Stabilizing Lasers Using Whispering Gallery Mode Resonators

Stabilizing Lasers using Whispering Gallery Mode Resonators

Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von Benjamin Sprenger aus Frankfurt am Main Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung: 2. 12. 2010

Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink

Erstberichterstatter: Prof. Dr. Lijun Wang

Zweitberichterstatter: Prof. Dr. Min Xiao To My Parents d Zusammenfassung

Laser mit ultraschmalen Linienbreiten werden in hoch-präzisen Wissenschaften unent- behrlich. Flüstergaleriemoden-Resonatoren (whispering gallery mode, auch "WGM") mit höheren und höheren Güten werden ständig gefertigt, wie z.B. Mikrokugeln aus Silikat, und neuerdings auch diamantgedrehte kristalline Scheiben aus Kalziumﬂuorid. Desweit- eren muss die Übertragung von ultraschmalen Frequenzen verstanden und perfektioniert werden. Diese Arbeit konzentriert sich auf die Stabilisierung zweier Laser durch WGM Resonatoren, und die Verteilung stabiler optischer Frequenzen durch die turbulente At- mosphäre. Wir produzieren Mikrokugeln mit 100 μm Durchmesser, und Güten von 108.Wir verwenden diese als frequenzselektive Elemente in einem Faser-Ring-Laser, der eine Erbium-dotierte Faser als Gain-Medium enthält. Wir koppeln mit verjüngten optischen Fasern in die schmalen Resonanzen der Kavität ein, und mit Winkel-polierten Fasern wieder auf der gegenüberliegenden Seite aus. Die Linienbreite des Lasers verringert sich dadurch um fünf Größenordnungen — gemessen durch eine heterodyne Schwebung mit einem stabilen Referenz Laser — und kommt auf 170 kHz volle Breite bei halber Höhe. Außerdem wird eine Rotverschiebung von 16 pm/μW durch die Pump Leistung beobachtet. Ein verbesserter Aufbau verwendet Prismakopplung auf beiden Seiten einer robusten Kalziumﬂuorid WGM Disk (5 mm Durchmesser), und einen Halbleiter-basierten optischen Verstärker als Gain-Medium. Die Linienbreite beträgt 13 kHz, gemessen durch die selbst-heterodyne Schwebung mit einer 45 km Faser Verzögerungsstrecke. Desweiteren wird die Allan Abweichung (10−11 mit 10 μs Mittelzeit) durch die “three-cornered-hat” Methode bestimmt, indem mit zwei Referenzlasern gleichzeitig Schwebungen erzeugt werden. Mit einem Interferometer auf dem Dach des Instituts wird das optische Phasenrauschen über 100 m Freiraum bestimmt. Heterodyne Detektion wird verwendet, und optischer Frequenz Transfer wird mit einem Radio-Frequenz-Amplituden modulierten Laser ver- ii Zusammenfassung glichen. Die Präzision optischen Transfers beträgt 1.68 × 10−13 bei 1 s Mittelung, und verringert sich auf bis zu 10−15 bei Mittelung über eine halbe Stunde. Wie erwartet ist der Radio Frequenz Transfer schlechter und beträgt 1.07 × 10−10 gemittelt über 1 s. Abstract

Ultra-narrow linewidth laser sources are becoming indispensable in high-precision scientific research. Whispering gallery mode (WGM) resonators are achieving higher and higher quality factors in fused silica microspheres, and, more recently, diamond-turned crystalline disks made of calcium fluoride are even superseding these. Furthermore, the dissemination of ultra-narrow frequencies must be understood and perfected. This work focuses on two lasers stabilized using WGM resonators, as well as the dissemination of stable optical frequencies through the turbulent atmosphere. The microspheres we fabricate, with diameters on the order of 100 μm, are measured to have quality factors around 108. We use such a microsphere as a frequency selective element in a fiber ring laser using erbium-doped fiber as a gain medium. We couple into the sharp modes of the microsphere by using tapered fiber coupling. Angle-polished fiber coupling is used on the other side, resulting in a narrow bandwidth filter. The laser’s linewidth is decreased by five orders of magnitude, as determined using the heterodyne beat technique with a stable reference laser, resulting in 170 kHz FWHM. Additionally, a red-shift of 16 pm/μW as a function of pump power is observed. An improved setup uses prism coupling on both sides of a rigid calcium fluoride WGM disk (5 mm in diameter), and a semiconductor optical amplifier as a gain medium. The linewidth is measured to be 13 kHz using the self-heterodyne beat technique with a 45 km fiber delay line. Furthermore, the Allan deviation gives 10−11 at 10 μs averaging — calculated using the three-cornered-hat method, by beating with two reference lasers simultaneously. Using an interferometer set up on the roof of the institute, the optical phase noise induced over 100 m of free space is measured. Heterodyne detection is used, and optical frequency transfer is compared to a radio frequency amplitude modulated laser beam. In optical transfer, the precision reaches 1.68 × 10−13 after 1 s of averaging, and improves down to 10−15 in half an hour of averaging. As expected, the radio frequency transfer is worse, giving 1.07 × 10−10 accuracy after 1 s. iv Abstract Acknowledgments

I want to express my deep gratitude to all the people that helped in making this thesis a success. Without their significant contributions this work would have been impossible. I would like to take this time to thank them. First of all, I am greatly indebted to my PhD advisor, Prof. Lijun Wang. He always inspired me, and managed to motivate me with his ingenious suggestions. I thank him for letting me follow my interests, and for providing insights and guidance when I required it. It was always my dream to make original contributions to science, and I am grateful that I had the possibility to do this in Prof. Wang’s group. I owe my eternal thanks to Dr. Harald G. L. Schwefel, whom I have been working with for two years now. We have been office mates for even longer, and he has been a friend, and a source of inspiration to me the entire time. Whenever I needed help, be it work-related or not, he would always take the time and encourage me with his positive attitude and ubiquitous knowledge. I also thank him for proof-reading my thesis. Many co-workers in Prof. Wang’s group have aided me in the understanding of physics, and influenced my life in a positive way. Furthermore, I have made many friends that I will never forget. I want to thank everyone for making the atmosphere in the group as pleasant as it was. Dr. Zehuang Lu always lent a helping hand, and took as much time as necessary to explain physical concepts to me and help me in publishing my first papers. I also thank him for proof-reading the atmospheric transfer chapter of the thesis. Dr. Jie Zhang was never too busy to answer my questions and motivate me to keep going with her benevolent attitude. My office mate Dr. Sergiy Svitlov and I always exchanged stories, and I thank him for helping in the zero-crossings calculations he performed. Dr. Vladimir Elman took a lot of time to aid us in the design of the external cavity diode laser. Dr. Rachit Sharma and Dr. Jan Schäfer were always there to encourage me, but also to distract me and talk about Life, the Universe and Everything when it was necessary. All of the group members contributed to this thesis in some way, and I really appreciate it. All my thanks to Sebastian Bauerschmidt, Dr. Marian Florentin Ciobanu, Prof. Dr. Gottfried vi Acknowledgments

Döhler, Simon Grams, Simon Heugel, Dr. Hua Hu, Dr. Tau Liu, Dr. Stefan Malzer, Dr. Jessica Mondia, Dr. Mingying Peng, Dr. Felix Müller, Dr. Sascha Preu, Dr. Christian Rothleitner, Dr. Alois Stejskal, Max Tillmann, Dr. Bo Wang, Dr. Jinxiong Wang, Dr. Jianwei Zhang, Dr. Quanzhong Zhao, and Dr. Yanning Zhao. I want to acknowledge and thank the supporting people in the Max Planck Institute. Kirsten Oliva, Lisa Spann, Dr. Sabine König, Nadine Danders, Anja Deckert, Caroline Edenharter, and Hildegard Porsch have all helped me significantly throughout the years. I always felt welcome, no matter whom I had to speak to. I also thank Dr. Carsten Schür and Heike Auer, the organizers of the International Max Planck Research School, who helped make my stay a pleasant one. Additionally, I want to thank the mechanical work- shop, specifically Bernhard Thomann, Robert Gall, and Thomas Spona, for constructing various holders and adapters, and for giving me important advice in my own construc- tions. Adam Käppel and Lothar Meier were always available for any electronics related questions that I had, and always designed and built electronic circuits when necessary — I really appreciate it. Thank you to Namvar Jahanmehr and Daniel Ploß for their help with the FIB and the SEM. Also, I am grateful for the IT support from Michael Zeller, Kristin Gregorius, and Benjamin Klier. I am also indebted to Florian Sedlmeir, Josef Fürst, Dr. Christoph Marquardt, and Dr. Dmitry Strekalov (NASA Jet Propulsion Laboratory, CA) for the fruitful discussions we had about whispering gallery modes. I also thank Dr. Markus Schmidt and Dr. Holger Hundertmark for their expert advice relating to fibers and other optics related questions. Thank you also to Sebastian Stark, who helped me with random questions about optics. Finally, I want to give a huge thanks and a hug to my parents, Ruth and Volker, for giving me the incredible opportunity to pursue a career in science. I don’t know how I can fully show my appreciation. My brother Thorsten, and my sister Rabea were always there for me, and influenced my life greatly. I have always looked up to them, and I always will. My family has always given me the support and motivation I needed. Without their appraisal I would never have made it this far. I am also greatly indebted to my girlfriend Daniela. I thank her for always being there for me when I was frustrated that things were not working, and moreover for being there to celebrate when things went just as I wanted them to. Contents

Zusammenfassung i

Abstract iii

Acknowledgments v

List of Figures xvi

1 Introduction 1 1.1 Laser Stabilization Techniques ...... 1 1.2 Whispering Gallery Modes ...... 4 1.3 Frequency Dissemination ...... 7 1.4 Motivation and Goal ...... 8 1.5 Outline of the Thesis ...... 9

2 Theoretical Background 11 2.1 Linewidth ...... 11 2.1.1 Schawlow-Townes Limit ...... 12 2.1.2 Bad Cavity Lasers ...... 14 2.1.3 Allan deviation ...... 15 2.2 Whispering Gallery Mode Resonators ...... 16 2.2.1 Ray Model ...... 17 2.2.2 Wave Model ...... 19 2.2.3 Quality Factor ...... 25 2.2.4 Coupling ...... 27 2.2.5 Intensity Dependence and Thermal Eﬀects ...... 32 2.2.6 Fundamental Limits ...... 34 2.3 Heterodyne Beating ...... 35 viii CONTENTS

2.3.1 Self-heterodyne Beating ...... 37 2.4 Three-Cornered Hat Measurement ...... 39 2.4.1 Correlation Removal ...... 40 2.5 Frequency Dissemination ...... 43 2.5.1 Satellite Transfer ...... 44 2.5.2 Fiber Transfer ...... 45

3 Experimental Details 47 3.1 Whispering Gallery Mode Resonators ...... 47 3.1.1 Microspheres ...... 48 3.1.2 Crystalline Resonators ...... 50 3.1.3 Prism Coupling ...... 51 3.1.4 Tapered Fiber Coupling ...... 54 3.1.5 Angle-Polished Fiber Coupling ...... 56 3.2 Littrow Grating Stabilized Diode Laser ...... 56 3.3 Fiber Lasers ...... 59

4 Microsphere Stabilized Fiber Laser 61 4.1 Setup ...... 61 4.2 Measuring the quality factor ...... 65 4.3 Heterodyne Beating and OSA results ...... 68 4.4 Pump Power and Temperature Dependence ...... 72

5 Calcium Fluoride Stabilized Fiber Laser 75 5.1 Setup ...... 75 5.2 Measuring the quality factor ...... 78 5.3 Self-Heterodyne Beating ...... 80 5.4 Three-Cornered Hat Measurement ...... 83

6 Atmospheric Transfer of Frequencies 91 6.1 Introduction to Frequency Dissemination ...... 91 6.2 Setup ...... 92 6.3 Allan Deviations, Optical Transmission ...... 96 6.4 Allan Deviations, Radio Frequency Transmission ...... 98 CONTENTS ix

7 Conclusion 101 7.1 Thesis Summary ...... 101 7.2 Future Outlook ...... 102

Bibliography 116

Curriculum Vitae 117

Publications 119 x CONTENTS List of Figures

1.1 Illustrations of a Fabry-Perot laser and a ring laser respectively...... 3

1.2 Illustrations of three grating stabilized lasers. Left, distributed feedback (DFB) laser. Middle, distributed Bragg reﬂector (DBR) laser. Right, external cavity diode laser (ECDL) in the Littrow conﬁguration...... 4

1.3 The Whispering Gallery in St. Paul’s Cathedral, London, UK...... 5

1.4 Schematics of WGM ﬁltering schemes...... 6

1.5 A frequency comb can be locked to an optical clock signal to stabilize it. Then a telecom laser can be locked to the frequency comb and transferred. 8

2.1 Schawlow-Townes linewidth with bad cavity modiﬁcation as a function of cavity loss per roundtrip in percent, assuming conventional Schawlow- Townes limit of 9 Hz as calculated before, and a resonator with a quality 8 factor of 10 .Oﬀset of frequency from gain center ν − ν0 is zero, and

assume a perfect four-level laser with Nsp equal to one...... 15

2.2 Slopes of Allan deviations on a log-log scale, indicating the noise types determined by the slope of the line...... 17

2.3 Left, rays inside a resonator can experience total internal reflection at flat angles, hence no light leaks out of the resonator. Right, specific frequencies and certain angles can have constructive interference after a complete roundtrip, so long photon confinement becomes possible...... 18

2.4 Bessel functions of the ﬁrst type with m = 0, 1, 2, 3...... 21

2.5 Bessel functions of the third type (Hankel functions) with m = 0, 1, 2, 3. . 22 xii LIST OF FIGURES

2.6 Fundamental and second order modes calculated using finite element software “Comsol Multiphysics”. The mode number m = 50 in both cases, and l = m in the left image, and l = m − 1 in the right image. The mode number n = 1 in both cases. The simulated diameter is twelve units, and the thickness is three units...... 24 2.7 Intensity distribution of whispering gallery mode with n = 1, and l = m = 13. The resonator boundary is indicated in white, and the decaying evanescent field outside of the resonator is visible...... 25 2.8 A thin waveguide, such as a tapered fiber, coupling to a WGM resonator. Light can couple from the fiber into the resonator and vice versa...... 28

2.9 Light is focused onto the back of the prism at angle θc. The refractive

index of the prism is nd, and ne for the WGM resonator...... 29 2.10 Side view of a WGM coupled using a prism. The disk radius is R, and the azimuthal (vertical) radius is r...... 30 2.11 Angle-polished fiber coupling. Light is totally internally reflected from the sharp fiber tip, and the evanescent field couples to the evanescent field

of the WGMs. The critical angle θc is indicated...... 31 2.12 Left, amplitude of two slightly oﬀset optical signals around 193 THz. Right, intensity on detector after frequency mixing. The frequency com-

ponents are | fa − fb|, and fa + fb...... 36 2.13 Delayed self-heterodyne method. A laser is split; one part is delayed in a long ﬁber, the other part is shifted in frequency using an AOM. The two are recombined, and the beat signal is observed on a spectrum analyzer. . 38

3.1 CO2 laser microsphere fabrication setup. The approximate beam path is indicated for clarity. Three mirrors steer the beam in the upward direction onto a ZnSe lens with a focal length of 50 mm. Objects can be placed into or near the beam focus using an x − y − z stage...... 48 3.2 A typical microsphere with a diameter of about 100 μm, and a stem of 20 μm. A tapered ﬁber can be seen behind the sphere...... 49 3.3 A thermoelectric cooler controls and stabilizes the microsphere temperature. A copper tube holds the stem of the microsphere using heat-conducting paste...... 50 LIST OF FIGURES xiii

3.4 The crystal disk polishing setup includes an air-bearing motor to hold and spin the resonator, as well as a motorized x − y stage to cut a speciﬁed shape into the disk using a sharp diamond tip...... 51 3.5 Scanning electron micrograph indicating the typical surface roughness of

a CaF2 resonator. The grooves are below 100 nm in depth and diameter. . 52

3.6 Prism coupling using SFL 11 glass and a CaF2 resonator...... 53 3.7 Fiber taper being held by clamps with gas flame shifting left to right. . . . 54 3.8 Signal on detector after transmission through the fiber while it is being tapered. Clear resonances are visible, until they subside when the diameter is narrow enough — about 1-2 μm...... 55 3.9 Bottom view of holder for angle-polished fibers. Nine fibers can be polished simultaneously...... 57 3.10 Schematic of angle-polished fiber coupling and scanning electron micrograph of polished fiber tip showing very little surface roughness...... 58 3.11 Littrow laser setup. An external cavity is formed using a grating in the Littrow configuration. The first order is reflected back into the laser diode. The zeroth order is reflected out. By varying θ the lasing frequency is shifted...... 58

4.1 Unstabilized fiber loop laser. A 980 nm DFB laser goes into a loop using a wavelength division multiplexer (WDM) and pumps some erbium-doped fiber. An isolator prevents lasing in the backwards direction. 10% of the 1550 nm emission is coupled out...... 62 4.2 Multimode emission from the unstabilized fiber laser. FWHM = 0.2 nm. . 63 4.3 Light couples into a microsphere using a tapered fiber. Transmission is coupled out using an angle-polished fiber...... 64 4.4 Stabilized fiber loop laser. A 980 nm DFB laser goes into a loop using a wavelength division multiplexer (WDM) and pumps some erbium-doped fiber. An isolator prevents lasing in the backwards direction. The tapered fiber, microsphere, and angle-polished fiber function as a narrow- linewidth filter. 10% of the 1550 nm emission is coupled out...... 65 4.5 Left, the spontaneous emission of the erbium-doped fiber after transmission through the tapered fiber. Red, dips due to resonances in the microsphere. Right, peaks at the same positions after coupling into the angle-polished fiber...... 66 xiv LIST OF FIGURES

4.6 The frequency of a tunable diode laser is scanned and the transmission through a tapered fiber is measured using a photodiode and recorded using an oscilloscope. On resonance a dip in the transmission appears...... 67 4.7 The frequency of the tunable diode laser is scanned over 600 Mhz. The Lorentzian shaped resonance shows a width of 1.7 MHz and 70% coupling into the microsphere...... 68 4.8 The linewidth of the laser is 0.01 nm, which is the limit of the resolution of the optical spectrum analyzer (Ando AQ6317B)...... 69 4.9 The lasing peak, as measured using a wavemeter, fluctuates around 1 GHz over seven hours. Generally the variation is on the order of hundreds of MHz, which is the resolution of the wavemeter...... 70 4.10 Heterodyne beat measurement setup. The stabilized fiber laser is spatially overlapped with an ECDL which is offset by a few MHz. The resulting beat is recorded using a detector and a spectrum analyzer...... 71 4.11 Heterodyne beat between microsphere stabilized fiber laser and ECDL reference laser, showing their combined frequency noise. The 3 dB width is 170 kHz, limited by the linewidth of the reference laser...... 73 4.12 There is a red-shift in the emission with increasing pump and output power. The slope in terms of the output power is 16 pm/μW...... 74

5.1 Left, spontaneous emission from the semiconductor optical amplifier with a FWHM of 94 nm. Right, lasing closing the loop of the SOA and coupling out 1%. Resonances are spaced 4 nm apart due to weak reflections from fiber connectors with a FWHM of 0.4 nm...... 76 5.2 Left, stabilized ring laser. A semiconductor optical amplifier (SOA) provides gain around 1550 nm. Polarization control (PC) is used before the coupling and before the SOA. An isolator prevents bi-directional lasing. Graded refractive index (GRIN) lenses are used for prism coupling. 1% of the emission is coupled out. Right, photograph showing the GRIN lenses and the resonator on top of a thermoelectric cooler...... 77 5.3 Left, 15 MHz resonance, corresponding to a Q factor of 107. Right, the black curve indicates the reflected light from the input prism, and the red curve indicates the transmission through the WGM disk and other prism and into the GRIN lens...... 78 LIST OF FIGURES xv

5.4 Left, top to bottom curves show a resonance with decreasing distance between the disk and the prism. Right, the red curve shows a decrease in coupling with increasing distance, as well as a decrease in FWHM of the resonance...... 80 5.5 Left, optical spectrum analyzer trace of stabilized laser emission. Line- width is 0.05 nm, limited by the resolution. Right, heterodyne beat with a Toptica DL Pro as a reference. Linewidth is 70 kHz, limited by the linewidth of the DL Pro...... 81 5.6 Self-heterodyne beat schematic. An acousto optic modulator (AOM) splits a laser signal. The 1st order is oﬀset by 100 MHz and delayed longer than the coherence length of the laser. Next, it is recombined using a ﬁber coupler and analyzed using a detector and an rf spectrum analyzer...... 82 5.7 Self-heterodyne beat measurements of homebuilt external cavity diode

laser (100 averages), Toptica DL Pro (10 averages), and CaF2 stabilized laser (70 averages). The 3 dB widths are 550 kHz, 58√ kHz, and 18 kHz respectively, as measured over 100 ms. Dividing by 2 gives the upper limit for the linewidths...... 82 5.8 Three input frequencies are mixed using six 3 dB ﬁber couplers...... 83 5.9 Typical sine-wave oscilloscope trace of a beat between two lasers. .... 85 5.10 Three simultaneous beats between the homebuilt ECDL, the Toptica DL Pro, and the WGM stabilized laser. The homebuilt ECDL is more noisy than the other two lasers. A 1.4 kHz harmonic was removed from both beats containing the ECDL...... 86 5.11 Allan deviations of the three combined beats. The DL Pro and ECDL beat

is very close to that of the CaF2 stabilized laser and the ECDL...... 87 5.12 Allan deviations calculated for the separate lasers using the three-cornered hat method...... 88 5.13 All Allan deviations calculated for the lasers using the three-cornered hat method. Invalid results are ignored...... 89

5.14 Correlation-removed Allan deviations with CaF2 laser as reference. .... 89 5.15 Correlation-removed Allan deviations with Toptica DL Pro laser as reference...... 90 5.16 Correlation-removed Allan deviations with ECDL as reference...... 90 xvi LIST OF FIGURES

6.1 Frequency noise measurement setup. An ECDL is split into two arms. One arm is transmitted over 50 m and reflected back by a retroreflector. The other arm is offset by 165 MHz using an acousto optic modulator (AOM). They are recombined on a photodiode (PD) and analyzed using an rf spectrum analyzer, an FFT spectrum analyzer, or a frequency counter. 93 6.2 Photograph showing the rooftop setup. Laser beam paths included for clarity...... 94 6.3 The 63.5 mm aperture gold retroreflector could be placed up to 50 m away from the optical setup...... 95 6.4 Left, beat signals showing an increase in frequency noise for 5 m, 20 m, 35 m, and 50 m. Right, beat signal 3 dB width as a function of distance showing a linear increase...... 96 6.5 Allan deviation for optical transfer of frequencies over 100 m first showing a τ−1/2 dependence, and later τ−1. Inset, histogram of beat signals with a Gaussian fit, FWHM = 70.5 Hz...... 97 6.6 Allan deviation for 80 MHz rf transfer of frequencies over 100 m. In- set, spectrum of the beat signal measured over 30 min with Gaussian fit, FWHM = 1.05 mHz...... 99 Chapter 1

Introduction

Lasers have the unique property that they emit light at a single frequency, which is not the case for classical light sources. This is, in fact, a simplification. Any real laser has some finite spread in frequency. For many applications it is desirable to minimize this spread. Stable narrow-linewidth lasers are indispensable in many scientific and techno- logical fields such as telecommunications, sensing, interferometric measurements, gas detection, and frequency metrology. Lasers can be locked to external cavities or atomic transitions of atoms, which are promising candidates for optical clocks that supersede the precision of microwave clocks. This thesis discusses the application of whispering gallery mode resonators as cavities for frequency stabilization and linewidth-narrowing in lasers, as well as the dissemination of ultrastable frequency sources in the turbulent atmosphere. Three distinct experiments are described, including two variations of whispering gallery mode resonator stabilized lasers, and an experiment to determine the phase noise induced by atmospheric conditions over 100 m of free space.

1.1 Laser Stabilization Techniques

In 1917 Albert Einstein already proposed the ingenious idea for stimulated emission in atoms [1]. The first experimental observation followed about ten years later by Ladenburg [2]. A few decades later, in the 1950s, the first devices were theoretically described and built which made use of this idea by amplifying the stimulated emission of microwave radiation to create the first masers [3, 4]. Efforts were made in the search for possibilities of devices with shorter wavelengths, and Schawlow and Townes theoretically described the concept of infrared and optical masers [5]. Finally in 1960 Maiman succeeded in 2 Introduction inducing the first stimulated optical output in Ruby crystals [6], which was the birth of the LASER (light amplification by stimulated emission of radiation). Subsequently research groups all over the world focused on using a multitude of gain media in their lasers, which included the invention of helium-neon lasers [7], semiconductor lasers [8], carbon dioxide lasers [9], dye lasers [10, 11], and later excimer lasers [12]. Incredible new uses for lasers were discovered in the last 50 years, primarily due to their unique properties such as their directionality, monochromaticity, and spatial coherence. Arguably their most important property is monochromaticity, implying a narrow spectral width. Various mechanisms lead to broadening of the linewidth though, so no laser is monochromatic as such, but over the decades numerous techniques have been invented to make the linewidth as narrow as possible. Lasers can run on multiple longitudinal modes at the same time, so additional efforts are needed to suppress all but one to allow single-mode emission of a laser. Likewise, the stability of the laser — essentially the drift rate of the lasing frequency — needs to be minimized for some applications. Narrow-linewidth lasers are required for long distance high data-rate transmission in telecommunications [13], since dispersion in optical fibers leads to signal loss if the linewidth is too broad or even multi-mode. Furthermore, various interferometric techniques require the use of narrow linewidth lasers, including nm precision distance measurements [14], accurate strain and temperature determination [15], measurement of the gravitational acceleration g [16], gas detection [17], and even the search for gravitational waves [18]. The basic layout of a laser will affect its noise properties and lasing characteristics considerably. In many cases it is preferable to use a cavity that is as small as possible, since detrimental temperature variations or vibrations tend to affect larger cavities more drastically. Furthermore, the separation of modes in a cavity is inversely proportional to the roundtrip time, so a smaller cavity leads to larger spacing, which facilitates selection of a single mode. In a standing-wave cavity, for example a Fabry-Perot resonator, the maxima and minima of excited states in the gain medium due to the standing wave will lead to spatial hole burning [19]. The result is that the lasing mode will experience gain saturation before the other cavity modes do, so mode competition occurs and other modes can lase. Addition- ally, a refractive index grating effect can lead to intra-cavity reflection of a lasing mode, which then detrimentally interferes with the counter-propagating mode. An alternative to avoid this is to use traveling wave cavities, often termed ring lasers or loop lasers. If uni-directional lasing in such a cavity is preferred, there will be no spatial hole burning, and single-mode emission can be achieved more easily [20]. One way to achieve this 1.1 Laser Stabilization Techniques 3 is by introducing a Faraday rotator and polarizing elements into the cavity to suppress a counter-propagating mode. The basic setup of a Fabry-Perot laser compared to a ring laser are shown in Fig. 1.1. In each setup, one mirror has less than 100% reflectivity, thus allowing the beam to exit at this point. The ring laser prevents a standing wave, and ensures uni-directional lasing with an optical isolator.

R = 100% R < 100% R = 100% R = 100%

Gain Medium Isolator

R = 100% R < 100%

Gain Medium (a) Fabry-Perot Laser (b) Ring Laser

Fig. 1.1: Illustrations of a Fabry-Perot laser and a ring laser respectively.

Over the years a multitude of methods has been conceived to stabilize lasers and decrease their linewidth, which can be grouped into active and passive stabilization techniques. Active stabilization generally requires some sort of feedback which adjusts the laser parameters (such as pump current or optical length in the cavity) in real-time. The two most common methods are to lock a laser onto a gas absorption line by measuring the transmission of the laser through a gas cell [21], or by phase-modulating the output beam and sending it into an external optical reference cavity [22]. The latter is commonly referred to as the Pound-Drever-Hall technique. Active stabilization can also refer to feedback which is used to stabilize the intensity of the output, the beam pointing, or the carrier-envelope phase in short laser pulses. Passive stabilization does not require an external feedback signal and usually requires some sort of frequency selective element that is part of the lasing cavity. Arguably the most basic passive linewidth narrowing device would be a thin glass plate inside a Fabry-Perot resonator [23]. If this is inserted, there are two resonance conditions that must be met by an oscillating mode, since the weak interface reflections from the glass front and back sides impose a second resonance condition. If tuned correctly (for example by adjusting the angle), single-mode emission can be possible. A common narrow mode selection scheme in a fiber laser is the use of a fiber Bragg grating. This is a periodic array of refractive index distributions which has a narrow reflection band. Thus it can be used to preferentially reflect the specified frequency and mode 4 Introduction competition will prevent other modes from receiving enough gain. This is analogous to using narrow-band Bragg mirrors in a standard Fabry-Perot setup, as in Fig. 1.1a. Often it is necessary to introduce a further mode selection element in such a cavity. Prisms with high dispersion can be inserted into cavities to select a single line that can receive gain. Diode lasers are frequently stabilized using some form of grating. Three types are illustrated in Fig. 1.2: the distributed feedback (DFB) laser has a grating structure throughout the gain region which only permits specific modes to exist. The distributed Bragg reflector (DBR) laser has a grating structure functioning as a back reflector behind the gain region. Finally, the external cavity diode laser (ECDL) in the Littrow configuration reflects the first order diffraction from a grating back into the diode to form an extended cavity. The angle can be changed in this case to scan over a large wavelength range. This type of external cavity stabilization was already implemented in dye lasers in 1972 [24]. The Littman-Metcalf configuration is similar to the Littrow configuration and includes another mirror for the reflection back along the grating [25].

Laser Diode Laser Diode

Bragg Grating Bragg Grating Lens Gain Section Grating Laser Diode Gain Section

(a) DFB (b) DBR (c) ECDL

Fig. 1.2: Illustrations of three grating stabilized lasers. Left, distributed feedback (DFB) laser. Middle, distributed Bragg reﬂector (DBR) laser. Right, external cavity diode laser (ECDL) in the Littrow conﬁguration.

These are just the main types of laser stabilization and linewidth-narrowing techniques that are used in the laser industry these days. The most promising results use a narrow- linewidth laser combined with some form of stabilization, such as a high Finesse optical cavity or gas cell stabilization as mentioned before. In this thesis two experiments will be described where a high Finesse whispering gallery mode resonator was used to stabilize a laser.

1.2 Whispering Gallery Modes

The Whispering Gallery in St. Paul’s Cathedral in London, UK already fascinated Lord Rayleigh a century ago. As in other round structures, such as City Hall in San Francisco and St. Peter’s Basilica in the Vatican City, the curved walls have the ability to guide 1.2 Whispering Gallery Modes 5 seemingly quiet sounds along the circumference, thus allowing a listener at the opposite end to vividly hear the whisperer. Lord Rayleigh’s famous paper detailed the physics of it in 1910 [26]. A similar eﬀect in circular shaped dielectrics can lead to an analogous eﬀect in optics.

Fig. 1.3: The Whispering Gallery in St. Paul’s Cathedral, London, UK.

When light in a medium impinges on a boundary with another medium which has a lower refractive index some of the light will be reflected, and some light will be refracted according to Snell’s law. Nevertheless, if the angle between the light beam and the boundary is steep enough, it will undergo total internal reflection. One can imagine a situation in which light inside a circular resonator (such as a sphere or cylinder) can impinge on the surface and be totally internally reflected, and at the right angle this can happen over and over again. If the light constructively interferes after a roundtrip, the light can be trapped inside the resonator. Due to time reversal it may seem quite difficult to get light into a resonator from which it cannot escape, but in fact there is an evanescent field outside of the medium due to the total internal reflection [27] which can be used to couple into the structure. These days, structures of this type are commonly termed whispering gallery mode (WGM) resonators. In fact, around the same time that Lord Rayleigh was describing the Whispering Gallery in London, Gustav Mie was investigating solutions with colloidal structures in- 6 Introduction side them and noticed sharp resonances when light passed through them [28]. Debye was making similar observations concerning light scattering from spheres [29]. A new field of study emerged and various scientists were investigating Mie scattering in different wavelength ranges [30, 31, 32]. The term morphological dependent resonance (MDR) was also used, but has generally been replaced with WGM more recently. Over the last few decades there has been a renewed interest in WGM resonators due to their compact sizes and high quality factors, which are proving to be very powerful for many applications. High quality factors and small mode volumes lead to very high intensities inside WGM resonators, which can be exploited for intensity-dependent applications such as low-threshold nonlinear optics [33]. Current WGM research includes biological sensing [34], optomechanics [35, 36], macroscopical ground-state cooling [37], cavity quantum electrodynamics [38], light storage and optical switching [39], microcavity lasers [40, 41], thresholdless lasers [42], on-chip frequency comb generation [43], and ultra-narrow filtering techniques [44].

(a) Microsphere (b) Crystalline Disk

Fig. 1.4: Schematics of WGM ﬁltering schemes.

The ultra-high quality factors, which originate from the long photon confinement, lead to very narrow filtering possibilities in WGM resonators. Fig. 1.4 illustrates two possibilities to create narrow band filters using WGM resonators. In one case, a tapered fiber is used to couple to a microsphere, and an angle-polished fiber couples the light out of the other side. In the other case two prisms are used to couple to a crystalline WGM disk. The result in each case is that narrow frequencies will be filtered out with quite large free spectral ranges since the resonators are so compact. These are just some of the possibilities that whispering gallery mode resonators offer. The future looks bright, and there will surely be many more surprises and new possibilities that will emerge. 1.3 Frequency Dissemination 7

1.3 Frequency Dissemination

Timing devices usually include something periodic in time that is counted. The most obvious choice is to count days as a means of time keeping. A sun dial may be used to get higher resolution if the position on earth is well known. Later in history clocks were built that rely on a pendulum swinging at Hz rate. Nowdays wrist watches with Quartz oscillators at 32 kHz are even more precise. The main conclusion to draw from these examples is that a higher frequency leads to a higher precision. The second is defined as the hyperfine transition in the cesium atom [45]: the transition is so precise that one can electronically count the oscillations (at a 9 GHz rate) and know with high precision how much time has passed. In fact, more recent developments focus on atomic transitions that have photon energies in the visible and ultraviolet regime. Even if precise time is known in one location, the time comparison of different clocks in different locations can be important. There has been great interest in disseminating frequencies in the last decades. The most common type of time dissemination was radio transmission of clock signals which could be received using clocks that interpreted the radio signals and synchronized to atomic clocks at various institutions. Arguably these days there are even more global positioning service (GPS) receivers than radio clocks, which all receive time information from atomic clocks on GPS satellites. Triangulation allows a receiver to determine its position by comparing the absolute differences of signals from multiple GPS satellites. Different distances from GPS satellites result in offset arrivals of time signals, so the position can be calculated. Many more applications of time dissemination exist and are emerging, for example intercomparison of optical and microwave atomic clocks, searches for variations in fundamental constants [46], very long baseline interferometry (VLBI) in radio astronomy [47], gravity wave searches, and accelerator physics. Precision measurements often require accurate timing, so dissemination of optical clock signals could even allow distant laboratories access to high precision clocks. A more promising candidate for time and frequency dissemination than satellite transfer is optical fiber dissemination. Modulated laser beams can be sent through long distances of fibers and received and used at the other end. Alternatively, even the frequency of a transmission laser itself can be stabilized to a high degree, and thereby allow a remote user to use the frequency in his laboratory, or compare it to one of his own frequency sources. A frequency comb is an octave spanning pulsed laser that has distinct narrow lines in the frequency domain. Fig. 1.5 illustrates a setup in which a frequency comb is 8 Introduction

Optical Telecom Clock Laser

Frequency Comb Stabilized Laser

Fig. 1.5: A frequency comb can be locked to an optical clock signal to stabilize it. Then a telecom laser can be locked to the frequency comb and transferred. locked to an optical clock transmission, and a further laser in the telecom range is locked to another line of the frequency comb. Hence the frequency of the telecom laser can also be stabilized to the atomic transition indirectly, and transferred into a fiber network. At the remote end, the same can be done in reverse to compare the clocks. The accuracy that can be achieved through hundreds of km of fiber supersedes the precision of the current best optical clocks [48, 49], so there is huge potential for future time dissemination through fibers.

1.4 Motivation and Goal

The objective of the work that is presented here is to combine the extraordinary quali- ties of whispering gallery mode resonators with a fiber laser system to stabilize it. There have been previous studies that use the back-reflection from whispering gallery mode resonators to stabilize lasers [50, 51, 52], but we have reason to believe that coupling through the resonator — in one side and out the other — leads to even better linewidth enhancement as compared to the passive linewidth of the resonator. This occurs if multiple roundtrips are made due to line narrowing. Passive linewidth stabilization as described here is advantageous since it can be very cost-effective and more robust. No external locking mechanism is required to stabilize the linewidth. The intra-cavity filtering mechanism will be described and tested. Furthermore, the use of a ring laser configuration in the setup can prevent spatial hole burning, thereby leading to narrower linewidth lasing due to less noise and unwanted 1.5 Outline of the Thesis 9 intra-cavity back-reflections. Two distinct experiments will be described involving an erbium-doped fiber laser setup, and a semiconductor optical amplifier as a gain medium within a fiber cavity. Various parameters need to be optimized to be able to fabricate microspheres, crystalline WGM disks, tapered fibers, and angle-polished fibers. Fine-tuning can lead to exceptional results, so these will be discussed in the following chapters. Ultra-narrow-linewidth lasers become quite difficult to measure and characterize. Com- mon techniques are heterodyne beating with a stable reference laser, and the self-heterodyne technique if no reference laser with lower linewidth is available. Results from these measurements will be demonstrated and explained. For full characterization it is essential to determine the absolute stability of a frequency source. More recently, the three-cornered-hat technique has been applied to narrow linewidth lasers to determine their linewidth and absolute stability, which was previously unknown except in the microwave domain for atomic clocks. We aim to apply this technique to a WGM resonator stabilized laser to evaluate its performance. Finally, the dissemination of stable frequencies through the turbulent atmosphere will be discussed. Previous research has focused on dissemination of radio waves over satellites, and optical frequency transmission through fibers. To our knowledge, we present the first experiments conducted using optical frequencies sent through the atmosphere to determine the phase noise induced. Such a system can be very practical when optical fiber links do not exist over short distances (between adjacent buildings for example). Instead, a telecom laser can be used with very little absorption in atmospheric conditions, and detected using a sophisticated setup. We intend to set up a system where the phase noise induced over a 100 m roundtrip can be measured and characterized.

1.5 Outline of the Thesis

The remaining six chapters of this thesis are divided into four sections: theoretical knowledge that is necessary for comprehension, experimental details that are relevant, separate chapters for the results of three distinct experiments, and finally the conclusion. Chapter 2 begins with a look at the fundamental linewidth limit in lasers, given by the Schawlow-Townes limit. This is followed by a discussion on “bad cavity” lasers, which can overcome this limit. Subsequently, the Allan deviation and its application to laser stability is explained. The chapter continues with the theory of whispering gallery mode 10 Introduction resonators. The basic ray model is described, and later a more in-depth view into the wave model is derived. Important quantities, such as the free spectral range and mode volume in WGMs, are presented. The quality factor of WGM resonators, and coupling conditions and techniques are explained, followed by some intensity-dependent and thermal effects that affect WGMs, and the fundamental limits on them. After presenting some theory on linewidth measuring techniques like the heterodyne beat method and the three-cornered- hat method, the chapter concludes with the theoretical understanding that is necessary to understand frequency dissemination over satellites and through optical fibers. Chapter 3 deals with the experimental details relevant to the thesis. The fabrication of WGM resonators and coupling techniques are described, including microspheres, crystalline resonators, prism coupling, tapered fiber coupling, and angle-polished fiber coupling. Subsequently, the homebuilt external cavity diode laser in the Littrow configuration is presented. Finally, some details on fiber lasers are given. Chapter 4 presents our investigations on an erbium-doped fiber laser, stabilized using a WGM microsphere. It starts with an overview of the setup and design of the laser, and then outlines the quality factor measurement technique for microspheres. The laser was measured using an optical spectrum analyzer, and later using the heterodyne beat technique with a reference laser, the results of which are both analyzed. Ultimately, the pump power and temperature dependence of microsphere laser stabilization is presented. Chapter 5 is devoted to a further experiment involving laser stabilization. In this case, a semiconductor optical amplifier is used as a gain medium, and prism coupling to a crystalline CaF2 resonator is used to stabilize the laser. To begin with, the basic layout is illustrated, followed by the quality factor measurement that was performed. The narrow linewidth was determined using the self-heterodyne method, and finally the novel three- cornered-hat technique results are presented. Chapter 6 focuses on an atmospheric phase noise measurement experiment. Initially, frequency dissemination is introduced. After a description of the experimental setup, the Allan deviation results are shown — including optical frequency transmission, and radio frequency transmission with an amplitude-modulated laser beam. Chapter 7 concludes this work, along with an outlook into future research. Chapter 2

Theoretical Background

This chapter explains the theoretical background and physics relevant to this thesis. Some aspects are discussed in detail, while others are merely mentioned and references to previous work are given, since a complete analysis would be beyond the scope of this thesis. Important fundamentals are described, and multiple references are given for further study of the subject matter. The chapter starts with some background on laser linewidth in section 2.1; including the Schawlow-Townes limit, “bad cavity” lasers, as well as evaluation of stability of a frequency source using the Allan deviation. Subsequently, whispering gallery modes are described using the ray model and the wave model in section 2.2, including environmental eﬀects on modes, coupling to modes, and the quality factor of a resonator. Next, section 2.3 details the heterodyne and self-heterodyne beat techniques used to measure linewidths of lasers. The three-cornered-hat technique is described in section 2.4. Finally, some aspects of frequency dissemination via global positioning satellites and ﬁber networks are discussed in section 2.5.

2.1 Linewidth

Narrow linewidth lasers have applications in many fields, including telecommunications [13], frequency metrology [53], sensors [54], interferometry [18], and gas detection [17]. Many lasers run multi-mode — meaning they run on various longitudinal modes and frequencies simultaneously. It is also possible to fabricate lasers that run on a single frequency, although they actually have a finite width too. There are various possibilities to ensure that a laser runs single-mode, as opposed to multi-mode. Diode lasers typically run multi-mode, unless they either have an additional internal frequency selective ele- 12 Theoretical Background ment (distributed feedback lasers, distributed Bragg reflector lasers), or have some sort of external feedback, from a grating for example. These techniques allow fairly narrow linewidths, but sometimes further stabilization is necessary to prevent drifting of the linewidth. Lasers may be locked to atom transitions in gas cells [21], or, as in this case, to an external cavity. Cavity locking has commonly been used by locking laser lines to Fabry-Perot type cavities [22], and, more recently, to whispering gallery mode resonators. A single-mode laser can be designed by using a gain material with a small bandwidth, as well as a resonator with a large free spectral range — implying a small cavity. On the other hand, if a cavity has a small free spectral range and a large gain bandwidth, many longitudinal modes can lase simultaneously. Whispering gallery mode resonators can be fabricated on the order of tens to hundreds of μm in microspheres, or a few mm for crystalline disks. For example, at 1550 nm a silica microsphere with a 100 μm diameter has a free spectral range of nearly 6 nm, thus offering a good potential for single-mode lasing.

2.1.1 Schawlow-Townes Limit

In most lasers the fundamental limit for the linewidth of a laser due to quantum noise is given by the Schawlow-Townes equation, as published in their famous paper in 1958; two years before the ﬁrst laser operated [5]. If Δν is the laser linewidth at FWHM, hν the photon energy, Δνc the resonator bandwidth at FWHM, and Pout the output power, then

π hν(Δν )2 Δν = c . (2.1) Pout

Often other environmental factors, such as vibrations and temperature fluctuations, will lead to much larger linewidths than those calculated using the Schawlow-Townes limit. The main reason for this finite limit in linewidth is spontaneous emission — essentially quantum noise — in a laser. It is important to note that the fundamental linewidth is only determined by the photon energy, the resonator bandwidth, and the output power. Ignoring increasing noise terms, a higher output power leads to a lower lasing linewidth. An excited atom has a finite lifetime, so either spontaneous emission can occur, or stimulated emission, which is more likely at higher pump and output power. Furthermore, the formula shows that it is beneficial to use a reference cavity within the laser cavity, as opposed to locking externally to a resonance in a cavity, to achieve a very narrow linewidth. One is limited by the transmission bandwidth of the reference cavity if one externally 2.1 Linewidth 13 locks to the reference cavity. If the reference cavity is used inside the laser cavity instead, an even lower lasing linewidth should be possible. Some typical numbers for the factors in the equation from our experiments are a photon energy of 0.8 eV at 1550 nm, a resonator bandwidth of 15 MHz FWHM in a CaF2 disk with a quality factor of 107 (more detail can be found in section 2.2.3), and an output power on the order of 10 μW. This results in a theoretical limit of just 9 Hz. In practice, various linewidth broadening noise terms come into play, leading to a higher linewidth. Various factors can contribute to atomic transition broadening, and hence laser output broadening. They are commonly divided into homogeneous and inhomogeneous broadening. If a broadening mechanism affects both absorbing and radiating atoms in the same way, it is referred to as homogeneously broadened. If the broadening mechanism has a different effect on absorbing and radiating atoms, however, it is inhomogeneously broadened [55]. Depending on the gain material, different broadening mechanisms dominate. For example, gas lasers are mainly affected by Doppler broadening, caused by different velocities of atoms in the gas, which is a type of inhomogeneous broadening. Fiber lasers are essentially a special variety of solid state lasers. In solid gain media, phonon broadening is often the dominant line broadening mechanism [56]. It is a type of homogeneous line broadening, in which high frequency vibrations in the material can affect the atomic transitions and lead to noise much higher than the quantum limit. In fiber lasers, both homogeneous and inhomogeneous broadening can lead to an increase in linewidth, although some commercial systems run quite close to the Schawlow-Townes limit. In a standing wave cavity, such as a Fabry Perot type laser, there are discrete minima and maxima of excited ions. This results in a periodic spatial modulation of the gain coefficient. A grating effect can occur, where some light moving in one direction can be scattered back into the other direction, thus overlapping with a π phase difference with the other light, leading to destructive interference. The coherence of the waves is disturbed, hence favoring multi-mode and broader emission. This effect, known as spatial hole burning, can prohibit single-mode and narrow linewidth emission in lasers. By using a traveling wave cavity — a ring laser for example — this problem can be overcome. A fiber loop or ring laser can be set up, and a directional loss can be implemented to ensure uni-directional lasing. In practice, an isolator is used, which only allows transmission of light in one direction by using a Faraday rotator and polarizing elements. To achieve single-mode emission it is important to either have a laser cavity with widely spaced modes and a narrow gain bandwidth, or some sort of selection of oscillating 14 Theoretical Background modes, by using a frequency selective element. In the later chapters, some experiments are described in which whispering gallery mode resonators are used to select single modes in a much larger fiber laser cavity with many densely spaced modes. Furthermore, the narrow transmission through such a resonator can lead to a very narrow linewidth of the single oscillating mode.

2.1.2 Bad Cavity Lasers

In a conventional laser, the assumption is that the gain bandwidth is much larger than the cavity loss rate. This assumption must be valid for the Schawlow-Townes limit to hold. In a “bad cavity” laser, the opposite is true. Contrary to the name, this actually allows a linewidth lower than that permitted by the Schawlow-Townes limit [57, 58].

The cold cavity loss rate in a Fabry Perot type laser is given by Γ0 = −(c/2L) ln(R1R2).

Here, L is the length of the cavity, c is the speed of light, and R1 and R2 are the reﬂectivities of the end-mirrors. In a loop laser the total loop length can be used, and R1R2 can be the amount of light that is not lost after a complete roundtrip. The Schawlow-Townes limit can be written as a function of Γ0.

hν Γ2 Δν = 0 , (2.2) 4π Pout where Δν is the laser linewidth, hν the photon energy, and Pout the output power. This equation is only valid assuming that the angular frequency gain bandwidth, 2γ = 2/T2,is much larger than the cavity loss rate. A modiﬁed version of this formula, as derived by Lax and Haken [59, 60], includes another factor for bad cavity lasers: ⎛ ⎡ ⎤ ⎞ ⎛ ⎞ ν Γ2 ⎜ ⎢ π ν − ν ⎥2⎟ ⎜ γ ⎟2 h ⎜ ⎢2 ( 0)⎥ ⎟ ⎜ ⎟ Δν = 0 N ⎝⎜1 + ⎣⎢ ⎦⎥ ⎠⎟ × ⎝⎜ ⎠⎟ . (2.3) 4π P sp γ + 1 Γ γ + 1 Γ out 2 0 2 0

Here, Nsp is the spontaneous emission factor equal to N2/(N2 − N1), where N1 and N2 are the populations of the lower and upper levels respectively, and ν − ν0 is the detuning of the lasing mode from the central gain frequency. Therefore, the modiﬁed bad cavity formula includes the linewidth enhancement from complete inversion, detuning from resonance enhancement, as well as the ﬁnal bad cavity factor — which goes to 1 in the good 1 Γ << γ cavity limit when 2 0 . This bad cavity limit also corresponds to the lifetime of the polarization being comparable to or larger than the lifetime of a photon inside the lasing cavity. It is apparent that the quantum-limited linewidth can reduce drastically due to the 2.1 Linewidth 15

γ/ γ + 1 Γ 2 [ ( 2 0)] factor, as shown in the example in Fig. 2.1. A more detailed analysis, ex- plaining the atomic memory eﬀect on the polarization, or an alternative way of looking at it in terms of a reduction of the group velocity, can be found in the paper by Kuppens et al. in Ref. [57].

0.1

0.01

1E-3

1E-4

1E-5

Bad Cavity Linewidth Limit (Hz)Bad Cavity

1E-6 0 50 100 Cavity Loss / Round Trip (%)

Fig. 2.1: Schawlow-Townes linewidth with bad cavity modiﬁcation as a function of cavity loss per roundtrip in percent, assuming conventional Schawlow-Townes limit of 9 Hz as calculated before, and a resonator with a quality factor of 108.Oﬀset of frequency from gain center ν − ν0 is zero, and assume a perfect four-level laser with Nsp equal to one.

Since the eﬀective gain region of a laser can be narrowed by using a whispering gallery mode resonator, it is possible to access the bad cavity regime. Furthermore, losses occur when coupling into and out of whispering gallery mode resonators, as described in sec- 1 Γ >γ tion 4.3, leading to a situation in which 2 0 .

2.1.3 Allan deviation

In the time domain a frequency source, such as an atomic clock or narrow linewidth laser, can be characterized by analyzing how its frequency varies over specific time intervals. By using a frequency counter, which records consecutive central frequency values in an electronic signal, a number of frequency points can be recorded, and statistical analysis 16 Theoretical Background can be performed on them. One might choose to perform common statistics, such as the standard deviation, on the set of data, but this will lead to problems since different kinds of noise, such as short term flicker noise and long term drift noise, diverge and lead to skewed results. In 1966 David W. Allan introduced a modified version of the common variance, commonly referred to as the Allan variance, or the two-sample variance [61]. By using different averaging times in the data, one can deduce the dominant noise processes at short and long time scales, although white frequency noise will lead to the same result as the common standard deviation [62]. The Allan variance is defined as

M−1 2 1 2 σ (τ) = [y + − y ] , (2.4) y 2(M − 1) i 1 i i=1 where σy(τ) is the Allan deviation for averaging time τ, M is the total number of points, and yi is the frequency value at point i. A good oscillator or frequency source will have a low Allan deviation value. After this summation is performed for a set of data, another sum can be calculated for longer averaging times. A new set of points is calculated by averaging two adjacent frequency values, thus resulting in a set with half as many values. Subsequently, four consecutive points will be averaged, then eight, and√ so on. The error or conﬁdence interval for a value is typically estimated to be ±σy(τ)/ N. Typically, the slope of the resulting line for increasing averaging times will be negative. The noise processes can be inferred from the slope of the line. For example, a τ−1 slope implies a white phase noise character, whereas a τ−1/2 implies a white frequency noise character [61, 64]. Plots of Allan deviations are usually on a log-log scale, thus sim- plifying the slope characterization. At some point further averaging might not increase the stability, known as the noise ﬂoor, after which a positive slope may occur due to drift or other types of noise. Fig. 2.2 shows some common noise types and the associated slopes that the Allan deviation graphs will show on a log-log scale [63].

2.2 Whispering Gallery Mode Resonators

The term whispering gallery mode resonator originates from the Whispering Gallery inside St. Paul’s Cathedral in London, UK. Lord Rayleigh published a paper in 1910 describing the curious eﬀect of sound traveling around a curved wall inside the cathedral [26]. In optics a similar eﬀect can occur. In a dielectric with a higher refractive index than its surrounding, say glass in air, a light ray inside the medium can experience total 2.2 Whispering Gallery Mode Resonators 17

Ĳ-1 Ĳ y Ĳ-1

Ĳ-1/2 Ĳ1/2 Ĳ0 $OODQ'HYLDWLRQı

white flicker white flicker random phase phase frequency frequency frequency 7LPH Ĳ noise noise noise noise walk

Fig. 2.2: Slopes of Allan deviations on a log-log scale, indicating the noise types determined by the slope of the line. Adapted from Ref. [63]. internal reflection if the angle is flat enough. If this medium has a circular shape, such as a sphere or a disk, certain frequencies can experience many total internal reflections and can constructively interfere after a complete roundtrip, which is known as a whispering gallery mode resonator. Whispering gallery mode resonators have applications in fields as diverse as biological sensing [34], metrology [43], optomechanics [35, 36], quantum electrodynamics [38], and in our case laser stabilization. Their high quality (Q) factors, which arise from their long photon confinement, make them ideal candidates for compact narrow frequency sources. Following, the intuitive ray model is described, and later the more precise wave model, as well as environmental effects that can perturb modes, coupling mechanisms, and finally the Q factors of resonators are discussed.

2.2.1 Ray Model

In circular shaped dielectric resonators, like spheres and disks, light can be confined around the circumference. Confinement is only possible for specific optical modes. In the limit of very large resonators as compared to the wavelength, a simple geometric 18 Theoretical Background model can be used to describe the resonances, which will be described here [65]. Fig. 2.3 illustrates such a resonator in two dimensions. Consider a ray inside the resonator which travels towards the circumference of the resonator. If the ray has a steep angle as in the top of the left image, it will be partly reflected at the same angle, and partly refracted out according to Snell’s law. Nevertheless, if the ray is incident on the surface at an angle −1 θinc larger than the total internal reflection angle sin n0/ni, all the light will be reflected and confined in the resonator. Due to the symmetry of the system, the ray will repeatedly reflect from the surface at θinc. This occurs if the resonator is circular or eliptical. One can imagine a situation in which NR reflections occur, and the ray once again overlaps with the starting position, as shown in Fig. 2.3b. If the ray is in phase again after such a roundtrip, which depends on the frequency of the light, a standing wave resonance will occur. This is known as a whispering gallery mode.

Į n n 0 0

Ȥ n Ȥ n i i

TIR TIR

(a) Total Internal Reﬂection (b) WGM Resonator

Fig. 2.3: Left, rays inside a resonator can experience total internal reflection at flat angles, hence no light leaks out of the resonator. Right, specific frequencies and certain angles can have constructive interference after a complete roundtrip, so long photon confinement becomes possible.

If NR is very large, the path of a complete roundtrip is close to the circumference of the circle. The phase diﬀerence that is generated after such a roundtrip must be a multiple of 2π. For a resonator with radius a, the phase diﬀerence will be k · 2πa · ni, where k is 2.2 Whispering Gallery Mode Resonators 19 givenby2π divided by the wavelength λ. The condition for resonance is thus π 2 · π · = · π, λ 2 a n l 2

→ n · 2πa = l · λ, (2.5) where l deﬁnes the mode number of the WGM. Higher order modes in a WGM resonator can be located further inside the resonator, so this approximation is only valid for low order modes in large resonators, since they travel very close to the circumference. The spacing of modes can be estimated to be Δν = c/(n · 2πa).

2.2.2 Wave Model

At the beginning of the 20th century, Mie investigated scattering from spherical particles [28]. He discovered that sharp resonances exist, dependent on the diameter of spheres. The sharp resonances were nothing else than whispering gallery modes, also sometimes known as morphology dependent resonances (MDRs). The optical eﬀect is analogous to the sound transfer along curved walls as described by Lord Rayleigh [26]. Here, a brief overview of the wave model for whispering gallery modes is given, and some important relationships and results are described. Since equatorial modes in spheres and disks that are large compared to the wavelength are our primary concern, we will solve the Helmholtz equation in a two-dimensional circle. The important quantities, such as wavelength spacing, in such a 2D model are very similar to those in spheres and disks. First we will start by deriving the Helmholtz equation from Maxwell’s equations. We consider a circular resonator with refractive index n > 1 surrounded by vacuum, with a radius R. The general Maxwell equations can be written as

∇·D = 0, (2.6) ∇·B = 0, (2.7) ∂ ∇×E = − B, (2.8) ∂t ∂ ∇×H = D. (2.9) ∂t

We will assume that the ﬁelds have a harmonic time dependence e−iωt, since any so- 20 Theoretical Background lution can be a Fourier superposition of harmonic functions [66]. This gives a simpliﬁed form of Maxwell’s equations:

∇· E = 0, (2.10) ∇·B = 0, (2.11) ∇×E = iωB, (2.12) ∇×B = −iωμ E. (2.13)

Now we will assume an index of refraction n2 = μ and a wavevector k = ω/c. The last two equations can be combined using the vector identity ∇×∇×A = ∇(∇·A) −∇2A to form the Helmholtz equation ⎧ ⎫ ⎪ ⎪ ⎨⎪E⎬⎪ ∇2 + n2k2 ⎪ ⎪ = 0. (2.14) ⎩B⎭

Let us only examine the electric ﬁeld E, and assume a two-dimensional circle. The equation then simpliﬁes to ∇2 + n2k2 E = 0. (2.15)

We will use the Ansatz of Bessel function solutions to the problem. Here, E< is the electric ﬁeld inside of the circle, and E> is the ﬁeld outside of the circle. The radius of the circular resonator is R, and the distance from the center is r.

< = α · imφ, Ez Jm(nkr) e (2.16) > = β − · imφ + γ + · imφ. Ez Hm(kr) e Hm(kr) e (2.17)

− + Here, Jm is the Bessel function of order m, and Hm and Hm are the Hankel functions (third order Bessel functions) of the incoming and outgoing fields respectively. Since we are only interested in resonances and not the scattering problem that Mie considered, we can ignore the incoming wave term. Since the component of the electric field E that is tangential to an interface of two dielectric media needs to be continuous, the following boundary condition must be met for continuity (the same is true for the H field): < = > , Ez R Ez R (2.18) 2.2 Whispering Gallery Mode Resonators 21 as well as the derivative: ∂ < = ∂ > . nEz R nEz R (2.19) + The Bessel functions Jm are dependent on nkR, and the Hankel functions Hm are dependent on kR. The first four orders for each type of function are shown in Fig. 2.4 and Fig. 2.5. The boundary conditions in Eq. (2.18) and Eq. (2.19) lead to these two relationships:

α = γ + , Jm(nkR) Hm(kR) (2.20) α∂ = γ∂ + . r Jm(nkR) rHm(kR) (2.21)

1.0

0.8 m = 0

0.6 m = 1 m = 2 0.4 m = 3

0.2

(nkR)

J 0.0

-0.2

-0.4

02 4681012 nkR

Fig. 2.4: Bessel functions of the ﬁrst type with m = 0, 1, 2, 3.

We can rewrite these equations in matrix form ⎛ ⎞ ⎛ ⎞ ⎜ J −H+ ⎟ ⎜α⎟ ⎝⎜ m m ⎠⎟ · ⎝⎜ ⎠⎟ = 0, (2.22) ∂ −∂ + γ r Jm rHm

= + = + where Jm Jm(nkR) and Hm Hm(kR). This relationship can only be true if the determi- 22 Theoretical Background

6 m = 3

(kR)

H 4 m = 2

m = 1 2 m = 0

0 012345 kR

Fig. 2.5: Bessel functions of the third type (Hankel functions) with m = 0, 1, 2, 3. nant of the matrix is also equal to zero, so we need to satisfy:

+∂ − ∂ + = . Hm r Jm Jm rHm 0 (2.23)

+ The partial derivatives of Jm and Hm are m ∂ J (nkR) = n · J − (nkR) − J (nkR) , (2.24) r m m 1 nkR m m ∂ H+(kR) = H+ − H+. (2.25) r m m−1 kR m

If we put Eq. (2.24) and Eq. (2.25) into Eq. (2.23), we ﬁnd the condition for resonance, or the dispersion relation:

· + = · + · . Jm(nkR) Hm−1(kR) n Hm(kR) Jm−1(nkR) (2.26)

This has an inﬁnite number of solutions for diﬀerent k values. They can be counted by m, and j values, where j is the jth number of zeros of the Bessel function. This condition is very similar to that derived in detail in Mie theory for spherical resonators [32, 67]. 2.2 Whispering Gallery Mode Resonators 23

The following expression can be derived in a similar manner to that above for a spherical resonator:

Jm+1/2(nkR) · Nm−1/2(kR) = n · Nm+1/2(kR) · Jm−1/2(nkR), (2.27) where Nm−1/2 is the second order Bessel function (also known as the Neumann function) where the order is a half-integral instead. In Ref. [67] a derivation from this expression gives the following approximation for the wavelength separation of fundamental same- polarization whispering gallery modes: √ arctan n2 − 1 Δx = √ , (2.28) n2 − 1 where x = 2πR/λ, and n is the refractive index of the dielectric sphere. The assumptions for this approximation are that x >> 1 (the resonator is large compared to the wavelength), m >> 1 (a large mode number, also implied by the previous expression), and x/m ∼ 1. A more useful form of the equation in terms of the wavelength λ is √ λ arctan n2 − 1 Δλ = · √ . (2.29) 2πR n2 − 1

For resonators that are large compared to the wavelength, this expression is quite accurate for circular, spherical, and cylindrical resonators. In a three-dimensional WGM resonator, such as a microsphere or crystalline disk [68], there are three numbers that define the precise mode that is resonating. This is analogous to the Schrödinger model for electron orbitals in the hydrogen atom [69]. The three parameters are commonly referred to as n, l, and m, which refer to the radial, polar, and azimuthal modes respectively [65]. The number n defines the number of maxima in the radial direction. The number of maxima in the azimuthal direction (around the sphere) is 2l, and the number of maxima in the polar direction is given by l − m + 1. From this relationship it is clear that the fundamental mode is n = 1 and l = m. A fundamental mode for a CaF2 disk resonator is shown in Fig. 2.6a, calculated using the finite element modeling tool “Comsol Multiphysics” [70, 71]. For comparison, a mode with l = m − 1 is shown in Fig. 2.6b, clearly indicating two maxima in the polar direction. The field of modes with n > 1 are generally deeper inside WGM resonators. Long confinement times combined with the extraordinarily small mode volume in whispering gallery mode resonators make them very attractive for many experiments that require high intensities, which would require large and expensive laser systems other- 24 Theoretical Background

(a) n = 1, l = m (b) n = 1, l = m − 1

Fig. 2.6: Fundamental and second order modes calculated using finite element software “Comsol Multiphysics”. The mode number m = 50 in both cases, and l = m in the left image, and l = m − 1 in the right image. The mode number n = 1 in both cases. The simulated diameter is twelve units, and the thickness is three units. wise. These can include ultra-low nonlinear responses in resonators, such as Raman lasing [72] and second-harmonic generation [73]. To calculate the intensities in a resonator, the approximate mode volume in a spherical whispering gallery mode resonator can be calculated by [74]: λ 3 √ V = 3.4π3/2 l11/6 l − m + 1, (2.30) ef f 2πn where Vef f is the effective mode volume, λ is the wavelength, n is the refractive index, and m and l are the mode numbers as defined previously. Let us assume a fundamental mode (l = m) at 1550 nm in a 100 μm diameter fused quartz resonator (n = 1.46). Since the resonator is large we can estimate l = πnd/λ ≈ 300. The resulting mode volume is then about 3 × 10−9 cm3. This is only 0.6% of the total volume of such a sphere (5 × 10−7 cm3). It may seem surprising that the spatial dimensions such as the diameter are not part of the formula, but this is explained by the fact that the l number combined with the wavelength and refractive index are the only values needed to define the geometry. The mode shown in Fig. 2.6 is quite large compared to the resonator, and corresponds to a 0.6 THz mode. An optical mode with l ≈ 300 occupies a much smaller space than is indicated. A top view of a fundamental mode (n = 1 and l = m) in a two-dimensional resonator is shown in Fig. 2.7. It is plotted by calculating Eq. (2.16) and Eq. (2.17), for m = 13, 2.2 Whispering Gallery Mode Resonators 25

Fig. 2.7: Intensity distribution of whispering gallery mode with n = 1, and l = m = 13. The resonator boundary is indicated in white, and the decaying evanescent field outside of the resonator is visible. and the first kR where the dispersion relation in Eq. (2.26) gives a zero. The plot shows the intensity distribution, and there are 26 maxima, corresponding to l = m = 13. The evanescent field of the modes is visible outside of the resonator boundary.

2.2.3 Quality Factor

The properties of a WGM resonator are often summed up by quoting the quality (Q) factor, which directly corresponds to the photon lifetime inside the cavity. This can be 26 Theoretical Background calculated in various ways. It relates to the Finesse by

F = 2 L, Q λ (2.31) here, λ is the wavelength, and L the length of the resonator, which can also be approximated by the circumference of the WGM resonator. In Fabry√ Perot type resonators and lasers the Finesse is often quoted. It is deﬁned as F = π R/(1 − R), where R is the reﬂectivity of the mirrors [75]. Furthermore, it gives the direct relationship between the free spectral range (FSR) and the linewidth of a resonance at FWHM: F = FSR/Δλ, where Δλ is the linewidth. This useful relationship can also be applied to and calculated for whispering gallery mode resonators.

Both Finesse and Q factor are important in WGM applications, but we are mainly concerned with the absolute linewidth of a resonance, so the Q factor is the defining quantity, which is also given by λ ν = 0 = 0 , Q Δλ Δν (2.32) where λ0 is the wavelength of a resonance, Δλ is its linewidth, and ν0 and Δν are the same in terms of optical frequency. For laser stabilization a narrow band filter is important, so a high Q factor of the resonator is desirable. Additionally, a high Finesse is indirectly also important, since a large free spectral range leads to only a few bands that transmit light over the gain bandwidth. Another more intuitive formula that defines the Q factor is [76]:

E 2πν T Q = ω stored = 0 rt , (2.33) Ploss I where ω is the angular frequency, Estored is the energy stored per cycle, Ploss is the power lost per cycle, ν0 is the optical frequency, Trt is the roundtrip time in the cavity, and I is the relative power loss in each roundtrip. From this formula it becomes apparent that the total internal reflection in whispering gallery mode cavities combined with low loss material, such as fused silica or even CaF2 crystal, can lead to long photon confinement, and hence low power loss in each roundtrip. The fundamental modes (n = 1 and l = m as explained in section 2.2.2) have the longest confinement and highest Q factors. A conventional Fabry Perot type cavity is often limited in Q factor by the reflective coatings on the end facets, thus only allowing high Q factors for fairly large cavities. Total internal reflection at a flat interface is theoretically perfect, but there is a slight radiative loss if the surface is strongly curved with respect to the wavelength. In resonators that are large compared to 2.2 Whispering Gallery Mode Resonators 27 the wavelength this is nearly negligible compared to the other loss factors. The following formula shows the total quality factor as a sum of the various types of losses:

−1 = −1 + −1 + −1 + −1 + −1 . Q0 Qabsorption Qscattering Qsurface QWGM Qexternal (2.34)

The first term is the loss due to absorption in the material, which is very low in fused silica and calcium fluoride, but forms the fundamental limit for whispering gallery modes. Scattering from the material itself, as well as surface roughness contribute the largest factor for less photon confinement in materials that have low absorption. Additional absorption due to water diffusion into fused silica can result in microspheres after a few weeks in normal atmosphere. WGM losses are due to the curvature and the resulting loss from total internal reflection. At flat angles in large cavities this is nearly negligible and really only dominates in cavities where the diameter is on the order of a few wavelengths. Further external loss occurs due to coupling using, for example, a prism or a tapered fiber in close proximity to the evanescent field of the whispering gallery modes. The main concern in fabrication of fused silica microspheres is thus the absorption of atmospheric water during the fabrication, as well as any further time the microsphere is kept in normal atmospheric conditions. Due to the surface tension induced surface, scattering losses due to roughness are minimal. The absorption is not an issue in calcium fluoride resonators, but instead the surface roughness must be kept at a minimum to reduce scattering. In most cases this is the limiting factor, although perfect surfaces still have more fundamental limits. Due to the extremely long photon confinement in high Q whispering gallery mode resonators, the ultimate limit arises from nonlinear absorption due to spontaneous and stimulated Raman scattering [77]. The highest values to date have been achieved by mechanical polishing and high temperature annealing of CaF2 disks, resulting in Q factors as high as 1011. This is still three orders of magnitude below the theoretical maximum of nearly 1014 [78, 79].

2.2.4 Coupling

High confinement due to total internal reflection in whispering gallery mode resonators also implies that it is quite difficult to get light into such a resonator. An intuitive approach is to use free space coupling, whereby a laser beam is focused onto a resonator, and some of the light will scatter and couple into the WGMs. This is often used in liquid drop WGM 28 Theoretical Background resonators [40, 41]. Free space coupling is convenient since the only alignment necessary is the focus of a laser beam, but the coupling efficiency can be quite low, often on the order of a few %. The most efficient coupling methods include prism coupling, tapered fiber coupling, and angle-polished fiber coupling, all of which can exceed 90% efficiency, with records around 99%. The principle of the latter three methods is the same. The evanescent field of a light beam is brought into proximity with the evanescent field of the WGMs. If the coupling conditions and the frequency of the light are just right, alignment can be achieved where light is coupled into a whispering gallery mode. After a complete roundtrip there is destructive interference with the light that tries to couple back out into the prism or waveguide.

Tapered Fiber

WGM Resonator

Fig. 2.8: A thin waveguide, such as a tapered ﬁber, coupling to a WGM resonator. Light can couple from the ﬁber into the resonator and vice versa.

In a tapered fiber, a standard telecom single mode fiber is locally tapered down to a few μm in diameter. In this region, the mode is guided by the silica-air interface of the fiber. The diameter is just 1-2 μm, so the initial core of the tapered fiber does not guide the light anymore [80]. The fabrication is described in section 3.1.4. At this point the evanescent field becomes very wide, and it can be overlapped with the evanescent field of the WGM in a microsphere, for an example see Ref. [81]. Some light will now couple into the microsphere, and be confined by total internal reflection. As described in the previous section, if the frequency is correct, it will experience constructive interference after a complete roundtrip thus forming a mode. If the distance between the tapered fiber and WGM resonator is chosen such that the light which couples back out of the resonator into the tapered fiber experiences destructive interference, there is no way for the light to leave the resonator. This is illustrated in Fig. 2.8, where a tapered fiber is in close proximity to a WGM resonator. With proper adjustment the light coupling from the WGM resonator 2.2 Whispering Gallery Mode Resonators 29 back out into the tapered fiber will destructively interfere. Very efficient excitement can result, with almost no transmission at this frequency along the rest of the tapered fiber. There are three distinct situations that can occur in the transmission through the tapered fiber: under-coupling, over-coupling, and critical coupling. In the under-coupled regime, the cavity decay rate exceeds the weak waveguide coupling, which generally occurs when the tapered fiber is placed at a large distance from the resonator. The result is that the losses of the resonator (due to absorption or scattering from surface roughness for example) exceed the coupling from the waveguide. In the over-coupled regime the opposite is the case. Cavity coupling exceeds the cavity decay rate, hence a lot of light is coupled into the cavity, but also out again. This occurs when the tapered fiber is too close to or even touching the WGM resonator. The ideal condition is critical coupling, in which the cavity decay rate is matched to the taper-WGM coupling. In this case the leaking field and the transmitted pump field have equal intensities and thus exhibit perfect destructive interference. In this case no light will be transmitted through the remainder of the fiber.

șc

Fig. 2.9: Light is focused onto the back of the prism at angle θc. The refractive index of the prism is nd, and ne for the WGM resonator.

Prism coupling works in a similar way to tapered fiber coupling. A light beam is fo- 30 Theoretical Background cused and totally internally reflected off the back side of a prism. Total internal reflection induces evanescent tunneling into the exterior medium (air in this case). This field can be overlapped with the evanescent field of the WGMs, and critical coupling can be achieved by varying the prism to resonator distance. It is important to note that there are more free parameters in this case. The angle of the incoming light beam needs to be phase-matched to the WGM, and the beam shape needs to be tailored for maximum overlap with the WGM by adjusting the focus. To match the phase velocities one can assume that the phase velocity of the beam in the prism is c/nd sin θc, and the WGM is c/ne [82]. The angle and refractive indeces are indicated in Fig. 2.9. Phase matching occurs when the two phase velocities are equal:

sin θc = ne/nd. (2.35)

This implies that the refractive index of the prism must be higher than that of the resonator for efficient phase matching. When a Gaussian beam is horizontally incident on an interface at an angle, at the contact point the beam will form an ellipse which is elongated along the horizontal axis. For efficient mode matching the coupling distance between prism and resonator must be matched to the shape of this spot, so this imposes a condition on the shape of the resonator itself. A spherical resonator, such as a microsphere, cannot be highly efficiently coupled using prism coupling since the radii in the azimuthal and the radial axes of the resonator are identical. By definition there will be a mode mismatch. In disk resonators the radius in the azimuthal direction can be chosen independently of the disk radius itself. Since the ellipse is elongated along the horizontal axis, the horizontal radius (radial as opposed to azimuthal) must be larger than the vertical radius of curvature of the disk edge.

Prism r 2R

Fig. 2.10: Side view of a WGM coupled using a prism. The disk radius is R, and the azimuthal (vertical) radius is r.

If the radial and azimuthal radii of the disk are R and r as indicated in Fig. 2.10, and θc is the critical coupling angle as before, the following relationship must be true for optimal 2.2 Whispering Gallery Mode Resonators 31 coupling [82]: r cos θ = . (2.36) c R Angle-polished fiber coupling relies on the same principles as prism coupling [83]. In- stead of using a prism and aligning a beam within it, a fiber is cut and polished at a sharp angle, in accordance with Eq. (2.35), as illustrated in Fig. 2.11. Light traveling along the fiber will then experience total internal reflection, and the evanescent field can be overlapped at critical coupling distance in the same way as in prism coupling. This method is useful since very compact fiber-pigtailed setups are possible. Coupling to high-index resonators, such as lithium niobate (n > 2), is problematic using angle-polished fibers or tapered fibers since high-index fibers are difficult to manufacture. Calcium fluoride disks and fused silica microspheres can be coupled to standard telecom fibers on the other hand. For lithium niobate resonators prisms made of high index materials such as diamond or silicon can be used instead.

Angle-polished Fiber

șc

WGM Resonator

Fig. 2.11: Angle-polished fiber coupling. Light is totally internally reflected from the sharp fiber tip, and the evanescent field couples to the evanescent field of the WGMs. The critical angle θc is indicated.

Eq. (2.35) implies that a sphere made of fused silica cannot be coupled to using an 32 Theoretical Background angle-polished fiber made of the same material. The effective refractive index for a whispering gallery mode is generally less than the refractive index of silica however. This can be understood intuitively since part of the WGM is traveling through air, and part through glass, so the average index is less than that of silica. Ilchenko et. al show in Ref. [83] that the effective refractive index of a WGM (defined as TElmq or TMlmq)isgivenby

cl neﬀ = , (2.37) aωlq where c is the speed of light, l the mode parameter of the WGM, a is the radius of the sphere, and ωlq is deﬁned as the following: nc P 3 ω = ν + 2−1/3α ν1/3 − + 2−2/3 lq a q (n2 − 1)1/2 10 − 2 1/3P(n2 − 2P2/3) × α2ν−1/3 − α ν−2/3 + O(ν−1) . (2.38) q n2 − 1 q

Here, n is the refractive index of the material, P = n for TE modes or P = 1/n for

TM modes, ν = l + 1/2, and αq is the qth root of the Airy function Ai(−z)(2.338 for q = 1). This works out to be an eﬀective refractive index of 1.39 for a 100 μm diameter sphere made of fused silica at 1550 nm. The corresponding critical angle calculated using Eq. (2.35)is71◦.

2.2.5 Intensity Dependence and Thermal Eﬀects

Intensity dependent effects, such as nonlinear optical phenomena, are greatly amplified in whispering gallery mode resonators due to the long confinement and small mode volume of the light field. Even at low pump powers very high intensities can build up, and power- dependent effects can occur at low thresholds. Even in fused silica or CaF2, which both have a very low nonlinear response, the high intensities can lead to various effects. Here, some power-dependent effects in fused silica microspheres will be discussed. If power builds up in a microsphere, there are three main effects that will lead to changes in the WGMs in the sphere. The nonlinear response, specifically the intensity- dependent refractive index due to the χ(3) nonlinearity, leads to an increase in the effective refractive index in a microsphere, which leads to a longer effective path length, and hence a red-shift in a resonance. Also, heating due to absorption of the light in the sphere will lead to a physical increase in the size of the cavity, which in turn will also lead to a red- 2.2 Whispering Gallery Mode Resonators 33 shift of a given resonance. Furthermore, the heating along the path of the WGM will lead to an increase in the refractive index due to the thermo-refractive index — the dependence of the refractive index on temperature in a given medium. All three effects thus lead to an increase in the optical path length, and thereby red-shift the WGM resonances. Nevertheless, the strength of each effect is quite different. The intensity-dependent refractive index, often referred to as the optical Kerr effect, arises from the χ(3) nonlinearity of a medium [84]. The resulting refractive indexn ˜ becomes

n˜ = n0 + n2I, (2.39) where n0 is the normal refractive index, I is the light intensity, and the nonlinear refractive index n2 can be calculated using

3 n = χ(3), (2.40) 2 2 4n0 0c where 0 is the permittivity of free space, and c is the speed of light. Typical values for fused silica are a normal refractive index of 1.46, χ(3) = 2.5×10−22 m2/V2, and a nonlinear −16 2 refractive index n2 of 3.2 × 10 cm /W[84]. For example, a typical circulating power in a WGM resonator might be 10 mW — at critical coupling the internal power would be identical to the pump power. The mode volume in a 100 μm microsphere is 3×10−9 cm3,as calculated in the previous section. The area of the mode can be approximated by dividing by the circumference of the microsphere, which gives 9.5 × 10−8 cm2. The intensity is then 105 kW/cm2. The resulting change in the refractive index works out to be 3 × 10−11, which is almost insignificant. Even though fused silica is highly transmissive at 1550 nm, a small fraction of the power is still absorbed, leading to an increase in the temperature. In a gas medium this leads to a decreased refractive index, but in various types of solid materials, including fused silica, this leads to an increase in the refractive index. If we assume that the temperature dependence of the refractive index in a given material is defined as (d n/ d T), and the temperature induced due to laser radiation is T˜1, then the effective refractive index becomes d n n˜ = n + n I + T˜ . (2.41) 0 2 d T 1

Here, the optical Kerr effect (n2I) is also included from Eq. (2.39). A more useful form for direct comparison of the strength of the effects would be a coefficient that can be multiplied by the intensity as opposed to the induced temperature. According to Sutherland 34 Theoretical Background

[85], this is given by ατ d n n(th) = · , (2.42) 2 ρC d T where α is the absorption in the material, τ is the thermal diffusion time constant in the steady state, ρ is mass density, and C is the specific heat. Let us assume, similar to typical fiber, absorption of 0.2 dB/km, then α ≈ 5 × 10−7 cm−1, τ ≈ 10−4 s[85], the temperature dependence of the refractive index (d n/ d T) ≈ 1.2 × 10−5 K−1 [84], and ρC ≈ 1.76 J/cm3. The effective nonlinear refractive index is then 2 × 10−10 cm2/W. This is six orders of magnitude more prominent than the Kerr nonlinearity. The change in the refractive index will be 2 × 10−5 with 10 mW of circulating power. From Eq. (2.41) we can calculate that the induced temperature is then 1.7 K. The coefficient of thermal expansion in fused silica is about 5.5 × 10−5 K−1 [86]. Assuming that the heat spreads evenly throughout the sphere, we can assume that the path length is changed by this factor for each Kelvin. To compare it to the previous two results, we will assume an induced temperature change of 1.7 K. This increases the path length by 9.4 × 10−5. The change is more severe than the thermal refractive index itself. Each of the three effects leads to a red-shift in the emission since the optical path length is increased by each of them in fused silica. The Kerr nonlinearity is the weakest effect with 3 × 10−11 at 10 mW of circulating power, the thermorefractive index is the much stronger with 2 × 10−5, corresponding to 1.7 K induced temperature. Assuming this temperature spreads evenly and heats the entire sphere by 1.7 K, expansion will increase the path length by 9.4 × 10−5. More realistically, there will be heat dissipation into the surroundings, so overall the thermally induced nonlinear refractive index will be the most prominent, unless the heat dissipation is prevented. On the other hand, if the temperature is externally induced by heating the entire sphere, the thermal expansion leads to a 5.5 × 10−5 K−1 increase in path length, and the thermal refractive index only to a 1.2 × 10−5 K−1 increase. To compare this to some numbers, a WGM at 193 THz (≈ 1550 nm) will decrease by 2.3 GHz due to the thermal refractive index, and over 10 GHz due to the thermal expansion if the ambient temperature increases by 1 K. The need for temperature stabilization of the environment becomes apparent.

2.2.6 Fundamental Limits

The degree of stabilization that can be achieved in a laser depends on the reference cavity that it is locked to. Generally, the higher the quality factor of the cavity, the better the 2.3 Heterodyne Beating 35 frequency reference it is. Fundamental quality factors of WGM resonators have been discussed elsewhere [77, 87, 88]. Before the thermal or even quantum limits [89] are reached however, the main limitation is the stability of the resonator due to environmental vibrations. Let us consider a cavity with size L, located in an environment characterized by the vibrational acceleration Δg. If the mass of the object is M we ﬁnd that

ΔF = M · Δg, (2.43) where ΔF is the force due to the vibration. For simplicity, let us assume M = ρ · A · L, where L is the length of the cavity, and A is the cross-sectional area. The force will also be equal to the change in length of the cavity Δl multiplied by A and Y, the typical stress modulus of the material (e.g. Young’s modulus), and divided by the length L. This leads us to the following: Δl ρ · L3 · Δg = Y · A · . (2.44) L Finally, the change in length divided by the length of the structure itself will be equal to the change in resulting frequency Δν divided by the central frequency ν0, so we ﬁnd that Δν Δl ρ · Δg · L = = . (2.45) ν0 L0 Y

This relationship shows us that the relative frequency stability can be greatly improved by reducing the structure size L. Whispering gallery mode resonators are thus an excellent candidate for high frequency stabilization and reference sources. They can be made very small compared to conventional Fabry-Perot type cavities.

2.3 Heterodyne Beating

Electronic frequency mixing has been used for a long time. A commonly observed fea- ture in acoustics, if two different frequencies are mixed together, a beat will occur at the difference frequency. In optics the heterodyne beat technique is of great importance, since it allows indirect measurements of optical frequencies. Since visible light has frequencies on the order of hundreds of THz it is impossible to detect these directly using electronics. Instead, frequency mixing of two lasers that are offset by a few kHz or MHz allows measurement of their beat frequency, which can be analyzed in various ways. 36 Theoretical Background

The basic principle of heterodyne beating relies on a definite phase relation of the two signals. Then the following trigonometric identity is true, where α and β are different values: 1 1 sin α · sin β = cos (α − β) − cos (α + β). (2.46) 2 2 It is evident that multiplying two sine waves together results in two separable waves with the difference frequency, and the summed frequency. In terms of optical frequencies, the sine terms would be 2π fat and 2π fbt, which also results in cosine terms with the summed and difference frequencies. In optics, the summed frequencies will be on the order of hundreds of THz again, but sufficiently close lasers can be overlapped to produce the cos 2π( fa − fb)t term on a conventional square-law photodetector, such as a photodiode. The beat frequency will thus be fbeat = | fa − fb|. The resulting linewidth of the beat signal will be proportional to the linewidth of each separate laser: Δ fbeat =Δfa +Δfb. Fig. 2.12 shows the amplitude of two slightly offset optical waves, and the resulting intensity as observed on a detector when the two are multiplied. The two frequency components are | fa − fb|, and fa + fb, but the latter is too fast for photodiodes to detect, so only the difference frequency will be recorded. Furthermore, frequency multiplication has the advantage that a very weak signal can be detected, since the output power is proportional to both of the input waves, so a powerful local oscillator combined with a weak signal will lead to a strong signal after all.

Amplitude

Intensity on Detector

0 100 200 300 400 500 0 100 200 300 400 500 Time (fs) Time (fs) (a) Amplitudes (b) Detector Signal

Fig. 2.12: Left, amplitude of two slightly oﬀset optical signals around 193 THz. Right, intensity on detector after frequency mixing. The frequency components are | fa − fb|, and fa + fb.

Now let us examine the behavior of Allan deviations performed on the beats between 2.3 Heterodyne Beating 37 two optical frequencies. From Eq. (2.4) we can obtain the following [90]:

M−1 1 σ2 = σ2 + σ2 − [y − y ] · [y − y ], (2.47) beat a b (M − 1) ai+1 ai bi+1 bi i=1 where σ2 refers to the Allan variance, M is the number of diﬀerences between frequency measurements or averages, yi is the frequency value at point i. The ﬁnal term in the equation is the correlation term, which goes to zero if the lasers are uncorrelated and many measurements are taken. Thus, the Allan variance of the beat of two uncorrelated lasers is simply the sum of the Allan variances of each individual laser, assuming they are uncorrelated. Hence, if a stable narrow-linewidth laser is used as a reference, the resulting linewidth of the heterodyne beat, or the Allan variance calculated from beat data will give an upper limit to the laser being probed. Alternatively two identical lasers can be used, assuming they have the same stability and linewidth, and the Allan variance and linewidth can simply be divided by two. Often two identical lasers are impractical or unavailable, so the next section describes a more common way to determine the absolute linewidth of a laser source.

2.3.1 Self-heterodyne Beating

Sometimes a stable reference laser might not be available to measure a less stable laser linewidth. In this case, one method to measure the linewidth of a laser is the self- heterodyne technique. The principle is that a laser beam is split into two halves, one of which is offset in frequency and delayed in space and time, and later recombined with the other and overlapped to be beat on a detector. If the delay time is sufficiently long, exceeding the coherence time of the laser, it can be assumed to be incoherent and a beat will be produced mimicking the use of two separate lasers. This provides a useful method to measure laser linewidths, if these are significantly below the resolution of an optical spectrum analyzer, and no stable reference laser is available with a similar frequency to use heterodyne beating. The method was published in 1980 by Okoshi et al. [91], and has been used to measure laser linewidths ever since. The interferometric technique relies on the fact that the laser beam has traversed enough distance such that the phase is uncorrelated. This will happen at a distance larger than the coherence length. In a Lorentzian shaped spectrum it is given 38 Theoretical Background by: = c Lcoh πΔν (2.48) where c is the speed of light, and Δν is the linewidth of the laser. For a narrow linewidth laser with a linewidth of 10 kHz the coherence length is thus 9.5 km. Traversing such a distance is impractical in free space, but fibers offer the advantage that they can be contained in a very small space if wound on a coil, and due to the refractive index in fused silica, the distance becomes about 6 km. To be able to beat the two frequencies, the reference which does not traverse a long distance must be shifted in frequency by using an acousto optic modulator (AOM). Such a device uses a high power sinusoidal voltage source which induces a standing wave in a crystal by the acousto optic effect. This standing wave acts as a grating for light, so multiple orders arise after transmission. The zeroth order will remain the same, and hence can be delayed in the long fiber. The first order will be shifted by the frequency that the AOM is running on, since a photon and a phonon combine. The negative first order will be shifted by the same amount in the opposite direction, so the photon looses some energy to a phonon. Part of the beam can be offset in this way, and recombination will result in a heterodyne beat signal as described in the previous section. Fig. 2.13 illustrates the delayed self-heterodyne method.

Delay Spectrum Analyzer

Laser AOM +100 MHz Detector

Fig. 2.13: Delayed self-heterodyne method. A laser is split; one part is delayed in a long ﬁber, the other part is shifted in frequency using an AOM. The two are recombined, and the beat signal is observed on a spectrum analyzer.

As shown by Richter et al. [92], shorter delays will result in a delta function with some side lobes. Some conclusions can be drawn using short delays as well, but using a delay of six times the coherence length will result in a single Lorentzian curve, showing the combined linewidth of the two laser signals. In fiber this corresponds to a delay of 36 km for a 10 kHz laser. Fiber spools with distances on the order of tens and hundreds of km are commercially available. The resulting beat signal will show the actual linewidth√ multiplied by a factor of 2 if the laser has a Lorentzian spectrum, or a factor of 2 if it is Gaussian [91]. As a matter 2.4 Three-Cornered Hat Measurement 39 of fact, the factor will be somewhere in-between these two values in most lasers. It is important to note that vibrations and temperature fluctuations in fibers can lead to excess noise. Long links are especially prone to phase noise fluctuations induced by environmental factors, so problems can arise if extremely narrow lasers are to be measured. The measured linewidth will be an upper limit to the actual linewidth of the laser. Furthermore, the self-heterodyne technique only measures the linewidth of a source. The stability can not be investigated using this technique. In a delay of 36 km, any drift that is slower than the roundtrip time of the light through the fiber (120 μs) will cancel out. The beat will always be right at the frequency of the AOM, since both the delayed signal and the reference signal will drift simultaneously by definition. Upper limit estimates can be made of the narrow linewidths of lasers in this way.

2.4 Three-Cornered Hat Measurement

A suitable solution for absolute frequency source characterization is the three-cornered hat technique. To perform an Allan deviation calculation, one needs to compare the oscillator in question with some sort of reference source. If the reference source is much more stable than the oscillator to be tested, one can assume that the resulting Allan deviation is very close to that of the oscillator. On the other hand, if comparable sources, such as two atomic clocks, are compared, one cannot be sure which source leads to a speciﬁc Allan deviation value. In 1974 Gray and Allan presented a method that has since been commonly used to determine the absolute stability of an individual oscillator [90], referred to as the three-cornered hat method. The concept is quite straight-forward: if three uncorrelated oscillators are available, they can each be compared with each other simultaneously, and the individual stability of each can be calculated, assuming they have similar performance. This method has been applied to microwave and other electronic frequency sources for decades. More recently, various groups have explored its potential in determining the absolute stability of laser sources as well. In 2006, López et al. measured the linewidth and stability of three diode lasers by producing three optical beats between them at electronic frequencies and comparing these [93]. In 2009, Liu et al. [94] and Zhao et al. [95] published papers detailing the three-cornered hat method as applied to ultrahigh stability reference cavities for use in optical clocks. Let us assume we have three uncorrelated frequency sources a, b, and c. Analogous 40 Theoretical Background to Eq. (2.47), we can ﬁnd three equations for the beats assuming they are all uncorrelated:

σ2 = σ2 + σ2, ab a b σ2 = σ2 + σ2, ac a c σ2 = σ2 + σ2, bc b c (2.49)

σ2 where ab refers to the Allan variance obtained from the beat between sources a and b, σ2 and a is the Allan variance of the individual source a. By rearranging these equations to obtain values for the separate Allan variances we get a new set of equations: σ2 = 1 σ2 + σ2 − σ2 , a 2 ab ac bc σ2 = 1 σ2 + σ2 − σ2 , b 2 ab bc ac 1 σ2 = σ2 + σ2 − σ2 . (2.50) c 2 ac bc ab

If three lasers are offsetbyaafewMHzinfrequency, three beats can be generated between each combination of lasers. Alternatively, three electronic frequency sources can be combined by detuning them by a few kHz or MHz and electronically mixing them. The Allan variances can be calculated at different averaging times from the beats as described before, and separated Allan variance values can be figured out using Eq. (2.50).

In some cases the three sources may not have similar stability. As mentioned in√ section 2.1.3, the error or conﬁdence interval for an Allan deviation is given by ±σy(τ)/ N, so it is proportional to the Allan deviation itself. From Eq. (2.50) it becomes apparent that a larger Allan deviation and hence error in one oscillator, e.g. oscillator a, will lead σ2 σ2 to large error in ab and ac. Since the separated Allan deviations for b and c are compar- atively small, their errors will be large due to the error induced by laser a. On the other hand, if lasers b and c are seen as stable reference lasers, the Allan deviation of laser a can be computed to a high degree of accuracy. Furthermore, in some cases negative values can result for the more stable lasers due to the relatively large uncertainty in laser a.

2.4.1 Correlation Removal

Sometimes three oscillators cannot be assumed to be uncorrelated. Premoli et al. proposed a method to remove the correlations between lasers from their beats by calculating a covariance matrix [96]. In their method, an arbitrary source can be chosen as a ref- 2.4 Three-Cornered Hat Measurement 41 erence and is then compared to the other two sources and combined into a covariance matrix. Here, this method is brieﬂy described. The same notation as in their paper is used for simplicity. The reference oscillator is laser 3, and it is compared with lasers 1 and 2. The 2 × 2 covariance matrix S contains the following known elements ⎡ ⎤ ⎢s s ⎥ S = ⎣⎢ 11 12⎦⎥ . (2.51) s21 s22

= σ2 = σ2 Here, s11 13, and s22 23. The other two elements are the correlation calculations — a modiﬁed version of the Allan variance calculation as given in Eq. (2.4):

M−1 1 s = s = σ (τ)σ (τ) = [y , + − y , ][y , + − y , ], (2.52) 12 21 13 23 2(M − 1) 13 i 1 13 i 23 i 1 23 i i=1 where σ13(τ)σ23(τ) is the correlated Allan deviation for averaging time τ for oscillator beats 13 and 23, M is the total number of points, y13,i is the frequency value at point i for the beat between 1 and 3. Let R be the 3 × 3 covariance matrix with the unknown

(co)variances r11, r22, r33, r12, r13, and r23: ⎡ ⎤ ⎢ ⎥ ⎢r11 r12 r13⎥ ⎢ ⎥ R = ⎢r r r ⎥ , (2.53) ⎣⎢ 12 22 23⎦⎥ r13 r23 r33 where r11, r22, and r33 are the Allan variances of the three oscillators with correlation eﬀects removed, and r12, r13, and r23 are the correlations between the cavity pairs. The covariance matrix S can also be written ⎡ ⎤ ⎢ r + r − 2r r + r − r − r ⎥ S = ⎣⎢ 11 33 13 12 33 13 23⎦⎥ . (2.54) r12 + r33 − r13 − r23 r22 + r33 − 2r23

The elements of the matrix can be rewritten to give the following three equations:

r11 = s11 − r33 + 2r13,

r12 = s12 − r33 + r13 + r23,

r22 = s22 − r33 + 2r23. (2.55)

The matrices S and R are both positive definite, so various conditions must be fulfilled 42 Theoretical Background by their respective elements in order to satisfy this criterion. The detailed calculation is beyond the scope of this thesis, but it can be found in [96]. Here, the necessary calculations are shown to find r13, r23, and r33. The following values are defined: c = 3 |S|s (s − s )(s − s ), 1 12 11 12 22 12 2 c2 = 2.25|S| + 2(s11 + s22 + s12)c1/(3 |S|), 3/2 c3 = 3|S| (s11 + s22) + c1/3,

c4 = |S| [1.5|S| + (s11 + s22 − s12)(s11 + s22 + s12)] , 3/2 c5 = |S| (s11 + s22), 2 c6 = |S| /4. (2.56)

These are the coeﬃcients of the sixth order equation

2 3 4 5 6 c1 f + c2 f + c3 f + c4 f + c5 f + c6 f = 0, (2.57)

for the unknown f . The unique minimum positive root fmin is chosen if it exists, if not the further calculations will be the same as the classical three-cornered hat method, with

fmin = 0. We define f = fmin to find the coefficients of a second order equation: b = |S| + 2(s + s − s ) f + 3 |S| f 2, 2 11 22 12 b = − |S|s − (2s2 + 3|S|/2) f − |S|(s + s ) f 2 −|S| f 3/2, 1 12 12 11 22 = | | 2 + 2 + + | | 2 2. b0 S s12 s12(s11 s22) f S s12 f (2.58)

Subsequently, the following four parameters are calculated a = 2 |S| + fs , 20 22 a = 2 |S| + fs , 02 11 a = |S|− fs , 11 12 a10 = a01 = |S|(2r33 + s12). (2.59) 2.5 Frequency Dissemination 43

Then the free parameters r13, r23, and r33 are found to be

r33 = −b1/b2, a (a − a ) r = r − 10 02 11 , 13 33 − 2 a20a02 a11 a (a − a ) r = r − 10 20 11 . (2.60) 23 33 − 2 a20a02 a11

Finally, from Eq. (2.55) the parameters r11, r12, and r22 can be calculated. The diagonal elements of matrix R give the Allan variances of oscillators 1, 2, and 3. Their respective Allan deviations are calculated by: √ σx = rxx. (2.61)

In principle, the choice of the reference cavity is arbitrary and correlation removed results should be equal. Nevertheless, if the errors in the Allan deviations of the beats are very large, as previously explained, this can lead to skewed results, even after correlation removal. The technique can be very powerful for measurements where three lasers are subject to the same environmental ﬂuctuations, such as long-term temperature drifts. Even though the calculations look complicated, they can easily be implemented into computer code, where raw frequency data from three beats can be input, and Allan deviations, correlations and so on can be calculated from them.

2.5 Frequency Dissemination

Precise dissemination of frequencies has been of interest for decades. In the 1980s, com- panies began selling clocks that synchronized to radio signals sent by stations with accurate atomic clocks [97]. This type of technology is only accurate to about 1 s. Nowdays, the most prominent frequency dissemination comes from the atomic clocks on the global positioning (GPS) satellites. Each satellite, at a height of over 20 000 km contains a very precise cesium atomic clock, which is referenced to a ground station. If multiple satellites send out a time signal (at frequencies of 1575.42 MHz or 1227.6 MHz), at any position where these signals are detectable, a receiver can determine its distance to each satellite by calculating the oﬀset of each clock due to the limited speed of light, and hence calculate the precise position in three dimensions. Furthermore, the GPS system is also used to disseminate and compare precise frequencies. More recently, optical atomic clocks 44 Theoretical Background have superseded the cesium atomic clock standard, so new dissemination techniques are needed. Here, the GPS method, as well as dissemination through ﬁbers will be described.

2.5.1 Satellite Transfer

In 1980, only a few years after the launch of the GPS satellites, Allan and Weiss published their paper on common-view time transfer using GPS satellites [98], which provided the possibility for time comparison with 10 ns accuracy. The principle is that two receivers track the same satellite at the same time, which limits the maximum distance across the earth, unless repeaters are used. One assumes that path-delay corrections are constant for both, as well as any error in the time of transmission from the satellite to the receiver. A receiver is then compared to a master clock — the reference ground station clock — then the error in transmission time can be transferred via conventional radio or internet link to the second receiver. Distances of thousands of km are thus possible. Common-view time transfer is a passive method, but active methods, in which a receiver also sends a signal to a satellite, have shown to be more accurate. Both one-way transfer, as well as two-way transfer have shown promising results [99]. In two-way satellite time and frequency transfer (TWSTFT) two institutes simultaneously transmit signals via a geostationary telecommunication satellite. Hence each facility requires a transmitter and a receiver. Uncertainties of less than 1 ns are common using this technique. Instead of using the time codes that GPS satellites emit, it is also possible to use the carrier itself as a means of transferring frequencies [100]. The coded time signals that are sent by GPS satellites are only at 1023 kHz, but the carrier frequency itself, as mentioned before, is about a factor of one thousand higher than that. A higher carrier frequency is better for encoding a frequency signal, hence its direct phase is used for the most accurate frequency dissemination using satellites. Time uncertainty of just 100 ps is possible, and frequency with a fractional uncertainty of just 10−15 after a day of averaging has been shown [101]. The best cesium fountain clocks have a stability of 5 × 10−16 [102]. Given a few days of averaging, GPS phase transfer can just about be used for dissemination of this kind of accuracy, or for comparison with such a clock. Current work is focusing on new frequency standards in the optical and ultraviolet regions however, where much higher carrier frequencies are prevalent. Recently, scientists from the National Institute of Standards and Technology (NIST) have demonstrated a clock based on optical transitions in Al+ with 8.6 × 10−18 stability [48]. To compare the various atoms and ions that are being used to 2.5 Frequency Dissemination 45 improve and replace the current cesium frequency standard, more accurate dissemination techniques are required.

2.5.2 Fiber Transfer

Fiber networks can be found all over the world these days. Single-mode telecommunication grade fibers exist between most major cities, countries, and even through all major oceans. They are the backbone of the internet, and allow huge amounts of data to be transferred from one end of the earth to the other at the speed of light. Research groups are exploring these fiber networks for their potential to disseminate frequencies at much higher accuracies than imaginable over GPS satellites. One of the first methods that was explored for microwave frequency dissemination was to amplitude modulate a laser at the microwave frequency. This could simply be sent through the fiber, and received at the other end using some sort of photodetector. Results using this technique show accuracies that are better than GPS phase dissemination, even at shorter time scales. With the advent of frequency combs however, new possibilities emerged [103]. The frequency of a laser itself can be stabilized to a frequency reference and then transferred. A frequency comb can be locked to a stable reference, such as an optical clock, and a further telecom laser at 1550 nm can be locked to another line of the frequency comb. Thus, the frequency of the telecom laser is essentially as good as the optical clock, so its frequency can be transferred through fiber. At the other end, another frequency comb can be used to measure the exact frequency of the laser, so precise frequency information can be disseminated using a carrier in the optical or infrared region [104]. Furthermore, this offers the possibility of direct comparison of optical clocks using two frequency combs, since optical clocks often take up entire scientific laboratories and are currently not easily portable. Unfortunately, various noise mechanisms in fibers can lead to phase fluctuations, which in turn degrade the accuracy of the transferred frequency. In 1994 Ma et al. published a paper that described a phase compensation scheme, which is implemented regu- larly today [105]. The idea is that a signal will accumulate some phase noise φ f when it is transferred through a fiber. At the remote end, a small fraction of the transmitted signal is reflected back to the sender, where it will have accumulated twice the phase noise

φrt = 2 × φ f . If a small portion of the outgoing signal at the sender is used as a local oscillator with phase φlo, it can be used to compare the phases between the local oscilla- 46 Theoretical Background

tor and the reflected φrt. In practice, an acousto-optic modulator (AOM) can be used to modify the outgoing phase such that the difference between φrt/2 and φlo is zero using a phase-locked loop. In experiments using fibers, application of this method commonly improves the accuracy of transmission by two orders of magnitude. Recent experiments have shown stabilities down to 3 × 10−15 at 1 s averaging, and down to 1 × 10−19 after 30 000 s in a 146 km fiber link [49]. The accuracy that GPS delivers with a day of averaging can thus be achieved in just 1 s using fibers, and accuracies well below the best optical clocks can be reached in fewer than nine hours. Free space frequency dissemination relies on the same principle as fiber frequency transfer. Either modulated carriers can be used, or, for more accuracy, an optical frequency itself can be sent. Atmospheric conditions like temperature drifts and pressure variations lead to more induced phase noise than in fibers, but for shorter distances it is feasible, using quite a similar setup to fiber frequency transfer. Chapter 3

Experimental Details

In this chapter, common experimental details that are relevant to later chapters are explained. Section 3.1 describes the fabrication of various whispering gallery mode resonators and tools. First, the fabrication of microspheres is explained in section 3.1.1, followed by section 3.1.2 which explains how CaF2 crystalline disks are polished. The various coupling mechanisms — prism coupling, tapered fiber coupling, and angle-polished fiber coupling — are described in sections 3.1.3, 3.1.4, and 3.1.5 respectively. Subse- quently, section 3.2 outlines the Littrow grating stabilized diode laser that was built for precision measurements. Finally, the fabrication of fiber lasers is discussed in section 3.3.

3.1 Whispering Gallery Mode Resonators

The most common types of whispering gallery mode resonators are microspheres [74], rare-earth doped microspheres [106], crystalline microdisks [107, 44], on-chip microdisks and toroids [108, 109], bottleneck resonators [110], sol-gel infused silica spheres[111], and liquid drops [112]. In our experiments, the first two types were fabricated and investigated. Coupling methods include free-space coupling, prism coupling, tapered fiber coupling, and angle-polished fiber coupling. Free-space coupling is useful for liquid microdrop excitation [40], but the efficiency is quite low. For solid resonators it is possible to get very close to the WGM resonators with prisms, tapered fibers, or angle-polished fibers. Coupling efficiencies with these methods can exceed 90%. These three methods were used in these experiments and will be discussed in this section. 48 Experimental Details

3.1.1 Microspheres

A common method for making whispering gallery mode resonators with small diameters is the microsphere fabrication technique [74]. Resulting diameters are around 50 to 150 μm, and Q factors can reach up to 109 in fused silica.

x Microsphere

ZnSe Lens

CO2 Laser

(a) Schematic (b) Top view

Fig. 3.1: CO2 laser microsphere fabrication setup. The approximate beam path is indicated for clarity. Three mirrors steer the beam in the upward direction onto a ZnSe lens with a focal length of 50 mm. Objects can be placed into or near the beam focus using an x − y − z stage.

First, we set up a CO2 laser (Synrad J48-1SW) at 10.6 μm such that mirrors and a lens focused the beam onto the same position as the center and focus of a microscope. This allowed us to position objects in the focus of the laser in a controlled way with a precision ofafewμm. The power could be adjusted and was usually set to 1 to 2 W. Fig. 3.1 shows a photograph of the setup. Next, stripped SMF-28 fibers were put into a holder on the left side, thus holding it slightly above the focus of the laser horizontally. By gently pulling on the right side with tweezers, the section above the beam focus was tapered from a 125 μm diameter down to around 10 μm. Next, the fiber was shifted to the right, thus allowing a quick laser burst to induce a right angled kink in the fiber. The tapered section, as well as the thicker chunk at the end were then vertical. Careful adjustment of the position and laser power permitted precise slow or quick melting of the chunk at the end of the tapered section. A sphere of liquid glass was created due to surface tension, which quickly hardened when the laser was turned off. The result was a sphere with a diameter of about 50 to 150 μm, determined by the positioning near the focus of the laser, as well as the power of the laser. Fig. 3.2 shows a microscope image of such a sphere. The highest Q factors we attained were on the order of 108. In literature [113], the 3.1 Whispering Gallery Mode Resonators 49

Fig. 3.2: A typical microsphere with a diameter of about 100 μm, and a stem of 20 μm. A tapered ﬁber can be seen behind the sphere.

9 highest Q factors are on the order of 10 . These are generally achieved by using a N2 filled environment during melting of the spheres. Atmospheric water can diffuse into silica spheres in standard air, by which the surface along which the WGMs travel is degraded. More absorption of light at the surface leads to a declination in the Q factor. In practice, we see this after some weeks in the lab. The Q factor can drop down to 107 if it has been in an air environment. To combat this, experiments were conducted inside of a transparent box with an N2 purge. Future experiments should include a new CO2 laser microsphere fabrication setup which involves an N2 environment during fabrication, as well as post- fabrication. As illustrated in Fig. 3.3, a thermoelectric cooler (Peltier element) was used to control and stabilize the temperature of microspheres. The thermorefractive index, as well as the physical expansion of the microcavity itself can lead to a shift in resonances, so stabilization is vital. The temperature coefficient of the refractive index for fused silica is 1.2 × 10−5 [84], and is a prominent effect in shifting of the resonance, as further described in Chapter 2. Although there was no direct contact between the thermoelectric cooler and the microsphere, the low heat conductivity of fused silica still allowed an improvement of 50 Experimental Details

Thermistor Copper Tube Microsphere

PID Copper controller Block TEC

Heat sink (a) Schematic (b) Photograph

Fig. 3.3: A thermoelectric cooler controls and stabilizes the microsphere temperature. A copper tube holds the stem of the microsphere using heat-conducting paste. long-term temperature drifts in the lab.

3.1.2 Crystalline Resonators

As of yet, calcium ﬂuoride (CaF2) resonators have been shown to have the highest intrinsic Q factors of all whispering gallery mode resonators. The highest reported to date is around 11 10 [107, 44]. In CaF2, Rayleigh scattering of inhomogeneities on the surface limits the Q factor, although for near-perfect surfaces the absorption becomes signiﬁcant after millions of roundtrips. In general, a special lathe can be used to cut and then polish crystalline resonators, resulting in a disk shaped resonator. The size of the resonators is another advantage over fused silica microspheres. Crystalline disks with diameters between tens of μm up to 10 mm can be manufactured [82]. WGM resonators on the order of mm are more rigid since they are not suspended using 10 μm stems as in microspheres. We used a motor with an air-bearing with a maximum speed of 6000 rpm. Rough disk shapes with diameters around 5 mm were cut from 2 mm thick high-grade UV windows made of CaF2 using a hollow drill. These were glued onto screw tops with a slightly smaller diameter, and then inserted into the motor. While spinning the motor around the maximum speed, a sharp diamond tip was moved using an x − y stage to cut a desired shape out of the crystal. The setup is shown in Fig. 3.4. Subsequently, the crystal edge was polished while still spinning using various diamond suspensions, starting at 9 μm, and moving down to sub-μm size grains. This resulted in surface roughness far below 1550 nm — the wavelength of the light used in the experiments — as shown in the scanning electron micrograph in Fig. 3.5. It was vital to remove any excess dust and other small particles by cleaning the surface carefully using acetone and methanol. If this was 3.1 Whispering Gallery Mode Resonators 51

Fig. 3.4: The crystal disk polishing setup includes an air-bearing motor to hold and spin the resonator, as well as a motorized x−y stage to cut a speciﬁed shape into the disk using a sharp diamond tip. not done, dust would cause scattering losses, leading to a degradation in the Q factor.

3.1.3 Prism Coupling

A common method to couple light into WGM resonators is prism coupling. A light beam is focused onto the back of a prism at an angle such that it is totally internally reflected. This results in an exponentially decaying evanescent field outside of the prism as described in Chapter 2. If the coupling conditions are just right, efficient energy exchange can occur. There are many degrees of freedom in prism coupling, making it challenging. The modal overlap between the totally internally reflected incoming beam and the whispering gallery mode needs to be optimized to ensure efficient coupling. Coupling becomes possible when the refractive index of the prism is higher than that of the WGM resonator. In this case the resonator is made of CaF2, which has a refractive index around 1.46 at 1550 nm. To ensure that coupling is feasible, we used a right-angled prism made of SFL 11 glass, which has a refractive index around 1.74. Fig. 3.6 illustrates 52 Experimental Details

Fig. 3.5: Scanning electron micrograph indicating the typical surface roughness of a CaF2 resonator. The grooves are below 100 nm in depth and diameter. our setup as well as a schematic showing the critical angle calculated using

ne θc = arcsin , (3.1) nd in which θc is the critical angle, ne is the refractive index of the WGM resonator, and nd is the refractive index of the prism. This leads to a critical angle of 32.9◦. A compact and convenient method to focus light onto the back of a prism is to use graded refractive index (GRIN) lenses, in which a small cylinder has an exponentially decaying refractive index from the center to the outside. This results in a longer path length for the light traveling in the center, hence a lensing effect can focus light that exits an SMF-28 fiber a few mm after the GRIN lens. We mounted the fiber and GRIN lens onto an x−y−z stage including the possibility to tilt x and y. To get a good first estimate for some of these parameters in the setup, we used a red 635 nm diode laser for alignment. By placing a camera above the disk and prism it was possible to adjust the focus of the GRIN lens, as well as the x-position and x-angle of the incoming light beam as shown in Fig. 3.6c. Prism coupling mainly excites the outer modes of WGM resonators, but inner modes 3.1 Whispering Gallery Mode Resonators 53

SFL11 șc

CaF2

(a) Schematic (b) Photograph (c) Top view

Fig. 3.6: Prism coupling using SFL 11 glass and a CaF2 resonator.

can also be coupled to in a spherical or thick resonator. To prohibit higher order transverse modes from being excited, the ideal radius of curvature for a single propagating mode in a disk resonator may be calculated by r n 2 = cos2 θ = 1 − sin2 θ = 1 − e , (3.2) R nd where r is the radius of curvature, R is the radius of the disk, θ is the coupling angle, ne is the refractive index of the WGM resonator, and nd is the refractive index of the prism. For a disk with a radius of 3 mm this works out to be approximately 1 mm. We attempted to achieve this value as accurately as possible in the fabrication of the resonator. For initial alignment in prism coupling we let the prism lightly touch the resonator to achieve over-coupling using red light. Since the prisms were mounted on piezo stages (AttoCube ANPx101), it was possible to slowly remove them when coupling in 1550 nm light while adjusting the other degrees of freedom to optimize coupling. The minimum step size was determined to be just under 50 nm, so precise position control was possible. To find a resonance, an external cavity diode laser (homebuilt as well as a Toptica DL Pro 1550) was tuned over a few GHz, while the reflection from the prism surface was monitored on a photodiode and analyzed using an oscilloscope. More details on this can be found in Chapter 5. On resonance, a dip in the reflected power would become apparent and could be optimized. The seven free parameters in the adjustment were the x, y, and z positions, x and y angles, the focus of the GRIN lens, and the distance between the prism and the WGM disk. Accurate alignment of all of these resulted in a precise modal overlap between the incoming light and the whispering gallery mode. 54 Experimental Details

3.1.4 Tapered Fiber Coupling

The principle of tapered fiber coupling is similar to prism coupling. An SMF-28 fiber is molten down from a diameter of 125 μmto1-2μm. The higher refractive index 8 μm core doped with germanium melts together with the outer cladding, but light can still be guided due to the refractive index difference between silica and air. Since the fiber now has a large evanescent field, it can be overlapped with the evanescent field of the whispering gallery modes of a resonator. If the coupling conditions are just right, a large portion of the incoming light can be coupled into a resonator. For a theoretical description please refer to Chapter 2. Fiber tapers were produced by stripping and cleaning a few cm of fiber and placing the fiber into clamps as seen in Fig. 3.7. An oxygen-butane gas flame was moved towards the fiber using a motorized x − y stage until it was adjacent to the fiber. Subsequently, the flame oscillated along 10 mm of the fiber at 1 Hz. Simultaneously, the fiber was symmetrically pulled apart at about 3 mm/s using another motor which pulled the clamps apart. The resulting fiber had an adiabatic profile and could guide over 90% of the light, although in practise 70% was common.

Slow pulling Slow pulling

Fiber Detector

Flame shifts Laser side to side

(a) Schematic (b) Photograph

Fig. 3.7: Fiber taper being held by clamps with gas ﬂame shifting left to right.

A diode laser at the functioning wavelength was used to monitor the throughput of the taper throughout the production. This ensured that the resulting taper did not support resonances. As the length of the fiber increased, the amplitude fluctuations of the transmission and their frequency increased as shown in Fig. 3.8. The cause of these fluctuations is the transition between guidance of the refractive index difference between core and cladding, to guidance by total internal reflection of the taper and air. In conventional single-mode

ﬁber, the only permissible hybrid mode is HE11. In contrast, the tapered ﬁber — which can be modeled as a single material cylinder in air — guides the HE11 mode, as well as 3.1 Whispering Gallery Mode Resonators 55

the HE12 mode, which has a minimum in the center of the fiber. There is a continuous energy exchange between these two modes, which can lead to a loss in transmission at the transition between taper to conventional fiber if the HE12 contains most of the energy at this point [114, 115]. In a perfectly adiabatic transition the HE12 should completely converge into the fundamental single mode. After a few mm of pulling, the fluctuations suddenly stopped, so the flame was quickly removed. The taper could now be glued onto a holder, or the clamps themselves could be moved to function in further experiments.

Fig. 3.8: Signal on detector after transmission through the ﬁber while it is being tapered. Clear resonances are visible, until they subside when the diameter is narrow enough — about 1-2 μm.

The large evanescent field outside of the waveguide meant it was prone to degradation. Dust and other small particles on the taper cause a significant decline in transmitted light due to scattering. Tapered fibers have to be replaced after a few weeks or months because of this. Furthermore, the 1-2 μm core is very delicate, so too much pressure or tension causes the fiber to break.

Critical coupling could only be achieved with precise alignment of the fiber next to the WGM resonator. In general, the taper was fixed, and the resonator was moved into the rough position using micrometer screws, and then finely adjusted using a piezo stage. If the taper was slightly loose, it would be attracted and stick to the microsphere due to electrostatic force. Careful control of the tension on the fiber and the positioning of the microsphere could prohibit this. 56 Experimental Details

3.1.5 Angle-Polished Fiber Coupling

Analogous to prism-coupling, angle-polished fibers exploit total internal reflection of light and the resulting tunneling of evanescent field components. The end of an SMF-28 fiber is polished at a sharp angle, thus inducing total internal reflection of the light guided along the fiber. By bringing the central portion close to a WGM resonator, the evanescent field outside of the fiber can overlap with the evanescent field of the modes and thus couple efficiently. The ease of use and compact size are advantageous. The angle needed for efficient coupling, as derived in [116], is given by

nsphere φ = arcsin , (3.3) nfiber where nfiber is the refractive index of the fiber, and nsphere is the effective refractive index describing azimuthally propagating whispering gallery modes. This is analogous to Eq. (3.1) known from prism coupling. The refractive index of SMF-28 fiber is 1.469, and the effective refractive index for a q = 1 mode in a 100 μm diameter sphere is about 1.39 at 1550 nm. More detail on the effective refractive index of a WGM can be found in section 2.2.4. This leads to a critical angle of 71◦. We polished multiple fibers simultaneously using a holder designed at the required angle as shown in Fig. 3.9. Stripped and cleaned fiber ends were glued into grooves inside the holder using mounting wax. This 100 mm diameter cylinder was placed upside down onto a polishing wheel, on which we applied various polishing solutions, ranging from a few μmAl2O3 suspension, to sub-μm suspension. This ensured fiber tips with very little surface roughness below the wavelength of light guided through it (see Fig.3.10). Before use, angle-polished fibers were inserted into thin glass tubes with a wider outer diameter for stability and simple handling. A piezo stage and micrometer screws were used to accurately position the fiber to ensure optimal coupling to microspheres. Dust on the surface of an angle-polished fiber was less problematic than on a tapered fiber due to its robustness, allowing it to be cleaned and sonicated after fabrication.

3.2 Littrow Grating Stabilized Diode Laser

Measuring the quality factor of whispering gallery mode resonators is essential to evaluate their performance and degradation in time due to dust accumulation on the surface or atmospheric water diﬀusion into the silica spheres. To be able to do this, a tunable narrow- 3.2 Littrow Grating Stabilized Diode Laser 57

Fig. 3.9: Bottom view of holder for angle-polished fibers. Nine fibers can be polished simultaneously. linewidth laser becomes necessary. Furthermore, since WGM resonators with diameters around tens of μm have free spectral ranges on the order of a few nm, the tunability of such a laser should be large enough to cover multiple resonances. We set up an external cavity diode laser using a holographic grating in the Littrow configuration to fulfill these tasks. The Littrow grating configuration was first used to stabilize dye lasers, and is now commonly used for narrow feedback into anti-reflection coated laser diodes [24]. These are often referred to as external cavity diode lasers (ECDL) [117]. The general principle of the Littrow ECDL is illustrated in Fig. 3.11a. Broad laser emission is collimated using a lens. When incident on the grating, a large portion of the light is reflected out (the zeroth order). The grating was aligned such that the first order was reflected directly back onto the lens and focused into the anti-reflection coated laser diode. Thus, the grating acted as an external filter, due to the wide angular spread of the wavelength components from the grating, so only a small portion was reflected back into the diode and could receive gain. By varying the angle of the grating, θ, the output wavelength could be selected. For coarse tuning, a precision screw was manually adjusted to vary the angle. For more precise and automated tuning, an additional piezo 58 Experimental Details

WGM Microsphere

Evanescent Field couples to Microsphere

Total Internal Reflection

(a) Schematic (b) SEM image

Fig. 3.10: Schematic of angle-polished ﬁber coupling and scanning electron micrograph of polished ﬁber tip showing very little surface roughness.

Lens Grating Laser Diode

(a) Schematic (b) Photograph

Fig. 3.11: Littrow laser setup. An external cavity is formed using a grating in the Littrow configuration. The first order is reflected back into the laser diode. The zeroth order is reflected out. By varying θ the lasing frequency is shifted. was integrated into the setup. To ensure accurate collimation and re-focusing of the light into the laser diode, another fine pitch screw was implemented. The whole setup was placed onto a thermoelectric cooler to guarantee temperature stability. A Toptica DC 110 controller was used for current and temperature control. When coupling into a fiber it was vital to pass the beam through an optical isolator to assure there was no further feedback into the cavity. Furthermore, an anamorphic prism pair was implemented for optimal coupling into the single-mode fiber. The ECDL could be tuned between 1528 and 1560 nm, with a linewidth of less than 200 kHz. Mode- hop free tuning was possible over a few GHz, depending on the gain region. It terms of performance, it was comparable to the Toptica DL 100. 3.3 Fiber Lasers 59

3.3 Fiber Lasers

Fiber lasers offer some distinct advantages over conventional types of lasers, including almost ideal Gaussian modes with M2 values close to 1, their compact sizes, and air- cooling operation as opposed to water-cooling in high-powered lasers. Generally, narrow linewidth can be achieved near a single frequency using additional frequency selective elements, such as fiber Bragg gratings [118]. We chose to use the ring laser setup since it prevents spatial hole burning due to standing waves, which simplifies single-frequency lasing. In its most basic form, a ring laser consists of a gain material (erbium-doped fiber), a pump (980 nm distributed feedback laser), a wavelength division multiplexer to allow circulation of the 1550 nm light and input of the 980 nm light, an isolator to prevent lasing in the opposite direction, and a fiber coupler to allow out-coupling of 1-10% of the light. Alternatively, as discussed in Chapter 5, a semiconductor optical amplifier (SOA) can be used in a fiber laser. It essentially consists of an electrically pumped laser diode with anti-reflection coatings on both sides. This results in spontaneous emission when a current is passed through it. On the other hand, if a seed within the gain region — such as another laser — is passed into one side, it will act as an optical amplifier. The atoms in the excited state will emit photons with the same properties as those from the input laser. Additionally, it is possible to set up a laser in a similar manner using such an amplifier. If the spontaneous emission is fed back into the laser in a loop configuration, mode competition will lead to certain modes receiving more gain. Above threshold, a laser cavity can be formed. To prevent standing waves inside the setups, we either used angled physical contact (APC) fibers which inhibit back-reflection, or spliced the components together. When inserting erbium-doped fiber, the length was kept such that no re-absorption and emission at a longer wavelength would occur. Fibers and all other components except for the WGM resonators themselves were not temperature stabilized. They were simply glued down to reduce vibrations. 60 Experimental Details Chapter 4

Microsphere Stabilized Fiber Laser

This chapter focuses on the development and testing of a ﬁber laser stabilized using the whispering gallery modes of a fused silica microsphere [119]. The chapter begins with the description of the ﬁber laser that was set up, including some results without stabilization. The integration of the microsphere is described step by step. Following this, section 4.2 describes the approach for measuring the quality factor of the microspheres. Section 4.3 explains how we characterized the linewidth and stability of the laser using heterodyne beating with a reference laser. Furthermore, some detail is given on the bad cavity regime, in which this laser functioned — implying the possibility of a lasing linewidth below the Schawlow-Townes limit. Finally, section 4.4 goes into the tuning and temperature dependence that was found in the laser.

4.1 Setup

Fiber lasers are becoming very popular due to their compact sizes, inexpensive components, low need for re-alignment and maintenance, high beam quality, high efficiency, and high average powers [120]. The most common configurations are either linear including fiber Bragg gratings at the desired wavelength [50, 51, 121], or in a ring configuration — typically also including a frequency selective element. We chose to use the ring laser configuration since it prevents spatial hole burning as in a standing wave cavity, and we included an isolator to ensure uni-directional lasing. This combination allows very narrow output frequencies to be produced [55, 20]. The unstabilized ring laser setup is illustrated in Fig. 4.1. A distributed feedback (DFB) laser at 980 nm with 200 mW maximum power is coupled into a fiber loop using 62 Microsphere Stabilized Fiber Laser

Splice 980nm WDM LD 980nm Er-doped 200mW fiber

Isolator

90%

90/10 10% Coupler 1550nm Output

Fig. 4.1: Unstabilized fiber loop laser. A 980 nm DFB laser goes into a loop using a wavelength division multiplexer (WDM) and pumps some erbium-doped fiber. An isolator prevents lasing in the backwards direction. 10% of the 1550 nm emission is coupled out. a wavelength division multiplexer (WDM). Approximately 70 cm of erbium-doped fiber (highly doped Er3+) are used as the gain medium. Emission occurs around 1530 nm. An isolator is included to ensure uni-directional lasing. A fiber coupler with a ratio of 90/10 is used to couple out the resulting emission. We used a fusion splicer to combine the elements to reduce losses and prevent standing waves between fiber ends. In this configuration, multiple roundtrips occur, and mode competition produces a few distinct lasing modes when the pump power is high enough. Fig. 4.2 shows an emission spectrum from the laser. Typically, three to four modes would lase simultaneously, with a full width at half maximum of 0.2 nm or higher, corresponding to 24 GHz. We fixed fibers onto an optical breadboard to ensure some stability, but temperature drifts, vibrations, and other environmental effects led to frequent mode hops overa4nmrange. Single transverse modes in fibers can be achieved by using single-mode fiber at a given wavelength. For 1550 nm single-mode fiber the only mode that can propagate is known as 4.1 Setup 63

1.0

0.8

0.6

0.4

0.2 Spectral Density (mW/nm)

0.0 1564 1568 1572 Wavelength (nm)

Fig. 4.2: Multimode emission from the unstabilized ﬁber laser. FWHM = 0.2 nm.

2 the HE11 hybrid mode, making single transverse mode lasing with a low M value trivial. Achieving a single longitudinal mode in a laser is more challenging. Generally fiber lasers use Bragg gratings to suppress gain for modes that are not at the specified wavelength [118]. This approach requires a fiber Bragg grating precisely within the output region of the gain material. The method that will be described here uses a different approach. A whispering gallery mode (WGM) resonator has sharp resonances. Hence, if a broadband source — such as spontaneous emission from erbium-doped fiber — is coupled into it, a drop-band filter is the result, in which sharp modes will be extracted from the spectrum. If another coupling mechanism is used simultaneously to extract the light on the other side, a compact narrow pass-band filter can be fabricated. In tapered fiber coupling [122], one can couple light into a resonator, while simultaneously monitoring the light which is not coupled into the resonator, as described in section 3.1.4. Once a fiber taper is fabricated, the only free parameters are the distance and position of the fiber with respect to the WGM resonator, as well as the input polarization. Generally, the thinnest central region of the fiber taper is used to couple into a 64 Microsphere Stabilized Fiber Laser resonator. The other two axes are adjusted while monitoring the output of the fiber taper while a tunable laser scans over a resonance. The coupling can thus be optimized. The WGM resonators used in this experiment were microspheres with a diameter of

50 to 100 μm. They were fabricated by melting the tips of SiO2 telecom ﬁbers using a

CO2 laser, detailed in section 3.1.1. The free spectral range of such a microsphere can be very large. For example, a 100 μm sphere has a mode separation of nearly 6 nm for the highest order whispering gallery modes.

Tapered fiber

Microsphere

Angle-polished fiber

(a) Schematic (b) Top view (c) Side view

Fig. 4.3: Light couples into a microsphere using a tapered ﬁber. Transmission is coupled out using an angle-polished ﬁber.

While coupling into such a microsphere using a tapered fiber, simultaneously an angle-polished fiber can be used [83], as described in section 3.1.5, to couple the light out of the other side of the resonator. Fig. 4.3 shows such a filter as used in our experiment. The left image illustrates a schematic of the setup, and the other two images show the top and side views of the WGM resonator respectively. Light is coupled into the WGM resonator using the tapered fiber, and after hundreds of thousands of roundtrips it is coupled out again using the angle-polished fiber. The coupling efficiency out of the microsphere should not be too efficient though — otherwise a drop in Q factor will result since the confinement of photons in the cavity is diminished. The stabilized ring laser configuration including the WGM filter is illustrated in Fig. 4.4. To achieve single-mode lasing, a fiber polarizer is inserted before the tapered fiber such that the coupling can be optimized for a single polarization mode. The only modes that can now receive gain in the ring laser are those that are transmitted through the WGM filter. The result is a frequency selective element that can be used over a broad range — only limited by the single-mode transmission of SMF-28 fiber, and the peak in spontaneous emission of the gain material used. The transmission of the tapered fiber is shown in the black line in Fig. 4.5a, recorded using an optical spectrum analyzer (Agilent 86140B). The red line in the same graph 4.2 Measuring the quality factor 65

Splice 980nm WDM LD 980nm Er-doped 200mW fiber

Isolator

PC Tapered Fiber Angle-polished Fiber 90%

Microsphere 90/10 Coupler 10% 1550nm Output

Fig. 4.4: Stabilized fiber loop laser. A 980 nm DFB laser goes into a loop using a wavelength division multiplexer (WDM) and pumps some erbium-doped fiber. An isolator prevents lasing in the backwards direction. The tapered fiber, microsphere, and angle- polished fiber function as a narrow-linewidth filter. 10% of the 1550 nm emission is coupled out. shows the transmission of the taper when it is brought into close proximity of the microsphere. Sharp dips appear, spaced a few nm apart as expected. At these wavelengths the light is coupled into the resonator and lost through other mechanisms, such as absorption or scattering. Accordingly, Fig. 4.5b shows the transmission through the microsphere at the end of the angle-polished fiber. The peaks are in the same positions as the troughs in the taper transmission graph as expected. These modes show the only regions of the spontaneous emission of the erbium-doped fiber that can receive gain after further roundtrips. The separation of the fundamental modes as calculated from the graph is 10 nm. At a 1550 nm wavelength this corresponds to a resonator diameter of 57 μm, which coincides with the microscope image. At least two further modes are visible in the diagram.

4.2 Measuring the quality factor

The quality (Q) factor of a cavity is a commonly quoted value which gives an indication of the linewidth of a resonance compared to the frequency, as well as the lifetime of a photon in the cavity. A high Q resonator requires a long conﬁnement of light in it, with little scattering or absorption. WGM resonators can have extremely high Q factors due 66 Microsphere Stabilized Fiber Laser

1.5 3

1.0 2

0.5 1

Spectral Density (μW/nm)

Spectral Density (nW/nm)

0.0 0 1520 1540 1560 1520 1530 1540 1550 1560 1570 Wavelength (nm) Wavelength (nm) (a) Taper transmission (b) Angle-polished ﬁber transmission

Fig. 4.5: Left, the spontaneous emission of the erbium-doped fiber after transmission through the tapered fiber. Red, dips due to resonances in the microsphere. Right, peaks at the same positions after coupling into the angle-polished fiber. to low absorption in carefully chosen materials, and the fact that they confine light by total internal reflection, which can be nearly lossless. The Q factor is often determined by measuring the linewidth of a resonance, since Q = ν0/Δν, where ν0 is the central frequency, and Δν is the FWHM of the resonance width. A common method to determine the precise width of the resonance of a WGM sphere is by monitoring the transmission of a tapered fiber as described before, while tuning a laser over the resonance [Fig. 4.6]. In this case, we used a homebuilt external cavity diode laser (ECDL) as detailed in section 3.2. The frequency can be scanned automatically using a piezo which shifts the angle of the Littrow grating used in the setup. The narrow frequency of the laser can thus be repetitively tuned over a few GHz mode-hop free. The offset of the scanning was adjusted when the taper was close to the microsphere, until a resonance was found. The coupling conditions — x, y, z position of the taper, as well as polarization — were adjusted to optimize the coupling into the resonator. Approximate positioning of the microsphere with respect to the taper was done using a 3-dimensional precision screw stage, and precise alignment was performed using a 3-dimensional piezo stage. Since the fabrication method of the microspheres produces a thin stem, this can be used as a holder for the microspheres, and precise positioning is simplified. Fig. 4.7 shows a typical oscilloscope trace of the intensity on the detector of such a measurement. The ECDL was repetitively scanned over 600 Mhz in this case. The sharp dip shows a Lorentzian shaped resonance as expected. The coupling efficiency for this scan was about 70%, and the full width at half maximum is 1.7 MHz. Furthermore, 4.2 Measuring the quality factor 67

Tunable Diode Laser Detector

Microsphere

Fig. 4.6: The frequency of a tunable diode laser is scanned and the transmission through a tapered ﬁber is measured using a photodiode and recorded using an oscilloscope. On resonance a dip in the transmission appears.

two smaller dips are seen at about +200 MHz offset, implying that higher order modes were excited as well. In the ring laser configuration, mode competition leads to the most prominent mode being amplified most in multiple roundtrips, so it is useful to find a mode that has a high level of coupling and to optimize it. Once the ring laser operates, the resonance position can be noted down, and then optimized by adjusting the coupling of a mode at the identical position using the tunable laser. Positioning of the taper with respect to the microsphere is vital. If the distance between the two is too large, under-coupling will prevent most of the light from entering the sphere. If the taper and the microsphere are too close or even touching, over-coupling will result, thus widening the linewidth by up to a factor of 10.

To determine the Q factor of a resonator, one can use the fact that Q = ν0/Δν as mentioned before. A carrier frequency of 193 THz and the linewidth of 1.7 MHz result in a Q factor of 1.1 × 108. In practice, the Q factor would degrade over a few weeks in the lab, until the value was just 107 due to diffusion of atmospheric water into the surface of the sphere, leading to higher absorption, as well as dust accumulation on the surface, leading to scattering. The highest Q factors in SiO2 microspheres found in literature are 9 around 10 [113]. They are fabricated in an N2 environment which prevents diffusion of atmospheric water while they are being melted. Degradation over time is difficult to suppress, but we enclosed our filter setup (tapered fiber, microsphere, and angle-polished

fiber) in a see-through box with an N2 purge to discourage water diffusion into the resonators. Furthermore, this reduced dust accumulation on the surface of the sphere, as well as the other sensitive components of the filter. 68 Microsphere Stabilized Fiber Laser

1.0

0.5 Normalized Intensity

0.0 -300 -200 -100 0 +100 +200 +300 Frequency Offset (MHz)

Fig. 4.7: The frequency of the tunable diode laser is scanned over 600 Mhz. The Lorentzian shaped resonance shows a width of 1.7 MHz and 70% coupling into the microsphere.

4.3 Heterodyne Beating and OSA results

For many applications, one of the most important parameters in a laser is its linewidth. Various methods exist for determining laser linewidths, depending on how narrow the lasing linewidth is. A simple approach is the use of an optical spectrum analyzer, although the limit in precision is on the order of tens of pm. A closer value can be determined for narrow linewidth lasers by beating the optical frequency with a stable reference laser. The stability of a laser is another matter of interest, and can be determined using a wavelength meter. All three approaches were pursued to evaluate the laser’s performance. After the coupling into and out of the microsphere were optimized, the WGM filter was inserted into the fiber ring laser as illustrated in Fig. 4.4. The output emission was studied using an optical spectrum analyzer. Further adjustment of the input polarization, and the x, y, and z positioning of the microsphere with respect to the tapered fiber, and the angle-polished fiber with respect to the microsphere allowed some tuning of the emission. 4.3 Heterodyne Beating and OSA results 69

Mode competition would typically choose an efficiently coupling mode around the peak of the spontaneous emission — in this case around 1530 nm. For stable lasing it was necessary to over-couple, so the taper, microsphere, and angle-polished fiber were all brought into contact. Electrostatic force ensured a fairly rigid setup. At critical coupling with a gap between the components, environmental effects became much more difficult to control, and mode hops and loss of lasing occurred frequently. If the height of the tapered fiber was varied with respect to the microsphere, it was also possible to get two or three stable modes to lase simultaneously, with up to 4 nm separation, corresponding to different polarization modes of WGMs of the microsphere. Changing the polarization tuned the intensity of the separate peaks.

1200

800

Spectral Density (nW/nm) 400

0 1528.2 1528.3 1528.4 1528.5 1528.6 Wavelength (nm)

Fig. 4.8: The linewidth of the laser is 0.01 nm, which is the limit of the resolution of the optical spectrum analyzer (Ando AQ6317B).

Single-mode emission, as shown in Fig. 4.8 could be achieved on a day-to-day basis within minutes. This measurement was done with an optical spectrum analyzer with a higher resolution (Ando AQ6317B). The resolution bandwidth of the device is 0.01 nm, and the full width at half maximum shows this exact value, so the measurement is resolution limited. Output power was on the order of a few μW. Once lasing was optimized, the output of the tapered fiber could also be used as a lasing signal. Since the coupling 70 Microsphere Stabilized Fiber Laser into the sphere was on the order of 70%, the transmitted power out of the other end was nearly 30%. The power at this fiber end was measured to be over 100 μW. In a finalized setup, the 90/10 coupler could be removed to reduce losses by a few percent. In practice, however, it is beneficial to have this output port to adjust the coupling of the tapered fiber and angle-polished fiber until lasing occurs. If the the output of the tapered fiber is monitored, results can be confusing when adjusting the angle-polished fiber coupling, until it is already optimized.

195.702

195.701

195.700

Frequency (THz)Frequency

195.699

10000 20000 30000 Time (s)

Fig. 4.9: The lasing peak, as measured using a wavemeter, ﬂuctuates around 1 GHz over seven hours. Generally the variation is on the order of hundreds of MHz, which is the resolution of the wavemeter.

Subsequently, the long-term stability of the lasing emission was studied using a wavemeter (Advantest Q8326, 100 MHz resolution). The fluctuation measured over seven hours is shown in Fig. 4.9. In total, the fluctuation is approximately 1 GHz, with nearly no drift. The standard deviation is on the order of a few hundred MHz, near the limit of the resolution of the wavelength meter. The temperature control, described in more detail in section 3.1.1, ensured that the lasing frequency could be kept constant over many hours. Slight temperature drifts in the lab would lead to a drift, and finally a loss in lasing or mode-hops if the temperature control was turned off. Additionally, a hard bump on the optical table would also lead to misalignment and mode-hops or a loss in lasing. 4.3 Heterodyne Beating and OSA results 71

Splice WDM 980nm LD 980nm Er-doped 200mW fiber Isolator

Tapered PC Fiber Angle-polished Fiber 90/10 90% Coupler 1550nm Output Microsphere 10% RF Spectrum Analyzer Detector Grating Stabilized Diode Laser 3dB Coupler

Fig. 4.10: Heterodyne beat measurement setup. The stabilized ﬁber laser is spatially overlapped with an ECDL which is oﬀset by a few MHz. The resulting beat is recorded using a detector and a spectrum analyzer.

A common method to measure the linewidth of a laser is to use a more stable reference laser, and observe the beat notes that are generated when the two lasers overlap on a detector. Specifically, the stable reference laser is offset by a few MHz in frequency, and the spatial overlap of the two beams on the detector generates a beat at the difference frequency fbeat = | fa− fb|. Commercial detectors can easily detect MHz levels, so these can be measured, since the direct optical frequency on the order of 1014 cannot be measured σ2 = σ2 + σ2 directly. The variance of the laser frequencies sums as: ab a b. If the noise of σ2 the laser to be measured is much larger than that of the reference laser, ab will give a close approximation of the true linewidth of the laser in question. We used our homebuilt external cavity diode laser (ECDL) as a reference laser. It was previously measured to have a linewidth just over 150 kHz by beating it with a fiber frequency comb. The setup is shown in Fig. 4.10. The two lasers are spatially overlapped by using a 3 dB fiber coupler, and neglecting one of the output ports. The output was focused onto a detector (MenloSystems FPD510), and results were analyzed using a radio frequency spectrum analyzer. The beat shifted over approximately 50 MHz in tens of seconds. The shift was slow enough to be able to simultaneously tune the precise frequency of the ECDL reference laser by hand, such that the beat frequency would remain within 1 MHz of its position. A typical measurement is shown in Fig. 4.11. The 3 dB width, corresponding to the FWHM on a linear scale, is 170 kHz. Since the ECDL was measured to be close 72 Microsphere Stabilized Fiber Laser to this value, 170 kHz is limited by linewidth of the reference laser. It corresponds to an effective lasing Q factor of over 109, or higher. This implies an enhancement factor of 10 from the single transmission through a microsphere.

A laser in which the cavity loss rate Γ0 is comparable to or even larger than the gain bandwidth is commonly referred to as a “bad cavity” laser [57, 58]. In a conventional laser, the angular frequency gain bandwidth (FWHM), commonly denoted as 2γ = 2/T2, is generally assumed to be much larger than the cavity loss rate Γ0. If this is not the case, the Schawlow-Townes limit [5] does not apply, and lasers can have narrower linewidths. The classical Schawlow-Townes limit is given by

Γ hν Γ2 Δν = 0 = 0 , (4.1) 4πS 4π Pout where Δν is the laser linewidth at FWHM, Γ0 is the empty cavity loss rate, S is the average number of photons in the lasing mode, hν is the energy per photon, and Pout is the total output power. The empty cavity loss rate in a linear cavity is

Γ0 = −(c/2L) ln(R1R2), (4.2)

where c is the speed of light, L the length of the cavity, and R1 and R2 are the reflectivities of the end mirrors. In our laser setup, the total losses — mainly due to coupling into and out of the microsphere — are relatively high around 0.75, so R1R2 can be assumed to be 0.25. The length of the ring laser L is about 7 m in fiber, so about 10.5 m of optical length. 7 −1 This results in a value for Γ0 of about 2×10 s . In the fiber ring laser, the gain bandwidth is essentially the transmission bandwidth through the microsphere, on the order of 106 Hz [Fig. 4.7]. Since the cavity loss rate is an order of magnitude higher, this implies that we are entering the bad cavity regime. In a bad cavity laser, the linewidth can be lower than the Schawlow-Townes limit.

4.4 Pump Power and Temperature Dependence

Since the conﬁnement of light in WGM resonators is very high in a small space, a high intensity can occur, leading to various intensity dependent eﬀects. For example, WGM resonators are often used for their low threshold behavior in nonlinear optics. Once the lasing threshold is reached, the output power of the ring laser increases lin- early with pump power from the 980 nm DFB laser as expected [Fig. 4.12]. Furthermore, 4.4 Pump Power and Temperature Dependence 73

-30

-60

Intensity (dBm)

-90 20 25 30 35 Frequency (MHz)

Fig. 4.11: Heterodyne beat between microsphere stabilized fiber laser and ECDL reference laser, showing their combined frequency noise. The 3 dB width is 170 kHz, limited by the linewidth of the reference laser. a red-shift with increasing pump power can also be seen. The thermorefractive index at high powers inside the cavity leads to an increase in the refractive index, hence a red- shift in the emission since the effective cavity length increases. The fundamental limit in the stability of a WGM resonator is determined by thermal fluctuations for the same reason [123, 87, 88]. More details and theoretical background on this can be found in section 2.2.5. In a high Q whispering gallery mode resonator, the photon lifetime is very long on the order of hundred thousands of roundtrips. The main limiting factor is the absorption, which in turn leads to heating along the path of the WGM. In general, the refractive index n˜ varies as d n n˜ = n + n · I + · T˜, (4.3) 0 2 d T where n0 is the refractive index at the central wavelength, n2 is the nonlinear refractive index due to the χ(3) nonlinearity at intensity I, and (d n/ d T) is the temperature dependence 74 Microsphere Stabilized Fiber Laser of the refractive index at a laser induced temperature change T˜ [84]. Additionally, a laser induced temperature change will also lead to heating and thus thermal expansion of the cavity itself. Each of these three effects will lead to an increase inn ˜, and hence a red-shift in the emission of the laser. The thermorefractive index and the thermal expansion of the cavity are the most prominent effects in this case, with a much lower contribution from the χ(3) nonlinearity. An order-of-magnitude calculation is given in chapter 2. The slope of the redshift as a function of output power as seen in Fig. 4.12b is 16 pm/μW. When pumping with 200 mW of power at 980 nm, and 10% out-coupled power of 20 μW, the refractive index changes by 3.1 × 10−4. Since the (d n/ d T) term is 1.2 × 10−5K−1 for fused silica [84], this implies an induced temperature change of 25.8 K. Rapid temperature tuning over hundreds of pm becomes possible, since the mass of the microsphere is very small, so a temperature equilibrium is reached quickly in the resonator. Additionally, the need for long-term temperature stability becomes apparent.

20 1529.4

10 :DYHOHQJWK QP

0 6SHFWUDO'HQVLW\ ȝ:QP

1529.2 Increasing Pump Power

1529.0 1529.2 1529.4 1529.6 1529.8 0 10 20 Wavelength (nm) 3RZHU ȝ: (a) Red-shift with power increase (b) Linear dependence

Fig. 4.12: There is a red-shift in the emission with increasing pump and output power. The slope in terms of the output power is 16 pm/μW.

In conclusion, an erbium-doped ﬁber ring laser was stabilized using the whispering gallery modes of a microsphere. The passive quality factor of the resonator was determined to be 108, but multiple roundtrips in the lasing cavity led to an enhancement factor of 10 in the ﬁnal lasing output. The linewidth was measured to be 170 kHz, at the limit of the resolution of the heterodyne beat measurement. Furthermore, a temperature dependence in the lasing linewidth was observed. The thermo-refractive index in the microsphere led to a red-shift of 16 pm/μW of output power. Passive WGM laser stabilization appears to be a better technique than actively locking to a resonance of a resonator due to the enhancement factor. Chapter 5

Calcium Fluoride Stabilized Fiber Laser

In this chapter, we present another whispering gallery mode (WGM) stabilized laser

[124]. In this case, a more stable macroscopic crystalline calcium ﬂuoride (CaF2) disk was used. Disks made of this material have been shown to achieve the highest Q factors in WGM resonators (up to 1011)[44], due to their extremely low absorption and high optical quality. The setup of the laser and implementation of the CaF2 disk are explained in section 5.1. Next, section 5.2 describes how the quality (Q) factors of the crystalline resonators were determined. Subsequently, section 5.3 gives detail on the determination of the linewidth of the stabilized laser using the self-heterodyne beat technique, by passing part of the laser through a 45 km long ﬁber delay line. Finally, section 5.4 describes further measurements undertaken to characterize the laser performance: a three-cornered hat measurement, in which two reference lasers were simultaneously beat with the stabilized laser to determine the absolute performance and Allan deviation of it.

5.1 Setup

The gain material used in the laser was a semiconductor optical amplifier (SOA) with broad gain around 1560 nm, which was integrated into a fiber laser. An SOA consists of an electrically pumped diode, similar to a laser diode. The end facets are anti-reflection coated however — contrary to a laser diode. If a current is passed through such an SOA, it will spontaneously emit photons with a broad linewidth. If, however, a seed source like a laser is incident on one end of the SOA, the signal will be amplified. The SOA used 76 Calcium Fluoride Stabilized Fiber Laser in this case was a Thorlabs S9FC1004P at 1560 nm, with an optical bandwidth of about 100 nm, and a small signal gain of up to 28 dB. The spontaneous emission is shown in Fig. 5.1a.

0.04 0.10

0.08 0.03

0.06

0.02

0.04

Spectral Density (mW/nm) 0.01 Spectral Density (mW/nm) 0.02

0.00 0.00 1400 1500 1600 1700 1600 1610 1620 1630 Wavelength (nm) Wavelength (nm) (a) SOA emission (b) SOA laser

Fig. 5.1: Left, spontaneous emission from the semiconductor optical amplifier with a FWHM of 94 nm. Right, lasing closing the loop of the SOA and coupling out 1%. Reso- nances are spaced 4 nm apart due to weak reflections from fiber connectors with a FWHM of 0.4 nm.

We set up a fiber ring laser by looping the emission from the output port of the SOA through an isolator and back into its input port. 1% of the resulting emission was coupled out using a fiber coupler, and resulted in multi-mode lasing as shown in Fig. 5.1b. The full width at half maximum (FWHM) of the peaks is 0.4 nm, and they are spaced 4 nm apart due to light reflections from fiber connectors. Angled physical contact (APC) fiber is polished at an angle of 8◦ to the fiber cladding, which results in very low back-reflection at fiber facets. APC fiber connectors were used in the stabilized setup to prevent standing waves like those mentioned due to reflections between connectors. In the previous section, a fiber laser using erbium-doped fiber as a gain material pumped with a 980 nm diode laser was described. The general principle was the same: a laser based on fibers was used, and a whispering gallery mode resonator filtered out narrow spectral components. The advantage of this setup is the more stable setup of the WGM resonator, as well as crystalline material, which can achieve higher Q factors. The whispering gallery mode resonator used in the setup of the stabilized laser was a calcium fluoride (CaF2) crystalline disk. The advantages over a microsphere are the possibility for ultra-high quality (Q) factors of up to 1011, as well the disks macroscopic rigidity; our microspheres are mechanically not as stable since they are on stems with diameters around 10-20 μm. It is easier to mechanically stabilize a macroscopic disk as 5.1 Setup 77 opposed to a microscopic sphere. The diameter of the disk was 5 mm, and the thickness 2 mm. The disk was drilled out of a high-grade UV window, and subsequently polished using a turning machine, as described in section 3.1.2. A common technique to couple to WGM resonators is the prism coupling technique. In it, a laser beam is focused onto the back of a prism at a totally internally reflecting angle, which is in close proximity to a WGM resonator. The evanescent field of the laser beam overlaps with the the evanescent field of the whispering gallery modes of the resonator. Provided that the angle is adjusted such that mode-matching between the two occurs, coupling will result. If all mechanical coupling conditions are carefully adjusted, and the incoming light is resonant in the cavity, a very efficient energy transfer into the resonator can occur. The angle of the light beam incident on the back of the prism depends on the refractive indeces of the prism, and the resonator. The material chosen for the prism was SFL 11 glass, which has a refractive index of 1.74, and is thus much higher than that of the

CaF2 resonator (1.46 at 1550 nm). The experimental details are described in section 3.1.3.

CaF2

GRIN

PC Coupler 1% 99%

SOA PC (a) Stabilized Laser schematic (b) Double prism ﬁlter

Fig. 5.2: Left, stabilized ring laser. A semiconductor optical ampliﬁer (SOA) provides gain around 1550 nm. Polarization control (PC) is used before the coupling and before the SOA. An isolator prevents bi-directional lasing. Graded refractive index (GRIN) lenses are used for prism coupling. 1% of the emission is coupled out. Right, photograph showing the GRIN lenses and the resonator on top of a thermoelectric cooler.

To use a WGM resonator as a band-pass filter, it is necessary to couple light into, and then out of the same resonator using a separate coupler. If only one coupler is used, a drop filter is the only possibility. In this case, two prisms were used — one on each side of the resonator. The resulting narrow linewidth filter was integrated into the SOA fiber laser as illustrated in Fig. 5.2. To ensure a single polarization WGM being excited, fiber polarization control was inserted before the resonator. Furthermore, the amplifier 78 Calcium Fluoride Stabilized Fiber Laser was polarization dependent, so an additional polarization control was inserted before it. To focus the beam onto the back of the prism, a graded refractive index (GRIN) lens was used, and a further one collected the frequency filtered light on the other side of the disk through the prism. Precise alignment of the x, y, and z axes, x and y angles, beam focus, and distance between prisms and resonator allowed single whispering gallery modes to achieve most gain in multiple roundtrips.

5.2 Measuring the quality factor

Precise determination of the quality (Q) factor was performed in a similar manner to that described in section 4.2. We scanned a commercial tunable external cavity diode laser (Toptica DL Pro at 1550 nm) over a few GHz repetitively. The emission was coupled into one side of the WGM resonator, and the reﬂection from the prism was focused onto a detector. Next, we connected an oscilloscope to the detector, and triggered it by the repetition frequency of the laser sweep. A typical resonance with a FWHM of 15 MHz is 7 shown in Fig. 5.3a. The Q factor in this case (ν0/Δν) is just over 10 . The limiting factor is scattering from dust particles, and surface roughness from the polishing procedure. We are currently investigating other techniques to increase the quality factor.

2.0 1.6

1.4

1.2 1.8 1.0 Reflection 0.8 Transmission

1.6 0.6

Intensity (V)

Intensity (V) 0.4

0.2

1.4 0.0

-0.2 -200 0 200 -200 -100 0 100 200 Frequency Offset (MHz) Frequency Offset (MHz) (a) Disk resonance (b) Resonance and transmission

Fig. 5.3: Left, 15 MHz resonance, corresponding to a Q factor of 107. Right, the black curve indicates the reﬂected light from the input prism, and the red curve indicates the transmission through the WGM disk and other prism and into the GRIN lens.

After the coupling into one side was optimized, the tunable laser was connected to the other side, and the procedure was repeated, to find the approximate position and angle to couple to whispering gallery modes. Different angles, positions of the lens, and dis- 5.2 Measuring the quality factor 79 tance between disk and WGM resonator lead to different modes being excited, but coarse alignment for the out-coupling prism was possible using this technique. To optimize a particular mode, the laser was connected to one GRIN lens and coupling was optimized, while simultaneously monitoring the light transmitted out the other prism and through the other GRIN lens using a further detector. A combination of signals is shown in Fig. 5.3b. In this case, a 47 MHz resonance was coupled into the disk. By adjusting in- and out- coupling parameters, the narrow line is transmitted. The result is a narrow-band filter. The depth of the resonance indicates that the in-coupling efficiency was up to 80%. The out- coupling efficiency was measured by measuring the power being transmitted out of the prism, which was about 50% of the in-coupled light. It was difficult to achieve optimum coupling of the light into the GRIN lens, so the overall loss was up to 9 dB through both GRIN lenses, prisms, and the WGM resonator. If coupling out of the resonator would have been further improved, a drop in the Q factor would have become apparent, since a loss mechanism drastically decreases the Q factor according to

2π Q = ν T , (5.1) 0 rt I where ν0 is the optical frequency, Trt is the roundtrip time in the optical resonator, and I is the fractional power loss per roundtrip. Hence, a higher roundtrip loss due to eﬃcient out-coupling would lead to a drastically decrease in the Q factor. The optimum trade-oﬀ between Q factor and little loss would be perfect in-coupling, and slightly under-coupled out-coupling. A source with a higher gain, or an additional SOA attached in series would lead to a bigger range of lasing in terms of pump power, allowing power dependence studies of the linewidth.

The disk to prism distance dependence of the coupling efficiency and FWHM of the resonance are shown in Fig. 5.4. When the distance between the disk and the prism decreases (using the AttoCube ANPx101 piezo stage), the coupling efficiency increases, but the FWHM increases as well. The optimum distance is a trade-off between high coupling efficiency, and low FWHM of the resonance. See section 2.2.4 for more detail on WGM coupling. Furthermore, in Fig. 5.4a it becomes apparent that over-coupling occurs when the distance between disk and prism is too low; unwanted higher order resonances are excited and leak into the WGM resonator. 80 Calcium Fluoride Stabilized Fiber Laser

80 100

1.2 80 60

0.8 60 40

Intensity (V)

Coupling (%) Coupling 40

FWHM (MHz) 20 0.4

0.0 0 -200 0 200 0.0 0.5 1.0 Frequency Offset (MHz) Distance from Disk (μm) (a) WGM resonance at varying distance (b) Distance dependence

Fig. 5.4: Left, top to bottom curves show a resonance with decreasing distance between the disk and the prism. Right, the red curve shows a decrease in coupling with increasing distance, as well as a decrease in FWHM of the resonance.

5.3 Self-Heterodyne Beating

An important indicator of the degree of stabilization of the fiber laser is the resulting linewidth when the WGM resonator filter is included. Generally, a mode at the center of the SOA gain peak which is most efficiently coupled will win mode competition. Single- mode lasing resulted, and was initially observed using an optical spectrum analyzer (Agi- lent 86140B). The linewidth measured 0.05 nm, at the limit of the OSA resolution similar to the microsphere stabilization experiment in the previous chapter [Fig. 5.5a]. With single-mode lasing operation on a narrow-linewidth, this filtering setup was already su- perior to the simple loop setup. The lasing threshold, depending on the precise coupling conditions, was generally around 300 mA. In theory, the higher the output power of the laser, the narrower the linewidth, so the SOA was set to its maximum pump current of 600 mA for further experiments. Higher resolution measurements were performed using the heterodyne beat technique. The Toptica DL Pro laser was set to a frequency offset by a few MHz from the WGM stabilized laser. The two were optically mixed in a fiber coupler, and the output was analyzed using a radio frequency spectrum analyzer. The difference frequency, fbeat =

| fa − fb|, also includes the combined noise term for the separate lasers. It is given by σ2 = σ2 +σ2 σ2 ab a b, where a is the variance of laser a. If a reference laser is used with a much σ2 narrower linewidth than the laser in question, ab gives the upper limit for the variance of the laser to be measured. The resulting beat can be seen in Fig. 5.5b. The 3 dB linewidth is 70 kHz measured over 100 ms, which corresponds to an eﬀective Q factor of over 109. The linewidth of the Toptica DL Pro is given in the manual as about 100 kHz, so we are 5.3 Self-Heterodyne Beating 81

-20

0.30 -30

-40

0.20 -50

-60

Intensity (dBm) -70 0.10

Spectral Density (μW/nm)

-80

-90 0.00 1558.8 1559.0 1559.2 1559.4 1559.6 46 48 50 52 54 Wavelength (nm) Frequency (MHz) (a) Optical spectrum analyzer (b) Heterodyne beat

Fig. 5.5: Left, optical spectrum analyzer trace of stabilized laser emission. Linewidth is 0.05 nm, limited by the resolution. Right, heterodyne beat with a Toptica DL Pro as a reference. Linewidth is 70 kHz, limited by the linewidth of the DL Pro. clearly at the resolution limit of this method. A common technique to measure very narrow linewidths is the self-heterodyne beat technique [91]. An acousto-optic modulator (AOM) is used to split the narrow linewidth laser signal into two. One remains unchanged, while the other is offset by some amount — in this case 100 MHz. An AOM uses the acousto-optic effect by producing a standing acoustic wave in a crystal, which acts as a diffraction grating for the incoming laser beam. In the 1st order of the occurring diffraction an acoustic phonon is added to each photon, resulting in a photon with an increased frequency ν = νoptical + νacoustic. Next, one of the two signals is delayed by at least six times the coherence length of the laser [92].

Assuming a Lorentzian spectrum, the coherence length is Lcoh = c/(πΔν), so a 10 kHz laser has a coherence length of 9.5 km, corresponding to just over 6 km in fiber. This leads to a required fiber loop of 36 km. In this case, a 45 km fiber delay line was used. Recombining the two allows a beat measurement using the laser signal, and a delayed copy of it. Fig. 5.6 shows the schematic of the self-heterodyne technique. Self-heterodyne measurements were performed on the homebuilt external cavity diode laser (ECDL) described in section 3.2, the Toptica DL Pro at 1550 nm, and our WGM disk stabilized fiber laser. The 3 dB width of the resulting beat signal at 100 MHz gives the initial linewidth√ of the laser multiplied by a factor of two for a Lorentzian shaped spectrum, and 2 for a Gaussian shaped spectrum [91]. Excess phase and frequency noise can accumulate in fibers due to environmental perturbations such as vibrations and temperature variations, so the result after a 45 km fiber delay line can be slightly higher 82 Calcium Fluoride Stabilized Fiber Laser

45 km delay 3 dB AOM coupler Laser 1st order Fiber 0th order Detector

Fig. 5.6: Self-heterodyne beat schematic. An acousto optic modulator (AOM) splits a laser signal. The 1st order is oﬀset by 100 MHz and delayed longer than the coherence length of the laser. Next, it is recombined using a ﬁber coupler and analyzed using a detector and an rf spectrum analyzer.

√ than the real linewidth. After dividing by 2, Fig 5.7 shows the upper limits of the linewidths of the ECDL, Toptica DL Pro, and WGM disk stabilized laser to be 390 kHz, 41 kHz, and 13 kHz respectively.

-78

-80 -35 -80

-82

-84 -45

-86

Intensity (dBm) -85

Intensity (dBm) Intensity (dBm) -88 -55

-90

-92 -65 -94 98 99 100 101 102 99.5 100.0 100.5 99.95 100.00 100.05 Frequency (MHz) Frequency (MHz) Frequency (MHz) (a) Homebuilt ECDL (b) Toptica DL Pro (c) WGM Stabilized Laser

Fig. 5.7: Self-heterodyne beat measurements of homebuilt external cavity diode laser (100 averages), Toptica DL Pro (10 averages), and CaF2 stabilized laser (70 averages). The 3 dB widths√ are 550 kHz, 58 kHz, and 18 kHz respectively, as measured over 100 ms. Dividing by 2 gives the upper limit for the linewidths.

This astounding result shows that the disk stabilized laser is significantly more stable than the other two lasers. The values confirm the heterodyne beat results in section 5.2. The effective Q factor of the 13 kHz WGM stabilized laser is thus at least 1.5 × 1010.The width of the initial WGM resonance was 15 MHz, implying an enhancement factor of 103 in lasing, caused by multiple roundtrips through the cavity. With a WGM resonator with an initial resonance width on the order of a few kHz (corresponding to Q = 1011), it should be possible to get sub-kHz lasing linewidths due to the enhancement from multiple roundtrips in a lasing cavity. This offers clear advantages over direct lasing in a WGM resonator, which is generally limited in Q factor by absorption. 5.4 Three-Cornered Hat Measurement 83

5.4 Three-Cornered Hat Measurement

The three-cornered hat technique has been used for some time now to determine the absolute stability of frequency sources, by comparing the respective relative beats of three sources [90]. It is frequently used for cesium clocks, hydrogen masers, and other frequency sources in the microwave region which is electronically accessible. More recently, the technique has been applied to optical sources, such as diode lasers stabilized to atoms in gas cells [93], and lasers stabilized to ultra-stable reference cavities [94, 95]. The three-cornered hat method relies on the beating between three lasers with similar stabilities; let us call them lasers a, b, and c. Optical frequencies of these lasers must be close together, separated by a frequency on the order of MHz that can be recorded using a photo-detector. Now, simultaneous beats must be recorded on three separate detectors — the optical beat between lasers a and b, lasers a and c, and lasers b and c. In practice, the overlapping and mixing of the optical frequencies was done using ﬁber optic couplers, as illustrated in Fig. 5.8. Six 3 dB couplers were spliced together, resulting in three input ports for the lasers, and three output ports to be connected to the detectors.

3 frequency mixer f ŇI -f Ň Laser A A B C Detector 1

f ŇI -f Ň Laser B B A C Detector 2

f ŇI -f Ň Laser C C A B Detector 3

(a) 3-frequency mixer (b) Photograph

Fig. 5.8: Three input frequencies are mixed using six 3 dB ﬁber couplers.

The frequency of a beat is characterized by fab = | fa − fb|, and similarly for the other two laser combinations. Furthermore, the variance of each beat is a combination of the variances of the two optical frequencies. The variance of each beat is given by

σ2 = σ2 + σ2, ab a b σ2 = σ2 + σ2, ac a c σ2 = σ2 + σ2, bc b c (5.2)

σ2 σ2 where ab is the variance of the beat between lasers a and b, a is the variance of laser a, and similarly for the other parameters. Assuming no correlation between the lasers, 84 Calcium Fluoride Stabilized Fiber Laser

σ2 σ2 σ2 we can solve for the three unknowns, a, b, and c, by rearranging and combining the equations. Finally, we arrive at a new set of equations: σ2 = 1 σ2 + σ2 − σ2 , a 2 ab ac bc σ2 = 1 σ2 + σ2 − σ2 , b 2 ab bc ac 1 σ2 = σ2 + σ2 − σ2 . (5.3) c 2 ac bc ab

Hence, by recording the three beats simultaneously, the stability of each individual laser can be calculated. A preferred method for calculating the stability of a frequency source is the Allan variance, or Allan deviation [61]. More details can be found in Chap- ter 2.Ifτ is the observation period, andy ¯n is the n-th fractional frequency average over time τ, the basic formulation for the Allan variance is 2 1 2 σ (τ) = (¯y + − y¯ ) . (5.4) y 2 n 1 n

In our experiments, three homemade detectors using fiber-coupled JDSU EPM 605 photodiodes were used. A Toptica DL Pro, a homemade ECDL, and the WGM stabilized ring laser were set to a similar wavelength — around 1559 nm — offset by 5 to 10 MHz. The three beats were simultaneously recorded using an oscilloscope, running at 100 MSamples/s. Up to 100 ms were recorded at a time. A typical sine wave of one of the beats is shown in Fig. 5.9. The resulting data was analyzed using the zero-crossings method [125], in which the sine wave is normalized to oscillate around zero, and the points of the crossing from positive to negative and vice versa are recorded [16]. Averaging was performed over 1 μsseg- ments, corresponding to eleven zero-crossings in a 5 MHz signal. Each frequency point was recorded, as in the 10 ms trace in Fig. 5.10. It is apparent that the beats including the ECDL are significantly noisier than the beat between the DL Pro and the WGM stabilized laser. Additionally, there was a harmonic at 1.4 kHz — most likely due to vibration — in the beats containing the ECDL. Since we are not as concerned with the absolute performance of the ECDL as the performance of the CaF2 disk stabilized laser, this harmonic was removed from the two beats containing the ECDL for further calculations. A self-written computer program performed the Allan deviation calculations as in Eq. (5.4), as well as the three-cornered hat technique according to Eq. (5.3) on the data points. The resulting Allan deviation σy(τ) in Fig. 5.11 shows, as expected, that the Allan 5.4 Three-Cornered Hat Measurement 85

0.13

0.12

Intensity (V)

0.11

5.94 5.96 5.98 Time (μs)

Fig. 5.9: Typical sine-wave oscilloscope trace of a beat between two lasers.

deviations of the data containing the ECDL are signiﬁcantly worse than that of the CaF2 stabilized laser and the DL Pro. In fact, the two ECDL beats nearly overlap in the graph. After performing three-cornered hat calculations for each combination of points at varying averaging times, the results were plotted in a similar fashion to show the separated Allan deviations of each laser. At 1 μs averaging, the ECDL performance is about 8 × 10−10, that of the DL Pro is 1.5 × 10−10, and the WGM stabilized ring laser has the best performance: 10−10. The highest performance is reached just after 10 μs averaging times in the disk stabilized laser, achieving down to 10−11. This implies a fast linewidth of just 2 kHz. At slightly longer averaging times the value of just below 10−10 corresponds to the 13 kHz linewidth measured over 100 ms using the self-heterodyne technique in the previous section. From Eq. (5.3) it becomes clear that due to the large errors imposed on each beat signal by the lower performance of the ECDL, the error in the separated variances becomes very large. In practice this meant that statistically, for some averaging times, the three-cornered hat method produced negative results in the Allan variance, resulting in imaginary numbers for the Allan deviation — clearly not a physical result. We calcu- 86 Calcium Fluoride Stabilized Fiber Laser

10 ECDL & CaF2

6 DL Pro & CaF2

Frequency (MHz) Frequency

DL Pro & ECDL 2

0 0510 Time (ms)

Fig. 5.10: Three simultaneous beats between the homebuilt ECDL, the Toptica DL Pro, and the WGM stabilized laser. The homebuilt ECDL is more noisy than the other two lasers. A 1.4 kHz harmonic was removed from both beats containing the ECDL. lated the Allan deviations of the beat frequencies over every possible averaging time up to 1 ms, and performed three-cornered hat calculations on each data point [Fig. 5.13]. The general trend of each laser is evident in the graph. The CaF2 laser shows the best performance at short averaging times, and compares to the Toptica DL Pro at longer averaging times. The ECDL is around an order of magnitude worse. These results agree with the self-heterodyne measurements for all three lasers. For more precise determination of the lasers performance, a more stable second reference laser would have been needed. The Toptica DL Pro has a linewidth and stability comparable to the disk stabilized laser — although not quite as good. The error due to the larger Allan deviations of the beats including the ECDL results in large uncertainty in the measurement of the more stable lasers. Over short measurement cycles of just 10 ms the three lasers should not be correlated. To be sure, a correlation removal method proposed by A. Premoli et al. [96]was implemented. Further theoretical details and calculations are explained in detail in sec- 5.4 Three-Cornered Hat Measurement 87

DL Pro & ECDL CaF2 & ECDL DL Pro & CaF2

1E-9

(τ)

Allan Deviation σ 1E-10

1 10 100 Time τ (μs)

Fig. 5.11: Allan deviations of the three combined beats. The DL Pro and ECDL beat is very close to that of the CaF2 stabilized laser and the ECDL. tion 2.1.3. In theory, an arbitrary laser of the three to be investigated can be chosen as a reference, whereby the covariance matrix between the other two sets of data is calculated. A computer program was written which could read in the raw frequency data and process all the information according to the proposed method. Fig 5.14, Fig 5.15, and

Fig 5.16 show the correlation-removed Allan deviations with the CaF2 stabilized laser as a reference, the DL Pro as a reference, and the ECDL as a reference respectively. If the three Allan deviations with correlation removal are compared, it becomes evident that the method fails if one of the lasers has a much lower stability than the other two. The values for the Allan deviations at varying averaging times for the ECDL are nearly identical, and coincide with the values previously calculated using the classical three-cornered hat method. If the disk stabilized laser is chosen as a reference, its performance is signiﬁcantly better than that of the DL Pro. Nevertheless, choosing the DL Pro as a reference results in it having the lowest deviation. Setting the ECDL as an arbitrary reference, once again, conﬁrms that the disk stabilized laser performs better.

Furthermore, outlying points can be observed, for example, in Fig. 5.14 on the CaF2 88 Calcium Fluoride Stabilized Fiber Laser

DL Pro 1E-9 ECDL CaF2

(τ)

1E-10

Allan Deviation σ

1E-11

1 10 100 Time τ (μs)

Fig. 5.12: Allan deviations calculated for the separate lasers using the three-cornered hat method. reference curve at 8 and 128 μs averaging times. In these cases, the correlation removal method results in zeros, hence the classical three-cornered hat method is automatically applied instead. In conclusion, the large uncertainty introduced by the ECDL results in very large uncertainty in separated Allan deviations for the better performing lasers. The classical three-cornered hat gives the best indication of the performance of each laser, and corresponds to the measurements conducted using the self-heterodyne technique in the previous section. Future experiments should include a further stable reference laser such as a DL Pro, as opposed to the ECDL. 5.4 Three-Cornered Hat Measurement 89

DL Pro 1E-9 ECDL CaF2

(τ)

1E-10

Allan Deviation σ

1E-11

1 10 100 1000 Time τ (μs)

Fig. 5.13: All Allan deviations calculated for the lasers using the three-cornered hat method. Invalid results are ignored.

1E-9 DL Pro MP 100

(τ) y CaF2

1E-10

Allan Deviation σ

1E-11 1 10 100 1000 Time τ (μs)

Fig. 5.14: Correlation-removed Allan deviations with CaF2 laser as reference. 90 Calcium Fluoride Stabilized Fiber Laser

1E-9

(τ)

DL Pro MP 100 CaF2

Allan Deviation σ 1E-10

1E-11 1 10 100 1000 Time τ (μs)

Fig. 5.15: Correlation-removed Allan deviations with Toptica DL Pro laser as reference.

1E-9

(τ)

DL Pro MP 100 CaF2

1E-10

Allan Deviation σ

1E-11

1 10 100 1000 Time τ (μs)

Fig. 5.16: Correlation-removed Allan deviations with ECDL as reference. Chapter 6

Atmospheric Transfer of Frequencies

This chapter describes an experiment conducted to disseminate ultra-stable optical frequencies through free-space [126]. For decades, stable frequencies have been transmitted through copper wires, but the noise accumulated after fairly short distances can be quite dramatic. More recently, the array of global positioning satellites (GPS) have been used to disseminate microwave atomic clock signals through the atmosphere. With enough averaging, it is sufficient for conventional atomic clock signal dissemination and comparison. With the advent of technology, clocks at optical frequencies are currently superseding microwave clocks in accuracy and stability. As a consequence, GPS dissemination is no longer sufficient. The best results have been achieved in dissemination of ultra-stable laser frequencies through optical fibers. A phase compensation technique can be applied, allowing dissemination of frequencies with noise significantly below the best microwave and optical clocks available to date. Often, there may not be a fiber network readily available, so dissemination of optical frequencies through the atmosphere can be of significant interest. Section 6.2 describes the setup of an interferometer to measure phase noise due to atmospheric turbulence over a 100 m roundtrip, set up on the top of the Max Planck Institute’s roof. Next, the results of transferring a stable frequency are described in section 6.3. Finally, the laser was intensity-modulated at a radio frequency, and its stable frequency was transferred. The results are reported in section 6.4.

6.1 Introduction to Frequency Dissemination

There are many important applications of long distance transfer of atomic and optical clock signals. Currently, an SI time unit second is defined as the duration of 9 192 631 770 92 Atmospheric Transfer of Frequencies periods of the oscillation of radiation emitted by the transition between the hyperfine splitting of the ground state of cesium 133 atoms at 0 K at rest [45]. The fractional frequency uncertainty of the best cesium fountain clocks is on the order of 5 × 10−16 [102]. Better results can be achieved with optical clocks. Recently scientists at the National Institute of Standards and Technology (NIST) have reported an optical clock based on Al+ transitions with fractional frequency inaccuracy of 8.6 × 10−18 [48]. Transitions in the visible and ultraviolet range in various atoms and ions have been shown to have extremely high accuracy and stability, superseding that of Cs atomic clocks. Generally, a higher carrier frequency allows a better stability. To date, optical clocks take up large amounts of space in a laboratory, and are not easily portable. Precise comparison between optical clocks is necessary, so the precise dissemination of clock signals has become of interest [127, 128]. Furthermore, accurate time dissemination would provide the possibility for less well equipped labs to have access to ultrastable frequencies without the need for an optical clock [129]. Additionally, there are applications in searching for time variations in fundamental constants [46], very long baseline interferometry (VLBI) in radio astronomy [47], sensitive gravity wave searches in the NASA Deep Space Network (DSN), and accelerator physics. Fiber networks have provided the possibility for accurate frequency and time transfer over very large distances. Phase compensation techniques can be applied to suppress phase noise incurred through vibrations and temperature fluctuations in fibers, allowing stabilities higher than the current best optical clocks [105]. Unfortunately, in some cases a fiber network might not be available, so there is a need for transfer of stable frequencies through free space. Very little research has been done in this field. Here, a setup is described in which stable frequencies are transferred over a 100 m roundtrip, and the stabilities of the transferred clock signals are measured.

6.2 Setup

One possibility to transfer frequencies is by intensity-modulating a laser at a radio frequency, and directly detecting the modulation after transfer. Another, more promising method is to directly lock the frequency of a laser to an ultra-stable source and transfer the frequency itself. This method has shown the best results in optical fibers, due to the higher carrier frequency. In practice, an optical frequency source may be in the optical or ultraviolet region of the electromagnetic spectrum, which may not be the ideal frequency for transfer through fibers or the atmosphere due to dispersion and absorption. There are 6.2 Setup 93 low absorption windows in both fibers and the atmosphere around 1.55 μm, so it is advantageous to use a 1.55 μm laser as a clock transfer laser. An optical frequency comb has a very broad spectrum spanning more than an octave, so the idea is to lock a spectral line of the comb to the optical clock, and then lock the 1.55 μm laser to the frequency comb. If the carrier envelope phase and the repetition rate are locked to the optical clock, and the transfer laser is locked to the comb, the frequency of the transfer laser can be as stable as the reference source itself. After transferring through fiber or the atmosphere, another frequency comb that is locked to another optical clock can be used to compare two optical clock signals. Initial experiments were conducted by setting up a Mach-Zehnder interferometer in the laboratory with a short reference arm, and a longer arm, which was perturbed using heat sources and fans. Some phase noise could be detected using a fan as well as a heat plate placed underneath the beam, although results do not necessarily simulate the outdoor atmospheric conditions. A more realistic experimental setup was necessary.

Diode Laser 50m 1550nm

Retroreflector AOM

Spectrum PD Analyzer

Fig. 6.1: Frequency noise measurement setup. An ECDL is split into two arms. One arm is transmitted over 50 m and reflected back by a retroreflector. The other arm is offset by 165 MHz using an acousto optic modulator (AOM). They are recombined on a photodiode (PD) and analyzed using an rf spectrum analyzer, an FFT spectrum analyzer, or a frequency counter.

Following, a technique is described to measure the phase noise induced by transfer through the turbulent atmosphere. A stable 1550 nm Littman-Metcalf stabilized diode laser is used (Tunics PRI 1550) in these experiments. The experimental setup, as illustrated in Fig. 6.1, was constructed on the roof of the Max Planck Institute building. A retroreflector was mounted on a tripod which could be moved over distances between 5 to 50 m from the rest of the optical setup on the roof. The rest of the indicated optics were mounted on an optical bench on a stable table inside of the top of the staircase, with the door open for experiments. The diode laser is fiber coupled, and a large commercial 94 Atmospheric Transfer of Frequencies collimating lens was used to collimate the beam. The 1/e2 intensity diameter is 7 mm, and about 14 mm after 100 m of propagation. Gold-coated mirrors with a two inch diameter were chosen to efficiently reflect 1550 nm, and to be large enough to capture the beam after long propagation. Beam splitting cubes were non-polarizing with a one inch side. The setup was a Mach-Zehnder interferometer: first, the collimated beam was split using a beam splitter. Half of the 5.5 mW radiation was used as a reference, and passed through an acousto optic modulator at 165 MHz to offset the frequency and make heterodyne detection possible. The other half was sent through the open door across the roof and into the gold-coated retroreflector, with an aperture of 63.5 mm in diameter. After a roundtrip of up to 100 m, the signals were recombined on another beam splitter, and then focused onto a fast photodiode (MenloSystems FPD510). The resulting beat signal indicated the phase noise accumulated from the long arm. The linewidth of the tunable diode laser was specified at 150 kHz, corresponding to a coherence length of over 600 m, hence should not affect the measurement seriously. All devices used for measurements were locked to a rubidium atomic clock (Stanford Research Systems FS725).

Fig. 6.2: Photograph showing the rooftop setup. Laser beam paths included for clarity.

A photograph showing the setup is indicated in Fig. 6.2. Laser beam paths are digitally 6.2 Setup 95 enhanced for clarity. The fiber connects to the collimator at the bottom right, and the beam is split in two parts. The recombined beams are overlapped on the detector, shown on the top left. A diode laser at 650 nm was also included with a flipper mirror between the last two exit mirrors to simplify alignment over large distances. At first the diode laser was coupled into a fiber and attached to the same collimator as the 1550 nm laser, but this proved to make things more difficult since the red beam focused after just 60 cm due to chromatic aberration in the collimating lens. Instead, the flipper mirror was included, and two pinholes were used to precisely overlap the 1550 nm beam with the 650 nm beam. To adjust this even more precisely, a target was set up 5 m away from the optical bench, and the two beams were overlapped. Once this was done, the beams were generally more or less overlapped after transmission over 100 m as well. Then, alignment at any position of the retroreflector could be performed by turning on the red laser, preferably in darkness after sunset, and adjusting the height and position of the retroreflector on the tripod such that the red beam entered it. Once a weak beat signal was visible, various parameters — mainly angles of the mirrors — were adjusted to maximize the 165 MHz beat signal when the detector was attached to a radio frequency spectrum analyzer (HP 4195A).

(a) Retroreﬂector (b) View from optical setup

Fig. 6.3: The 63.5 mm aperture gold retroreﬂector could be placed up to 50 m away from the optical setup.

Fig. 6.3 shows a photograph of the retroreﬂector, mounted onto the tripod, as well as a view from the optical bench out of the door toward the end of the roof. After 100 m of propagation, beam sway due to refractive index variations from wind and temperature currents became apparent. In total, the beam swayed back and forth approximately 10 mm at a frequency on the order of a few Hz. Nonetheless, the large mirrors and large focusing lens used for the detector generally allowed a fairly stable beat signal to be visible. On sunny days, the heat currents emitted from the dark roof would cause signiﬁcant 96 Atmospheric Transfer of Frequencies

-70

20 -80

-90 3dB width (kHz)

Intensity (dBm) 10

-100

0 164.95 165.00 165.05 0 1020304050 Frequency (MHz) Distance (m) (a) Beat signals (b) Linear increase

Fig. 6.4: Left, beat signals showing an increase in frequency noise for 5 m, 20 m, 35 m, and 50 m. Right, beat signal 3 dB width as a function of distance showing a linear increase. beam sway, so these were avoided in experiments. Also, wind caused significant fluctuation of the beam, and led to wider beat signals. Future experiments in atmospheric transfer need to include active beam pointing stabilization, especially if signals are to be transferred from one building to another due to sway in tall buildings. The experimental data in Fig. 6.4 was taken late on a windless evening to prevent thermal fluctuations, and make the alignment of the beam easier. On the left, four beat signals are shown as recorded using an rf spectrum analyzer at distances of 5 m, 20 m, 35 m, and 50 m (times two for the roundtrip). The FWHM at 3 dB for each of these, and further data points at 5 m intervals are shown on the right. The slope corresponds to about 250 Hz per meter.

6.3 Allan Deviations, Optical Transmission

The Allan deviation is a commonly quoted value to describe the stability of an oscillator or a clock [61]. Diﬀerent averaging times can be used to evaluate whether the short-term or long-term performance of a frequency standard is better. More details on the Allan deviation can be found in Chapter 2. Following is the basic formulation for the Allan variance. 2 1 2 σ (τ) = (¯y + − y¯ ) , (6.1) y 2 n 1 n where τ is the observation period, andy ¯n is the nth fractional frequency average over time τ. It can give a good indication of the types of noise processes involved in the instability. 6.3 Allan Deviations, Optical Transmission 97

Here, we use the Allan deviation as a measure of the performance of the transmission of frequencies. A frequency counter (Agilent 53181A) was used with a gate time of 1 s. Hence, the counter gave an average value of the frequency over one second; determined using zero-crossings counting. Values were recorded over approximately one hour, and Allan deviation calculations were performed on them. Ideally, a frequency dissemination technique should have a better stability than the frequency standards it is meant to disseminate. If this is not the case, extremely long averaging times might be needed to reach the required stability level for clock comparison. Experiments in ﬁbers have shown stabilities down to 3 × 10−15 at 1 s, and down to 1 × 10−19 after 30 000 s [49]. This precision would be enough for comparing even the best optical clocks to date, over a distance of 146 km between Hannover, Germany, and the PTB (Physikalisch Technische Bundesanstalt) in Braunschweig, Germany.

10 -12 500

Counts 250 10 -13 (τ) y 0 -100 0 100 Frequency Offset (Hz) 10 -14 Allan Deviation σ 10 -15 10 0 10 1 10 2 10 3 10 4 Averaging Time τ (s)

Fig. 6.5: Allan deviation for optical transfer of frequencies over 100 m ﬁrst showing a τ−1/2 dependence, and later τ−1. Inset, histogram of beat signals with a Gaussian ﬁt, FWHM = 70.5 Hz.

In Fig. 6.5 the Allan deviation for free-space optical transfer with no phase compensation is shown. At one second averaging, the precision is already 1.68 × 10−13, and the 98 Atmospheric Transfer of Frequencies highest precision is reached after about 2000 s of averaging at 10−15. Further phase compensation and a lower linewidth laser should allow even better precision. The slope of the line up to 500 s follows a τ−1/2 dependence, and later follows a τ−1 dependence. The underlining noise processes can be inferred from these slopes [64]. At shorter averaging times up to 500 s this implies a white frequency noise character, and at longer averaging times the steeper τ−1 dependence implies a white phase noise character. The inset shows a histogram with a Gaussian ﬁt with a FWHM of 70.5 Hz, which corresponds quite well with the short term linewidth of about 32.4 Hz, calculated using the 1.68 × 10−13 Allan deviation at 1 s and the 193 THz carrier frequency.

6.4 Allan Deviations, Radio Frequency Transmission

Another common technique that has been used to disseminate frequencies is to modulate the intensity of a laser at a radio frequency and collect the signal on a photodiode. Before frequency combs became commercially available, this technique allowed direct sending and collecting of radio frequencies, so it was the preferred method. In this setup, the diode laser was amplitude modulated using an external frequency synthesizer (Rhode & Schwarz SMH) at 80 MHz; once again locked to the rubidium reference clock. An interferometric technique was not necessary, so the reference arm and acousto optic modulator were blocked. The modulated laser beam was reflected after 50 m from the retroreflector, and focused onto the photodiode as before. The 80 MHz signal was detected and electronically beat with an 81 MHz reference signal (from another frequency generator) to produce a 1 MHz beat, which could be processed using a counter. The resulting Allan deviation is shown in Fig. 6.6. At one second of averaging the result is about three orders of magnitude worse than the optical frequency transfer experiment, at 1.31 × 10−10. This corresponds to previous results reported in fibers [128, 130, 131], in which the lower carrier frequency performs worse than using the optical frequency directly. It is important to note that this result is quite close to the limit of the rubidium reference clock. When the two frequency generators were beat directly (an 80 MHz electronic signal with an 81 MHz signal), and the resulting 1 MHz beat was analyzed using the counter, the resulting Allan deviation was only marginally better than the dissemination result. At 1 s, the Allan deviation was 1.07 × 10−10, compared with 1.31 × 10−10, and similarly for the further points. For a more precise experiment, a better reference, such as a cesium clock or hydrogen maser, would have been needed. 6.4 Allan Deviations, Radio Frequency Transmission 99

10 -9 0.012

0.006 (τ) y 10 -10 0.000 Intensity (mW) -4 -2 0 2 4 Frequency Offset (mHz)

10 -11 Allan Deviation σ

10 0 10 1 10 2 10 3 Averaging Time τ (s)

Fig. 6.6: Allan deviation for 80 MHz rf transfer of frequencies over 100 m. Inset, spectrum of the beat signal measured over 30 min with Gaussian ﬁt, FWHM = 1.05 mHz.

Additionally, the 50 m transmitted beat was analyzed using a fast fourier transform analyzer (Stanford Research Systems SR760). For this experiment, the 80 MHz signal was mixed down to 10 kHz with the HP 8656B frequency synthesizer, and the highest resolution bandwidth (0.476 mHz) was used. A thirty minute frequency spectrum is shown in the inset of Fig. 6.6. The FWHM is 1.05 mHz — close to the resolution limit of the FFT spectrum analyzer — and corresponds to a stability of 1.3 × 10−11. In conclusion, a free space laser link is feasible, although much longer distances can be traversed through ﬁbers. Even unstabilized free space links over tens of meters can have stability down to 10−15 with some averaging. At 1 s the optical transmission had an Allan deviation of 1.68 × 10−13, and using amplitude modulation it was 1.31 × 10−10. For many applications this precision may be enough, and for higher precision a locked link could be implemented. A more stable laser may have been useful, since the 100 m roundtrip was approaching the coherence length of the laser. Recently, other groups have focused on optimizing free space links, including optimized telescope designs, normally used for astrophysics, stabilized and locked lasers, as well as high precision atomic clocks 100 Atmospheric Transfer of Frequencies

[132, 133]. Results are promising, and free space links will become more commonly used in the future, as alternatives to costly ﬁber networks. Chapter 7

Conclusion

7.1 Thesis Summary

The work presented in this thesis dealt with the stabilization of two laser systems using whispering gallery mode resonators, as well as the dissemination of stable frequency sources in the optical regime through the turbulent atmosphere. The WGM resonators were used as filters by coupling into and out of different resonators using prism coupling, tapered fiber coupling, and angle-polished fiber coupling. Two laser systems were set up in a ring configuration using different gain media — both in the 1550 nm telecom region. Various linewidth measurement techniques were applied and investigated. Furthermore, a rooftop experiment was set up with a 100 m roundtrip phase noise measurement setup, and optical versus radio frequency dissemination was examined. An erbium-doped fiber ring laser pumped by a 980 nm diode laser was set up. To stabilize the frequency, a microsphere was fabricated by melting the tip of a telecom fiber 8 into a spherical shape. The quality factor (ν0/Δν) was determined to be 10 . We coupled into the microsphere using a tapered fiber with a large evanescent field due to the 1 μm diameter, and out of the other side of the microsphere using an angle-polished fiber to create a frequency selective element. By adjusting the coupling parameters we induced single-mode lasing and studied the linewidth with an optical spectrum analyzer, as well as the heterodyne beat technique using a stable homebuilt external cavity diode laser. The linewidth was determined to be 170 kHz, which is an order of magnitude better than the passive linewidth of the microcavity. Furthermore, the pump power and temperature dependence was investigated. An increasing pump power lead to the optical Kerr effect, the thermo-refractive index, as well as physical cavity expansion — all resulting in a total 102 Conclusion red-shift of 16 pm/μW of pump power. Subsequently, an improved laser system was designed and built. In this case, a more rigid crystalline calcium fluoride (CaF2) disk was fabricated using a diamond turning machine, and prism coupling was implemented. The quality factor of the disk was measured to be 107. Since heterodyne beating with a Toptica DL Pro external cavity diode laser was resolution limited (70 kHz linewidth), we implemented the self-heterodyne technique for a more precise linewidth determination. Using a 45 km fiber delay line and an acousto-optic modulator the upper limit for the linewidth was shown to be 13 kHz — an enhancement of one thousand compared to the passive linewidth. Further stability measurements were conducted using the three-cornered-hat measurement, in which three lasers were cross-beat to determine the absolute stability of the WGM stabilized laser — a novel technique in the optical regime. The separated Allan deviation result was an astounding 10−11 after 10 μs of averaging. Finally, the phase noise induced in atmospheric transfer of stable laser frequencies was presented. An interferometer was devised and set up. One arm was offset in frequency using an acousto-optic modulator, and the other was transmitted over 50 m and reflected back for a 100 m roundtrip, then recombined on a fast photodiode for heterodyne detection. This way the optical phase noise, which was induced by temperature and pressure variations in the air, could be measured. At 1 s averaging the precision was already 1.68 × 10−13, and at 2000 s it reached 10−15. This stability is sufficient for dissemination of most atomic clocks. For comparison, the outgoing laser beam was amplitude modulated and the radio frequency signal was detected directly and electronically beat with a reference signal (locked to a rubidium reference clock). The resulting Allan deviation had a stability of 1.07 × 10−10 at 1 s averaging, which is in accordance with previous experiments in fibers, in which optical transmission has higher stability than radio frequency amplitude modulated transmission.

7.2 Future Outlook

The future for whispering gallery mode resonators looks bright, and it is likely that more laser stabilization experiments will make use of them. The two stabilized laser setups presented here were essentially prototypes, which can still be reﬁned in various aspects. The atmospheric transfer experiment already led to some compelling results, but longer distances and phase compensation can be examined in the future. 7.2 Future Outlook 103

The quality factors that we achieved in the calcium fluoride disks were 107, so there 11 is still a lot of room for improvement. State of the art CaF2 disks can have up to 10 quality factors [44]. Current work is focusing on an upgraded diamond turning machine that should create better disks to start with, and new techniques for polishing and cleaning the disks should decrease scatterers and surface roughness on the surface of the disks, and thereby lead to a much longer photon confinement. Currently, the linewidth is about 15 MHz, but due to multiple roundtrips in the ring cavity configuration the resulting linewidth is just 13 kHz — a factor of 1000 better. A disk with a quality factor of 1010 or even 1011 should allow sub-kHz linewidth laser stabilization. Furthermore, active stabilization, such as gas cell locking, combined with this setup should decrease the linewidth even more. More isolation from the environment should decrease the intensity fluctuations. Currently the filters were in a normal air environment with little isolation from air currents and temperature variations. A more compact setup could use two angle-polished fibers coupled to a crystalline resonator. Atmospheric transfer experiments are being conducted in various research laboratories these days. Further improvements to the current setup should include a laser with a narrower linewidth, to ensure no self-heterodyne type broadening. Longer distances need to be explored, which would involve a better telescope setup for a larger beam with less divergence. Finally, a phase compensation scheme would need to be applied, which can improve performance by two orders of magnitude in optical fibers [49], and is likely to have a similar effect in free space frequency dissemination. In conclusion, the results presented in this thesis aid the understanding of WGM filtering devices as a means of passive laser stabilization, as well as the advancement of frequency dissemination in free space using laser frequency transfer. 104 Conclusion Bibliography

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Allan variance, 84, 96 microspheres, 65 optical transmission, 96 theory, 25 rf transmission, 98 Schawlow-Townes limit, 12, 72 theory, 15 theory, 12 Atmospheric Transfer, 91 Self-heterodyne beating, 37, 80 Experimental setup Stabilization techniques, 1 atmospheric transfer, 92 Stabilized laser crystalline resonator stabilization, 75 calcium ﬂuoride, 75 microsphere stabilization, 61 microsphere, 61

Fabrication Temperature dependence angle-polished fibers, 56 microspheres, 72 crystalline resonators, 50 Theory, 11 microspheres, 48 Three-cornered hat, 83 tapered fibers, 54 correlation removal, 40, 86 Frequency dissemination, 43, 91 theory, 39 fibers, 45 Whispering gallery modes GPS, 44 coupling, 27 history, 7 experimental, 47 Heterodyne beating, 35, 68, 80 fundamental limits, 34 history, 4 Lasers Kerr effect, 33 bad cavity, 14, 72 prism coupling, 51 fiber, 59 ray model, 17 linewidth, 11 theory, 16 Littrow stabilized, 56 thermal expansion, 34 thermal refractive index, 33 Quality factor wave model, 19 CaF2, 78 Curriculum Vitae

Personal Information

Name: Benjamin Sprenger Born: 6. June, 1985 Gender: Male Nationality: German

Education

10/2007–present: PhD: Doctorate Thesis with Prof. Dr. Lijun Wang. Max Planck Institute for the Science of Light, Erlangen, Germany 10/2006–09/2007: MSc (Hons): Master of Science in Optics and Photonics. Physics Department, Imperial College London, UK 10/2003–06/2006: BSc (Hons): Bachelor of Science in Physics. Physics Department, Imperial College London, UK 08/2001–06/2003: Bilingual International Baccalaureate Diploma (IB). Frankfurt International School, Germany 118 Curriculum Vitae Publications

Journal Articles

• B. Sprenger, J. Zhang, Z. H. Lu, and L. J. Wang, “Atmospheric transfer of optical and radio frequency clock signals,” Opt. Lett. 34, 965–967 (2009).

• B. Sprenger, H. G. L. Schwefel, and L. J. Wang, “Whispering-gallery-mode-resonator- stabilized narrow-linewidth ﬁber loop laser,” Opt. Lett. 34, 3370–3372 (2009).

• B. Sprenger, H. G. L. Schwefel, Z. H. Lu, S. Svitlov, and L. J. Wang, “CaF2 whispering-gallery-mode resonator stabilized narrow-linewidth laser,” Opt. Lett. 35, 2879–2872 (2010).

Conference Contributions

• B. Sprenger, H. G. L. Schwefel, and L. J. Wang, “Whispering gallery mode resonator stabilized ﬁber loop laser,” Proc. CLEO Europe 2009 (Poster).

• B. Sprenger, H. G. L. Schwefel, and L. J. Wang, “Spherical microcavity stabilization of a ﬁber loop laser,” Proc. Frontiers in Optics 2009 (Oral).

• B. Sprenger, H. G. L. Schwefel, and L. J. Wang, “Frequency stable ﬁber ring laser based on whispering gallery modes,” Proc. DPG 2010 (Oral).

• B. Sprenger, H. G. L. Schwefel, and L. J. Wang, “Auto-stabilization of ring lasers based on whispering gallery mode resonators,” Proc. CLEO 2010 (Oral).