[Paper Review] Generation of Kerr Frequency Combs in Resonators with Normal GVD

Andrey B. Matsko, Anatoliy A. Savchenkov+1 more|arXiv (Cornell University)|Nov 16, 2011

Advanced Fiber Laser Technologies3 citations

TL;DR

This paper demonstrates via numerical simulations that Kerr frequency combs can stably form in microresonators with normal group velocity dispersion (GVD), contrary to prior assumptions that anomalous GVD is required. The comb arises from resonant modulation instability driven by a continuous-wave pump, exhibiting a hard excitation threshold and forming a single, localized optical pulse in the cavity with a stable, periodic spectral comb structure.

ABSTRACT

We show via numerical simulation that Kerr frequency combs can be generated in a nonlinear resonator characterized with normal group velocity dispersion (GVD). We find the spectral shape of the comb and temporal envelope of the corresponding optical pulses formed in the resonator.

Motivation & Objective

• To investigate whether Kerr frequency combs can be generated in microresonators with normal group velocity dispersion (GVD), challenging the prevailing belief that anomalous GVD is necessary.
• To resolve discrepancies in prior theoretical and numerical studies that failed to capture stable comb states or hard excitation dynamics in normal GVD regimes.
• To provide a rigorous numerical model of comb formation in a CaF2 whispering gallery mode resonator with experimentally realizable parameters, including small normal GVD.
• To characterize the spectral and temporal properties of the generated comb, including its stability and threshold behavior.

Proposed method

• Numerical simulation of a nonlinear coupled-mode system involving 21 optical modes in a microresonator with normal GVD, using a slow-amplitude formalism to model the intracavity field evolution.
• Incorporation of second-order dispersion and cubic nonlinearity via a Hamiltonian interaction term, with parameters derived from CaF2 material properties and resonator geometry.
• Use of normalized slow amplitudes and dimensionless pumping strength f to analyze stability and bifurcation behavior across pump power and frequency detuning.
• Application of a fixed-point iteration and continuation method to trace stable and unstable solution branches, identifying the onset of comb generation.
• Verification of solution robustness by varying the number of modes and initial conditions, confirming consistent spectral and temporal behavior.
• Analysis of time-domain evolution to confirm the formation of a single, localized optical pulse corresponding to the frequency comb.

Experimental results

Research questions

• RQ1 Can Kerr frequency combs be generated in microresonators with normal group velocity dispersion, despite the conventional expectation of requiring anomalous GVD?
• RQ2 What is the nature of the excitation threshold for comb generation in normal GVD systems—does it exhibit a hard or soft threshold?
• RQ3 How do the spectral and temporal characteristics of the comb, such as pulse shape and sideband spacing, depend on pump power and frequency detuning?
• RQ4 Can stable, steady-state solutions for the comb be numerically obtained in normal GVD systems, and how do they compare to previous simulations that assumed weak sidebands?

Key findings

• A stable, dynamically robust Kerr frequency comb is generated in a CaF2 microresonator with normal GVD (β₂ ≈ 0.055 ps²/km), demonstrating that comb formation is possible even when GVD is positive.
• The comb exhibits a hard excitation threshold: the onset of comb generation occurs via a discontinuous jump in mode amplitudes when the pump power exceeds a critical value, consistent with the absence of soft modes in normal GVD.
• The temporal profile of the intracavity field forms a single, complex-shaped optical pulse with a duration and amplitude dependent on pump power, confirming coherent pulse formation.
• The spectral comb spans 10 red- and 10 blue-detuned sidebands around the pump, with a free spectral range (FSR) of 100 GHz, and is stable under parameter variations.
• The simulation results are robust to changes in the number of modes and initial conditions, confirming the stability of the attractor and the physical relevance of the solution.
• The system's behavior is captured by a dimensionless pumping strength f = 2.244, with a normalized pump detuning of (ω - ω₁₁)/2πγ₀ = -4, within a well-defined parameter region for comb generation.