Dynamical Systems of Oscillating Ultrashort Pulse Pairs

. We propose a theoretical method to model the complex phenomenon of oscillating ultrashort pulse pair molecules. Using a phenomenological viewpoint, we construct e ff ective dynamical systems, whose degrees of freedom are the inter-pulse timing and overall phase. The e ff ective dynamical system is characterized by a limit cycle attractor that is fit to the experimentally measured soliton oscillation using data-driven methods. Good agreement is achieved between the dynamical system orbits and experimental observations made in a mode-locked fiber laser.


Soliton pairs in mode locked fiber lasers
Passively mode locked lasers with anomalous dispersion produce ultrashort pulses, also called dissipative solitons, with quantized energy [1].When pumped beyond the soliton-energy threshold, a cascade of instabilities leads to laser waveforms with two or more solitons, which are subject to a complex variety of interaction mechanisms [2].Compared with single-soliton operation, multisoliton waveforms can exhibit a much richer and more versatile dynamics: Even the simplest multisoliton waveform, the soliton pair molecule, can be stationary, undergo regular oscillations [3], and even exhibit chaos [4].
However, in spite of the great potential, multisoliton mode locked lasers are barely used in ultrafast technology applications.One of the main hurdles is the fine tuning of the laser components and operating parameters needed to achieve a specific multisoliton output, and the lack of a systematic theory for it.* Corresponding author: omrigat@mail.huji.ac.ilState of the art models of mode locked lasers are able to reproduce many of the experimentally observed mutlipulse phenomena, but their usefulness is limited by the need to fine-tune model parameters, that mirrors the analogous problem in laser experiments.In principle, the properties of multisoliton steady states and their stability can be derived directly from laser models, but this task is difficult and has not yet been carried out.Instead, we put forward a phenomenological description of an oscillating soliton pair molecules as limit cycles of dynamical systems.The degrees of freedom of the soliton-pair dynamical system Phase portrait of an oscillating soliton pair molecule in the plane of relative time and phase quadratures.Green points are experimental data and the thick dark curve is the periodic orbit fit to the data, as shown in figure 1.The stream lines and orange curve are trajectories of the polynomial dynamical system are (1) the time delay τ between the two solitons, and (2) the shift ϕ between their overall phases.The equations of motion of the dynamical system are derived by a datadriven protocol that we describe next.

The pair dynamical system
We performed experiments in an erbium-doped fiber laser mode locked with the nonlinear polarization evolution technique.Polarization controllers and pump current were adjusted to produce an oscillating soliton pair molecule, and the inter-pulse timing and phase were captured using the time-stretch dispersive Fourier transform technique [3].
Figure 1 shows the quadratures x = (τ − κ) cos ϕ, y = (τ − κ) sin ϕ, where κ = 6.2 ps is a constant shift, captured at cavity roundtrip intervals.The quadratures follow roughly a periodic trajectory that constitutes a limit cycle.Our first step is to express the limit cycle z t = (x t , y t ) (where t is time measured in roundtrip units) as a trigonometric polynomial of degree 3, with coefficients obtained by least-square fit, as shown in figure 1.
Our main goal is to derive equations of motion for the quadratures, which have z t as a limit cycle.We seek equations of polynomial form where z (2) and z (3) are vectors of 2nd and 3rd degree monomials (respectively) in x and y, and M k are 2-by-k + 1 constant matrices.The parameters of the model are the matrix elements M k,mn ; when the pair-oscillation limit cycle z t is substituted in equation ( 1) one obtains an overdetermined linear system for the Fourier harmonics from 0 to 9, that we solve for the model parameters by least-square fitting.The steps described so far produce a dynamical system which has z t as an orbit.The last piece of our model construction is needed to make it an attractor, i.e. a limit cycle that is reached from arbitrary initial pulse configurations after a short-lived transient.However, the experimental data that is measured on the limit-cycle steady state does not provide direct information on the process of approach to steady state; instead, we impose a phase-space contraction condition where J is the Jacobian matrix of the equation of motion (1) evaluated on the limit cycle, and γ < 0 is a dissipation parameter determining the rate of transient spiraling to the limit cycle.This construction yields a candidate soliton-pair dynamical system for each choice of γ, with a corresponding candidate limit cycle orbit.Our final step is to choose the value of γ for which the dynamical-system limit cycle best approximates the experimentally observed pulse oscillations.The results of this optimization are demonstrated in figure 2, which shows the experimental observations (green points) and limit-cycle fit (thick dark curve) of figure 1 on the phase space of the two quadratures.The stream lines are tangent to the trajectories of soliton pairs obtained from the equation of motion (1), and the orange curve shows a trajectory that converges to the limit cycle of the dynamical system after an initial transient, as expected.

Conclusions and outlook
The simple dynamical system (1) is seen to reproduce with high accuracy the complicated nonlinear oscillations of an ultrashort pulse pair molecule.We were able to apply the protocol on several oscillating-pair configurations of mode locked fiber lasers, based on erbium as well as thulium gain media.In contrast with the standard laser modeling which depends on structure parameters, like the transmissivity of the saturable absorber, that are often difficult to measure, the parameters of soliton-pair dynamical systems are derived directly from acquisition of lasergenerated data.
Beyond a description of the steady state, the dynamical system can make predictions about the response of the soliton molecule to external perturbation and noise.A systematic mapping of the coefficients in the equations of motion as laser parameters are varied can become a valuable design tool for tailored pulse interactions, that lead toward on-demand generation of complex multipulse waveforms.

Figure 1 .
Figure 1.Measured trajectory (points) and periodic fit (curves) of the quadratures of of the relative timing and phase of an oscillating pair soliton molecule.