A vacuum integrated fibre Fabry-Pérot cavity setup for optical and optomechan- ical characterisation Master’s thesis in Physics Johanna Örgård DEPARTMENT OF MICROTECHNOLOGY AND NANOSCIENCE CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2025 www.chalmers.se www.chalmers.se Master’s thesis 2025 A vacuum integrated fibre Fabry-Pérot cavity setup for optical and optomechanical characterisation Johanna Örgård Department of Microtechnology and Nanoscience Quantum Technology Laboratory Quantum Sensing and Foundations Lab Chalmers University of Technology Gothenburg, Sweden 2025 A vacuum integrated fibre Fabry-Pérot cavity setup for optical and optomechanical characterisation Johanna Örgård © Johanna Örgård, 2025. Supervisor: Hannes Pfeifer, Department of Microtechnology and Nanoscience - MC2 Examiner: Witlef Wieczorek, Department of Microtechnology and Nanoscience - MC2 Master’s Thesis 2025 Department of Microtechnology and Nanoscience Quantum Technology Laboratory (QTL) Quantum Sensing and Foundations Lab Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Picture of a fibre Fabry-Pérot cavity in the vacuum integrated setup for optomechanical characterisation explained in Chapter 3. Typeset in LATEX, template by Kyriaki Antoniadou-Plytaria Printed by Chalmers Reproservice Gothenburg, Sweden 2025 iv A vacuum integrated fiber Fabry-Pérot cavity setup for optical and optomechanical characterisation Johanna Örgård Department of Microtechnology and Nanoscience Chalmers University of Technology Abstract Fibre Fabry–Pérot cavities are miniaturised Fabry–Pérot cavities integrated on opti- cal fibres. Their small size, open mode volume and intrinsic integration with optical fibres make them well suited for optomechanical experiments in which one studies the interaction between light and mechanical motion. The short cavity length enables enhancement of the vacuum optomechanical coupling strength and the open mode volume allows for the easy insertion of mechanical resonators or other quantum sys- tems directly into the cavity. These features make fibre Fabry–Pérot cavities suited for many applications such as levitation of nano particles and quantum-enhanced inertial and force sensing. In this master’s thesis, an experimental platform was developed with the goal to enable optical and optomechanical characterisation of fibre Fabry–Pérot cavities consisting of a fibre mirror and a reflective mechanical resonator on a chip. To this end, an existing experimental setup for optical characterisation was improved upon through automatisation and a vacuum-integrated setup for optomechanical experi- ments was built and tested. The resulting experimental platform enables measurements of the resonance linewidth and free spectral range of fibre Fabry–Pérot cavities. Within the vacuum-integrated setup the cavity length can be locked to the drive laser. This setup will in fu- ture work be used for optomechanical experiments measuring the optomechanical coupling strength and controlling mechanical resonator modes utilising the optical spring effect and optomechanical damping. Keywords: optomechanics, fibre Fabry–Pérot cavity, photonic crystal, distributed bragg reflector, nanoscience v Acknowledgements I would like to thank Prof. Witlef Wieczorek for giving me the opportunity to work in such a supportive and stimulating research team and for giving me so much support during these past few months. I would also like to thank my supervisor Hannes Pfeifer for guiding through every step of this project and teaching me so much about this existing research field. I am grateful to the optomechanics team as a whole, including Anastasiia Ciers and Alexander Wolfgang Martin Jung, for always being there to answer any questions I have had and giving me so many new insights. Thank you to the rest of the people at the Quantum Sensing and Foundations Lab: Paul Nicaise, Achintya Paradkar, Fabian Resare, Alireza Hashemi and Somiya Islam Soke; you have all been very welcoming to me I have always felt like there was someone to turn to if I needed help. Johanna Örgård, Gothenburg, June 2025 vii List of Acronyms Below is the list of acronyms that have been used throughout this thesis listed in alphabetical order: AC alternating current DBR distributed Bragg reflector DC direct current EOM electro-optic modulator FC/APC angled physical contact for fibre patch cord FFPC fibre Fabry-Pérot cavity FPC Fabry-Pérot cavity HV high vacuum PDH Pound–Drever–Hall SoF side-of-fringe ix Nomenclature Below is the nomenclature of recurring parameters and variables used throughout this thesis. â photon annihilation operator ↠photon creation operator âin the stochastic quantum field incident on the incoupling mirror âout the field reflected from a Fabry-Pérot cavity A parameter for the lineshape of the reflection from a FFPC b̂ phonon annihilation operator b̂† phonon creation operator c speed of light f̂in the stochastic quantum field incident on other incoupling channels than âin F finesse of an optical cavity F (ω) complex reflection amplitude at an optical cavity G optical frequency shift per mechanical displacement g0 vacuum optomechanical coupling strength g g = g0 √ n̄cav, light-enhanced optomechanical coupling strength ℏ Planck’s constant Lcav cavity length meff effective mass of mechanical resonator n̄cav average number of photons circulating in an optical cavity N translational mode number of optical cavity Pin the power incident on the optical cavity Pout = Pref the power reflected back from the optical cavity Qopt optical quality factor Qm mechanical quality factor Sϵ noise spectral density for the error signal for a cavity lock xi Sνν noise spectral density for the cavity resonance frequency νcav x(t) global amplitude of mechanical motion xZPF zero-point fluctuation amplitude of mechanical resonator β amplitude of the EOM signal Γm mechanical damping rate Γopt damping of the mechanical oscillator due to radiation pressure forces ∆ ∆ = ω − ωcav, laser detuning with respect to the cavity mode δΩm shift of mechanical frequency resulting from optical spring effect ϵ the error signal for a cavity lock ηdip parameter expressing the resonance depth of FPCs ηr parameter for the lineshape of the reflection from a FFPC ηL parameter for the lineshape of the reflection from a FFPC κ full cavity decay rate κex cavity decay rate due to transmission through the incouplng mirror κin cavity decay rate which is not due to transmission through the incouplng mirror λcav wavelength of a mode inside an optical cavity νFSR Free spectral range in terms of frequencies, equal to ωFSR νcav frequency of a mode inside an optical cavity, equal to ωm/2π ν ν = (ω−ωcav)/κ, variable expressing the lineshape of the reflection from a FFPC or FPC νlaser frequency of the main tone of the laser drive, equal to ω/2π νlaser1 and νlaser2 frequencies of the sideband tones τ average lifetime of a cavity photon ωcav angular frequency of a mode inside an optical cavity ωFSR Free spectral range in terms of angular frequencies ω angular frequency of the laser Ωm angular frequency of mechanical resonator Ω angular frequency of the EOM signal xii Contents List of Acronyms ix Nomenclature xi List of Figures xv 1 Introduction 1 1.1 Goal and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Theory 3 2.1 Fabry-Pérot cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Input-output theory for FPCs . . . . . . . . . . . . . . . . . . 5 2.1.2 Fibre Fabry-Pérot cavities . . . . . . . . . . . . . . . . . . . . 6 2.2 Cavity optomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Mechanical resonators . . . . . . . . . . . . . . . . . . . . . . 8 2.2.2 Optomechanical coupling . . . . . . . . . . . . . . . . . . . . . 9 2.2.2.1 The optical spring effect and optomechanical damping 11 3 Methods 13 3.1 Optical characterisation of FFPCs . . . . . . . . . . . . . . . . . . . . 13 3.1.1 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Data collection and analysis . . . . . . . . . . . . . . . . . . . 15 3.2 Optomechanical measurements . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . 18 3.2.2 Pound-Drever-Hall locking for optomechanical characterisation 22 3.2.3 Side-of-fringe locking and the optical spring effect . . . . . . . 24 4 Results 27 4.1 Optical characterisation . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1.1 Sweeping of the cavity length . . . . . . . . . . . . . . . . . . 27 4.1.2 Measurements of cavity finesse . . . . . . . . . . . . . . . . . . 28 4.1.3 Polarisation dependency of the reflected power . . . . . . . . . 35 4.2 Experiments with the vacuum integrated optomechanical setup . . . . 37 5 Conclusion & Outlook 41 A Placement of the structures on the photonic crystal chip I xiii Contents xiv List of Figures 1.1 A conceptual sketch of a fibre Fabry-Pérot cavity that is explored in this thesis. It consists of a fibre mirror and a reflective structure on a chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 A conceptual sketch of Fabry-Pérot cavity with the cavity length Lcav. The fields in the input-output description of a FPC are illustrated. They are â, the complex field amplitude of a mode inside the cavity, âout, the rate of light leaving the cavity from Mirror 1 and âin and f̂in which are the rates with which light arrives at Mirror 1 and Mirror 2 respectively. Also illustrated are the photon cavity de- cay rates, κex, which describes the rate at which cavity photons leave the cavity out through Mirror 1, and κin, the decay rate associated with other channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Sketch of the expected reflection signal from a FFPC based on Equation 2.18. The total reflected power (red line) is the sum of a Lorentzian part (dotted green line) and a dispersive part (dash-dotted blue line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 A conceptual sketch of a fibre Fabry-Pérot cavity illustrating the relevant spatial modes of light propagating in the fibre and the cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Plot of δΩm (a) and Γopt (b) as a function of laser detuning ∆ when κ ≫ Ωm for g = 2π · 100 kHz, Ωm = 2π · 1 MHz and κ = 2π · 800 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1 Overview of the setup for optical characterisation of FFPCs consisting of a continuously tunable laser source, that via single-mode optical fibre sends a signal through a paddle wheel polarisation con- troller to an electro-optic modulator, a 10:90 fibre-optic splitter, a photodetector and through another paddle wheel polarisation con- troller to the FFPC to be characterised. The fibre-end of the FFPC is mounted on a shear plate actuator. This enables continuous modu- lation of the cavity length by a piezo controller which in turn gets its signal from an oscilloscope. The other end of the FFPC consists of a chip that can be tilted and moved in any direction with micrometer precision, see Figure 3.2. The oscilloscope is also used to read out the signal from the photodetector. . . . . . . . . . . . . . . . . . . . . . . 14 xv List of Figures 3.2 Sketch of the structure holding up and aligning the FFPCs during the optical characterisation. The fibre-end of the FFPC is mounted on a shear plate actuator, enabling continuous modulation of the cavity length. The other end of the FFPC consists of a chip that is mounted on a tiltable mirror mount that is in turn mounted on a micrometer translation stage. . . . . . . . . . . . . . . . . . . . . 16 3.3 Sketch of the expected shape of the reflection signal around resonance. The signal features three dips, one from the main signal and two from the sidebands created with the EOM. . . . . . . . . . . 16 3.4 Sketch of a cross section of the vacuum chamber, showing the structure holding and aligning the FFPCs during the optomechanical experiments. The chip with the mechanical res- onator is mounted on a tiltable mirror mount which is in turn mounted on top of two nanopositioners enabling movement of the chip in two dimensions. An aluminium holder was made to mount the mirror mount to the nanopositioners. The aluminium holder is designed such that the weight is equally distributed on the nanopositioners to avoid torque that might damage them. The fibre is mounted on a third nanopositioner enabling alignment of the cavity in the third dimension. On top of this nanopositioner, a shear piezo element is mounted which is used for fine-positioning of the fibre and thereby locking of the cavity. The fibre is glued to the shear piezo using UV epoxy. The position of the end of the fibre is stabilised in two di- mensions by having it go through a glass ferrule. A USB microscope is mounted outside the vacuum chamber by a viewport, providing a visual image of the fibre and the chip with the help of a silver mirror (not shown in this sketch) which is mounted inside of the vacuum chamber. Also not shown in this sketch are wires sending signals to the tiltable mirror mount, nanopositioners and shear piezo connected to the outside through ports on the side of the chamber as well as the fibre feedthrough on the top of the chamber. . . . . . . . . . . . . . . 19 3.5 Overview of the setup for the optomechanical experiments consisting of the same optical components as in the setup for optical characterisation with some additional electrical components to enable the locking of the cavity length to the drive laser. These additional components are a multiplicative mixer, a low-pass filter, a resistor, a bias tee and a high-voltage amplifier and a RedPitaya. The three circuit breakers illustrate which path is used for PDH or SoF locking. The RedPitaya controls a shear piezo (here illustrated as a brown rectangle) used for fine positioning during PDH and SoF locking. A nanopositioner from Attocube (here illustrated as a dark green rectangle) controlled by an AMC100 is used to move the fibre along greater distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.6 Expected values for δΩm/2π (a) and Γopt (b) as a function of laser power for g0 = 2π · 10 kHz, Ωm = 2π · 1 MHz, κ = 2π · 800 MHz and ∆ = 2π · 400 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . 25 xvi List of Figures 4.1 Applied modulation voltage at resonance for a number of wavelengths for two FFPCs consisting of a fibre mirror from LAY- ERTEC and two different dielectric DBR chips. The different markers in the plots represent different series of measurements. The change in the vertical bias of the voltage profiles between different series of measurements indicates a creep in the mechanics of the setup. . . . . 28 4.2 Examples of reflection dips fitted to Equation 3.5. The blue graphs indicate the reflected power as recorded by the photodetector whilst the orange dashed lines are the fit of the reflection data to Equation 3.5. The reflection is normalised to the maximum reflection according to the fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 Measured finesse for a FFPC consisting of a fibre mirror from LAYERTEC and a macroscopic mirror with the same reflective coating, alternating layers of SiO2 and Ta2O5 of varying thicknesses. . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.4 Reflectivity of the fibre mirror before etching, taken from the specification sheet from LAYERTEC. . . . . . . . . . . . . . . . . . . 31 4.5 Measured finesse for two FFPCs consisting of a fibre mir- ror from LAYERTEC and dielectric DBRs. The two dielectric DBRs making up the macroscopic mirror in either FFPC are chips made from a wafer with varying layers of 111 nm silicon and 270 nm silicon dioxide. The different markers indicate different series of mea- surement. c) illustrates the structure of the two FFPCs and indicates where on the DBR wafer the two chips where taken. . . . . . . . . . . 32 4.6 The measured finesse for a FFPC consisting of a fibre mirror from LAYERTEC and a macroscopic mirror, the structure of which is shown in the see inset. . . . . . . . . . . . . . . . . . 34 4.7 The measured finesse for a FFPC consisting of a fibre mirror from LAYERTEC and a macroscopic mirror, the structure of which is shown in the see inset. . . . . . . . . . . . . . . . . . 34 4.8 The measured finesse for a FFPC consisting of a fibre mirror from LAYERTEC and a macroscopic mirror, the structure of which is shown in the see inset. . . . . . . . . . . . . . . . . . 35 4.9 Simulated reflectivities for a Gaussian beam, with waist ra- dius 15µm, perpendicular to the photonic crystal structure of the membrane in Figure 4.6 . . . . . . . . . . . . . . . . . . . 36 4.10 a) shows the simulated reflectivity for a Gaussian beam, with waist radius 15µm, perpendicular to the ideal version of the InGaP on DBR structure in Figure 4.7. b) shows 2π 1−R where R is the reflection coefficient in a). . . . . . . . . . . . . . . . . . 36 4.11 A series of reflection dips showcasing birefringence in the cavity mirrors. The x-axis is time and the y-axis reflected power, each with arbitrary units. The labels indicate the laser frequency for the corresponding measurement. . . . . . . . . . . . . . . . . . . . . . 38 4.12 Noise power spectral density of the error signal for a SoF lock of a FFPC for two different noise frequency ranges. . . . . . . 40 xvii List of Figures A.1 Where on the chip with the photonic crystal mechanical res- onators that the measurements were taken, marked by the red circle. a) corresponds to the measurement in Figure 4.6. b) corresponds to the measurement in Figure 4.7. c) corresponds to the measurement in Figure 4.8. . . . . . . . . . . . . . . . . . . . . . . . . I xviii 1 Introduction In cavity optomechanics, one studies the interaction between the light in an optical cavity and mechanical motion. Optical cavities coupled to mechanical systems have many potential applications such as quantum-enhanced inertial and force sensing [1]. They can also be used to create non-classical states of light and mechanics as well as entangled states of light [2]. Furthermore, cavity optomechanics has been proposed as a key technology to establish long distance links in quantum information networks, for example by interconverting information stored in microwave domain solid-state qubits with information stored in flying photonic qubits [3]. One type of system used to achieve an optomechanical interaction is fibre Fabry–Pérot cavities (FFPCs). FFPCs are miniaturised Fabry–Pérot cavities (FPCs) integrated on optical fibres. They offer many advantages over classic FPCs such as intrinsic integration with optical fibres. In addition, their small size allows for reduction of the cavity length, which enhances the optomechanical coupling strength [3, 4]. FFPCs also have advantages over other platforms of similar or smaller size, such as whispering gallery mode resonators or optomechanical crystals. FFPCs can have larger intra cavity photon numbers without facing issues due to heating or thermal instability [5]. Additionally, the open mode volume of FFPCs allows for the easy insertion of other systems, such as membranes or levitated nano particles, to act as the mechanical element. 1.1 Goal and objectives This masters thesis tackled the experimental realisation of a FFPC-based optome- chanical testing platform in the ”Quantum sensing and foundations lab” at Chalmers. The goal of this testing platform was to enable optical characterisation of, and op- tomechanical experiments with, FFPCs consisting of a fibre mirror and a reflective structure on a chip, see Figure 1.1. In the optomechanical experiments, these re- flective structures take the form of mechanical resonators. The testing platform would enable manipulation and high precision read-out of their vibrational modes through optomechanical interaction with light in the cavity. To achieve this, an existing experimental setup for the optical characterisation of FFPCs was improved upon by automating the measurements using Python. This was then used to char- acterise some FFPCs by measuring the signal reflected from them and extracting their finesse. 1 1. Introduction In addition to this, an experimental setup was constructed to enable optical readout and control of on-chip mechanical resonators. This was integrated into a vacuum chamber so that the mechanical damping rate is not limited by the surrounding air. With this, initial test measurements where done to determine the amount of measurement noise in the setup and find ways to reduce it. Chapter 2 of this report covers the theory which is necessary to understand FFPCs and the optomechanical experiments proposed in this thesis. Chapter 3 then explains the two experimental setups and the experiments they can be used for. Finally, Chapter 4 presents and discusses results of the experiments conducted with these setups. Figure 1.1: A conceptual sketch of a fibre Fabry-Pérot cavity that is explored in this thesis. It consists of a fibre mirror and a reflective structure on a chip. 2 2 Theory This chapter introduces Fabry-Pérot cavities and Fibre Fabry-Pérot cavities, giv- ing a basic theoretical framework and introducing relevant properties. Then, the basic concepts of cavity optomechanics will be introduced and some important con- sequences of optomechanical coupling in Fabry-Pérot cavities will be highlighted. 2.1 Fabry-Pérot cavities A Fabry-Pérot cavity (FPC) is an optical cavity consisting of two highly reflective mirrors facing each other. Between the mirrors, light propagates in free space. The following introduction to the basic theory of FPCs and their properties will be based on [3]. A FPC contains a series of resonances with angular frequencies ωcav, N = 2πνcav, N ≈ Nπc Lcav ⇔ Nλcav,N ≈ 2Lcav (2.1) where N is an integer that is called the mode number, Lcav is the distance between the mirrors and c is the speed of light in the medium between the mirrors. These resonances, or longitudinal modes, are separated in angular frequency by the free spectral range ωFSR = πc Lcav = 2πνFSR. (2.2) An electromagnetic wave can only couple into a FPC if its frequency is close to some ωcav,N. To understand this, imagine an electromagnetic wave with another frequency being reflected back and forth between the mirrors. The waves travelling back and forth would cancel each other out due to destructive interference. An electromagnetic wave with angular frequency ω ≈ ωcav,N that couples into a FPC will be reflected back and fourth between the mirrors and constructively interfering. this results in the field between the mirrors being stronger than the field coupling into the cavity. The enhancement of the electromagnetic power relative to the power coupled into the cavity is 1 2π times the finesse of the cavity F = ωFSR κ (2.3) where κ is the cavity photon decay rate. κ is a measure of how quickly a photon leaves the cavity. This can happen via transmission through or absorption by the 3 2. Theory Figure 2.1: A conceptual sketch of Fabry-Pérot cavity with the cavity length Lcav. The fields in the input-output description of a FPC are illustrated. They are â, the complex field amplitude of a mode inside the cavity, âout, the rate of light leaving the cavity from Mirror 1 and âin and f̂in which are the rates with which light arrives at Mirror 1 and Mirror 2 respectively. Also illustrated are the photon cavity decay rates, κex, which describes the rate at which cavity photons leave the cavity out through Mirror 1, and κin, the decay rate associated with other channels. mirrors and scattering out through the sides of the cavity. Note that κ can be defined differently in different texts, sometimes κ is defined such that F = ∆νFSR κ . Throughout this report κ is defined such that Equation 2.3 holds true. The optical quality factor of the cavity Qopt = ωcavτ, (2.4) where ωcav is the angular frequency of the particular mode in question and τ = 1 κ is the average lifetime of a cavity photon, measures how well the cavity confines energy. As the average time before a photon leaves the cavity is τ = 1 κ = FLcav πc (2.5) and the time it takes for a photon to travel once back and forth in the cavity is τR = 2Lcav c , (2.6) the average number of trips back and forth that a photon does before leaving the cavity is τ τR = F 2π (2.7) which is the enhancement of electromagnetic power for the intra-cavity field. One often needs to distinguish between energy leaving the cavity in ways necessary for measurements and energy leaving the cavity through other, often unwanted, channels. In this project, single-sided cavities are used, that is to say cavities where light is coupled into the cavity through Mirror 1, see Figure 2.1, and transmission out through Mirror 2 is not measured. The photon cavity decay rate due to transmission through the incoupling Mirror 1 is monitored and will be referred to as κex. The rest of κ, including scattering out through the sides of the cavity, absorption by both mirrors and transmission through Mirror 2 will here be referred to as κin = κ− κex. 4 2. Theory 2.1.1 Input-output theory for FPCs Light in an optical cavity can be described like a driven harmonic oscillator by input- output theory. Consider the situation where a field from a coherent, monochromatic free space laser with angular frequency ω couples into a one-sided cavity. Input- output theory describes the time evolution of the annihilation operator for the intra- cavity field, â, by the Heisenberg equation of motion ˙̂a = −κ 2 â+ i∆â+ √ κexâin + √ κinf̂in (2.8) in a frame of reference which rotates with the laser frequency such that âlab = e−iωtâ. Here ∆ = ω − ωcav is the laser detuning with respect to the cavity mode, âlab is the annihilation operator in the reference system of the lab and â is the same annihilation operator in the reference system that is assumed in Equation 2.8 which rotates relative to the lab. ↠is the creation operator for the intra-cavity field and the two operators have the commutation relation [â, â†] = 1. (2.9) âin is a stochastic quantum field incident on Mirror 1, consisting of the laser field and the vacuum electric field coupling into the cavity through Mirror 1. âin is defined such that ⟨↠inâin⟩ is the rate with which photons arrive at Mirror 1 meaning that the input power at Mirror 1 is Pin = ℏω⟨↠inâin⟩, (2.10) Correspondingly, f̂in refers to the field incident on any other incoupling channel such as Mirror 2 and the open sides of the cavity. The field reflected from the FPC is given by âout = âin − √ κexâ (2.11) defined such that Pout = ℏω⟨↠outâout⟩ (2.12) is the power of the field going back from Mirror 1. See Figure 2.1 for a conceptual sketch of these fields. In this project, it is sufficient to go to the classical case by replacing â with its time average ⟨â⟩ and similarly letting âout 7→ ⟨âout⟩, âin 7→ ⟨âin⟩ and f̂in 7→ ⟨f̂in⟩ = 0. In the classical case ⟨â⟩ = √ κex⟨âin⟩ κ/2 − i∆ (2.13) is the steady state solution to Equation 2.8. The average number of photons circu- lating in the cavity is then given by n̄cav = |⟨â⟩|2 = κex|⟨âin⟩|2 ∆2 + (κ/2)2 = κexPin [∆2 + (κ/2)2] ℏω (2.14) and the reflection amplitude at the cavity is given by F (ω) = ⟨âout⟩ ⟨âin⟩ = ( ⟨âin⟩ − κex⟨âin⟩ κ/2 − i∆ ) /⟨âin⟩ = (κin − κex)/2 − i∆ (κin + κex)/2 − i∆ . (2.15) 5 2. Theory For macroscopic FPCs that are coupling to free space light, the power reflected back from the cavity is then given by Pout(ω) = ℏω|⟨âout⟩|2 = |F (ω)|2Pin (2.16) if the incident field is mode matched to the cavity field. When ∆ ≪ ωFSR, |F (ω)|2 = Pout(ω) Pin = 1 − ηdip 1 1 + ν2 (2.17) where ν = ∆/κ = (ω − ωcav)/κ [6]. ηdip depends on the fraction κex κin and reaches its maximum for critical coupling, κex = κin. ηdip then decreases symmetrically for equally overcoupled (κex > κin) and undercoupled (κex < κin) cavities. ηdip further depends on the spatial mode matching efficiency at Mirror 1 which can be maximised by adjusting the relative alignment of the mirrors, maximising the resonance depth for a set value of κex κin , for more details see [6]. 2.1.2 Fibre Fabry-Pérot cavities Fibre Fabry–Pérot cavities (FFPCs) are miniaturised Fabry–Pérot cavities where one or both mirrors are on the facet of an optical fibre. The light coupling into the cavity is then not free space light, but fibre-guided light. They offer many advantages over classic FPCs such as high local field strength, small size, intrinsic integration with optical fibres and an open mode volume. They do however present some challenges compared to macroscopic FPCs. One such issue is that mechanical deformation of the fibre can cause noise when measuring the reflection from a FFPC. Another issue is that when measuring reflected power from a fibre end of a FFPC, |F (ω)|2 cannot be described by Equation 2.17. The reflection at FFPCs and its deviation from the classic FPCs described above is explored in [6]. The reason for this difference is that in order for the light reflected at a FFPC to reach a photodetector it needs to couple back into a guided mode of the fibre. Thus, the transverse mode shapes of the light in the cavity and fibre need to be considered. The problem is analogous to a macroscopic free space FPC with a mode-filter between the cavity and the photodetector. The light from the cavity that does not couple into the guided mode of the fibre is lost to the fibre cladding and not recorded by the photodetector. As a result, the reflection from the FFPC becomes a sum of a Lorentzian function, similar to the macroscopic case, and a dispersive function which amounts to |F (ν)|2 = Pout(ν) Pin = ηr − ηL ( 1 1 + ν2 − A ν 1 + ν2 ) (2.18) see Figure 2.2. ηr, ηL and A depend on the transverse spatial overlap amplitudes between the mode in the cavity and the guided mode of the fibre as well as between the mode directly reflected at the incoupling mirror and the guided mode of the fibre. For a more in-depth description of ηr, ηL and A consider the forward and backward propagating spatial modes in the fibre, |ψ+ f ⟩ and |ψ− f ⟩, and in the cavity, |ψ+ cav⟩ and 6 2. Theory Figure 2.2: Sketch of the expected reflection signal from a FFPC based on Equation 2.18. The total reflected power (red line) is the sum of a Lorentzian part (dotted green line) and a dispersive part (dash-dotted blue line). Figure 2.3: A conceptual sketch of a fibre Fabry-Pérot cavity illustrating the relevant spatial modes of light propagating in the fibre and the cavity. 7 2. Theory |ψ− cav⟩, together with the spatial mode of the reflection of |ψ+ f ⟩ at Mirror 1, |ψr⟩, see Figure 2.3. Following [6] one can define ζ = ⟨ψ+ cav |ψ+ f ⟩ (2.19) and ξ = ⟨ψ− f |ψr⟩ (2.20) and then notice that ⟨ψ− cav |ψr⟩ = ⟨ψ+ cav|R†R|ψ+ f ⟩ = ⟨ψ+ cav |ψ+ f ⟩ = ζ, (2.21) where R is the operator for reflection at mirror 1, and ⟨ψ− cav |ψ− f ⟩ = (⟨ψ− f |ψr⟩)∗ = ζ∗. (2.22) It is then possible to show that ηr = |ξ|2, (2.23) ηL = 4T1 T1 + T2 + L1 + L2 ( Re[ξ(ζ2)∗] − |ζ|4 T1 T1 + T2 + L1 + L2 ) (2.24) and A = Im[ξ(ζ2)∗] Re[ξ(ζ2)∗] − |ζ|4 T1 T1+T2+L1+L2 (2.25) where Ti is the transmission coefficient for Mirror i and Li is the losses for Mirror i due to absorption and scattering. Note that Equation 2.18 is not symmetric around ν = 0 and that |F (ν)|2 does not reach its minimum for ν = 0. Furthermore, the resonance depth for a FFPC (the depth of the reflection dip) is not maximised by maximising the mode matching efficiency between the cavity mode and the fibre guided mode, |ζ|2. The resonance depth can still be used as an approximation when aligning the cavity mirrors. For exact mode matching efficiencies, one would need to fit the reflection signal to Equation 2.18 and extract ζ and ξ from equations 2.23, 2.24 and 2.25. 2.2 Cavity optomechanics Optomechanics is the study of how light interacts with mechanical motion through radiation pressure. This section will start with a brief background on mechanical resonators before explaining the optomechanical interaction in optical cavities based on derivations in [3]. 2.2.1 Mechanical resonators The theoretical description of vibrational movement is based on the linear theory of elasticity which describes vibrational movement in an object as a set of normal modes with corresponding frequencies [3]. The following description will focus on one normal mode with frequency Ωm. The displacement of any part of the object is given by the displacement field u⃗(r⃗, t) = x(t) · u⃗(r⃗) where the time-evolution of 8 2. Theory the global amplitude of motion, x(t), can be described as a harmonic oscillator with effective mass meff such that meff dx2(t) dt2 +meffΓm dx(t) dt +meffΩ2 mx(t) = Fex(t). (2.26) Γm is called the mechanical damping rate and can be expressed in terms of the mechanical quality factor Qm = Ωm/Γm (2.27) and Fex(t) is the sum of all external forces acting on the mechanical oscillator. The energy of the mechanical oscillator is described by the Hamiltonian Ĥ = ℏΩmb̂ †b̂+ 1 2ℏΩm (2.28) where b̂† and b̂ are the phonon creation and annihilation operators such that x̂ = √ ℏ 2meffΩm (b̂+ b̂†) = xZPF(b̂+ b̂†) (2.29) and [b̂, b̂†] = 1. (2.30) xZPF is the zero-point fluctuation amplitude which describes the variance in position for the mechanical oscillator in the mechanical ground state, meaning that ⟨0|x̂2|0⟩ = x2 ZPF (2.31) where |0⟩ is the mechanical ground state. 2.2.2 Optomechanical coupling Mechanical and optical modes can affect each other through various radiation- pressure forces [3]. This project concerns FFPCs where one of the cavity mirrors is on the facet of an optical fibre whilst the other takes the form of a reflective mem- brane on a chip which acts as a mechanical resonator. In such a system, the primary optomechanical interaction is the direct momentum transfer between photons and the mechanical resonator due to reflection. For such a system it is sufficient to first consider the Hamiltonian Ĥoptomech. = ℏωcav(x)â†â+ ℏΩmb̂ †b̂ (2.32) (here the zero-point energies are excluded since they do not affect the optomechanical coupling). The displacement of the vibrating mirror changes the length of the optical cavity, Lcav, and thus also the cavity frequency such that ωcav(x) = ωcav + x ∂ωcav ∂x + x2∂ 2ωcav ∂x2 + ..... (2.33) where it is sufficient to keep only the linear term. Defining G = −∂ωcav ∂x as the optical frequency shift per displacement, which for our simple case is G = ωcav Lcav , the 9 2. Theory Hamiltonian for the system containing a mechanical and optical resonator can be written as Ĥoptomech. = ℏ(ωcav −Gx̂)â†â+ ℏΩmb̂ †b̂. (2.34) In accordance with Equation 2.29 this can be written as Ĥoptomech. = ℏ(ωcav −GxZPF(b̂+ b̂†))â†â+ ℏΩmb̂ †b̂. (2.35) The interaction part of the Hamiltonian is then Ĥint = −ℏg0â †â(b̂+ b̂†) (2.36) where g0 = GxZPF (2.37) is called the vacuum optomechanical coupling strength or the single-photon optome- chanical coupling strength. Only considering Equation 2.36 the equations of motion for â and b̂ would be the coupled differential equations ˙̂a = i ℏ [Ĥint, â] = ig0(b̂+ b̂†)(̂a) = iGx̂â (2.38) and ˙̂ b = i ℏ [Ĥint, b̂] = ig0â †â. (2.39) In practice, both the cavity photons and the phonons of the mechanical resonator interact with more than just each other. The Hamiltonian describing the full system includes terms describing how energy is added into the cavity through a laser drive. It also includes phenomena such as photon cavity decay and mechanical damping as a result of the optical and mechanical resonators coupling to the rest of their environment. The effects on the optical mode from the full system is treated with input-output theory. In a reference system which, as described in Section 2.1.1, rotates with the laser frequency ω, input-output theory yields the full equation of motion for the optical field amplitude ˙̂a = −κ 2 â+ i(∆ +Gx̂)â+ √ κexâin + √ κinf̂in (2.40) by adding Equation 2.38 to Equation 2.8. b̂, like â, behaves as a harmonic oscillator and the equation of motion for b̂ without optomechanical coupling is analogous to Equation 2.8, replacing κ with Γm and ∆ with −Ωm. The full equation of motion for b̂, including optomechanical coupling, then becomes ˙̂ b = ( −iΩm − Γm 2 ) b̂+ ig0â †â+ √ Γmb̂in (2.41) as long as Ωm ≫ Γm, which can be assumed throughout this project. Both equa- tions of motion are non-linear, with Equation 2.40 containing x̂â and Equation 2.41 containing â†â. This makes cavity optomechanics a possible platform to study non- linear quantum effects. However, this requires large values for g0 compared to κ. FFPCs is a possible platform for achieving higher values for g0 because in a FPC g0 = GxZPF = ωcavxZPF Lcav = ωcav Lcav √ ℏ 2meffΩm (2.42) 10 2. Theory and the small spatial dimensions of FFPCs make it possible to achieve small values for Lcav. Furthermore, as the cavity beam focus of a FFPC is small, it becomes possible to use small mechanical resonators with small meff. As of now, for most applications, it is sufficient to go to a linearised approximation of the interaction Hamiltonian Ĥ (lin) int = −ℏg0 √ n̄cav(δ↠+ δâ)(b̂+ b̂†) (2.43) where δâ = â − ⟨â⟩ = â − ᾱ and n̄cav = ᾱ2 is the average number of photons circulating in the cavity. Applying input-output theory to this linearised Hamiltonian (or approximating Equation 2.40 around a steady state ᾱ by setting â = ᾱ + δâ and only keeping linear terms) yields the linear coupled equations of motion δ ˙̂a = ( i∆ − κ 2 ) δâ+ ig(b̂+ b̂†) + √ κexδâin(t) + √ κinf̂in(t) (2.44) and ˙̂ b = ( −iΩm − Γm 2 ) b̂+ ig(δâ+ δâ†) + √ Γmb̂in (2.45) where g = g0 √ n̄cav is the optomechanical coupling strength in the linearised regime. These linearised equations of motion are sufficient to explain the optomechanical effects that can be observed in the experiments constructed in this thesis. 2.2.2.1 The optical spring effect and optomechanical damping Solving equations 2.44 and 2.45 in frequency space, and comparing the solution for the optomechanical coupled mechanical oscillator with equations expressing the behaviour of uncoupled mechanical oscillators, shows how the mechanical oscillator responds to the optomechanical coupling. The full derivation can be found in [3] and shows that for ∆ ̸= 0, the laser drive shifts the frequency of the mechanical oscillator with δΩm(Ωnew) = g2 Ωm Ωnew [ ∆ + Ωnew (∆ + Ωnew)2 + κ2/4 + ∆ − Ωnew (∆ − Ωnew)2 + κ2/4 ] (2.46) and adds a term Γopt(Ωnew) = g2 Ωm Ωnew [ κ (∆ + Ωnew)2 + κ2/4 − κ (∆ − Ωnew)2 + κ2/4 ] (2.47) to the damping of the mechanical oscillator so that the total damping becomes Γm,total = Γm,intrinsic + Γopt(ω). (2.48) Here Ωm is the original unperturbed mechanical frequency while Ωnew is the current mechanical frequency. In the, for this thesis, experimentally relevant limit of κ ≫ Ωm (the so called unresolved-sideband regime) it is sufficient to approximate this around Ωnew = Ωm and these equations simplify to δΩm|κ≫Ωm = g2 2∆ κ2/4 + ∆2 = g2 0n̄cav 2∆ κ2/4 + ∆2 (2.49) 11 2. Theory (a) (b) Figure 2.4: Plot of δΩm (a) and Γopt (b) as a function of laser detuning ∆ when κ ≫ Ωm for g = 2π · 100 kHz, Ωm = 2π · 1 MHz and κ = 2π · 800 MHz. and Γopt(ω)|κ≫Ωm = g2 [ κ (∆ + Ωm)2 + κ2/4 − κ (∆ − Ωm)2 + κ2/4 ] . (2.50) See Figure 2.4 for some values for δΩm|κ≫Ωm and Γopt(ω)|κ≫Ωm as functions of detun- ing ∆. This shift of the mechanical frequency is called the optical spring effect and the shift in the damping is referred to as optomechanical damping (for Γopt(ω) > 0) or antidamping (for Γopt(ω) < 0). 12 3 Methods The optomechanical testing platform developed in this master’s thesis consists of two parts. Part one is an experimental setup which enables optical characterisation of FFPCs measuring their reflection and free spectral range. Part two is an ex- perimental setup in high vacuum which enables optomechanical experiments using FFPCs and on-chip mechanical resonators. The FFPCs consists of a single mode optical fibre with a concave mirror on the facet and a chip with a reflective coating or reflective structures that are free to vibrate. The details of the two experimental setups are described in this chapter. 3.1 Optical characterisation of FFPCs For optical characterisation under ambient conditions, an experimental setup and a Python-based control code was developed to measure finesse and coupling depth of FFPCs. 3.1.1 Experimental apparatus An overview of the experimental setup is shown in Figure 3.1. First a laser tone is generated by a continuously tunable single-frequency diode laser. Three lasers1 are available in the lab and are used depending on the desired wavelength range, together covering the range 1420-1620 nm. The laser light is coupled into fibre patch cord which passes through a paddle wheel polarisation controller to a phase modulating electro-optic modulator (EOM). The EOM is polarisation sensitive and the optimal polarisation depends on the wavelength of the light. The paddle wheel polarisation controller allows for the optimization of the polarisation going into the EOM. The EOM2 is controlled by a signal generator3 that adds a time dependent phase, φ = β sin Ωt, to the light going into the EOM. Thus, the signal exiting the EOM, which is the signal incident on the FFPC, becomes Einc = E0e iωt+iβ sin Ωt. (3.1) 1CTL 1470, CTL 1500 and CTL 1550 from TOPTICA photonics controlled by digital laser controllers DLC pro from TOPTICA photonics 2MPZ-LN-10 from Exail 3SynthHD Mini 13 3. Methods Figure 3.1: Overview of the setup for optical characterisation of FFPCs consisting of a continuously tunable laser source, that via single-mode optical fibre sends a signal through a paddle wheel polarisation controller to an electro-optic modulator, a 10:90 fibre-optic splitter, a photodetector and through another paddle wheel polarisation controller to the FFPC to be characterised. The fibre-end of the FFPC is mounted on a shear plate actuator. This enables continuous modulation of the cavity length by a piezo controller which in turn gets its signal from an oscilloscope. The other end of the FFPC consists of a chip that can be tilted and moved in any direction with micrometer precision, see Figure 3.2. The oscilloscope is also used to read out the signal from the photodetector. 14 3. Methods According to [7], this expression can be expanded to first order using Bessel functions to Einc ≈ E0 [ J0(β)eiωt + J1(β)ei(ω+Ω)t − J1(β)ei(ω−Ω)t ] . (3.2) Consequently, the EOM adds two additional tones, called sidebands, to the light guided in the fibre, detuned by ±Ω to the main tone. The sidebands and the main tone, is then led into Port 1 of a 10:90 fibre-optic splitter4 with four ports, see Figure 3.1. 10% of the light exits the fibre-optic splitter at Port 3 and goes through another paddle wheel polarisation controller whilst the remaining 90% is lost to a beam dump at Port 2. This signal loss is compensated for by increasing the laser power. After the polarisation controller, the light is led into a copper coated single-mode optical fibre5 with a convex mirror on its facet constituting one end of the FFPC. Port 4 of the fibre-optic splitter is coupled to a photodetector6 which measures the signal reflected from the fibre-end of the FFPC. The reflected power is displayed on an oscilloscope7 which can be interfaced with the computer. The fibre-end with the mirror is mounted on a shear plate actuator8, using a v- groove fibre holder9, see Figure 3.2. The shear plate actuator enables continuous modulation of the cavity length by a piezo controller10. The output voltage of the piezo controller is modulated by a signal generated by the oscilloscope’s wave generator output. The other end of the FFPC consists of a reflective structure on a chip that is mounted on a tiltable mirror mount that is in turn mounted on a micrometer translation stage11. The translation stage enables measurements on several parts of the chip. The tiltable mirror mount makes it possible to optimise the tilt of the chip for cavity alignment indicated by an optimal coupling of the optical mode in the fibre with the main optical mode of the cavity. A USB microscope12 is used to get a visual image when adjusting the placement and tilt of the chip. 3.1.2 Data collection and analysis The laser controller, radio frequency signal generator and oscilloscope are connected to a computer, enabling collection and analysis of data. The oscilloscope, through the piezo controller, applies voltage on the shear plate actuator as a sinusoidal function of time. The shear plate actuator subsequently puts the fibre end of the FFPC into a periodic forwards and backwards motion making the cavity length follow a periodic function in time. The amplitude of this curve is greater than half of the wavelength of the laser in air. This ensures that the cavity length at some point during the oscillation is such that the cavity is resonant with the laser. Around resonance, the reflection signal, measured with the photodetector, will have three dips as illustrated in Figure 3.3. The centre dip corresponds to the resonance of the main tone. On each side of this main dip there will be one additional dip that corresponds to the resonance for each of the sidebands created with the EOM. If 4FOBC-2-15-10-L-1-S-2 from AFW technolo- gies 5Cu1300 from IVG 6PDA015C InGaAs amplified detector from Thorlabs 7EDUX1002G from Keysight 8NFL5DP20 from Thorlabs 9HFV001 from Thorlabs 10MDT693B from Thorlabs 11PT3 from Thorlabs 12Dino-lite digital microscope 15 3. Methods Figure 3.2: Sketch of the structure holding up and aligning the FFPCs during the optical characterisation. The fibre-end of the FFPC is mounted on a shear plate actuator, enabling continuous modulation of the cavity length. The other end of the FFPC consists of a chip that is mounted on a tiltable mirror mount that is in turn mounted on a micrometer translation stage. Figure 3.3: Sketch of the expected shape of the reflection signal around resonance. The signal features three dips, one from the main signal and two from the sidebands created with the EOM. 16 3. Methods these dips occur when the movement of the fibre end can be approximated as linear then the time-axis is proportional to change in cavity length. This length can be expressed as Lcav = Lo + ∆L where Lo is the cavity length such that the cavity is resonant with the main tone of the laser. ∆L corresponds to a change in the resonance frequency of the cavity, δν = νlaser − νcav, in accordance to Lo = Nc 2νcav ⇒ Lo + ∆L = Nc 2(νlaser + δν) . (3.3) Close to resonance this can be expanded to Lo + ∆L ≈ Nc 2 ( 1 νlaser + 1 ν2 laser δν ) ⇒ δν ≈ 2ν2 laser Nc (Lo + ∆L) − νlaser (3.4) where N is the order of the cavity and c is the speed of light in the cavity medium. This is a linear function of ∆L which in turn is approximately a linear function of time. Therefore, as the frequency differences between the main signal and the sidebands are known, the sidebands facilitate the rescaling of the x-axis to the frequency domain. According to [6] the shape of one of these reflection dips can be described by Equation 2.18. The total reflective line shape can thus be described by Pout(ν) Pin = ηr −Dmain −DSB1 −DSB2, where Dmain = ηL ( 1 1 + ν2 − A ν 1 + ν2 ) , DSB1 = SSBηL ( 1 1 + ν2 SB1 − A νSB1 1 + ν2 SB1 ) and DSB2 = SSBηL ( 1 1 + ν2 SB2 − A νSB2 1 + ν2 SB2 ) . (3.5) Dmain corresponds to the dip in the reflected power that the resonance of the main tone gives rise to. DSB1 and DSB2 corresponds to the resonances with the sidebands. Therefore νSB1 = 2π(νlaser1 − νcav)/κ and νSB2 = 2π(νlaser2 − νcav)/κ (3.6) where νlaser1 and νlaser2 are the frequencies of the sidebands. SSB is a scale factor between the main dip and the sideband dips owing to the fact that the main tone and the sidebands do not make up equal fractions of the total power transmitted in the fibre. The reflection data is fitted to Equation 3.5 using SciPy [8]. How the cavity length is regulated can cause distortions of the measured reflection signal. The data analysis described here assumes that Lcav changes as a linear function of time. In this project, Lcav was varied using a sinusoidal control signal for the shear 17 3. Methods plate actuator and only measuring the finesse on reflection dips that occur when the voltage applied on the shear plate is within the middle 62.5 % of the voltage range in an attempt to stay within the relatively linear part of the sinus curve. The visibility of the sideband dips can be optimised by changing the frequency of the sidebands with the EOM and changing the polarisation going into the EOM with the first paddle wheel polarisation controller. The second paddle wheel po- larisation controller makes it possible to test if a frequency splitting of differently polarised cavity modes occurs for the reflection from the FFPC, for example due to a polarisation dependent reflection from the chip-surface. The changing of wavelength with the laser controller as well as the process of retriev- ing data from the oscilloscope was automated using the application programming interface VISA (virtual instrument software architecture) via the Python libraries toptica-lasersdk [9] and PyVISA [10]. This makes it possible to quickly cycle through and measure κ for several wavelengths. If the reflective surface of the chip can be approximated as a plane mirror, this can further be used to identify the wavelength dependent reflection properties of the chip. The free spectral range, νFSR, of the FFPC is determined by recording the voltage applied to the shear plate actuator at resonance, ”the resonance voltage”, while changing the frequency of the main laser tone. As the cavity length is a function of this voltage, νFSR is the smallest non-zero difference in frequency between two laser signals with the same resonance voltage. κ and νFSR is then used to calculate the finesse, F = 2πνFSR κ , at every wavelength step. 3.2 Optomechanical measurements In order to perform optomechanical experiments with an FFPC, the cavity, the gray square in Figure 3.1, needs to be in high vacuum (HV). This is so that the quality factor of the mechanical resonator, as described in Section 2.2.1, is not limited by the damping from the surrounding air. To accomplish this, a FFPC setup was integrated into a HV chamber. In order to measure and control the frequency of the mechanical resonator as well as measure the optomechanical coupling strength, Pound-Drever-Hall (PDH) and side-of-fringe (SoF) locking were implemented. This section covers these locking techniques, how they were implemented in this project and how they can be used in optomechanical experiments. 3.2.1 Experimental apparatus In order for the FFPC to be in vacuum, the shear plate actuator, tiltable mirror mount and translation stage is exchanged for vacuum compatible and remote con- trolled alternatives as shown in Figure 3.4. The chip is mounted onto a vacuum compatible, tiltable mirror mount13. The mirror mount is in turn mounted on two nanopositioners14 that enable movement of the chip in the plane parallel to its sur- 13AG-M100 from Newport 14ECSx5050/StSt/NUM+/UHV from at- tocube 18 3. Methods Figure 3.4: Sketch of a cross section of the vacuum chamber, showing the structure holding and aligning the FFPCs during the optomechanical ex- periments. The chip with the mechanical resonator is mounted on a tiltable mirror mount which is in turn mounted on top of two nanopositioners enabling movement of the chip in two dimensions. An aluminium holder was made to mount the mir- ror mount to the nanopositioners. The aluminium holder is designed such that the weight is equally distributed on the nanopositioners to avoid torque that might dam- age them. The fibre is mounted on a third nanopositioner enabling alignment of the cavity in the third dimension. On top of this nanopositioner, a shear piezo element is mounted which is used for fine-positioning of the fibre and thereby locking of the cavity. The fibre is glued to the shear piezo using UV epoxy. The position of the end of the fibre is stabilised in two dimensions by having it go through a glass ferrule. A USB microscope is mounted outside the vacuum chamber by a viewport, providing a visual image of the fibre and the chip with the help of a silver mirror (not shown in this sketch) which is mounted inside of the vacuum chamber. Also not shown in this sketch are wires sending signals to the tiltable mirror mount, nanopositioners and shear piezo connected to the outside through ports on the side of the chamber as well as the fibre feedthrough on the top of the chamber. 19 3. Methods Figure 3.5: Overview of the setup for the optomechanical experiments consisting of the same optical components as in the setup for optical characterisation with some additional electrical components to enable the locking of the cavity length to the drive laser. These additional components are a multiplicative mixer, a low- pass filter, a resistor, a bias tee and a high-voltage amplifier and a RedPitaya. The three circuit breakers illustrate which path is used for PDH or SoF locking. The RedPitaya controls a shear piezo (here illustrated as a brown rectangle) used for fine positioning during PDH and SoF locking. A nanopositioner from Attocube (here illustrated as a dark green rectangle) controlled by an AMC100 is used to move the fibre along greater distances. face. This is all mounted on the bottom of a HV chamber. A mounting post is attached to the top of the chamber. Attached at the bottom end of this mounting post is a slip-stick piezo-based nanopositioner15 used to move the fibre end of the FFPC up and down. On top of the nanopositioning stage there is a holder with a shear piezo chip16 on which the fibre is glued using UV epoxy. This shear piezo is controlled by the output from a voltage amplifier which in turn is controlled by the RedPitaya (a type of single-board computer) and used for fine positioning when scanning for resonances and for locking the cavity. The end of the fibre is going through a glass ferrule to stabilise it in two dimensions. For extra stabilisation in the third dimension, two metal springs are added at the end of the structure holding the fibre. During measurements, these springs are in contact with the chip holder, suppressing vibrations along the cavity axis that would otherwise change the cavity length. A USB microscope is mounted outside the vacuum chamber to provide a visual image of the inside. Figure 3.5 shows an overview of the setup for the optomechanical experiments in- cluding the electrical components. The optical components are the same as for the 15ECSx3050/StSt/HL/NUM/HV controlled by an AMC100 from attocube 16PL5FBP3 from Thorlabs 20 3. Methods optical characterisation described above. To enable SoF locking the RedPitaya was added to function as a proportional–integral–derivative (PID) controller and spec- trum analyser. The RedPitaya is controlled using the open-source software PyRPL [11]. For the optomechanical experiments, the RedPitaya controls the oscillating piezo, here the shear piezo chip, instead of the oscilloscope and the piezo controller. Since the RedPitaya cannot put out higher voltage than 1 V a high-voltage ampli- fier17 is added between the RedPitaya and the piezo chip, amplifying the voltage by a factor of 100. Additionally, a 18 kΩ resistor is added after the high-voltage amplifier. This resistor, together with the capacitance of the shear piezo chip, acts as a RC low-pass filter with a cutoff frequency of about 5.5 kHz that filters out the high frequency electrical noise from the RedPitaya. This noise would otherwise dis- tort the reflection measurement by adding too much electrical noise to the control signal for the shear piezo chip. At the same time, the cut-off frequency for the filter cannot be too low as that would limit the lock bandwidth, making it unable to compensate for fast fluctuations in the cavity length. To determine the required resistor, a potentiometer, a type of variable resistor, was first used in its place. In the final setup however, the potentiometer is replaced with a normal resistor as the inductance of the potentiometer created a resonator circuit with the capacitance in the shear piezo chip, which distorted the reflection signal. To enable PDH locking, a multiplicative mixer18 is added, mixing the signal from the photodetector with the control signal for the EOM. To the same end, a low-pass filter19 is also added between the mixer and the RedPitaya. In addition to this, a bias tee20 is added between the photodetector and the mixer, splitting the low frequency component of the reflection signal from the high frequency components relevant for the PDH locking. This enables simultaneous visualization of the reflection dip on the oscillo- scope during PDH locking. Whether this mixer, low-pass filter and bias tee is used depends on which type of locking is to be used, this is illustrated with the three circuit breakers in Figure 3.5. It is noted in [6] that optical interference along the fibre between the EOM and the cavity, as well as between the cavity and the photodetector, can cause drifts in the cavity lock. An example of such optical interference is reflection at the connection between the FFPC fibre and the rest of the fibre. In [6], they tested different alternatives for connecting these two fibres to minimise said reflection and found that the best result was achieved by fusion splicing the single mode FFPC fibre with commercially available FC/APC connected single mode fibres. In this project the FFPC fibre was similarly connected to FC/APC connected single mode fibres. To further minimise the optical interference along the beam path, the fibre was taped down onto the optical table where possible. 17A600 from pendulum 18ZX05-42MH-S+ from Mini-Circuits 19EF5.. from Thorlabs 20ZFBT-4R2GW+ from Mini-Circuits 21 3. Methods 3.2.2 Pound-Drever-Hall locking for optomechanical char- acterisation The slope of the power of a signal reflected from a FFPC is approximately symmetric around resonance, see Figure 3.3. This means that looking only on the reflected power one cannot differentiate between the cavity being slightly too long and the cavity being slightly too short when trying to lock on resonance. Therefore, the reflected power cannot directly be used as an error signal when locking the cavity at resonance. This is solved by PDH locking. The PDH locking technique was originally designed to improve frequency stabilisation in lasers by locking the laser frequency to a stable Fabry-Pérot cavity but it can just as well be used to lock a Fabry-Pérot cavity to a stable laser signal, as will be shown through the following derivation based on [7]. As the expression in Equation 3.2 describes the light incident on the cavity, the light reflected at the fibre end of the cavity can be expressed as Eref = E0 [ F (ω)J0(β)eiωt + F (ω + Ω)J1(β)ei(ω+Ω)t − F (ω − Ω)J1(β)ei(ω−Ω)t ] (3.7) where F (ω) is the reflection coefficient at the cavity for light with angular frequency ω. The photodetector measures the reflected power Pref =|Eref|2 = Pmain|F (ω)|2 + PSB { |F (ω + Ω)|2 + |F (ω − Ω)|2 } + 2 √ PmainPSB{Re[F (ω)F ∗(ω + Ω) − F ∗(ω)F (ω − Ω)] cos Ωt + Im[F (ω)F ∗(ω + Ω) − F ∗(ω)F (ω − Ω)] sin Ωt} + (terms with frequency 2Ω) (3.8) where Pmain = J2 0 (β)|E0|2 (3.9) is the power of the incident main tone and PSB = J2 1 (β)|E0|2 (3.10) is the power of each of the incident sidebands. The signal from the photodetector is mixed with the control signal for the EOM, β sin Ωt+ ϕ with a multiplicative mixer. Here ϕ represents the phase shift between the two signals due to unequal delays in the two signal paths. ϕ can be adjusted by cable length, a phase shifter or the choice of Ω to be ϕ = 0 rad or ϕ = π/2 rad so that the signal from the photodetector mixes with a pure sin Ωt or a pure cos Ωt signal. Then, since sin Ωt sin Ωt = 1 2(1 − cos 2Ωt), (3.11) sin Ωt cos Ωt = 1 2 sin 2Ωt (3.12) and cos Ωt cos Ωt = 1 2(1 + cos 2Ωt), (3.13) the signal exiting the mixer consists of a DC term and AC terms with the frequencies Ω and 2Ω. The AC signal is filtered out with the low-pass filter and what is left is ϵ = √ PmainPSBIm[F (ω)F ∗(ω + Ω) − F ∗(ω)F (ω − Ω)] (3.14) 22 3. Methods if ϕ = 0 rad or ϵ = √ PmainPSBRe[F (ω)F ∗(ω + Ω) − F ∗(ω)F (ω − Ω)] (3.15) if ϕ = π/2 rad. Notice how the DC part of the reflected power, see in Equation 3.8, is filtered out with the mixer and low-pass filter before it reaches the RedPitaya. In other words, it does not affect the locking and might as well be used for something else. In this setup, the DC part of Pref is diverted to the oscilloscope using a bias tee and used to visualise the reflection signal while locking. In the regime where Ω ≪ κ, F (ω)F ∗(ω + Ω) − F ∗(ω)F (ω − Ω) ≈ d|F (ω)|2 dω Ω (3.16) which is purely real so ϕ = π/2 is desired. The cavities in this project have a linewidth, κ/2π, on the order of GHz whilst Ω/2π ≈ 300 MHz. Then ϵ ≈ √ PmainPSB d|F (ω)|2 dω Ω ≈ Pinβ 2 d|F (ω)|2 dω Ω (3.17) which would be linear around resonance and zero at the resonance frequency for a cavity whose reflection signal is symmetric around resonance. This makes it suitable as an error signal for a PID controller locking the cavity length of a FPC at resonance with a stable laser signal. Similar arguments can be made for when Ω ≫ κ. When the cavity is locked to the laser, the mechanical resonator, making up the other mirror in the FFPC, will cause a periodic change in the cavity length which will cause a periodic change in the error signal. The frequency of this periodic change, Ωm/2π ≈ 1 MHz, will be far outside the locking bandwidth of the PID controller. Thus, this periodic change will cause a peak in the noise spectral density of ϵ, Sϵ, at the frequency of the mechanical resonator, allowing one to measure it. However, FFPCs are not symmetric and d|F (ω)|2 dω ̸= 0 at resonance, see Section 2.1.2. So locking an FFPC with ϵ as error signal means that the cavity is blue-detuned and the mechanical resonator will be slightly spring hardened (see Equation 2.49) which affects the accuracy of the vibration measurements. To mitigate this, a bias, δϵ, is added to ϵ to form the error signal. This bias can be estimated as δϵ ≈ − √ PmainPSBΩ d|F (ω)|2 dω ∣∣∣∣∣ ω=ωcav = − √ PmainPSBΩ 1 |E0|2 dPref dω ∣∣∣∣∣ ω=ωcav (3.18) using Equation 3.17. Then using Equation 2.18 where ν = 2π(νlaser − νcav)/κ = (ω − ωcav)/κ (3.19) to express the reflected power around resonance results in 1 |E0|2 dPref dω ∣∣∣∣∣ ω=ωcav = ηL κ ( − 2ν (1 + ν2)2 − A 1 − ν2 (1 + ν2)2 )∣∣∣∣∣ ω=ωcav = −ηLA κ . (3.20) Thus δϵ ≈ PinβΩηLA 2κ . (3.21) 23 3. Methods ηL, A and κ one obtains from fitting the reflection signal to Equation 2.18 as de- scribed in Section 3.1. To get a more accurate value for ∆ϵ the fact that κ ≫ Ωm is utilised. In this regime, δΩm, caused by the optical spring effect, is linearly de- pendent on n̄cav, see Equation 2.49, and there by also on the power of the laser, see Equation 2.14. Thus, measuring the dependency that Ωm has on the laser power for different biases, δϵ, and plotting this in a graph yields straight lines that bisect when the laser power approaches zero. The value for Ωm where these lines bisect is the value for Ωm unaffected by the optical spring effect. The bias that results in this value independently of the laser power is the exact bias to add to the error signal to lock the cavity at resonance. When the cavity is locked, one can use the noise spectral density of ϵ for the locked cavity, Sϵ, to extract the optomechanical coupling strength, g0. According to [12, 13], the peak in the noise spectral density for the cavity resonance frequency, νcav, caused by the mechanical resonator can be expressed as Sνν(f) = 2g2 0ΩmΓmkBT π2ℏ [(Ω2 noise − Ω2 m)2 + Γ2 mΩ2 noise] (3.22) where f = Ωnoise/2π is the noise frequency, Γm/2π is the mechanical linewidth, kB is Boltzmann’s constant and T is the temperature of the mechanical resonator. Getting g0 is then a question of getting from Sϵ to Sνν and then fitting Sνν to Equation 3.22. The first step of getting from Sϵ to Sνν is to measure ϵ as a function of δν = νlaser − νcav. Measuring ϵ as a function of time whilst changing the cavity length, one can get ϵ as a function of δν the same way as in Section 3.1.2. Close to resonance, ϵ has an approximately linear dependency on δν. Therefore it is simple to extract dϵ dδν from the measurement after rescaling the time axis to the frequency domain. Then one can calculate Sνν = Sϵ ( dϵ dδν )−2 (3.23) and fit its peak around Ωnoise = Ωm to Equation 3.22 to get g0. 3.2.3 Side-of-fringe locking and the optical spring effect In SoF locking, the cavity is locked to one of the slopes in the reflection signal on either side of resonance. In the middle of such a slope, the reflected power has an approximately linear dependence on a change in cavity length. This means that, for SoF locking, the reflected power is in itself suitable as an error signal for a PID controller. As such, the signal from the photodetector is led directly to the RedPitaya without passing the mixer or low-pass filter. Locking the cavity to the side of resonance causes a frequency shift in the mechanical resonator as expressed by Equation 2.46, and a change in the mechanical damping rate, as expressed by Equation 2.50 for standard optomechanical setups21. Figure 3.6 shows δΩm and Γopt using equations 2.46 and 2.50 for g0 = 2π ·10 kHz, Ωm = 2π ·1 MHz, κ = 2π ·800 MHz 21Here, ”standard optomechanical setups” refers to FFPCs where one optical mode inter- acts with one mechanical mode. 24 3. Methods (a) (b) Figure 3.6: Expected values for δΩm/2π (a) and Γopt (b) as a function of laser power for g0 = 2π · 10 kHz, Ωm = 2π · 1 MHz, κ = 2π · 800 MHz and ∆ = 2π · 400 MHz. and ∆ = 2π · 400 MHz, values of the orders of magnitude expected in this project. These values are large enough that both the optical spring effect and optomechanical damping should be observable through this setup. As such, this setup not only enables measurements of mechanical modes but also control of them. 25 3. Methods 26 4 Results In this chapter, the results of the experiments conducted during this master’s thesis are presented and discussed. 4.1 Optical characterisation This section deals with the experimental results obtained with the setup for optical characterisation of FFPCs. This setup was used to analyse reflective properties of FFPCs. 4.1.1 Sweeping of the cavity length Sweeping the cavity length, by applying a sinusoidally varying voltage on the shear plate actuator, whilst simultaneously measuring the reflection from the FFPC, re- sults in a number of dips in the reflection signal. Recording the voltage applied when these reflection dips occur results in plots like those shown in Figure 4.1. These re- flection dips correspond to a resonance between a mode in the FFPC and the laser drive. As such, a data point in Figure 4.1 indicates the voltage necessary to modu- late the cavity length such that ωcav = ω for a laser frequency ω. The different types of markers for data points in the plots represent different series of measurements. As shown in Equation 2.1, the length of the cavity has an approximately linear dependence on the wavelength. Measurements of the voltage applied to the shear plate actuator, as exemplified by Figure 4.1, show a corresponding, almost linear, relationship between the wavelength and corresponding voltage at resonance. This indicates that there is an almost linear relationship between the voltage applied on the shear plate actuator and the cavity length. The figure shows several actuator voltages resulting in resonance for every wavelength step. The two lines of data points that are close to each other (<0.1 V) for the same measurement series are the result of the hysteresis of the shear plate actuator. This hysteretic behaviour makes it so that Lcav does not have the exact same dependency on the modulation voltage for the backwards and forwards movements [14]. As such, one of the corre- sponding reflection dips occurs as Lcav increases and one occurs as Lcav decreases. The existence of more than one pair of close lines for each series of measurement shows that the shear plate actuator modulates Lcav with more than half a wave- length. Therefore, the laser, during one sweep with the fibre mirror back and forth, becomes resonant with more than one longitudinal cavity mode, indicated by the 27 4. Results (a) (b) Figure 4.1: Applied modulation voltage at resonance for a number of wavelengths for two FFPCs consisting of a fibre mirror from LAYERTEC and two different dielectric DBR chips. The different markers in the plots represent different series of measurements. The change in the vertical bias of the voltage profiles between different series of measurements indicates a creep in the mechanics of the setup. mode number N in Equation 2.1. This means that, for a large enough depth of the reflection dip, one will always see a reflection dip whilst sweeping Lcav in this way. Furthermore, it is possible to approximate the free spectral range of the cavity by measuring the horizontal distance between these different groups of lines. During the different measurement series, indicated by the different marker types in Figure 4.1, the vertical bias of the voltage profiles varies despite the fact that the measurement setup was not changed in between these measurement series. This indicates a creep in the mechanics of the setup which changes the cavity length. This creep could for example be the result of thermal fluctuations in the setup. 4.1.2 Measurements of cavity finesse Figure 4.2 shows three examples of reflection dips for the same FFPC but for different wavelengths. In all three instances, the sideband frequency applied by the EOM were the same and so was the alignment of the cavity mirrors. As such it is clear that the FFPC had a higher finesse for wavelength 1570 nm than for 1500 nm as the width of the dip, directly proportional to κ, is smaller for 1570 nm. The figure also shows fits of the reflection signals to Equation 3.5. The fit was successful for Figures 4.2a and 4.2b but not for 4.2c. Failure of the fitting algorithm was caused by poor estimations of start parameters, in particular the start parameters for νlaser1 and νlaser2. This happened approximately once per 70 wavelengths steps and these measured values for κ could be manually discerned as unreliable by looking at the fit or plotted approximation of the sideband placement. In the future, the fitting algorithm could be improved upon to sort out failed fits automatically. Figures 4.3, 4.5, 4.6, 4.8 and 4.7 show the measured finesse, F , for different FFPCs as a function of wavelength c ω . As before, the different markers indicate different 28 4. Results (a) (b) (c) Figure 4.2: Examples of reflection dips fitted to Equation 3.5. The blue graphs indicate the reflected power as recorded by the photodetector whilst the orange dashed lines are the fit of the reflection data to Equation 3.5. The reflection is normalised to the maximum reflection according to the fit. 29 4. Results Figure 4.3: Measured finesse for a FFPC consisting of a fibre mirror from LAYERTEC and a macroscopic mirror with the same reflective coating, alternating layers of SiO2 and Ta2O5 of varying thicknesses. instances of measurement. Figure 4.3 shows the measured finesse for a FFPC con- sisting of a DBR (distributed Bragg reflector) fibre mirror from LAYERTEC with a macroscopic mirror from LAYERTEC. Both the fibre mirror and the macroscopic mirror had reflective coatings consisting of alternating layers of SiO2 and Ta2O5 of varying thicknesses. Some of the layers of the fiber mirror had been etched away with hydrofluoric acid to decrease reflectivity. Figure 4.4 shows the reflectivity for the fibre mirror before etching according to LAYERTEC. The reflectivity is relatively flat in the wavelength range 1500-1650 nm. The finesse of a FFPC is, F = ωFSR κ = 2π Ltot = 2π∑ i(1 −Ri) (4.1) where Ltot is the total losses in the system and Ri is the the reflectivity of cavity mirror i. Therefore, the finesse is also expected to have a relatively flat response for the same wavelength range. However, that does not appear to be the case in Figure 4.3. F ≈ 2π 1−Rfibre because the reflection from the macroscopic mirror was much larger than from the fibre mirror. So an increase of finesse from F = 8000 to F = 12000 corresponds to a reflection increase of about R|F=12000 −R|F=8000 R|F=8000 = 1 − 2π 12000 − ( 1 − 2π 8000 ) 1 − 2π 8000 ≈ 0.026 % (4.2) which is a very small change and could be the result of roughness or imperfections of the fibre mirror caused by the etching. It is also possible that just the act of removing layers from the fibre mirror through etching changed its properties such 30 4. Results Figure 4.4: Reflectivity of the fibre mirror before etching, taken from the specification sheet from LAYERTEC. that it has a less flat reflection curve then it did before. Two cavity mirrors with the reflectivity displayed in Figure 4.4 would have a finesse F ≈ 21000 for a wavelength of 1575 nm so it is clear that the etching had effect. It is interesting though that the spread of the measured finesse is so large in the range 1540-1560 nm and that the scattering of the Finesse values in this range seems systematic. This discrepancy is not only between different measurements series. The finesse also varies quickly in that range even for the same measurement set. It could be that, for this range of wavelengths, resonances are excited between the layers in the DBRs. How much of these modes are excited could have a strong dependency to the relative alignment between the mirrors and thus would be sensitive to the vibrational noise in the setup. As a result, the finesse value would be more sensitive to the vibrational noise of the setup. The error margins of the measurement scheme need more careful exploration in order to say to which degree this finesse feature is caused by noise in the setup and to which degree it is an inherent property of the cavity mirrors. In order to investigate this one could try different scan speeds. Slow noise processes have a decreasing effect with increased scan speed. One could also explore the hysteresis of the shear plate actuator as that might distort the measured reflection signal. One could also measure the finesse many times for the same wavelength and plot the spread of those values as a function of time. Figure 4.5 shows finesse measurements for two different FFPCs consisting of a fi- bre mirror from LAYERTEC and two different dielectric DBRs. The two dielectric DBRs making up the macroscopic mirror in either FFPC are chips made from dif- ferent parts of a wafer where the composition aimed for was alternating layers of 270 nm silicon dioxide and 111 nm silicon. The reflection coefficient was much lower for the macroscopic mirrors than for the fibre mirror. Therefore, the wavelength de- pendency of the finesse for these FFPCs can be assumed to be due to the wavelength dependency of the reflectivity from the macroscopic mirrors. For one of the DBRs (see Figure 4.5b) the finesse continued to increase for higher wavelengths throughout 31 4. Results (a) (b) (c) Figure 4.5: Measured finesse for two FFPCs consisting of a fibre mirror from LAYERTEC and dielectric DBRs. The two dielectric DBRs making up the macroscopic mirror in either FFPC are chips made from a wafer with varying layers of 111 nm silicon and 270 nm silicon dioxide. The different markers indicate different series of measurement. c) illustrates the structure of the two FFPCs and indicates where on the DBR wafer the two chips where taken. 32 4. Results the entire wavelength range of the lasers. This indicates that the highest reflection peak for this chip was shifted to outside the laser range probably due to slightly thicker silicon and silicon dioxide layers on that part of the wafer. For the other DBR (see Figure 4.5a), maximum finesse was reached at a wavelength of about 1540 nm after which the finesse approximately flattened out with the exception of a dip at around 1555 nm. Similar to Equation 4.2, this dip in finesse corresponds to a change in reflection with around 0.025 % which is not explained with the theoretical prediction for reflection from the DBR. However, this feature was closely reproduced for two different measurement series so it does not appear to be a measurement error. Perhaps energy is coupled to another optical mode within this wavelength range, for example guided modes of the DBR, which would guide energy away from the cavity. For a plane wave incident perpendicular on an ideal DBR, this would not happen as the k-vector for the plane wave and the k-vectors for the guided DBR modes would be perpendicular. FFPC modes however have more lateral spread in k-space and can therefore couple more easily to modes perpendicular to one of the cavity mirrors. Furthermore, the ideal case for the DBR assumes completely smooth layers which will not be the case for a real DBR. That the behaviour is different for the different DBR chips despite them being made from the same wafer can be explained by uneven thickness of the layers across the wafer. Figures 4.6, 4.8 and 4.7 show the finesse measured for FFPCs consisting of the fibre mirror in Figure 4.3 and different structures on a chip. Where on the chip those structures were is shown in Figure A.1. Figure 4.6 shows the measured finesse for a FFPC where the macroscopic mirror consisted of a suspended membrane of a pho- tonic crystal made from InGaP patterned with microscopic holes. This membrane was suspended over a thin layer of unpatterned InGaP on top of a DBR consisting of alternating layers of GaAs and AlGaAs as illustrated in the inset of the graph. Outside the wavelength range 1480-1510 nm, the resonance dips were too shallow and the finesse could not be measured which indicates that the finesse outside that range was lower than the finesse inside that range. Figure 4.7 shows the finesse measured on the same chip but without the suspended photonic crystal membrane. There the structure of the chip was instead a layer of unpatterned InGaP, on top of a layer of GaAs, on top of a thin layer of InGaP, on top of the DBR. In Figure 4.8 the finesse was measured on a part of the chip where the top InGaP layer and the GaAs had been removed. For comparison, Figure 4.9a and 4.9b show the theoretical reflection curve for a Gaussian beam from the photonic crystal simulated with S4 [15]. Figure 4.10a similarly shows the simulated reflectivity of a Gaussian beam from the DBR with the InGaP layers in the ideal case. The waist for the Gaussian beam was assumed to be 15µm for both simulations whilst the waist in the FFPCs was estimated to be about 7µm, this difference is not expected to cause significant change in the reflectivity. The simulation did not take into account the phase of the reflected light which might affect the reflectivity due to constructive and destructive interference. Figure 4.10b shows 2π 1−RDBR where RDBR is the reflection coefficient in Figure 4.10a. The reflectivity of the fibre mirror was much higher than the reflectivity of the DBR chip. Because of that, the reflectivity of the fibre mirror did not have much effect on the finesse value in comparison to the effect 33 4. Results (a) Figure 4.6: The measured finesse for a FFPC consisting of a fibre mirror from LAYERTEC and a macroscopic mirror, the structure of which is shown in the see inset. (a) Figure 4.7: The measured finesse for a FFPC consisting of a fibre mirror from LAYERTEC and a macroscopic mirror, the structure of which is shown in the see inset. 34 4. Results (a) Figure 4.8: The measured finesse for a FFPC consisting of a fibre mirror from LAYERTEC and a macroscopic mirror, the structure of which is shown in the see inset. of the DBR chip, see Equation 4.1. Therefore, the finesse for the FFPC, consisting of the fibre mirror and the DBR with the InGaP layers, should in theory be similar to Figure 4.10b. Although, the reflection coefficient in Figure 4.10a represents an ideal case of this DBR, in reality the reflectivity of the DBR was much lower and therefore the actual finesse was also expected to be lower than 2π 1−RDBR . But the wavelength dependency of 2π 1−RDBR and the finesse of the FFPC in Figure 4.7 should in theory be similar and they are. The difference can be explained by slight differences in layer thicknesses, shifting the reflection maxima to a higher wavelength for the real DBR. The wavelength dependency of the finesse for the photonic crystal on the other hand does not correspond to the simulated reflectivity. This is a more complex system then the other FFPCs explored in this project. The photonic crystal and the DBR underneath form an additional optical cavity. There are also additional modes that energy can couple to in the photonic crystal that are not accounted for in the simple model for the FFPC [16]. As such, F ∝ 1∑ i (1−Ri) no longer holds as this equation assumes independent reflectivities. 4.1.3 Polarisation dependency of the reflected power Figure 4.11 shows a series of reflection dips for different wavelengths for an FFPC consisting of the fibre mirror in Figure 4.3 and a DBR consisting of two alternating layers of InGaP and GaAs on top of alternating layers of AlGaAs and GaAs. Several of the panels, for example Figure 4.11b, show two overlapping reflection dips. This 35 4. Results (a) (b) Figure 4.9: Simulated reflectivities for a Gaussian beam, with waist radius 15µm, perpendicular to the photonic crystal structure of the membrane in Figure 4.6 (a) (b) Figure 4.10: a) shows the simulated reflectivity for a Gaussian beam, with waist radius 15µm, perpendicular to the ideal version of the InGaP on DBR structure in Figure 4.7. b) shows 2π 1−R where R is the reflection coefficient in a). 36 4. Results is because the reflection at the FFPC was polarisation dependent. This was shown by the fact that changing the polarisation of the light entering the FFPC with the paddle wheel polarisation controller shifted the balance between the two peaks, reducing them to a single peak for the right polarisation. This happens because the effective cavity length (including penetration into the mirrors as well as the distance between the mirrors) differs for different polarisations if one or both mirrors are birefringent. The fibre mirror did not cause noticeable birefringence so the double dips here are due to birefringence in the macroscopic mirror. Observing birefringence in strained crystals is not surprising and has been observed for both InGaP [17] and GaAs [18]. The next step to investigate these birefringent properties would be to measure how the fibre changes the polarisation of the light. This can be done by putting a polarisation filter between the facet of the fibre and a photodetector and measuring the power on the photodetector for different wavelengths. This will then make it possible to discern whether the birefringence is wavelength dependent. It is possible that the wavelength dependency observed in 4.11 is because the polarising properties of the fibre are wavelength dependent. Further, one could use sidebands to rescale the x-axis in the reflection plots to frequency as described in Section 3.1.2. The the frequency difference between the two reflection dips could then be used to calculate the difference in refractive index between the two polarisation states. As νcav ∝ νFSR, see equations 2.1 and 2.2, the fraction between the two frequencies would be νe νo = νFSR,e νFSR,o = Lcav effective,o Lcav effective,e = Lcav + Lpo Lcav + Lpe (4.3) where Lpo is the effective penetration depth into the DBR chip for one polarisation state and Lpe correspondingly for the perpendicular polarisation state. After some algebra this can be expressed as ∆fLcav = (fe − fo)Lcav = foLpo − feLpe = Lp,geo(fono − fene) (4.4) where Lp,geo is the actual, geometric, penetration depth into the chip and no and ne are the refractive indices for the two polarisation states. 4.2 Experiments with the vacuum integrated op- tomechanical setup The tests done on the vacuum integrated setup for optomechanical experiments concerned decreasing the noise in the setup to enable PDH and SoF locking, the results of which will be presented in this section. The electrical noise from the RedPitaya was decreased by adding a resistor after the voltage amplifier to act as a low-pass filter together with the capacitance of the shear plate actuator, see Figure 3.5. Without this extra resistor, the reflection signal was distorted too much by the electrical noise. Attempts to lock were made with a 66 kΩ and a 18 kΩ resistor. The lock was more stable with the 18 kΩ resistor indicating that the bigger resistor made the cut-of frequency for the low-pass filter low enough that it inhibited the ability of 37 4. Results (a) 1549.5 nm (b) 1550.0 nm (c) 1550.5 nm (d) 1551.0 nm (e) 1551.5 nm (f) 1552.0 nm (g) 1552.5 nm (h) 1553.0 nm (i) 1553.5 nm (j) 1554.0 nm (k) 1554.5 nm (l) 1555.0 nm Figure 4.11: A series of reflection dips showcasing birefringence in the cavity mirrors. The x-axis is time and the y-axis reflected power, each with arbitrary units. The labels indicate the laser frequency for the corresponding mea- surement. 38 4. Results the PID controller to compensate for high frequency noise in the system. Attempts were also made to lock with and without the metal springs, see Figure 3.4. Without the springs, locking was not possible as the resonance dip in the reflection signal drifted too much. This indicates the presence of mechanical drift or low frequency oscillations in the setup, causing a change in the cavity length over time. The springs managed to decrease this by creating an additional mechanical connection between the structure holding the fibre mirror and the structure holding the chip mirror. Noise in the reflection signal was further decreased by taping the patch cord fibre to the optical table. The vacuum pump created sinusoidal oscillations in the background reflection signal, inhibiting locking when the pump was on. This can be reduced by turning the vacuum pump off during measurements or by replacing the pumping system with one having lower vibrations. Figure 4.12 shows the noise spectral density for the error signal, Sϵ, of a SoF lock of a FFPC consisting of a fibre mirror and a reflective structure on chip that was not free to vibrate. This measurement was taken before the setup was put into vacuum. The noise is expected to be less in vacuum. The noise that is most relevant for the application is the noise in the frequency range close to the expected mechanical frequencies, Ωm/2π, of a few MHz. The magnitude of Sϵ in that range needs to be small compared to the additional noise caused by the mechanical resonator motion in order to extract the resonator frequency and g0 from Sϵ with the help of equations 3.22 and 3.23. That will be the subject of future studies. 39 4. Results (a) (b) Figure 4.12: Noise power spectral density of the error signal for a SoF lock of a FFPC for two different noise frequency ranges. 40 5 Conclusion & Outlook In this thesis, an experimental platform for optical characterisation of fibre Fabry- Pérot cavities (FFPCs) has been implemented. This setup allows one to study their properties by facilitating measurements with high wavelength resolution. The high wavelength resolution makes it possible to discern small features in the reflectivity spectrum of on-chip structures and fibre mirrors. The experimental platform has been used to study the reflective properties of a fibre mirror and various reflective on-chip structures, like DBR coatings and pho- tonic crystal structures. The results showed that DBRs consisting of InGaP on top of alternating layers of AlGaAs and GaAs can be birefringent with polarisation dependent properties. The results also demonstrated how small imperfections in fabrication can affect the optical properties of DBRs. For example, chips from the same wafer, and thus theoretically with the same composition, had different reflec- tive properties. Further, the results suggested that the Gaussian mode of a FFPC, consisting of a fibre mirror and a dielectric DBR, can couple to optical modes guided in the DBR leading to additional features in the expected reflection curves of the DBR. The optical properties of a FFPC consisting of a fibre mirror and a photonic crystal suspended over a DBR were examined. The optical properties of that FFPC could not be modelled only by the reflectivities of the fibre mirror and the photonic crystal. This demonstrated that a more complex model is needed to theoretically explain that system, a finding which is consistent with [16]. This experimental plat- form can in future works be an important tool to facilitate the testing of theoretical models for the reflection from FFPCs with photonic crystals. One could increase the reliability of such measurements by studying the hysteresis of the shear plate ac- tuator and how to better modulate the cavity length to minimise hysteresis-induced distortions of the reflection signal. Additionally, a vacuum integrated experimental setup was constructed that in fu- ture work will be used to conduct optomechanical experiments with FFPCs, allow- ing optical readout and control of mechanical resonators on chips. Integrating the setup with a vacuum chamber allows for optomechanical experiments without the quality factor of the mechanical resonators being limited by the damping from the surrounding air. In future work, the vacuum integrated setup will be used for optomechanical ex- periments with FFPCs consisting of a fibre mirror and photonic crystal mechanical resonators. The small size of the FFPC will allow for small cavity lengths, Lcav. 41 5. Conclusion & Outlook Using photonic crystals will make it possible to achieve smaller effective masses, meff, of the membranes compared to if one would use for example DBRs. This will make it possible to achieve higher values for the single photon coupling strength g0 At the same time, FFPCs have been shown to achieve small values for κ [19]. Together, this makes it possible to achieve higher values for g0 κ and approach the strong coupling regime, g ∼ κ, where the optical and mechanical modes hybridise. 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