ABSTRACT Title of Dissertation: NONLINEAR AND STOCHASTIC ANALYSIS OF MINIATURE OPTOELECTRONIC OSCILLATORS BASED ON WHISPERING-GALLERY MODE MODULATORS Helene Nguewou-Hyousse Doctor of Philosophy, 2021 Dissertation Directed by: Professor Yanne K. Chembo Department of Electrical and Computer Engineering Optoelectronic oscillators are nonlinear closed-loop systems that convert opti- cal energy into electrical energy. We investigate the nonlinear dynamics of miniature optoelectronic oscillators (OEOs) based on whispering-gallery mode resonators. In these systems, the whispering-gallery mode resonator features a quadratic nonlin- earity and operates as an electrooptical modulator, thereby eliminating the need for an integrated Mach-Zehnder modulator. The narrow optical resonances eliminate as well the need for both an optical ber delay line and an electric bandpass lter in the optoelectronic feedback loop. The architecture of miniature OEOs therefore appears as signicantly simpler than the one of their traditional counterparts, and permits to achieve competitive metrics in terms of size, weight, and power (SWAP). Our theoretical approach is based on the closed-loop coupling between the optical intracavity modes and the microwave signal generated via the photodetection of the output electrooptical comb. In the rst part of our investigation, we use a slowly-varying envelope approach to propose a time-domain model to analyze the dynamical behavior of miniature OEOs. This model takes into account the interactions among the intracavity modes, as well as the coupled interactions with the radiofrequency (RF) microstrip. The stability analysis allows us to determine analytically and optimize the critical value of the feedback gain needed to trigger self-sustained oscillations. It also allows us to understand how key parameters of the system such as cavity detuning or cou- pling eciency inuence the onset of the radiofrequency oscillation. Furthermore, we determine the threshold laser power needed to trigger oscillations in amplier- less miniature OEOs based on WGM modulators. This latter architecture, while also improving on the size, weight, performance and cost (SWAP-C) constraints, is intended to reduce noise in the system. In the second part of our investigation, we use a Langevin approach to perform a stochastic analysis of our miniature OEO. We propose a stochastic mathematical model to describe the system dynamics and analyze the stochastic behavior below threshold. We also propose a normal form approach for the noise power density and the phase noise spectrum. Our study is complemented by time-domain simulations for the microwave and optical signals, which are in excellent agreement with the analytical predictions. In the third part of our study, we discuss our preliminary results in the analysis of the eects of dispersion in a microcomb oscillator with optical feedback. For this purpose, we propose a closed-loop miniature optical oscillator. The output signal is optically amplied before being coupled back into the cavity using a prism coupling. Using a Lugiato-Lefever approach, we propose a spatiotemporal nonlinear partial dierential equation to describe the dynamics of the total intracavity eld. We perform temporal and spatial analysis and derive the bifurcation maps in anomalous and normal dispersion regimes. NONLINEAR AND STOCHASTIC ANALYSIS OF MINIATURE OPTOELECTRONIC OSCILLATORS BASED ON WHISPERING-GALLERY MODE MODULATORS by Helene Nguewou-Hyousse Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulllment of the requirements for the degree of Doctor of Philosophy 2021 Advisory Committee: Professor Yanne K. Chembo, Chair/Advisor Professor Thomas E. Murphy Professor Kevin Daniels Professor Mario Dagenais Professor Rajarshi Roy ? Copyright by Helene Nguewou-Hyousse 2021 Dedication To my parents Dr and Mrs. Nguewou. To my husband Patrick Hyousse, and our daughters Johanna, Einya, Yahnisse and Yahleen. ii Acknowledgments Wu m? nk? ngu y? l? nz? ng? m? nz? sa'.1 Bamena Proverb This journey has been one of faith, hard-work, friendship, and self-discovery. I could never have done it without my Lord, but also the support and love of many people whom I would like to acknowledge here. First and foremost, I would like to thank the most awesome adviser Professor Yanne Chembo for taking me on as one of his graduate students. As a member of his group, I have been challenged to think critically, use my skills, and gain new ones to solve stimulating problems in the area of microwave photonics. As a result, I have learned tremendously about the eld and I have equipped myself with many valuable skills that make me a better engineer and researcher. In addition, his mentoring has also impacted me in so many ways. For all these reasons, he is one of my role models, and I feel honored to be a member of the Photonics Systems Lab for Aerospace and Communication Engineering (PSACE Lab). I would also like to thank my dissertation committee Prof. Tom Murphy, Prof. Kevin Daniels, Prof. Raj Roy, and Prof. Mario Dagenais for taking the time out of their busy schedule to review this manuscript and serve on my thesis 1As long as you carry a child, he doesn't know that the road is long. iii committee. I am also indebted to President Daryll Pines, Prof. Ankur Srivastava, Prof. Percy Pierre, Prof. Gil Blankenship, Prof. Andre Tits, Prof. Derek Paley, and Melanie Prange for mentoring. I would also like to thank the ECE graduate oce and IREAP sta for continued support and taking care of the administrative duties so that I could focus on my research. Thank you Dorothea (Dottie) Brosius for not only developing the UMD Latex thesis template, but also for stopping by my oce often to inquire about my well-being. Many thanks also to Ms. Parker and the UMD Center for Minorities in Science and Engineering for nancial support through NSF LSAMP fellowship in my early years, and for always being a place of learning and sharing. I would like to take this opportunity to express my gratitude to Prof. Asamoah Nkwanta, Prof. Gerald Rameau, Prof. Craig Scott and Prof. Iheyani Eronini at Morgan State University. You have taught and mentored me through the years, and you have laid the foundations of the engineer and person that I am. I am proud to call myself a Bear. Professor Pablo Iglesias, your classes at Johns Hopkins University introduced me to nonlinear feedback control, and this has become my universe. Thank you for being a wonderful mentor and a friend throughout the years. I have been inspired in this journey by my older academic siblings, those who have walked this walk before me at College Park and have been great friends and examples: Dr. Jarred Young, Dr. Lina Castano, Dr. Sarah Mburu, Dr. James Lankford, Dr. Mukul Kulkarni & Devika, Dr. Vidya Raju, Dr. Proloy Das and Dr. Richard (Ricky) Brewster. I also want to acknowledge my friends who are still in the iv journey, but have contributed in giving me a sense of family here at College Park. Thank you Franklin Presi Nouketcha, Landry Horimbere, Nehemiah Emaikwu, Reza Hadadi, and in particular my colleagues at the PSACE Lab: Fengyu Liu, Meenwook Ha and Haoying Dai. I have received great support and balance from outside academia. To this wonderful circle, I would like to express my sincere gratitude: my wonderful mother Jeannette Nguewou who sacriced so much to see me through this season, the Hebou family, my siblings: Patrick, Yvette, Stephane and Nadia, my sisters from other mothers Carine Ndamfeu and Christelle Njomgang, my bros Hermann Nganwa and Yannick Tchatchoua, Minister Edith Tengen, Evang. Patrick Kamdem and my other family the Stevens. And last but certainly not least, to my husband Patrick: this journey required your sensitivity, love, patience and strength. Thank you for taking it with me, and I love you. An African proverb says, It takes a village to raise a child. I would like to apologize to those I have inadvertently left out, as I am sure they are many. To all, receive my heartfelt gratitude! v Table of Contents Dedication ii Acknowledgements iii Table of Contents vi List of Tables x List of Figures xi List of Abbreviations xvii List of Publications xviii Chapter 1: General Introduction 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Broadband Optoelectronic Oscillators . . . . . . . . . . . . . . . . . . 4 1.2.1 Modeling Ikeda-like OEOs . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Broad Bandpass Filter OEOs . . . . . . . . . . . . . . . . . . 6 1.3 Narrowband OEOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3.1 Early Architectures . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Modeling Delayed-based Narrowband OEOs . . . . . . . . . . 9 1.4 Applications of Optoelectronic Oscillators . . . . . . . . . . . . . . . 11 1.4.1 Ultrapure Microwave Generation . . . . . . . . . . . . . . . . 11 1.4.2 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Motivation for this Work . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 2: Whispering-Gallery Mode-Based Modulators 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Mach-Zehnder Electro-Optic Modulator . . . . . . . . . . . . . . . . . 17 2.2.1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Techonological Applications of Mach-Zehnder Interferometers 18 2.3 Components of the WGM-Based Electro-Optic Modulator . . . . . . 19 2.3.1 Whispering-Gallery Mode Resonator . . . . . . . . . . . . . . 20 2.3.2 Photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.3 Radio Frequency Amplier . . . . . . . . . . . . . . . . . . . . 26 vi 2.4 WGM-Based Electro-Optic Modulator System . . . . . . . . . . . . . 27 2.4.1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 Quantum Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.2 Hamiltonian Equations . . . . . . . . . . . . . . . . . . . . . . 31 2.5.3 Time-Domain Equations . . . . . . . . . . . . . . . . . . . . . 33 2.6 Semi-Classical Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6.2 Slowly-Varying Envelope Approach . . . . . . . . . . . . . . . 34 2.7 Output Fields and Transmission Functions . . . . . . . . . . . . . . . 35 2.7.1 Output Optical and Microwave Field . . . . . . . . . . . . . . 36 2.7.2 Optical and Microwave Transmission Function . . . . . . . . . 38 2.8 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.8.1 Optical Modes Temporal Dynamics . . . . . . . . . . . . . . . 41 2.8.2 RF Cavity Temporal Dynamics . . . . . . . . . . . . . . . . . 41 2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Chapter 3: Miniature Optoelectronic Oscillators 44 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.1 Conventional OEO . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.2 Miniature OEO based on WGM modulator . . . . . . . . . . . 46 3.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.1 Open-Loop Semi-Classical Model . . . . . . . . . . . . . . . . 48 3.3.2 Closed-Loop Model . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Numerical Simulation of the Temporal Dynamics . . . . . . . . . . . 51 3.4.1 Voltage inside the RF Resonator Strip . . . . . . . . . . . . . 51 3.4.2 Optical and Microwave Output Power . . . . . . . . . . . . . . 52 3.5 Stability Analysis and Threshold Gain . . . . . . . . . . . . . . . . . 56 3.5.1 Trivial Equilibrium Points . . . . . . . . . . . . . . . . . . . . 56 3.5.2 Pertubation Analysis . . . . . . . . . . . . . . . . . . . . . . . 56 3.5.3 Reduced Jacobian Matrix . . . . . . . . . . . . . . . . . . . . 58 3.5.4 Routh-Hurwitz Analysis and Critical Gain . . . . . . . . . . . 60 3.6 Optimization: System Parameters Leading to the Smallest Threshold Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6.1 Optimal Laser Detuning from Resonance . . . . . . . . . . . . 67 3.6.2 Resonator Coupling Coecient . . . . . . . . . . . . . . . . . 68 3.6.3 Explicit Analytical Approximation of Critical Gain . . . . . . 70 3.6.4 Optimal Resonator Coupling Coecient . . . . . . . . . . . . 71 3.7 Threshold Laser Power in the Amplierless Miniature OEO . . . . . . 73 3.7.1 Input Laser Power and Critical Gain . . . . . . . . . . . . . . 74 3.7.2 Critical Laser Power in Amplierless Miniature OEO . . . . . 75 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 vii Chapter 4: Stochastic Analysis of Miniature Optoelectronic Oscillators 78 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Noise in Miniature OEOs . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.1 Sources and Eects . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.3.1 Noiseless System . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3.2 Noisy System . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4 Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.1 Stochastic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.4.3 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . 88 4.5 Stochastic Analysis Under Threshold . . . . . . . . . . . . . . . . . . 89 4.5.1 Pertubation Analysis and Reduced Flow Dynamics . . . . . . 89 4.5.2 Fourier Transform and Jacobian . . . . . . . . . . . . . . . . . 90 4.5.3 Microwave Output RF Power . . . . . . . . . . . . . . . . . . 92 4.6 Numerical Simulation of the Stochastic Dynamics . . . . . . . . . . . 94 4.6.1 Optical and Microwave Temporal Dynamics . . . . . . . . . . 94 4.6.2 Noise Power Density Below Threshold . . . . . . . . . . . . . 96 4.7 Normal Form Approach for Stochastic Analysis and Phase Noise . . . 97 4.7.1 Normal Form Approach for Stochastic Analysis . . . . . . . . 97 4.7.2 Normal Form Approach for Phase Noise . . . . . . . . . . . . 100 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Chapter 5: Miniature Optical Oscillator Based on Whispering-Gallery Mode Resonator 104 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 Miniature Optical Oscillator based on WGM Modulator . . . . . . . 106 5.3.1 Open-Loop Model . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.3.2 Closed-Loop Model . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4 Temporal Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 111 5.4.1 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4.2 Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.4.3 Temporal Behavior . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5 Spatial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5.1 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.5.2 Spatial Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . 115 5.5.3 Bifurcation Maps . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6 Ongoing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.6.1 Supercritical and Subcritical Turing Patterns . . . . . . . . . . 121 5.6.2 Number of Rolls in Turing Patterns in Anomalous Dispersion Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Chapter 6: Conclusions and Outlook 126 viii Bibliography 129 ix List of Tables 2.1 Measured Q-factor for various material and resonator shape. The Q-factor varies according to the fabrication material and resonator shape. Source: ref. [58, 59, 60] . . . . . . . . . . . . . . . . . . . . . 24 5.1 Eigenvalues and spatial bifurcations in the Lugiato-Lefever model . . 122 x List of Figures 1.1 Schematic of an optotelectronic oscillator. The system converts ligth energy from a laser pump into electrical energy. The system can out- put signal in both optical (? 50? 500 THz) and electrical frequency ranges (? 0? 100 GHz). The time-delay corresponds to a round-trip time. The system is nonlinear, dissipative and autonomous. . . . . . 2 1.2 First optoelectronic design proposed by Yao and Maleki in 1994 [10]. System consists of optical source that is pumped into a modulator. Optical output travels through an optical bre before detection by a photodiode; the electrical output is then amplied and ltered before being fed back into the modulator through an RF electrode. . . . . . 4 2.1 Intensity modulation using a Mach-Zehnder interferometer. Source: ref. [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Ray of light propagating inside WGMR through TIR. Here ` = 10 . Source: ref. [58]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 (a) Right: 3D representation of a WGMR cavity with l = 30 modes, n = 1 and l?|m| = 0. Right: 2D representation of the same dielectric cavity. (b) Examples of 2D transverse eld distributions for dierent radial and polar eigennumbers n and m. Source: ref. [42]. . . . . . . 21 2.4 WGMR is coupled to the light source through evanescent coupling. (a) Coupling through prism. (b) Waveguide side coupling. (c) Waveg- uide tip-coupling. Source: ref. [90]. . . . . . . . . . . . . . . . . . . . 22 2.5 WGM-based electro-optic modulator. Ain and Cin are the optical and microwave pump eld. Aout is the output photon ux; PD: Photodi- ode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 xi 2.6 Frequency-domain representation of photonic up- and down-conversion in a WGM resonator with ?(2) nonlinearity. These two processes can be leveraged to translate microwave energy to the optical do- main inside the WGM resonator. When belonging to the same fam- ily, the eigenmodes of the resonator with free-spectral range ? are R quasi-equidistantly spaced as ?l ' ?0 + l? , where l = ` ? `0 isR the reduced azimuthal eigenumber, and ?0 is the pumped resonance. (a) Photonic upconversion (stimulated): An infrared photon annihi- lates a microwave photon and is upconverted as h??l + h?? ? h??R l+1. (b) Photonic downconversion (stimulated or spontaneous): An in- frared photon emits a microwave photon and is downconverted as h??l ? h??l?1 + h?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31R 2.7 Output optical and microwave power of the rst harmonic as a func- tion of the detuning between the laser pump frequency and theWGMR's resonant pump frequency. (a) As ? approaches 0, almost all the emit- ted light goes into the WGMR and Popt,out approaches 0. (b) A small0 fraction of the emitted light is detected by the PD ans converted into RF signal. Fig. 2.7(a) is computed with Eq. (2.29), and Fig. 2.7(b) is computed with Eq. (2.33). . . . . . . . . . . . . . . . . . . . . . . 36 2.8 Optical power transmission of the rst three harmonics. The max- imum power of the rst harmonic approaches 1. The transmission power is computed with Eq. (2.34). . . . . . . . . . . . . . . . . . . . 38 2.9 Normalized microwave power transmission versus input microwave power P . Higher transmission is achieved at lower input cavity eld M . The transmission output power is computed with Eq. (2.35). . . . 39 2.10 Demodulated microwave power versus input microwave power P . M The absolute value of the demodulated RF signal is about 20 dB less than the input microwave power. The demodulated power is computed with Eq. (2.33). . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 Number of photons inside the cavities of a WGMR. Simulations were made for N = 3 modes and few photons initially. As time increases, more photons are found in the sidemode cavities, with an equal dis- tribution between modes l =| l |. | A |2?1 is overshadowed by | A 21 | . This graph is computed from Eqs. (2.25) and (2.26). . . . . . . . . . 41 2.12 Voltage inside the resonator strip. Simulation was carried for N = 3 modes and few photons initially. As time increases, the microwave volatge increases to a steady value determined by Eq. (2.36). . . . . 42 xii 3.1 Comparison between the architectures of conventional and miniature OEOs. The optical paths are in red, and the electric paths in black. Polarization controllers between the lasers and the modulators are generally necessary, but have been omitted here for the sake of sim- plicity. (a) Conventional OEO. MZM: Mach-Zehnder modulator; DL: Delay line; PD: Photodiode; PS: Phase shifter; BPF: Narrowband bandpass lter; Amp: RF amplier. (b) Miniature OEO. WGMR: Whispering-gallery mode resonator; The other acronyms are the same as in (a). Note that in the miniature OEO, the WGMR is a single component that replaces the MZM, the DL and the BPF in the con- ventional OEO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Amplitude of the closed-loop microwave signal at ? = 0.5, with Vcav being the voltage inside the RF resonator strip. The results were simulated using Eqs. (3.5) and (3.6). (a) ? = 10. (b) ? = 12. . . . . 51 3.3 Time domain dynamics for the optical and microwave power, ob- tained via the numerical simulation the model presented in Eqs. (3.5) and Eqs. (3.6) for ? = 0.5 and ? = 0. The dierent columns cor- respond to dierent values of the feedback gain. The top row dis- plays the temporal dynamics of some output optical modes Popt,out,l = h?? |A 2 L out,l| , while the bottom row displays the temporal dynamics of the microwave signal Prf,out = ? 2|M 21| /2Rout at the output of the RF amplier. The critical value of the gain below which there is asymptotically no sidemode and RF oscillation is ?cr ' 10.97. . . . . 55 3.4 Evolution of the particular determinants ?i as a function of the gain ?. Routh-Hurwitz condition for stability requires ?i > 0 for all i. ?3 and ?4 become negative at ? = 11.97. The plots were computed from the ?i, i = 1, ? ? ? , 6 in Eq. (3.27). . . . . . . . . . . . . . . . . . 60 3.5 Variation of the critical feedback strength ?cr as a function of ?. The symbols are obtained via the numerical simulation of the time-domain OEO model presented in Eqs. (3.5) and (3.6), while the solid line corresponds to the analytical solution provided in Eq. (3.36). It can be seen that the stability analysis permits to determine the threshold gain needed to trigger microwave oscillations with exactitude. It also appears that minimum gain is achieved for ? ' ?1. . . . . . . . . . 64 3.6 Bifurcation diagrams for the optical output signals Popt,out,l, for the microwave power Prf,1 generated by the photodiode (before the RF amplier), and for the RF power Prf,out generated at the output of the RF amplier. The parameters of the system are the same as those of Fig. 3.3, with ? = 0.5 and ? = 0. The critical value of the gain below which there is no OEO oscillation is ?cr ' 10.97, in agreement with Fig. 3.5. Note that as the gain ? is increased, there are optical mode power switches within a given sidemode pair ?l 6= 0, while the pumped optical mode l = 0 and the RF signals are varying smoothly. 66 xiii 3.7 ?opt as a function of ? = ?e/?. We see that ?opt ' 1 for almost all coupling regime. This gure is simulalted from Eq. (3.39) and using the relationships of Eqs. (3.42) and (3.43) . . . . . . . . . . . . . . . 69 3.8 ?cr as a function of ? = ?e/?. The optimal ?cr is achieved in critical coupling (? = 0.5). This result is simulated from Eqs. (3.36), (3.42) and (3.43) where we express K1, K2 and K3 as a function of ?. . . . 71 3.9 Amplierless miniature OEO. . . . . . . . . . . . . . . . . . . . . . . 73 4.1 Miniature OEO based on WGMR modulator. The optical and elec- tronic components are assumed to be noiseless. PD: photodiode; Amp: amplier; PS: phase shiftter. . . . . . . . . . . . . . . . . . . . 81 4.2 Miniature OEO based on WGMR modulator. Noise sources are opti- cal and electronic. The optical source noise arise at each mode l of the resonator; the electronic noise arise at the PD and the microwave RF strip. The noises become additive as we go around the closed-loop. We negelct the eect of the multiplicative noise. PD: photodiode; Amp: amplier; PS: phase shiftter. . . . . . . . . . . . . . . . . . . . 84 4.3 Variation of the optical power Poptout,l under threshold gain ?cr for the noisy miniature OEO. In noiseless system Poptout,l = 0, ? l 6= 0. However, in noisy system, Poptout,l uctuates randomly according to the noise. The results were obtained by simulating the dynamics of Eqs. (4.8) and (4.9) with ?a = ?c = 1 and computing the microwave output signal with Eq. (2.29) of Chapter 2. The value of ? is 6, and is about half the threshold gain ?cr. . . . . . . . . . . . . . . . . . . 94 4.4 Variation of the microwave power Prf1 under threshold gain ?cr for the noisy miniature OEO. In noiseless system Prf1 = 0, ? l 6= 0. However, in noisy system, Prf1 uctuates randomly according to the noise. The results were obtained by simulating the dynamics of Eqs. (4.8) and (4.9) with ?a = ?c = 1 and computing the microwave output signal with Eq. (2.33) of Chapter 2. The value of ? is 6, and is about half the threshold gain ?cr. . . . . . . . . . . . . . . . . . . . . . . . . . . 95 xiv 4.5 Variation of the noise power P 2rf,out = |Mout| /2Rout, when the nor- malized gain ? ? ?/?cr is increased under threshold. We have ? < 1, so that the gain in dB is 20 log ?, and is negative. The plots from left to right correspond to noise amplitudes ?a,c = 1, 10, and 100 respectively. The blue dot symbols stand for the numerical results obtained using Eq. (4.33), via the time-domain simulation of the stochastic dierential Eqs. (4.8) and (4.9). The continuous black lines stand for analytical results obtained via Eq. (4.36). The dashed red lines stand for the scaling behavior as predicted by the normal form theory in Eq. (4.40). The dotted gray lines indicate the microwave noise power corresponding to a gain of ?4.18 dB, which directly gives the amplitude of the driving Gaussian white noise power in the nor- mal form model (from left to right, pout = m 2/2Rout = ?71, ?51, and ?31 dBm, respectively). One can note the excellent agreement between numerical simulations and analytical predictions. . . . . . . 96 4.6 Variation of the microwave power Prf,out = |Mout|2/2Rout, when the normalized gain ? ? ?/?cr is increased above threshold (? > 1). The blue dot symbols stand for the numerical results obtained using the time-domain simulation of the stochastic dierential Eqs. (4.8) and (4.9). The dashed red lines stand for the scaling behavior as predicted by the normal form theory in Eq. (4.38). The microwave power has been normalized to an arbitrary reference power P in REF order to evidence the scaling ? ? ? 1 above threshold predicted by Eq. (4.38) for Prf,out ? |M|2. One can note the good agreement between numerical simulations and analytical predictions. The linear scaling of the power with the gain above threshold is expected to break down when ?  1 because of the higher-order nonlinear terms neglected in the normal form approach are then becoming dominant. 102 5.1 Eigenmodes of WGMR. The real location of the eigenfrequencies with anomalous or normal dispersion is represented by solid lines, while the dashed lines represent the location of the eigenfrequencies with normal or anomalous dispersion if the dispersion were null (perfect equidistance). The enlarged gure shows the relationship between the laser frequency ?0 (?L in our work), the frequnecy of the pumped mode 0 ?`0 , the detuning frequency ? and the loaded linewidth ??tot (?? in our work) [105]. . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 Open-loop conguration for the optical oscillator. . . . . . . . . . . . 107 5.3 Closed-loop conguration for the optical oscillator. The ouptut op- tical signal is amplied and fedback into WGMR. Amp: optical am- plier. WGMR: Whispering-gallery mode resonator. . . . . . . . . . 110 5.4 Evolution of the number of nontrivial equilibria. The critical equilib- rium is equal to the the detuning frequency ? and is achieved when ? = 1. This gure is an illustration of Eqs. (5.16) and (5.18). . . . . 113 xv 5.5 Eigenvalue bifurcation diagram (not to scale) for the case of anoma- lous dispersion (? < 0). The areas are labeled using Roman numerals (I, II, and III), and area II is subdivided into two subareas (II1 and II2). The lines are labeled using capital letters, with line A standing for the limit ?2 = (? ? 1)2 (dashed red line in the gure); B stands for the critical line ?2 = 1, and is also subdivided into two rays B1 and B2. Finally, the points are labaled into lower case letters. We only have one point a which is the critical point at which ? = ? and ?2 = 1.The system has three equilibria in area I, II1 and II2; it has two equilibria along the lines B1 and B2, and only one equilibrium in area III. The eigenvalue pictogram are in black when they lead to a bifurcation and in grey otherwise. . . . . . . . . . . . . . . . . . . . 120 5.6 Eigenvalue bifurcation diagram (not to scale) for the case of anoma- lous dispersion (? < 0). The areas are labeled using Roman numerals (I, II, and III), and area II is subdivided into two subareas (II1 and II2). The lines are labeled using capital letters, with line A standing for the limit ?2 = (? ? 1)2 (dashed red line in the gure); B stands for the critical line ?2 = 1, and is also subdivided into two rays B1 and B2. Finally, the points are labaled into lower case letters. We only have one point a which is the critical point at which ? = ? and ?2 = 1. The system has three equilibria in area I, II1 and II2; it has two equilibria along the lines B1 and B2, and only one equilibrium in area III. The eigenvalue pictogram are in black when they lead to a bifurcation and in grey otherwise. . . . . . . . . . . . . . . . . . . . 121 5.7 This gure shows the pictogram of the eigenvalues leading to bifur- cation, as well as the location of the bifurcations in Fig. 5.5 and 5.6. The lines are labeled using capital letters, with line A standing for the limit ?2 = (? ? 1)2. Point a is the critical point at which ? = ? and ?2 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 xvi List of Abbreviations BOD Bistable optical device CaF2 Calcium uoride CW Continuous-wave FSR Free-spectral range IR Infrared IREAP Institute for Research in Electronics and Applied Physics LiNbO3 Lithium niobate MgF2 Magnesium uoride MZM Mach-Zehnder modulator OEO Optoelectronic oscillator PD Photodiode SWAP Size, weight and power TIR Total internal reection WGM Whispering-gallery mode WGMR Whispering-gallery mode resonator xvii List of Publications To Date Publications Related to this Thesis 1. H. Nguewou-Hyousse and Y. K. Chembo, Nonlinear dynamics and stability analysis of self-starting Kerr comb oscillators, In preparation. 2. H. Nguewou-Hyousse and Y. K. Chembo, Stochastic analysis of miniature op- toelectronic oscillators based on whispering-gallery mode electrooptical mod- ulators, IEEE Photon. J. Accepted for publication (2021). 3. H. Nguewou-Hyousse and Y. K. Chembo, Dynamical analysis of miniature optoelectronic oscillators based on whispering-gallery mode modulators with quadratic nonlinearity, in Proc. SPIE 11672, Laser Resonators, Microres- onators, and Beam Control XXIII, San Francisco, CA March 2021. 4. H. Nguewou-Hyousse and Y. K. Chembo, Nonlinear dynamics of miniature optoelectronic oscillators based on whispering-gallery mode electrooptical mod- ulators, Opt. Express 28, 3065630674 (2020). Other Publications 1. H. Nguewou-Hyousse, W. Scott and D. A. Paley, Distributed Control of a Planar Discrete Elastic Rod Model for Caterpillar-Inspired Locomotion, in Proc. ASME Dyn. Syst. and Control Conf., October 2019, Park City, UT. 2. H. Nguewou-Hyousse, G. Franchi and D. A. Paley, Microuidic circuit dy- namics and control for caterpillar-inspired locomotion in a soft robot, in Proc. IEEE Conf. Control Technol. Appl., August 2018, Copenhagen, DN. 3. H. Nguewou and A. Nkwanta, A Perl algorithm for computing RNA folding rates, Intl. J. Evol. Equ. 9, 110 (2014). xviii Chapter 1: General Introduction 1.1 Overview Optoelectronic oscillators (OEOs) are microwave photonic systems that com- bine an optical and electronic branch in a closed feedback loop (Fig. 1.1). They are nonlinear, autonomous and dissipative systems. As long as the system satises the Barkhausen conditon, it is expected to oscillate. The Barkhausen condition states that in order to sustain steady-state oscillations, the gain of the amplifying element must outweigh the losses, so that the loop gain inside the feedback loop must be at least unity. The concept of combining an electrical path to an optical path found its origin in the late 1960s, when researchers noted that continuous-wave lasers where out- puting an oscillatory optical output that needed to be stabilized. In that eort, they realized that converting this output to an electrical signal and feeding it back to the laser stabilized both the optical and radio-frequency signals [1, 2]. These early concepts lacked controllability and tunability because the laser was used as the source of nonlinearity. In 1977, Smith and Turner from Bell Labs proposed to control the nonlinearity by using a lithium niobate (LiNbO3) crystal placed inside a Fabry-Perot resonator [3]. The idea was to leverage on the nonlinear refractive 1 Optical branch E/O O/E Delay conversion conversion Electrical branch Figure 1.1: Schematic of an optotelectronic oscillator. The system converts ligth energy from a laser pump into electrical energy. The system can output signal in both optical (? 50?500 THz) and electrical frequency ranges (? 0?100 GHz). The time-delay corresponds to a round-trip time. The system is nonlinear, dissipative and autonomous. index in the LiNbO3 by feeding it back with a fraction of the output light from the Fabry-Perot resonator. The resulting system showed bistability with respect to the input laser power. In addition, the Fabry-Perot system requires a low optical power for operation owing to its resonance. Its main drawbacks, however, are mainly twofold: rstly, the switching time is increased because the optical eld needs time to build inside the cavity; secondly, the reected light from the cavity may couple back into the laser, and thus the system needs an optical isolator. These inconve- niences were circumvented in 1978 when Garmire et al. showed that bistable optical devices (BOD) with an electrical feedback do not require a Fabry-Perot resonator. They replaced the resonator with a LiNbO3 waveguide modulator, thus opening the door to multimode bistable optical devices [4, 5]. Okada et al. later showed that LiNbO3 waveguide-based systems can also achieve multistability [6]. It is notewor- thy to mention that simultaneously with Garmire et al., Feldman proposed in 1978 a bistable optical device architecture in which the Fabry-Perot resonator was re- 2 placed by Pockels cells [7, 8, 9]. Although using LiNbO3 waveguides or Pockels cells as modulators reduced the size of the system and allowed possible integration into electronic chips, both systems required high operating voltage (? 50?100 V). This issue was solved in 1979 when Ito et al. and Schnapper et al. introduced a LiNbO3 Mach-Zehnder modulator (MZM) as a low-operating voltage mean to control the nonlinearity in a laser system with electrical feedback [8, 9]. A MZM typically has an operating voltage of about ? 3? 15 volts. The modern concept of optoelectronic oscillator can be traced back to 1994 when Yao and Maleki from the NASA Jet Propulsion Laboratory proposed an oscil- lator where there was a continuous-wave (CW) laser feeding into a MZM [10]. The optical output traveled through an optical ber (a few km) before being converted into an RF signal by a photodetector; this signal was amplied and ltered before being fed back into the MZM. Figure 1.2 is the original schematic proposed by Yao and Maleki. The idea of using an optical ber and storing optical energy instead of electrical energy allowed to improve the stability and purity of the microwave signal. This multimode system exhibited bistable, oscillatory, or chaotic behavior, and was used for ultrapure microwave generation. This system became subsequently known as an optoelectronic oscillator (OEO). Since the work of Yao and Maleki, OEOs have become the focus of many research activities, and are some of the most studied systems in microwave photonics. They are used to study the properties of nonlinear time-delayed systems [11]. They also have technological applications, such as optical chaos communication and radar- frequency generation, just to name a few [12]. The aim of this chapter is to give 3 only free parameter. The solid line in Fig. 2 is achieved with 8 = eter B of an MZ modulator is presented. Theory and experiment 0.75 and shows excellent agreement with the measurements for have shown good agreement. In the example presented, the MZ aMz. modulator has a chirp of 4 = 0.75, which is reasonable for this type of modulator. ?MZ 0 IEE 1994 18 July 1994 Electronics Letters Online No: 19941005 M. Schiess (Laboratory of Photonics and Microwave Engineering Department of Electronics, Royal Institute of Technology ( K T H ) Electrum 229, S-164 40 Kista, Sweden) H. CarldCn (Teliu AB Telia Research Sfo S-136 80 Haninge, Sweden) References KOYAMA. F., and IGA. K.: ?Frequency chirping in external -*t 7 I m modulators?, J. Lightuave Technol.. 1988, LT-7, ( I ) , pp. 87-93 - 3 1 I DJUPSJOBACKAA. . : ?Residual chirp in integrated-optic modulators?, IEEE Photonics Technol Left . , 1992, 3. (1). pp. 4143 -4f 1 I ELREFAIE.A.F., WAGNER. R.E. . ATLASD.A ., and DAUT. D.G.: ?Chromatic dispersion limitations in coherent lightwave 5 transmission systems?, J. Lightwave Technol.. 1988, LT-6, (S), pp. Fig. 2 a parameter for M Z modulator as function of applied bias volt- 704~709 age GNAUCK. A.H.. KOROTKY. S.K., VESELKA. J.J., NAGEL, J . , W experimental results KEMMERER. C.T., MINFORD, W J.. and MOSER. D.T.: ?Dispersion __ fitted theoretical value with chirp parameter 8 = 0.75 penalty reduction using an optical modulator with adjustable chirp?, IEEE Photonics Technol. Lett., 1991, 3, (IO), pp. 91fL918 Conclusions: Based on known theory about the a parameter of DEVAUX. F., SOREL. Y , a nd KERDiLEs. J F.: ?Simple measurement of MZ modulators [2] and using the measurement method for it [5], a fiber dispersion and of chirp parameter of intensity modulated light simple and accurate method for the evaluation of the chirp param- emitter?, J. Lightwave Technol., 1993, LT-11, 112). pp. 1937-1940 High frequency optical subcarrier generator tro-optic oscillation will start. Because both optical and electrical processes are involved in the oscillation, both optical and electrical X.S. Yao and L. Maleki signals will be generated simultaneously. We built two such E/O oscillators using two different moduba- tors. In the first oscillator. the Mach-Zehnder modulator has a Indexing terms: Elertro-optical devices, Oprical modulation bandwidth of 8GHz and a half-wave voltage V, of -17V. It has an internal bias control circuit that automatically sets the modula- The authors descnbe an electro-optical oscillator capable of tor bias at 50% of the transmission peak. The photoreceiver has a generating high stability optical signals at frequencies up to bandwidth of 12GHz and a responsivity of -0.35AiW. The ampli- 70GHz. Signals as high as 9.2GHz were generated with an optical fier has a total electrical power gain of 50dB. a bandwidth of wavelength of 1310nm using the oscillator, and a comb of stable 5GHz centred around 8GH2, and an output IdB compression of frequencies was produced by modelocking the oscillator. 20dBm. The input and output impedances of all electrical compo- nents in the loop are 50R. The loop length is -9m. In advanced photonic analogue communication systems, high fre- quency optical suhcamer generation is essential for photonic sig- Studies [3] have shown that depending on the biasing point of nal up and down conversions [I]. In this Letter, we report a novel the modulator, the E/O oscillator may he bistable, oscillatory, or optical subcarrier generator, called an electro-optic (E/O) oscilla- chaotic. However, the E/O modulator used above has a fixed bias tor, that is capable ofgenerating an up to 70GHz (limited by the point that cannot be adjusted. To investigate the effect of bias speed of E/O modulator and photoreceiver) high stability optical point on the EJO oscillator, we built another E/O oscillator with a subcarrier. By modelocking the oscillator, a comb of stable high modulator that has an independently controlled bias electrode. frequencies can also be generated. However, this modulator is slower ( 1 GHz bandwidth) and has a half-wave voltage of -10V. E/O modulator optical optical output rF O t I 9.22 GHz. 5dBrn low noise 2 -201 laser 9 5 -LO!- ., . ..- 8...5 9..~0 9.5 10.0 Fig. 1 Construction of electro-optic modulator frequency.GHz Figure 1.2: First optoelectronic design proposed by Yao and Maleki in 199F4ig[. 120 G].enerated 9.22Hz oscillation observed on an RF spectrum analyser System consists oTfheo pEt/Oic aolscsiolluatrocre [t2h] aist diesscpriubmedp iend Fiingt. o1.a Lmighotd furolamt oorn.eO ofp tical output travels throughthae nouotpptuitc paolrtsb orfe tbhee fmoroedudleattoerc tisi odnetebcyteda pbyh othteo dphioodtoed;ettehcetoer lectricalWouitth- the first E/O oscillator, we demonstrated in the laboratory and then is amplified, filtered. and fed hack to the electrical input put is then amplied and ltered before being fed back into the modulator ththreo ufigrsht high frequency electro-optic oscillator that generated an port of the modubator. If the modulator is properly biased and the an RF electrode. optical subcarrier and the accompanying electric signal up to open loop gain of the feedback loop is properly chosen, self-elec- 9.2GHz, using a diode pumped YAG laser at 1310nm. The gener- an introductionELtEoCthTeRtOopNiIcCoSf LOEEVOEsRanSd la1ystt hSeepfotuenmdbateior n7s9t9o4 undVeorsl.t a3n0d thNeo. 78 1525 work presented in the following chapters. Therefore, this chapter is organized as follows: Section 1.2 will introduce the notion of broadband OEOs, while Sec. 1.3 will present the narrowband counterparts. Section 1.4 will discuss some technological applications of OEOs. The motivation for our research will be exposed in Sec. 1.5, and Sec. 1.6 will present the outline of this thesis. 1.2 Broadband Optoelectronic Oscillators Broadband OEOs have a broad bandpass that is the result of the overlap between the MZM bandwidth and the photodiode (PD) and the lter bandwidths. The bandwidth range of such systems is typically in the gigahertz (GHz) range. Broadband OEOs can be used to investigate a wide variety of complex phenomena ranging from multistability to chaos. In this section we will introduce the theoretical concepts needed to model time-delay systems and we will present the mathematical equation governing the delay dynamics of broad bandpass OEOs. 4 1.2.1 Modeling Ikeda-like OEOs Delayed systems are systems in which the dynamics at a given instant t de- pends on the state of the system at an anterior time t ? T , where T is the time delay. Ikeda-like OEOs feature a nonlinear function, a time-delay, a linear gain and a linear lter inside a closed-loop feedback. If we dene x(t) as the signal variable, then the delayed signal is x = x(t ? T ), with t ? [?T, 0]. Using a signal process- T ing approach in the Fourier domain, we can dene the input-output relationship in Ikeda-like systems as X(?) = H(?)?F (?)e?i?T , (1.1) NL where X(?) is the Fourier transform of x(t), ? is the linear gain of the amplier, F NL is the Fourier transform of the nonlinear function f [(x(t)], and H(?) is Fourier NL transform of the impulse response to the linear ltering done on the nonlinear input signal f [(x(t)] [13]. Applying H?1(?) to both sides of Eq. (1.1) and taking the NL inverse Fourier transform yields the following time-domain equation H?{x(t)} = ?f [x ], (1.2) NL T where H?{x(t)} is a linear integrodierential operator. Equation (1.2) is an Ikeda-like equation [1]. 5 1.2.2 Broad Bandpass Filter OEOs Broad bandpass lter OEOs are characterized by the fact that the low and high cut-o frequencies are very distant one from each other. As such, the broad bandpass can be viewed as a combination of a high-pass and low-pass lters with respective low and high cuto frequencies fL, and fH. Let ? and ? dene respectively the high and low cuto response times of the broad bandpass OEO dened as 1 ? = (1.3) 2?f H 1 ? = . (1.4) 2?f L Broad bandpass OEOs can be modeled as Ikeda-like equations in the form of Eq. (1.2) as [12, 27] ( ) ? t H?{x(t)} ? ? 11 + x+ ? x?+ x(s)ds = ?f [x ]. (1.5) ? ? NL Tt0 Very often, f  f , so that we use the approximation (1 + ?/?)x ' x. This system L H has only one xed point which is globally stable for small feedback gain but loses its stability and enters bifurcation as the gain is increased [12, 15]. 1.3 Narrowband OEOs The ltering process in both the broadband and narrowband OEOs is done in the RF branch. However, unlike their broadband bandpass counterparts, narrow- 6 band OEOs have a very narrow bandwidth, which makes the system highly frequency selective and suitable for the generation of ultrapure microwave signals. In this sec- tion, we will discuss the evolution of the narrow-band OEOs' architecture. We will segment this evolution in three generations, according to the improvement towards meeting the constraints of size, weight and power (SWAP). We will also introduce the theoretical framework to describe the deterministic dynamics of narrowband OEOs. 1.3.1 Early Architectures Although the idea of optoelectronic oscillator can be traced back as early as the late 1960s, the most popular architecture of OEO was proposed in 1994 by Yao and Maleki [10, 16, 17]. This design, presented in Fig.1.2, modeled the rst generation of narrowband OEO and consisted of three main components performing each a particular function: a MZM modulator for the system's nonlinearity; a few km-long optical ber for optical energy storage; and a RF lter for bandwidth selection. At this date, this system is well understood and mathematical models have been derived to describe its dynamics, mostly using an Ikeda-like approach [18, 19]. Using commercial-o-the-shelf (COTS) components permits to achieve remarkable phase noise performances, down to a record ?163 dBc/Hz at 6 kHz oset from a 10 GHz carrier [37]. The main advantage of this rst generation of narrowband OEO is a low-phase noise; however, the main drawback is their bulkyness (mainly due to the length of the optical ber), their heavyness, and their energy-greed due to many 7 components. The second-generation of OEOs was then proposed by Volyanskiy et al., in 2010 [44]. It improved on the previous designs by replacing both the delay-lines and electric bandpass lters with whispering-gallery mode (WGM) resonators, which are low-loss dielectric cavities capable of trapping photons for long durations via total internal reection [38, 39, 40, 41, 42, 43]. This is a few millimeter-sized- radius microcavity capable of performing the lightwave storage and RF ltering because of the narrow bandwidth of the resonator modes. Therefore, because they could perform both photon storage and narrowband ltering in the linear regime, millimetric or sub-millimetric WGM resonators have been successfully inserted on OEO loops, and they have permitted a signicant reduction of the oscillators in terms of size  see for example refs. [44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. In 2013, Coillet et al., proposed a deterministic model to study the dynamics of this system [47], and Nguimdo et al., proposed a stochastic model to study the phase noise in 2015 [50]. While the size of this type of narrowband OEO was signicantly reduced from the rst generation, they still required a modulator for the nonlinearity. Miniature optoelectronic oscillators based on a whispering gallery-mode mod- uator are the third generation of OEOs. They were rst introduced by Matsko et al., in 2003 [54], and feature a simpler architecture in which the modulator, optical ber and RF lter are all replaced by a WGM resonator with a RF cavity strip. This system can achieve ultrapure microwave generation owing to the high photon storage capability of the resonator coupled to its narrow bandwidth. In addition to having one component perform the nonlinearity, optical storage and RF lter- 8 ing, this system oers the best SWAP performance for OEOs [55]. Despite these advantages, there is a lack of understanding of the nonlinear interactions inside the microcavity and in the feedback loop. 1.3.2 Modeling Delayed-based Narrowband OEOs As mentioned in section 1.3.1, the dynamics of delay-based narrowband OEOs is well understood today. A deterministic model for the temporal behavior of these systems has been introduced in 2007 [18]. The particularity of this model is that it is derived from the Ikeda-like equation of the broadband OEOs in Eq. (1.5) by redening the characteristic timescale variables ? and ? as 1 ? = (1.6) ?? ?? ? = , (1.7) ?20 such that the dynamics of narrowband OEO can be described as below: 2 ? t H?{x} ? 1 ?x+ x?+ 0 x(s)ds = ?f [x ], (1.8) ?? ?? NL Tt0 where ?0 = 2?F0 is the angular central frequency of the RF lter and ?? = 2??F is its narrow bandwidth. The variable x(t) varies rapidly and the ratio between the fastest and slowest timescales is generally high (?0/?? ? 100). Therefore, Eq. (1.8) is not suitable to study the dynamics of narrowband OEOs. We can instead dene a slowly-varying complex envelope A(t) for the modulated signal x(t). Only signal 9 frequencies that are close to the RF lter central frequency ?0 will carry through the closed-loop because of the lter narrow bandwidth. Therefore, we can represent x(t) as a signal of central carrier frequency ?0 modulated by A(t) as 1 1 x(t) = A(t)ei?0t + A?(t)e?i?0t, (1.9) 2 2 where A = |A|ei? is the slowly-varying complex envelope. If the MZM nonlinearity can be expressed as f ? cos2 (x + ?), the slowly-varying complex envelope A NL T obeys A? = ?A? 2??e?i?Jc1[2|A |]A , (1.10)T T where ? ? ??/2 is the eective half-bandwidth of the RF lter, ? = ?0T is the microwave round-trip phase, ? = ? sin 2? is the eective gain of the feedback loop, and Jc1 = J1(x)/x is the Bessel-cardinal function. A stability analysis of the delay narrowband OEOs using the slowly-varying complex envelope model of Eq. (1.10) was done in [19]. The results showed that the microwave signal has a single xed point which is globally stable for gain |?| < 1. Beyond this threhold gain value, the signal undergoes a primary Hopf bifurcation and oscillates with constant amplitude; a secondary Hopf bifurcation (Neimark-Sacker bifurcation) occurs as we further increase the gain, and the signal operates in torus-shape with oscillations with two constant amplitudes. As we increase even more the loop gain, the system becomes chaotic. 10 1.4 Applications of Optoelectronic Oscillators Optoelectronic oscillators can output signal in both optical (? 50? 500 THz) and electrical frequency ranges (? 0 ? 100 GHz), and as such they have found numerous applications in lightwave and microwave technology. In this section, we will mainly discuss their application in ultrapure microwave generation. We will also briey discuss other applications such as optical communication, analog computing, and sensing. 1.4.1 Ultrapure Microwave Generation The main application of narrowband OEOs is ultrapure microwave genera- tion [1]. Ultrapure microwaves are needed in mobile telecommunications, where they are used as carriers to be modulated by information-bearing signals. They are also useful in radar and lightwave technology, where the signal frequency needs to be of high accuracy. Time-frequency metrology is another area where ultrapure mi- crowave can be used to measure time and frequency with precision, calibrate systems for high resolution, or as reference oscillators in clock-driven systems. In microwave photonics, ultrapure microwave are used to study the intractions between microwave and optical signals. 1.4.2 Other Applications Chaos synchronization and communications is one of the othe main applica- tions of broadband OEOs, which are used to embed a signal in a chaotic optical 11 carrier, and retrieve the signal via chaos synchronizaton. Chaotic systems are very sensitive to initial conditions and are unpredictable in the long term. This prop- erty can be used to encode information in the physical layer with chaotic laser light [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Reservoir computing is an emerging eld at the intersection of microwave photonics and electronics. It builds on the promise of computing at light speed. OEOs seem a good candidate to achieve this goal because they combine an optical and electrical path. Moreover, broadband OEO can process a large amount of information, and their nonlinearity make them good candidate for machine learning applications such as neuromorphic photonic computing [31, 32, 33, 34, 35, 36]. Narrowband OEOs are highly frequency-selective, and as such can act as op- tical sensor. In that conguration, the modulation of their signal output (temper- ature, pressure,...) is a result of the detection. OEOs sensors have been used to detect magnetic eld and refraction index, load and strain, temperature and pres- sure; they have also been used for distance, rotation and vibration sensisng, as well as multiphysics sensing [1]. 1.5 Motivation for this Work Miniature OEOs based on WGM modulators couple a microwave strip cavity to a WGM resonator with ?(2) nonlinearity, which can then play the role of an elec- trooptical modulator and eliminate the need for its Mach-Zehnder equivalent [54]. In this case, the three tasks of photon storage, narrowband ltering and nonlinearity 12 can be performed by the WGM resonator. The main interest of this approach is that it eectively leads to the best SWAP performance for OEOs. However, to the best of our knowledge, there is no theoretical model available to analyze the nonlinear dynamics and stability of miniature OEOs. Indeed, understanding the dynamical behavior of miniature OEOs requires an anal- ysis of the electrooptical conversion phenomena that are taking place in a WGM cavity pumped by both a resonant laser and coupled to a RF strip cavity pumped by a microwave signal. These intracavity processes, which involve microwave and optical photons interacting quantum-mechanically, are the fundamental phenomena enabling the concepts of electrooptical WGM modulators [61, 62, 63, 64, 65] and ultra-sensitive microwave photonic receivers [66, 67, 68, 69, 70, 71, 72, 73, 74] . Most works related to electrooptical WGM resonators are restricted to the three-modes operation involving the pump, signal and idler modes. A noteworthy exception is for example the work of Ilchenko et al. in ref. [65], where they analyzed the intracavity dynamics for an arbitrary number of modes. The multimode analysis is indispensable for the understanding and characterization of the miniature OEO, as these cascaded intracavity interactions contribute to the saturation nonlinearity in the feedback loop, thereby dening the amplitude of the stationary microwave and lightwave oscillations. Microwave purity is generally dened by phase noise. However, there is no analysis available to understand how the optical and electrical noise sources in the optoelectronic loop of the miniature OEO are converted into microwave phase noise. Such an analysis is indispensable to gain a deep understanding of the metrological 13 performances of this oscillator. To this eect, a Langevin approach has been used with remarkable success for ber-based OEOs, where it was shown that it can pro- vide an excellent agreement with experimental phase noise spectra [50, 93, 95, 99]. The objective of this thesis is therefore to study the deterministic and stochas- tic behaviors of miniature OEOs based on WGM modulators; in particular, we want to propose mathematical frameworks accounting for all nonlinear interactions in miniature OEOs based on electrooptical WGM modulators. We also aim at per- forming the temporal and spatial stability analyses, and determine the conditions leading to improve the system stability and microwave purity. 1.6 Thesis Outline This work is organized as follows: In Chapter 2 we discusss the time-domain dynamics of the open-loop WGM electrooptical modulators. The analysis of the open-loop system is a necessary preliminary to the study of its closed-loop coun- terpart. Chapter 3 introduces the closed-loop feedback and proposes a full-time deterministic model accounting for the intracavity dynamics. We will also investi- gate the stability conditions of the closed-loop system as well as its optimization. In Chapter 4, we use a Langevin approach to derive the stochastic model describ- ing the noisy dynamics of the miniature OEO with random noise; we investigate the stochastic behavior below threshold and propose a normal form aproach for stochastic analysis and phase noise analysis. Chapter 5 introduces a new topic as we present our preliminary results in the analysis of a closed-loop miniature optical 14 oscillator based on whispering-gallery mode resonator. We use a Lugiato-Lefever approach and then perform the spatiotemporal analysis to determine the conditions leading to spatial bifurcations. We conclude by sharing some nal remarks and future perspectives in Chapter 6. 15 Chapter 2: Whispering-Gallery Mode-Based Modulators 2.1 Introduction In the previous chapter we have discussed traditional optoelectronic oscilla- tors. These systems have been studied thoroughly and mathematical models have been derived to describe their dynamics. The main disadvantage of such systems is their bulkiness, so that a new and smaller system using whispering-gallery mode resonators as modulators has been proposed. WGM resonators have applications in modern nonlinear optics, where they create high nonlinear responses to weak electromagnetic eld [57]. In this chapter we will propose a time-domain mathematical model to de- scribe the dynamics of the WGM electrooptic modulator. The chapter is organized as follows: Section 2.2 will present the Mach-Zehnder modulator whish is the most common type of modulators used in optoelectronic oscillators. In Sec. 2.3, we will present the components of the new system. This section will present the whispering- gallery mode resonator and its main characteristics. Section 2.4 will introduce the WGM electrooptic modulator system. Section 2.5 will present the quantum equa- tions for the system while Sec. 2.6 will discuss the classical formalism of the dynam- ics' equations. In Sec. 2.7, we will analyze the optical and microwave output eld 16 Figure 2.1: Intensity modulation using a Mach-Zehnder interferometer. Source: ref. [12]. as well as the transmission functions. Section 2.8 will then present some numerical simulations before concluding in Sec. 2.9. 2.2 Mach-Zehnder Electro-Optic Modulator The Mach-Zehnder modulator (MZM) is the most known type of electro-optic modulator. In this section we will describe the system and present some of applica- tions of Mach-Zehnder interferometers. 2.2.1 System The Mach-Zehnder modulator is an interferometer made from a material that features strong electro-optic eect (such as LiNbO3) so that an applied electric eld causes a change in the refractive index, resulting in intensity modulation [12, 100]. A light beam that goes into the modulator is divided into two equal parts which are routed into two dierent optical paths. The rst optical path applies an electric 17 eld to create a phase modulation of the incoming signal, while the second path applies an electric eld to create a phase shift to the light amplitude. The signals from the two paths are then recombined to yield an intensity-modulated output signal dened as [12] [ ] ?V (t) ?VB Pout = P cos 2 + , (2.1) in 2V? 2VRF ?DC where Pin (Pout) is the input (output) power; V? and V? are the half-wave volt-DC RF ages, VB is the DC bias voltage and V (t) is the input voltage. 2.2.2 Techonological Applications of Mach-Zehnder Interferometers MZ interferometers have many applications in optical communication [100]. It may be used in optical sensing where the change in output signal is induced by the measurand; it can also be used in optical communication as an optical add- drop multiplexer (OADM) for ber-based optical networks [100, 101]. Another application in optical communication is as an optical switch for ultrafast signal processing; this is achieved by creating a phase dierence while passing the signal through the two branches of the MZI [100, 103, 104]. Finally, the Mach-Zehnder interferometer can be used as a modulator (as described in Subsec. 2.2.1) for high- speed optical communication. As such, they dene the bandwidth and minimize the eect of dispersion, thus increasing the performance of high speed bre-optic communication systems [1, 100]. Although the MZM is used in most OEOs for electrical to optical conversion 18 Figure 2.2: Ray of light propagating inside WGMR through TIR. Here ` = 10 . Source: ref. [58]. and nonlinearity, it is relatively bulky and only performs one the three main tasks completed by an OEO (nonlinearity), thus leaving the need for other components in the system. Therefore there is a need for a new type of modulator that operates in a dierent way. 2.3 Components of the WGM-Based Electro-Optic Modulator The system under study consists of a laser source which emits photons in the IR light spectrum; the optical photons are then trapped in a whispering-gallery mode cavity while also ltering the photon frequency. The output ux of photon is then detected by a photodiode, converted into a RF signal and amplied before being analyzed. In this section, we will present the dierent components and give an overview of how they operate. 19 2.3.1 Whispering-Gallery Mode Resonator The term whispering-gallery mode was rst used by Lord Rayleigh in 1896 [56] to explain a phenomenon by which one could hear a whisper across the other side of the dome of St Paul's cathedral in London. Lord Rayleigh used a ray interpretation to explain the total internal reection (TIR) of the acoustic waves across the internal periphery of the dome. A whispering-gallery mode resonator is a microcavity that serves the purpose of optical resonator because photons are trapped inside the cavity by TIR where they complete round trips [57, 58]. Figure 2.2 shows a light ray trapped inside a WGM cavity with ` = 10. The number of roundtrips is proportional to the loss property of the material used to manufacture the WGM. 2.3.1.1 Eigenspectrum and Eigenmodes AWGM is has an eigenspectrum which satises the following Helmholtz equa- tion: [ ( ) ] ? 2? ? + (r) ??(r) = 0, (2.2) c where ?? is the eigenfrequency associated with the eigenmode ??(r) and (r) is the spatially dependent permittivity. ? is a quadruplet of eigennumbers {`,m, n, p}, of which ` is the most important and represents the azimuthal eigennumber. It is 20 Figure 2.3: (a) Right: 3D representation of a WGMR cavity with l = 30 modes, n = 1 and l ? |m| = 0. Right: 2D representation of the same dielectric cavity. (b) Examples of 2D transverse eld distributions for dierent radial and polar eigennumbers n and m. Source: ref. [42]. associated with the resonance condition 2?ang = `??, (2.3) where a is the radius of the resonant cavity, ng is the group velocity index of the ma- terial, and ?? is the wavelength associated with the eigenfrequency. Equation (2.3) imposes that the number of TIR (optical path) in a round trip inside the WGMR must be an integer multiple of the wavelength. The polar eigennumberm is bounded as | m |? 1 and is such that the condition `?|m|+1determines the number of nodes in the perpendicular direction relative to the equatorial plane of the resonant cavity. The radial eigennumber m determines the number of lobes in the radial direction, 21 Figure 2.4: WGMR is coupled to the light source through evanescent coupling. (a) Coupling through prism. (b) Waveguide side coupling. (c) Waveguide tip- coupling. Source: ref. [90]. while the eigennumber p is the polarization which is either TM or TE. Figure 2.3 shows a WGM cavity with ` = 30 modes. As a convention, a family of modes is dened by a xed n, m and p, with a varying `. Moreover, a WGM microcavity is characterized by three main parameters: the loss factor Q, the free spectral range (FSR) ?R, and the nonlinearity. 2.3.1.2 Quality Factor The Q-factor or loss factor of a WGMR is a dimensionless parameter used to quantify the capacity of that resonator to keep phtons inside it dielectric walls. It is dened by the intracavity loss Qi and the extrinsic (excitation, coupling) loss Qe, which are dened as (refs. [38, 57, 58, 91]) ?0 Qi,e = , (2.4) 2?i,e where ?0 is the resonant frequency of the WGMR at mode `0, and ?i,e is the intrinsic (respectively extrinsic) half-linewidth contribution of the resonance. The intrinsic 22 loss corresponds to the uncoupled WGMR; it is a fabrication parameter of the resonator, and is characterized by the intracavity volumic loss (Qvol), the loss due to surface scattering (Qsurf), and the radiative loss (Qrad). Qi is therfore dened as Q?1 = Q?1 +Q?1in vol surf +Q ?1 rad, (2.5) We note here that the radiative loss Qrad varies with the eigenmode ` and is quasi-innite for `  1; as a result, its contribution to the intracavity loss is negligible [39, 57, 91]. Qe on the other hand occurs when the resonator is coupled to the light source. This is achieved through evanescent coupling [Fig. 2.4]. We therefore dene the loaded Q-factor as Q?1 = Q?1i +Q ?1 e , and (2.6) ?0 Q = = ?0? , (2.7) ?? ph where ?? is the linewidth of the optical resonance around ?0, and ?ph is the photon lifetime inside the cavity. The three regimes of coupling are undercoupled (Qe < Qi), overcoupled (Qe > Qi), and critically coupled (Qe = Qi). WGM resonators have high quality factors because reection loss and photon absorption can be very low. The linewidth ?? is the bandpass frequency of the output signal. Equation (2.7) shows that as Q increases, ?? decreases, leading to signals of high spectral purity. Q-factors of WGM resonators can be as high as 1010 at 1550 nm with a linewidth of the order of 100 kHz for Q ? 109 [47, 57, 58]. The 23 choice of the materials and the shape of a microcavity often aects its Q-factor. Table 2.1 shows the Q-factor of some material and resonator shape. 2.3.1.3 Free-Spectral Range The FSR is the distance in optical frequency between two successive spectral lines generated in an optical resonator. It is expressed as [57, 58]: c 2? ?R = = , (2.8) ang TR where a is the radius of the resonator, ng is the group velocity index of the material, and TR is the photon round-trip time. We note that the FSR is inversely proportional to the radius of the resonator and inversely proportional to the photon round-trip time. 2.3.1.4 Nonlinearity The nonlinear behavior of whispering-gallery mode dielectric cavities is a re- sult of the high Q-factor coupled with the high photon density. It is caused by the Shape of resonator Material Q-factor Sphere silica 108109 Torus silica 108 Truncated spheroid CaF 112 10 Truncated spheroid MgF2 10 8 Truncated spheroid LiNbO3 > 10 8 Table 2.1: Measured Q-factor for various material and resonator shape. The Q- factor varies according to the fabrication material and resonator shape. Source: ref. [58, 59, 60] 24 resonant enhancement of the low nonlinear interactions in the resonator. The non- linearities in the fabrication material arise from various sources such as the thermal dependence of the index of refraction, or its electric dependence. The work in this thesis focuses on the latter, so that this subsection will expand only on it. The electro-optic eect is an optical property through which the refraction index of some material can be modied by applying an electric eld E [12]. Assuming a scalar electric eld, the Taylor expansion of the dependence of the refraction index n(E) is given as [ ] [ ] dn 1 d2n n(E) = n0 + ? E + ? E2 +O(E3), (2.9) dE 2 dE2 E=0 E=0 Where only the rst and second terms of the Taylor expansion are written explicitely. Our work assumes linear nonlinearity, which means the refraction index depends on the sign of E. This nonlinearity is called the Pockels eect [12, 38, 57, 58], and ?(2) denotes the second-order susceptibility of the material. The work in this chapter and the subsequent chapters of this thesis characterize the conditions under which the ?(2) nonlinearity in miniature OEOs may induce second-harmonic generation and optical oscillations. 2.3.2 Photodiode A photodiode is an electronic component that converts optical energy into electrical energy. Upon absorption of the photons, a photocurrent is generated and provides an output signal. The photodiode is generally characterized by its 25 responsivity R in the unit A/W. If we couple the responsivity to a transimpedence amplier with gain g in the unit V/A, we can then characterize the photodiode by its conversion factor S which determines the electrical power following the equation Vout(t) = SVin, (2.10) where Vin is the optical signal to the PD and Vout is the electrical signal. S = Rg is in the unit volts per watt (V/W). 2.3.3 Radio Frequency Amplier A radio frequency (RF) amplier is an electronic component that amplies a low power RF signal. It has some nonlinearity that is often disregarded, and its main characteristic is the gain G, used to dene the output voltage through the following relationship: Vout = GVin. (2.11) The RF gain is futher dened as the product of the gains and attenuations (losses) in the system, so that we may write G = GAGL. The main disadvantage of an RF amplier is that it also amplies the noise in the system; therefore, our work will investigate the conditions to have an amplierless OEO. 26 Laser A A PDin out Popt,out Cin Figure 2.5: WGM-based electro-optic modulator. Ain and Cin are the optical and microwave pump eld. Aout is the output photon ux; PD: Photodiode. 2.4 WGM-Based Electro-Optic Modulator System In this section we will introduce the WGM-based electro-optic modulator. We will also discuss the parameters that were used to analyze the system. 2.4.1 System The WGM modulator under study is displayed in Fig. 2.5. The WGM res- onator is a lithium niobate (LN) disk of main radius a, that is used as a resonant electrooptical modulator. This modulator has an optical input, an RF input, and an optical output. The optical input is a telecom laser signal at power P with L wavelength ? ' 1550 nm, and the corresponding angular frequency is ? = 2?c/? L L L 27 with c being the velocity of light in vacuum. The WGM resonator has a free-spectral range that can be determined as ? = c/ang = 2?/T , where ng is the group ve-R R locity index of the lithum niobate at the pump wavelength, and T is the photon R round-trip time in the optical cavity. The WGM cavity has a loaded quality factor dened as ? Q = L , (2.12) 2? where ? = ?i + ?e is the loaded half-linewidth of the resonances at telecom wave- length, while ?i = ? /2Qi and ?e = ? /2Qe correspond to the intrinsic and extrinsicL L (i.e., coupling) contributions, respectively [42]. The WGMs of the resonator that are involved in this process belong to the same mode family. Therefore, they can be unambiguously labelled by their az- imuthal order `. Since the pumped mode has an azimuthal order `0, it is useful to introduce the reduced azimuthal order l = ` ? `0 so that the WGMs involved in the system's dynamics can now be symmetrically labeled as l = 0,?1,?2, ? ? ?, with l = 0 being the pumped mode which has a resonant frequency ?0. The pump frequency ? is very close to the resonant frequency ?0 of the pumped mode, theL detuning factor ? being dened as ? = ? ? ?0. (2.13)A L It is convenient to introduce the normalized optical detuning ? dened as ??? = A , (2.14) ? 28 which is such that resonant pumping translates to |?| ? 1. The RF strip resonator coupled to the WGM disk has a resonance frequency that matches the FSR of the optical cavity. It has a loaded quality factor Q dened M as ? Q = R , (2.15) M 2? where ? is the half-linewidth of the loaded RF cavity resonance. The microwave input with power PM has a frequency ? very close to ? , with the RF detuningM R factor ? dened as ? = ? ? ? . (2.16) C M R Here also, we dene the normalized RF detuning ? as ??? = C , (2.17) ? which is within the resonance when |?| ? 1. The second-order susceptibility ?(2) of the lithium niobate crystal is a nonlin- earity that mediates the coherent interaction between the microwave photons h?? M fed to the RF strip cavity and the optical photons h??l circulating inside the WGM cavity. At the photon level, the intensity of this nonlinear interaction is weighted by a normalized coupling parameter g ? ?(2), which has the dimension of an angular frequency [65, 70, 74, 75]. Interestingly, the ratio between the energy of the optical photons comparatively to their microwave counterparts is approximately equal to their azimuthal eigenumber ` ' ?l/? , which would be here of the order of a fewR 29 thousands. The output optical signal of the WGM resonator is an electrooptical frequency comb whose intermodal frequency is an RF signal corresponding to the FSR of the cavity. This comb is sent to a photodetector (with sensitivity S), that retrieves this beating intermodal frequency and outputs a microwave signal, which may be subsequently amplied before being analyzed. The two main tasks to undertake are now (i) to build a time-domain model to describe the dynamics of this oscillator, and (ii) to determine the optical and microwave transmission functions. 2.4.2 Parameters Unless otherwise stated, we will consider the following parameters for our system throughout this chapter, without loss of generality: P = 1 mW; ? = L L 1550 nm; ? /2? = 10 GHz; S = 20 V/W; g/2? = 20 Hz; Q = 5 ? 107i andR Qe = 10 7 (this denes all the ? coecients); Q = ? /2? = 100; and nally, the M R RF line is impedance-matched with the modulator input electrode with Rout = 50 ? and ?i = ?e = ?/2. 2.5 Quantum Formalism 2.5.1 Phenomenology The interactions inside the WGM generator involve microwave photons of en- ergy h?? , and and optical photons of energy h??l. As explained in Fig. 2.6, theR second-order susceptibility ?(2) mediates two dierent processes in the resonator. 30 (a) Photon upconversion (b) Photon downconversion Energy Energy diagram diagram -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 ??0 + ??R ? ??1 ???1 ? ???2 + ??R Absorbed Emitted microwave photon microwave photon Figure 2.6: Frequency-domain representation of photonic up- and down-conversion in a WGM resonator with ?(2) nonlinearity. These two processes can be leveraged to translate microwave energy to the optical domain inside the WGM resonator. When belonging to the same family, the eigenmodes of the resonator with free- spectral range ? are quasi-equidistantly spaced as ?l ' ?0 + l? , where l = `? `R R 0 is the reduced azimuthal eigenumber, and ?0 is the pumped resonance. (a) Photonic upconversion (stimulated): An infrared photon annihilates a microwave photon and is upconverted as h??l + h?? ? h?? . (b) Photonic downconversion (stimulated orR l+1 spontaneous): An infrared photon emits a microwave photon and is downconverted as h??l ? h??l?1 + h?? .R The rst one is parametric upconversion following h??l + h?? ? h??l+1. This inter-R action is always stimulated, i. e., it can only occur when the WGMR is RF-pumped. The second process is parametric downconversion, following h??l ? h??l?1 + h?? .R This downconversion can be either stimulated (does only occur in presence of RF pumping) or spontaneous (does always occur regardless of RF pumping), with both processes having dierent microwave photon production rates. 2.5.2 Hamiltonian Equations The interaction between optical and microwave photons is best decribed from the quantum-mechanical view point. In that framework, the intracavity elds are described by the annihilation operators a?l for the optical modes and c? for the mi- crowave eld, as well as by the corresponding creation operators a?? and c??l . All these 31 operators commute, except [a? , a??] = 1 and [c?, c??l l ] = 1. The operators n? = a? ?a? and l l l n? = c??c? stand for the photon numbers in the optical and microwave elds, respec- C tively. The optical and microwave input signals are treated as quantum coherent states [76]. The total Hamiltonian of the open-loop system can be explicitly dened as H?tot = H?int + H?free + H?pump (2.18) where ? H?int = h?g {c? a? ? ? ?ma?m+1 + c? a?ma?m+1} (2.19) m is the interaction Hamiltonian corresponding to the quadratic nonlinearity of the WGM resonator, ? H?free = h?? c? ?c? + h?? a??ma?m (2.20)C A m is the free Hamiltonian corresponding to the cavity frequency detunings, and ? ? H? ? ? ? ?pump = ih? 2?e(Aina?0 ?Aina?0) + ih? 2?e(Cinc? ? Cinc?) , (2.21) is the Hamiltonian that accounts for the optical and microwave pump elds Ain and 32 Cin, which are dened as ? ? A P PL Min = and Cin = . (2.22) h?? h?? L M We can therefore compute the total Hamiltonian in Eq. (2.18) and use it to de- rive the time-domain equations describing the interactions between the optical and microwave photons. 2.5.3 Time-Domain Equations We can now use the total Hamiltonian H?tot to obtain the following equations for the annihilation operators in the Heisenberg picture: 1 ? { ? } a?? l = [a?l, H?tot] + ? ?sa?l + 2?s V?s,l ih? s=i,e ? = ??(1 + i?)a?l ? ig(c?a? + c??l?1 a?l+1) + ?(l) 2?eAin ? ? + 2?i V?i,l + 2?e V?e,l (2.23) 1 ? { ? } c?? = [c?, H?tot] + ? ?sc? + 2?s W?s ih? s=i,e? ? = ??(1 + i?)c?? ig a?? ? ? m a?m+1 + 2?e Cin m + 2?i W?i + 2?e W?e , (2.24) where the temporal vacuum uctuations associated with losses have been explicitly introduced using the operators V?i,l (V?e,l) for the intrinsic (extrinsic) optical losses for the mode l, and W?i (W?e) for the intrinsic (extrinsic) microwave losses, respec- tively. These operators have zero expectation value and obey the commutation rules 33 [V? (t), V?? (t?s,l s?,l? )] = ?s,s? ?l,l? ?(t ? t?) and [W? (t), W?? (t?s s? )] = ?s,s? ?(t ? t?), with the V? and W? operators uniformly commuting as well. 2.6 Semi-Classical Formalism 2.6.1 Motivation The quantum formalism is required when certain phenomena such as sponta- neous parametric down conversion need to be investigated in depth. In our system, we are only interested in the macroscopic and deterministic behavior of these intra- cavity elds, and therefore, only the stimulated eects are of interest. In that case, the approach where the elds are treated semiclassically is appropriate and provides sucient accuracy. 2.6.2 Slowly-Varying Envelope Approach Passing from the quantum to the semiclassical model corresponds to transfor- mations where the creation and annihilation operators are transformed into complex- valued, slowly-varying envelopes variables, following a? ? ?l ? Al, a?l ? Al , c?? C, and c?? ? C?. By analogy to the photon number operators a??l a?l and c??c?, the real-valued quantities A?lAl ? |Al|2 correspond to the number of optical photons in the mode l, while C?C ? |C|2 is the number of microwave photons in the RF strip cavity. Both these photon number quantities are dimensionless, and so are Al and C. However, one should note that while Al and C are cavity elds, the input elds Ain and Cin are propagating elds: They are such that |Ain|2 and |Cin|2 correspond to photon 34 uxes (i. e., number of photons per second) entering the modulator when the optical and microwave input powers are P and P , respectively. Therefore, the unit of the L M input elds Ain and Cin is s?1/2. In our analysis, we are only interested in the deterministic dynamics of the in- tracavity elds, and therefore we can disregard the quantum uctuations (along with any other stochastic inuence). Consequently, the quantum Eqs. (2.23) and (2.24) can now be rewritten under the following semiclassical form: ? A?l = ??(1 + i?)Al ? ig[CAl?1 + C?Al+1] + ?(l) 2?eAin (2.25) ? ? C? = ??(1 + i?)C ? ig A?mAm+1 + 2?e Cin , (2.26) m where the new dynamical variables of the system are the complex-valued cavity eld envelopes Al and C, of respective carrier frequencies ? + l? and ? . This equationL R R ignores the time delay from the detector to the electroptic modulator because it is negligible in comparison to the cavity photon lifetime ? . ph 2.7 Output Fields and Transmission Functions In this section we have a discussion on the derivation and eects of the output optical and microwave elds, as well as their transmission functions. 35 1 (a) 0.5 0.5 (b) ?0 10 ?5 0 5 10 ? Figure 2.7: Output optical and microwave power of the rst harmonic as a function of the detuning between the laser pump frequency and the WGMR's resonant pump frequency. (a) As ? approaches 0, almost all the emitted light goes into the WGMR and Popt,out approaches 0. (b) A small fraction of the emitted light is detected by0 the PD ans converted into RF signal. Fig. 2.7(a) is computed with Eq. (2.29), and Fig. 2.7(b) is computed with Eq. (2.33). 2.7.1 Output Optical and Microwave Field The output optical elds are expressed as ? Aout,l = ?Ain ?(l) + 2?eAl . (2.27) for each mode l, and the total output eld is ? A = A eil? tout out,l R . (2.28) l Note that Aout is a propagating eld like Ain (and not a cavity eld like Al), and consequently, its square modulus |A 2 ?1out| is also a photon ux with units of s . The corresponding optical output power in units of watts is P 2opt,out = h?? |Aout| (2.29)L 36 Prf,1 (?W) Popt,out (mW)0 As far as the microwave output power is concerned, we note that an innite- bandwidth photodetector would output a RF signal proportional to the incoming optical power, and we can write V (t) = SPopt,out = h?? S|A 2out| , (2.30)PD L where V (t) is in volts, while S is the sensitivity of the photodiode in units of V/W. The generated microwave would be a multi-harmonic signal, and would feature spectral components of frequency n ? ? , with n = 0, 1, 2, . . . The voltage output R of the photodiode can therefore be Fourier-expanded as 1 +?? [ ]1 +?? V (t) = M0 + Mn exp(in? t) + c.c. ? V ,n(t) , (2.31)PD 2 2 R PDn=1 n=0 where c.c. stands for the complex conjugate of the preceding terms, and ? M ?n = 2h?? S AL out,mAout,m+n (2.32) m is the complex slowly-varying envelope corresponding to the microwave spectral component VPD,n(t) of frequency n ? ? (in volts). The microwave power for theR harmonic of frequency n? ? can then be evaluated as R |M |20 |M 2n| Prf,0 = and Prf,n = for n ? 1 , (2.33) 4Rout 2Rout where Rout is the characteristic load resistance in the RF branch. 37 1 | T 2opt,1 | 2 ? | T2 opt,2 |10 | T 2opt,3 | 10?4 10?6 10?5 10?4 10?3 10?2 10?1 PM (mW) Figure 2.8: Optical power transmission of the rst three harmonics. The maximum power of the rst harmonic approaches 1. The transmission power is computed with Eq. (2.34). The optical and microwave output power give us a prole of the input laser power that goes inside the WGMR cavity as a function of the alignment between the laser pump frequency and the WGMR's resonant frequency of the pumped mode. We note that when the laser frequency is perfectly aligned with the resonant frequency mode of the WGMR (i.e ? = 0), almost all the pumped photons go inside the WGMR cavity. As a result, the output optical ux drops sharply [Fig(2.7)(a)] and only a small fraction of the emitted photons is photodetected and converted into RF signal, causing a decrease in Prf1 [Fig(2.7)(b)]. 2.7.2 Optical and Microwave Transmission Function The optical and microwave power transmission give us a prole of dependance of the optical harmonics and output microwave powers on the input microwave pump power. 38 | T 2opt,l | 1 10?3 10?6 10?9 10?6 10?5 10?4 10?3 10?2 10?1 PM (mW) Figure 2.9: Normalized microwave power transmission versus input microwave power P . Higher transmission is achieved at lower input cavity eld . The transmission M output power is computed with Eq. (2.35). The optical power transmission coecient of the modulator is dened as P |A |2|T 2 opt,out outopt| = = h?? (2.34) P Lin PL We note that |T 2opt| ? [0, 1]. In comparison, the transmission coecient for a typical Mach-Zehnder electrooptical modulator is dened instead as |T 2opt| = P 2out/Pin = cos [x + ?] ? [0, 1], where x and ? are the suitably normalized RF and bias voltages. In this work, we only computed the optical power transmission of the rst 3 harmonics [Fig. 2.34]. The normalized maximum power of the rst harmonic approaches unity. The microwave power transmission coecient of the modulator is dened as 2 |T |2 Prf1,out |M1|rf = = (2.35) Pin 2RoutPM We note that higher transmsission of the Prf,out signal is achieved at lower micowave 39 | T 2rf,out | ?60 ?80 10?6 10?5 10?4 10?3 10?2 10?1 PM (mW) Figure 2.10: Demodulated microwave power versus input microwave power P . The M absolute value of the demodulated RF signal is about 20 dB less than the input microwave power. The demodulated power is computed with Eq. (2.33). input eld [Fig. 2.35]. The normalized maximum power of the rst harmonic is less than unity. Finally, we see from Fig., 2.10, that the absolute value of the demodulated RF signal is about 25 dB less than the input microwave power. 2.8 Numerical Simulations In this section we will present the time-domain simulations of the intracavity number of photons as well as the voltage inside the microwave RF strip. All simu- lations were performed using a fourth-order Runge-Kutta algorithm on Eqs. (2.25) and (2.26). We used a xed time-step that is proportional to the smallest time-scale of the system. For the purpose of the simulations presented in this chapter, the xed step is ?e/60. 40 Demodulated PRF , dB ?107 1 | A?1 |2 | A |20 | A |20.5 1 0 0 0.05 0.1 0.15 Time (?s) Figure 2.11: Number of photons inside the cavities of a WGMR. Simulations were made for N = 3 modes and few photons initially. As time increases, more photons are found in the sidemode cavities, with an equal distribution between modes l =| l |. | A 2?1 | is overshadowed by | A 21 | . This graph is computed from Eqs. (2.25) and (2.26). 2.8.1 Optical Modes Temporal Dynamics Simuations of Eq. (2.26) shows that when the system is excited with an input photon ux Ain and microwave photon ux Cin, the sidemodes also get excited so that the number of photons in modes l = ?1, 1 becomes higher than that of the central mode; furthermore, the photons are equally distributed between the modes | l | of the optical cavity [Fig. 2.11]. 2.8.2 RF Cavity Temporal Dynamics We may also gain a better insight into the open-loop dynamics of the mi- crowave by looking at the voltage in the resonator strip Vcav instead of the microwave eld C. This voltage is calculated as: ? 2Routh??M Vcav = | C | . (2.36) TR 41 | A |2l 6 4 2 0 0 0.05 0.1 0.15 0.2 Time (?s) Figure 2.12: Voltage inside the resonator strip. Simulation was carried for N = 3 modes and few photons initially. As time increases, the microwave volatge increases to a steady value determined by Eq. (2.36). Because the microwave and optical leds dynamics are coupled, the nal value of Vcav depends on both the optical (Ain) and microwave (Cin) pump elds [Fig. 2.12]. We also note from Eq. (2.36) that the microwave voltage is inversely proportional to the square root of the photon round-trip TR. 2.9 Conclusions In this chapter we have introduced the fundamental concepts needed to un- derstand our model. We have presented the whispering-gallery mode resonator and the WGM-based elctro-optic modulator system. We have proposed a deterministic model that allows us to understand the dynamics of the system. In particular, we presented a quantum approach followed by its semi-classical formalism. We also de- rived the output optical and microwave elds and transmission functions. Finally, we presented some numerical simulations to validate our model. The next chapter will focus on the closed-loop dynamics of such system that corresponds to the situation where the microwave output signal is fed back into the WGMR. The resulting system is a miniature optoelectronic oscillator based on 42 Vcav (V) whispering-gallery mode modulator. We will focus on determining and optimizing the parameters that lead to the sustained oscillations of this type of OEO. 43 Chapter 3: Miniature Optoelectronic Oscillators 3.1 Introduction In the previous chapter we derived a deterministic model to describe the dy- namics of a whispering-gallery mode-based electro-optical modulator. We started with the quantum formalism and derived the semi-classical equivalent. In this chap- ter, we will study the closed-loop system that results when the microwave output signal is fed back into the whispering-gallery mode modulator; such system is a miniature optoelectronic oscillator based on the WGM modulator. The objective of this chapter is therefore to propose a full time-domain model accounting for all nonlinear interactions in miniature OEOs based on electrooptical WGM modula- tors. We also aim at performing an analytical stability study that will permit the determination of the threshold value of the feedback gain beyond which self-starting oscillations are triggered. This chapter is organized as follows. Section 3.2 is devoted to the description of the miniature OEO under study. The time-domain equations governing the dy- namics of the miniature OEO are derived in Sec. 3.3. Section 3.4 presents some numerical simulations of the temporal dynamics of the system. The stability anal- ysis to determine the threshold gain for the self-oscillations is presented in Sec. 3.5 44 (a) Conventional OEO (b) Miniature OEO Laser MZM Laser Photon The WGMR A PDin Aout storage DL simultaneously Bias achieves: ? PS - NonlinearityBPF PD Amp Nonlinearity - Filtering ?? - Photon storage Cin ? Amp Filtering PS Figure 3.1: Comparison between the architectures of conventional and miniature OEOs. The optical paths are in red, and the electric paths in black. Polariza- tion controllers between the lasers and the modulators are generally necessary, but have been omitted here for the sake of simplicity. (a) Conventional OEO. MZM: Mach-Zehnder modulator; DL: Delay line; PD: Photodiode; PS: Phase shifter; BPF: Narrowband bandpass lter; Amp: RF amplier. (b) Miniature OEO. WGMR: Whispering-gallery mode resonator; The other acronyms are the same as in (a). Note that in the miniature OEO, the WGMR is a single component that replaces the MZM, the DL and the BPF in the conventional OEO. and the optimization analysis is led in Sec. 3.6. Section 3.7 analyzes the important case of amplierless miniature OEOs. The last section concludes the chapter. 3.2 System While the most common type of OEOs feature a time-delayed feedback, we will present and analyze a miniature OEO based on WGM modulator. 3.2.1 Conventional OEO The rst generation of OEO is displayed in Fig. 3.1(a). It features a light source, a modulator to enhance the optical nonlinearity, an optic ber for optical energy storage, and a lter for spectral purity. The length of the optic ber incurs 45 a time delay and is inversely proportional to the FSR. Recalling Sec. 1.3 of Chapter 1, the dynamics of the microwave oscillations in a time-delayed OEO may be investigated with Eq. (1.9). Using the assumption that the ratio between the fastest and slowest time-scale of narrowband OEOs is too big (Q ? 100), we can also describe the microwave dynamics with Eq. (1.10). This RF latter equation is known as the slowly-varying complex envelope dynamics. The analysis of time-delayed OEOs is computationally intensive and requires delay-based algoritghm. Moerover, such systems are very bulky because of the long delay line and additional electronic components. Therefore, there is a need to investigate other technologies that may bypass the time-delay and be of smaller size. One such technology is the miniature OEO based on a WGM modulator. 3.2.2 Miniature OEO based on WGM modulator The miniature OEO under study is displayed in Fig. 3.1(b). The WGM res- onator is a lithium niobate (LN) disk of main radius a, that is used as a resonant electrooptical modulator. This modulator has an optical input, an RF input, and an optical output. The optical input is a telecom laser signal at power P with L wavelength ? ' 1550 nm, and the corresponding angular frequency is ? = 2?c/? L L L with c being the velocity of light in vacuum. The WGM resonator has a free-spectral range that can be determined as ? = c/ang = 2?/T , where ng is the group ve-R R locity index of the lithum niobate at the pump wavelength, and T is the photon R round-trip time in the optical cavity. 46 As a reminder from Subsec. 2.4.1 in Chapter 2, we consider the reduced az- imuthal order l = `? `0 so that the WGMs involved in the system's dynamics can now be symmetrically labeled as l = 0,?1,?2, ? ? ?, with l = 0 being the pumped mode which has a resonant frequency ?0. The pump frequency ? is very closeL to the resonant frequency ?0 of the pumped mode, the normalized detuning being equal to ? = ?(? ? ?0)/? (Eqs. 2.13 and 2.14).L The RF strip resonator coupled to the WGM disk has a resonance frequency that matches the FSR of the optical cavity. It has a loaded quality factor Q dened M in Eq. 2.15. The microwave input with power PM has a frequency ? very close toM ? , with normalized RF detuning factor ? = ?(? ? ? )/? (Eqs. 2.16 and 3.7). R M R The two main tasks to undertake are now (i) to build a time-domain model to describe the dynamics of this oscillator, and (ii) to perform the stability analysis of this model in order to determine the threshold gain leading to the self-oscillatory behavior. Unless otherwise stated, the parameters of our system are the same as in Chapter 2 and are, without loss of generality: P = 1 mW; ? = 1550 nm; ? /2? = L L R 10 GHz; S = 20 V/W; g/2? = 20 Hz; Q = 5?107i and Qe = 107 (this denes all the ? coecients); Q = ? /2? = 100; and nally, the RF line is impedance-matched M R with the modulator input electrode with Rout = 50 ? and ?i = ?e = ?/2. 47 3.3 Model In this section we will build on the open-loop model described in Chapter 2 to derive the full-time domain model describing the deterministic dynamics of minia- ture OEO based on WGM modulators. We use a slowly-varying envelope approach. 3.3.1 Open-Loop Semi-Classical Model Recalling Eqs. (2.25) and (2.26) from Chapter 2 and using semi-classical for- malism, the open-loop model of our system can be written as ? A?l = ??(1 + i?)Al ? ig[CAl?1 + C?Al?+1 ] + ?(l) 2?eAin (3.1) ? C? = ??(1 + i?)C ? ig A?mAm+1 + 2?e Cin , (3.2) m where the dynamical variables of the system are the complex-valued cavity eld envelop?es Al and C, of respec?tive carrier frequencies ? + l? and ? . The variablesL R R Ain = PL/h??L and Cin = PM/h??M are respectively the optical and microwave pump elds. 3.3.2 Closed-Loop Model The miniature OEO corresponds to the closed-loop system where the output microwave signal of the photodetector is used to feed the RF electrode of the WGM electrooptical modulator. In order to mathematically describe this physical proce- dure, we assume that only the fundamental toneM1 [see Eq. (2.32)] with frequency 48 ? of the photodetected optical signal is fed back to the RF electrode of the mod- R ulator, while the DC and higher-harmonic tones are ltered out. In order to close the oscillation loop, the corresponding voltage signal is subsequently amplied and phase-shifted before being injected in the RF electrode of the electrooptical mod- ulator. The envelope of the normalized microwave signal at the input port of the WGM modulator is now dened as C i? 1in,OEO = ?e [2Rout h?? ]? 2M1 , (3.3)R where ? ? 0 is the real-valued dimensionless feedback gain, which is controlled by an RF amplier just after the photodiode. All the loop losses are lumped into the feedback term ? as well (including the portion of the RF signal that is outcoupled for technological utilization, but excluding the strip and WGM resonator losses). We can therefore express the gain as ? = GAGL , (3.4) where GA (? 1) is the RF amplier gain, while GL (? 1) is the loss factor of the electric branch. The parameter ? stands for the microwave rountrip phase shift, that can be adjusted to any value (modulo 2?) using the in-loop RF phase shifter. By replacing Cin by Cin,OEO in Eq. (3.2), we obtain the closed-loop model for 49 the miniature OEO as ? A?l = ??(1 + i?)Al ? ig[CAl?1 + C?Al+1] + ?(l) 2?eAin (3.5) ? C? = ??(1 + i?)C ? ig A? { m Am+1 ? m ? } +?ei? ? 2?e A?mAm+1 ? Ain 2?e(A??1 +A1) , (3.6) m where the dimensionless constant ? 1 2?e ? = 2h?? S (3.7) L 2Rout h??R is a characteristic optoelectronic parameter of the oscillator (' 3.5 ? 10?3 in our case). Obviously, this eciency coecient ? is larger when the photodetector sen- sitivity S is increased; it increases as well when ? is decreased, that is, when R the resonator is enlarged. This is due to the fact that the electrical energy yields more microwave photons when their individual energy quantum is lower. This phe- nomenology indicates that high-Q mm-size WGM resonators, which are character- ized by GHz-range FSRs, are the most suitable form that perspective. The reader can note that the overall electrical gain of the feedback loop is in fact the parameter ? dened as ? = ??ei?. (3.8) This parameter weights the eciency of the process that retrieves microwave en- ergy from the output electrooptical comb generated by the WGM modulator via photodetection, and feeds it back as an electrical signal inside the RF strip cavity 50 ?10?5 0.5 0.5 (a) (b) 0.25 0.25 0 0 0 2.5 5 0 2.5 5 Time (?s) Time (?s) Figure 3.2: Amplitude of the closed-loop microwave signal at ? = 0.5, with Vcav being the voltage inside the RF resonator strip. The results were simulated using Eqs. (3.5) and (3.6). (a) ? = 10. (b) ? = 12. of the modulator. Also note that since our input optical eld Ain is real-valued, we can drop the calligraphic notation and simply write it as Ain: it means that we have arbitrarily set its phase to 0, and as a consequence, the optical phase to all the intracavity elds Al is determined with regard to the pump laser eld. 3.4 Numerical Simulation of the Temporal Dynamics Unless otherwise stated, all simulations of Eqs. (3.5) and (3.6) presented in this chapter were performed using a fourth-order Runge-Kutta algorithm with a xed time-step that is ?e/20. 3.4.1 Voltage inside the RF Resonator Strip Equations (3.5) and (3.6) govern the dynamics of the miniature OEO, and permit to undertake a complete theoretical analysis of that closed-loop system. In particular, they allow us to achieve a deep understanding of the system's temporal dynamics via numerical simulation as the gain ? is varied. Figure 3.9 displays 51 Vcav (V) numerical simulations performed with the fourth-order Runge-Kutta algorithm, and we have considered a total of 41 modes (l = ?20, . . . , 20). The initial conditions are set such that there are a few photons in the optical modes and in the RF cavity (|A (0)|2l ? |C(0)|2 ? 1), and the eld variables have random phases. The laser detuning is set at ? = 0.5, and the loop phase shift is ? = 0. Figure 3.9(a) shows a low microwave signal when the gain ? = 10; this signal eventually decays to 0. On the other hand, Figure 3.9(b) shows a growing microwave signal at ? = 12; such signal eventually settles to a non-zero equilibrium value. We note here that the change of behavior we observe is an evidence of the existence of a critical gain ?cr, beyond which our system oscillates at a constant amplitude. 3.4.2 Optical and Microwave Output Power Analogously to Eq. (2.22), we can determine that the microwave photon ux after the photodetector is Prf,1/h?? , where Prf,1 is the power of the fundamentalR tone as dened in Eq. (2.33). From the technological perspective, it is useful to note that the output optical signal (electrooptical comb) of the miniature OEO is proportional to Aout, while the microwave output signal is proportional toM1. In the later case, the RF power at the output of the photodiode is Prf,1, while the microwave power of the signal after the amplier is 2 P = h?? |C 2 2 |M1| 2rf,out R in,OEO| = ? = ? Prf,1 , (3.9)2Rout 52 and it corresponds to the maximal RF power generated in the miniature OEO feedback loop. Figure 3.3 displays numerical simulations performed with the fourth-order Runge-Kutta algorithm, and we have considered a total of 41 modes (l = ?20, . . . , 20). The parameters are the same as in Subsec. 2.4.2, and the initial conditions are the same as in Subsec. 2.8 The top row displays the time-domain dynamics of some output optical modes Popt,out,l, described by the equation below: P 2opt,out,l = h?? |Aout,l| , (3.10)L where Aout,l is dened in Eq. (2.27). We have numerically observed, as expected, that the dynamics of a given mode l is of the same order of magnitude (but not identical) to the one of its mirror mode ?l: for that reason, we have only plotted the modes l ? 0 in order to avoid crowding the gures with redundant plots. The bottom row displays the temporal dynamics of the RF signal at the output of the amplier, i. e. Prf,out as dened in Eq. (3.9). For the chosen parameters, numerical simulations asymptotically yield a non- null value for the pumped mode l = 0, but a null amplitude for the sidemodes l 6= 0 when ? < 10.97, leading to a null RF output as well. Once the feedback gain ? is set to a value higher than 10.97, the sidemodes dynamics eventually leads to constant non-zero amplitudes, and an RF signal is generated. We have not observed here metastable (unusually long) transient behavior as it can sometimes be the case in conventional OEOs (see ref. [77]). 53 When ? = 12, Fig. 3.3(a) shows that the pumped mode becomes depleted and exchanges energy with the modes l = ?1, which subsequently settle to a non-null constant value. The dynamics of the other sidemodes (|l| ? 2) is still negligible at this point. As shown in Fig. 3.3(d), this process generates a RF signal at the same timescale, with Prf,out ' 0.04 mW. When the gain is increased to ? = 20 [Fig. 3.3(b)], the energy exchange from the pump to the sidemodes is more pronounced, and eventually leads to the situation where the output power in the sidemodes l = ?1 is higher than the one in the pumped mode l = 0 (note however that these are output elds, and not intracavity elds). The sidemode pair l = ?2 starts to have a noticeable amplitude as well. The RF signal dynamics displays a transient behavior qualitatively similar to the one of the optical modes, before settling to a steady- state value Prf,out ' 0.3 mW [Fig. 3.3(e)]. As shown in Fig. 3.3(c), further increase of the gain to ? = 40 leads to higher complexity in the pump-to-sidemode power conversion, so that the sidemode pair l = ?3 starts to display sizable oscillations as well. Accordingly, the RF signal settles to a higher value with Prf,out ' 0.9 mW [Fig. 3.3(f)]. Several trends can be outlined in the OEO dynamics as the feedback gain ? is increased. We can rst observe that the output optical modes always have a power that is of the order of the laser pump (here, P = 1 mW), and that the benet of L increasing the feedback gain is to improve the conversion eciency from the pump to the sidemodes (up to a certain extent). The top row consistently shows the exci- tation of additional pairs of sidemodes as the gain is increased, thereby conrming that the WGM resonator plays the role of a dynamical frequency converter. The 54 ? = 12 ? = 20 ? = 40 1.0 l = 0 1.0 l = 0 1.0 l = 0 (a) l = 1 (b) l = 1 (c) l = 1 l = 2 l = 2 l = 3 0 0 0 0.1 1.0 5.0 (d) (e) (f) 0 0 0 0 2 4 0 0.5 1 0 0.25 0.5 Time (?s) Time (?s) Time (?s) Figure 3.3: Time domain dynamics for the optical and microwave power, obtained via the numerical simulation the model presented in Eqs. (3.5) and Eqs. (3.6) for ? = 0.5 and ? = 0. The dierent columns correspond to dierent values of the feedback gain. The top row displays the temporal dynamics of some output optical modes P 2opt,out,l = h?? |Aout,l| , while the bottom row displays the temporal dynamicsL of the microwave signal P 2 2rf,out = ? |M1| /2Rout at the output of the RF amplier. The critical value of the gain below which there is asymptotically no sidemode and RF oscillation is ?cr ' 10.97. second observation is that while the optical power is only redistributed amongst the side modes, the RF power steadily increases with the gain. The third observation is that when the gain becomes larger, the transient dynamics is shortened while re- maining in the ?s timescale (set by the ? photon loss rates). However, this shortened transient dynamics induces pronounced, sharply peaked relaxation oscillations. In the next sub-section, we will investigate the stability properties of our time-domain model and dene the conditions under which self-starting oscillations are triggered in the miniature OEO. 55 Prf,out (mW) Popt,out (mW) 3.5 Stability Analysis and Threshold Gain When ? is very low, conventional wisdom from self-oscillators theory (con- rmed by our numerical simulations in Subsection 3.4.1) suggests that none of the sidemodes with l 6= 0 is excited. However, as the gain is increased, there should be a critical value ?cr beyond which self-sustained oscillations are obtained, with asymptotic values C =6 0 and Al 6= 0. The objective of this section if to nd ?cr analytically. 3.5.1 Trivial Equilibrium Points When the gain parameter ? is null, the system receives no RF excitation and the steady state solution of Eqs. (3.5) and (3.6) can be straightforwardly derived as ????? ?2?e? Ain if l = 0C ?(1+i?)= 0 and Al = ??? . (3.11)0 if l 6= 0 This solution is the trivial equilibrium of our oscillator, and it corresponds to a situation where only the central mode l = 0 is excited. Our objective is to study the stability of this trivial equilibrium. 3.5.2 Pertubation Analysis In order to determine the linear stability of the trivial xed point of Eq. (3.11), we need to nd the Jacobian of the ow corresponding to Eqs. (3.5) and (3.6). We 56 achieve this by applying a pertubation Al + ?Al and Cl + ?C to our system. If we consider an electrooptical comb with 2N + 1 sidemodes, the variables of the perturbation ow are ?Al with l = ?N, ? ? ? , N and ?C, such that the dimensionality of the resulting ow is 2N+2 and the Jacobian around the trivial solution is an (2N+ 2)? (2N +2) complex-valued matrix. However, one can note that the perturbations ?Al with |l| ? 2 are of second order and do not inuence the eigenvalue spectrum of this Jacobian. This is due to the fact that the rst sidemodes to be excited in electrooptical combs are necessarily the ones adjacent to the pumped mode, with l = ?1, and from there the comb sequentially grows outwards in the frequency domain. In other words, the sidemodes l = ?2,?3,?4, ? ? ? are excited through a cascaded mechanism that require the modes l = ?1,?2,?3, ? ? ? to be excited beforehand. This phenomenology is similar to the one observed in WGM OEOs with Mach-Zehnder modulators (see ref. [47]), but quite dierent from the one observed in Kerr comb formation where the rst modes to be excited via modulational instability are not necessarily adjacent to the pumped mode [42, 78]. Along with the perturbations ?Al with |l| ? 2, the perturbation ?A0 of the pumped mode is also irrelevant for the stability analysis, because it is a neutrally stable with a null eigenvalue. Therefore, stability analysis is drastically reduced from 2N + 2 to 3 perturbation variables, namely ?A?1, ?A1 and ?C, which obey the linearized autonomous ow ?A???1 = ?? (1? i?) ?A??1 + igA?0?C (3.12) ?A?1 = ?? (1 + i?) ?A1 ? igA0?C (3.13) 57 [ ? ] ?C? = ?? (1 + i?) ?C + (2?e? ? ig)A0 ? ? 2?eA ?A?[ in ?1? ] + (2?e? ? ig)A?0 ? ? 2?eAin ?A1 , (3.14) where A0 is explicitly dened via Eq. (3.11), while ? = ??ei? is the overall gain parameter in the electrical branch. The Barkhausen phase condition for autonomous oscillators imposes that ? should be real-valued, i.e. the phase shifter should be set such that ? = 0 or ? (modulo 2?)  as we will see later on, the appropriate sign for ? will actually depend on the sign of ?. 3.5.3 Reduced Jacobian Matrix The complex-valued ow in Eq. (3.14) can be rewritten under the matrix form as ?X? = J ??X, where ?X = [?A? T?1, ?A1, ?C] is the perturbation vector and J is the 3? 3 Jacobian whose eigenvalues will decide the stability of the trivial xed point. This Jacobian is given as ? ? ???? ??(1? i?) 0 igA ? 0 ??? J = ????? ? 0 ??(1 + i?) ?igA ?0 ???? (2?e? ? ig)A0 ? ? ? ? 2?eAin (2? ? ? ig)A?e 0 ? ? 2?eAin ??(1 + i?) (3.15) From the analytical point of view, it is mathematically dicult investigate the spectral stability of a three-dimensional Jacobian when it is complex-valued. How- ever, this task is mathematically more tractable for real-valued Jacobian matrices. For this reason, we transform the complex-valued ow of Eqs. (3.12) through (3.14) 58 into a real-valued one by decomposing the perturbation vector and the Jacobian into their real and imaginary parts, following ?X = ?Xr+i?Xi, and J = Jr+iJi. As a con- sequence, by plugging these decompositions into the autonomous ow ?X? = J ? ?X, we nd that Eq. (3.14) can now be rewritten under the form of a six-dimensional real-valued ow following ?? ? ? ? ? ???? ?X?r ???? ?? ?Xr ?= J ? ? Jr ?Ji ?ri = ?? ?? with Jri = ??? ??? (3.16) ?X?i ?Xi Ji Jr being the expanded Jacobian, while the sub-matrices Jr and Ji are explicitly dened as ?? ???? ?? 0 pg? ? ??? Jr = ???? 0 ?? p ? g ?? (3.17)?? (p + pg) (p ? p ) ??? ? g and ? ? ???? ?? 0 qg ???? J = ???i ? 0 ??? ?q ???? (3.18)? g ? ?(qg ? q ) ?(q? g + q ) ???? < A ?with qg = g ( 0), pg = g=(A0), p = ?[2?e<(A0)? 2?eAin], and q = 2?e?=(A0).? ? Without loss of generality, we will simplify the calculations in the remainder of the article by considering that the microwave signal fed back to the RF strip resonator 59 Figure 3.4: Evolution of the particular determinants ?i as a function of the gain ?. Routh-Hurwitz condition for stability requires ?i > 0 for all i. ?3 and ?4 become negative at ? = 11.97. The plots were computed from the ?i, i = 1, ? ? ? , 6 in Eq. (3.27). is resonant, i.e. ? = 0. 3.5.4 Routh-Hurwitz Analysis and Critical Gain The trivial xed point of Eq. (3.11) is linearly stable (i.e., the OEO does not oscillate) when the real parts of all the eigenvalues of the Jacobian matrix Jri are strictly negative. These eigenvalues are solution of the 6-th order characteristic polynomial ?6 det[Jri ? ?I6] = m k6?k? = 0 , (3.19) k=0 60 where the real-valued polynomial coecients are explicitly dened as m0 = 1 (3.20) m1 = 2 (2?+ ?) (3.21)( ) m2 = ?4p p + 4q q + 2 3 + ?2 ?2 + 8??+ ?2 (3.22){ g ? g ? ( ) ( ) m3 = 4 qgp???+ pgq???+ 1 + ? 2 ?3 + 3 + ?2 ?2? } +??2 ? pgp? (3?+ [?) + qgq? (3?+ ?) (3.23)( ) ( ) m = 4p2p2 + 4q2q2 + ?2 2 2 2 2 4 g ? g ? 1 + ? ? ? + 8 1 + ? ??( ) ] [ ( ) +2 3 + ?2 ?2 + 4q 2g? q? 3 + ? ?+ 3q?? +p?? (2?+ ?)]? 4pg [2q( ( ) g p?q? )] +? p? 3 + ? 2 ?+ 3p??? q?? (2?+ ?) (3.24)[ ( ) ] m5 = 2? 2qg (q? + p??)? 2pg (p? ? q??) + 1 + ?2 ??[ ( )] ?{?2pgp? + 2qgq? + ? ?+ ? 2?+ 2? (3.25) m6 = ? 2 4q4g? 2 + 4p4g? 2 + 4q2g (q? + p ?) 2 [ ( ? ) ] +4p2 2g p? ? 2 ? p?q?? + 2q2 + q2g ? ?2( ) ( )2 +4q 2 2 2 2g (q? + p??) 1 + ? ??+ 1 + ? ? ?[ ( ) ] } ?4 ? pg (p? ? q??) 2qg (q? + p??) + 1 + ?2 ?? (3.26) The Routh-Hurwitz theorem states a necessary and sucient condition for all the eigenvalues of the characteristic polynomial of Eq. (3.19) to have strictly 61 negative real parts is to fulll the inequalities ???? ?? ?? ?? m1 m0 0 0 0 ? ? ? 0 0 ?? ??? ??? m m m m 0 ? ? ? ? ? ? ? ? ? ??3 2 1 0??? ?? ???m5 m4 m3 m2 m1 ? ? ? ? ? ? ? ? ?? ??? ??? ?? m7 m6 m5 m4 m3 ? ? ? ? ? ? ? ? ? ?? ? ?i = ?? ?? > 0 for i = 1, ? ? ? , 6 . (3.27)??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???? ??? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?mi?1 mi?2 ?? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? m ?i+1 mi ? The numerical computation of the determinants ?i as ? is varied shows that the lowest-order critical determinant that fails to fulll this inequality is ?3 (see Fig 3.4). The direct numerical computation of the eigenvalue spectrum for both J and Jri conrms that at least one eigenvalue transversely crosses the imaginary axis when ?3 = 0. The critical gain value ?cr needed to trigger the oscillations is therefore a root of the algebraic equation ?3 = m1m2m3 ?m21m4 ?m0m23 +m0m1m5 = a[?ei?]2 + b[?ei?] + c = 0 (3.28) with a = 128?2?2?2g2A4 (?? ?e) (?+ 2?e) e in { (3.29)?2 (1 +( ?2)2 ) b = ?16??gA2 ?ein 4 1 + ?2 ?4 + ?3 (?+ 2?e)? (1 + ?2) 62 [ ( )] +2??2 (3?+ 4?e) + 2? 3 8?? 2?e }?3 + ? 2 [( ) ( )] +?2? 17 + ?2 ?+ 2? 2e 9 + ? (3.30)[ ( ) ( ) c = 8? 8 1 + ?2 ?5 + 29 + 14?2 + ?4 ?4? ( ) ( ) ] +8 5 + ?2 ?3?2 + 2 13 + ?2 ?2?3 + 8??4 + ?5 . (3.31) The solution to the quadratic Eq. (3.28) is [ ] K1 1 1 ? i?cr?e = ? K2 2 ? | |K3 (3.32)Ain ? ? where ? (1 + ?2)? K1 = (3.33) 16?g(?e(??)?e)(?+ 2?e) K = 4?4 1 + ?2 + ?32 (?+ 2?e) + 2?? 2 (3?+ 4? ) [ ] [ ( ) e ( )] +2?3 8?? 2? (?2{ e ? 3) + ? 2? ? 17 + ?2 + 2? 2e 9 + ? (3.34)( ) K3 = (2?+ ?) ? 2?3 1 + ?2 + ?2 (?}+ 2?e) + 2?? (2?+ 6?[ ( ) ( )] e ) +?2 ? 1 + ?2 + 2? 2e 5 + ? (3.35) Equation (3.32) involves two branches of solutions, the rst one of the equations being ?K (K ?K ) /(?A21 2 3 in) and the second one being ?K1 (K2 +K3) /(?A2in). However, the second branch yields solutions that are about two orders of magnitude larger than the rst one in absolute value: these solutions are unphysical and can be discarded in our current conguration. 63 30 20 10 Analytical Simulated 0?4 ?2 0 2 4 ? Figure 3.5: Variation of the critical feedback strength ?cr as a function of ?. The symbols are obtained via the numerical simulation of the time-domain OEO model presented in Eqs. (3.5) and (3.6), while the solid line corresponds to the analytical solution provided in Eq. (3.36). It can be seen that the stability analysis permits to determine the threshold gain needed to trigger microwave oscillations with exacti- tude. It also appears that minimum gain is achieved for ? ' ?1. Therefore, we nally obtain the following formula for the critical feedback gain ???? ? K ? ? = 0 if ? > 0 1 ?cr = (K2 ?K3) > 0 with ???? . (3.36)A2in?ei? ? = ? if ? < 0 The miniature OEO is expected to oscillate when the feedback gain is such that ? > ?cr, and we observe that the feedback phase ? has to be adjusted dierently depending on the sign of ?, i.e., depending on the direction of the detuning from resonance in the pumped mode. Although general to all miniature OEOs, Eq. (3.36) that gives us the analytical formula of the critical gain is mathematically involved because of the computations of K1 through K3. Hence, we will dene an approximation of this formula in the coming sections. 64 ?cr Figure 3.5 displays the variations of ?cr as a function of the optical detuning ?. One can observe that the curve is symmetric with regard to axis of symmetry at ? = 0. Moreover, ?cr diverges when ? ? 0 and when ? ? ??. This can be understood in rst aproximation via the variations of the output eld Aout,0 given below: [ ] A 2?eout,0 = Ain ? 1 . (3.37) ? (1 + i?) On the one hand, when ? ? 0, the pump is resonant and accordingly Aout,0 is weak, so that the comb photodetection voltage is low  thus requiring a high gain ? to oset this power decit. On the other hand, when ? ? ??, the coupling is weak and so is A0, so that the electrooptical comb generation is poor and the photodetection signal is low as well. Therefore, it appears that optimal operation of the miniature OEO (i.e., low threshold feedback gain ?cr) requires to detune the pump laser in between these two asymptotic cases. Figure 3.6 shows the bifurcation diagrams for the optical output signals Popt,out,l, for the microwave power Prf,1, and for the RF power Prf,out generated at the out- put of the RF amplier as the gain ? is varied. The rst salient feature is that the optical power in paired modes ?l 6= 0 displays a switching behavior, with Popt,out,l 6= Popt,out,?l: However, the power in the pumped mode l = 0 and in the RF signals varies smoothly with the gain. Decreasing the algorithm time-step will lead to a sharper switching behavior between modes ?l. This behavior is quite dierent from the one observed in Kerr optical frequency combs, for example, where paired 65 0.6 (a) Optical power l = 0 of the WGMs (output) l = ?1 l = 1 l = ?2 0.4 l = 2 l = ?3 l = 3 0.2 0 1.0 (b) RF power before the amplifier 0 4.0 (c) RF power after the amplifier 0 0 50 100 150 200 Gain ? Figure 3.6: Bifurcation diagrams for the optical output signals Popt,out,l, for the microwave power Prf,1 generated by the photodiode (before the RF amplier), and for the RF power Prf,out generated at the output of the RF amplier. The parameters of the system are the same as those of Fig. 3.3, with ? = 0.5 and ? = 0. The critical value of the gain below which there is no OEO oscillation is ?cr ' 10.97, in agreement with Fig. 3.5. Note that as the gain ? is increased, there are optical mode power switches within a given sidemode pair ?l =6 0, while the pumped optical mode l = 0 and the RF signals are varying smoothly. modes typically have the same power [40, 41, 42]. The second observation that can be made is that quantitatively, Prf,1  Prf,out, with a ratio that can grow up to four orders of magnitude in our simulations. The third note is that qualitatively, the RF power Prf,1 at the output of the photodiode does not increase steadily, while the 66 Prf,out (mW) Prf,1 (?W) Popt,out (mW) power Prf,out after the amplier always does. 3.6 Optimization: System Parameters Leading to the Smallest Thresh- old Gain In this section, we determine the optimal conditions leading to the smallest value of the critical gain ?cr for the feedback gain. 3.6.1 Optimal Laser Detuning from Resonance We rst need to nd the optimal detuning ?opt for which the gain becomes minimal. We look for the roots of the algebraic equation d(?cr)/d? = 0 for ? > 0, and we are led to the equation: ( ) ( ) ( ) ?1 + 2?2 + 3?4 ?4 + 2 ?1 + ?2 ??+ ?1 + ?2 ?2 = 0 , (3.38) which is bi-quadratic in ?. There are two roots ?2 2opt,?; The solution ?opt,? has to be discarded for being negative (and thus, unphysical), while the other solution yields the desired results as ? [ ] ?1 [ ]2 ? 2 2 2 2 2 2opt,+ ? ?opt = ?? ? ? + (?+ ?) + ? + (?+ ?) + 12? (?+ ?) 6? (3.39) The formula above can be simplied: indeed, the miniature OEO is generally cong- ured in a way that the loaded optical resonance linewidth 2? is much smaller than 67 the loaded RF resonance linewidth. If we write this condition as |?/?|  1 and using this ratio as a smallness parameter, a Taylor expansion of Eq. (3.39) yields the following expression for the optimal detuning: ? ( ) ?2 ? ?opt ' ? 1? 1 ? ? ' ? ?1 when ? 0 . (3.40) 2 ? ? It therefore appears that the laser driving miniature OEO should ideally be detuned to the edge of the optical resonance, since ? = ?1 translates to ? = ??. This is A conrmed in Fig. 3.5 where it can be seen that the critical gain ?cr is minimal (' 9) around ? = ?1. We note that here, despite the fact that we have a relatively high ratio ?/? (' 0.23), the approximation ?opt = ?1 already appears to be very good, since the exact value given by Eq. (3.39) is 0.94. As noted above, the precision of this approximation ?opt = ?1 is expected to increase as ?/? ? 0, i. e., when the optical resonance becomes increasingly narrower than the microwave one. From a technological perspective, it is interesting to note that this requirement is fortunately not stringent, as the minimum appears to be relatively at: in other words, a deviation of ?5% with regard to ?opt still yields a close-to-minimum critical gain value. 3.6.2 Resonator Coupling Coecient The objective here is to dene a tunable parameter for the optimization of the system. One should keep in mind that the intrinsic coupling coecient ?i is a property of the resonator and cannot be tuned. However, ?e can be viewed as 68 1 0.75 ?i constant 0.5 0 0.5 1 ? = ?e/? Figure 3.7: ?opt as a function of ? = ?e/?. We see that ?opt ' 1 for almost all coupling regime. This gure is simulalted from Eq. (3.39) and using the relationships of Eqs. (3.42) and (3.43) a coupling eciency parameter that is indeed tunable, for example by varying the distance (a few ? ) between the prism and the resonator in Fig. 3.1. It results that L the loaded linewidth 2? can be varied by the same token. The critical gain dened in Eq. (3.36) is written as a function of ? and ?e, which in inconvenient in the present case because both parameters are coupling dependent. We therefore need to rewrite that equation in a way that a single parameter becomes responsible for the variations in coupling strength. For that purpose, it is convenient to introduce the parameter ?e ? = ? [0, 1], (3.41) ? which is the ratio between outcoupling and total losses in the resonator. The res- onator is in the regime of undercoupling when 0 < ? < 1 (most losses are intrinsic), 2 overcoupling when 1 < ? < 1 (most losses are extrinsic), and critical coupling when 2 ? = 1 . The limit case ? = 0 corresponds to the situation where the resonator is 2 uncoupled (all losses are intrinsic), while the limit case ? = 1 corresponds to the situation where the intrinsic losses are null (the intrinsic Q-factor is innite and 69 ?opt all losses are coupling-induced). The critical gain dened in Eq. (3.36) can now be rewritten as a function of the intrinsic loss parameter ?i, and the coupling ratio ? whose variations from 0 to 1 scan all the possible coupling congurations. It is noteworthy that this coecient ? plays a major role in the quantum applications of WGM resonators [79]. From Eq. (3.41), we can express ?e and ? as functions of ? and ?i as ??i ?e = ? ? [0,?[, and (3.42)1 ? ?i ? = ? ? [?i,?[, (3.43)1 ? The formulations of Eqs. (3.42) and (3.43) are particularly interesting because they allow us to describe the critical gain as a function of ?i which is a fabrication parameter of the microresonator. The critical gain of Eq. (3.36) can therefore be expressed as a function of ? where the coecients K1, K2 and K3 in Eqs. (3.333.35) are also dened as functions of ?. Figure (3.8) is a simulation of Eq. (3.36) with the parameters dened in Eqs. (3.333.35) and ? = 0.5. It shows that the optimal gain is achieved around critical coupling (? = 0.5). 3.6.3 Explicit Analytical Approximation of Critical Gain In OEO systems in general, we observe that the optical resonances are much narrower than the microwave ones; that is, the loaded half-linewidth of the optical resonance is much smaller than the half-linewidth of the loaded RF cavity resonance. 70 30 ?i constant 20 10 0 0 0.5 1 ? = ?e/? Figure 3.8: ?cr as a function of ? = ?e/?. The optimal ?cr is achieved in critical coupling (? = 0.5). This result is simulated from Eqs. (3.36), (3.42) and (3.43) where we express K1, K2 and K3 as a function of ?. We can therefore consider the following assumption: ?  1 (3.44) ? Using the assumption of Eq. (3.44) to simplify Eq. (3.36) yields a much simpler and explicit formula for ?cr ' 1 1 + ? 2 ??i ?cr ?(1? . (3.45)?) 2|?| g?A2in Equation (3.45) indicates that ?cr is grows unboundedly as ? becomes small, and null optical detuning should be avoided as it leads to prohibitively large critical gain values. 3.6.4 Optimal Resonator Coupling Coecient The objective here is to nd the optimal value ?e,opt for the resonator coupling coecient ?. In order to nd the theoretical value of the optimal ?opt (or equiva- 71 | ?cr | lently, the optimal ?e,opt), one has to insert Eq. (3.39) into Eq. (3.36), and obtain the optimal value as the solution of the algebraic equation d?cr/d? = 0. However, this procedure would be cumbersome because the equations involved are algebraically long and complicated. Nevertheless, these calculations can be signicantly simpli- ed if we straightforwardly consider the approximation |?/?|  1, thus considering Eq. (3.45), along with ?opt ' ?1. This gives accurate results as shown in Sec. 3.6.1 dealing with the optimal laser detuning. In that case, the formula for the critical gain can be approximated as ' ??i ??cr ? when ? 0 . (3.46)?(1 ?)g?A2in ? The formula above yields ?cr ' 8 with our parameters, a value that approximates quite well the minimum that is obtained in Fig. 3.5. Equation (3.46) also clearly indicates that the critical gain needed to trigger the microwave oscillations in the miniature OEO increases when the resonator becomes too undercoupled (? ? 0) or too overcoupled (? ? 1). The optimal value ?opt leading to a minimum critical gain is readily found by solving the algebraic equation d?cr/d? = 0, which therefore leads to the approximation: ' 1?opt , (3.47) 2 corresponding to critical coupling (?i ' ?e and Qi ' Qe). Numerical simulations indicate that the critical coupling condition is not stringent, and a deviation of ?5% 72 Laser PD Ain Aout Cin ? PS Figure 3.9: Amplierless miniature OEO. with regard to ?opt still yields a close-to-minimum critical gain value. This optimal value permits to nd the absolute minimum for the critical gain as ? ' 4??i ? ? h?R 1? = L R outmin ? ? . (3.48) g?A2in gSP QL i QM gSP Qi QL M For our parameters, we obtain ?min ' 4.4, which is then the absolute minimum gain needed to trigger oscillations in our miniature OEO. The formula from Eq. (3.48) indicates that the threshold gain can be lowered by increasing the nonlinearity, photodetector sensitivity, and optical power, which was expected; but more impor- tantly, it indicates that increasing the intrinsic Q-factor of the WGM resonator is more eective than increasing the Q-factor of the microwave strip cavity. 3.7 Threshold Laser Power in the Amplierless Miniature OEO In the preceding sections, we have analyzed an architecture of miniature OEO where an amplier is inserted in the electrical branch, and the role of the stability 73 analysis was to nd the feedback strength ?cr needed to self-start the microwave oscillation. We had implicitly assumed that the amplier had a tunable gain, while the optical power was xed. However, it is possible to have instead an amplier with xed gain, while the pump laser is power-tunable. The question in this case is to nd the critical laser power PL,cr that is needed to trigger RF oscillations. 3.7.1 Input Laser Power and Critical Gain We can use the results from Subsec. 3.5.4 to nd an analytical formula for the critical laser power leading to RF oscillations in amplierless miniature OEOs. In this case, we need to express the input laser power as a function of the critical gain. Hence, considering the relationships A2in = P /h?? and ? = GAGL, we canL L use Eq. (3.36) to express the critical gain ?cr as a function of the input laser power PL as ?K1h??L?cr = (K2 ?K3) (3.49) PL?ei? We can derive the critical laser power from Eq (3.49) as ?(?, ?) K1 PL,cr = with ?(?, ?) ? ?h?? (K ?K ) > 0 (3.50) G G L?ei? 2 3 A L where K1, K2, K3, and ? are the same as in Eq. (3.36). Once again we are able to derive a simpler approximation of P by using L 74 Eq. (3.45) ' 1 1 + ? 2 ??i P L ?(1? | | h?? , (3.51)?) 2 ? g?? Lcr It results that high gain amplication allows for lower laser powers, and vice versa. For example, Ilchenko et al. have reported in ref. [80] a miniature OEO where the laser power was around 70 ?W while the amplier had a gain of 45 dB (i. e., GA ? 180). However, on the other hand, higher laser power permits to use ampliers with lower gain: In fact, if the optical power is high enough, it is even possible to get rid of the amplier, thereby leading to an amplierless miniature OEO. The reader can note that amplierless OEOs have already been demonstrated with conventional ber-based architectures (see for example ref. [81]). 3.7.2 Critical Laser Power in Amplierless Miniature OEO In our system, eliminating the amplier mathematically corresponds to set GA = 1 in Eq. (3.50). As a consequence, the OEO architecture of the miniature OEO presented in Fig. 3.1 is signicantly simplied. The critical laser power needed to trigger RF oscillations in the amplierless miniature OEO can be exactly calculated as ?(?, ?) PL,cr = . (3.52) GL From this analysis, we can now dene the absolute minimal optical power that is needed to trigger microwave oscillations in an amplierless OEO. The procedure 75 for doing so is to consider negligible electrical losses (GL,opt = 1), optimal laser detuning (? = ?opt) and optimal coupling (? = ?opt), so that this absolute minimal laser power can be calculated as ( ) ?(?opt, ?opt) 1 PL,min = ' ???1,GL,opt 2 ' 4??i ' ? ? h?Rh?? L R out . (3.53) L g? gSQi QM For our parameters, this value is corresponds to 4.4 mW. The reader can also note that the last approximations in Eq. (3.53) can be readily obtained from the the numerical approximation Eq. (3.51) by setting the gain ?cr = 1. It should be noted that if amplierless miniature OEOs have the great advantage to simplify the architecture of the system, they require a careful management of the thermal eects induced in the WGM resonator by the higher laser power [82, 83, 84]. 3.8 Conclusion In this chapter, we have proposed a mathematical framework to study the time- domain nonlinear dynamics of miniature OEOs based on nonlinear WGM resonators. Our model uses time-domain equations to track the dynamics of the complex-valued envelopes of the optical and microwave elds. We have performed a stability analysis that permitted to calculate analytically the threshold value of the feedback gain that is needed to self-start the microwave oscillations. An optimization analysis has also been performed, and led us to the conclusion that the system should ideally be 76 operated at the edge of the optical resonance and close to critical coupling. Further investigation has shown that beyond a certain laser power, RF amplication is not needed anymore and the miniature OEO can become amplierless. A next step to consider in the study of the technology related to miniature OEOs is the eect of noise on the dynamics of this system. 77 Chapter 4: Stochastic Analysis of Miniature Optoelectronic Oscillators 4.1 Introduction In Chapter 3 we proposed a full-time deterministic model to describe the dy- namics of the miniature optoelectronic oscillator based on whispering-gallery mode modulators. We pertubed the system from its trivial equilibrium and derived a reduced Jacobian. We then performed the stability analysis of the Jacobian and derived the analytical formula for the critical gain leading to oscillations in the system. Finally, we optimized the critical gain and derived the critical laser input power for oscillations in an amplierless miniature OEO. The interest in studying the amplierless case rests in the fact that it decreases the noise in the system. Indeed, the stochatic dynamics of miniature OEO is linked to their spectral purity and therefore it is important to have a full model that accounts for the eects noise. However, there is no analysis available to understand how the optical and electrical noise sources in the optoelectronic loop are converted into microwave phase noise. The objective of this chapter is therefore to propose a stochastic model accounting for the optical and electronic noise in miniature OEOs based on whispering-gallery mode modulators. We will use the Langevin approach to add random noise to the deterministic model and then perform a dynamical analysis of the stochastic system. 78 This approach has already been used with remarkable success for ber-based OEOs, where it was shown that it can provide an excellent agreement with experimental phase noise spectra [50, 93, 95, 99]. This chapter is organized as follows. In Sec. 4.2 we present the various sources of noise in the miniature OEO. In Sec. 4.3 we describe the miniature OEO under study, as well as the deterministic model that rules its nonlinear temporal dynamics. We also explain the phenomenology that occurs once we add stochastic noise to our system. Section 4.4 is devoted to the derivation of the stochastic model, with an emphasis on sources of noise originating from the resonant WGM/RF cavities and from the active electronic elements (photodiode and amplier). The dynamics of the system under threshold is investigated in Sec. 4.5, while Sec. 4.6 presents the numerical simulation of the temporal dynamics. Section 4.7 presents a stochastic normal form approach that allows us to analyze the eect of noise below and above threshold. The last section concludes the article. 4.2 Noise in Miniature OEOs The object of this section is to explain the causes of noise in the oscillator, and their eects on the model's parameters. 4.2.1 Sources and Eects Noise in miniature optoelectronic oscillators arise both in the optical and elec- tronic components. It may be due to thermal uctuations, in addition to electrical 79 or optical sources. The optical noise in the whispering-gallery mode modulator is due to the laser uctuations, as well as the intracavity eld uctuations; on the other hand, the electrical noise in the photodiode is called shot noise and is caused by the random uctuations of the electric current due to the discrete nature of the electrons; shot noise can be modeled as a Poisson process [99]. Finally, the electrical noise in the amplier is mainly electronic circuitry noise, although we can have other noises such as icker and resistance eects. We should also note that as the noisy signal travels through the closed-loop, it is amplied by the amplier, inducing a multiplicative noise element. Noise aects the system's parameters in a such a way that the optical loss ? is replaced by ? + ??(t); the PD sensitivity S becomes S + ?S(t), and the gain ? becomes ? + ??(t). These noises are converted into microwave phase noise and as they increase, the signal quality decrease. In addition, noise set a limit to the smaller signal power that can be recovered. In our treatment of the noisy system in this chapter, we will neglect the noise from the PD and the gain, which are assumed to be smaller than the others. In addition, we will use the Langevin approach and use white Gaussian noise to characterize the optical and RF noises. 4.3 System In this section we briey revisit the dynamics of the system in the absence of noise; in particular, we will look into the microwave modal eld and RF output power, and review the dependance of the system's behavior on the gain. We will 80 Laser The WGMR A PDin Aout simultaneously achieves: - Nonlinearity Amp - Filtering - Photon storage Cin ? PS Figure 4.1: Miniature OEO based on WGMR modulator. The optical and electronic components are assumed to be noiseless. PD: photodiode; Amp: amplier; PS: phase shiftter. then introduce the random noise and derive the stochastic dierential equations ruling the system dynamics. We will also investigate the stochastic behavior under threshold and propose a normal form theoretical formula to compute the phase noise spectrum. 4.3.1 Noiseless System 4.3.1.1 Deterministic Model Let us recall Eqs. (3.5) and (3.6) from Chapter 3 describing the time domain dynamics of the miniature optoelectronic oscillators based on whispering-gallery 81 mode modulator: ? A?l = ??(1 + i?)Al ? ig[CA ?l?1 + C Al+1] + ?(l) 2?eAin (4.1) ? C? = ??(1 + i?)C ? ig A?mA{ m+1? m ? } +?ei? ? 2?e A?mAm+1 ? Ain 2? ?e(A?1 +A1) , (4.2) m ? where the dimensionless constant ? = 2h?? S ?e/RL outh?? is a characteristic op-R toelectronic parameter of the oscillator, ? is the microwave rountrip phase shift, ? = ?(? ? ?0)/? is the normalized optical detuning between the laser and theL pumped mode resonance, and ? = ?(? ? ? )/?, is the normalized detuning M R between the RF signal and the strip cavity resonance (set to 0 in this study). A key parameter of the oscillator is the real-valued dimensionless feedback gain ? = GAGL ? 0 where where GA (> 1) is the RF amplier gain while GL (< 1) is the overall loss f?actor of the electric branch. The laser pump eld of the WGM resonator is Ain = P /h?? : It is a real-valued envelope (null phase), and for thatL L reason it plays the role of reference for all the intracavity elds Al. 4.3.1.2 Microwave Modal Field and Ouput RF Power ? The overall optical eld exiting the WGM resonator is A = A eil? tout l out,l R , ? with Aout,l = ?Ain ?(l) + 2?eAl being the modal output elds [Note that they are propagating eld like Ain and their square modulus is therefore also a photon ux in units of s?1]. The microwave signal of interest is the output of the RF amplier, which is 82 dened by an envelope ? Mout = ?M1 = 2?h?? S A?L out,mAout,m+1 (4.3) m and power 2 |M1|2Prf,out = ? , (4.4) 2Rout where Rout is the characteristic load resistance in the RF branch, and M1 = ? 2h?? S mA?out,mAout,m+1 is the complex-valued envelopes of the microwave har-L monics of frequency ? [Eq. (2.32) in Chapter 2]. R 4.3.1.3 Dependence on the Gain In Subsection 3.5.4 of Chapter 3 we established the dependence of the dynamics of the system described by Eqs. (4.1) and (4.2) on the feedback gain parameter ? > 0. When the feedback gain is below a given critical ?cr, only the pumped mode A0 is excited (by the pump laser) and no microwave is generated. However, when ? > ?cr, a cascaded process leads to the excitation of the sidemodes Al with l 6= 0, thereby leading to the formation of an optical frequency comb that generates a self-sustained microwave oscillation in the electric branch. The analytical value of ?cr could be determined exactly analytically, but in Subsection 3.6.3 of Chapter 3 we used the approximation ?/? 1 to express explicitly the critical gain as 1 1 + ?2 ??i ?cr ' ? h?? , (4.5)?(1 ?) 2|?| g?P L L 83 Laser A PDin Aout WGMR and laser noise RFA ?a,l (t) M ?out RF strip PS cavity noise ?c (t) Figure 4.2: Miniature OEO based on WGMR modulator. Noise sources are optical and electronic. The optical source noise arise at each mode l of the resonator; the electronic noise arise at the PD and the microwave RF strip. The noises become additive as we go around the closed-loop. We negelct the eect of the multiplicative noise. PD: photodiode; Amp: amplier; PS: phase shiftter. where ? = ?e/? ? [0, 1] is the coupling ratio between extrinsic and total losses in the resonator. ?cr is minimized by critical coupling (? = 1) and edge-of-resonance 2 detuning (? = ?1), respectively. In addition, the roundtrip ? phase shift has to be set to 0 when ? > 0, and to ? when ? < 0. 4.3.2 Noisy System The output signal of a noiseless miniature OEO system is a sinusoid with fre- quency around ?R and a constant phase ?. However, as we take into consideration the noise generated in the intramodal cavities of the WGMR and that induced in the RF microstrip cavity, the phase no longer remains constant. Rather, it becomes 84 a function of time ?(t). As ?(t) varies, it may cause a frequency shift in the output signal in the time-domain, which is translated into a phase drift in the frequency domain. Unless otherwise stated, we will consider the following parameters for our sys- tem throughout this article, without loss of generality: P = 1 mW; ? = 1550 nm; L L ? /2? = 10 GHz; S = 20 V/W; g/2? = 20 Hz; Qi = 5 ? 107 and Qe = 107R (this denes all the ? coecients); Q = ? /2? = 100; and nally, the RF line M R is impedance-matched with the modulator input electrode with Rout = 50 ? and ?i = ?e = ?/2. 4.4 Stochastic Model The object of this section is to identify the most relevant sources of noise in the oscillator, and dene how they should be accounted for in the stochastic model. These random noise terms either have an additive of multiplicative eect on the system's dynamics. 4.4.1 Stochastic Noise In this work, we will only focus on the additive noise source terms, which in our context are dominant. Moreover, for the sake of simplicity, these random noise signals will be assumed to be Gaussian and white. Therefore, depending on their real- or complex-valued nature, these random signals will always be proportional to either a real-valued Gaussian white noise ?(t) with ??(t)?(t?)? = ?(t ? t?), or to a 85 complex-valued Gaussian white noise ?(t) with ??(t)??(t?)? = ?(t? t?). The miniature OEO has two cavities (optical WGM resonator and RF strip resonator), which are driven by external optical and radiofrequency signals, respec- tively. Indeed, the oscillator is unavoidably submitted to the inuence of various random noise sources, which end up driving the stochastic uctuations of the intra- cavity elds. The optical elds Al(t) are driven by a modal random eld normalized as ? za,l(t) = ?a 2? ?a,l(t), (4.6) which has to be added in the right-hand side of Eq. (4.1). One should note that the noise term za,l(t) will create ? 2 a optical photons on average in the mode l [79, 97]. Analogously, The intracavity microwave eld C(t) is driven by a random signal normalized as ? zc(t) = ?c 2? ?c(t) (4.7) to be added in the right-hand side of Eq. (4.2), that will generate ?2c microwave photons on average inside the RF strip cavity. 4.4.2 Model The sources of noise in the miniature OEO can now be added to the core deterministic Eqs. (4.1) and (4.2) to obtain the following stochastic model: ? A?l = ??(1 + i?)A ?l ? ig[CAl?1 + C Al+1] + ?(l) 2?eAin ? +?a 2? ?a,l(t) (4.8) 86 ? C? = ??(1 + i?)C ? ig A?mA{ m+1 + ? m ? } ?ei?? 2? A?e mAm+1 ? Ain 2?e(A?? ?1 +A1) m +?c 2? ?c(t) , (4.9) with the noise correlations ??a,l(t)?? ?a,l?(t )? = ?l,l??(t? t?) and (4.10) ?? (t)??(t?c c )? = ?(t? t?). (4.11) The noisy output microwave signal is stillMout(t) = ?ei?M1(t) and the out- put RF power is still determined by Prf,out = |M 2out| /2R 2 2out = ? |M1| /2Rout. One can note that when the sources of noise are discarded, the stochastic Eqs. (4.8) and (4.9) degenerate into the deterministic Eqs. (4.1) and (4.2). In this work, the stochastic dierential Eqs. (4.8) and (4.9) will be numerically simulated using the Milstein algorithm (see ref. [92]). 87 4.4.3 Equilibrium Points 4.4.3.1 Trivial Equilibrium Points As we recall from Subsection 3.5.1 in Chapter 3, the trivial equilibrium of Eqs. (4.1) and (4.2) can be straightforwardly derived as ????? ?2?e? Ain if l = 0C A ? ?(1+i?)= 0 and l = ?? . (4.12)0 if l =6 0 This solution corresponds to a situation where only the central mode l = 0 is excited. 4.4.3.2 Non-Trivial equilibrium Points On the other hand, the nontrivial equilibrium of Eqs. (4.1) and (4.2) corre- sponds to ? ?(1 + i?)Al + ig[CA C?l?1 Al+1] = ?(l) 2?eAin , and (4.13) ? { ? ?(1 + i?)C + ig A? A = ?ei?m m+1 ? 2?e A?mAm+1 m m ? ?A ?in 2?e(A?1 +A1) (4.14) This solution corresponds to a situation where all the modes are excited and enter a stationary state characterized by oscillations of xed amplitude. 88 4.5 Stochastic Analysis Under Threshold In this section, we aim at calculating the microwave power generated under threshold, that is, when ? < ?cr. The sub-threshold dynamics is generally over- looked in the literature, however, previous studies have shown that the sub-threshold stochastic dynamics is important for the characterization of the various sources of noise in the system (see for example ref. [93]). 4.5.1 Pertubation Analysis and Reduced Flow Dynamics In the case of the miniature OEO under threshold, it is possible to develop an analytical method to compute the noise power density. The starting point is to note that the stochastic model displayed in Eqs. (4.8) and (4.9) can be simplied using two assumptions. The rst one is that stochastic eects in the intracavity elds Al and C can be accounted for via the output microwave eld M1. The second one is that below threshold, there is no self-sustained microwave oscillation and as a ? consequence, only the mode l = 0 is excited with A0 = 2?eAin/?(1 + i?). The intracavity elds A?1 and C are of rst order of smallness and can be linearized around zero, while the elds Al with |l| > 1 can be outright neglected for being of higher order of smallness. Using these two simplifying assumptions, the stochastic Eqs. (4.8) and (4.9) are now reduced to ? ?A???1 = ??(1? i?)?A??1 + igA?0?C + ?a 2? ?a,?1(t) (4.15) 89 ? ? ?A?0 = ??(1 + i?)?A0 + 2?eAin + ?a 2? ?a,0(t) (4.16) ? ?A?1 = ??(1 + i?)?A1 ? igA0?C + ?a 2? ?a,1(t) (4.17) ?C? = ??(1 + i?)?C ? ig[A ?A? +A?0 ?1 0?A1]{ ? } +? 2?e[A ?A? +A? ?? 0 ?1 0 ?A1]? 2?eAin(?A?1 + ?A1) +?c 2? ?c(t) (4.18) where ? = ??ei? is the overall electrical gain of the OEO. 4.5.2 Fourier Transform and Jacobian The perturbation ?A0 is independent of the other ones and is irrelevant in the subsequent analysis for being permanently dominated by the non-null amplitude A0. Therefore, we can ignore the corresponding equation and rewrite the remaining three linear equations in the Fourier domain to obtain ?? ? ?? ? ? ?? ?A? ? ?1(?) ???? [ ] ???? ? ? a 2? Z?a,?1(?) ? ?1 ???? ?A? (?) ???? = i?I ? J ?? ? ? 1 ? 3 ?? ? ? ? ?? 2? Z? (?) ???? , (4.19)a a,1 ? ?C? ?(?) ?c 2? Z?c(?) where I3 is the three-dimensional identity matrix and ? ? ???? ??(1? i?) 0 igA ? 0 ???? J = ? ???? 0 ??(1 + i?) ?igA ? ?? (4.20)0 ? ? ?? 2?e?A ? ?out,0 ? igA0 2?e?Aout,0 ? igA0 ??(1 + i?) 90 is the Jacobian of the linear ow, with the variables Z?(?) being the Fourier transform of their stochastic counterparts ?(t). In the Fourier domain, Eq. (4.19) permits to determine explicitly the three stochastic variables of interest as a linear combination of the intracavity noise terms Z?a,c(?). In principle, the time-domain solutions could be recovered via an inverse Fourier transform. The analytical formulation of the inverse matrix is: ?? ?[ ] ? a a a ??11 12 13?1 ???? ?i?I3 ? J = ??? ? a ? , where (4.21)21 a22 a23 ???? a31 a32 a33 ? = 1/{ (1 + i?) (i+ ?)? [(1? i?)?+ i?] [(1 + i?)?+ i?] ? [(?(?1 + ?) + ?)] + 2g?e [??? (4??+ 2i??)] } + (?? i?? + i?) (pg ? iqg) (pg ? p? + i (qg + q?)) (4.22) a11 = ?? { ((1 + i?)?+ i?) (?+ i??+ i?) + (pg ? iqg) (pg ? p? + i (qg + q?)) } (4.23) a12 = ?? { ? (pg + iqg) (pg ? p? + i (qg + q?)) } (4.24) a13 = ?? { ? ((1 + i?)?+ i?) (?pg ? iqg) } (4.25) a21 = ?? { (pg ? iqg) (pg + p? ? i (qg ? q?)) } (4.26) a22 = ?? { ((i (1) + ?)?? ?) ((?i+ ?)?+ ?) ? (pg + iqg) (pg + p? ? i (qg ? q?)) } (4.27) a23 = ?? { ? ((1? i?)?+ i?) (?pg + iqg) } (4.28) 91 a31 = ?? { ? ((1 + i?)?+ i?) (?pg ? p? + i (qg ? q?)) } (4.29) a12 = ?? { ? (pg + iqg) (pg ? p? + i (qg + q?)) } (4.30) a32 = ?? { ? ((1? i?)?+ i?) (pg ? p? + i (qg + q?)) } (4.31) a33 = ?? { ((1 + i?)?+ i?) (((1? i?)?+ i?) } (4.32) Equation (4.19) allows us to solve for ?A???1(?), ?A?1(?) and ?C?(?) to which we can apply the inverse Fourier transform to obtain the time-domain corresponding signals. 4.5.3 Microwave Output RF Power The microwave power generated under threshold can be obtained via the nu- merical simulation of Eqs. (4.8) and (4.9), whose output can be suitably averaged to give 1 ?2 Prf,out = ?|Mout(t)|2? = ?|M1(t)|2? . (4.33) 2Rout 2Rout However, these numerical simulations do not give any theoretical insight into why the subthreshold noise increases the way it does with the gain. Using Eq. (2.32) of Chapter 2, which gives the formula for the microwave modal eld of the rst harmonic, it appears that the output microwave signalM1(t) is now a linear combination of ?A1(t) and ?A??1(t). This linearity can be translated in the 92 Fourier domain following ? { } M?1(?) = 2h??LS 2?e Aout,0?A?? ??1(?) +Aout,0?A?1(?) , (4.34) where Aout,0 is the output optical eld at the central mode and is computed from Eq. (3.37) in Chapter 3. Equation (4.34) implies that M?1(?) is also a linear com- bination of the intracavity noise terms Z?a,c(?). The frequency-domain integral of the microwave output spectral energy is given as ? ? [ ? ] { ? ? |M?1(?)|2 2 = 2h??LS 2?e | A 2out,0 | ?(2? 2? [a ? ?a 11a + a a ?? ?? 11 12 12 ? + a21a ? + a22a ? ]?d? + 2? 2 ? ? 21 22 c ? [a13a + a13 23a ]d?)?? 23 ? + (Aout,?0) 2(2?2? [a ? ?a 11a + a12a ]d? ?? 21 22 ? ? ? + 2?2 ?c? [a13a ]d??) + (A ? )2out,0 (2? 2 ? a? [a a21 ?? 23 ? } ?? 11 + a? a22 ]d? + 2? 2 c? [a ? a23 ]d? , (4.35)12 ?? 13 We can now dene the sub-threshold microwave output power after the RF amplier as { } ?2 1 ? +? Prf,out = |M?1(?)|2 d? , (4.36) 2Rout 2? ?? where we are using Parseval's theorem since we explicitly know M?1(?) via Eq. (4.34). 93 0.3 Poptout,?2 0.2 Poptout,?1 Poptout,1 0.1 Poptout,2 0 0 0.1 0.2 Time (?s) Figure 4.3: Variation of the optical power Poptout,l under threshold gain ?cr for the noisy miniature OEO. In noiseless system Poptout,l = 0, ? l 6= 0. However, in noisy system, Poptout,l uctuates randomly according to the noise. The results were obtained by simulating the dynamics of Eqs. (4.8) and (4.9) with ?a = ?c = 1 and computing the microwave output signal with Eq. (2.29) of Chapter 2. The value of ? is 6, and is about half the threshold gain ?cr. 4.6 Numerical Simulation of the Stochastic Dynamics In this section we present the results of the temporal simuations of the noisy miniature OEO. We rst look at the optical output elds and then the microwave modal eld output; nally, we look at the ouptut RF power evolution under thresh- old. 4.6.1 Optical and Microwave Temporal Dynamics In the absence of noise and below the threshold gain for oscillations, only the central mode l is excited. As a result, the optical output power Poptout,l is zero at all modes except mode l = 0. When we account for the optical and microwave cavity noises below threshold, we notice that the side modes have a random photon ux, which yields to a nonzero uctuating number of output photons and optical output power. This power is very small in magnitude compared to the optical output power 94 Poptout,l (nW) 3 2 1 0 0 0.1 0.2 Time (?s) Figure 4.4: Variation of the microwave power Prf1 under threshold gain ?cr for the noisy miniature OEO. In noiseless system Prf1 = 0, ? l 6= 0. However, in noisy system, Prf1 uctuates randomly according to the noise. The results were obtained by simulating the dynamics of Eqs. (4.8) and (4.9) with ?a = ?c = 1 and computing the microwave output signal with Eq. (2.33) of Chapter 2. The value of ? is 6, and is about half the threshold gain ?cr. at mode l = 0 (? 6 orders of magnitude smaller). As we see in Fig. 4.3, it is in the order of nanowatts (nW). These results were obtained by simulating the dynamics of Eqs. (4.8) and (4.9) with ?a = ?c = 1 and computing the microwave output signal with Eq. (2.29) of Chapter 2. Although |Aout,0|  |Aout,l| for all l 6= 0, the sum of beatings between the successive output cavity elds below threshold gain becomes noisy random uctu- ation so that the ouptut RF power PRF1 is a noise. Figure 4.4 shows the optical output power around half the threshold gain (? = 6). The result is obtained by sim- ulating the dynamics of Eqs. (4.8) and (4.9) with ?a = ?c = 1 and computing the microwave output signal with Eq. (2.33) of Chapter 2. We note that the microwave output power is in the picowatts (pW). 95 Prf1 (pW) (a) ?a,c = 1 (b) ?a,c = 10 (c) ?a,c = 100 ?50 ?30 ?10 Numerical Numerical Numerical ? Analytical ? Analytical Analytical60 Normal form 40 Normal form ?20 Normal form ?70 ?50 ?30 ?80 ?60 ?40 ?90 ?70 ?50 ?10?0 ? ?8030 20 ?10 0 ?30 ?20 ? ?610 0 ?0 30 ?20 ?10 0 ? = ?/?cr (dB) ? = ?/?cr (dB) ? = ?/?cr (dB) Figure 4.5: Variation of the noise power P 2rf,out = |Mout| /2Rout, when the nor- malized gain ? ? ?/?cr is increased under threshold. We have ? < 1, so that the gain in dB is 20 log ?, and is negative. The plots from left to right correspond to noise amplitudes ?a,c = 1, 10, and 100 respectively. The blue dot symbols stand for the numerical results obtained using Eq. (4.33), via the time-domain simulation of the stochastic dierential Eqs. (4.8) and (4.9). The continuous black lines stand for analytical results obtained via Eq. (4.36). The dashed red lines stand for the scaling behavior as predicted by the normal form theory in Eq. (4.40). The dotted gray lines indicate the microwave noise power corresponding to a gain of ?4.18 dB, which directly gives the amplitude of the driving Gaussian white noise power in the normal form model (from left to right, p = m2out /2Rout = ?71, ?51, and ?31 dBm, respectively). One can note the excellent agreement between numerical simulations and analytical predictions. 4.6.2 Noise Power Density Below Threshold Figure 4.5 displays the comparison between the analytical formula of Eq. (4.36) and the numerical simulations using Eq. (4.33) via the time-domain stochastic dif- ferential Eqs. (4.8) and (4.9). The normalized gain ? ? ?/?cr is increased under threshold (? < 1) and the variation of the noise power P 2rf,out = |Mout| /2Rout is determined for various values of the noise amplitudes ?a,c. One can note that the analytical formula predicts accurately the growth of noise power as the gain increased. 96 Prf,out (dBm) 4.7 Normal Form Approach for Stochastic Analysis and Phase Noise The deterministic dynamics of the miniature OEO as described in Eqs. (4.1) and (4.2) is high-dimensional and non-trivial. For example, it was shown in ref. [97] that above threshold, symmetric modes with eigenumbers ?l do not have the same amplitude, and are therefore beyond any tractable analytical approximation. How- ever, the microwave signal is only two-dimensional (complex-valued envelope car- rying information about amplitude and phase), and certainly more amenable to mathematical analysis across a gain range covering the regimes below and above threshold. Moreover, having a dierential equation for the microwave variable would enable us to investigate analytically its phase noise properties. 4.7.1 Normal Form Approach for Stochastic Analysis One could try to obtain an exact equation for M?1 through the time derivation of Eq. (2.32) in Chapter 2. This operation would result in expressing M?1 as a nonlinear expansion of terms A??lAl+1 and A?l A?l+1, but would not yield a closed- form dierential equation that only depends onM1. However, from the nonlinear dynamics systems point of view, the onset of the microwave oscillation can be viewed as the result of a Hopf bifurcation. As a consequence, normal form theory states that there is a closed-form equation for the microwave valid at least close to the vicinity of the bifurcation, with arbitrarily high precision. The bifurcation will be characterized by a linear parameter a, and a nonlinear parameter b. In general, a large (but nite) sequence of involved mathematical operations 97 are needed to determine the parameters a and b, even for low-dimensional systems (see for example ref. [96]). In our case, the minimum number of optical modes considered above threshold is 11 (with l varying from ?5 to 5), so that Eqs. (3.5) and (3.6) are at least 24-dimensional: Under these conditions, following the standard mathematical protocol to derive the normal form coecients is practically dicult to carry out. However, we will show that in the stochastic regime, using the nor- mal form approach will provide the scaling behaviors of interest below and above threshold. In this study, we will write the stochastic normal form equation for the mi- crowave as ? M? = ?aM+ ?[aM+m 2a ?m(t)]? ab|M|2M , (4.37) whereM = |M|ei? ?Mout is the complex-valued microwave envelope of interest (in V), a stands for the linear damping of the microwave (in rad/s), b stands for the nonlinear saturation (in V?2), m stands for the root-mean-square amplitude (in V) of the driving Gaussian white noise, which is delta-correlated as ??m(t)?? ?m(t )? = ?(t ? t?). The parameter ? ? ?/?cr > 0 is the normalized feedback gain, which is here aecting both the microwave and the random noise. In its deterministic version (m = 0), the normal form in Eq. (4.37) yields the 98 following solution: ?????? ?0 when ? < 1|M| = ??? , (4.38)(? ? 1)/b ? ?Mb ? ? 1 when ? > 1 ? where Mb = 1/ b can be interpreted as the characteristic amplitude ofM (in V). In other words, the trivial solution is stable when ? < 1, while the nontrivial (i.e. oscillatory) solution is stable when ? > 1. However, when noise is accounted for, the stochastic behavior deviates substantially from the deterministic one. In the stochastic sub-threshold case (m 6= 0 and ? < 1), the linear terms are of rst order of smallness, while the nonlinear term is of third order of smallness and can then be neglected. Equation (4.37) is therefore reduced to the well-known Ornstein- Uhlenbeck process, whose stationary properties can be obtained analytically. We rst rewrite Eq. (4.37) in the Fourier domain as ? M? ?m 2a(?) = (1? Z?m(?) (4.39)?)a+ i? from which we calculate the corresponding power using again Parseval's theorem, leading to { } 1 ?|M |2? 1 1 ? +? |M? |2 ? 2 m2 Prf,out = (t) = (?) d? = . (4.40) 2Rout 2Rout 2? ?? 1? ? 2Rout It appears that the microwave power under threshold should distinctively scale as ?2/(1??), and the coecient of proportionality is the noise power p 2out = m /2Rout. 99 Interestingly, Prf,out = pout (i.e., the power of the output signal and input noise are ? equal) when ?2 = 1 ? ?, that is, when ? = ( 5 ? 1)/2 ' 0.618 (or ?4.18 dB). This property is useful in order to retrieve the parameter m via pout from the sub- threshold power variation as a function of gain. Figure 4.5 displays the comparison between the numerical simulations using Eq. (4.33) and the scaling law predicted by the normal form theory in Eq. (4.40). The excellent agreement conrms the validity of the scaling behavior predicted by the normal form theory. 4.7.2 Normal Form Approach for Phase Noise 4.7.2.1 Wiener Process Dynamics The Wiener process, also called Brownian motion, is a real-valued continuous- time stochastic dynamics used to model the time-evolution of random Gaussian white noise. It is dened as ??(t) = D?(t), (4.41) where ?(t) is the real-valued Gaussian white noise and D2 is the variance of the noise. The signal described by Eq. (4.41) has a phase noise which can be analytically determined by the following equation: D2|?(?)|2 = , (4.42) ?2 100 where ? = 2?f is the frequency oset. The main advantage to describe a system with a Wiener process is that it allows for a straight-forward analytical computation of the phase noise using Eq. (4.42). 4.7.2.2 Analytical Formula for Phase Noise The stochastic dynamics above threshold corresponds to m 6= 0 and ? > 1. In this case, it is customary to neglect amplitude noise in comparison to phase noise [?t|M| ' 0], so that the amplitude of the microwave is still considered constant and given by Eq. (4.38). As a consequence, the stochastic Eq. (4.37) is reduced to M? = [?t|M|+ i??|M|] ei? ' i??|M| ei? ? ' ?m 2a ?m(t) . (4.43) from which we straightforwardly derive the phase noise spectrum as [ ] | |2 1 a? 2m2 ?2 abm2 ??(?) = |M| ' ? (4.44)?2 2 ? 1 ?2 in units of rad2/Hz. This phase noise spectrum displays the usual f?2 dependence for oscillators driven by white noise, and the normal form analysis provides two key elements for phase noise optimization. The rst one is that the diusion coecient of the phase noise isD = abm2: In other words, it depends on the three parameters that characterize the stochastic normal form Eq. (4.37). From the physical viewpoint, we nd as expected that the phase noise is reduced by lower cavity losses (a ? 0) 101 2 Numerical Normal form 1 0 0 1 2 3 ? = ?/?cr Figure 4.6: Variation of the microwave power Prf,out = |M 2out| /2Rout, when the normalized gain ? ? ?/?cr is increased above threshold (? > 1). The blue dot symbols stand for the numerical results obtained using the time-domain simulation of the stochastic dierential Eqs. (4.8) and (4.9). The dashed red lines stand for the scaling behavior as predicted by the normal form theory in Eq. (4.38). The microwave power has been normalized to an arbitrary reference power P in or- REF der to evidence the scaling ? ? ? 1 above threshold predicted by Eq. (4.38) for P 2rf,out ? |M| . One can note the good agreement between numerical simulations and analytical predictions. The linear scaling of the power with the gain above threshold is expected to break down when ?  1 because of the higher-order non- linear terms neglected in the normal form approach are then becoming dominant. and lower driving noise (m2 ? 0). The intuition that larger microwave signals improve the phase noise performance is recovered from the condition b ? 0, which corresponds to a large characteristic voltage for the oscillator. The second one is that since phase noise scales as ?2/(??1), increasing the gain leads to a deterioration of the phase noise performance by a factor ? ? when ?  1. Therefore, increasing the microwave signal to decrease phase noise via a larger ? will not be successful for miniature OEOs  instead, as indicated above, large signals should be obtained by design with the lowest b possible (i.e., the highest characteristic voltageMb possible) . 102 Prf,out/PREF 4.8 Conclusion In this chapter, we have investigated the stochastic dynamics of an architec- ture of miniature OEO. We have rst introduced the stochastic dierential equations ruling the dynamics of the system when driven by white noise sources, and provided an analytical framework to determine the power of the generated microwave. We have also proposed a stochastic normal form approach to extract the scaling be- havior of the microwave power as the gain is increased below and above threshold. The analytical results were found to be in excellent agreement with the numerical simulations. 103 Chapter 5: Miniature Optical Oscillator Based on Whispering-Gallery Mode Resonator 5.1 Introduction In previous chapters of this thesis we studied the miniature optoelectronic oscillator based on a whispering-gallery mode modulator. This system consists of an optical branch an an electrical branch. We derived the deterministic and stochastic dynamics of both the open and closed-loop system, and did the stability analysis on both models and determined the critical values leading to system's optimization as well as the parameters aecting its performance. In this chapter we present our preliminary results on the analysis of a miniature optical oscillator based on a whispering-gallery mode resonators. The key dierence between an optical oscillator and an OEO is that the former only has optical input and output, while the latter has both an optical and electrical output. The aim of this work is to determine a spatiotemporal model for the closed-loop optical oscil- lator, and derive the critical conditions leading self-starting oscillations. Therefore, this chapter is organized as follows: in Section 5.2 we wll give a brief overview of dispersion analysis; in Section 5.3 we will rst present the Lugiato-Lefever formal- 104 ism for the open-loop system with dispersion. This model fully is well explained in Godey et al.,in ref. [105]. Next, we will design a feedback scheme and derive the equations governing the dynamics of the system. Section 5.4 presents a temporal analysis of the system while Sec. 5.5 studies the conditions leading to bifurcation. Finally, Sec. 5.6 presents our preliminary results in the analysis of Kerr-comb gen- eration. Although this work is still an ongoing eort, our main contributions presented in this chapter are: (1) The derivation of a spatiotemporal model to describe the closed-loop dynamics. (2) The temporal and spatial analysis of the system and derivation of the conditions for stability. 5.2 Dispersion Let us consider that the pumped mode of a WGMR is `0. Each mode ` represents a family of eigenmodes and is characterized by an eigenfrequency and a modal linewidth ??. A photon sees a dierent eigenfrequency at each mode, therefore ` also represents the number of internal reections that a photon does in that mode during a rountrip. Dispersion occurs when the refraction index depends on the frequency. In a WGMR without dispersion, the modes are equidistant, whereas in the presence of dispersion, the distance between the modes grows as we are getting away from the pumped mode `0 (Fig. 5.1). Dispersion can be anomalous or normal. In an anomalous dispersion regime, the eigenmodes ` are shifted to the right, while they are shifted to the left in a normal dispersion regime. 105 Figure 5.1: Eigenmodes of WGMR. The real location of the eigenfrequencies with anomalous or normal dispersion is represented by solid lines, while the dashed lines represent the location of the eigenfrequencies with normal or anomalous dispersion if the dispersion were null (perfect equidistance). The enlarged gure shows the relationship between the laser frequency ?0 (?L in our work), the frequnecy of the pumped mode 0 ?`0 , the detuning frequency ? and the loaded linewidth ??tot (?? in our work) [105]. In this chapter, the dispersion parameter is characterized by ?. A dispersion is characterized normal GVD when ? > 0; otherwise (? < 0), it is an anomalous GVD dispersion. 5.3 Miniature Optical Oscillator based on WGM Modulator In this section we will start with the Lugiato-Lefever formalism for the open- loop system. We will then derive the closed-loop dynamics and analyze the spa- 106 Laser Ain Aout Figure 5.2: Open-loop conguration for the optical oscillator. tiotemporal stability to derive the bifurcation maps in anomalous and normal dis- persion regime. 5.3.1 Open-Loop Model The system under study is shown in Fig. 5.2. It consists of a laser pump that emits photons that are trapped in a WGMR though evanescent coupling. In Subsec. 2.3.1 of Chapter 2, we introduced the notion of intrinsic and extrinsic Q- factor of a WGMR. In a similar way to Eq. (2.7), we dene the intrinsic and extrinsic linewidths, respectively ??i and ??e, as ?0 ??i = , (5.1) Qi ?0 ??e = (5.2) Qe 107 where Qi and Qe are respectively the intrinsic and extrinsic (coupled) Q-factor of the WGMR, and ?0 is the resonant frequency of the WGMR; it is the frequency of the pumped mode `0. The total linewidth of the WGMR ?? is dened as ?? = ??i + ??e, (5.3) The modal linewidth ?? can be seen as a measure of the total cavity loss in the resonator, to which it is inversely proportional. Indeed, the average photon lifetime ?ph is 1 ? = (5.4) ph ??tot If we consider that Al(t) is the slowly-varying envelope equation of the cavity eld at mode `, then we can normalize A 2l such that |Al| is the number of photons inside the mode `. Furthermore, we can dene the total intracavity eld A as the sum of the modal elds Al, that is ? A = Aleil?, (5.5) l where ? ? [??, ?] is the azimuthal angle along the circumference of the microres- onator. Using Eqs. (2.7) and (2.12), we have a relationship between the cavity linewidth and the total loss ?? = 2? (5.6) 108 Let's consider the normalized optical detuning factor ? dened in Eq. (2.14). With Eq. (5.6), we can express ? as a function of ?? as ?2?? = A , (5.7) ?? where ? = ? ? ?0 and ? is the laser frequency. The interest in expressingA L L ? as in Eq. (5.7) is that it shows the dependence of the frequency detuning on the loaded linewidth of the central cavity mode. Let's assume a moving frame so that we can dene ?(t) as the total intracavity eld dynamics in the moving frame. We use a spatiotemporal LLE to describe the dynamics of the normalized total intracavity eld as ?? 2 = ?(1 + i?)? + i|?|2 ? ? ? ?? i + ? , (5.8) ?? 2 ??2 in where ? is the dimensionless input cavity pump, ?(?, ?) is the complex envelope of in the total intracavity eld, ? ? [??, ?] is the azimuthal angle along the circumference of the resonator, and ? = t/(2?ph) is the dimensionless time, where ?ph is dened in Eq. (5.4), and ? is the overall dispersion parameter dened as ? 2?2? = , (5.9) ?? where ?2 is the second order dispersion. We note that Eq. (5.8) has periodic bound- ary conditions 109 Ain Amp Aout Figure 5.3: Closed-loop conguration for the optical oscillator. The ouptut optical signal is amplied and fedback into WGMR. Amp: optical amplier. WGMR: Whispering-gallery mode resonator. 5.3.2 Closed-Loop Model The closed-loop conguration of the system under study is shown in g. 5.3. We feed back the amplied optical output into the WGMR cavity. Therefore, we dene the input cavity eld as ? = ?ei??, (5.10) in where ? ? 0 is the real-valued dimensionless feedback gain; it is controlled by an optical amplier; all the losses are lumped into the feedback term ?. The parameter ? stands for the optical round trip phase shift and can be adjusted to any value. Therefore, the equation describing the dynamics of the closed-loop optical oscillator based on WGMR is ?? [ ] 2 = ? ? ? ?(1 + i?) + ?ei? ? + i|?|2? ? i , (5.11) ?? 2 ??2 110 where ?ei? is always positive as dened below: ?????? ? > 0 if ? = 0??? . (5.12)? < 0 if ? = ? Although it can be adjusted to any value (modulo 2?), without any loss of generality, we will only consider ? = 0 (modulo 2?) in our subsequent analysis, so that ?ei? = ? > 0. 5.4 Temporal Stability Analysis In this section we will derive the equilibrium points of the system and analyze their temporal and spatial stability. We will also derive the bifurcation map. 5.4.1 Equilibrium Points The equilibria of system described in Eq. (5.11) obey the relationship 2 [?(1 + i?) + ?]? + i| ? ? ??|2? ? i = 0, (5.13) 2 ??2 so that we have a trivial and nontrivial equilibria given as ?e = 0 (5.14)( )2 ?2 = 1 + ?? |? |2e (5.15) 111 5.4.2 Critical Point Let |?e|2 = ?, such that ? is non-negative. We can rewrite the nontrivial stability condition of Eq. (5.15) as a quadratic equation with two possible solutions G(?, ?) ? ?2 ? 2??+ ?2 ? ?2 + 1 = 0, (5.16) so that the nontrivial equilibrium is achieved for ?? , which are the solutions to Eq. (5.16) and given by ? ?? = ?? ?2 ? 1 (5.17) Equation (5.16) has only one critical point which is computed by taking the deriva- tive of G(?, ?) with respect to ?, yielding the condition ?cr = ?, (5.18) As such, the minimum nontrivial equilibrium point G(?) is achieved when ? = ?1 (5.19) For all other values of ?, there exists two distinct nontrivial equilibria points ?? and ?+ . 112 Figure 5.4: Evolution of the number of nontrivial equilibria. The critical equilibrium is equal to the the detuning frequency ? and is achieved when ? = 1. This gure is an illustration of Eqs. (5.16) and (5.18). 5.4.3 Temporal Behavior From the analysis in the preceding subsection, we see that the critical equilib- rium ?cr = ? is achieved when the gain ? = ?1. Above that gain, we always have two equilibria determined by Eq. (5.17); these equilibria are such that ?? < ?cr < ?+ . We can prove that only one equilibrium is stable while the other is not. Figure 5.4 shows the evolution of the number of equilibria satisfying eq. (5.16). 5.5 Spatial Analysis In order to study the spatial stability of the system, we set the temporal derivative ??/?? = 0 so that the resulting spatial equilibria obey Eq. (5.13). 113 5.5.1 Jacobian Let's decompose ? into its real and imaginary component, following ? = ?r + i?i; the second order partial derivative of ? with respect to ? is therefore ?2? ?2? 2r ? ?i = + i . (5.20) ??2 ??2 ??2 By plugging this decomposition in Eq. (5.13) we obtain a two-dimensional real- valued ow as ?2? [ ]r 2 = (?2 + ?2i ? ?)?r ? ?i(1? ?) (5.21)??2 ? ?2? 2 [ ]i = (?2 + ?2i ? ?)?i + ?r(1? ?) (5.22)??2 ? We cannot derive the Jacobian of the 2-D ow described by Eqs. (5.25) because the it cannot be written in matrix form. In order to circumvent this issue, we dene an intermediate variable Xr,i as ??r,i Xr,i = , (5.23) ?? so that Eqs. (5.25) can be transformed into a 4-D real-valued nonlinear ow which state space representation is ??r Xr = (5.24) ?? ?X ?2r ? 2 [ ]r = = (?2 + ?2i ? ?)?r ? ?i(1? ?) (5.25)?? ??2 ? 114 ??i Xi = (5.26) ?? ?X ?2? 2 [ ]i i = = (?2 + ?2i ? ?)?i + ?r(1? ?) (5.27)?? ??2 ? The equilbria of this system, ?e = ?e,r + i?e,i, still obeys Eq. (5.13), so that the linearized 4-D ow around ?e can be written under the matrix form as ? ? ? ? ? ? ??? A ???? = J ???? A ???? , (5.28)?? ?A ?A ?? ?? where the state space vector is two-dimensional and exprssed as A = [?r, ?i] T and the Jacobian J is ? ? ???? 0 0 1 0 ??? ? ? ????? 0 0 0 1 ?J = ?? ?? ? (5.29) ?? 2 [3?2 2 2e,r + ?,e,i ? ?] [?e,r?,e,i ? (1? ?)] 0 0 ?? ? ? ???? 2 [?e,r?,e,i + (1? ?)] 2 [3?2e,r + ?2,e,i ? ?] 0 0? ? 5.5.2 Spatial Bifurcations The eigenvalues ? of the Jacobian described in Eq. (5.29) obey the 4th-order characteristic equation 4 4 [ ] ?4 ? (2?? ?)?2 + 3?2 ? 4??+ ?2 + (?? 1) = 0 (5.30) ? ?2 We note that Eq. (5.30) is bi-quadratic, thus it will always have four eigenval- 115 ues which are pairwise opposite (when real), or pairwaise conjugated. The paired solutions obey ? 4 (2?? ?)? ? ?2 ?? = , (5.31)2 where ? is the discriminant and is dened as 16 [ ] ? = ?2 ? (?? 1)2 (5.32) ?2 The solutions described by Eq. (5.31) will fall into one the two following cases: ????? ? ? 0 =? ?2? ? are pairwise opposite??? (5.33)? < 0 =? ?2? are pairwise conjugate We note that the pair of eigenvalues (?2 , ?2? ) is real when ? ? 0 and complex+ otherwise. This in turn may lead to a variety of behaviors for the eigenvalues (?1,2 , ?3,4) = (??? ,??+). 5.5.2.1 First Case: ? > 0 The eigenvalues are real, pairwise opposite and are dened by Eq. (5.31). The product of paired solutions is [ ] ?2 2 4 ?? = 3? 2 ? 4??+ ?2 + (?? 1)2 ? F(?, ?) (5.34) + ?2 Subcase a: F(?, ?) > 0. Both eigenvalues have the same sign. We can rewrite 116 the characteristic polynomial of Eq. (5.30) as ?4 ? 4 (2?? ?)?2 + F (?, ?) = 0 (5.35) ? (1) ? < 0. If 2? ? ? < 0, the characteristic polynomial is of the form (?2 ? a2)(?2 ? b2) = 0, so that the eigenvalues can be written as (?1,2 ;?3,4) = (?a;?b) (5.36) If 2??? > 0, the characteristic polynomial is of the form (?2+a2)(?2+b2) = 0, so that the eigenvalues can be written as (?1,2 ;?3,4) = (?ia;?ib) (5.37) (2) ? > 0. If 2? ? ? < 0, the characteristic polynomial is of the form (?2 +a2)(?2 + b2) = 0, so that the eigenvalues is of the form (?1,2 ;?3,4) = (?ia;?ib). If 2??? > 0, the characteristic polynomial is of the form (?2?a2)(?2?b2) = 0, so that the eigenvalues can be written as (?1,2 ;?3,4) = (?a;?b). Subcase b: F(?, ?) = 0. One of the eigenvalues is null, and the other one is ei- ther positive or negative. We can rewrite the characteristic polynomial of Eq. (5.30) as [ ] ?2 ?2 ? 4 (2?? ?) = 0 (5.38) ? 117 (1) ? < 0. If 2?? ? < 0, the eigenvalues can be written as (?1,2 ;?3,4) = (?a; 0) (5.39) If 2?? ? > 0, the eigenvalues can be written as (?1,2 ;?3,4) = (0;?ib) (5.40) (2) ? > 0. If 2??? < 0, the eigenvalues is of the form (?1,2 ;?3,4) = (0;?ib). If 2?? ? > 0, the eigenvalues can be written as (?1,2 ;?3,4) = (?a; 0). Subcase c: F(?, ?) < 0. The two eignevalues have opposite signs. Regardless of the sign of ?, the eigenvalues can be written as (?1,2 ;?3,4) = (?a;?ib) (5.41) 5.5.2.2 Second Case: ? = 0 Equation (5.31) has a double root given by 2 ?2? = (2?? ?) (5.42)? (1) ? < 0. If 2?? ? < 0, the eigenvalues can be written as (?1,2 ;?3,4) = (?a;?a) (5.43) 118 If 2?? ? > 0, the eigenvalues can be written as (?1,2 ;?3,4) = (?ia;?ia) (5.44) If ? = 2?, the eigenvalues can be written as (?1,2 ;?3,4) = (0; 0) (5.45) (2) ? > 0. If 2? ? ? < 0, the eigenvalues is of the form (?1,2 ;?3,4) = (?ia;?ia). If 2?? ? > 0, the eigenvalues can be written as (?1,2 ;?3,4) = (?a;?a). If ? = 2?, the eigenvalues can be written as (?1,2 ;?3,4) = (0; 0). 5.5.2.3 Third Case: ? < 0 This leads to ? < ??1. Equation (5.31) has two complex and conjugate roots given by [ ] 2 2 ? ?? = | | (2?? ?)? i (?? 1) 2 ? ?2 (5.46) ? Regardless of the sign of ?, the eigenvalues can be written as (?1,2 ;?3,4) = (a? ib; c? id) (5.47) 119 ? < 0 I A1 A2 II II1 2 B B1 2 III ? a Figure 5.5: Eigenvalue bifurcation diagram (not to scale) for the case of anomalous dispersion (? < 0). The areas are labeled using Roman numerals (I, II, and III), and area II is subdivided into two subareas (II1 and II2). The lines are labeled using capital letters, with line A standing for the limit ?2 = (? ? 1)2 (dashed red line in the gure); B stands for the critical line ?2 = 1, and is also subdivided into two rays B1 and B2. Finally, the points are labaled into lower case letters. We only have one point a which is the critical point at which ? = ? and ?2 = 1.The system has three equilibria in area I, II1 and II2; it has two equilibria along the lines B1 and B2, and only one equilibrium in area III. The eigenvalue pictogram are in black when they lead to a bifurcation and in grey otherwise. 5.5.3 Bifurcation Maps The stability analysis that performed in the previous subsection can be sum- marized into two bifurcation maps. The rst one presented in Fig. 5.5 considers the case of anomalous dispersion, while Fig. 5.6 shows the bifurcation map in the case of normal dispersion. We note that the bifurcation map for the normal dispersion is a symmetry along the y-axis of that in the anomalous dispersion. A summary of the type of eigenvalues encountered is presented in Table 5.1 below. The table presents the 120 line area line area area point line area line ? ? 0 I A1 A2 II1 II2 B B1 2 III ? a Figure 5.6: Eigenvalue bifurcation diagram (not to scale) for the case of anomalous dispersion (? < 0). The areas are labeled using Roman numerals (I, II, and III), and area II is subdivided into two subareas (II1 and II2). The lines are labeled using capital letters, with line A standing for the limit ?2 = (? ? 1)2 (dashed red line in the gure); B stands for the critical line ?2 = 1, and is also subdivided into two rays B1 and B2. Finally, the points are labaled into lower case letters. We only have one point a which is the critical point at which ? = ? and ?2 = 1. The system has three equilibria in area I, II1 and II2; it has two equilibria along the lines B1 and B2, and only one equilibrium in area III. The eigenvalue pictogram are in black when they lead to a bifurcation and in grey otherwise. physicist and the mathematician nomenclatures. Figure 5.7 shows the pictogram of of the eigenvalues leading to a bifurcation, as well as the location of the bifurcation in the dierent maps. 5.6 Ongoing Work 5.6.1 Supercritical and Subcritical Turing Patterns The main dierence between a super- and a subcritical pitchfork depends on how the comb emerges around the limit ?2 = (? ? 1)2. Thus by plugging this 121 line area line area area point line area line Pictogram Bifurcation Location in Fig. 5.5 Location in Fig. 5.6 Figure 5.7: This gure shows the pictogram of the eigenvalues leading to bifurcation, as well as the location of the bifurcations in Fig. 5.5 and 5.6. The lines are labeled using capital letters, with line A standing for the limit ?2 = (?? 1)2. Point a is the critical point at which ? = ? and ?2 = 1 . condition into Eq. (5.16) while recalling that ? > 0 yields ?2 ?th = + 1 (5.48) 2(? + 1) 5.6.2 Number of Rolls in Turing Patterns in Anomalous Dispersion Regime W e are interested in the number of rolls in the Turing pattern arising from the (i?)2 bifurcation at ? = ??1. Recalling Eq. (5.11) that models the dynamics of the closed-loop system, a pertbation ??(?, ?) of the equilibrium ?e obeys the linearized Type Nomenclature Eigenvalues (?1,2;?3,4) Bifurcation 1 (?a,?b) 2 Quadruple-zero (0;0) 04 3 (?ia;?ib) 4 Takens-Bogdanov (?a;0) 02 5 Takens-Bogdanov-Hopf (0; ?ib) 02(i?) 6 (?a;?ib) 7 Hamiltonian-Hopf (?ia;?ia) (i?)2 8 (?a;?a) 9 (a? ib;c? id) Table 5.1: Eigenvalues and spatial bifurcations in the Lugiato-Lefever model 122 equation ? ? ?2 [??] = [?(1 + i?) + ?] ?? + 2i|?e |2?? + i?2??? ? i [??] (5.49)?? e 2 ??2 We recall that ? is the total intracavity eld and modulated sum of the modal cavity eld ? , where l ? `? ` l 0 corresponds to the eigennumber of the WGMs with respect to the eigenmode `0. In a similar way, we dene the ansatz ? ??(?, ?) = ??l(?)e il?, (5.50) l and insert it into Eq. (5.49) so that we obtain the expansion. To project the ex- ? pansion unto a given mode l?, we multiply that equation by eil ? and integrate the product with respect to ? from ?? to ?. The results of the projection on mode l and ?l are [ ] ? ? [??l] = [?(1 + i?) + ? + 2i|? 2 e| ? i l2] [?? ] + i? 2[??? ] (5.51) ??[ ] 2 l e ?l ? ??? ? = ?(1? i?) + ?? 2i|? |2 ? i l2 [??? ]? i?2e [?? ] (5.52)?? ?l 2 ?l e l Equations (5.51) and (5.52) represents the two-dimensional complex ow de- scribing the dynamics of the modes pertubation from equilibrium. It can be written in matrix form as ?? ?? ??? ?? ???? ????? M N ? ?? ? ??????? ??l l=?? ??? ? , (5.53) ??? N ? M? ??? ?l ?l 123 whereM and N are given as M = ? ?(1 + i?) + ? + 2i|? |2e + i l2 (5.54)2 N = i?2 (5.55) e The stability analysis of the four-dimensional Jacobian of Eq. (5.53) shows that the real-part of the leading eigenvalue is { ? ( )} G(l) = Re ? 11 + ? + ?2 ? ?? 2?? ?l2 (5.56) 2 where ? = |?e |. G(l) represents the excitation gain of the mode following the pertubation from equilibrium. At the threshold, there is no gain (G(l) = 0), while know that ? = ? ? 1. Solving equation Eq. (5.56) for the threshold mode l , we th nd ? 2 l = [?? 2(?? 1)] (5.57) th ? 5.7 Conclusions In this chapter we have presented our investigations of a closed-loop optical oscillator based on a whispering-gallery mode modulator. Starting with a Lugiato Lefever model for the open-loop sytem, we designed a feedback and derived a spa- tiotemporal model describing the dynamics of the total intracavity eld in the minia- ture closed-loop optical oscillator. We then performed a temporal stability analysis and a spatial analysis, and derived the bifurcaion maps in normal and anomalous 124 dipersion regime. On-going eorts are aiming at characterizing the Kerr-comb gen- eration in both dispersion regimes. 125 Chapter 6: Conclusions and Outlook The work presented in this thesis has focused on the investigation of the non- linear dynamics in miniature optoelectronic oscillators based on whispering-gallery mode modulators. We studied the time-domain nonlinear dynamics and introduced the stochastic dierential equations ruling the dynamics of the system when driven by white noise sources. In the rst part of our work, we have proposed a full time- domain deterministic model to describe the nonlinear dynamics of the complex- valued envelopes of the optical and microwave elds. This model takes into account the intracavity eld interactions inside the whispering-gallery mode resonator. We have performed a stability analysis to determine the stability conditions and have derived an analytical formula for the threshold gain leading to oscillatons (primary Hopf bifurcation) in the system. We also performed an optimization analysis and concluded that the system operates in optimal regime when the laser frequency is at the edge of the optical resonance and around the critical resonator coupling. We then investigated the idea of an amplierless miniature OEO based on WGM modulator; this idea may be an additional step toward meeting the constraints of SWAP-C, in addition to improving the phase noise performance. We optimize the system and determined a theoretical formula for the threshold laser power leading 126 to oscillations in the amplierless miniature oscillators. This threshold power is higher than the power needed for the amplied miniature OEO. After proposing the deterministic model, we used the Langevin approach to derive the full time-domain stochastic model describing the dynamics of miniature OEOs with random noise. We provided an analytical framework to determine the power of the generated mi- crowave. We have also proposed a stochastic normal form approach to extract the scaling behavior of the microwave power as the gain is increased below and above threshold. The analytical results were found to be in excellent agreement with the numerical simulations. The last chapter of this thesis investigated the nonlinear dynamics of miniature optical oscillators based on whispering-gallery mode modulator. In our preliminary eorts presented here, we have proposed a Lugiato-Lefever model to study the eects of dispersion and proposed bifurcation maps in the case of anomalous and normal dispersion. There are still several areas of investigation with regard to technology related to miniature OEOs. Starting from the normal form presented in Chapter 4, we have yet to extract the scaling factors that will lead to an analytical formula of the power density spectrum. 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