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1. Dynamical Methods for Studying Stability and Noise in Frequency Comb Sources

   Menyuk, Curtis R. ; Courtright, Logan ; Qi, Zhen ; Wang, Shaokang ( October 2021 , AMS Fall Western Virtual Sectional Meeting 2021)

   Frequency combs, invented in 2000, have revolutionized frequency measurement and thereby impacted a host of applications. These include applications to military systems, medicine, environmental sensing, astrophysics, and basic physics. The sources have improved dramatically in the past decade, evolving from laboratory-size lasers to fiber lasers to microresonators on a chip. However, the theoretical input to these developments has been surprisingly small. The key problem in designing frequency combs is to determine where in the experimentally-adjustable parameter space stable solutions exist, to determine how to access them, and to determine the impact that noise has on them. While analytical approaches to answer these questions exist, computational tools to implement these approaches in realistic settings have been lacking. Our research has developed computational tools to address these issues, focusing on fiber laser and microresonator combs. In this talk, we will review our progress to date and discuss open problems. 

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2. Dynamical Methods for Studying Stability and Noise in Frequency Comb Sources

   Curtis R. Menyuk, Brandon Wilson ( May 2021 , SIAM Conference on Applied Dynamical Systems)

   null (Ed.)

   Frequency combs, invented in 2000, have revolutionized frequency measurement and thereby impacted a host of applications. These include applications to military systems, medicine, environmental sensing, astrophysics, and basic physics. The sources have improved dramatically in the past decade, evolving from laboratory-size lasers to fiber lasers to microresonators on a chip. However, the theoretical input to these developments has been surprisingly small. The key problem in designing frequency combs is to determine where in the experimentally-adjustable parameter space stable solutions exist, to determine how to access them, and to determine the impact that noise has on them. While analytical approaches to answer these questions exist, computational tools to implement these approaches in realistic settings have been lacking. Our research has developed computational tools to address these issues, focusing on fiber laser and microresonator combs. In this talk, we will review our progress to date and discuss open problems. 

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3. Stability and Noise in Frequency Combs: Harnessing the Music of the Spheres

   Menyuk, Curtis R. ( January 2019 , Colloquium talk: Boston University, Columbia University, Cornell University, University of West Virginia)

   Frequency combs have revolutionized the measurement of time and frequency and impacted a wide range of applications spanning basic physics, astrophysics, medicine, and defense. The key theoretical issues in understanding and designing frequency combs are finding regions in the adjustable parameter space where combs operate stably, determining their noise performance, and optimizing them for high power, low noise, and/or large bandwidth. Here, we present a unique set of computational tools that we have developed that allow us to efficiently address these issues. These tools combine 400-year-old dynamical systems theory with modern computational methods, and they are 3–5 orders of magnitude faster than standard evolutionary methods and provide important physical insight. We have applied these tools to frequency combs from passively modelocked lasers with fast and with slow saturable absorbers and to frequency combs from microresonators. Our methods predict improved operating regimes for combs that are produced from both the passively modelocked lasers and the microresonators. 

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4. Dynamical Methods for Studying Noise in Frequency Comb Sources

   Menyuk, Curtis R. ; Wang, Shaokang ( March 2022 , Twelfth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory,)

   Frequency combs have revolutionized the measurement of time and frequency since their invention in 2000, and have a wide array of applications to applications that range from basic science applications, to a wide array of sensing applications, to commercial applications, to military applications, and the list goes on. Noise poses a fundamental limit to these systems, and calculating its impact play a critical role in system design. Frequency combs are created by modelocked laser systems that emit a periodic train of short pulses. Laser systems are complex nonlinear systems and the usual method for determining the impact of noise is to carry out computationally-expensive Monte Carlo methods. That limits the parameter range over which it is possible to study the noise impact. We have developed a new approach based on dynamical systems methods. In our approach, we determine a stationary state of the laser system as parameters vary solving a root-finding problem [Wang1]. Starting from a stationary state, we determine all the eigenvalues and eigenvalues of the linearized system. The variance of the amplitudes of the eigenvalues obey either random walk of Langevin equations [Menyuk]. Starting from that point, we can determine the power spectral density of the key laser parameters (amplitude jitter, timing jitter, frequency jitter, phase jitter) [Wang2]. We applied this approach to SESAM lasers and found that we were able to reproduce a computation that took 20 minutes on a cluster with 256 cores with a computation that took less than 4 minutes on a desktop computer. 

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5. Influence of Mode Structure on the Generation of Phononic Frequency Combs

   Zhen Qi, Curtis R. ( March 2020 , ArXivorg)

   The mechanical analog of optical frequency combs, phononic frequency combs, has recently been demonstrated in mechanical resonators and has been attributed to coupling between multiple phonon modes. This paper investigates the influence of mode structure on comb generation using a model of two nonlinearly coupled phonon modes. The model predicts that there is only one region within the amplitude-frequency space where combs exist, and this region is a subset of the Arnold tongue that describes a 2:1 autoparametric resonance between the two modes. In addition, the location and shape of the comb region are analytically defined by the resonance frequencies, quality factors, mode coupling strength, and detuning of the driving force frequency from the mechanical resonances, providing clear conditions for comb generation. These results enable comb structure engineering for applications in areas as broad as sensing, communications, and quantum information science. 

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