1.2 Overview of the goals of interferometer design

As set out in very many works, gravitational-wave detectors strive to pick out signals carried by passing gravitational waves from a background of self-generated noise. The principles of operation are set out at various points in the review, but in essence, the goal has been to prepare many photons, stored for as long as practical in the ‘arms’ of a laser interferometer (traditionally the two arms are at right angles), so that tiny phase shifts induced by the gravitational waves form as large as possible a signal, when the light leaving the appropriate ‘port’ of the interferometer is detected and the resulting signal analysed.

The evolution of gravitational-wave detectors can be seen by following their development from prototypes and early observing systems towards the Advanced detectors, which are currently in the final stages of planning or early stages of construction. Starting from the simplest Michelson interferometer [18Jump To The Next Citation Point], then by the application of techniques to increase the number of photons stored in the arms: delay lines [31], Fabry–Pérot arm cavities [16Jump To The Next Citation Point17] and power recycling [15]. The final step in the development of classical interferometry was the inclusion of signal recycling [4130], which, among other effects, allows the signal from a gravitational-wave signal of approximately-known spectrum to be enhanced above the noise.

Reading out a signal from even the most basic interferometer requires minimising the coupling of local environmental effects to the detected output. Thus, the relative positions of all the components must be stabilised. This is commonly achieved by suspending the mirrors etc. as pendulums, often multi-stage pendulums in series, and then applying closed-loop control to maintain the desired operating condition. The careful engineering required to provide low-noise suspensions with the correct vibration isolation, and also low-noise actuation, is described in many works. As the interferometer optics become more complicated, the resonance conditions, i.e., the allowed combinations of inter-component path lengths required to allow the photon number in the interferometer arms to reach maximum, become more narrowly defined. It is likewise necessary to maintain angular alignment of all components, such that beams required to interfere are correctly co-aligned. Typically the beams need to be aligned within a small fraction (and sometimes a very small fraction) of the far-field diffraction angle, and the requirement can be in the low nanoradian range for km-scale detectors [4421]. Therefore, for each optical component there is typically one longitudinal (i.e., along the direction of light propagation), plus two angular degrees of freedom (pitch and yaw about the longitudinal axis). A complex interferometer can consist of up to around seven highly sensitive components and so there can be of order 20 degrees of freedom to be measured and controlled [357].

Although the light fields are linear, the coupling between the position of a mirror and the complex amplitude of the detected light field typically shows strongly nonlinear dependence on mirror positions due to the sharp resonance features exhibited by cavity systems. However, the fields do vary linearly or at least smoothly close to the desired operating point. So, while well-understood linear control theory suffices to design the control system needed to maintain the optical configuration at its operating point, bringing the system to that operating condition is often a separate and more challenging nonlinear problem. In the current version of this work we consider only the linear aspects of sensing and control.

Control systems require actuators, and those employed are typically electrical-force transducers that act on the suspended optical components, either directly or – to provide enhanced noise rejection – at upper stages of multi-stage suspensions. The transducers are normally coil-magnet actuators, with the magnets on the moving part, or, less frequently, electrostatic actuators of varying design. The actuators are frequently regarded as part of the mirror suspension subsystem and are not discussed in the current work.

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