Analytical theory of frequency-modulated combs: generalized mean-field theory, complex cavities, and harmonic states

TL;DR

This paper provides a comprehensive analytical framework for frequency-modulated combs in lasers, deriving properties, effects of cavity imperfections, and stable states using mean-field and eigenvalue theories.

Contribution

It introduces a general method to construct mean-field theories for laser master equations and applies it to analyze FM combs, including effects of cavity defects and stability of various states.

Findings

Derived an expression for FM chirp in Fabry-Perot cavities

Showed how gain curvature affects FM combs

Characterized stable laser states using eigenvalue formulation

Abstract

Frequency-modulated (FM) combs with a linearly-chirped frequency and nearly constant intensity occur naturally in certain laser systems; they can be most succinctly described by a nonlinear Schr\"odinger equation with a phase potential. In this work, we perform a comprehensive analytical study of FM combs in order to calculate their salient properties. We develop a general procedure that allows mean-field theories to be constructed for arbitrary sets of master equations, and as an example consider the case of reflective defects. We derive an expression for the FM chirp of arbitrary Fabry-Perot cavities$-$important for most realistic lasers$-$and use perturbation theory to show how they are affected by gain curvature. Lastly, we show that an eigenvalue formulation of the laser's dynamics can be useful for characterizing all of the stable states of the laser: the fundamental comb, the continuous-wave solution, and the harmonic states.