Coating thermal noise of a finite-size cylindrical mirror

Advancement in the reduction of technical noise and isolation of seismic vibration has let a high-precision measurement device like the interferometric gravitational-wave detectors [1] be so sensitive that tiny thermal fluctuation of the measurement surface can limit the sensitivity. It is important to develop a method to estimate the thermalnoise level as accurately as possible. Our study with a finite-size mirror is an upgrade of previous works for coating thermal noise with some approximations. For mirrors currently planned to be used in the next-generation gravitational-wave detectors, the difference is a few percent between the results with our finite-size analysis and with a conventional infinite-size analysis. Besides, our analysis, for the first time, includes the effect of temperature fluctuations in the substrate and the coatings coherently summed up, with which the estimation of thermal noise will be more accurate at lower frequencies. The difference from the previous results can be larger if the mirror is thin. Thermal noise is also important in cold damping experiments [2], where the purpose is to reach a quantum limit with a low-mass mirror, which may tend to be thin. In this paper, we show calculation results of coating thermal noise with a broad range of aspect ratios, which agree to the previous results with an infinite-size mirror in the thick limit, and also agree to the results with a thin-plate that are calculated using the modalexpansion method. It is important to know the thermalnoise level in the middle range so that an appropriate mirror can be used in the various experiments. There are two different ways that non-zero temperature causes fluctuation of the surface of a mirror. The first one is via volume fluctuation under fixed temperature; called Brownian thermal noise. Brownian thermal noise in the power spectrum density (m/ √ Hz) is proportional to the square-root of temperature √ T , besides the mechanical loss has some temperature dependence. The second one is via temperature fluctuation that converts into the surface fluctuation through the thermal expansion and through the change of the refraction index. Thermal noise through the expansion is called thermoelastic noise [3] and thermal noise through the change of the refraction index is called thermorefractive noise [4][5]; the coherent sum of thermoelastic noise and thermorefractive noise is called thermo-optic noise in Ref. [6]. Thermooptic noise in the power spectrum density is linearly proportional to T , besides some parameters like the thermal conductivity or the thermal expansion depend on the temperature. Brownian thermal noise is related to the mechanical loss angle φ. A current gravitational-wave detector employs a mirror made of silica coated by tantala-silica doublets, and the loss angle of the silica substrate is several orders lower than that of the coatings [7][8]. Thermooptic noise is related to the heat flow in the r-direction (transverse to the beam) and in the z-direction (along the beam) of the cylindrical mirror. The both contributions are to be taken into account in the case without coatings [9], while the latter becomes dominant with coatings according to the difference of the mechanical parameters of the materials [10]. In this paper, we focus on the derivation of Brownian thermal noise in the coatings and thermo-optic noise in the z-direction. Historically, Brownian thermal noise of a mirror had been analyzed using a so-called modal-expansionmethod [11].