## Abstract

Interdisciplinary concept of dissipative soliton is unfolded in connection with ultrafast fibre lasers. The different mode-locking techniques as well as experimental realizations of dissipative soliton fibre lasers are surveyed briefly with an emphasis on their energy scalability. Basic topics of the dissipative soliton theory are elucidated in connection with concepts of energy scalability and stability. It is shown that the parametric space of dissipative soliton has reduced dimension and comparatively simple structure that simplifies the analysis and optimization of ultrafast fibre lasers. The main destabilization scenarios are described and the limits of energy scalability are connected with impact of optical turbulence and stimulated Raman scattering. The fast and slow dynamics of vector dissipative solitons are exposed.

### Keywords

- Ultrafast fibre laser
- mode-locking
- dissipative soliton
- non-linear dynamics
- vector solitons
- optical turbulence
- stimulated Raman scattering

## 1. Introduction

Over the last decades, ultrafast fibre laser technologies have demonstrated a remarkable progress. By definition [1–4], these technologies concern generation, manipulation and application of optical pulses from a fibre laser or a laser-amplifier system with (i) peak power _{0} exceeding substantially an average laser power _{ave} and (ii) pulse widths * τ* which are much lesser than a laser round-trip period

*. Such a definition can be re-interpreted in terms of a laser*T

*, which means a phase-locked interference of the*mode-locking

*laser eigenmodes producing an equidistant train of ultrashort pulses. Then,*M

*∝ 1/*τ

*(*Mδω

*is a number of locked eigenmodes and*M

*is an inter-mode frequency interval defined by*δω

*∝ 1/*T

*) and*δω

P

_{0}∝

MP

_{ave}[5,6]. It means that a mode-locked laser generates a comb of equidistant optical frequencies comprising the broad spectral range

*∝*∆

*. It is clear that the substantial enhancement of*Mδω

P

_{0}(by the factor of

*∝ 1/*M

*~10*τδω

^{5}÷ 10

^{6}, i.e. upto over-MW level [7,8]) and

*–reduction (~*τ

*, i.e. down to sub-100 fs level [9]) promise an outlook for different applications [10] including non-linear and ultrasensitive laser spectroscopy [11–14], biomedical applications [15–19], micromachining [20,21], high-speed communication systems [22], metrology [23] and many others [24,25]. The extraordinary peak powers in combination with the drastic pulse width decrease bring the high-field physics on tabletops of a mid-level university lab [26–29]. Moreover, the over-MHz pulse repetition rates*T/M

*provide the signal rate improvement factor of 10*δω

^{3}÷ 10

^{4}in comparison with that of classical chirped-pulse amplifiers [26]. As a result, the signal-to-noise ratio enhances substantially, as well.

Another aspect of the ultrafast laser applications is connected with studying non-linear phenomena [30]. Ultrafast lasers became an effective platform for investigation of general non-linear processes such as instabilities and rogue waves [31,32], self-similarity [33] and turbulence [34]. A coherent self-organization in such non-linear systems [35,36] is the keystone of this review, and it will be considered below in detail. But here, we have to point at the multidisciplinary context of our topic. The ultrafast fibre lasers can be treated as an ideal playground for exploring of non-linear system phenomenology as a whole [37]. Such a playground spans gravity and cosmology [38], condensed-matter physics and quantum field theory [39–41], biology, neurosciences and informatics [42,43]. The advance of ultrafast laser technology is that the theoretical insights promise to become directly testable, controllable and, on the other part, the theory can be urged by new precise measurable experimental challenges.

To date, the solid-state lasers allowed generating shortest pulses with highest peak powers directly from an oscillator with high repetition rates (* δω* > 1 MHz) [44–49]. The main advantages of solid-state laser systems are (i) broad gain bands (i.e. very large

*) allowing generation of extremely short pulses (*M

*approaches one optical cycle for such media as Ti:Sp or Cr:chalcogenides) and (ii) covering the spectral range from visible (Ti:Sp) through infra-red (Ti:Sp, Cr:forsterite and Cr:YAG) to mid-infrared (Cr:chalcogenides) wavelengths, as well as (iii) possibility of independent and precise dispersion [50] and non-linearity [46,48] control. Nevertheless, fibre lasers have unprecedented prospects [51] due to (i) possibility of mean power scaling provided by large gain, (ii) high quality of laser mode, (iii) reduced thermo- and environment-sensitivity, (iv) compactness and integrity of laser setup. Additionally, one has to point at broader gain bands of fibre media in comparison with the energy-scalable, thin-disk, solid-state oscillators operating within analogous wavelength ranges [9,52]*τ

The width of a gain band is not a decisive factor per se because both pulse width and its spectrum are affected by various factors including higher-order dispersions, non-linearity, etc. [53,54].

and possibility to break into deep-UV and mid-IR optical spectral ranges [55,56].In this review, we will concern the concepts of a mode-locking and a dissipative soliton in a nutshell.

## 2. Mode-Locking

The concept of mode-locking is universal and closely connected with a principle of synchronisation of coupled oscillators [57–64]. A laser is, in fact, the interferometer which possesses a set of eigenmodes (longitudinal modes) separated by * δω* = 2π/

*. Simultaneously, it is an*Τ

*resonator, which means an amplification ∆*active

*of mode amplitude*Α

*during the resonator round-trip in the vicinity of the maximum gain frequency*Α

ω

_{0}as

*(*g

ω

_{0}) is a gain at the frequency

ω

_{0,}

*is a net-loss coefficient and*ℓ

ω

_{0}defined by a gain bandwidth

*Ω. In such an oversimplified model, only*δ

*mode with the maximum net-gain*one

*(i.e. a gain is*A

*that results in a mode competition or*saturable

*, Figure 1).*mode selection

In the case of * active mode-locking*, a periodic external modulation with the frequency of

*excites the ±*δω

*sidebands for each mode in the comb so that the modes*δω

*(ω),*A

*(ω ±*A

*) become coupled. In the framework of our oversimplified model, a steady-state regime Δ*δω

*0 is described by the equation in time domain [59–61]:*A =

which is the classical equation for an oscillator in the potential defined by * ν* ∝

δω

^{2}. This equation has a trivial solution in the form of a Gaussian pulse [59–63]:

A

_{0}is defined by the condition of energy balance of

*(*A

*) in the form of Hermitian–Gaussian solutions of Equation (1) is possible. Since the pulse width is defined by*t

*so that*ν

*-value. The situation can be changed in the presence of the self-phase modulation (SPM) [65] and the dynamic gain saturation. Then*δω

Using the non-linear processes such as SPM, loss and gain saturation allows generating the ultrashort pulses due to mechanism of the so-called * passive mode-locking* [60]. Periodical perturbations caused by transitions through non-linear laser elements such as saturable absorber or gain medium enrich the spectrum with new components

*1, 2,...,*m =

*), which becomes locked through non-linear interaction [64]:*M

where a four-wave non-linear process defined by non-linear susceptibility χ^{3} mixes the frequencies _{1}, _{2}, _{3} and _{4} during the propagation through a non-linear medium along the * z*-coordinate (

*is the speed of light,*c

*is the frequency-dependent refractive index).*n

Both active and passive mode-locking concepts can be easily united from the point of view of spice-time duality [64,67–69]. For instance, let’s consider * heat diffusion* equation:

where a heat radiated at * x* = 0 by the point source

*diffuses along*σ

*-axis and is absorbed by cooler with the parabolic ‘cooling potential’. The replacements*x

t

*and*z

x

*result in an equation for ‘diffusion’ of light describing an*t

*(see Eq. (1)):*active amplitude mode-locking

Eq. (4) is clearly understandable in the Fourier domain:

The space-time duality can be extended further with the help of * diffraction-dispersion* duality:

where k and β_{2} are the wave number and group-delay dispersion coefficients, respectively. Both processes describe the beam/pulse spreading with propagation which is accompanied by phase * ϕ* profile distortion, i.e. by appearance of the

chirp

*from this point of view*active phase modulation

(compare with (4)) looking as * ϕ*(

*) is parabolic in this case.*t

The transition to a * passive mode-locking* looks straightforward, but one has to be careful in this case. The spice-time duality suggests a simple way to realize the temporal focusing like that in space domain: combination of phase modulation (‘time lens’) from Eq. (6) with dispersion (‘time diffraction’) from Eq. (5) allows compressing a pulse. Therefore, a replacement of time focusing by a time self-focusing (SPM) would provide a laser pulse self-trapping like the effect of laser beam self-trapping:

which is the famous * non-linear Schrödinger equation* describing propagation of optical solitons in a fiber (

β

_{2}< 0 corresponds to an anomalous dispersion,

*is a SPM-coefficient) [35,70,71].*γ

It is appropriate to mention here that the space-time duality * x * allows extending the physical context of consideration beyond scopes of optics. For instance,

*,*A

*= ∫d*E

*|*x

*|*A

^{2}, and ϕ can be related for a mean-field amplitude, number of particle (mass of condensate) and momentum (wave number) for a Bose–Einstein condensate [39]. Then, it is clear that the dispersion and SPM terms in Eq. (7) describe the kinetic energy and four-particle interaction potential for gas of bosons. Such an interpretation opens a road to a quantum theory of solitons [72–74].

Following the same procedure for Eq. (4), describing the active amplitude mode-locking results in the simplest version of equation for a passive mode-locking, so-called * cubic non-linear Ginzburg*–

*[35,36,75]:*Landau equation

This equation describes a combined action of saturated net-gain (σ), spectral dissipation (α) and non-linear gain (κ). The last term results from loss saturation in a non-linear absorber with the response time much lesser than the pulse width. As will be shown below, such an assumption is valid for a broad class of fibre mode-locking mechanisms. Physics of passive mode-locking resembles that of active one: self-focusing in time domain causes a spectrum broadening which is balanced by spectral dissipation. Loss and energy-dependent gain are required for developing and stabilizing the mode-locking (all these factors are included in * σ*-term which is < 0 for a steady-state pulse). Eqs. (7) and (8) have a similar solution

*(DSs, see next section) [36]. Combining Eqs. (7) and (8) gives the famous complex cubic non-linear Ginzburg–Landau equation (cubic CNGLE) [35–37,42]:*dissipative solitons

which is a playground for study of DSs. Equation (9) allows a number of further generalizations such as: (i) description of non-distributed evolution due to dependence of the equation coefficients on * z* [77]; (ii) generalization of non-linearity type aimed first of all to adequate description of different mode-locking mechanisms (see below); (iii) taking into account the higher-order dispersions, i.e.

*-dependence of*ω

β

_{2}[68]; (iv) taking into account the vector nature of light, i.e. transition to a system of coupled two-component CNGLEs [68,78–81], etc.

Now let us consider the mode-locking mechanisms for fibre lasers in more detail. Active mode-locking can be utilized for DS generation from a fibre laser [82–84], but the widespread mechanism is based on the * non-linear polarization rotation* (NPR) which uses the effect of intensity-dependent polarization mode coupling in a fibre [85–88]. There is voluminous literature concerning the experimental realization of NPR mode-locking in fibre lasers; therefore, our selection of references is rather subjective and concerns the DS context [95–111].

It is known [70] that an ideal single-mode fibre supports two degenerate orthogonally polarized modes. However, a real fibre has inherent birefringence caused by core asymmetry or mechanical stress (Figure 2).

Since SPM as well as cross-phase modulation (XPM) contribute to refractivity index with the strength defined by field intensity, such a contribution will change the state of polarization (SOP, Figure 3) [60,70,89] that can be described by coupled equations for two orthogonal (* x* and

*) polarization components [70]:*y

where the dissipative factors from Eq. (9) are taken into account and * L* is a beat-length). As was shown in [81,91–94,179], the multi-scale averaging technique allows reducing Eq. (10) to the modified scalar non-linear Ginzburg–Landau equation (

_{b}

*–*so-called sinusoidal Ginzburg

*) in which the self-amplitude modulation term (SAM, last term in Eq. (9)) is replaced by*Landau equation

*is a complex function defined by birefringence and settings of laser wave plates and polarizer. Such an approach opens a way to multi-parametrical optimization of fibre lasers mode-locked by NPR.*Q

Despite its relative simplicity in principle as well as possibility of all-fibre-integrity of a laser, NPR in the form presented in Figure 3 is too sensitive to laser setup, uncontrollable perturbations and requires a precise manual tuning. The modified SAM setup, which can utilize both NPR and scalar SPM, is shown in Figure 4. It is the so-called * non-linear optical loop mirror* or

*(Figure 4) [112,113,265]. In principle, this setup is an all-fibre realization of additive-mode-locking [82,114] with inherently adjusted linear optical propagation lengths for counter-propagating beams. The main control parameter here is the beam splitting ratio*figure eight laser

*controlling the mutual intensities of counter-propagating beams.*ρ

The unique property of this SAM setup is its ability to utilize different types of non-linearities for mode-locking (e.g. see [115–118]). Different modifications of this mode-locking mechanism have been used in DS fibre lasers [119–125]. Nevertheless, a fibre loop defining SAM remains environment- and tuning-sensitive.

There is a class of alternative approaches utilizing non-fibre well-controllable non-linearities for mode-locking by the cost of broken fibre-integrity of a laser. Such an alternative was provided by development of high-non-linear * semiconductor saturable absorber mirror* (SESAM) [126–135]. The point is to put a semiconductor layer into a composed multi-layer mirror with the well-controllable spectral characteristics as well as with the adjustable intensity concertation of penetrating field within a semiconductor layer. In fact, it is an advanced non-linear Fabri–Perot interferometer with the reflectivity coefficient depending on the incident intensity (or energy) [136]. Interaction of light with a semiconductor layer can be characterized roughly as excitation of carriers from a valence band of semiconductor to its conduction band. Excited carriers thermalize inside a conduction band with the character time ~100 femtoseconds. This time defines a fastest response of SESAM to a laser radiation. Then, the thermalized carriers can relax into valence band or intra-band trapping states with the characteristic times from picoseconds to nanoseconds. Thus, SAM due to SESAM is

*in comparison with that due to NPR or SPM because the response times of the lasts are defined by intra-atom polarization dynamics, i.e. these times belong to femtosecond diapason. Additionally, the*slow

*of SESAM response is substantially squeezed in comparison with that of pure electronic non-linearities due to resonant character of SESAM non-linearity. This can trouble the mode-locking within a spectral range exceeding the SESAM bandwidth. But the reverse side of the SESAM-band squeezing is that a non-linear response of SESAM becomes resonantly enhanced. This means that SESAM can provide more easily starting, stable and controllable mode-locking. The key characteristics of SESAM are [127]*spectral diapason

loss saturation fluence

*is a photon energy,*hν

*is an absorption cross section),*σ

_{a}

modulation depth

*is a density of states in semiconductor),*N

relaxation (recovery) time

*,*T

_{r}

*,*unsaturable loss

*and level of*saturable loss bandwidth

*.*two-photon absorption

Akin mode-locking methods providing full fibre-integrity, broadband absorption, sub-picosecond response time and avoiding a complex multi-layer mirror weaving use nanotube and graphene saturable absorbers [30,137–143] and other low-dimensional structures [144].

From the theoretical point of view, the response of saturable absorber (SESAM or other quantum-size structures) to a laser field can be very complicate. In principle, one has to take into account finite loss bandwidth, its dispersion, dependence of refractive index on carrier’s (or exciton’s) density (so-called linewidth enhancement), complex kinetics of excitation and relaxation, etc. However, the praxis demonstrated that a simple model of two-level absorber is well working [145]:

with some possible modifications (e.g. see [146]). Since DSs, as a rule, have over-picosecond widths (see next section), one may use an adiabatic approximation for (11) so that the expression for SAM coefficient in the last term in Eq. (9) has to be replaced:

where

One may propose a hypothesis that an analogue of Kerr-lens mode-locking, which is a basic mechanism for generation of femtosecond pulses from solid-state lasers [60,85,147], can be realized in a fibre laser as well. Such an insight is based on possible enhancement of the laser beam spatial-trapping induced by non-linearity in a medium with spatially inhomogeneous gain/loss or refractivity [148–152]. The model for analysis of such phenomena can be based on extension of dimensionality of Eq. (9), with taking into account the diffraction and transverse inhomogeneity of gain, loss or/and refractive index (the last can work as SAM due to the waveguide leaking loss) [153]:

where cylindrical symmetry is assumed, * x* is a radial coordinate and

*is a coefficient (complex in general case) which describes a transverse inhomogeneity of a fibre. Figure 5 shows the net-gain profiles (*κ

*) and the intra-laser pulse energies (*a

*) as function of an effective aperture size obtained on the basis of variational approach for Eq. (13) [153]. The results demonstrate a principal feasibility of the Kerr-lens mode-locking regime for a DS fibre laser.*b

All these mode-locking techniques are realizable for both soliton proper and DS fibre lasers (excluding the Kerr-lens mode-locking which requires sufficiently high pulse energies provided by only a DS laser). Now let’s consider the DSs fibre lasers proper.

## 3. DS concept: Theory and experiment

A ‘classical’ soliton can be formally defined as a solution of non-linear evolution equation belonging to discrete spectrum of the inverse scattering transform [71,76,154]. The non-linear equations, which can be solved by inverse scattering transform, are ‘exactly integrable’. This means that they are akin to linear equations in some sense. In particular, they obey the superposition principle and, as a result, can be canonically quantized [155,156]. One has to note that integrability of a non-linear evolution equation and non-dissipative (non-Hamiltonian) character of the latter are not equivalent because there are both non-integrable Hamiltonian systems and integrable dissipative ones [36]. The point is that the DS concept is not connected with ‘integrability’; therefore, DSs are not ‘true’ solitons in a mathematical sense. However, many properties of DSs, in particular, their stable localization, robustness in the processes of scattering and interaction, well-organized internal structure, etc., resemble the properties of ‘true’ solitons. Formally, one may define * DS* as a

*[36]. DSs are abundant in the different natural systems ranging from optics and condensed-matter physics to biology and medicine. In this sense, one may paraphrase that DSs “are around us. In the true sense of the word they are absolutely everywhere” [157]. Therefore, the concept of DS became well established in the last decade [36,37,42,158].*localized and stable structure emergent in a non-linear dissipative system far from the thermodynamic equilibrium

Stability of a DS under condition of strong non-equilibrium can be achieved only due to well-organized energy exchange with environment and subsequent energy redistribution within a DS. It results in energy flux inside a DS and, thereby, in DS phase inhomogeneity [36]. For a simplest case of Eq. (9), which has a DS solution in the form of

as well as the spectrum _{2} (the data are based on an approach of [160]). One can see that the spectrum broadening transfers the action of spectral dissipation on a pulse ‘in whole’ into well-structured energy exchange: inflow at pulse centrum and outflow on its wings. The key characteristic of a dissipation inhomogeneity is a * chirp*, i.e. an inhomogeneity of phase. In absence of the chirp, the spectral dissipation acts on a pulse in whole that, in particular, induces a multi-pulse instability [161]. However, a power-dependent chirp causes inhomogeneity of energy transfer (Figure 7). Energy flows in the region closer to central wavelength where the gain is maximal. This region is located in the vicinity of pulse maximum. Energy flows out from the spectrum wings which are located on the wings of pulse, that is, the pulse localization is supported by spectral dissipation through non-linear mechanism of chirping [42,160,162]. One has to note that a direction of energy fluxes inside a DS depends on parameters and can be inversely related to the direction shown in Figures 6,7 (i.e. energy can flow from wings to centre). The corresponding structure was named

*[210].*dissipative anti-soliton

Thus, an additional mechanism of SAM (in addition to mechanisms considered in the previous section) appears, which provides unique robustness of DSs (i.e. DS exists within a broad range of laser parameters [163,164]).

Below, we will consider a chirp as the essential characteristic of DS [210]. One of the reasons is that the chirp allows DS to accumulate energy * energy-scalable* [37,164–167]. The last statement does not mean that a chirp-free pulse is not energy-scalable. However, the energy-scalability of such pulses can be provided by only fine-tuned and separated control of SPM and GDD that can be achieved in solid-state oscillators [26,168] or in large mode area (LMA) fibre lasers [169]. For fibre lasers such an approach entails issues of full-fibre integrability, higher-order mode control [170,171]

However, namely LMA and photonic-crystal fibres could realize a Kerr-lens mode-locking in a fibre laser [152,153].

and thermo-effects impact [172].In the terms of space-time duality (see above), the mechanism of formation of time window, within which a DS is localized, resembles a phenomenon of total internal reflection from some ‘borders’ created by phase discontinuity. Such borders are formally defined by the equivalence of the wave number of out-/in-going radiation

where * E* is a DS energy. Eq. (15) is valid for the

*in which SAM has a form of*cubic-quintic CNGLE

*= 0 (see Eq. (9)). The asymptotic*σ

*DS or to a phenomenon of*perfectly energy-scalable

*(DSR) [37,44,165–167,175–183], which is sufficiently robust, exists in different SAM environments and even within the anomalous GDD range [178,182]. Important property of DSR is that the DS energy*dissipative soliton resonance

*can be scaled without loss of stability by plain scaling of laser average power or/and its length*E

*[44,180,184]. The chirp scales with length as well. As a result, the DS peak power and spectrum width tend to a constant for fixed parameters of Eq. (9) (i.e. fixed*L

*,*α

β

_{2},

*and*σ

*) and the energy scaling is provided by DS stretching in time domain.*κ

This ideology of energy scaling by the pulse stretching goes back to the so-called wave-breaking-free or stretched pulse fibre lasers where the propagation within the anomalous-dispersion fibre sectors alternates with the propagation under normal GDD action [96,101,185–187]. As a result of pulse stretching, the non-linear effects in such systems are reduced, which allows increasing an energy and suppressing a noise. As an alternative approach, one can exclude an anomalous GDD at all and to realize a so-called * similariton* regime, when a pulse accumulates an extremely large chirp and, thereby, an energy [33,103,188]. However, a self-similar regime is not soliton-like one in nature; therefore, we will focus on the all-normal-dispersion fibre lasers (ANDi) which produces DSs possessing a high stability within a broad range of laser parameters [97,101,185,186,189,190]. In Figure 8, the energy-scalable DS lasers are sub-divided into three main types: (1)

*, (2)*all-fibre

*and (2) LMA including rod and photonic-crystal fibre PCF.*fibre with a free-space sector

The advantage of the * first type* of lasers is their integrity, which does not require an operational alignment, includes potentially compressing and delivering sections, and provides environment insensitivity and easy integrability with fibre-amplifier cascades [200]. The last advantage is especially attractive because it allows a direct seeding of DS into chirp-pulse amplifier without preliminary pulse stretching. The

*of the DS fibre lasers can be considered as a testbed for development of the first type. No wonder that the results achieved here are more impressive (Figure 8). At last, the*second type

*of the DS fibre lasers is most akin to the thin-disk solid-state ones with simultaneous advantage of the broad gainbands. Such lasers provide the DS energy scalability by scaling of laser beam area in combination with the scaling of laser period and average power. Nevertheless, one has to keep in mind that both LMA and PCF technologies have some disadvantages (see above) which make them similar to solid-state lasers.*third type

The diversity of the results obtained (Figure 8) needs a comprehension from a * unified viewpoint*; therefore, let’s survey briefly some theoretical aspects relevant to the DS fibre lasers. There is vast literature regarding the theory of DSs. Some preliminary systematization can be found in [44]. However, it is necessary at first to declare the stumbling block of this theory:

*. There exist unbroken walls between the circles of scientific community exploiting and exploring the DS concept: walls between the solid-state and fibre laser representations of the theory, condensed-matter one, numerical and analytical approaches, etc. Briefly and conditionally, the relevant theoretical approaches can be divided into (1) numerical, (2) exact analytical and (3) approximated analytical. The last includes the models based on (3.1) perturbative, (3.2) adiabatic models (AM) as well as those based on (3.3) phase-space truncation (i.e. variational approximation (VA) and method of moments (MM)).*absence of a unified viewpoint

As was emphasized repeatedly, both linear and non-linear dissipations are crucial for the DS formation. The simplest and most studied models for such a type of phenomena are based on the different versions of CNGLE (e.g. Eq. (9)).

Extensive * numerical study* of DSs of the cubic-quintic CNGLE has been carried out by N. N. Akhmediev with co-authors [35–37,42,157,166,175,176,178,182]. The simulations have allowed finding the DS stability regions for some two-dimensional projections of CNGLE parametrical space. The summarizing description of the results obtained is presented in [44]. Most impressive results are: (i) parametric space of DS has a reduced dimensionality resulting, in particular, in the appearance of DSR; (ii) DSR remains in a model with lumped evolution that is typical for the most of fibre lasers; (iii) DSR and, correspondingly, DS exist within the anomalous GDD region as well. However, the main shortcomings of the numerical approaches are: the parametrical space under consideration is not physically relevant, and the true dimensionality of DS parametric space is not identified. It is clear that the only advanced and self-consistent

*theory of DS would provide, in particular, a true representation of DS parametric space and DSR conditions.*analytical

As was mentioned above, the evolution equations describing DSs are not-integrable. The efforts based on the algebraic techniques [62,213,214] and aimed to finding the generalized DS solutions of CNGLE were not successful to date. Nevertheless, few exact partial DS-solutions are known. For instance, * sole known exact analytical DS-solution* of cubic-quintic CNGLE is [110,166,176,182,189,211,212]:

where _{0}, * B*, τ, ψ and

*are real constants [189]. This solution belongs to a fixed-point solution class, which means that it exists only if some constraints are imposed on the cubic-quintic CNGLE parameters. Solution (16) provides with important insights into properties of DSs. In particular, the systematical classification of DS spectra (truncated concave, convex, Lorentzian and structured spectra) and DS temporal profiles (from*ϕ

*-shaped*sech

*(*A

*) to tabletop one) is possible in the framework of analytical approach. The transition to a DSR-regime reveals itself in the ‘time-spectral’ duality shown in Figure 9 [141,189]. The sense of this ‘duality’ will be explained below from the point of view of adiabatic theory of DSs.*t

The crucial shortcoming of the approach based on few exact DS solutions of evolution equations is that the strict restrictions are imposed on the equation parameters. As a result, the DS cannot be traced within a broad multidimensional parametric range and the picture obtained is rather sporadic and is of interest only in the close relation with the numerical results and experiment. Some additional information can be obtained on the basis of perturbation theory which provides with a quite accurate approximation for a low-energy DS [215–217].

Most powerful approaches to the theory of DSs have been developed in the framework of * approximated techniques* (for review see [44]): AM [165,167,217–221], VA [77,177,222–224,225] and MM [175,210,222,226]. The most impressive results obtained are: (i) physically relevant representation of DS parametric space was revealed (it is a so-called

*, see [44] for review and Figure 10); (ii) such a representation allows understanding the structural properties of DS and its energy-scaling laws (i.e. DSR conditions) for different mode-locking techniques; (iii) DS dynamics and an issue of optimal arrangement of laser elements providing the maximum DS stability and energy have been explored [224,225,227,228]; (iv) vectorial extension of VA concerning a vector DS (VDS) was endeavoured [229].*master diagram

Both AM and VA demonstrate two-dimensional representation of DS parametric space in the form of master diagram. Dimensionality can grow with complication of CNGLE non-linearity when SPM becomes saturable so that the cubic non-linear term in Eq. (9) has to be replaced by

The master diagram is a manifold of isogains (i.e. curves with * σ* = 0) corresponding to the DS stability limit (upper curves) as well as the borders between ‘energy-scalable’ and ‘energy-non-scalabe’ branches of DS (lower curves, see [44,167] for a formal definition

Energy-non-scalable branch has two distinguishing characteristics: it turns into solution of Eq. (9) with

which agrees with experimental observations of linear growth of DS energy with bandwidth [107,108] as well with a rule

In the case of unsaturable SAM corresponding to SESAM, some nanotube and graphene absorbers, Kerr-lensing, etc. (see Eq. (12)), the energy scaling requires scaling of the control parameter * C*. In this case, the asymptotic

*for*energy scaling law

The spectral properties of DS are described clearly in the frameworks of AM [44,167]. In the simplest case of cubic-quintic CNGLE, the DS spectrum _{L} and is truncated at frequencies ±∆ [44,167,218]:

where H is a Heaviside function. The DS energy is

and

Here, we trace the zero-isogain σ = 0. The DS time-profile is defined by an implicit expression:

with the DS width of

It is clear that in this ‘low-energy’ sector the DS time-profile is bell-like and its spectrum has tabletop form (

that is, a DS in the DSR sector has a flattop temporal profile and a Lorenzian spectrum (

## 4. DS spectrum and stability

As was explained, the dual balances in frequency domain:

are formative for DS existence and stabilization. No wonder that the spectrum of DS is benchmark of its inherent properties.

Prior to consider the aspects of interweaving of spectral and stability properties of DSs, one has to point to a possibility of multi-wavelength multi-pulsing DSs provided by DS robustness. As was demonstrated in [234] theoretically, the multi-DSs compounds in a mode-locked laser can be stabilized at multiple frequencies. Experimentally, such multi-frequency DS compounds can be realized by birefringence filters with a periodical (interference-like) dependence of transmission on wavelength under conditions of sufficiently broad gainband and powerful pump [235–239]

A multi-porting configuration of a DS laser supports even simultaneous generation of conventional and dissipative wavelength-separated solitons [240].

.As was demonstrated in previous section, DS has non-trivial internal structure due to energy fluxes inside it. The elements of this structure (* internal modes*) can be excited that causes pulsating or chaotic dynamics of DS with preservation of its temporal and spectral localization [241]. The spectral envelope acquires a shape of ‘glass with boiling water’ (Figure 11). Appearance of such perturbations is understandable in frameworks of the DS perturbation theory in spectral domain [242]. One has to note that such perturbations take a place inside the DS stability region where σ < 0 (below the corresponding upper curves in Figure 10). Above the stability boarder (‘no DS’ region in Figure 10), there are three main destabilization scenarios [243]. For small energies and in the vicinity of stability border (point

*in Figure 10), DS is*A

*(Figure 12), which means its aperiodic disappearance with excitation of continuous waves and subsequent DS recreation [37,244–248]. With the energy growth (point*exploding

*in Figure 10), the*B

*DSs develop (Figure 13) [37,249–253]. Such a regime can be interpreted as DS structural chaotization, that is, generation of multiple DSs with strong interactions causing extreme dynamics.*rogue

For sufficiently large energies in the vicinity of stability border (point * C* in Figure 10), the typical destabilization scenario is the generation of multiple DSs (Figure 14). The source of this destabilization is the growth of spectral dissipation caused by DS spectral broadening with approaching to stability border so that the DS splitting becomes more energy advantageous [161]. Moreover, the DS splitting can be enhanced by its phase inhomogeneity because the gain (energy in-flow) is maximum at the points of stationary phase

The numerical simulations of cubic-quintic CNGLE with taking into account a quantum noise validated the fact of inconsistency of spectral condensation and absence of temporal thermolization that breaks the DS energy scalability (see previous section) [243]. As a result, the DS stability region breaks abruptly with energy growth (dashed curve in Figure 10) and multitude of turbulent scenarios of DSs evolution develops (Figure 15) [34,243].

Serious limitations on power and energy scalability of DSs in fibre lasers arise from stimulated Raman scattering (SRS) [2,109]. The stability border of DS under action of SRS is shown in Figure 10 by dot blue curve (DS is stable on the left of this curve) [260]. As was found, SRS enhances the tendency to multi-pulsing with energy growth caused by enhancement of spectral dissipation due to SRS [260]. Simultaneously, generation of anti-Stokes radiation causes chaotization of DS dynamics and irregular modulation of DS temporal and spectral profiles [261] (Figure 16). DS profile remains localized, but it is strongly cut by colliding dark and grey soliton-like structures [34].

As was shown, the DS dynamics can be regularized by formation of * dissipative Raman soliton* (DRS). DRS can exist in the form of DS which is Stokes-shifted due to self-Raman scattering (Figure 17,

*) [260] or as bound DS–DRS complex (Figure 17,*a

*) [262]. In the last case, stabilization is achieved by feedback, i.e. reinjection of Stokes signal through a delay line [263]. In the absence of a feedback, the Raman pulse is noisy [264] (see dash line spectrum in Figure 16,*b

*).*b

## 5. Vector DSs

As was pointed above, SOP can play leading role in a fibre laser dynamics. In particular, it can contribute to mode-locking or/and spectral filtering. However, diapason of polarization phenomena in a DS fibre laser spreads essentially broader. As was found, intrinsic fibre birefringence (Figure 2) can lead to DS splitting into two independent SOPs [78]. This phenomenon is used to realize the NPR mode-locking mechanism where a DS SOP evolves (or remains locked) as a whole during propagation [265-269]. The polarization dynamics can be fast (* T*) and vary from regular (with possible period multiplication or harmonic mode-locking) to chaotic one [270,271]. There are evidences of ultrafast SOP evolution when SOP changes across a DS profile [272].

The specific multiple pulse instability of vector dissipative solitons (VDSs) leads to generation of the bound states of DSs with different SOPs (* vector soliton molecules*) which are locked by a non-linear coupling [273,274] or by a group-velocity locking produced by spectrum shift between DSs with different SOP [275]. As was shown experimentally (Figure 18) [276], the dynamics of VDS molecules can be highly non-trivial and demonstrate both fast and slow periodic switching between fixed SOPs as well as SOP procession, which is especially interesting for fibre laser telecommunications based on polarization multiplexing.

The important breakthrough in the recent theory of VDSs is the demonstration of insufficiency of approaches based on the coupled CNGLEs (like (10)) for adequate description of DS polarization dynamics. It was demonstrated that an active medium polarizability contributes to DS dynamics substantially [277]. As was shown, the SOP-sensitive interaction between DS and a slowly relaxing active medium with taking into account the birefringence of fibre laser elements and light-induced anisotropy caused by elliptically polarised pump field change the SOP at a long time scale that results in fast and slowly evolving SOPs of VDSs (Figure 19).

The non-trivial contribution of active medium kinetics and polarizability with taking into account the pump SOP and SPM demonstrates a complex dynamics including spiral attractors and dynamic chaos (Figure 20) [278]. One may assume that such a non-trivial polarization dynamics is of great importance for DS energy scaling, in particular, due to vector nature of SRS [279]. These topics remain unexplored to date.

## 6. Conclusion

The recent progress in development of ultrafast fibre lasers and advances in exploring of DS are interrelated. DSs allowed scoring a great success in ultrashort pulse energy scalability that is defined by unprecedented stability and robustness of DS. At this moment, it is possible to achieve over-MW peak powers for sub-100 fs pulses directly from a fibre laser at over-MHz repetition rates. New spectral diapasons became reachable owing to development of mid-IR active fibres and using the frequency-conversion directly in a laser. Development of new mode-locking techniques, especially based on using of SESAMs, graphene and another quantum-sized structure allowed improving a laser stability, integrity and environment insensitivity. A great advance has been achieved in the theory of DSs. New powerful analytical techniques based on extensive numerical simulations and experimental advances extended understanding of the DS fundamental properties and revealed new prospects in improvement of characteristics of ultrafast fibre lasers. Based on achieved results, one may outline some unresolved problems. As was found, there are stability limits for a DS energy scaling imposed by optical turbulence and SRS. Deeper insight into the nature of these phenomena could allow to overcome these limits without substantial complication of laser setup. Simultaneously, control of intra-laser spectral conversion is a direct way to broadening of spectral range. Then, the dynamics and properties of VDSs remain scantily explored. Recent studies demonstrated a multitude of polarization phenomena, which cannot be grasped in frameworks of existing models. In particular, polarizability and kinetics of an active fibre in combination with birefringence of a laser in a whole can contribute non-trivially to a laser dynamics. As an additional aspect of further development, one may point at the development of new mode-locking techniques, which could improve DS stability and integrity of a fibre laser, decrease pulse width and extend a diapason of pulse repetition rates. At last, one has to remember that a fibre laser is an ideal playground for study of complex non-linear phenomena and, undoubtedly, new bridges between different fields of science will be built with a further progress of ultrafast fibre lasers.

## Acknowledgments

This work was supported by FP7-PEOPLE-2012-IAPP (project GRIFFON, No. 324391).

## References

- 1.
Fermann ME, Hartl I. Ultrafast fiber laser technology. IEEE J Sel Topics in Quantum Electron 2009;15(1):191–206. DOI: 10.1109/JSTQE.2008.2010246. - 2.
Fermann ME, Galvanauskas A, Sucha G, Harter D. Fiber-lasers for ultrafast optics. Appl Physics B 1997;65:259–75. - 3.
Fermann ME, Hartl I. Ultrafast fiber lasers. Nature Photonics 2013;7:868–74. DOI: 10.1038/NPHOTON.2013.280. - 4.
Limpert J, Röser F, Schreiber Th, Tünnermann A. High-power ultrafast fiber laser systems. IEEE J Sel Top Quantum Electron 2006;12(2):233–44. DOI: 10.1109/JSTQE.2006.872729. - 5.
Lamb WE. Theory of an optical laser. Phys Rev 1964;134:A1429–50. - 6.
Siegman AE. Lasers. Sausalito: University Science Book; 1986. 1283 p. - 7.
Lefranҫois S, Kieu K, Deng Y, Kafka JD, Wise FW. Scaling of dissipative soliton fiber lasers to megawatt peak powers by use of large-area photonic crystal fiber. Opt Lett 2010;35(10):1569–71. - 8.
Baumgartl M, Lecaplain C, Hideur A, Limpert J, Tünnermann C. 66 W average power from a microjoule-class sub-100 fs fiber oscillator. Opt Lett 2012;37(10):1640–2. DOI: 10.1364/OL.37.001640. - 9.
Chong A, Renninger WH, Wise FW. Route to the minimum pulse duration in normal-dispersion fiber lasers. Opt Lett 2008;33(22):2638–40. DOI: 10.1364/OL.33.002638. - 10.
Sucha G. Overview of industrial and medical applications of ultrashort pulse lasers. In: Fermann ME, Galvanauskas A, Sucha G. (Eds.) Ultrafast Lasers: Technology and Applications. New York: Marcel Dekker, Inc.; 2003. pp. 323–358. - 11.
Xu C, Wise FW. Recent advances in fibre lasers for nonlinear microscopy. Nature Photonics 2013;7:875–82. DOI: 10.1038/NPHOTON.2013.284. - 12.
Müller M, Squier J. Nonlinear microscopy with ultrashort pulse lasers. In: Fermann ME, Galvanauskas A, Sucha G. (Eds.) Ultrafast Lasers: Technology and Applications. New York: Marcel Dekker, Inc.; 2003, pp. 661–97. - 13.
Kalashnikov VL, Sorokin E. Soliton absorption spectroscopy. Phys Rev A 2010;81:033840. DOI: 10.1103/PhysRevA.81.033840. - 14.
Kalashnikov VL, Sorokin E, Sorokina IT. Chirped dissipative soliton absorption spectroscopy. Opt Express 2011;19(18):17480–92. - 15.
Kurtz RM, Sarayba MA, Juhasz T. Ultrafast lasers in ophthalmology. In: Fermann ME, Galvanauskas A, Sucha G. (Eds.) Ultrafast Lasers: Technology and Applications. New York: Marcel Dekker, Inc.; 2003, pp. 745–65. - 16.
Clowes J. Next generation light sources for biomedical applications. Optic Photonic 2011;3(1):36–8. - 17.
Fujimoto JG, Brezinski M, Drexler W, Hartl I, Kärtner F, Li X, Morgner U. Optical coherence tomography. In: Fermann ME, Galvanauskas A, Sucha G. (Eds.) Ultrafast Lasers: Technology and Applications. New York: Marcel Dekker, Inc.; 2003, pp. 699–743. - 18.
Drexler W, Fujimoto JG. (Eds.) Optical Coherence Tomography. Berlin: Springer-Verlag; 2008. 1346 p. - 19.
Lanin AA, Fedotov IV, Sidorov-Biryukov DA, Doronina-Amitonova LV, Ivashkina OI, Zots MA, Sun C-K, Ilday FÖ, Fedotov AB, Anokhin KV, Zheltikov AM. Air-guided photonic-crystal-fiber pulse-compression delivery of multimegawatt femtosecond laser output for nonlinear-optical imaging and neurosurgery. Appl Phys Lett 2012;100:101104. DOI: 10.1063/1.3681777. - 20.
Osellame R, Cerullo G, Ramponi R. (Eds.) Femtosecond Laser Micromachining. Heidelberg: Springer; 2012. 483 p. - 21.
Gattass RR, Mazur E. Femtosecond laser micromachining in transparent materials. Nat Photon. 2008;2:219–25. DOI: 10.1038/nphoton.2008.47. - 22.
Nakazawa M. Ultrahigh bit rate communication system. In: Fermann ME, Galvanauskas A, Sucha G. (Eds.) Ultrafast Lasers: Technology and Applications. New York: Marcel Dekker, Inc.; 2003. pp. 611–660. - 23.
Udem Th, Holzwarth R, Hänsch TW. Optical frequency metrology. Nature 2002;416:233–7. DOI: 10.1038/416233a. - 24.
Liu Y, Tschuch S, Rudenko A, Dürr M, Siegel M, Morgner U, Moshammer R, Ullrich J. Strong-field double ionization of Ar below the recollision threshold. Phzs Rev Lett 2008;101:053001. DOI: 10.1103/PhysRevLett.101.053001. - 25.
Sciaini G, Miller RJD. Femtosecond electron diffraction: heralding the era of atomically resolved dynamics. Rep Prog Phys 2011;74:096101. DOI: 10.1088/0034-4885/74/9/096101. - 26.
Südmeyer T, Marchese SV, Hashimoto S, Baer CRE, Gingras G, Witzel B, Keller U. Femtosecond laser oscillators for high-field science. Nat Photon 2008;2:599–604. DOI: 10.1038/nphoton.2008.194. - 27.
Krausz F, Ivanov M. Attosecond physics. Rev Mod Phys 2009;81(1):163–234. DOI: 10.1103/RevModPhys.81.163 - 28.
Pfeifer T, Spielmann C, Gerber G. Femtosecond x-ray science. Rep Prog Phys 2006;69(2):443–505. DOI: 10.1088/0034-4885/69/2/R04. - 29.
Mourou GA, Tajima T, Bulanov SV. Optics in the relativistic regime. Rev Mod Phys 2006;78(2):309–71. DOI: 10.1103/RevModPhys.78.309. - 30.
Martinez A, Sun Z. Nanotube and graphene saturable absorbers for fibre lasers. Nat Photon 2013;7:842–5. - 31.
Lecaplain C, Grelu Ph, Soto-Crespo JM, Akhmediev N. Dissipative rogue waves generated by chaotic bunching in a mode-locked laser. Phys Rev Lett 2012;108:233901. - 32.
Dudley JM, Dias F, Erkintalo M, Genty G. Instabilities, breathers and rogue waves in optics. Nat Photon 2014;8:755–64. - 33.
Dudley JM, Finot Ch, Richardson DJ, Millot G. Self-similarity in ultrafast nonlinear optics. Nat Phys 2007;3:597–603. - 34.
Turitsyna EG, Smirnov SV, Sugavanam S, Tarasov N, Shu X, Babin SA, Podivilov EV, Churkin DV, Falkovich G, Turitsyn SK. The laminar–turbulent transition in a fibre laser. Nat Photon 2013;7:783–6. DOI: 10.1038/NPHOTON.2013.246. - 35.
Akhmediev NN, Ankiewicz A. Solitons: Nonlinear Pulses and Beams. London: Chapman & Hall; 1997. - 36.
Akhmediev NN, Ankiewicz A. (Eds.) Dissipative Solitons. Berlin: Springer-Verlag; 2005. - 37.
Grelu Ph, Akhmediev N. Dissipative solitons for mode-locked lasers. Nat Photon 2012;6:84–92. DOI: 10.1038/nphoton.2011.345 - 38.
Faccio D, Belgiorno F, Cacciatori S, Gorini V, Liberati S, Moschella U. (Eds.) Analogue Gravity Phenomenology. Analogue Spacetimes and Horizons, from Theory to Experiment. Heidelberg: Springer; 2013. p. 438. DOI: 10.1007/978-3-319-00266-8. - 39.
Kevrekidis PG, Frantzeskakis DJ, Carretero-González R. (Eds.) Emergent Nonlinear Phenomena in Bose-Einstein Condensates. Berlin: Springer-Verlag; 2008. - 40.
Yang Y. Solitons in Field Theory and Nonlinear Analysis. New York: Springer; 2001. - 41.
Abdulaev FK, Konotop VV. (Eds.) Nonlinear Waves: Classical and Quantum Aspects. Dordrecht: Kluwer Academic Pub.; 2004. - 42.
Akhmediev NN, Ankiewicz A. (Eds.) Dissipative Solitons: From Optics to Biology and Medicine. Berlin: Springer-Verlag; 2008. - 43.
Naruse M, (Ed.) Nanophotonic Information Physics. Berlin: Springer-Verlag; 2014. - 44.
Kalashnikov VL. Chirped-pulse oscillators: route to the energy-scalable femtosecond pulses. In: Al-Khursan A. (Ed.) Solid State Laser. InTech, 2008; pp. 145–184. DOI: 10.5772/37415. - 45.
Baer CRE, Heckl OH, Saraceno CJ, Schriber C, Kränkel C, Südmeyer T, Keller U. Frontiers in passively mode-locked high-power thin-disk laser oscillators. Opt Express 2012;20:7054–65. DOI: 10.1364/OE.20.007054. - 46.
Saraceno CJ, Emaury F, Heckl OH, Baer CRE, Hoffmann M, Schriber C, Golling M, Südmeyer Th, Keller U. 275 W average output power from a femtosecond thin disk oscillator operated in a vacuum environment. Opt Express 2012;20:23535–41. DOI: 10.1364/OE.20.023535. - 47.
Zhang J, Brons J, Lilienfein N, Fedulova E, Pervak V, Bauer D, Sutter D, Wei Zh, Apolonski A, Pronin O, Krausz F. 260-megahertz, megawatt-level thin-disk oscillator. Opt Lett 2015;40:1627–30. DOI: 10.1364/OL.40.001627. - 48.
Brons J, Pervak V, Fedulova E, Bauer D, Sutter D, Kalashnikov V, Apolonskiy A, Pronin O, Krausz F. Energy scaling of Kerr-lens mode-locked thin-disk oscillators. Opt Lett 2014;39:6442–5. DOI: 10.1364/OL.39.006442. - 49.
Naumov S, Fernandez A, Graf R, Dombi P, Krausz F, Apolonski A. Approaching the microjoule frontier with femtosecond laser oscillators. New J Phys 2005;7:216. DOI: 10.1088/1367-2630/7/1/216. - 50.
Fedulova E, Fritsch K, Brons J, Pronin O, Amotchkina T, Trubetskov M, Krausz F, Pervak V. Highly-dispersive mirrors reach new levels of dispersion. Opt Express 2015;23:13788–93. DOI: 10.1364/OE.23.013788. - 51.
Richardson DJ, Nilsson J, Clarkson WA. High power fiber lasers: current status and future prospects. J Opt Soc Am B 2010;27:B63–B92. - 52.
Sraceno CJ, Heckl OH, Baer CRE, Schriber C, Golling M, Beil K, Kränkel ST, Huber G, Keller U. Sub-100 femtosecond pulses from a SESAM modelocked thin disk laser. Appl Phys B 2012;106:559–62. DOI: 10.1007/s00340-012-4900-5. - 53.
Zhang J, Brons J, Seidel M, Pervak V, Kalashnikov V, Wei Z, Apolonski A, Krausz F, Pronin O. 49-fs Yb:YAG thin-disk oscillator with distributed Kerr-lens mode-locking. In: CLEO/Europe-EQEC Scientific Programme; 21–25 June; Munich, Germany. 2015. p. 166 (PD-A.1 WED). - 54.
Sorokin E, Kalashnikov VL, Naumov S, Teipel J, Warken F, Giessen H, Sorokina IT. Intra- and extra-cavity spectral broadening and continuum generation at 1.5 \mu m using compact low energy femtosecond Cr:YAG laser. Applied Phys B 2003;77(2–3):197–204. - 55.
Joly NY, Nold J, Chang W, Hölzer P, Nazarkin A, Wong GKL, Biancalana F, Russel P. St J. Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber. Phys Rev Lett 2011;106:203901. - 56.
Jackson SD. Towards high-power mid-infrared emission from a fiber laser. Nat Photon 2012;28:423–31. DOI: 10.1038/NPHOTON.2012.149. - 57.
Acerbron JA, Bonilla LL, Vicente CJP, Ritort F, Spigler R. The Kuramoto model: a simple paradigm for synchronization phenomena. Rev Mod Phys 2005;77:137–85. - 58.
Pikovsky A, Rosenblum M, Kurths J. Synchronization: a universal concept in nonlinear sciences. Cambridge: Cambridge University Press; 2001. - 59.
Kuizenga DJ, Siegman AE. FM and AM mode locking of the homogeneous laser - Part I: Theory. IEEE J Quantum Electron 1970;6(11):694–708. DOI: 10.1109/JQE.1970.1076343. - 60.
Haus HA. Mode-locking of lasers. IEEE J Sel Topic Quantum Electron 2000;6(6):1173–85. - 61.
Haus HA. Short pulse generation. In: Duling IN, III (Ed.) Compact Sources of Ultrashort Pulses. Cambridge: Cambridge University Press; 1995. pp. 1–56. - 62.
Kalashnikov VL. Mathematical Ultrashort-Pulse Laser Physics [Internet]. 15/09/2000 [Updated: 29/03/2002]. Available from: http://lanl.arxiv.org/abs/physics/0009056v3. - 63.
Kuizenga DI, Siegman AE. Modulator frequency detuning effects in the FM mode-locked laser. IEEE J Quantum Electron 1970;QE-6:803–8. - 64.
Akhmanov SA, Vysloukh VA, Chirkin AS. Optics of Femtosecond Laser Pulses. New York: AIP; 1992. - 65.
Haus HA, Silberberg Y. Laser mode locking with addition of nonlinear index. IEEE J Quantum Electron 1986;QE-22(2):325–31. - 66.
Kalashnikov VL, Poloyko IG, Mikhaylov VP. Generation of ultrashort pulses in lasers with external frequency modulation. Quantum Electron 1998;28(3):264–8. - 67.
Kolner BH. Space-time duality and the theory of temporal imaging. IEEE J Quantum Electron 1994;QE-30(8):1951–63. - 68.
van Howe J, Xu Ch. Ultrafast optical signal processing based upon space-time dualities. IEEE J Lightwave Technology 2006;24(7):2649–62. DOI: 10.1109/JLT.2006.875229. - 69.
Salem R, Foster MA, Gaeta AL. Application of space-time duality to ultrahigh-speed optical signal processing. Adv Optics Photon 2013;5:274–317. DOI: 10.1364/AOP.5.000274. - 70.
Agrawal GP. Nonlinear Fiber Optics. Third Edition ed. San Diego: AP; 2001. 466 p. - 71.
Newell AC. Solitons in Mathematics and Physics. Philadelphia: SIAM; 1985. 246 p. - 72.
Lai Y, Haus HA. Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation. Phys Rev A 1989;40:844–53. - 73.
Lai Y, Haus HA. Quantum theory of solitons in optical fibers. II. Exact solution. Phys Rev A 1989;40:854–66. - 74.
Yoon B, Negele JW. Time-dependent approximation for a one-dimensional system of bosons with attractive δ-function interactions. Phys Rev A 1977;16:1451. - 75.
Haus HA. Theory of mode-locking with a fast saturable absorber. J Appl Phys 1975; 46:3049–58. - 76.
Ablowitz MJ, Clarkson PA. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press; 1991. 516 p. - 77.
Turitsyn SK, Bale BG, Fedoruk MP. Dispersion-managed solitons in fibre systems and lasers. Phys Rep 2012;521:135–203. DOI: 10.1016/j.physrep.2012.09.004. - 78.
Menyuk CR. Nonlinear pulse propagation in birefringent optical fibers. IEEE J Quantum Electron 1987;23(2):174–6. - 79.
Menyuk GR. Pulse propagation in an elliptically birefringent Kerr medium. IEEE J Quantum Electron 1989;25(12):2674–82. - 80.
Hasegawa A. Optical Solitons in Fibers. Berlin: Springer-Verlag; 1990. 79 p. - 81.
Ding E, Renninger WH, Wise FW, Grelu Ph, Shlizerman E, Kutz JN. High-energy passive mode-locking of fiber lasers. Int J Optics 2012;2012(ID354156):1–17. DOI: 10.1155/2012/354156. - 82.
Wang R, Dai Y, Yan L, Wu J, Xu K, Li Y, Lin J. Dissipative soliton in actively mode-locked fiber laser. Optics Express 2012;20(6):6406–11. - 83.
Koliada NA, Nyushkov BN, Ivanenko AV, Kobtsev SM, Harper P, Turitsyn SK, Denisov VI, Pivtsov VS. Generation of dissipative solitons in an actively mode-locked ultralong fibre laser. Quantum Electron 2013;43(2):95–8. DOI: 10.1070/QE2013v043n02ABEH015041. - 84.
Wang R, Dai Y, Yin F, Xu K, Li J, Lin J. Linear dissipative soliton in an anomalous-dispersion fiber laser. Optics Express 2014;22(24):29314–20. DOI: 10.1364/OE.22.029314. - 85.
Haus HA, Fujimoto JG, Ippen EP. Analytic theory of additive pulse and Kerr lens mode locking. IEEE J. Quantum Electron 1992;28(10):2086–96. - 86.
Fermann ME, Andrejco MJ, Silberberg Y, Stock ML. Passive mode locking by using nonlinear polarization evolution in a polarization-maintaining erbium-doped fiber. Optics Lett 1993;18(11):894–6. DOI: 10.1364/OL.18.000894. - 87.
Hofer M, Ober MH, Haberl F, Fermann ME. Characterization of ultrashort pulse formation in passively mode-locked fiber lasers. IEEE J Quantum Electron 1992;28(3)DOI:720–8. - 88.
Fermann ME. Nonlinear polarization evolution in passively modelocked fiber lasers. In: Duling IN, III (Ed.) Compact sources of ultrashort pulses. Cambridge: Cambridge University Press; 1995. pp. 179–207. - 89.
Winful HG. Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices. Optics Lett 1986;11(1):33–5. - 90.
Kalashnikov VL, Kalosha VP, Mikhailov VP. Self-mode locking of continuous-wave solid-state lasers with a nonlinear Kerr polarization modulator. J Opt Soc Am B 1993;10:1443–6. - 91.
Ding E, Kutz JN. Operating regimes, split-step modeling, and the Haus master mode-locking model. J Opt Soc Am B 2009;26(12):2290–300. - 92.
Ding E, Shlizerman E, Kutz JN. Generalized master equation for high-energy passive mode-locking: the sinusoidal Ginzburg-Landau equation. IEEE J Quantum Electron 2011;47(5):705–14. - 93.
Komarov A, Leblond H, Sanchez F. Multistability and hysteresis phenomena in passively mode-locked fiber lasers. Phys Rev A 2005;71(5):053809. - 94.
Komarov A, Leblond H, Sanchez F. Quantic complex Ginzburg-Landau model for ring fiber lasers. Phys Rev E 2005;72(2):025604. - 95.
Matsas VJ, Newson TP, Richardson DJ, Payne DN. Selfstarting passively mode-locked fibre ring soliton laser exploiting nonlinear polarization rotation. Electron Lett 1992;28(15):1391–3. - 96.
Tamura K, Ippen EP, Haus HA, Nelson IE. 77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser. Optics Lett 1993;18(13):1080–2. - 97.
Chong A, Buckley J, Renninger W, Wise F. All-normal-dispersion femtosecond fiber laser. Optics Express 2006;14(21):10096–100. - 98.
Zhao LM, Tang DY, Wu J. Gain-guided soliton in a positive group-dispersion fiber laser. Optics Lett 2006;31(12):1788–90. - 99.
Cabasse A, Ortac B, Martel G, Hideur A, Limpert J. Dissipative solitons in a passively mode-locked Er-doped fiber with strong normal dispersion. Optics Express 2008;16(23):19323–9. - 100.
Kieu K, Renninger WH, Chong A, Wise FW. Sub-100 fs pulses at watt-level powers from a dissipative-soliton fiber laser. Optics Lett 2009;34(5):593–5. - 101.
Ruehl A, Wandt D, Morgner U, Kracht D. Normal dispersive ultrafast fiber oscillators. IEEE J Sel Topic Quantum Electron 2009;15(1):170–81. - 102.
Wu X, Tang DY, Zhang H, Zhao LM. Dissipative soliton resonance in an all-normal-dispersion erbium-doped fiber laser. Optics Express 2009;17(7):5580–4. - 103.
Renninger WH, Chong A, Wise F. Self-similar evolution in an all-normal-dispersion laser. Phys Rev A 2010;82:021805(R). - 104.
Zhao L, Tang D, Wu X, Zhang H. Dissipative soliton generation in Yb-fiber laser with an invisible intracavity bandpass filter. Optics Lett 2010;35(16):2756–8. - 105.
Chichkov NB, Hausmann K, Wandt D, Morgner U, Neumann J, Kracht D. High-power dissipative solitons from an all-normal dispersion erbium fiber oscillator. Optics Lett 2010;35(16):2807–9. - 106.
Baumgartl M, Ortac B, Lecaplain C, Hideur A, Limpert J, Tuenermann A. Sub-80 fs dissipative soliton large-mode-area fiber laser. Optics Lett 2010;35(13):2311–3. - 107.
Lecaplain C, Baumgartl M, Schreiber T, Hideur A. On the mode-locking mechanism of a dissipative-soliton fiber laser. Optics Express 2011;19(27):26742–51. - 108.
Zhang Z, Dai G. All-normal-dispersion dissipative soliton Ytterbium fiber laser without dispersion compensation and additional filter. IEEE Photon J 2011;3(6):1023–9. DOI: 10.1109/JPHOT.2011.2170057. - 109.
Kharenko DS, Podivilov EV, Apolonski AA, Babin SA. 20 nJ 200 fs all-fiber-highly chirped dissipative soliton oscillator. Optics Lett 2012;37(19):4104–6. - 110.
Li X, Wang Y, Zhao W, Liu X, Wang Y, Tsang YH, Zhang W, Hu X, Yang Zh, Gao C, Li Ch, Shen D. All-fiber dissipative solitons evolution in a compact passively Yb-doped mode-locked fiber laser. J Lightwave Tech 2012;30(15):2502–7. - 111.
Duan L, Liu X, Mao D, Wang L, Wang G. Experimental observation of dissipative soliton resonance in an anomalous-dispersion fiber laser. Optics Express 2012;20(1):265–70. - 112.
Doran NJ, Wood D. Nonlinear-optical loop mirror. Optics Lett 1988;13(1):56–8. - 113.
Ippen EP, Haus HA, Liu LY. Additive pulse modelocking. J Opt Soc Am B 1989;6:1736–45. - 114.
Duling IN III, Dennis ML. Modelocking of all-fiiber lasers. In: Duling IN III, (Eds.) Compact Sources of Ultrashort Pulses. Cambridge: Cambridge University Press; 1995. pp. 140–178. - 115.
Mark J, Liu LY, Hall KL, Haus HA, Ippen EP. Femtosecond pulse generation in a laser with a nonlinear external resonator. Optics Lett 1989;14(1):48–50. - 116.
Kalashnikov VL, Kalosha VP, Mikhailov VP, Poloyko IG. Multi-frequency continuous wave solid-state laser. Optics Commun 1995;116(4–6):383–8. - 117.
Kalashnikov VL, Kalosha VP, Mikhailov VP, Poloyko IG, Demchuk MI. Efficient self-mode locking of continuous-wave solid-state lasers with resonant nonlinearity in an additional cavity. Optics Commun 1994;109:119–25. - 118.
Kalashnikov VL, Kalosha VP, Mikhailov VP, Poloyko IG, Demchuk MI. Self-mode-locking of cw solid-state lasers with a nonlinear antiresonant ring. Quantum Electron 1994;24(1):35–9. - 119.
Richardson DJ, Laming RI, Payne DN, Matsas V, Phillips MW. Selfstarting, passively modelocked Erbium fibre laser based on amplifying Sagnac switch. Electronics Lett 1991;27(6):542–3. - 120.
Nicholson JW, Andrejco M. A polarization maintaining, dispersion managed, femtosecond figure-eight fiber laser. Optics Express 2006;14(18):8160–7. - 121.
Yun L, Liu X, Mao D. Observation of dual-wavelength dissipative solitons in a figure-eight erbium-doped fiber laser. Optics Express 2012;20(19):20992–7. - 122.
Wang S-K, Ning Q-Y, Luo A-P, Lin A-B, Luo Z-C, Xu W-C. Dissipative soliton resonance in a passively mode-locked figure-eight fiber laser. Optics Express 2013;21(2):2402–7. - 123.
Zhao LM, Bartnik AC, Tai QQ, Wise FW. Generation of 8 nJ pulses from a dissipative-soliton fiber laser with a nonlinear optical loop mirror. Optics Lett 2013;38(11):1942–4. - 124.
Lin H, Guo Ch, Ruan Sh, Yang J. Dissipative soliton resonance in an all-normal-dispersion Yb-doped figure-eight fibre laser with tunable output. Laser Phys Lett 2014;11:085102. - 125.
Xu Y, Song Y, Du G, Yan P, Guo Ch, Zheng G, Ruan Sh. Dissipative soliton resonance in a wavelength-tunable Thulium-doped fiber laser with net-normal dispersion. IEEE Photonics J 2015;7(3):1502007. DOI: 10.1109/JPHOT.2015.2424855. - 126.
Keller U. Recent developments in compact ultrafast lasers. Nature 2003;424(14):831–8. - 127.
Keller U. Semiconductor Nonlinearities for Solid-State Laser Modelocking and Q-switching. In: Garmire E, Kost A. (Eds.) Nonlinear Optics in Semiconductors II (Semiconductors and Semimetals, Vol. 59). San Diego: AP; 1999. pp. 211–86. - 128.
Keller U, Weingarten KJ, Kaertner FX, Kopf D, Braun B, Jung ID, Fluck R, Hoenninger C, Matuschek N, Aus der Au J. Semiconductor saturable absorber mirrors (SESAM's) for femtosecond to nanosecond pulse generation in solid-state lasers. IEEE J Sel Top Quantum Electron 1996;2(3):435–53. - 129.
Kaertner F. Lecture Notes: Introduction to Ultrafast Optics, Chapter 8 [Internet]. 2005. Available from: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-977-ultrafast-optics-spring-2005/lecture-notes/chapter8.pdf. - 130.
Chong A, Renninger WH, Wise FW. Environmentally stable all-normal-dispersion femtosecond fiber laser. Optics Lett 2008;33(10):1071–3. DOI: 10.1364/OL.33.001071. - 131.
Cabasse A, Martel G, Oudar JL. High power dissipative soliton in an Erbium-doped fiber laser mode-locked with a high modulation depth saturable absorber mirror. Optics Express 2009;17(12):9537–42. - 132.
Tang M, Wang H, Becheker R, Oudar J-L, Gaponov D, Godin T, Hideur A. High-energy dissipative solitons generation from a large normal dispersion Er-fiber laser. Optics Lett 2015;40(7):1414–7. - 133.
Gumenyuk R, Vartianen I, Tuovinen H, Okhotnikov OG. Dissipative dispersion-managed soliton 2mm thulium/holmium fiber laser. Optics Lett 2011;36(5):609–11. - 134.
Lecourt J-B, Duterte C, Narbonneau F, Kinet D, Hernandez Y, Giannone D. All-normal dispersion, all-fibered PM laser mode-locked by SESAM. Optics Express 2012;20(11):11918–23. DOI: 10.1364/OE.20.011918. - 135.
Jiang K, Ouyang C, Wu K, Wong JH. High-energy dissipative soliton with MHz repetition rate from an all-fiber passively mode-locked laser. Optics Commun 2012;285(9):2422–5. DOI: 10.1016/j.optcom.2012.01.033. - 136.
Poloyko IG, Kalashnikov VL. Semiconductor saturable absorber mirrors as mode-locking device for femtosecond lasers: nonlinear Fabri-Perot resonator approach. Optics Commun 1999;168:167–75. - 137.
Sun Z, Hasan T, Ferrari AC. Ultrafast lasers mode-locked by nanotubes and graphene. Physica E 2012;44:1082–91. DOI: 10.1016/j.physe.2012.01.012. - 138.
Zhang H, Tang D, Knize RJ, Zhao L, Bao Q, Loh KP. Graphene mode locked, wavelength tunable, dissipative soliton fiber laser. Appl Phys Lett 2010;96:111112. - 139.
Zhao LM, Tang DY, Zhang H, Wu X, Bao Q, Loh KP. Dissipative soliton operation of an Ytterbium-doped fiber laser mode locked with atomic multilayer graphene. Optics Lett 2010;35(21):3622–4. - 140.
Cui YD, Liu XM, Zeng C. Conventional and dissipative solitons in a CFBG-based fiber laser mode-locked with a graphene-nanotube mixture. Laser Phys Lett 2014;11:055106. - 141.
Cheng Zh, Li H, Shi H, Ren J, Yang Q-H, Wang P. Dissipative soliton resonance and reverse saturable absorption in graphene oxide mode-locked all-normal-dispersion Yb-doped fiber laser. Optics Express 2015;23(6):7000–6. DOI: 10.1364/OE.23.007000. - 142.
Liu X, Cui Y, Han D, Yao X, Sun Zh. Distributed ultrafast laser. Sci Rep2014;5:9101. DOI: 10.1038/srep9101. - 143.
Im JH, Choi SY, Rotermund F, Yeom D-I. All-fiber Er-doped dissipative soliton laser based on evanescent field interaction with carbon nanotube saturable absorber. Optics Express 2010;18(21):22141–6. - 144.
Du J, Wang Q, Jiang G, Xu Ch, Zhao Ch, Xiang Y, Chen Y, Wen Sh, Zhang H. Ytterbium-doped fiber passively mode locked by few-layer molybdenum disulfide (MoS _{2}) saturable absorber functioned with enhanced field interaction. Sci Rep 2014;4:6346. DOI: 10.1038/srep06346. - 145.
Paschotta R, Keller U. Passive mode locking with slow saturable absorbers. Appl Phys B 2001;73:653–62. DOI: 10.1007/s003400100726. - 146.
Haus HA, Silberberg Y. Theory of mode locking of a laser diode with a multiple-quantum-well structure. J Opt Soc Am B 1985;2(7):1237–43. - 147.
Kaertner F. Lecture Notes: Introduction to Ultrafast Optics. Chapter 7 [Internet]. 2005. Available from: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-977-ultrafast-optics-spring-2005/lecture-notes/chapter7.pdf. - 148.
Lam C-K, Malomed BA, Chow KW, Wai PKA. Spatial solitons supported by localized gain in nonlinear optical waveguides. Eur Phys J Special Topics 2009;173:233–43. DOI: 10.1140/epjst/e2009-01076-8. - 149.
Sakaguchi H, Malomed BA. Stable two-dimensional solitons supported by radially inhomogeneous self-focusing nonlinearity. Optics Lett 2012;37(6):1035–7. DOI: 10.1364/OL.37.001035. - 150.
Borovkova OV, Kartashov YV, Vysloukh VA, Lobanov VE, Malomed BA, Torner L. Solitons supported by spatially inhomogeneous nonlinear losses. Optics Express 2012;20(3):2657–67. DOI: 10.1364/OE.20.002657. - 151.
Kartashov YV, Konotop VV, Vysloukh VA. Two-dimensional dissipative solitons supported by localized gain. Optics Lett 2011;36(1):82–4. DOI: 10.1364/OL.36.000082. - 152.
Kalosha VP, Chen L, Bao X. Feasibility of Kerr-lens mode locking in fiber lasers. In: Vallée R, Piché M, Mascher P, Cheben P, Côté D, LaRochelle S, Schriemer HP, Albert J, Ozaki T. (Eds.) Proc SPIE 7099, Photonics North 2008; 2–4 June 2008; Montréal, Canada. Bellingham: SPIE; 2008. p. 70990S. DOI: 10.1117/12.807415. - 153.
Kalashnikov V, Apolonski A. Simulation of a Kerr fiber laser. In: Proc. Meeting on Russian Fiber Lasers; 27–30 March; Novosibirsk, Russia. 2012. pp. 113–114. - 154.
Shaw JK. Mathematical Principles of Optical Fiber communications. Philadelphia: SIAM; 2004. p. 93. - 155.
Kaup DJ. Exact quantization of the nonlinear Schroedinger equation. J Math Phys 1975;16:2036–41. - 156.
Thacker HB, Wilkinson D. Inverse scattering transform as an operator method in quantum field theory. Phys Rev D 1979;19:3660–5. - 157.
Akhmediev NN, Ankiewicz A. Solitons around us: integrable, Hamiltonian and dissipative systems. In: Porsezian K, Kuriakose VC. (Eds.) Optical Solitons: Theoretical and Experimental Challanges. Berlin: Springer-Verlag; 2002. pp. 105–126. - 158.
Kuszelewicz R, Barbay S, Tissoni G, Almuneau G. Editorial on dissipative optical solitons. Eur Phys JD 2010;59:1–2. DOI: 10.1140/epjd/e2010-00167-7. - 159.
Martinez OE, Fork RL, Gordon JP. Theory of passively mode-locked lasers for the case of a nonlinear complex-propagation coefficient. J Opt Soc Am B 1985;2(5):753–60. DOI: 10.1364/JOSAB.2.000753. - 160.
Haus HA, Fujimoto JG, Ippen EP. Structures for additive pulse mode locking. J Opt Soc Am B 1991;8(10):2068–76. - 161.
Kalashnikov VL, Sorokin E, Sorokina IT. Multipulse operation and limits of the Kerr-lens mode locking stability. IEEE J Quantum Electron 2003;39(2):323–36. - 162.
Proctor B, Westwig E, Wise F. Characterization of a Kerr-lens mode-locked Ti:sapphire laser with positive group-velocity dispersion. Optics Lett 1993;18(19):1654–6. DOI: 10.1364/OL.18.001654. - 163.
Chen S, Liu Y, Mysyrowicz A. Unusual stability of one-parameter family of dissipative solitons due to spectral filtering and nonlinearity saturation. Phys Rev A 2010;81:061806(R). - 164.
Kalashnikov VL. Chirped-pulse oscillators: route to the energy-scalable femtosecond pulses. In: Al-Khursan AH. (Ed.) Solid State Laser. InTech; 2012. pp. 145–184. DOI: 10.5772/37415. - 165.
Kalashnikov VL, Apolonski A. Chirped-pulse oscillators: a unified standpoint. Phys Rev A 2009;79:043829. - 166.
Akhmediev N, Soto-Crespo JM, Grelu Ph. Roadmap to ultra-short record high-energy pulses out of laser oscillators. Phys Lett A 2008;372:3124–8. DOI: 10.1016/j.physleta.2008.01.027. - 167.
Kalashnikov VL. Chirped dissipative solitons. In: Babichev LF, Kuvshinov VI. (Eds.) Nonlinear Dynamics and Applications. Minsk: 2010. pp. 58–67. - 168.
Brons J, Pervak V, Fedulova E, Bauer D, Sutter D, Kalashnikov V, Apolonskiy A, Pronin O, Krausz F. Energy scaling of Kerr-lens mode-locked thin-disk oscillators. Optics Lett 2014;39(22):6442–5. DOI: 10.1364/OL.39.006442. - 169.
Hu M-L, Wang Ch-L, Tian Zh, Xing Q-R, Chai L, Wang Ch-Y. Environmentally stable, high pulse energy Yb-doped large-mode-area photonic crystal fiber laser operating in the soliton-like regime. IEEE Photon Technol Lett 2008;20(13):1088–90. DOI: 10.1109/LPT.2008.924300. - 170.
Ramachandran S, Fini JM, Mermelstein M, Nicholson JW, Ghalmi S, Yan MF. Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers. Laser Photon Rev 2008;2:429–48. DOI: 10.1002/lpor.200810016. - 171.
Vukovic N, Healy N, Peacock AC. Guiding properties of large mode area silicon microstructured fibers: a route to effective single mode operation. J Opt Soc Am B 2011;28(6):1529–33. DOI: 10.1364/JOSAB.28.001529. - 172.
Jansen F, Stutzki F, Otto H-J, Eidam T, Liem A, Jauregui C, Limpert J, Tünnermann A. Thermally induced waveguide changes in active fibers. Optics Express 2012;20(4):3997–4008. DOI: 10.1364/OE.20.003997. - 173.
Kalashnikov VL, Sorokin E. Dissipative Raman soliton. Optics Express 2014;22(24):30118–26. DOI: 10.1364/OE.22.030118. - 174.
Chernykh AI, Turitsyn SK. Soliton and collapse regimes of pulse generation in passively mode-locking laser systems. Optics Lett 1995;20(4):398–400. DOI: 10.1364/OL.20.000398. - 175.
Chang W, Ankiewicz A, Soto-Crespo JM, Akhmediev N. Disssipative soliton resonances. Phys Rev A 2008;78:023830. DOI: 10.1103/PhysRevA.78.023830. - 176.
Chang W, Ankiewicz A, Soto-Crespo JM, Akhmediev N. Dissipative soliton resonances in laser models with parameter management. J Opt Soc Am B 2008;25(12):1972–7. - 177.
Kalashnikov VL, Apolonski A. Energy scalability of mode-locked oscillators: a completely analytical approach to analysis. Optics Express 2010;18(25):25757–70. - 178.
Grelu Ph, Chang W, Ankiewicz A, Soto-Crespo JM, Akhmediev N. Dissipative soliton resonance as a guideline for high-energy pulse laser oscillators. J Opt Soc Am B 2010;27(11):2336–41. - 179.
Ding E, Grelu Ph, Kutz JN. Dissipative soliton resonance in a passively mode-locked fiber laser. Optics Lett 2011;36(7):1146–8. - 180.
Kharenko DS, Shtyrina OV, Yarutkina IA, Podivilov EP, Fedoruk MP, Babin SA. Generation and scaling of highly-chirped dissipative solitons in an Yb-doped fiber laser. Laser Phys Lett 2012;9(9):662–8. DOI: 10.7452/Japl.201210060. - 181.
Cheng Zh, Li H, Wang P. Simulation of generation of dissipative soliton, dissipative soliton resonance and noise-like pulse in Yb-doped mode-locked fiber lasers. Optics Express 2015;23(5):5972–81. DOI: 10.1364/OE.23.005972. - 182.
Chang W, Soto-Crespo JM, Ankiewicz A, Akhmediev N. Dissipative soliton resonances in the anomalous dispersion regime. Phys Rev A 2009;79:033840. DOI: 10.1103/PhysRevA.79.033840. - 183.
Zh-Ch, Ning Q-Y, Mo H-L, Cui H, Liu J, Wu L-J, Luo A-P, Xu W-Ch. Vector dissipative soliton resonance in a fiber laser. Optics Express 2013;21(8):109910204. DOI: 10.1364/OE.21.010199. - 184.
Smirnov SV, Kobtsev SM, Kukarin SV, Turitsyn SK. Mode-locked fibre lasers with high-energy pulses. In: Jakubczak K. (Ed.) Laser Systems for Applications. InTech; 2011. pp. 39–58. - 185.
Wise FW, Chong A, Renninger WH. High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion. Laser Photon Rev 2008;2(1–2):58–73. DOI: 10.1002/lpor.200710041. - 186.
Renninger WH, Chong A, Wise FW. Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers. IEEE J Sel Top Quantum Electron 2012;18(1):389–98. DOI: 10.1109/JSTQE.2011.2157462. - 187.
Nelson LE, Fleischer SB, Lenz G, Ippen EP. Efficient frequency doubling of a femtosecond fiber laser. Optics Lett 1996;21(21):1759–61. DOI: 10.1364/OL.21.001759. - 188.
Oktem B, Ülgüdür C, Ilday FÖ. Soliton–similariton fibre laser. Nat Photon 2010;4:307–11. DOI: 10.1038/nphoton.2010.33. - 189.
Renninger WH, Chong A, Wise FW. Dissipative solitons in normal-dispersion fiber lasers. Phys Rev A 2008;77:023814. DOI: 10.1103/PhysRevA.77.023814. - 190.
Chong A, Renninger WH, Wise FW. Properties of normal-dispersion femtosecond fiber lasers. J Opt Soc Am B 2008;25(2):140–8. - 191.
Schultz M, Karow H, Prochnow O, Wandt D, Morgner U, Kracht D. Optics Express 2008;16(24):19562–7. DOI: 10.1364/OE.16.019562. - 192.
Im JH, Choi SY, Rotermund F, Yeom D-I. All-fiber Er-doped dissipative soliton laser based on evanescent field interaction with carbon nanotube saturable absorber. Optics Express 2010;18(21):22141–6. DOI: 10.1364/OE.18.022141. - 193.
Kieu K, Wise FW. All-fiber normal-dispersion femtosecond laser. Optics Express 2008;16(15):11453–8. DOI: 10.1364/OE.16.011453. - 194.
Yang H, Wang A, Zhang Zh. Efficient femtosecond pulse generation in an all-normal-dispersion Yb:fiber ring laser at 605 MHz repetition rate. Optics Lett 2012;37(5):954–6. DOI: 10.1364/OL.37.000954. - 195.
Chichkov NB, Hausmann K, Wandt D, Morgner U, Neumann J, Kracht D. 50 fs pulses from an all-normal dispersion erbium fiber oscillator. Optics Lett 2010;35(18):3081–3. DOI: 10.1364/OL.35.003081. - 196.
Ruehl A, Kuhn V, Wandt D, Kracht D. Normal dispersion erbium-doped fiber laser with pulse energies above 10 nJ. Optics Express 2008;16(5):3130–5. DOI: 10.1364/OE.16.003130. - 197.
Lhermite J, Machinet G, Lecaplain C, Boullet J, Traynor N, Hideur A, Cormier E. High-energy femtosecond fiber laser at 976 nm. Optics Express 2010;35(20):3459–61. DOI: 10.1364/OL.35.003459. - 198.
Buckley J, Chong A, Zhou Sh, Renninger W, Wise FW. Stabilization of high-energy femtosecond ytterbium fiber lasers by use of a frequency filter. J Opt Soc Am B 2007;24(8):1803–6. DOI: 10.1364/JOSAB.24.001803. - 199.
Buckley JR, Wise FW, Ilday FÖ, Sosnowski T. Femtosecond fiber lasers with pulse energies above 10 nJ. Optics Lett 2005;30(14):1888–90. DOI: 10.1364/OL.30.001888. - 200.
Renninger WH, Chong A, Wise FW. Giant-chirp oscillators for short-pulse fiber amplifiers. Optics Lett 2008;33(24):3025–7. DOI: 10.1364/OL.33.003025. - 201.
Chong A, Renninger WH, Wise FW. All-normal-dispersion femtosecond fiber laser with pulse energy above 20 nJ. Optics Lett 2007;32(16):2408–10. DOI: 10.1364/OL.32.002408. - 202.
Ortaç B, Lecaplain C, Hideur A, Schreiber T, Limpert J, Tünnermann A. Passively mode-locked single-polarization microstructure fiber laser. Optics Express 2008;16(3):2122–8. DOI: 10.1364/OE.16.002122. - 203.
Lefrancois S, Sosnowski ThS, Liu Ch-H, Galvanauskas A, Wise FW. Energy scaling of mode-locked fiber lasers with chirally-coupled core fiber. Optics Express 2011;19(4):3464–70. DOI: 10.1364/OE.19.003464. - 204.
Lecaplain C, Ortaç B, Hideur A. High-energy femtosecond pulses from a dissipative soliton fiber laser. Optics Lett 2009;34(23):3731–3. DOI: 10.1364/OL.34.003731. - 205.
Lecaplain C, Chédot C, Hideur A, Ortaç B, Limpert J. High-power all-normal-dispersion femtosecond pulse generation from a Yb-doped large-mode-area microstructure fiber laser. Optics Lett 2007;32(18):2738–40. DOI: 10.1364/OL.32.002738. - 206.
Ortaç B, Schmidt O, Schreiber T, Limpert J, Tünnermann A, Hideur A. High-energy femtosecond Yb-doped dispersion compensation free fiber laser. Optics Express 2007;15(17):10725–32. DOI: 10.1364/OE.15.010725. - 207.
Lhermite J, Lecaplain C, Machinet G, Royon R, Hideur A, Cormier E. Mode-locked 0.5 μJ fiber laser at 976 nm. Optics Lett 2011;36(19):3819–21. DOI: 10.1364/OL.36.003819. - 208.
Lecaplain C, Ortaç B, Machinet G, Boullet J, Baumgartl M, Schreiber T, Cormier E, Hideur A. High-energy femtosecond photonic crystal fiber laser. Optics Lett 2010;35(19):3156–8. DOI: 10.1364/OL.35.003156. - 209.
Ortaç B, Baumgartl M, Limpert J, Tünnermann A. Approaching microjoule-level pulse energy with mode-locked femtosecond fiber lasers. Optics Lett 2009;34(10):1585–7. DOI: 10.1364/OL.34.001585. - 210.
Ankiewicz A, Devine N, Akhmediev N, Soto-Crespo JM. Dissipative solitons and antisolitons. Phys Lett A 2007;370:454–8. DOI: 10.1016/j.physleta.2007.06.001. - 211.
van Saarlos W, Hohenberg PC. Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations. Physica D 1992;56:303–67. - 212.
Soto-Crespo JM, Akhmediev NN, Afanasjev VV, Wabnitz S. Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion. Phys Rev E 55;4:4783–96. - 213.
Conte R. (Ed.) The Painlevé Property: One Century Later. New York: Springer-Verlag; 1999. 810 p. - 214.
Greco AM. (Ed.) Direct and Inverse Methods in Nonlinear Evolution Equations. Berlin: Springer; 2003. 282 p. - 215.
Kivshar YS, Malomed BA. Dynamics of solitons in nearly integrable systems. Rev Mod Phys 1989;61(4):763. DOI: http://dx.doi.org/10.1103/RevModPhys.61.763. - 216.
Malomed BA, Nepomnyashchy AA. Kinks and solitons in the generalized Ginzburg-Landau equation. Phys Rev A 1990;42:6009. - 217.
Kalashnikov VL. Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation. Phys Rev E 2009;80:046606. - 218.
Podivilov E, Kalashnikov VL. Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation. JETP Lett 2005;82(8):467–71. - 219.
Kalashnikov VL, Podivilov E, Chernykh A, Apolonski A. Chirped-pulse oscillators: theory and experiment. Appl Phys B 2006;83(4):503–10. - 220.
Ablowitz MJ, Horikis ThP. Solitons in normally dispersive mode-locked lasers. Phys Rev A 2009;79:063845. - 221.
Kharenko DS, Shtyrina OV, Yarutkina IA, Podivilov EV, Fedoruk MP, Babin SA. Highly chirped dissipative solitons as a one-parameter family of stable solutions of the cubic-quintic Ginzburg-Landau equation. J Opt Soc Am B 2011;28(10):2314–9. - 222.
Malomed BA. Variational methods in nonlinear fiber optics and related fields. In: Wolf E. (Ed.) Progress in Optics, Vol. 43. North-Holland: Elsevier; 2002. pp. 71–193. - 223.
Ankiewicz A, Akhmediev N, Devine N. Dissipative solitons with a Lagrangian approach. Optical Fiber Technol 2007;13(2):91–7. - 224.
Bale BG, Boscolo S, Kutz JN, Turitsyn SK. Intracavity dynamics in high-power mode-locked fiber lasers. Phys Rev A 2010;81:033828. - 225.
Bale BG, Kutz JN. Variational method for mode-locked lasers. J Opt Soc Am B 2008;25(7):1193–202. DOI: 10.1364/JOSAB.25.001193. - 226.
Tsoy EN, Ankiewicz A, Akhmediev N. Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation. Phys Rev E 2006;73:036621. - 227.
Bale BG, Kutz JN, Chong A, Renninger WH, Wise FW. Spectral filtering for high-energy mode-locking in normal dispersion fiber lasers. J Opt Soc Am B 2008;25(10):1763–70. DOI: 10.1364/JOSAB.25.001763. - 228.
Bale BG, Boscolo S, Turitsyn SK. Dissipative dispersion-managed solitons in mode-locked lasers. Optics Lett 2009;34(21):3286–8. DOI: 10.1364/OL.34.003286. - 229.
Ding E, Kutz JN. Stability analysis of the mode-locking dynamics in a laser cavity with a passive polarizer. J Opt Soc Am B 2009;26(7):1400–11. DOI: 10.1364/JOSAB.26.001400. - 230.
Kalashnikov VL. Dissipative soliton energy scaling. Phys. Rev. A. Forthcoming. - 231.
Liu X. Pulse evolution without wave breaking in a strongly dissipative dispersive laser system. Phys Rev A 2010;81(5):053819. - 232.
Shen X, Li W, Zeng H. Polarized dissipative solitons in all-polarization-maintained fiber laser with long-term stable self-started mode-locking. Appl Phys Lett 2014;105:101109. - 233.
Chen Sh, Liu Y, Mysyrowicz A. Unusual stability of one-parameter family of dissipative solitons due to spectral filtering and nonlinearity saturation. Phys Rev Lett 1997;79:4047. - 234.
Farnum ED, Kutz JN. Multifrequency mode-locked lasers. J Opt Soc Am B 2008;25(6):1002–10. - 235.
Zhang H, Tang DY, Wu X, Zhao LM. Multi-wavelength dissipative soliton operation of an erbium-doped fiber laser. Optics Express 2009;17(15):12692–7. - 236.
Yun L, Liu X, Mao D. Observation of dual-wavelength dissipative solitons in a figure-eight erbium-doped fiber laser. Optics Express 2012;20(19):20992–7. - 237.
Zhang ZX, Xu Z, Zhang L. Tunable and switching dual-wavelength dissipative soliton generation in an all-normal-dispersion Yb-doped fiber laser with birefringence fiber filter. Optics Express 2012;20(24):26736–42. - 238.
Xu ZW, Zhang ZX. All-normal-dispersion multi-wavelength dissipative soliton Yb-doped fiber laser. Laser Phys Lett 2013;10:085105. - 239.
Huang S, Wang Y, Yan P, Zhao J, Li H, Lin R. Tunable and switchable multi-wavelength dissipative soliton generation in a graphene oxide mode-locked Yb-doped fiber laser. Optics Express 2014;22(10):11417–26. - 240.
Mao D, Liu X, Han D, Lu H. Compact all-fiber laser delivering conventional and dissipative solitons. Optics Lett 2013;38(16):3190–3. - 241.
Kalashnikov VL, Chernykh A. Spectral anomalies and stability of chirped-pulse oscillators. Phys Rev A 2007;75:033820. - 242.
Kalashnikov VL. Dissipative solitons: perturbations and chaos formation. In: Skiadas CH, Dimotikalis I, Skiadas C. (Eds.) Chaos Theory: Modeling, Simulation and Applications. Singapore: Worlds Scientific Publishing; 2011. pp. 199–206. - 243.
Kalashnikov VL. Dissipative solitons in presence of quantum noise. Chaotic Model Simulat 2014;(1):29–37. - 244.
Soto-Crespo JM, Akhmediev N, Ankiewicz A. Pulsating, creeping, and erupting solitons in dissipative systems. Phys Rev Lett 2000;85(14):2937–40. - 245.
Soto-Crespo JM, Akhmediev N. Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation. Math Computers Simulat 2005;69(5–6):526–36. DOI: 10.1016/j.matcom.2005.03.006. - 246.
Cundiff ST, Soto-Crespo JM, Akhmediev N. Experimental evidence for soliton explosions. Phys Rev Lett 2002;88(7):073903. - 247.
Crtes C, Descalzi O, Brand HR. Exploding dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in one and two spatial dimensions. Eur Phys J Special Topics 2014;223:2145–59. DOI: 10.1140/epjst/e2014-02255-2. - 248.
Crtes C, Descalzi O, Brand HR. Noise can induce explosions for dissipative solitons. Phys Rev E 2012;85(015205(R)). DOI: 10.1103/PhysRevE.85.015205. - 249.
Arecchi FT, Bortolozzo U, Montina A, Residori S. Granularity and inhomogenety are the joint generators of optical rogue waves. Phys Rev Lett 2011;106:153901. - 250.
Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi FT. Rogue waves and their generating mechanisms in different physical contexts. Phys Rep 2013;528:47–89. - 251.
Solli DR, Ropers C, Koonath P, Jalali B. Optical rogue waves. Nature 2007;450:1054–7. - 252.
Soto-Crespo JM, Grelu Ph, Akhmediev N. Dissipative rogue waves: Extreme pulses generated by passively mode-locked lasers. Phys Rev E 2011;84(1):016604. - 253.
Zavyalov A, Egorov O, Iliew R, Lederer F. Rogue waves in mode-locked fiber lasers. Phys Rev A 2012;85:013828. - 254.
Komarov A, Sanchez F. Structural dissipative solitons in passive mode-locked fiber lasers. Phys Rev E 2008;77:066201. - 255.
Zakharov V, Dias F, Pushkarev A. One-dimensional wave turbulence. Phys Rep 2004;398:1–65. - 256.
Yun L, Han D. Bound state of dissipative solitons in a nanotube-mode-locked fiber laser. Optics Commun 2014;313:70–3. - 257.
Amrani F, Haboucha A, Salhi M, Leblond H, Komarov A, Sanchez F. Dissipative solitons compounds in a fiber laser. Analogy with the states of the matter. Appl Phys B 2010;99:107–14. DOI: 10.1007/s00340-009-3774-7. - 258.
Kalashnikov VL. Dissipative solitons: structural chaos and chaos of destruction. Chaotic Model Simulat 2011;(1):51–9. - 259.
Zhang L, Pan Zh, Zhuo Zh, Wang Y. Three multiple-pulse operation states of an all-normal-dispersion dissipative soliton fiber laser. Int J Optics 2014;(169379). DOI: 10.1155/2014/169379. - 260.
Kalashnikov VL, Sorokin E. Dissipative Raman solitons. Optics Express 2014;22(24):30118–126. DOI: 10.1364/OE.22.030118. - 261.
Kalashnikov VL. Chaotic dissipative Raman solitons. Chaotic Model Simulat 2014;(4):403–10. - 262.
Babin SA, Podivilov EV, Kharenko DS, Bednyakova AE, Fedoruk MP, Kalashnikov VL, Apolonski A. Multicolour nonlinearly bound chirped dissipative solitons. Nature Commun 2014;5:4653. DOI: 10.1038/ncomms5653. - 263.
Bednyakova AE, Babin SA, Kharenko DS, Podivilov EV, Fedoruk MP, Kalashnikov VL, Apolonski A. Evolution of dissipative solitons in a fiber laser oscillator in the presence of strong Raman scattering. Optics Express 2013;21(18):29556–64. DOI: 10.1364/OE.21.02056. - 264.
Kharenko DS, Bednyakova AE, Podivilov EV, Fedoruk MP, Apolonski A, Babin SA. Feedback-controlled Raman dissipative solitons in a fiber laser. Optics Express 2015;23(2):1857–62. DOI: 10.1364/OE.23.001857. - 265.
Haus JW, Shaulov G, Kuzin EA, Sanchez-Mondragon J. Vector soliton fiber lasers. Optics Lett 1999;24(6):376–8. - 266.
Barad Y, Silberberg Y. Polarization evolution and polarization instability of solitons in a birefringent optical fiber. Phys Rev Lett 1997;78(17):3290–3. - 267.
Lei T, Tu Ch, Lu F, Deng Y, Li E. Numerical study on self-similar pulses in mode-locking fiber laser by coupled Ginzburg-Landau equation model. Optics Express 2009;17(2):585–91. - 268.
Cundiff ST, Collings BC, Akhmediev NN, Soto-Crespo JM, Bergman K, Knox WH. Observation of polarization-locked vector solitons in an optical fiber. Phys Rev Lett 1999;82(20):3988–91. - 269.
Akhmediev N, Buryak A, Soto-Crespo JM. Elliptically polarised solitons in birefringent optical fibers. Optics Commun 1994;112(5–6):278–82. DOI: 10.1016/0030-4018(94)90631-9. - 270.
Wu J, Tang DY, Zhao LM, Chan CC. Soliton polarization dynamics in fiber lasers passively mode-locked by the nonlinear polarization rotation technique. Phys Rev E 2006;74:046605. DOI: 10.1103/PhysRevE.74.046605. - 271.
Zhang H, Tang DY, Zhao LM, Wu X, Tam HY. Dissipative vector solitons in a dispersion-managed cavity fiber laser with net positive cavity dispersion. Optics Express 2009;17(2):455–60. - 272.
Kong L, Xiao X, Yang Ch. Polarization dynamics in dissipative soliton fiber lasers mode-locked by nonlinear polarization rotation. Optics Express 2011;19(19):18339–44. - 273.
Mesentsev VK, Turitsyn SK. Stability of vector solitons in optical fibers. Optics Lett 1992;17(21):1497–9. - 274.
Zhang H, Tang D, Zhao L, Bao Q, Loh KP. Vector dissipative solitons in graphene mode locked fiber lasers. Optics Commun 2010;283:3334–8. DOI: 10.1016/j.optcom.2010.04.064. - 275.
Luo Zh-Ch, Ning Q-Y, Mo H-L, Cui H, Liu L, Wu L-J, Luo A-P, Xu W-Ch. Vector dissipative soliton resonance in a fiber laser. Optics Express 2013;21(8):10199–204. DOI: 10.1364/OE.21.010199. - 276.
Tsatourian V, Sergeyev SV, Mou Ch, Rozhin A, Mikhailov V, Rabin B, Westbrook PS, Turitsyn SK. Polarization dynamics of vector soliton molecules in mode locked fibre laser. Sci Rep 2013;3:3154. DOI: 10.1038/srep03154. - 277.
Sergeyev SV. Fast and slowly evolving vector solitons in mode-locked fibre lasers. Philos Transac Royal Soc A 2014;372:20140006. DOI: 10.1098/rsta.2014.0006. - 278.
Sergeyev SV, Mou Ch, Turitsyna EG, Rozhin A, Turitsyn SK, Blow K. Spiral attractor created by vector solitons. Light Sci Applic 2014;3:e131. DOI: 10.1038/lsa.2014.12. - 279.
Lin Q, Agrawal GP. Vector theory of stimulated Raman scattering and its application to fiber-based Raman amplifiers. J Opt Soc Am B 2003;20(8):1616–31.

## Notes

- The width of a gain band is not a decisive factor per se because both pulse width and its spectrum are affected by various factors including higher-order dispersions, non-linearity, etc. [53,54].
- However, namely LMA and photonic-crystal fibres could realize a Kerr-lens mode-locking in a fibre laser [152,153].
- Energy-non-scalable branch has two distinguishing characteristics: it turns into solution of Eq. (9) with ζ, χ→0 (‘Schrödinger limit’ [218]) and is unstable in absence of dynamic gain saturation, i.e. if σ is not energy-dependent [221].
- A multi-porting configuration of a DS laser supports even simultaneous generation of conventional and dissipative wavelength-separated solitons [240].