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We overview our recent developments in the theory of dispersion-managed (DM) solitons within the context of optical applications. First, we present a class of localized solutions with a period multiple to that of the standard DM soliton in the nonlinear Schrödinger equation with periodic variations of the dispersion. In the framework of a reduced ordinary differential equation-based model, we discuss the key features of these structures, such as a smaller energy compared to traditional DM solitons with the same temporal width. Next, we present new results on dissipative DM solitons, which occur in the context of mode-locked lasers. By means of numerical simulations and a reduced variational model of the complex Ginzburg-Landau equation, we analyze the influence of the different dissipative processes that take place in a laser.

Nonlinear distributed systems with periodic variations of one or several key parameters present a very important branch of nonlinear science with a range of practical applications in solid-state physics, optics, plasma physics, hydrodynamics, wave physics and other fields. The impacts of nonlinear effects on wave propagation or modifications of the nonlinear wave properties by a medium with periodically varying parameters are two important fundamental problems that have been actively studied in the past decades. It is well accepted nowadays that very often, nonlinear models governing rather different physical phenomena engaging nonlinearity and periodicity of a medium characteristics can be quite similar mathematically. Therefore, the analysis of certain generic nonlinear models might be interesting in a range of physical contexts. One of the important examples of such generic nonlinear models is the complex Ginzburg-Landau equation (CGLE), which was originally proposed as a phenomenological approach in the context of phase transitions [

The emergence of localized, particle-like structures that are called solitary waves or solitons is a widespread phenomenon occurring in a variety of physical problems. Classical optical solitons (i.e., solitons of the NLSE with uniform dispersion) are formed as a result of a continuous balance between the effects of linear dispersive pulse broadening and nonlinearity. An interesting modification of the classical soliton occurs in optical signal transmission along dispersion-managed (DM) links. In fibre optic communications, a periodic dispersion management—alternation of the fibre spans with positive and negative linear dispersion—is an established technique to improve system performance and has been used to achieve ultra-long haul transmission systems [

Recent developments in the field of nonlinear optics and optical communications are partially stimulated by the never-stopping demand of further increase of the capacity of fibre transmission systems. Efficient growth of the capacity of communication systems can be achieved by an increase in the channel bit rate—the speed at which information bits are transmitted. Increasing the channel rate assumes the utilization of shorter time slots allocated for each information bit and, consequently, of shorter carrier pulses. The propagation of ultrashort pulses is then strongly affected by the fibre dispersion, which results in large temporal broadening of the carrier pulses. Because of the temporal broadening during propagation, the carrier pulse power spreads over many time slots and so the accumulated effect of the instantaneous fibre nonlinearity tends to get averaged out. Signal transmission using very short optical pulses, often referred to as the quasilinear regime [

In the context of powerful laser systems, we analyze mode-locked lasers and study the properties of periodic (DM) solutions. The evolution of an optical pulse inside a laser cavity implemented either through a ring laser configuration or using a Fabry-Perot cavity design has an inherent periodicity related to the resonator round trip. Effects of gain and loss, dispersion, nonlinearity, gain dispersion (spectral filtering), and saturable absorption all can be mathematically treated as periodic variations of the system parameters. Solitary wave solutions of the CGLE for such systems are pulses that simultaneously balance the phases accumulated from dispersion and nonlinearity as well as the amplitude modulations from gain, linear and nonlinear (saturable) loss, and spectral filtering. Soliton solutions in a system with loss and gain are referred to as dissipative solitons [

The optical pulse propagation along a fibre link with varying dispersion is governed by the normalized NLSE (see, e.g., [

The traditional DM soliton [

Breathing dynamics of a DM soliton

In general, DM solitons are very stable structures that exist for a wide range of parameter space, in bound-state configurations [

In this section, we present a class of multiple-period solutions of the NLSE with varying dispersion [

Along with the standard solutions with period

As mentioned before, in the limit when dispersion dominates nonlinearity, the pulse solution of the NLSE (

We note that system (

Without loss of generality, we consider the TM system (

Here, we briefly recall the properties of the simple periodic solutions of the TM sytem. Let us consider the trajectory of one such solution in the

Dependence of minimum

To characterize multiple-period solutions, it is convenient to define the transformation

Trajectories of the points on the

The solutions of the TM system corresponding to the stable points

Evolution of

It is customary to describe the properties of DM solitons in terms of map strength and normalized peak power

Dependence of the normalized power

It is evident from the structure of the trajectories of the phase-plane points under

The multiple-period solutions of the previous section have less energy when compared with the single-period solutions of (

The evolution of electromagnetic energy in the laser cavity is subject to various dissipative effects such as attenuation, gain saturation, gain dispersion, and saturable absorption (intensity-discrimination). Haus proposed that these different elements could be averaged together resulting in the cubic-quintic Ginzburg-Landau evolution [

The CGLE (

The governing equation (

Inspired by numerical simulations and previous work done on DM solitons [

To illustrate the mode-locking dynamics we consider a physically realizable two-step dispersion map (

To understand the effects gain saturation and SAM have on the governing equation (

Pulse evolution obtained from solving the evolution equation (

Figure

Pulse evolution obtained from solving the evolution equation (

The two-norm of the difference between consecutive pulses which are located in the middle of the anomalous dispersion segment. Stable periodic evolution occurs as this quantity approaches zero. (Inset) Phase plane diagram showing the Poincaré section of consecutive pulses located in the middle of the anomalous dispersion segment.

A comparison between the solution to the reduced differential equations (

Evolution of the intensity, FWHM, and chirp parameter over one map period in the case where only gain saturation and SAM are present. The results from both the full evolution equation (

Phase plane dynamics over

Although the amplitude modulations from the gain saturation and saturable absorber create a stable attracting state, the overall structure of the steady state is similar to the DM soliton. Specifically, at the center of each segment in the map there exists a peak-amplitude and minimum pulse-width corresponding to the zero-chirp points. Further, the side-lobe structure observed for DM solitons at maximum compression exists as well. Thus gain saturation along with SAM selects the exact dissipative DM soliton that will satisfy the energy constraints. In general, gain saturation and SAM allow for high map-strength evolution to persist resulting in higher-energy pulses than those considered in the previous section.

When the pulse bandwidth is comparable to the gain bandwidth, we must include the effects of bandwidth-limited gain. For a gain bandwidth which can vary from

One-map period evolution in the case where gain saturation, saturable absorber, and gain bandwidth are included (

The dissipative perturbations considered selects the exact DM soliton that will satisfy particular energy constraints. Figure

Numerically calculated gain from three different simulations of (

Since the reduced system provides an excellent description of the full model, it can be used to calculate relevant physical quantities. For example, the chirp parameter

It should be noted that all Hamiltonian terms (with

We have discussed some recent developments in the theory of DM solitons within the context of photonic applications. We have considered two directions of further development of the DM soliton theory and corresponding practical applications in optical signal transmission/processing and mode-locked laser systems. In the context of optical communications, it is desirable to keep the energy of the carrier pulses below a certain level, which is imposed by power consumption and safety conditions. In the framework of a reduced ODE-based model, we have examined a class of periodic localized solutions of the NLSE with periodic variations of the dispersion. Such solutions have multiple periods and lower energies compared to traditional DM solitons of the same temporal width. The multiple-period DM solitons described in this work experience larger broadening during propagation and, therefore, they much more closely mimic the widely used in practice quasilinear transmission regime compared to traditional DM solitons.

In the context of powerful laser systems, we have examined mode-locked lasers and studied the properties of periodic dissipative DM solitons with high energies. Note that in powerful laser systems, one of the important goals is to generate stable optical pulses with as high energy as possible. By means of numerical simulations and a reduced variational model of the governing CGLE, we have analyzed the influence of the different dissipative processes that take place in the laser cavity. The reduced model introduced here rather accurately describes the key characteristics of the dissipative DM solitons, such as pulse width, chirp, peak power and energy. This model might be very useful for multi-parametric optimization of complex laser systems.