Closed-form solutions and scaling laws for Kerr frequency combs

TL;DR

This paper presents a unified analytical framework for Kerr frequency combs, including closed-form solutions and scaling laws, offering new insights and design guidelines for soliton and wavetrain-based combs.

Contribution

It introduces the first closed-form solutions for Kerr-comb behaviors, including a new area theorem and pump-detuning relation, advancing understanding and design of Kerr frequency combs.

Findings

Derived a single closed-form solution for Kerr-comb behaviors.

Developed a Kerr-comb area theorem and pump-detuning relation.

Identified new parameter regimes for enhanced comb performance.

Abstract

A single closed-form analytical solution of the driven nonlinear Schr\"{o}dinger equation is developed, reproducing a large class of the behaviors in Kerr-comb systems, including bright-solitons, dark-solitons, and a large class of periodic wavetrains. From this analytical framework, a Kerr-comb area theorem and a pump-detuning relation are developed, providing new insights into soliton- and wavetrain-based combs along with concrete design guidelines for both. This new area theorem reveals significant deviation from the conventional soliton area theorem, which is crucial to understanding cavity solitons in certain limits. Moreover, these closed-form solutions represent the first step towards an analytical framework for wavetrain formation, and reveal new parameter regimes for enhanced Kerr-comb performance.