Frequency comb formation in doubly resonant second-harmonic generation

TL;DR

This paper theoretically investigates frequency comb generation in doubly resonant second-harmonic generation cavities, revealing a new modulation instability regime and providing simplified models for understanding comb formation.

Contribution

It introduces a coupled mean-field model for doubly resonant SHG, uncovers a novel modulation instability regime, and simplifies the analysis to a single mean-field equation.

Findings

Identification of a new temporal modulation instability regime due to walk-off.

Numerical simulations show emergence of coherent optical frequency combs.

Reduction to a single mean-field equation provides analytical insight.

Abstract

We theoretically study the generation of optical frequency combs and corresponding pulse trains in doubly resonant intracavity second-harmonic generation (SHG). We find that, despite the large temporal walk-off characteristic of realistic cavity systems, the nonlinear dynamics can be accurately and efficiently modelled using a pair of coupled mean-field equations. Through rigorous stability analysis of the system's steady-state continuous wave solutions, we demonstrate that walk-off can give rise to a new, previously unexplored regime of temporal modulation instability (MI). Numerical simulations performed in this regime reveal rich dynamical behaviours, including the emergence of temporal patterns that correspond to coherent optical frequency combs. We also demonstrate that the two coupled equations that govern the doubly resonant cavity behaviour can, under typical conditions, be reduced to a single mean-field equation akin to that describing the dynamics of singly resonant cavity SHG [F. Leo et al., Phys. Rev. Lett. 116, 033901 (2016)]. This reduced approach allows us to derive a simple expression for the MI gain, thus permitting to acquire significant insight into the underlying physics. We anticipate that our work will have wide impact on the study of frequency combs in emerging doubly resonant cavity SHG platforms, including quadratically nonlinear microresonators.