Theory of Kerr frequency combs in Fabry-Perot resonators

Abstract

We derive a spatiotemporal equation describing nonlinear optical dynamics in Fabry-Perot (FP) cavities containing a Kerr medium. This equation is an extension of the equation that describes dynamics in Kerr-nonlinear ring resonators, referred to as the Lugiato-Lefever equation (LLE) due to its formulation by Lugiato and Lefever in 1987. We use the new equation to study the properties of Kerr frequency combs in FP resonators. The derivation of the equation starts from the set of Maxwell-Bloch equations that govern the dynamics of the forward and backward propagating envelopes of the electric field coupled to the atomic polarization and population difference variables in a FP cavity. The final equation is formulated in terms of an auxiliary field $\psi (z,t)$ that evolves over a slow time $t$ on the domain $-L\le z\le L$ with periodic boundary conditions, where $L$ is the cavity length. We describe how the forward and backward propagating field envelopes are obtained after solving the equation for $\psi$. This formulation makes the comparison between the FP and ring geometries straightforward. The FP equation includes an additional nonlinear term relative to the LLE for the ring cavity, with the effect that the value of the detuning parameter $\alpha$ of the ring LLE is increased by twice the average of $|\psi {|}^2$. This feature establishes a connection between the stationary phenomena in the two geometries. For the FP-LLE, we discuss the linear stability analysis of the flat stationary solutions, analytic approximations of solitons, Turing patterns, and nonstationary patterns. We note that Turing patterns with different numbers of rolls may exist for the same values of the system parameters. We then discuss some implications of the nonlinear integral term in the FP-LLE for the kind of experiments which have been conducted in Kerr-nonlinear ring resonators.

arXiv comment16 pages, 8 figures

References in corpus (14)

• Optical frequency comb generation from a monolithic microresonator
• Temporal solitons in optical microresonators
• An Integrated-Photonics Optical-Frequency Synthesizer
• Photonic chip based optical frequency comb using soliton induced Cherenkov radiation
• Mode interaction aided excitation of dark solitons in microresonators constructed of normal dispersion waveguides
• Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model
• Spatiotemporal Model for Kerr Comb Generation in Whispering Gallery Mode Resonators
• Stability Analysis of the Lugiato-Lefever Model for Kerr Optical Frequency Combs. Part I: Case of Normal Dispersion
• Soliton crystals in Kerr resonators
• Universal scaling laws of Kerr frequency combs
• Broadband dispersion engineered microresonator on-a-chip
• Writing and erasing of temporal cavity solitons by direct phase modulation of the cavity driving field
• Routes to spatiotemporal chaos in Kerr optical frequency combs
• Spontaneous creation and annihilation of temporal cavity solitons in a coherently-driven passive fiber resonator

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• Theory and Application of Cavity Solitons in Photonic Devices
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• Dual-Frequency Comb in Fiber Fabry-Perot Resonator
• Optimization of a fiber Fabry-Perot resonator for low-threshold modulation instability Kerr frequency combs
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• Controlling microresonator solitons with the counter-propagating pump
• Frequency-modulated combs as phase solitons