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.
- Ultrafast fibre laser
- dissipative soliton
- non-linear dynamics
- vector solitons
- optical turbulence
- stimulated Raman scattering
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
Another aspect of the ultrafast laser applications is connected with studying non-linear phenomena . Ultrafast lasers became an effective platform for investigation of general non-linear processes such as instabilities and rogue waves [31,32], self-similarity  and turbulence . 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 . Such a playground spans gravity and cosmology , 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 (
In this review, we will concern the concepts of a mode-locking and a dissipative soliton in a nutshell.
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
In the case of
which is the classical equation for an oscillator in the potential defined by
Using the non-linear processes such as SPM, loss and gain saturation allows generating the ultrashort pulses due to mechanism of the so-called
where a four-wave non-linear process defined by non-linear susceptibility χ3 mixes the frequencies
where a heat radiated at
Eq. (4) is clearly understandable in the Fourier domain: where an external (i.e. active) modulation defined by the ν-coefficient ‘diffuses’ (i.e. broadens) a field spectrum and such diffusion is compensated by spectral dissipation defined by the α-coefficient. In the time-domain, the spectrum broadening corresponds to a light pulse shortening due to parabolic potential action which is balanced by spectral dissipation causing a pulse widening.
The space-time duality can be extended further with the help of
where k and β2 are the wave number and group-delay dispersion coefficients, respectively. Both processes describe the beam/pulse spreading with propagation which is accompanied by phase
(compare with (4)) looking as in the Fourier domain describes a ‘diffraction’ (dispersion) in the frequency domain inspired by phase modulator which is balanced by spectral dissipation. The main difference from (4) is that the phase modulation in (6) distorts the phase and thereby produces chirp like the action of thin lens but in the time domain. In other words, the phase modulation in (6) pushes the spectral components out of the point of stationary phase , adding the frequency shift (Doppler shift) which enhances the spectral dissipation on the pulse wings and thereby forms a pulse like the active amplitude modulator. But the phase profile
The transition to a
It is appropriate to mention here that the space-time duality
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
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
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
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
It is known  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 (
where the dissipative factors from Eq. (9) are taken into account and describes a ‘strength’ of linear birefringence (
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
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
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 .
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 :
with some possible modifications (e.g. see ). 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 is a loss saturation power.
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) :
where cylindrical symmetry is assumed,
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 . 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
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 . For a simplest case of Eq. (9), which has a DS solution in the form of ( is a dimensionless chirp parameter) [85,159,160], the DS energy generation 
as well as the spectrum are shown in Figure 6 in dependence of
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 . One of the reasons is that the chirp allows DS to accumulate energy , which means that DS is
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 (wave number of dispersive linear wave) and the DS wave number : , where is a DS peak power and a DS spectral width is . Since a system is dissipative, the above phase equilibrium has to be supplemented by loss compensation condition: spectral loss has to be compensated by non-linear gain . Combination of above criteria gives a definition of the parametric limits for DS [44,173]:
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
The advantage of the
The diversity of the results obtained (Figure 8) needs a comprehension from a
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)).
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,
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
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 . This effect can appear in a fibre laser with NPR (e.g. see  where such a completely cubic-quintic CNGLE is connected with the NPR mode-locking technique). In this case, DS soliton exists in both normal and anomalous GDD regions [175,182,233].
The master diagram is a manifold of isogains (i.e. curves with ). Figure 10 demonstrates the zero-isogains ( Energy-non-scalable branch has two distinguishing characteristics: it turns into solution of Eq. (9) with (‘Schrödinger limit’ ) and is unstable in absence of dynamic gain saturation, i.e. if σ is not energy-dependent .
Energy-non-scalable branch has two distinguishing characteristics: it turns into solution of Eq. (9) with (‘Schrödinger limit’ ) and is unstable in absence of dynamic gain saturation, i.e. if σ is not energy-dependent .
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
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 in the limit of is a Lorentzian profile which has a characteristic width Ω
where H is a Heaviside function. The DS energy is
Here, we trace the zero-isogain σ = 0. The DS time-profile is defined by an implicit expression:
with the DS width of Now, there are the following limiting cases:
It is clear that in this ‘low-energy’ sector the DS time-profile is bell-like and its spectrum has tabletop form (). In the DSR limit, one has:
that is, a DS in the DSR sector has a flattop temporal profile and a Lorenzian spectrum (). Eqs. (24) demonstrate that asymptotical growth of DS energy leads to a spectral condensation () without a parallel temporal thermalization (), which means an inevitable destabilization of a plain energy-scalability . This conclusion does not mean a participial impossibility of DS energy-scaling in the frameworks of cubic-quintic CNGLE model. For instance, a saturable SPM allows DSs with tabletop profiles and on the pulse edges. Such a DS possesses enhanced energy scalability and was observed experimentally .
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  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 .
A multi-porting configuration of a DS laser supports even simultaneous generation of conventional and dissipative wavelength-separated solitons .
As was demonstrated in previous section, DS has non-trivial internal structure due to energy fluxes inside it. The elements of this structure (
For sufficiently large energies in the vicinity of stability border (point
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) . 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) . As was found, SRS enhances the tendency to multi-pulsing with energy growth caused by enhancement of spectral dissipation due to SRS . Simultaneously, generation of anti-Stokes radiation causes chaotization of DS dynamics and irregular modulation of DS temporal and spectral profiles  (Figure 16). DS profile remains localized, but it is strongly cut by colliding dark and grey soliton-like structures .
As was shown, the DS dynamics can be regularized by formation of
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 . 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 () or slow (>>
The specific multiple pulse instability of vector dissipative solitons (VDSs) leads to generation of the bound states of DSs with different SOPs (
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 . 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) . 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 . These topics remain unexplored to date.
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.
This work was supported by FP7-PEOPLE-2012-IAPP (project GRIFFON, No. 324391).
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- 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’ ) and is unstable in absence of dynamic gain saturation, i.e. if σ is not energy-dependent .
- A multi-porting configuration of a DS laser supports even simultaneous generation of conventional and dissipative wavelength-separated solitons .