Using silicon disk resonators to measure mechanical losses caused by an electric field.

Several projects of the next generation gravitational-wave detectors use the high purity monocrystalline silicon test masses. The electric field of the actuator that is applied to correct the position of the silicon test mass causes additional mechanical losses and associated noise. Disk mechanical resonators are widely used to study mechanical losses in multilayer optical coatings that are deposited on the test masses of gravitational-wave detectors. We use silicon disk resonators to study losses caused by an electric field. In particular, the dependence of mechanical losses on the resistivity of silicon is investigated. The resonator is a thin commercial silicon wafer in which a low frequency nodal diameter mode is excited. A DC voltage is applied between the wafer and a nearby electrode. We use two measurement configurations. In the first configuration, the dependence of losses on the resistance in the voltage supply circuit is investigated. The dependence of losses on the resistivity of silicon is investigated in the second configuration. We propose a model that relates the electric field induced mechanical loss in disk resonators to the resistivity of the material. Measurements are carried out for low and high resistivity silicon wafers. The measurement results are compared with calculations. Based on these studies, it is possible to estimate the loss and noise of the test masses of gravitational-wave detectors associated with electrostatic actuators.

Measurement of fluctuations of electrostatic force acting between a dielectric plate and an electrostatic drive.

A setup for the measurement of the noise associated with the interaction of an electrostatic field produced by an electrostatic drive with a fused silica plate is presented. The fused silica plate is a part of a 63 Hz high-Q torsional oscillator. Its oscillations are measured by an optical interferometric sensor. The measurements are PC-controlled and fully automated. A digital post-processing scheme is described, allowing the calculation of the plate rotation angle fluctuations and the evolution of the charge distribution on the surface of the fused silica plate. The characteristic surface charge relaxation time has been purposely kept small on the order 103-104 s. The upper limit of the investigated noise has been obtained. We used this result to estimate the electrostatic drive noise in the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) gravitational wave detector at frequencies of about 18 Hz. The obtained upper limit of the strain amplitude spectral density is (1±0.13)⋅10-22Hz-1/2.

Enhanced interaction between a mechanical oscillator and two coupled resonant electrical circuits.

This paper reports result of calculation and experimental realization of an electromechanical system that consists of a high-Q mechanical oscillator parametrically coupled in the manner of a capacitive transducer with a radio frequency (RF) circuit, which is in turn inductively coupled with another RF circuit. The system operates in the resolved sideband regime when the mechanical oscillator's frequency is larger than the electrical circuits' bandwidths. Using two coupled RF circuits allowed one to enhance the interaction between them and the mechanical oscillator which is one of flexural vibrational modes of a free-edge circular silicon wafer. Such a coupled electromechanical system can be used as a high-sensitive capacitive vibration sensor.

Surface vibrational modes in disk-shaped resonators.

The natural frequencies and distributions of displacement components for the surface vibrational modes in thin isotropic elastic disks are calculated. In particular, the research is focused on even solutions for low-lying resonant vibrations with large angular wave numbers. Several families of modes are found which are interpreted as modified surface modes of an infinitely long cylinder and Lamb modes of a plate. The results of calculation are compared with the results of the experimental measurements of vibrational modes generated by means of resonant excitation in duraluminum disk with radius of ≈90 mm and thickness of 16 mm in the frequency range of 130-200 kHz. An excellent agreement between the calculated and measured frequencies is found. Measurements of the structure of the resonant peaks show splitting of some modes. About a half of the measured modes has splitting Δfsplit/fmode at the level of the order of 10(-5). The Q-factors of all modes measured in vacuum lie in the interval (2…3)×10(5). This value is typical for duraluminum mechanical resonators in the ultrasonic frequency range.