Tabla de parámetros

WGM resonators have a spectrum of resonances, described in the introduction, and ex- perimentally excited in this chapter. The set-ups built during this Ph.D. for fabrication of the microsphere-cantilever and the tapered fibre used to couple light to the microsphere is described. The taper coupled microsphere-cantilever system was experimentally investi- gated, exciting narrow WGM resonances of 5 MHz FWHM linewidth with Qopt> 5 × 107.

The external coupling rate γe that mediates the exchange of photons between taper

and microsphere depends exponentially on the coupling distance, d. This was verified in this chapter, and the WGM coupling regime was tuned from undercoupled, to critical coupling (d ≈ 150 nm), to overcoupled.

The high intensity generated within the microsphere leads to bulk heating where the refractive index and sphere dimensions change with increased powers coupled to the WGM. Such an effect presents itself as thermal bi-stability, seen as linewidth broadening and nar-

Figure 2.25: Counterpropagating WGMs are excited using the strong beam (black) and a detuned transduction beam (red), detuned by 20 MHz for a 40 MHz FWHM mode. T denotes the transmission measured by PD 1 (strong beam), and PD 2 (transduction beam). The error signal, formed using the strong beam, is decoupled from the transduction beam so that the zero crossing coincides with the WGM excited by the strong beam.

rowing when the laser is scanned approaching from the blue-detuned side or red-detuned side respectively. Despite this thermal bi-stability, the active PDH locking demonstrated in this chapter can stabilise the laser on-resonance with the WGM. A passive thermal lock- ing equilibrium is successfully applied, where no active feedback electronics are required to keep the WGM excited on the blue-detuned side, useful for cases where a purposely large amplitude modulation of intracavity intensity is required that can break the PDH lock. This is crucial for chapter 4 and chapter 6 where large modulations of the strong beam Pc, and driven displacements are implemented respectively. Lastly, the input-output

mechanism of the tapered fibre allows for excitation of the counterpropagating WGM (i.e. the ±m degenerate modes), for the purposes of the active cooling presented in chapter 4, where one beam is dedicated for transduction and the other for maintaining laser stabil- isation to the WGM resonance (or modulating optical forces). The transduction beam is investigated in chapter 3.

Chapter 3

Transduction of Mechanical

Motion using Whispering Gallery

Modes

3.1

Introduction

This chapter investigates the optomechanical coupling between whispering gallery modes (WGMs) and the thermal motion of the microsphere-cantilever and tapered fibre. The coupling allows for optical transduction of mechanical motion by detecting the fluctuations imprinted onto the WGM field coupled to that motion. Measuring displacements with WGMs is essential for the sensing and cooling of thermal motion explored in this thesis. For example, to use the microsphere-cantilever for accelerometry, as detailed in chapter 6, it is important to characterise the mechanical parameters that predict its response to stimuli, such as mechanical frequency, mechanical quality factor, spring constant and effective mass. These can be deduced by analysing the power spectral density (PSD) of the WGM transmission i.e. from the dedicated transduction beam created in section 2.6.1 of chapter 2. In active feedback cooling, demonstrated in chapter 4, a feedback signal formed from the transduction beam is required to drive actuating forces to apply damping.

Transduction is defined as the process by which a device converts one type of quan- tity (e.g. energy) into another. The first transducers were materials that could convert electrical and magnetic fields into mechanical motion, such as via Joule magnetostriction which causes the constriction of iron under the influence of a magnetic field. The Curie brothers discovery in the late 18th century that an electric charge is produced by applying

an external mechanical force to quartz crystals.

The transduction of mechanical motion using optical fields can be realised when the displacement of a mechanical oscillator causes perturbations in the optical intensity [135], polarisation [136] (i.e. via the strain-optic effect), wavelength [4, 137, 138], and/or phase [31, 139]. Detection schemes can be as simple as measuring the changing shadow cast by a movable object in front of a light source [135], or can be enhanced by adding a resonant cavity structure such as a Fabry-Perot (F-P) cavity [137, 138]. The modulation to the photon density can be measured by monitoring the intensity directly, or through interfero- metric or homodyne methods to monitor the phase or frequency shifts. When mechanical motion shifts a cavity resonance frequency, it is said to be dispersively coupled to the op- tical field, governed by a linear scale-factor called the dispersive coupling parameter, gom,

in units of Hz m−1. Cavities that have high optical quality factors Qopt, such as the WGM

employed in this thesis, can enhance gom, and in some cases provide an interferometric

calibration between displacement and frequency [21].

Cavity enhanced optomechanical coupling using optical WGMs has been used to trans- duce and implement cooling of the radial breathing modes of a toroid [4], the flapping mo- tion of two stacked disks [15], and mechanical modes of nanostrings in the near-field [1]. The gom associated with WGMs is often larger than those for F-P cavities, owing to the

higher Qopt and the mode confinement. For the system studied in this thesis, comprising

of a microsphere-cantilever coupled to a waveguide, the use of the WGM around the mi- crosphere to detect the motion of its own cavity has been studied by one other group [45]. In 2015 they derived a theoretical framework for this transduction, showing that both dispersive and dissipative (motion is coupled to the loss of the WGM with dissipative coupling parameter γom) coupling enhances the displacement measurement. The inclusion

of γom differentiates this system from traditional optomechanics experiments which have

negligible γom. This is discussed in chapter 5.

The sensitivity of a measurement is defined by the imprecision noise of the detector (i.e. noise of the light field plus electronic noise and photodetector noise), and depends on the measurement acquisition time since averaging white noise over long timescales decreases the uncertainty [140]. Displacement sensitivity is measured in units of m Hz1/2, and defines the linear relationship between the resolution and the measurement time. This is also the square root of the PSD that characterises the power distribution across frequencies [141].

Imprecision noise defines the noise floor of the PSD, and places a cooling and measurement limit. The benefits of using optical fields to detect motion are related to the lower noise levels easily obtainable in often quantum-limited sources, which can also be squeezed in amplitude or phase [53, 142, 143].

As cooling progresses to a point where kBT < ~Ωm (Ωm is the mechanical frequency,

T is the effective bath temperature), quantum zero-point fluctuations of the mechanical oscillator (ZPF) will play a dominant role over thermal fluctuations. Conversely, the light field that probes this motion will have quantum noise that defines the precision of measuring the ZPF, as well as heating the motion through radiation pressure backaction1. A discussion of the quantum treatment of noise and the ultimate limits for transduction will be presented.

In summary, this chapter will present:

• The theory and experimental demonstration of WGM transduction, showing detec- tion of the thermal motion of the microsphere-cantilever and tapered fibre.

• Analysis of the classical power spectral density (PSD), created from the transduction signal, allowing calculation of the mode temperature, and structural mechanical properties.

• The identification of actuating forces to control the motion of the microsphere- cantilever and tapered fibre.

• The limitations of continuous displacement measurements.