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This work evaluates the suitability of the finite difference methods and the pseudo-spectral methods for validating the pulse propagation problem in an optical fiber, which is modeled by the nonlinear Schrödinger equation (NLSE) represented in a classical electromagnetic version. In particular, the finite difference methods have been reported as excellent schemes for solving the nonlinear Schrödinger type-equations represented in multiple fields of study. However, a rigorous analysis of the finite difference methods for solving specifically the NLSE in fiber has not been reported yet. On the other hand, the pseudo-spectral methods are reported as optimal schemes to integrate the NLSE in fiber. Thus, four schemes of finite difference methods and three schemes of pseudo-spectral methods are analyzed by the validation of the propagation of a fundamental soliton, which demands a high level of convergence and stability to reproduce the complex behavior involved in this pulse propagation problem. As a result, we observe that the multiple phenomenologies modeled by the NLSE in fiber are reproduced numerically with the best degree of convergence and stability by the pseudo-spectral methods, whereas the finite difference methods are not suitable to validate this pulse propagation problem due to a loss of convergence and a high computational cost.

In the present work, the temporal dynamics of an all-normal-dispersion ytterbium-doped fiber ring laser operating in regimes intermediate between continuous-wave operation and mode locking are studied experimentally. Exploiting the segmented memory data acquisition possibilities of an ultrafast digital oscilloscope and through the use of a specifically developed algorithm, the real-time waveform evolution anchored in absolute time could be retrieved, without relying on a fixed reference carried by the signal itself, which does not exist in these regimes. By controlling the time spacing between the successively acquired frames, several dynamics taking place over different time scales could be evidenced and described. These measurements highlight in particular the evolution cycles of intense localized structures including spikes that emerge, grow, decay, interact and describe peculiar trajectories in phase-space diagrams, whereas their temporal positions evolve, driven in particular by gain dynamics. In spite of their dramatic variability, these structures are found to be enduring features of these regimes. Analysis of these data helps assessing the degree of partial mode locking associated with these dynamics. The time-domain mapping technique optimized using segmented memory data acquisition is shown to be useful to characterize precisely highly dynamical evolutions such as those revealed in this work, which are dominated by structures that present large and complex variations in amplitude, shape and position, and develop over different time scales.

We present the suitability of two optimized split-step methods for validating the femtosecond pulse propagation problem in the nonlinear fiber optics formalism that is governed by an extended nonlinear Schrödinger equation. In particular, the embedded split-step Fourier method (embedded-SSFM) and the embedded symmetrized split-step Fourier method (embedded-SymSSFM), which are optimized by the implementation of a step size control algorithm, are tested in terms of the femtosecond soliton fission phenomenology to analyze the numerical stability. As a result, it is possible to validate that these numerical methods introduce the appropriate resolution as function of the accuracy order that is needed at different stages of the soliton fission dynamics, which involves very large spectral shifts driven by the interaction between the nonlinear response and the high-order dispersion contribution of the optical fiber. Thus, the presented numerical methods can be used to validate accurately complex processes such as the development of supercontinuum spectra in the current formalism.

This work explains in detail the role of the dispersion contribution provided by a single-mode optical fibre (SMF) in the pulse propagation problem modelled by the nonlinear Schrödinger equation. Firstly, the basics of chromatic dispersion in fibre are presented, including a modified Sellmeier equation for representing the dispersion contribution of the fibre core material (in an illustrative case of fused silica doped with germanium dioxide) that is modified by the waveguide dispersion contribution. At the same time, the contribution of the second-, third- and fourth-order dispersion parameters are illustrated for a SMF. Finally, a numerical analysis of the propagation problem is presented in order to understand the dispersion effects for a sech2 pulse profile, as well as the interplay of its contribution with the nonlinear Kerr effect, which is also involved in the pulse propagation. For a better understanding of the numerical methods applied to develop the propagation problem, we include a guide detailing the stages to follow in the numerical approach, considering how to select the initial parameters as well as the iterative steps of the numerical method.

We report an extensive experimental study of a dual-wavelength (1530 and 1560 nm) passively mode-locked fiber laser in a ring configuration with strict control of the polarization state. Different types of laser operation are identified, which are classified into ten mode locking regimes (solitons-noise-like pulsing, solitons–solitons, solitons-CW, bound solitons-CW, solitons (1530 nm), CW-solitons, CW-bound solitons, Q-switched-like noise-like pulsing, steady noise-like pulsing at 1530 nm and intermittent mode locking), which are mapped in terms of the polarization settings (ellipticity and azimuth). The resulting map establishes, in a clear and repeatable way, the relation existing between each mode locking regime and the polarization adjustment, which allows controlling at the same time intracavity losses, the gain profile, the cavity spectral transfer function, as well as the nonlinear switching characteristic.

We study numerically noise-like pulsing in a long passively mode-locked figure-eight fibre laser with a polarization-imbalanced nonlinear optical loop mirror (NOLM) and a strong dispersion map. Contrary to the common scenario where a single noise-like pulse develops, it is found that multiple small bunches tend to form in the cavity, sharing roughly the same characteristics, which are determined by the dispersion parameters. The wave packets are subject to ample periodic variations of their duration and peak intensity as they pass through sections of opposite dispersion signs, displaying a breathing dynamics analogous to the one undergone by solitons in stretched-pulse fibre lasers. Multiple noise-like pulsing operation can be understood by considering the strong compression and corresponding intensity increase of small bunches as they travel through the NOLM, which yields higher transmission through the NOLM and thus smaller cavity losses for these small units than for larger wave packets. The present results could serve as a basis for a deeper study of the dynamics taking place in partially mode-locked fibre sources, which is particularly complex when multiple bunches of radiation coexist and interact in the cavity.

In this paper, we report an experimental study of different dynamics taking place in a 20 m long passively mode-locked fibre ring laser in dual-wavelength operation, at 1531?nm and 1558?nm. For different polarization adjustments, self-starting mode locking is obtained, yielding different types of emission: bunches of solitons in quasi-stationary regime, a compact bunch of solitons coexisting with loose bunches of solitons, a noise-like pulse coexisting with bunches of solitons and a noise-like pulse displaying quasi-periodic fluctuations. In each regime, we extract information on the pulse dynamics from measurements of the temporal profile evolution using a 16 GHz real-time oscilloscope and, at the same time, we propose a phase-space diagram representation of the intensity versus the energy of the temporal profile of the pulses; the latter allows evidencing patterns that could not be identified using conventional measurement techniques.

We present a systematic experimental study of supercontinuum (SC) spectra produced by propagating noise-like pulses (NLPs) from an erbium-doped figure-eight laser through sections of different lengths of standard single-mode fibre (SMF) and of high-nonlinearity fibre (HNLF), as well as their combinations. With an average input power that does not exceed 35 mW, very broad and smooth SC spectra extending over several hundreds of nm are typically observed, even when only SMF is used. However, maximal broadening and flatness are obtained through a section of about 300 m of SMF followed by 50 m of HNLF; the spectrum then spans from ~1260 nm up to ~2070 nm, with a bandwidth of 431 nm at 5.7 dB from the maximum (over 800 nm at 20 dB). Our analysis indicates that nonlinear processes are still operating even after propagation through several hundreds of meters of fibre, extending and flattening the spectrum, its maximal bandwidth being eventually limited by the strong fibre attenuation in the 2 µm region.

Passively mode-locked fiber lasers (PML-FLs) are versatile sources that are capable of generating a broad variety of short and ultrashort optical pulses. Besides conservative solitons, PML-FLs allow the generation of different kinds of dissipative structures, usually called dissipative solitons, a concept that also encompasses more complex structures and collective behaviors such as soliton molecules, gas, rain of solitons, etc. In addition to this, PML-FLs are also able to generate even more complex objects, the so-called noise-like pulses (NLPs). A few recent research results revealed a connection between NLPs and solitons, a sign that deterministic ingredients enter into the composition of NLPs, whose nature is traditionally assumed to be random. Although it is usual that a fiber laser is able to generate either solitons or noise-like pulses, depending on pump power and adjustments in the cavity, these two regimes are rarely observed simultaneously. In this paper, a PML-FL in a ring configuration is presented, in which it is possible to observe and verify experimentally the simultaneous presence of NLPs and solitons. Interestingly, these two components are found in different spectral regions, which greatly facilitates their separation and individual study and characterization.

We propose the use of a figure eight fiber laser cavity in fundamental and harmonic mode-locking noise-like pulsing regimes, with output pulse energies as high as 302.8 nJ, as an improved supercontinuum source. The absence of a polarizer in the laser setup, so that the state of polarization along the cavity is not restricted, allows enhanced performances compared to other similar schemes. Noise-like pulses with durations of a few tens of ns were generated and a coherence length in the order of ps. For proper adjustments, self-starting passive mode-locking is obtained and a supercontinuum spectrum is observed at the laser output. We observe that the spectrum flatness is significantly improved in the harmonic mode-locking regime of the fiber laser. These results were achieved without the need to insert any high-nonlinearity propagation medium in the setup or at the laser output.

We report on the dynamics of noise-like pulses at the ns scale in a passively mode-locked fibre laser, which grow in complexity as wave retarder adjustments are performed. We can observe that the laser operating in the fundamental mode can be tuned to get different shapes of the noise-like pulse. Following a regime of a very stable waveform, regimes characterized by a much more variable (but still compact) waveform are observed. Then we can get the fragmentation of the main bunch and expulsion of sub-packets and, finally, a variety of puzzling dynamics with increasing complexity are evidenced. Although the collective behaviour of the multiple waveforms is at first sight random we can observe some well-defined patterns in the kinematics of light bunches at the global cavity scale. These results may be useful to unravel the subtle mechanisms at play in complex dissipative nonlinear systems such as passively mode-locked fibre lasers.