scale on the surface by a microscope. The result (Fig. 4) shows that the thermal conductivity decreased to half. Therefore, the mean free path of a phonon depends on not only the fiber diameter, but also on the quality of the fiber surface. 3 Estimation of heat transfer along a sapphire fiber Since the thermal conductivity of a thin sapphire fiber is proportional to the diameter below the temperature at the thermal conductivity peak (about 40 K), the heat transfer along the fiber is proportional to the cube of the diameter of the sample. Figure 5 shows the relation between the diameter of a sapphire fiber and the maximum heat transfer along it to keep the mirror temperature at 20 K. In 5

6 m Fig. 5. Maximum heat transfer along a fiber limited by the fiber diameters (left axis). The right axis shows the allowable optical absorption coefficient in the mirror calculated from the maximum heat transfer along the fiber. We assumed a fiber length of 25 cm, two loops suspension, the medium mass at 10 K, the mirror at 20 K, laser power into the mirror of 2.5 kw and a mirror length of 10 cm. this figure, we assume that the suspension fiber length is 25 cm, the mirror is suspended by two loops, the temperature of the medium mass (an end of the fibers) is 10 K, the laser power into the mirror is 2.5 kw (injection laser power, 100 W; power recycling gain[17], 50; two arms) and the mirror length is 10 cm. If we suspend the mirror by two loops to bend the fiber normally, such as in Fig. 1, the diameter of the sapphire fiber must be at largest 550 µm for a mirror diameter of 30 cm for the LCGT, since the minimum bending diameter of φ550 µm sapphire fiber is about 30 cm[16]. Thus maximum heat transfer along the fiber is 57 mw (the allowable optical absorption coefficient in the sapphire mirror is at most 2.3 ppm/cm). 4 Discussion To cool the mirror to 20 K, we need to reduce its optical absorption and to make the heat transfer along the fiber greater than the estimated value given in section 3. A simple method to increase the heat transfer along the fiber is to use a thick fiber. However, there is a limitation for the diameter of a suspension fiber from the requirements concerning other noises. A most important limitation of the fiber diameter comes from a reduction of the pendulum Q[18] due to increasing the elasticity of the fiber. A reduction of the pendulum Q causes the thermal noise of the suspension pendulum to increase. To achieve the LCGT 6

7 sensitivity, which requires a pendulum Q of , the fiber diameter must be at most 1 mm. The only issue in using such thick fibers is how to suspend the mirrors. A bonding technique between a sapphire mirror and sapphire fibers, such as silicate bonding[19], is one solution. A plastic bending technique of the fiber[20] is an another solution. When we use sapphire fibers of 1 mm in diameter by two loops, the maximum heat transfer along the fibers is 340 mw (allowable optical absorption coefficient of 14 ppm/cm) in the sapphire mirror. An effective method to reduce the optical absorption is Resonant Sideband Extraction (RSE)[21], which is a configuration of an interferometer. This configuration can reduce the laser power into the mirrors without reducing the shot-noise sensitivity. When we adopt the RSE without power recycling, we can use φ700 µm sapphire fibers, even in the case that the optical absorption coefficient in the sapphire mirror substrate is the present value. Here, the optical absorption at the reflective coating is assumed to be 0.5 ppm. If a φ550 µm sapphire fiber is used, half of the value of the optical absorption used in the above estimation is required for the mirror substrate and the reflective coating. In these estimations, we ignore the thermal resistance between the fiber and the mirror. A reduction in the thermal resistance is also an important issue. Since these estimations include the effect of phonon reflection on the fiber surface, we must treat the fiber surface carefully. 5 Conclusion We showed that the thermal conductivity of a thin sapphire fiber is limited by the diameter in the temperature range below 40 K and that the maximum heat transfer along the fiber is proportional to the cube of the diameter of the fiber. Since the allowable optical absorption coefficient is very small (at largest 2.3 ppm/cm) when sapphire fibers are normally bent, it is important not only to improve the quality of the sapphire mirror substrate, but also to develop cooling techniques, such as bonding between sapphires, bending fiber, reducing the thermal resistance between the fiber and the mirror, and the RSE configuration. 7

8 Acknowledgement This study was supported by the Joint Research and Development Program of KEK and by a grant-in-aid prepared by Ministry of Education, Culture, Sports, Science and Technology. References [1] K. Kuroda et al., Proceedings of the international conference on gravitational waves: source and detectors, Singapore, (1997) 100. ; K. Tsubono, Proceedings of the first Edoardo Amaldi Conference on Gravitational Wave Experiments, Singapore, (1995) 112. [2] M. Ando et al., Phys. Rev. Lett. 86 (2001) [3] P. R. Saulson, Phys. Rev. D 42 (1990) [4] V. B, Braginsky et al., Phys. Lett. A 264 (1999) 1. [5] T. Uchiyama et al., Phys. Lett. A 261 (1999) 5. [6] M. Cerdonio et al., Phys. Rev. D 63 (2001) [7] K. Kuroda et al., Int. J. Mod. Phys. D 8 (1999) 557. [8] M. A. Barton et al., Rev. Sci. Instrum. 70 (1999) [9] A. Takamori et al., Proceedings of the 4th Edoardo Amaldi Conference on Gravitational Waves, Class. Quantum Grav. 19 (2002) [10] Thermophysical Properties of Matter, Vol. 2, Thermal Conductivity- Nonmetallic Solids, The TPRC Data Series, IFI/PLENUM New York- Washington (1970). [11] T. Uchiyama et al., Phys. Lett. A 273 (2000) 310. [12] T. Uchiyama et al., Phys. Lett. A 242 (1998) 211. [13] T. Tomaru et al., Phys. Lett. A 283 (2001) 80. [14] M. Fejer et al., [15] H. M. Rosenberg, Low temperature solid state physics: some selected topics, Oxford : Clarendon Press, (1963) ; Also J. Hough pointed out the size limitation of the thermal conductivity in thin sapphire fiber. (Private communication at 2000 Aspen Winter Conference of GW Detection.) [16] [17] R. W. P. Drever, The detection of gravitational wave, Ed: D. G. Blair, Cambridge University Press, (1991)

9 [18] P. R. Saulson, Fundamentals of interferometric gravitational wave detectors, World Scientific Publishing, (1994) [19] S. Rowan et al., Phys. Lett. A 246 (1998) 471. [20] L. Tong et al., Appl. Opt. 39 (2000) 494. [21] J. Mizuno et al., Phys. Lett. A 175 (1993)