Dissipative cnoidal waves (Turing rolls) and the soliton limit in microring resonators

Zhen Qi,Curtis R. Menyuk,Giuseppe D’Aguanno,José Jaramillo-Villegas,Minghao Qi,Shaokang Wang,Thomas F. Carruthers,Andrew M. Weiner

doi:10.1364/optica.6.001220

Abstract

Single solitons are a special limit of more general waveforms commonly referred to as cnoidal waves or Turing rolls. We theoretically and computationally investigate the stability and accessibility of cnoidal waves in microresonators. We show that they are robust and, in contrast to single solitons, can be easily and deterministically accessed in most cases. Their bandwidth can be comparable to single solitons, in which limit they are effectively a periodic train of solitons and correspond to a frequency comb with increased power. We comprehensively explore the three-dimensional parameter space that consists of detuning, pump amplitude, and mode circumference in order to determine where stable solutions exist. To carry out this task, we use a unique set of computational tools based on dynamical system theory that allow us to rapidly and accurately determine the stable region for each cnoidal wave periodicity and to find the instability mechanisms and their time scales. Finally, we focus on the soliton limit, and we show that the stable region for single solitons almost completely overlaps the stable region for both continuous waves and several higher-periodicity cnoidal waves that are effectively multiple soliton trains. This result explains in part why it is difficult to access single solitons deterministically.

Highlights

It has long been known that single solitons are a special limit of a more general family of stationary nonlinear waves that are almost universally referred to as cnoidal waves in the plasma and fluid physics communities [1,2,3,4,5,6]
In the nonlinear Schrödinger equation (NLSE) approximation, solitons can be analytically expressed in terms of hyperbolic-secant functions [sech (x)], while more generally cnoidal waves can be expressed in terms of Jacobi elliptic functions [cn (x), sn (x), dn (x)] [12]
While the stable region for single solitons (Nper = 1 cnoidal waves) is large, it almost entirely overlaps the regions of stability for several higher periodicity cnoidal waves in the parameter range that we show, as well as with continuous waves

Summary

INTRODUCTION

It has long been known that single solitons are a special limit of a more general family of stationary nonlinear waves that are almost universally referred to as cnoidal waves in the plasma and fluid physics communities [1,2,3,4,5,6]. The dynamical methods are computationally rapid, making it possible to determine the stability boundaries for all the cnoidal wave solutions within a broad parameter range. In this case, the Nper = 8 cnoidal wave appears to be a modulated continuous wave, rather than a periodic train of solitons. That can be important for applications where the power in a single comb line must be made large

THE SOLITON LIMIT

Findings

CONCLUSION