Engineering the optical spring via intra-cavity optical-parametric amplification

Abstract

The 'optical spring' results from dynamical back-action and can be used to improve the sensitivity of cavity-enhanced gravitational-wave detectors. The effect occurs if an oscillation of the cavity length results in an oscillation of the intra-cavity light power having such a phase relation that the light's radiation pressure force amplifies the oscillation of the cavity length. Here, we analyse a Michelson interferometer whose optical-spring cavity includes an additional optical-parametric amplifier with adjustable phase. We find that the phase of the parametric pump field is a versatile parameter for shaping the interferometer's spectral density.Introduction -Electromagnetic dynamical back-action was first observed in radio-frequency systems and its existence predicted for optical Fabry-Perot cavities by Braginsky and his colleagues more that 50 years ago [1,2]. The first proposal of using dynamical backaction to improve the sensitivity of laser-interferometric gravitational-wave detector was made in 1997, again by Braginsky and co-workers [3]. The new scheme was called 'optical bar', since the light's radiation pressure force rigidly connects two far separated mirrors, which are suspended as pendula but quasi-free otherwise. This way, a gravitational-wave signal is transformed into an acceleration of mirrors with respect to the local frame. The interferometric topologies that are considered in [3] as well as in related work [4] are different from the Michelson topology having a balanced beam splitter, and were not experimentally realised so far. Recently, a more practical design was proposed [5]. The second proposal was made in 2002 by Buonanno and Chen [6] and was called 'optical spring'. It targets the sensitivity improvement of Michelson-type gravitationalwave detectors having a signal-recycling (SR) cavity [7,8] or signal-extraction (SE) cavity, also called resonantsideband extraction (RSE) [9,10]. For the purpose of utilizing the optical spring in a Michelson interferometer operated on dark output port, these cavities need to be detuned from carrier light resonance. If the frequency of the carrier light is blue-detuned with respect to the cavity, the lower sidebands of phase modulations that are produced by gravitational waves and that are matching the detuning frequency get optically enhanced while the corresponding upper sidebands are suppressed. Due to energy conservation, the mechanical (pendulum) motion of the suspended mirror is enhanced [11,12]. The overall process corresponds to optomechanical parametric amplification and results in optical heating of the mechanical motion, i.e. the opposite of optical cooling [13]. The radiation pressure of the light not only results in an optomechanical parametric amplification of the pendulum motion but also to an additional (optical) spring constant that increases the pendulum resonance frequency from typically 1 Hz to an opto-mechanical resonance of up to about 100 Hz. Around this frequency the mechanical response of the GW detector is significantly enhanc...

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