Demonstration of an optical spring in the 100 g mirror regime

Optical rigidity will play an important role in future generations of gravitational wave (GW) interferometers which employ high laser power in order to reach and exceed the standard quantum limit. Several experiments have demonstrated the optical spring effect for very low weight mirror masses. In this paper we extend this to a mass and frequency regime more directly applicable to GW detectors. Using a 100 g ?> end mirror mass we demonstrate an optical spring resonant at 500 Hz ?> and a stiffness of 9.4 ×105 N m−1. The 100 g ?> to 1 kg ?> mass regime may also be useful for the application as a readout mirror for optical bar or optical lever configurations.


Introduction
The first generation of laser-interferometric gravitational wave (GW) detectors (LIGO [1], Virgo [2], GEO 600 [3] and Tama [4]) have successfully demonstrated the long-term operation km-scale Michelson interferometers, delievering observational data of unpreceeded strain sensitivity. The next generation of GW detectors is currently under construction and commissioning (advanced LIGO [5], GEO-HF [6], KAGRA [7] and advanced Virgo [8]) and is expected to achieve about a ten-fold increased sensitivity. Over most of their frequency Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
range these advanced instruments will be limited by so-called quantum noise, which consists of photon shot noise at the high-frequency end and photon radiation pressure noise at the low frequency of the observing band. Optical spring techniques [9] have various applications in further reducing quantum noise, and in the long term future even in surpassing the standard quantum limit (SQL) [10]. Detuned signal recycling [11] produces an optical spring, and the resulting change to the opto-mechanical dynamics of the interferometer provides one example of how the SQL may be beaten [12]. Such an optical spring resonance with detuned signal recycling was observed at the Caltech 40 m prototype [13]. Other interferometer topologies based on the principle of optical rigidity include optical bar [14], optical lever [15] and local readout configurations [16]. All of these techniques are currently under consideration [17,18] for upgrades of Advanced LIGO or third generation observatories such as ET [19,20].
In this article we describe the demonstration of an optical spring in a three-mirror coupled cavity. Our experiments extend the range in which optical springs have been demonstrated for Fabry-Pérot cavities, from 1 g [21] up to a mass range of 100 g and to a 10 m prototypesystem scale, bringing it therefore one step closer to application in future GW observatories.

Optical spring generation
To observe opto-mechanical coupling between the mirrors of an optical cavity, the radiation pressure force exerted by the internal cavity field must be comparable to, or greater than, the mechanical restoring force of the pendulums [9]. Since the radiation pressure force is inherently weak, this requires high-finesse cavities to maximise the amount of stored light. In our system two cavities were combined in a three-mirror coupled cavity configuration, analogous to that of the power recycling and arm cavities used in GW detectors, to form a coupled system of high-finesse and allowing high power build up in the 'arm cavity (AC)' (see figure 1).
As the stored light in a three mirror coupled cavity interacts with the mirrors, it imparts a radiation pressure force and modifies the dynamics of the suspended mirrors. We relate the cavity length, or more specifically its detuning, to the optical power with coupling considered to be optimal when there is maximum power in the AC. The steady power in the AC P 0 is given by and P laser is the laser power impinging on the recycling cavity mirror. If we arrange that the recycling cavity is exactly on resonance and held such that the power recycling cavity resonance condition is decoupled from the AC resonance [25], equation (1) can then be expressed in terms of the AC detuning θ, as where F c is the coefficient of finesse for the coupled cavity system and g ca and g ab are the amplitude gains of the recycling and arm cavities respectively. The recycling cavity can thus be considered simply as an additional source of power gain and the system is reduced in complexity such that the optomechanical coupling behaves like that of a simple cavity. Such systems have been considered in detail [9,26] and the dynamics of a detuned cavity of linewidth, γ, are well known. The quasi-static optical spring constant for such systems (where the optical spring frequency is f opt g  ), can be defined as  =´with a corresponding optical spring frequency of f 496 Hz opt = . Using the same parameters the linewidth of the coupled cavity is 3305 Hz) and is therefore sufficiently narrow that any meaasurement process to obtain the optomechanical response will induce motion at comparable frequencies. Hence, the assumption of quasi-static detuning is not sufficient for the experimental system under consideration and the cavity response time must be included in the analysis.
The frequency dependent optical rigidity has been studied [9,27], and upon simplification can be shown to be The complete description of the frequency dependent optical rigidity is complex with the real components attributed to rigidity and imaginary parts describing a velocity dependent damping force. The full form of the optical spring constant is valid for all mirror frequencies, and we see that for sufficiently slow mirror motion ( 0 w  ) the imaginary term disappears resulting in the expression obtained earlier for static detuning. Figure 2 (right) shows the expected change to the strength of the optical spring constant with respect to detuning and frequency of observation, based on the numerical model. At frequencies greater than the cavity linewidth the spring constant is predicted to change sign. In other words there is a fundamental limit to the observation of an optical spring that is directly related to the response time of the cavity.

Optical layout of three-mirror coupled cavity
The three-mirror coupled cavity configuration was installed in the Glasgow 10 m prototype interferometer laboratory. Each mirror was suspended as a multi-stage pendulum to provide isolation from seismic vibrations, and housed in a vacuum system to eliminate the effects of acoustic noise and air-currents.
The Glasgow system typically houses 2.7 kg mirrors (test masses) and utilises lower circulating power such that the effects from radiation pressure are negligible, but in this experiment we wanted to accentuate the spring effect. To this end, the AC end mirror was reduced to 100 g to bring the optical spring resonance into an appropriate frequency band to allow unambiguous observation (around 500 Hz) and extend the mass range from those explored in previous experiments [9,22,23].
The cavity mirrors were suspended as triple pendulums, providing considerable isolation against ground motion above the fundamental longitudinal pendulum resonance (at around 0.6 Hz). Through careful design choices regarding masses, wire thickness, length and positioning of break-off points it was possible to keep the rigid body modes below 30 Hz, leaving an unobstructed region near the target optical spring frequency range at around a few hundred Hz.

Coupled cavity control scheme
It is necessary to control the relative positions of the mirrors to maintain the resonant condition for maximum power build up in the AC. In this experiment it was desirable to decouple the length sensing signals from the two cavities and so a scheme was implemented whereby the AC and power recycling cavity length sensing and control signals were separately detected using a combination of the conventional Pound Drever Hall approach and a doubly resonant amplitude-modulated sub-carrier. This orthogonalised control scheme allows for large deviations of each cavity from the centre of resonance, while maintaining decoupling between the coupled cavity control signals. A detailed explanation of the approach used is described in [25].
To introduce detuning, and hence create an optical spring, a DC offset was introduced at the error point of the AC loop, producing a static frequency offset between the laser and the AC. As the AC feedback control is provided by the laser frequency stabilisation loop and has a high bandwidth (100 kHz), this loop maintains the lock and thus ensures stability of the system during detuning.
To measure the response of the optical spring to a perturbation of the system, a test signal was added to the DC offset in the AC control loop. The effect of the optomechanical spring can subsequently be observed by measuring the cavity transfer function.
It should be noted that both the optical spring strength and cavity control signals are dependant on the optical power stored in the cavity. Thus any effect which can alter the stored power during the course of the measurement, such as long-term drifts in cavity alignment or Sidle-Sigg effects [28], will cause fluctuations away from the intended detuning as determined by the DC offset. From observation of power changes on the stored light power during measurements the detuning uncertainty is approximately 0.03 d g . Figure 3 presents the largest observed optical spring frequency, f opt , located at around 496 Hz. As the reduced mass of the coupled two mirror system is known, m 96.4 g r = , we can determine the rigidity for an optical spring at this frequency by =´ [29], indicating that our investigations on optical rigidity with a fully suspended coupled cavity enabled a effective coupling medium of about the same stiffness as diamond.
To investigate variation of the stiffness of the optical spring with cavity detuning we operated the system with several different DC offsets and estimated the spring frequency in each case. It was found that the clearest marker of the resonant frequency, in the presence of measurement noise, was the point at which the phase trace crossed 90°. Comparison of five measured resonant frequencies plotted as a function of detuning is shown with the numerical model in figure 4.

Summary and future work
The largest optical spring observed at f 496 Hz opt = corresponds to an optical spring constant of K 9.4 10 N m opt 5 1 =´-, for which an effective coupling medium between the cavity mirrors would have a Young's modulus essentially that of diamond.
The main limitation to our measurements was due to residual laser frequency noise. In future experiments, lowering this noise, for example by using a pre-stabilised laser, would allow much shorter measurement times and reduced sensitivity to drift. Lowering mirror losses and increasing the injected optical power would allow even stiffer springs, more appropriate to the 10 kg-scale masses employed in GW detectors.