The results of the simple model in section 5.3, and the measured optomechanical coupling rates of the taper coupled, microsphere-cantilever WGM system in section 5.4 indicate a worthwhile pursuit of dissipative cooling in a future experiment. However, three improve- ments are required:
1. A suitably high frequency mechanical mode (Ωm> 100 kHz) must be used, which
could belong to a torsional mode of the tapered fibre [136], or the microsphere-cantilever itself if the cantilever is tailored to obtain a high Ωm(described in chapter 7).
2. The measured gom is too large and must be suppressed with the aid of active
feedback stabilisation of the laser frequency, similar to the PDH stabilisation implemented in chapter 2. However, if a high Ωmis used, further optimisation of the feedback is required
such that it is fast enough to counteract the dispersive shift of the WGM mode.
3. The predicted laser powers required to demonstrate dissipative cooling are less than 1 W. However, the current fabrication and storage methodology for the tapered fibres in this thesis (chapter 2) results in tapers that typically melt at input laser powers over 5 mW at atmospheric pressure, and less than 1 mW at 0.5 mBar. Switching to a hydrogen flame reduces surface contaminates, and working within a clean room can help reduce dust adhering onto the taper during installation into the vacuum chamber. This will allow for more light to be coupled into the taper, and thus the WGM.
It should be noted, that if successful dissipative cooling of the c.o.m. motion of the microsphere-cantilever is achieved, the system can be further optimised for reaching the
quantum ground state and testing quantum phenomena, described in chapter 7. The microsphere can be untethered from the cantilever, and levitated using an optical and/or ion trap, whilst a tapered fibre optically couples light to the WGM circulating within it13.
5.6
Conclusion
In conclusion, this chapter broadly introduces passive cooling methods utilising optome- chanical coupling between mechanical motion and optical fields. Both dispersive and dis- sipative coupling exists between the tapered fibre and the microsphere-cantilever, whereby the motion of either oscillator shifts the WGM resonance (dispersive coupling rate gom),
and alters the WGM decay rate (dissipative coupling rate γom). The experimentally mea-
sured values of γom and gom for the taper coupled microsphere-cantilever system, in the
range of 10 MHz nm−1, are sufficient to obtain dispersive cooling, dissipative cooling or a combination of both. The latter two cooling schemes would herald new results, currently only achieved by one set-up [211].
A simple classical model using the coupled equations of motion to describe disper- sive and/or dissipative cooling has been implemented. The classical and quantum model of dispersive cooling has been actively studied for many years, and as such, no new in- formation is presented here. However, dissipative cooling is relatively new, with fewer experimental demonstrations. Although the original quantum analysis of dissipative cool- ing predicts cooling to the quantum ground state without requiring the sideband resolved regime, classical modelling is useful for investigating the validity of this claim when the oscillator contains many phonons. It is shown in section 5.3 that dissipatively cooling a high frequency (Ωm > 100 kHz) mechanical mode of either the microsphere-cantilever or
the tapered fibre could be demonstrated with relatively achievable laser powers. Signature behaviours predicted by the quantum theory of dissipative cooling is validated through the classical model, such as the existence of 2 heating and 2 cooling regimes dependent on the laser detuning with respect to the cavity resonance. Unlike dispersive cooling, where cooling is obtained with red-detuned light, dissipative cooling allows for cooling when blue-detuned. Of course, to obtain more accurate predictions of the damping rate, and the final cooled mode temperature, the simple model presented here should be modified to include the stochastic force responsible for Brownian motion.
13
Chapter 6
A Whispering Gallery Mode
Accelerometer
6.1
Introduction
The cumulative work of chapters 2 & 3 can be combined into an optical whispering gallery mode (WGM) accelerometer, unlike any commercially available device.
Accelerometers are sensors that can measure the acceleration of a moving or vibrating body. Such devices play an integral role in precise navigation [228, 229], gravity gradiom- etry [135, 230], and structural health monitoring. There are two types of accelerometer; those using a test-mass on a spring, or free-fall accelerometers. The most common use a test-mass on a spring (i.e. cantilevers or electrodes), and measure the extension of the spring as the ‘g-force’ (also referred to as proper acceleration) is applied. The spring elongates until the restoring forces within the spring pull the test-mass back. This mea- surement of acceleration is relative to the Earth’s reference frame, such that a constant reading of −g (g=9.81 m s−2) is obtained in the y-axis. Free-fall accelerometers such as atom interferometers [231] measure the time it takes for the test-mass to fall a set distance. This cancels out the measurement of the Earth’s gravitational field, and are considered more accurate but are larger in size.
It will come as no surprise that within the last twenty years many optical systems have revolutionized the field of inertial sensing. Optical accelerometers operate by measuring the interaction of a moving test-mass with an optical field, for example, the change in light coupling efficiency into an optical fibre [23, 232], the optical mode coupling between two
photonic-crystal zipper beams [12]1, or the dispersive coupling of a movable mirror in a Fabry-Perot cavity [21]. Optical sensors offer significant improvements over their electrical counterparts2, as they are non-conductive, immune to electromagnetic interference and can reach sensitivities limited only by the quantum fluctuations of light [2, 71, 179] (and the mechanics of the test-mass).
The microsphere-cantilever fabricated and studied in chapter 2 offers both a test-mass and an optical resonance in the form of the WGM. Similar devices use WGMs to detect a moving test-mass placed in the WGM evanescent field [19, 24]. However, in this thesis, the test-mass is the WGM resonator itself, which has been studied previously by Haus et al. [20] using a WGM resonator coupled to a MEMS waveguide. Here, the coupling is pro- vided by a tapered fibre (the fabrication and coupling of WGMs using the tapered fibre is shown in chapter 2). The circulating WGM within the microsphere can transduce its own motion on the cantilever, which is optimised and described in chapter 3. In that chapter, the thermal motion of the microsphere-cantilever could be detected, which demonstrates the feasibility of transducing a resultant deflection of the cantilever in response to accel- eration3. An internal force known as stress (force per unit area) causes the cantilever to bend such that the deflection (related to strain) is proportional to acceleration. The the- ory that describes the relationship between stress, strain, and deflection will be presented in this chapter, in order to compare experimental measurements with prediction.
The limit to sensing, i.e. the minimum resolvable acceleration, is set by the noise floor in the displacement power spectral density (PSD). Typical displacement PSD’s obtained in chapter 3 using the WGM transduction show a displacement sensitivity ≈ 10−12m Hz−1. For the purposes of measuring acceleration, this displacement sensitivity corresponds to an acceleration sensitivity, called the noise equivalent acceleration aNEA, such that an
aNEA≈ 1 µg Hz−1/2 implies a measurement of 1 µg within a sample time of 1 s. It will be
shown that the WGM sensor studied in this chapter offers the lowest reported aNEA from
a WGM device.
1
A photonic zipper beam comprises of a clamped-clamped beam whose cross section is patterned for strong optical localisation into a small mode volume at the centre of the beam. Two nanobeams placed within the near-field of the other results in strong optomechanical coupling to the relative motion between the beams.
2
Capacitive accelerometers measure the capacitance change between a fixed electrode and a movable electrode. They are prone to electromagnetic disturbance and jamming, often requiring significant invest- ment and engineering to measure accelerations ≤ µg.
3
The taper can also function as an accelerometer but it is 20 times lighter therefore will have an acceleration sensitivity which is a factor of√20 smaller compared with a microsphere-cantilever with similar mechanical frequency and mechanical quality factor, see section 6.4.4 on the noise equivalent acceleration.
The sensing range is entirely determined by the material and geometry of the microsphere- cantilever, as well as the physical separation distance to the taper. This DC coupling distance between the taper and the microsphere is referred to as the ‘null position d0’
in accordance with accelerometer terminology, and large displacements away from d0 can
eventually cause the microsphere to touch the taper, setting a maximum acceleration limit. The maximum sensing range for this type of system has not been previously reported, and no indication of the stability using the microsphere-cantilever has been studied. Stability in particular refers to drifts that may offset the measurement, providing a false signal. The so-called vibration rectification error (VRE) affects all test-mass sensors due to non- linearity caused by asymmetric clamping or geometry (i.e. the time averaged deflection is biased in one direction), and to a lesser extent, asymmetric damping. The VRE is troublesome in high vibration environments as it causes a false bias in the output.
The lack of experimental data characterising the WGM sensor in terms of drift, VRE, and the sensing range (i.e. linear range vs. non-linear range) motivates the majority of the experimental investigations presented here.
This chapter will:
• Discuss the theory behind the WGM microsphere-cantilever accelerometer.
• Present measurements of the WGM acceleration sensing range, compared with a commercial accelerometer.
• Present measurements of the noise equivalent acceleration, and measure the mini- mum resolvable acceleration over a range of driving frequencies.
• Investigate drift, including vibration rectification errors.