Suspension and Control for Interferometric Gravitational Wave Detectors

The detection of gravitational radiation is one of the most exciting current endeavours in experimental physics. One method of sensing gravitational waves from astronomical events is to use an interferometer to detect the perturbations in the distance between inertially free masses. This thesis describes some of the work involved in the design of the mechanical suspensions in the UK/German GEO 600 interferometric detector. The weak interactions between gravitational waves and matter results in very small signal. For detectors on Earth, great care must be taken to reduce the displacement noise of the mirrors of the interferometer. Specifically, these mirrors must be isolated from the seismic motion of the ground and suspended in such a manner that the unavoidable Brownian motion does not exceed the size of the signals to be measured. In order to predict the performance of a candidate suspension, a computer model of a multiple-stage pendulum has been developed, based on a Lagrangian formulation. This model includes sufficient detail to predict the dynamic and thermal noise performance of the pendulum. The code uses a minimum of assumptions, allowing asymmetric suspensions or suspensions with limited degree of freedom to be analyzed. This feature allows the model to be used to test the robustness of a pendulum design against perturbations in the mechanical parameters which may occur during construction. The model has specifically been designed to include the effect of blade springs which can be used to achieve the necessary additional vertical vibration isolation not provided by a simple wire pendulum. The predictions of the code have been compared to the results of a force model, written by Dr. Calum Torrie, and to experimental results for a single stage wire pendulum and for a triple stage pendulum using two sets of cantilevers. These comparisons confirm the validity of the Lagrangian model. The Lagrangian code was then extended to provide a more detailed analysis of the suspension system. This included the proper performance of the blade springs, which has been analyzed and interpreted in terms of the centre of percussion of the springs. Additionally, the code has been used to analyze the cross-coupling between input vertical motion and resulting horizontal output motion due to small imperfections in the pendulum construction. This analysis confirms that the GEO 600 assumption that the crosscoupling would not exceed 0.1% is appropriate. Taking into account this factor, it is predicted that the amount of transmitted seismic noise which will be observed at 50 Hz, the low frequency corner of the detection band, in the measured direction is 2.4 X 10-20m√Hz. This is approximately a factor of 3 below the dominant noise at that frequency, the internal thermal noise of the test mass, which is expected to be 7 X 10-20m/√Hz. This seismic noise is dominated above ~10 Hz by the vertical noise coupling to the measurement direction. The Lagrangian model has also been used to predict the design of suspension which will result in the lowest pendulum thermal noise. Specifically, a suspension which uses two wires to suspend the optic, one wire in front of the other, has been compared to a four wire suspension equivalent to two loops of wire. This confirms that the four wire suspension is the best choice for GEO 600. The model is also used to compare more general cases, including other suspension geometries which are not suitable for use in the GEO 600 control scheme. The modelling code also allows the full pendulum thermal noise of the triple pendulum observed in the sensed direction at 50 Hz to be predicted at a level of 1.4 x 10-20m/√Hz. The thermal noise is reduced away from the resonant frequencies of the pendulum, the reductions being larger the higher the Q of the resonance. These resonant peaks cause unwanted amplification of low frequency motion. The amplitude of these peaks is reduced by application of electronic feedback control. This "local control" must damp the resonances without adding additional noise within the operating frequency band of the system. The successful performance of this local controller has been demonstrated in all degrees of freedom. Additional modelling has been done to examine other control problems associated with interferometric gravitational wave detection. While the local control can be relatively low bandwidth (~5 Hz), the "global control" used to maintain the entire interferometer on a dark fringe may require substantially higher bandwidth. A standing wave model of a wire pendulum has been developed to examine the possiblity of using higher bandwidth actuation in a non-collocated fashion. While this model does not have all the details of the full multi-stage pendulum model, it allows more accurate prediction of performance in the presence of the internal modes of the suspension wires (the 'violin modes'). Finally, one additional question is considered, namely the suitability of the standard RF error signal for locking a resonant optical cavity which has long optical storage times. A model of the cavity fields in a Fabry-Perot cavity which expands upon the standard error signal to include the effects of a moving end mirror has been experimentally verified. This model may be useful in designing a more robust locking algorithm. In conclusion, the design of the main suspension for GEO 600 has been verified, the parameters of which are presented in this thesis