Elliptic-type soliton combs in optical ring microresonators

Abstract

Soliton crystals are periodic patterns of multi-spot optical fields formed from either time or space entanglements of equally separated identical high-intensity pulses. These specific nonlinear optical structures have gained interest in recent years with the advent and progress in nonlinear optical fibers and fiber lasers, photonic crystals, wave-guided wave systems and most recently optical ring microresonator devices. In this work an extensive analysis of characteristic features of soliton crystals is carried out, with emphasis on their one-to-one correspondance with Elliptic solitons. In this purpose we examine their formation, their stability and their dynamics in ring-shaped nonlinear optical media within the framework of the Lugiato-Lefever equation. The stability analysis deals with internal modes of the system via a 2 × 2-matrix Lamé type eigenvalue problem, the spectrum of which is shown to possess a rich set of boundstates consisting of stable zero-fequency modes and unstable decaying as well as growing modes. Turning towards the dynamics of Elliptic solitons in ring-shaped fiber resonators with Kerr nonlinearity, first of all we propose a collective-coordinate approach, based on a Lagrangian formalism suitable for Elliptic-soliton solutions to the nonlinear Schródinger equation with an arbitrary perturbation. Next we derive time evolutions of Ellipticsoliton parameters in the specific context of ring-shaped optical fiber resonators, where the optical field evolution is tought to be governed by the Lugiato-Lefever equation. By solving numerically the collective-coordinate equations an analysis of the amplitude, the position, the phase of internal oscillations, the phase velocity and the energy is carried out and reveals a complex dynamics of the Elliptic soliton in ring-shaped optical microresonators.

Soliton-comb structures in ring-shaped optical microresonators: generation, reconstruction and stability

“…Formula (24) suggests that the repetition rates θ d,c,s 0 are all proportional to the width of soliton ccomponents in the soliton-crystal structures Eqs. (19), (20) and (21), and inversely proportional to their amplitudes. Hence ES envelopes should be broadened as the amplitude grows, thereby reducing overlaps of eigenfunctions of the soliton crystals.…”

“…collisionless) self-organized ensemble of co-propagating solitons forms having the crystallographic structure typical of soliton crystals. However, unlike the soliton crystals in previous works [20,21] the later ones present no defect-like vacancies and shifted pulses (or kinks), since we neglected the gain and loss in the cavity. In fact, formula (27)…”

“…where a(θ) is the ES envelope, and u(θ) and v(θ) are amplitudes of the twomode noise field having a common frequency ω [21,33]. Inserting Eq.…”

“…In the context of soliton combs, ESs exhibit interesting features among which the dependence of their amplitude on both the coupled resonance width and the pulse repetition rate [21]. Moreover the existence and stability of ESs can be determined via the pulse repetition rate of solitons: the smaller the pulse repetition rate the larger the pulse amplitude [18,21]. Also the ES amplitude is inversely proportional to the cavity resonance width [21], meaning that the soliton envelope should broaden as the amplitude grows thereby reducing overlaps and collisions between pulses in the soliton-comb structure.…”

“…alternating kink and anti-kink solitons), have captured no or only a very little theoretical attention. In particular the inner structures of these other possible soliton-comb structures have not been unveiled, and their stability properties are still not well understood as is the case for brigh-soliton lattice patterns [21].…”