Mechanical quality factor of a sapphire fiber at cryogenic temperatures

Abstract

A mechanical quality factor of 1.1×10⁷ was obtained for the 199 Hz bending vibrational mode in a monocrystalline sapphire fiber at 6 K. Consequently, we confirm that pendulum thermal noise of cryogenic mirrors used for gravitational wave detectors can be reduced by the sapphire fiber suspension.

Introduction

A cryogenic interferometric gravitational wave detector (Large scale Cryogenic interferometric Gravitational wave Telescope: LCGT) is planned in Japan [1]. The most outstanding characteristic of LCGT is the use of cryogenic techniques to reduce thermal noise. LCGT has fiber suspended pendulum-like mirrors similar to other detectors, such as LIGO [2], VIRGO [3], GEO [4] and TAMA [5]. These suspended mirrors behave as test masses for the detection of gravitational waves. The gravitational waves are detected by measuring the difference of displacement between mirrors.

Thermal noise is the thermally excited vibration of the mirrors [6]. Major components of the vibration are the pendulum motion (pendulum thermal noise) and the elastic vibration of the mirror itself (mirror thermal noise). Typically, the pendulum thermal noise is the dominant noise source of the detector in a frequency range from approximately 10 Hz to 100 Hz and the mirror thermal noise is dominant from 100 Hz to 500 Hz.

The amplitude of the thermal vibration obeys the equation,

$$\text{〈x(ω)}{}^2\text{〉∝T·φ(ω),}$$

where T is the temperature and φ(ω) the dissipation angle. Mechanical quality factor (Q-factor) is defined as the inverse of φ(ω) at the resonant frequency ω₀, i.e. Q=1/φ(ω₀). Phenomenologically, φ(ω) is constant in the frequency range of interest when the dominant source of dissipation is internal friction of the suspension fibers or the mirror substrate. This is known as structure damping [6]. For measurement analysis, we applied the structure damping model and focused attention on the Q-factor.

According to Eq. (1), the reduction of the thermal noise is accomplished both by lowering the temperature and by improving the Q-factors of both the pendulum and mirror acoustic modes. Our strategy to reduce the thermal noise includes cooling the whole mirror suspension system to cryogenic temperature (below 30 K at the mirror), along with the use of low acoustic loss material for the mirror and suspension fiber. We have chosen sapphire for both the mirror and fiber, since it has shown to have a high Q-factor [7] and high thermal conductivity [8] at cryogenic temperatures. We have already presented results of the cryogenic mirror suspension system with a sapphire test mass which was cooled down to cryogenic temperatures and exhibited a Q-factor of 10⁸ [9], [10]. These results suggests that the cryogenic mirror suspension system could possibly reduce the mirror thermal noise.

To enhance the sensitivity of the detector, we must also reduce the pendulum thermal noise which is estimated by the Q-factor of the pendulum motion (Q_pend) with temperature. Since Q_pend depends on the dissipation in the suspension fibers, we must construct them of low loss materials. Since direct measurement of Q_pend is not an easy task, we measured the frequency dependence of Q-factor of the sapphire fiber itself (Q_fiber). Q_pend can be obtained by applying the result of the dissipation dilution theorem to Q_fiber [6].

The measurement was undertaken at cryogenic temperatures (6 K and 78 K) with one end of the fiber fixed in a clamp and the other end free. We chose the lowest frequency vibration bending mode of the clamped sapphire fiber, since the deformation of this mode is similar to that of the pendulum motion. Applying the structure damping model, the dissipation in the sapphire fiber due to bending vibration reflects that due to pendulum motion, even though the resonant frequencies are different (the resonance of the pendulum is 1 Hz, whereas that of the fiber is 200 Hz to 500 Hz).

Room temperature measurement of Q_fiber of the sapphire fiber represents 5×10³ at 200 Hz [11]. This Q-factor measurement was limited by the thermoelastic effect [12]. Since the thermoelastic effect of the sapphire fiber is negligibly small in cryogenic temperature, Q_fiber is expected to be higher for this measurement. To our knowledge, this paper represents the first cryogenic measurement of sapphire fiber Q-factor.

In this letter, we show that the measured Q_fiber of the sapphire fiber at 6 K was as high as that exhibited at room temperature by a fused silica fiber considered to be the highest Q-factor fiber at room temperature.