Temperature coefficient of refractive index of sapphire substrate at cryogenic temperature for interferometric gravitational wave detectors

Transcription

1 Temperature coefficient of refractive index of sapphire substrate at cryogenic temperature for interferometric gravitational wave detectors T. Tomaru, T. Uchiyama, C. T. Taylor, S. Miyoki, M. Ohashi, K. Kuroda Institute for Cosmic Ray Research (ICRR), The University of Tokyo, Kashiwanoha, Kashiwa, Chiba, , Japan T. Suzuki, A. Yamamoto, T. Shintomi High Energy Accelerator Research Organization (KEK), Oho, Tsukuba, Ibaraki, , Japan We present the cryogenic measurement of temperature coefficient of refractive index dn/dt of a mono-crystalline sapphire at µm wavelength of light, which will be used as mirror substrates for next generation interferometric gravitational wave (GW) detectors. We measured dn/dt on an average between 5 K and 40 K by measuring the change of refractive angle due to the change of the sample temperature. We found that dn/dt of sapphire at cryogenic temperature was at least two order of magnitude smaller than that of room temperature. This result implies that the thermal lensing in interferometric GW detectors becomes negligible by cooling sapphire mirrors. I. INTRODUCTION The sensitivity of interferometric gravitational wave (GW) detectors such as TAMA [1], LIGO [2], VIRGO [3] and GEO [4] are limited by seismic noise, thermal noise of mirrors and suspensions, and photon shot noise. Since it is thought that GW event rate of coalescing neutron star binary systems is about 10 6 in a matured galaxy in a year, it is very difficult to detect GW signals by these detectors in a limited observation time. Therefore we need to improve these sensitivity limits. Among these sensitivity limits, the thermal noise depends on material properties of mirrors and suspensions, which are mechanical Q and temperature of materials [5]. It is thought that mono-crystalline sapphire is the most suitable as mirror substrate. To reduce the thermal noise more drastically by using cryogenic sapphire mirror and suspension systems, Large-scale Cryogenic Gravitational wave Telescope (LCGT) was planned [6], and R&D experiments were carried forward, where positive results were obtained [7,8]. However, sapphire crystals have relatively large optical loss at present. To reduce the photon shot noise, very high power laser ( W) and optical amplification techniques (Fabry-Perot cavities or Delay-Line paths and optical power recycling [9]) are used in these GW detectors. These techniques require extremely low loss mirrors. Especially, optical absorption loss is harmful because it may cause thermal lensing [10,11] in very high optical power. The thermal lensing distorts wave-front of laser beam and reduces power density in the cavity in the end. It comes out by temperature distribution in a mirror caused by the optical absorption, which results in difference of optical path due to the temperature dependence of refractive index of the mirror. We had already measured the optical absorption coefficient in sapphire at cryogenic temperature. This result showed that the optical absorption in sapphire had to 13-1 c 2000 American Institute of Physics

2 be reduced by more than an order of magnitude for the LCGT [12]. As the next study, we measured temperature coefficient of refractive index dn/dt of sapphire crystal at cryogenic temperature. Since a direct measurement of refractive index at different temperatures, which is a general method, is very difficult at cryogenic temperature, we measured only the change of refractive index of sapphire at cryogenic temperature by changing the sample temperature. We present this measurement method and the result in this paper. II. MEASUREMENT PRINCIPLE The principle of measurement is to measure the change of refractive angle by changing the temperature of the sample. Figure 1 shows optical path at the sample. If the laser beam is injected into the sample with injection angle θ (corresponding refractive angle is φ) and the refractive index of the sample is changed by its temperature change, the displacement of the transmitted beam spot δx is described as δx = sin( π 2 θ) l δφ, cos φ (1) l = d cos φ, (2) δφ = tan φ δn n, (3) where d shows the length of the sample, l shows the length of optical path in the sample, δφ shows the change of the refractive angle, n shows the refractive index of the sample at temperature T and δn shows the change of the refractive index. The signs of the displacement of the beam spots were defined in this figure. This equation means that the displacement of the transmitted beam spot is maximum when beam injection angle θ is between 50 degree and 60 degree in the case of sapphire. For example, when we set the injection angle θ to 60 degree and applied dn/dt of K 1, which is a value of sapphire at room temperature [13], to this equation, the displacement of the transmitted beam will be 0.17 µmk 1. Where we used refractive index of sapphire of 1.75 at µm wavelength of light and the length of the sample d of 60 mm. Since we input the heat into the sample to change its temperature, the sample may deform by thermal stress. The thermal deformations of the sample cause shifts of the transmitted beam spot. In following, we consider several thermal deformations of the sample. 1.Translation: Figure 2 shows a shift of optical path by a translation of the sample. The translation of the sample 13-2

3 causes only a shift of the reflective beam spot. If the translation of the sample is and injection beam angle is θ, the displacement of the reflective beam spot δx trans is described as δx trans = sin(π 2θ). (4) cos θ 2.Thermal expansion: Linear thermal expansion ratio α of sapphire at cryogenic temperature T is described as α =(7.5 ± 1) T 3, (5) [14]. If the sample temperature change from T 0 to T 1, the thermal expansion length of the sample δd is described by δd = d (7.5 ± 1) (T 4 1 T 4 0 ). (6) Figure 3 shows a shift of optical path by the thermal expansion of the sample. Thermal expansion of the sample causes both shifts of the transmitted beam spot and the reflective beam spot. The displacement of the transmitted beam spot δx exp is described as δx exp =2 cos θ(tan θ tan φ) δd 2. (7) On the other hand, the displacement of the reflective beam spot δx exp is described as δx exp = δd/2 sin(π 2θ). (8) cos θ 3.Rotation: Figure 4 shows a shift of optical path by a rotation of the sample. Rotation of the sample causes both shifts of the transmitted beam spot and the reflective beam spot. This shift of the transmitted beam is composed by three parts of deformation, a shift of beam injection point, a change of beam injection angle and a shift of beam transmission point. If the center of rotation is agree with the center of the sample and the rotation angle δθ is very small, the shift of beam injection point can be approximately described as the rotation at the center of surface. Then the component of the displacement of the transmitted beam spot δx 1 caused by the shift of the beam injection point is described as δx 1 = cos θ(tan θ tan φ)r i δθ, (9) where r i shows the length between the center of the sample surface and the beam injection point. The component of the displacement of the transmitted beam spot δx 2 caused by the change of the beam injection angel is described as 13-3

4 δx 2 = cos θ l(δθ δφ), cos φ (10) δφ = 1 cos θ δθ. n cos φ (11) Though the component of the displacement of the transmitted beam δx 3 caused by the shift of the beam transmission point is similar to δx 1, it has inverse sign; δx 3 = + cos θ(tan θ tan φ)r t δθ, (12) where r t shows the length between the center of the sample surface and the beam transmission point, which is described by r t = r i d tan φ. (13) From the foregoing, we can describe the displacement of the transmitted beam spot δx rot caused by the rotation of the sample as δx rot = δx 1 + δx 2 + δx 3 = cos θ [(tan θ tan φ)(r i r t )+ l cos φ (1 cos θ )]δθ. (14) n cos φ On the other hand, the shift of the reflective beam spot δx rot is described as δx rot = L 2δθ, (15) where L is the distance between the sample surface and a position sensitive detector (PSD). III. EXPERIMENTAL SET UP Figure 5 shows the experimental setup. We set the sample in the cryostat and other components outside of the cryostat, since it is not easy to adjust components in it. The size of sapphire sample is φ mm and produced by Crystal Systems Incorporated. This sample was set to be beam injection angle θ of 60 degree. To change the temperature of the sample, a manganin film heater was put on the side of the sample. A Carbon-Glass Resistance thermometer was mounted on the sample. The sample was thermally connected with helium reservoir using pure aluminum wires to cool it, against the radiation from 300 K through the optical windows. We isolated thermally between the sample and the optical table in the cryostat, not to deform this table. Since the thermal conductivity 13-4

5 of sapphire at cryogenic temperature is extremely large, we could ignore the temperature gradient in it. A Nd:YAG laser (1.064 µm) was used in this measurement. The displacement of the transmitted beam spot was measured by a quadrant photo detector (QPD). The QPD was made of silicon and its detection diameter 2a was 1.0 mm. The QPD was tightly set on a X-Z stage, which was fixed on an aluminum blockand could be adjusted by piezo actuators. Since the seismic vibration is a important noise source, we measured it on the aluminum blockby an accelerometer. The result showed that the seismic vibration on the aluminum blockwas 10 times lager than the ground from 0.1 Hz to 1 Hz. The displacement of the transmitted beam spot on the QPD was calibrated by a contact-type displacement sensor. The beam diameter on the QPD w 0 was 66 µm. Since the distribution of the beam power is Gaussian and the displacement of the transmitted beam spot is much smaller than the beam diameter, we can approximate the relation between the change of detected beam power δp and the displacement of the beam spot δx as linear; δp (1 exp( 2a2 w0 2 ))δx, (16) where we set the center of beam spot to that of the QPD. In preliminary measurements, we observed a large drift of the displacement of the transmitted beam, which was caused by inhomogeneous thermal expansion of the experimental setup such as the laser and the optical table outside of the cryostat. We reduced this effect by circulating nitrogen gas around the laser and surrounded all of the setup by thermal insulators in the final measurement. To examine the deformation of the sample caused by thermal stress, we monitored the displacement of the reflective beam spot from the sample surface using a position sensitive detector (PSD). The resolution of the PSD was 0.1 µm. The distance L between the PSD and the sample surface was 1 m. The output of the PSD was also calibrated by a contact-type displacement sensor. IV. RESULT AND DISCUSSION We tried several times measurements of dn/dt of sapphire from 5 K to 40 K, which is a temperature region included a peakof thermal conductivity of sapphire and supposed to be used in the LCGT. Figure 6 shows the one of the measurement data. We could not find a displacement of the transmitted beam spot even if rising the sample temperature until 40 K. On the other hand, the deformation of the sample was detected by the PSD. Though we could not identify the value of dn/dt of sapphire at cryogenic temperature, we could obtain its upper limit, which is an important information for interferometric GW detectors. In following, we examine the several deformations 13-5

6 described in section 2 and obtain upper limit of dn/dt. 1.Translation: Though the translation of the sample don t affect to the displacement of the transmitted beam spot, it affects to the reflective beam spot. There are two possibilities about the source of the translation of the sample, thermal expansion of the table and tilt of the sample for pitch direction (z-direction). Since the table in the cryostat is thermally isolated with the sample, we could ignore the thermal expansion of the table. The tilt of the sample for pitch direction corresponds a translation for the displacement of the transmitted beam spot δx. If the sample tilts, the transmitted beam spot also displace to z-direction. However, we could not find such displacement and this effect was at least an order smaller than detected δx level. Therefor, we could ignore the translation of the sample in this measurement. 2.Thermal expansion: Since we rose the temperature of the sample from 5 K to 40 K, the length of the thermal expansion of the sapphire sample is m. This corresponds to δx exp of m and we could not ignore this effect. This displacement has only plus direction. On the other hand, since δx exp is m and the detected δx meas is about m, we could ignore this effect for the displacement of the reflective beam. 3.Rotation: Since the contribution of the other deformation of the sample to the displacement of the reflective beam spot was relatively small, the detected displacement of the reflective beam spot was caused by the rotation of the sample. Since the sample holder cramped only the points of side of the sample along its center axis, we could regard the center of rotation as the center of the sample. Then the displacement of the transmitted beam spot δx rot is about m. To consider above three deformations, we can describe the displacement of the transmitted beam spot δx dn/dt caused by the change of refractive index due to the change of the sample temperature as δx dn/dt = δx meas (δx exp + δx rot ) =( ) 10 8 m, (17) on an average of all measurements, where δx meas means the measured displacement of the transmitted beam spot. Finally, we could obtain the upper limit of the temperature coefficient of refractive index of sapphire on an average between 5 K and 40 K; dn , (18) dt where we used refractive index of sapphire of 1.75 at 1.064µm wavelength of light. This refractive index was obtained 13-6

7 by the power reflectivity R measurement from the sapphire sample at cryogenic temperature and calculated by This value is the same as that of room temperature in a few percent error. n = 1+ R 1 R. (19) The deformation of wave-front caused by the thermal lensing depends on the factor; ɛ dn κ dt, (20) where ɛ shows the optical absorption coefficient and κ shows thermal conductivity of the mirror. Table I shows the comparison of several parameters about thermal lensing and relative thermal lensing for fused silica, sapphire (300 K) and sapphire ( 40 K). Since cryogenic sapphire has extremely large thermal conductivity and extremely small temperature coefficient of refractive index, its thermal lensing becomes at least four order of magnitude smaller than others. Therefore, thermal lensing in cryogenic sapphire mirror will be unharmed. V. CONCLUSION We measured temperature coefficient of refractive index of sapphire at cryogenic temperature to measure the change of refractive angle, and obtained the absolute value of at largest. This value is at least two order of magnitude smaller than that of fused silica and sapphire at room temperature. From these results, we concluded that relative thermal lensing of cryogenic sapphire is at least four order of magnitude smaller than that of others. Therefor, cryogenic sapphire mirror technique is very effective method for thermal lensing in interferometric gravitational wave detectors. ACKNOWLEDGMENTS This study was supported by the Joint Research and Development Program of KEK and by a grant-in-aid prepared by Ministry of Education, Science, Sports and Culture. We appreciate Dr. Y. Higashi for a lot of advises. [1] K. Kuroda et al., Proceedings of the international conference on gravitational waves: source and detectors, I. Ciufolini, F. Fidecaro, ed., World Scientific, Singapore, (1997) 100. [2] A. Abramovici et al., Science 256, (1992) 325. [3] C. Bradaschina et al., Nucl. Instrum. Methods Phys. Res. A 256, (1990)

8 [4] K. Danzmann et al., Proceedings of the first E. Amaldi Conference on Gravitational Wave Experiments, E. Coccia, G. Pizzella, F. Ranga, ed., World Scientific, Singapore, (1995) 100. [5] P. R. Saulson, Phys. Rev. D 42, (1990) [6] K. Kuroda et al., Int. J. Mod. Phys. D 8 (1999) 557. [7] T. Uchiyama et al., Phys. Lett. A 261, (1999) 5. [8] T. Uchiyama et al., Phys. Lett. A 273, (2000) 310. [9] M. Ando et al., Phys. Lett. A 248, (1998) 145. [10] P. Hello and J. Y. Vinet, Phys. Lett. A 178, (1993) 351. [11] W. Winkler et al., Phys. Rev. A 44, (1991) [12] T. Tomaru et al., submitted to Phys. Lett. A. [13] I. H. Malitson, J. Opt. Soc. Am. 52, (1962) [14] C. T. Taylor et al., Opt. Commun. 131, (1996) 311. [15] V. B. Braginsky et al., Phys. Lett. A 271, (2000)

9 φ δφ δ θ FIG. 1. The layout of optical path at the sample. θ: Injection beam angle, φ: Refractive angle, δφ: Change of refractive angle, d: Length of the sample, l: Optical path length in the sample, QPD: Quadrant photo detector, PSD: Position sensitive detector, L: Distance between the sample surface and the PSD. The signs of the displacement at the QPD and the PSD are described in this figure, respectively. x +δ θ φ z FIG. 2. Shift of optical path in the case of a translation of the sample. : Translation length, δx trans: Displacement of the reflective beam spot caused by the translation of the sample. 13-9

10 +δ +δ θ φ δ δ FIG. 3. Shift of optical path in the case of the thermal expansion of the sample. δd: Thermal expansion length, δx exp: Displacement of the transmitted beam spot caused by the thermal expansion of the sample, δx exp: Displacement of the reflective beam spot caused by the thermal expansion of the sample. O C +δ φ δ θ C 2 δθ δθ FIG. 4. Shift of optical path in the case of a rotation of the sample. δθ: Rotation angle, δx rot: Displacement of the transmitted beam spot caused by the rotation of the sample, δx rot: Displacement of the reflective beam spot caused by the rotation of the sample, O: Center of the sample, C: Center of the sample surface

11 FIG. 5. Experimental setup. 1: Nd:YAG laser, 2: Lens, 3: Optical window, 4: Cryostat, 5: Table, 6: Sample, 7: Manganin film heater, 8: QPD, 9: X-Z stage, 10: Aluminum block, 11: Displacement sensor, 12: PSD, 13: Thermal insulator, 14: Nitrogen gas. The size of sapphire sample is φ mm. This sample was set to be beam injection angle of 60 degree

12 FIG. 6. One of the measurement data. (1): The displacement of the transmitted beam spot detected by the QPD, (2): The displacement of the reflective beam spot detected by the PSD, (3): The temperature of the sample

13 TABLE I. Comparison of several parameters about thermal lensing for fused silica, sapphire (300 K) and sapphire ( 40 K). Fused Silica (300 K) Sapphire (300 K) Sapphire ( 40 K) Absorption coefficient ɛ ppm/cm [12] Thermal conductivity κ W/m K dn/dt K [15] [13] Relative thermal lensing ɛ dn κ dt (2 20) 10 5 (1 4)