4.3 Thermal noise

Thermal noise associated with the mirror masses and the last stage of their suspensions is the most significant noise source at the low frequency end of the operating range of initial long baseline gravitational wave detectors [276Jump To The Next Citation Point]. Advanced detector configurations are also expected to be limited by thermal noise at their most sensitive frequency band [211Jump To The Next Citation Point, 238Jump To The Next Citation Point, 168Jump To The Next Citation Point, 121]. Above the operating range there are the internal resonances of the test masses. The thermal noise in the operating range comes from the tails of these resonant modes. For any simple harmonic oscillator such as a mass hung on a spring or hung as a pendulum, the spectral density of thermal motion of the mass can be expressed as [276Jump To The Next Citation Point]
4k T ω2ϕ (ω ) x2 (ω ) = ------2--B--202----4--2----, (3 ) ωm [(ω0 − ω ) + ω0ϕ (ω )]
where kB is Boltzmann’s constant, T is the temperature, m is the mass and ϕ(ω) is the loss angle or loss factor of the oscillator of angular resonant frequency ω 0. This loss factor is the phase lag angle between the displacement of the mass and any force applied to the mass at a frequency well below ω0. In the case of a mass on a spring, the loss factor is a measure of the mechanical loss associated with the material of the spring. For a pendulum, most of the energy is stored in the lossless gravitational field. Thus, the loss factor is lower than that of the material, which is used for the wires or fibres used to suspend the pendulum. Indeed, following Saulson [276Jump To The Next Citation Point] it can be shown that for a pendulum of mass m, suspended on four wires or fibres of length l, the loss factor of the pendulum is related to the loss factor of the material by
√ ----- ϕ (ω) = ϕ (ω )4--T-EI-, (4 ) pend mat mgl
where I is the moment of the cross-section of each wire, and T is the tension in each wire, whose material has a Young’s modulus E. In general, for most materials, it appears that the intrinsic loss factor is essentially independent of frequency over the range of interest for gravitational-wave detectors (although care has to be taken with some materials in that a form of damping known as thermo-elastic damping can become important for wires of small cross-section [246] and for some bulk crystalline materials [102]). In order to estimate the internal thermal noise of a test mass, each resonant mode of the mass can be regarded as a harmonic oscillator. When the detector operating range is well below the resonances of the masses, following Saulson [276], the effective spectral density of thermal displacement of the front face of each mass can be expressed as the summation of the motion of the various mechanical resonances of the mirror as also discussed by Gillespie and Raab [153]. However, this intuitive approach to calculating the thermally-driven motion is only valid when the mechanical loss is distributed homogeneously and, therefore, not valid for real test-mass mirrors. The mechanical loss is known to be inhomogeneous due to, for example, the localisation of structural defects and stress within the bulk material, and the mechanical loss associated with the polished surfaces is higher than the levels typically associated with bulk effects. Therefore, Levin suggested using a direct application of the fluctuation-dissipation theorem to the optically-sensed position of the mirror substrate surface [211]. This technique imposes a notional pressure (of the same spatial profile as the intensity of the sensing laser beam) to the front face of the substrate and calculates the resulting power dissipated in the substrate on its elastic deformation under the applied pressure. Using such an approach we find that Sx(f) can then be described by the relation
2k T W Sx(f) = --B----di2ss, (5 ) π2f2 F 0
where F0 is the peak amplitude of the notional oscillatory force and Wdiss is the power dissipated in the mirror described as,
∫ Wdiss = ω 𝜖(r)ϕ(r)∂V , (6 )
where 𝜖(r ) and ϕ(r) are the strain and mechanical loss located at specific positions within the volume V. This formalisation highlights the importance of where mechanical dissipation is located with respect to the sensing laser beam. In particular, the thermal noise associated with the multi-layer dielectric mirror coatings, required for high reflectivity, will in fact limit the sensitivity of second-generation gravitational-wave detectors at their most sensitive frequency band, despite these coatings typically being only ∼ 4.5 µm in thickness [168]. Identifying coating materials with lower mechanical loss, and trying to understand the sources of mechanical loss in existing coating materials, is a major R&D effort targeted at enhancements to advanced detectors and for third generation instruments [224].

In order to keep thermal noise as low as possible the mechanical loss factors of the masses and pendulum resonances should be as low as possible. Further, the test masses must have a shape such that the frequencies of the internal resonances are kept as high as possible, must be large enough to accommodate the laser beam spot without excess diffraction losses, and must be massive enough to keep the fluctuations due to radiation pressure at an acceptable level. Test masses currently range in mass from 6 kg for GEO600 to 40 kg for Advanced LIGO. To approach the best levels of sensitivity discussed earlier the loss factors of the test masses must be ≃ 3 × 10–8 or lower, and the loss factor of the pendulum resonances should be smaller than 10–10.

Obtaining these values puts significant constraints on the choice of material for the test masses and their suspending fibres. GEO600 utilises very-low–loss silica suspensions, a technology, which should allow detector sensitivities to approach the level desired for second generation instruments [105, 266Jump To The Next Citation Point, 268], since the intrinsic loss factors in samples of synthetic fused silica have been measured down to around 5 × 10–9 [68]. Still, the use of other materials such as sapphire is being seriously considered for future detectors [104, 195, 266] such as in LCGT [234Jump To The Next Citation Point, 247Jump To The Next Citation Point].

The technique of hydroxy-catalysis bonding provides a method of jointing oxide materials in a suitably low-loss way to allow ‘monolithic’ suspension systems to be constructed [267]. A recent discussion on the level of mechanical loss and the associated thermal noise in advanced detectors resulting from hydroxy-catalysis bonds is given by Cunningham et al. [123]. Images of the GEO600 monolithic mirror suspension and of the prototype Advanced LIGO mirror suspension are shown in Figure 8View Image.

View Image

Figure 8: Monolithic silica suspension of (a) GEO600 6 kg mirror test mass suspended from 4 fibres of thickness 250 µm and (b) prototype monolithic suspension for Advanced LIGO at LASTI (mirror mass of 40 kg, silica fibre thickness 400 µm).

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