# Dissipative solitons for mode-locked lasers

Philippe Grelu, Nail Akhmediev

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Philippe Grelu, Nail Akhmediev. Dissipative solitons for mode-locked lasers. Nature Photonics, 2012, 6 (2), pp.84-92. 10.1038/nphoton.2011.345. hal-04901011

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Dissipative solitons for mode-locked lasers

Philippe Grelu¹* & Nail Akhmediev²

¹ Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 5209 CNRS, Université de Bourgogne, BP 47870, 21078 Dijon Cedex, France

² Optical Sciences Group, Research School of Physics and Engineering, Institute of Advanced Studies, The Australian National University, Canberra ACT 0200, Australia.

*e-mail: philippe.grelu@u-bourgogne.fr
**Abstract**

*Dissipative solitons* are localized formations of an electromagnetic field that are balanced through the energy exchange with the environment in presence of nonlinearity, dispersion and/or diffraction. Their growing use in the area of passively mode-locked lasers is remarkable: the concept of a dissipative soliton provides an excellent framework for understanding complex pulse dynamics and stimulates innovative cavity designs. Reciprocally, the field of mode-locked lasers serves as an ideal playground to test the concept and reveal unusual dynamics. This Review provides basic definitions of the dissipative soliton concept, summarizes its implications for the design of high-energy mode-locked fibre laser cavities, highlights striking emerging dynamics, such as dissipative soliton *molecules*, pulsations, *explosions*, *rain*, and finally provides the outlook for dissipative light bullets.

**The concept of a dissipative soliton**

Introduced in 1965 by Zabusky and Kruskal, the term *soliton* was originally attributed to the localized solutions of integrable nonlinear systems. Such solutions have the following remarkable properties: they maintain their shape and velocity after colliding with each other, and they remain intact when interacting with radiation waves. In optics, integrable - *conventional* - solitons result from the *single balance between nonlinearity and dispersion*, or *diffraction*. Since then, other types of localized solutions in nonlinear science have been dubbed solitons as well, despite some resistance from the adepts of strict mathematical definitions¹. A dramatic turning point occurred at the beginning of the 90s, when physicists realized that solitary waves did exist in a wide range of non-integrable and non-conservative systems. Thus, new terminology had to be introduced, and solitary waves in nonlinear optical systems with nonlinear gain or loss mechanisms were called *dissipative solitons*²,³. Another
term, autosoliton, developed in the frame of plasma physics and describing reaction-diffusion
systems⁴, had also been used in the early optical context⁵. Reaction-diffusion systems involve
supply and decay of matter⁶, whereas in optics, we consider the balance of energy. Hence,
dissipative optical solitons in a nonlinear medium are confined wave-packets of light whose
existence and stability crucially depend on the energy balance. In 2005, the major principles
of dissipative solitons were unified and presented in a single volume⁷. Although most of its
contributions were related to phenomena observed or predicted in nonlinear optics, the new
concept appeared sufficiently general to be applied in other disciplines, such as biology⁸.

While being a fundamental extension to the **conventional** soliton theory, the dissipative soliton concept incorporates the powerful notions developed in the theory of nonlinear dynamics, such as attractors and bifurcations. The concept also stems from the seminal ideas of Prigogine on open systems far from equilibrium⁹: "dissipative systems", through the perpetual and bi-directional exchange of energy with their environment, can manifest conditions for self-organization and become a fascinating area of research. Thus, solitons, nonlinear dynamics and dissipative systems constitute basically the three sources and component parts of the dissipative soliton concept⁸. In practice, this extension of the soliton concept implies that the single balance between nonlinearity and dispersion is replaced by a composite balance between several effects, as sketched in Fig. 1a. An important point here is that the balance between gain and loss, which should be exact in order to produce stationary localised solutions, plays a dominant role in the dynamics. As a consequence, and compared to the case of conventional (conservative) solitons, the impact of dissipative effects reveals unusual situations where solitons can be found, such as bright dissipative solitons in the normal dispersion regime¹⁰, or stable multi-soliton complexes, also known as soliton molecules¹¹⁻¹³.
The stability of a dissipative soliton is directly linked to the stability of the associated fixed point, using the terminology of nonlinear dynamics. A strong attractor naturally provides a high degree of stability. In contrast to the conservative case, where a given set of parameters for the governing propagation equation generally leads to a family comprising an infinity of soliton solutions, the attractor in a dissipative system is a fixed localised solution. Namely, the dissipative soliton energy, profile and chirp are predetermined by the equation parameters, rather than by the initial conditions. These features of dissipative solitons offer highly desirable properties for applications, such as generation of stable trains of laser pulses by mode-locked cavities, or in-line regeneration of optical data streams.

Although the profile of a dissipative soliton is fixed for a given set of equation parameters, there is a wide range of profiles that can be obtained by tuning these parameters. This way, one can witness various transformations of dissipative solitons, either smooth or in the form of bifurcations. One well-known type of bifurcation transforms a fixed point into a limit cycle, which is an oscillatory attracting state. Thus, a previously stationary soliton becomes a pulsating localised formation¹⁴. Further transformations in the parameter space may produce irregular and chaotic soliton dynamics. The optical field being a function of continuous spatial variables, the related dynamical system possesses an infinite number of degrees of freedom. Hence, an extended exploration in the parameter space is propitious to reveal large numbers of dissipative soliton varieties and bifurcations. This constitutes a fertile ground for investigations, as it is currently the case in the exploration of mode-locked laser dynamics. The motivation for the latter is twofold. The field of mode-locked lasers can be considered as an ideal playground for exploring nonlinear dissipative dynamics. Varieties of accessible laser configurations, generated pulse shapes and their dynamics have expanded very quickly over the past few years, particularly within the fibre laser technology. Another motivation is at the applied level. The implementation of dissipative soliton concepts can be
highly useful in the development of compact, efficient, and reliable sources of short and ultra-
short light pulses, with unprecedented pulse energy or repetition rate, and improved output
stability.

Before developing these ideas further, let us point at one significant difference
between conventional, or conservative, and dissipative solitons. In order to be stationary,
solitons in dissipative systems need to have distinct internal regions which extract energy
from an external source, as well as other internal regions where energy is lost into the
environment. Thus, a stationary dissipative soliton results from the continuous energy
exchange with the environment along with its *live* redistribution between the various parts of
the soliton. Technically, the internal energy flow corresponds to a non-uniform phase across
the dissipative soliton profile, in marked contrast with the constant nonlinear phase profile
across a stationary conservative soliton. At a given part of the pulse, the energy flow is
proportional to the first temporal derivative of the phase, while the density of energy
generation is proportional to the second temporal derivative of the phase. Interestingly,
pulsating dissipative solitons that undergo a periodic evolution along their direction of
propagation generalize this idea further into the spatial domain: locations where the soliton
mostly extracts energy are followed by locations where losses dominate, providing an average
balance inside a period of evolution. This typical situation is met in all laser cavities.

An art of producing dissipative solitons - A world of pulse profiles out of lasers

In the following, we consider passive mode locking, which uses nonlinearity to promote a
saturable absorber effect and allows the generation of ultra-short optical pulses. However, the
concept of dissipative solitons can also be applied in the context of active mode locking¹⁵,
*although this relationship has not yet been precisely investigated*. There are different levels of
modelling a given laser system. Since lasers are made of several components, an accurate
model should involve consecutive sets of propagation equations. Models can be vectorial when the polarization nature of light is involved. They can include the delayed response of the saturable absorber and that of the gain medium. However, dealing with too many parameters and lengthy numerical calculations hampers the exploration of original dynamics and their classification. In addition, all parameters are not always precisely known or even relevant. For these reasons, and taking part to the underlying quest of a "mother of all mode-locking theories", it is beneficial to start the exploration of dissipative soliton dynamics with a master equation that incorporates the main physical ingredients at play in mode-locked lasers, in a distributed way. This initiative is inspired by the work of H. Haus, who, from the mid-70s, developed the master equation approach to mode locking<sup>16,17</sup>. It turns out that the main master equation studied by Haus is similar to the cubic complex Ginzburg-Landau equation, a universal equation describing systems in the vicinity of bifurcations<sup>18-20</sup>. However, the inclusion of a quintic saturating term in the Ginzburg-Landau was shown to be essential for the stability of pulsed solutions<sup>21-23</sup>. Seen from another point of view, the complex cubic-quintic Ginzburg-Landau equation (CGLE) is the extension of the nonlinear Schrödinger equation to higher-order and dissipative terms. The CGLE has been used to describe a wide range of nonlinear optical systems, such as passively mode-locked lasers with fast saturable absorbers, parametric oscillators, wide aperture lasers, and nonlinear optical transmission lines<sup>1,8</sup>. Its normalized form, in the (1+1)-dimensional case, reads:

$$i\psi_z + \frac{D}{2}\psi_{tt} + |\psi|^2\psi + v|\psi|^4\psi = i\delta\psi + i\epsilon|\psi|^2\psi + i\beta\psi_{tt} + i\mu|\psi|^4\psi.$$

The optical field envelope $\psi$ is a function of $t$, the retarded time in the frame moving with the pulse, and $z$, the propagation distance. The left-hand-side of the equation contains the conservative terms, where $D$ is for the anomalous dispersion propagation regime if positive (normal, if negative), and $v$ applies to a quintic Kerr effect. The right-hand-side of the
equation includes all dissipative terms: $\delta$, $\varepsilon$, $\beta$ and $\mu$ are the coefficients for linear loss (if negative), nonlinear gain (if positive), spectral filtering, and saturation of the nonlinear gain (if negative), respectively. These are indeed the basic physical effects required to build a mode-locked laser, with the expected signs of their contributions as suggested above. In particular, linear loss and nonlinear gain account for the required saturable absorber effect. Using an averaging method, such as the multi-scale analysis, it is possible to relate, although in an approximate way, all the coefficients of the CGLE to the physical parameters of the laser cavity$^{24-26}$. This procedure reveals surprising features: for instance, although the quintic saturation of the Kerr effect is not reached in most laser materials under practical operating conditions, an *effective* quintic Kerr term arises from the reduction of a lumped laser propagation model to a distributed one$^{25}$. Numerical comparisons show that the generally good qualitative picture provided by the distributed CGLE model becomes quantitative when the pulse alterations inside each cavity roundtrip are not too large$^{26}$. Naturally, gain saturation delayed dynamics is outside the scope of the CGLE model that assumes an instantaneous response$^{27,28}$.

The CGLE is non-integrable, which means that general analytical solutions are not available. Only for specific relations between the parameters of the equation, selected analytical solutions can be found. Several approximate methods, such as the Lagrangian approach or the method of moments can be used to approximate simple localised solutions$^{29-31}$. By projecting the initial evolution problem into a low-dimensional subspace of collective variables – the pulse width, amplitude, chirp, etc. – or pulse moments, one can obtain approximate relations between the soliton solution and the equation parameters. Whereas the search for a unified picture of mode-locking regimes presents a grand challenge, direct comparisons with numerical simulations remain essential to ensure the validity of approximate methods.
Generally, the variety of stable soliton profiles is significantly wider than we can imagine and approximate, as illustrated in Fig. 1b. Here, varying the nonlinear gain parameter $\varepsilon$ transforms the nearly-hyperbolic-secant pulse shape into a flat-top profile. The latter can be considered as two mirror-imaged interacting fronts. At certain values of the CGLE parameters, the combination of two fronts and a plain pulse, called a *composite pulse*, becomes stable too (Fig. 1c). It follows from the phase profile that energy flows from the centre of the structure to the wings of the composite pulse. For lasers, the implication is that varying the amount of nonlinear dissipative effects strongly influences the pulse shape, even for given amounts of dispersion and self-phase modulation.

The large and multiple regions of parameter domains of the CGLE where stable dissipative solitons can be found is a valuable gift for laser physicists: it signifies that there is substantial room for exploring innovative cavity designs. For instance, the fact that highly stable bright dissipative solitons have been found within the normal dispersion regime$^{32,33}$ means that the goal of achieving stable pulses with increased energy can be gradually released from the constraint of dispersion compensation. Actually, playing with dispersion compensation, also known as dispersion management, gradually shifted the interest of laser physicists to the net normal dispersion regime. This was stimulated by the dispersion-managed fibre laser configurations tested in the mid-90s in the group of H. Haus$^{34}$. Whereas the mode-locked fibre laser configurations tested in the early 90s favoured conventional, “Schrödinger soliton” pulse shaping by using predominant anomalous fibre segments, the stretched-pulse mode-locked fibre laser combines fibres of opposite dispersion signs, introducing a huge intra-cavity temporal breathing dynamics as soon as there is an approximate compensation between anomalous and normal dispersions. This circulation, in most of the cavity, of highly-stretched pulses of reduced amplitude entitled to reach pulse energy an order of magnitude higher, thus approaching the nano-joule range.
To understand the challenge of producing high-energy ultra-short pulses out of laser oscillators, let us remind the reader that the combined action of the Kerr nonlinearity and the chromatic dispersion generally leads to the pulse break-up, once the accumulated nonlinear phase exceeds a certain level. Overdriving the ultra-fast saturable absorber effect, as in the popular nonlinear-polarization-evolution mode locking, is also a directly related issue. These effects can be circumvented using a significant temporal stretching of the pulse inside of the cavity, thus reducing the peak power while maintaining a large spectral content. In essence, this strategy relates to incorporating chirped-pulsed amplification inside the laser oscillator³⁵-³⁷.

In the beginning of the 2000s, laser physicists found alternative cavity designs to achieve stable stretched-pulse operation with higher pulse energy. In 2004, the group of F. Wise successfully implemented the self-similar propagation of chirped parabolic pulses in the passive part of the laser cavity, a normally dispersive fibre. This led to an increased pulse energy performance with nearly 10 nJ per pulse³⁸. The implementation of self-similarity inside a cavity relies on a delicate balance. Normal dispersion and nonlinearity, which act together in the passive fibre, need to be balanced by spectral filtering, nonlinear gain and anomalous dispersion in the subsequent portion of the laser cavity, thus performing a precise self-consistent pulse reshaping prior to re-injection into the passive fibre. The use of self-similarity is currently being revisited in the light of the dissipative soliton concept³⁹,⁴⁰. Indeed, the acceptance of unusually large intra-cavity gain and losses, through spectral filtering in particular, seems to be the clue to enter specific pulse regimes, such as self-similarity. Self-similar pulses, or similaritons, are forever expanding pulses that do not match with the notion of a soliton. However, they possess a useful attracting asymptotic state that can be exploited – even though partially, since the propagation distance is limited – in the design of a mode-locked laser cavity. This way, an improved stability performance can be obtained, as was
recently reported with the development of a soliton-similariton fibre laser³⁹. This laser
features *active* self-similarity –obtained in the gain fibre – followed by conventional soliton
reshaping in the passive anomalous fibre. Interestingly, such laser operation brings about the
idea of “management of dynamics” inside one cavity roundtrip, confirming further that the
balance between strong physical effects can be solved at the scale of the entire cavity, and
providing *original* dissipative solitons dynamics hardly accessible to distributed models. The
transition from similariton to soliton dynamics is performed by a spectral filter, which
truncates the pulse exiting the gain fibre in both spectral and temporal domains. Indeed,
filtering spectrally a highly chirped pulse has a significant clipping effect on the temporal
pulse wings. This clipping effect was also evidenced a few years ago as an efficient key
allowing mode-locked operation in all-normal-dispersion fibre lasers⁴¹⁻⁴³.

In all-normal-dispersion fibre lasers, highly chirped pulses do not breathe. After the pulsed operation has been promoted by the saturable absorber effect, the composite balance between spectral filtering on the first hand, and nonlinearity, dispersion and gain on the second hand, dominates the cavity dynamics, which tends to produce M-shaped optical spectra. An example of such laser cavity and its output features are illustrated in Figs. 2 a,b. Ytterbium-doped mode-locked fibre lasers employing such design could produce pulses with energy above 20 nJ using standard fibre technology⁴¹⁻⁴³. There is a current trend to call these lasers "dissipative-soliton lasers" ⁴³⁻⁴⁵. Such name is justified in the sense that strongly dissipative effects per roundtrip are at the heart of the pulse formation and stability. However, this can be confusing if one would think that dissipative solitons are exclusively found in all-normal-dispersion lasers. As we shall see in the following sections, dissipative solitons are also found in lasers with anomalous dispersion, clearly revealing dynamics unusual to Schrödinger solitons.
There are currently several trends in developing higher-pulse-energy mode-locked fibre laser oscillators. Perhaps, the most efficient one relies on advanced technology: the development of large-mode-area gain fibres via micro-structuring enables to reach microjoule-level pulse energies$^{46}$. Another trend is to shift mode-locking operation further into the normal dispersion regime, increasing the pulse chirping effect$^{35,47}$. To remain compatible with the available pumping power and heat-management, these strategies are usually supplemented with an increased cavity length to reduce the pulse repetition rate.

Recently, an alternative conceptual approach, called dissipative soliton resonance (DSR), was proposed to scale up the pulse energy$^{48-50}$. Found initially in the frame of the CGLE master equation, it describes dissipative solitons acquiring theoretically infinite energy when the equation parameters approach a specific hyper-surface in the parameter space. In practice, this implies that mode-locked laser setups could be designed where the pulse energy would increase linearly with the pumping power without suffering pulse break-up, and within a narrow range of laser parameters. An illustration of the DSR phenomenon is presented on Fig. 2c. Here, within tiny adjustments of the dispersion parameter, the pulse energy increases by orders of magnitude. Pulses close to the DSR share common features, such as a clamped peak power, and a finite bandwidth. Thus, the increase of energy results in the increase of the duration of the chirped pulse (Fig. 2d). A similar approach utilizes a trial function to reduce the CGLE dynamics to a master diagram where scalable pulse energy can be found$^{51,52}$. Although high-order physical effects might be flattening the resonance effect, a numerical analysis of the DSR phenomenon in a realistic fibre laser setup model, taking into account gain saturation, provides the guidelines to test the concept experimentally$^{50}$. Meanwhile, experimental indications for the existence of the DSR phenomenon have been reported$^{53,54}$.
Dissipative soliton molecules

Soliton molecules in dissipative systems and higher-order soliton solutions of the nonlinear Schrödinger equation (NLSE) are different formations¹¹. Higher-order NLSE solitons are nonlinear superpositions of fundamental solitons, which breathe periodically in both temporal and spectral domains. They are easily destroyed by external perturbations or additional effects such as third-order dispersion or Raman scattering. The Nth-order soliton requires a total energy proportional to N². In a marked contrast, dissipative soliton molecules are robust multi-soliton structures that can be perfectly stationary, and whose energy remains proportional to the number of single-soliton constituents. In analogy with the physics of matter, soliton molecules can be formed by the interaction of initially separate single solitons. The energy of a soliton molecule being different from the sum of the energies of its soliton constituents, the formation of a soliton molecule requires dissipative processes, and entails the existence of a binding energy. As the interaction of optical pulses through their tails is phase-sensitive⁵⁵,⁵⁶, self-phase locking of these pulses can produce soliton molecules. Stable soliton pairs, characterized by a quadrature-phase difference, have been predicted in 1997 on the basis of the CGLE model⁵⁷. They were found in a fibre laser experiment a few years later⁵⁸. These results were obtained in the case of anomalous dispersion. Stable soliton pairs have also been found in diverse laser experimental configurations, dispersion regimes, emission wavelengths, and confirmed in series of numerical simulations using various models⁵⁹⁻⁶³. Consequently, soliton pairs represent the central feature of the collective dynamics of dissipative solitons⁷,¹¹. They also bring a frame for understanding some of the early experimental observations of multiple pulsing, particularly in the case of *pulse bunching*⁶⁴⁻⁶⁷. The possibility of forming a stationary soliton pair is linked to the existence of a point attractor in the dynamical system (Fig. 3a). The attractor properties do vary with the model and the parameters. For instance, bound pulses tend to become out-of-phase in the normal
dispersion regime$^{59,61,63,68}$, while cavity discreteness in the anomalous regime entails the existence of a large necklace of attractors and the subsequent quantization of allowed temporal separations$^{69}$. Experimentally, the direct observation of soliton molecules on an oscilloscope trace, as shown on Fig. 3b, challenges the bandwidth of fast electronics when the temporal separation falls below 20 ps. In addition, a full characterization requires proof of the self-phase locking. For these reasons, soliton molecules are better characterized via the analysis of the spectral fringes and using the optical autocorrelation, as illustrated with Fig. 3c. From nonlinear dynamics, we know that Hopf-type bifurcations can occur when the system parameters are altered, transforming a fixed-point attractor into a limit-cycle. In the case of infinite-dimensional systems, these bifurcations provide pulsating dissipative solitons$^{12,70,71}$, and vibrating soliton pairs$^{72,73}$. Chaotic bound states are further generalization of the notion of dissipative soliton pairs$^{74}$.

Another way to enter complexity is through the increase of the number of interacting solitons. Dealing with three pulses, several "isomers" can be considered$^{75}$, whereas the complexity of the dynamics becomes challenging. For instance, periodic pulse collisions resulting from the mutual exchange between a soliton pair and a third interacting soliton were reported experimentally and analysed theoretically. These phase sensitive interactions open the possibility of control over synthesis and dissociation of soliton molecules$^{76,77}$. In the anomalous dispersion regime, macromolecules comprising up to several hundreds of bound dissipative solitons can be formed. Various types of soliton patterns can be self-assembled (see Fig. 3b) while regular and stable large soliton molecules are not generic. Soliton crystal is the most ordered self-arrangement observed experimentally$^{78}$. There could be a fascinating conceptual bridge between soliton macromolecules, crystals on one side, and modulational-instability laser dynamics reinterpreted within dissipative four-wave mixing on the other side$^{79}$.
Complex dynamics: pulsations, explosions and soliton rain

Robust pulsations require a stable limit-cycle attractor, which exists only in nonlinear dissipative systems. Since a laser cavity possesses an internal structure, its overall dynamics converges to a single point attractor when, at any given location, the pulse parameters recover the same values after every round-trip. As the pulse is observed through a given output port, the experimentalist witnesses, in a stroboscopic way, the pulse *macroevolution* at multiples of the round-trip time. Pulsations can be divided into two categories: short-period pulsations, whose periodicity remains comparable to the round-trip time, and long-period pulsations. Both have been predicted and observed in mode-locked lasers$^{70}$. Pulsations are usually found, in the space of parameters, close to the boundary between the domains of stationary solitons and domains of unstable non-vanishing solutions. Apart from the archetypal cascade of period-doubling bifurcations leading to chaotic pulse dynamics, other types of bifurcation, such as period-tripling for instance, can be found. Combination of short and long period pulsations is another possibility.

At certain values of the cavity parameters, pulsating solitons can become chaotic: the macroevolution of their profiles never repeats, although they remain localized. When such dynamics starts for a wide range of localised initial conditions, the *strange soliton attractor* terminology becomes appropriate. One of the most striking species of strange soliton attractors is that of *soliton explosions*$^{80,81}$. These are soliton solutions that intermittently suffer explosive instabilities, but return to their original shapes after bursting in both temporal and spectral domains. Such a soliton has extended intervals of almost stationary propagation. However, instability develops until the *explosion* occurs, followed by a recovery of the pseudo-stationary profile. All successive explosions have similar features, but being chaotic, they are not identical. As a strange attractor, the dynamics attracts neighbouring trajectories in
the phase space. A particular illustration of soliton explosion is depicted on Fig. 4. The experiment is based on a Kerr-lens mode-locked titanium-sapphire laser (Fig. 4a). To record the explosions, which take place at the microsecond time scale, the laser output is dispersed by a diffraction grating across an array of six fast detectors, providing a real-time spectrogram. The spectral signature of one soliton explosion is prominent in the spectrogram of Fig. 4b. A numerical simulation based on a parameter-managed CGLE model corroborated this typical event⁸⁰.

Chaotic pulsations are also one of the forms of evolution of multi-soliton complexes, although they may be even more difficult to detect. One such example is the prediction of *shaking soliton pairs*, which shows similarities with soliton explosions. These also reveal intermittent instability development followed by recovery⁷³.

Among complex dynamics of large sets of dissipative solitons, the *soliton rain* is one particularly spectacular illustration of self-organization⁸². This dynamics was discovered in an erbium-doped mode-locked fibre laser experiment, where the formation of numerous intra-cavity pulses was favoured by operating in the anomalous dispersion regime, under Watt-level pumping power. The existence of a relatively intense noisy background together with tens of soliton pulses aggregated in a condensed soliton phase constitutes a necessary condition for the appearance of the effect. The temporal dynamics, recorded in real time (see Fig. 5), displays new soliton pulses that form spontaneously from background fluctuations and drift at a constant velocity until they reach the main peak component of the temporal intensity. This peak stands in fact for a compact, or condensed, soliton phase. Indeed, zooming in with a 30-GHz sampling oscilloscope reveals a sub-structure of bound jittering solitons. The soliton rain terminology originates from a compelling analogy with the evaporative cycle of water. First, similar to a process of water evaporation, the condensed soliton phase emits large amounts of radiation that move forward to shorter times. Radiation, superimposed on pre-existing cw
modes, produces a noisy inhomogeneous background. Second, at random times and relative
locations, fluctuations exceed a threshold over which a new dissipative soliton can be formed.
Third, as raindrops, randomly appearing solitons drift back to the condensed phase. The
whole scenario repeats endlessly as a quasi-stationary process, which can also be viewed as an
illustration of self-excitability. The soliton rain dynamics addresses the largely unexplored
fields of nonlinear dissipative dynamics in the case of large soliton numbers<sup>83</sup> and that of
dissipative soliton interactions in presence of a fluctuating background<sup>84</sup>.

Spatio-temporal dynamics: towards dissipative light bullets

The generalization of stable dissipative solitons to higher space-time dimensionality can be
readily obtained from the theoretical and numerical points of view<sup>85-87</sup>. In contrast to solitons
in Kerr media whose instability in more than two dimensions is known to lead to either
expansion or collapse<sup>88,89</sup>, dissipative solitons remain stable. One of the major challenges in
nonlinear optics is the experimental demonstration of *light bullets*, beautiful objects of
fundamental science that could also be used in parallel optical data processing<sup>90,91</sup>. The light
bullet concept assumes a complete spatio-temporal soliton, for which confinement in the three
spatial dimensions as well as localization in the temporal domain are achieved owing to the
balance between the focusing nonlinearity and the spreading due to both chromatic dispersion
and angular diffraction, while propagating in a homogeneous medium (Fig. 6a). The use of
dissipative nonlinearities is widely considered to be the most promising strategy for the
experimental implementation of stable light bullets<sup>91-94</sup>, although other approaches are also
possible<sup>95-97</sup>. In practice, the experiment could take the form of a wide-aperture laser cavity,
where saturable absorption would simultaneously provide transverse localisation and mode-
locking<sup>86,92</sup>. Semi-conductor devices offer many advantages in this prospect: based on highly
nonlinear materials, most of their geometrical and physical properties can be engineered, and
scaled for integration. One important step in the realization of laser light bullets has been the
experimental observation and manipulation of two-dimensional spatial solitons in optically
driven semiconductor micro-cavities. Dissipative solitons in driven nonlinear optical cavities
have been called cavity solitons⁹³,⁹⁸. Recently, devices generating cavity solitons without the
need for permanent external driving fields, namely cavity soliton lasers, have been
demonstrated⁹⁹,¹⁰⁰. One example is provided on Fig. 6b: it consists of two vertical cavity
surface emitting semiconductor lasers (VCSELs) coupled in a self-imaging configuration.
One of the lasers is below threshold to perform as a saturable absorber. A transient writing
beam enables on and off switching of cavity solitons (Fig. 6c). The implementation of mode-
locking in addition to spatial localisation in semiconductor devices remains a practical
challenge. On the fabrication side, preparation of uniform samples allowing homogeneous
injection of the electrical current density is complicated for devices whose transverse section
is of the order of 100 µm or larger. Composite cavities, with separate sections for gain and
saturation, need to have adequate lengths, with respect to the available gain bandwidth and the
relaxation time of electrical carriers. Despite these difficulties, the observation of transient
and pulsating cavity soliton dynamics¹⁰¹ provides reasonable hope that mode-locked cavity
soliton lasers could be realized in the near future, opening up new prospects for fundamental
studies, technological breakthroughs and all-optical data processing applications.

**Conclusion: the bright future of dissipative solitons**

While the whole field of passively mode-locked laser dynamics is being influenced by the
ideas of dissipative solitons, to the extent that some of the systems are called dissipative
soliton lasers, we feel that a timely clarification and dissemination of the underlying
principles of dissipative solitons would be of interest to the community of optical scientists.
We tried to make it clear that the variety of dissipative soliton dynamics is potentially
enormous. It is not critically limited by the particular physical properties such as the sign of chromatic dispersion, by the type of mode-locking mechanism or by the range of the pumping power. Large areas of knowledge in this field remain potentially unexplored. The notions of attractors and their transformations in the space of laser parameters have been central in making the dissipative soliton concept a functional one. Stability of these formations is an important attribute for applications. The development of original interdisciplinary tools using the principles of statistical physics and nonlinear science shall be required to tackle the behaviour of large groups of dissipative solitons. Such views are likely to create a fascinating new chapter of dissipative soliton dynamics. Moreover, appearance of *rogue waves* in dissipative nonlinear systems may also attract significant attention¹⁰². While we have concentrated on the close interplay between the dissipative soliton concept and mode-locked laser dynamics, one should keep in mind the broader application of the concept in nonlinear optics in general, as well as in many other scientific fields.

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## Acknowledgements

Ph.G. is supported by the Agence Nationale de la Recherche (project ANR-2010-BLANC-0417-01). N.A. acknowledges the support of the Australian Research Council (Discovery Project DP0985394).

## Author contributions

Ph.G. and N.A. have contributed equally to this Review article.

## Additional information

The authors declare no competing financial interests.
FIGURE LEGENDS

a

<figure><img src="image_2.png" /></figure>

b

<figure><img src="image_3.png" /></figure>

c

<figure><img src="image_4.png" /><figcaption>Figure 1</figcaption></figure>

**Figure 1. Illustration of the dissipative soliton concept.**

a. Qualitative differences between the solitons in Hamiltonian and dissipative systems, once the equation parameters are fixed.

b. Examples of dissipative solitons that are stable attractors of the CGLE master equation. They can take wide range of profiles: roughly hyperbolic-secant, flat-top, and (c) composite profiles. The system parameters are shown inside each figure. *Adapted from ref. 1.*
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 2</figcaption></figure>

**Figure 2. High-energy dissipative solitons.**

a. Experimental setup of an all-normal, dispersion-compensation-free, fibre laser cavity operating at a 70 MHz repetition rate and generating 31-nJ pulses. PBS, polarising beam splitter; BQP, birefringent quartz plate; $\lambda/4$, quarter wave plate, $\lambda/2$, half wave plate.

b. Characterization of the output pulses. Autocorrelation is performed after de-chirping outside the laser cavity, achieving 80 fs pulses with 200 kW of peak power. From ref. 43.

c. Illustration of the dissipative soliton resonance (DSR). The CGLE model with parameters shown inside the plot predicts that the DSR appears when the dispersion parameter approaches 0.697.

d. The amplitude and phase profiles of the stable pulse solutions for the values of D indicated in fig. 2.c by dots of corresponding colours. From ref. 49.
<figure><img src="image_10.png" /><figcaption>Figure 3</figcaption></figure>

<caption>**Figure 3. Variety of dissipative soliton molecules.**</caption>

a. Direct interaction between two dissipative solitons may lead to various types of soliton pairs in the CGLE model, depending on the values of the equation parameters.

b. Direct recording of stable self-assembled soliton molecules in a fibre laser (photograph upper left) experiment: (upper right) a soliton pair, (lower left) an irregularly-spaced 13-soliton molecule, and (lower right) a $1(011)_{N=14}$ macromolecule. A 30-GHz sampling oscilloscope is used.

c. The optical spectrum and the intensity autocorrelation (inset) of a stable soliton triplet. Stable spectral fringes are a direct consequence of self-phase locking of the three pulses with $+\pi/2$ and $-\pi/2$ consecutive phase differences, at a 1.2 ps relative separation. *Adapted from Ref. 76.*
<figure><img src="image_11.png" /><figcaption>Figure 4. Soliton explosions in a mode-locked Ti:sapphire laser.</figcaption></figure>

a. The experimental setup.

b. Recording of fast spectral dynamics that displays a soliton explosion event.

From ref. 80.
<figure><img src="image_12.png" /><figcaption>**Figure 5.** *Soliton rain in a mode-locked fibre laser*</figcaption></figure>

a. Sketch of an all-fibre erbium laser setup. EDF, erbium-doped fibre; WDM, 980/1550 wavelength multiplexer; ISO, optical isolator; PC1,2, polarization controllers; DCF, dispersion compensation fibre; OC, output coupler; PBS, integrated polarising beam splitter; PhD, 5.6-GHz photodiode.

b. Stroboscopic recording of the output optical intensity trace using a 6-GHz real-time oscilloscope, showing three main field components: a noisy background, drifting solitons, and a condensed phase of bound solitons (highest peak). A temporal magnification of the condensed phase is performed with a 30-GHz sampling oscilloscope. *Adapted from ref. 82.*
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

Figure 6

<caption>Figure 6. Towards a mode-locked cavity soliton laser</caption>

a. Illustration of the notion of light bullets.

b. Sketch of a cavity soliton laser experiment. VCSEL 1,2, vertical cavity surface emitting semiconductor lasers that are coupled in a self-imaging configuration; BS, beam splitter; CCD 1,2: imaging devices; WB, writing beam.

c. Cavity solitons switched on and off at selected locations with the transient application of the external writing beam (WB). After ref. 100.