Abstract
Instabilities of uniform states are ubiquitous processes occurring in a variety of spatially extended nonlinear systems. These instabilities are at the heart of symmetry breaking, condensate dynamics, selforganisation, pattern formation, and noise amplification across diverse disciplines, including physics, chemistry, engineering, and biology. In nonlinear optics, modulation instabilities are generally linked to the socalled parametric amplification process, which occurs when certain phasematching or quasiphasematching conditions are satisfied.
In the present review article, we summarise the principle results on modulation instabilities and parametric amplification in nonlinear optics, with special emphasis on optical fibres. We then review stateoftheart research about a peculiar class of modulation instabilities (MIs) and signal amplification processes induced by dissipation in nonlinear optical systems. Losses applied to certain parts of the spectrum counterintuitively lead to the exponential growth of the damped mode themselves, causing gain through losses. We discuss the concept of imaging of losses into gain, showing how to map a given spectral loss profile into a gain spectrum. We demonstrate with concrete examples that dissipationinduced MI, apart from being of fundamental theoretical interest, may pave the way towards the design of a new class of tuneable fibrebased optical amplifiers, optical parametric oscillators, frequency comb sources, and pulsed lasers.
Modulation instabilities (MIs)
In this article, the state of the art of peculiar dissipationinduced MIs in nonlinear optics is reviewed and contextualised with respect to other known instabilities. New possible applications are suggested, and potential future directions for this research field are indicated.
Local growth initiated by instabilities underpins the dynamic evolution of a vast majority of natural phenomena, ranging from living organisms to cosmological and geological processes. These underlying mechanisms are widely exploited in the operation of various engineering systems and devices. Many patternforming and selforganisation processes that dramatically modify the behaviour of physical, chemical, or biological systems can be explained in terms of nonlinear instabilities. Classical conceptual examples of such instabilities include the onset of turbulence^{1}; formation of coherent structures in lasers from a continuous wave (CW) state^{2,3}; emergence of coherent structures in chemical reactions, biological systems, ocean, plasmas, and gases; and pattern formation in a variety of physical systems far from equilibrium^{4}.
The ubiquity of the growththroughinstability paradigm is linked to the universal nature of the underlying mathematical models, such as the complex Ginzburg–Landau equation^{5} (CGLE) or the nonlinear Schrödinger equation (NLSE). The NLSE is one of the basic mathematical models describing nonlinear dispersive wave interactions in a number of natural and engineering systems^{6,7}.
An enormous number of different MIs have been studied in nonlinear science. In this paper, we will describe some of the most relevant instabilities associated with selforganisation processes and signal amplification in nonlinear optics.
From a methodological point of view, it is important to stress that while we discuss here the instabilities of spatial or temporal homogeneous states and their associated pattern formation, we will address neither the stability properties of the patterns generated through the associated selforganisation process nor the stability of coherent structures in general, such as solitons or light pulses.
Possibly the most known and studied MI is the Benjamin–Feir instability, socalled after the names of its first discoverers in fluids. It consists of the spontaneous modulation of the uniform state solution of the NLSE, CGLE, and their generalisations^{8} and has been deeply studied in the context of plasmas^{8} and hydrodynamics^{9} and in nonlinear optics^{10,11}.
Benjamin–Feir instability can be understood in terms of a synchronisation process between the homogenous mode and two symmetrically detuned spectral sidebands occurring for certain specific parameters of the system. When synchronisation occurs, the spectral sidebands experience exponential amplification, leading to a modulation of the homogeneous state. In single mode optical fibres, Benjamin–Feir instability occurs owing to the interplay between Kerr nonlinearity and anomalous dispersion, but it can also arise due to crossphase modulation between two copropagating light beams^{12} or due to polarisation effects^{13,14,15}. Benjamin–Feir instability occurs in fibre amplifiers^{16,17,18}, is the initiating mechanism of supercontinuum formation in photonic crystal fibres^{19} and manifests itself in twodimensional optical systems, where it induces pattern formation^{20}. It is also important to stress its nonlinear stage connection to soliton formation^{21}, breather dynamics^{22,23,24} and rogue waves generation^{25}.
A second paradigmatic MI is the Faraday instability. Faraday instability occurs due to the periodic temporal modulation of one parameter of a spatially extended system. It was observed for the first time in a vertically shaken mercury layer by Michael Faraday in 1832^{26}. A general theory of the process was obtained many years later by Benjamin and Ursell^{27}.
Faraday instability can be understood in terms of a synchronisation process mediated by the periodic forcing between the homogeneous mode and small amplitude modulation waves oscillating on top of a finite field background. The periodic forcing excites a series of parametric resonances occurring at integer multiples of half of the forcing frequency. The associated spatial pattern formation occurs with a wavenumber determined by the parametrically excited temporal frequency through the dispersion relation of the particular system being considered^{28}. Faraday instability has been observed in a variety of physical systems, including chemical reactions^{29}, granular media^{30}, plasmas^{31}, and Bose–Einstein condensates^{32,33}. In nonlinear optics, Faraday instability has been observed in dielectric layered media^{34}, in lasers^{35} and in optical fibres. In nonlinear fibre optics, the spatial evolution of the electric field defined in a comoving temporal reference frame occurs along a spatial coordinate. Spatial and temporal coordinates are interchanged compared to the case of spatially extended systems. Hence, a periodic modulation of the dispersion or nonlinearity coefficient along the spatial evolution coordinate may trigger Faraday instability^{36,37,38,39,40,41}; the associated pattern formation can be observed in fibre resonators^{42,43}.
The Turing instability is another example of an MI that has a significant role in nonlinear optics, especially concerning patternformation processes. It was proposed originally by Alan Turing in a milestone paper aiming at understanding the processes underlying morphogenesis in biological systems through coupled nonlinear reaction–diffusion equations. Turing instability is based on the combined action of two principles: local selfenhancement and lateral inhibition^{44}. Turing instability and its associated pattern formation can also be generalised to nonlinear optics. Local selfenhancement can be provided, for instance, by the Kerr effect, while diffraction or dispersion play a role analogous to diffusion^{45,46}. The spatial Turing instability may occur in an optical cavity containing a Kerr medium, while the temporal one can be observed in externally driven passive fibre resonators. Turing patterns can also arise in OPOs or in lasers^{3,47}.
Cavity boundary conditions^{48,49} and the internal dynamics of the field solutions^{50} in nonlinear fibre optics may also be responsible for inducing MI, while more sophisticated forms of instabilities may arise in multimode fibres^{51,52,53,54}.
Concluding this brief classification of instabilities in nonlinear optics, it is important to stress that a variety of instabilities affect the CW solution of lasers not only in the most diverse operational modes, leading to turbulent dynamics^{55,56}, but also to selforganised spontaneous locking among oscillating modes, with the consequent generation of a regular pulse train; the latter case is exemplified by the Risken–Nummedal–Graham–Haken instability^{3,57,58,59} or by the socalled MI laser^{60}.
Parametric amplification
The invention of the laser has dramatically changed the science of lightmatter interaction, allowing the deep study of nonlinear optics and initiating a scientific and technological revolution. Intense light beams generated by lasers and propagating through material media are able to trigger electronic oscillations in atoms and molecules that in turn reradiate an electric field, hence determining a selfaction of the propagating light on itself. Such nonlinear selfaction of light on itself, mediated by matter, is what allows interaction among photons with consequent generation of new frequencies of the electromagnetic field starting from vacuum oscillator fluctuations. A variety of nonlinear processes are thus possible, depending on the properties of the material medium and of the light waves involved in the interaction, including second and third harmonic generation, optical phaseconjugation, Raman and Brillouin scattering and many more^{61}.
One of the most important forms of wave interaction in nonlinear optics is fourwave mixing^{6,61}. Fourwave mixing allows energy transfer between interacting waves in a medium with thirdorder nonlinearity, given that certain phasematching conditions corresponding to energy and momentum conservation are satisfied^{6,61,62,63,64,65}. Indeed, fourwave mixing can be considered as the underlying physical mechanism responsible for the previously described Benjamin–Feir instability.
In general, two pump fields oscillating at frequencies Ω_{p1} and Ω_{p2} amplify two sidebands with frequencies Ω_{s} and Ω_{i}; this is the socalled nondegenerate fourwave mixing. If Ω_{p1} = Ω_{p2}, the process is said to be degenerate. The latter phenomenon can be routinely observed in an optical fibre, where phasematching conditions are achieved in case of anomalous dispersion: a sufficiently intense pump field amplifies two spectral sidebands symmetrically located with respect to the pump wave frequency, which, in this context, are called the signal and idler. From a fundamental point of view, the process corresponds to the annihilation of two quanta of energy in the pump field with the simultaneous creation of two quanta, one in each of both sideband fields. Sideband amplification thus leads to homogeneous state destruction through modulation of the homogeneous field solution. In the field of optical communications, such MI is the underlying mechanism of the socalled fibreoptics parametric amplifiers, which are promising devices for signal amplification^{66,67,68,69,70,71,72}.
The physics of nonlinear interaction also includes the possibility of an inverse fourwave mixing process, where energy flows backwards from two powerful sidebands to the pump frequency^{73,74}.
When a great number of light waves having different frequencies interact, propagating through a nonlinear medium, specific techniques from the field of wave turbulence, such as wave kinetics or other statistical approaches^{1,75,76,77,78} (see also references therein), are needed to successfully describe fourwave mixing and MI processes in the presence of incoherence.
From a crossdisciplinary point of view, it is interesting to mention that the dynamics of an atomic Bose–Einstein condensate, in the mean field approximation, is described by the Gross–Pitaevskii equation, which is formally identical to the NLSE with interchanged spatial and temporal coordinates. Hence, for Bose–Einstein condensates, four wavemixing occurs in momentum space where nonlinearity is provided by interactions among particles^{79,80,81}.
Gain through losses (GTL): the concept
If we consider the classical Benjamin–Feir instability of CW radiation, dissipation, consisting in the symmetric damping of both sideband waves, is normally believed to reduce their growth rate, stabilising the already unstable modes but slightly extending the region of unstable frequencies^{82}. Intuitively, one could expect that damping would reduce the growth rate of modes amplified by MI. There are, however, physical systems in which the presence of dissipation can induce MIs. Dissipationinduced MIs have been studied extensively in classical mechanics and in fluid dynamics, where friction or viscosity can cause the instability of equilibrium states in many finitedimensional physical systems, such as gyroscopes, the levitron, and the rotating shaft, and in infinitedimensional ones, with the baroclinic instability as a key example^{83}.
However, here, we are addressing a very different concept of dissipationinduced MIs, e.g., instabilities induced by an asymmetric dissipation profile for signal and idler waves.
The first example of dissipationinduced MI in nonlinear optics has been studied analytically and experimentally by Tanemura and coauthors^{84}. A reduced threewave model (originally introduced in ref. ^{85} to capture the nonlinear stage of Benjamin–Feir instability) describes the interacting pump, signal and idler waves. It was shown that in a defocussing NLSE with unbalanced losses for the signal and idler, both sidebands can be amplified. In the same article, theoretical predictions were validated experimentally for CW radiation propagating in a normal dispersion optical fibre, where a counterpropagating probe beam stimulates Brillouin scattering. Choosing the idler in the vicinity of the Brillouin gain frequency corresponds to inducing losses at the idler frequency, and consequently, it was experimentally shown that the signal wave suffered amplification upon propagation due to such asymmetric spectral dissipation.
An alternative theoretical derivation of the gain in such amplification processes was obtained later by other authors, again starting from the reduced threewave model^{86}.
Rather counterintuitively, unbalanced (asymmetric in spectrum) losses for signal and idler waves in a modulationally stable system result in a nonlinear dissipative process that eventually leads to homogeneous state destruction at the expense of substantial signal and idler amplification. In particular, when strong dissipation is applied to frequencies in the vicinity of the signal frequency Ω_{s}, both signal and idler waves, the latter having a frequency relative to the pump Ω_{i} = −Ω_{s}, experience a significant net growth. Losses applied on a given spectral mode are converted into a net growth of the mode amplitude itself.
It is important to stress the difference between the results published in ref. ^{84} and other nonphasematched parametric amplification processes, where a spectrally asymmetric dissipative susceptibility is present, for instance, when Raman scattering is involved^{87,88}. In the latter case, the asymmetry of the dissipative part of the susceptibility allows the amplification of the Stokes wave and would, at first glance, entail a damping of the antiStokes wave. The antiStokes component can be ultimately amplified owing to a fourwave mixing interaction with the previously amplified Stokes wave itself. However, the gain experienced by the Stokes wave makes amplification of the otherwise lossy antiStokes wave possible. In dissipationinduced MI^{84}, it is the presence of losses that is responsible for triggering the amplification of the lossy modes themselves.
In the following sections of this paper, we show, through some paradigmatic examples, that the concept originally suggested by Tanemura and coauthors^{84} is indeed very general in nonlinear optical systems and can occur in different setups and configurations. To unify the descriptions of the different but related phenomena, we propose to define such dynamics as the GTL process.
We will illustrate in the following part of this paper how the GTL process can lead to the intriguing imaging of losses into gain (ILG), where the resulting MI gain spectrum is a replica of the spectral profile of the dissipation function used. The GTL process and ILG concepts are illustrated pictorially in Fig. 1.
The GTL process is a nontrivial example of dissipationinduced MI in nonlinear optical systems, where the damping acting on perturbation modes causes a selective energy transfer from the homogeneous state to the damped frequencies, inducing a substantial amplification. We can anticipate that this idea can be potentially exploited for fibrebased optical amplifiers, for generation of pulses in resonators and to achieve gain in nonphasematched parametric oscillators. Before entering into a detailed characterisation of these various applications and describing the connection of the GTL process with other MIs discussed in the literature, we will first describe some relevant theoretical results.
Theory of GTL
In refs. ^{84,86}, simplified models for the GTL process in fibre optics were introduced, allowing us to derive expressions for the gain of the process, which show that its efficiency increases with the nonlinearity and with the asymmetry in the loss strength for signal and idler waves.
We provide here new results derived from a more general theory that allows us to include the impact of a generic absorber, rigorously satisfying the Kramers–Kronig relations and hence physical causality. To provide a general insight into the GTL process, we first consider the evolution equation for the electromagnetic field slowly varying envelope A(z,τ), defined in the local time reference frame τ and propagating along the spatial direction denoted by the coordinate z of an optical fibre with normal group velocity dispersion (GVD) (β_{2} > 0) and focusing Kerr nonlinearity (γ > 0), coupled to an absorbing twolevel system (2LS) that acts as a distributed spectral filter (see Supplementary Information for the derivation):
Here, \(\theta = {\cal F}^{  1}\left\{ {\chi _{2{\rm LS}}\left({\Omega} \right)} \right\}\), where \({\cal F}^{  1}\) is the inverse Fourier transform and * denotes convolution.
The complex susceptibility:
contains both a dissipative, Re[χ_{2LS}(Ω)], and a dispersive, Im[χ_{2LS}(Ω)], contribution and is characterised by the resonance frequency Ω_{a}, spectral width γ_{⊥} and strength g.
A linear stability analysis can be performed using the following ansatz for the electric field: \(A\left( {z,t} \right) = A_{\rm s}\left[ {1 + a_ + \left( z \right){\rm e}^{  {\rm i}\Omega \tau } + a_  \left( z \right){\rm e}^{{\rm i}\Omega \tau }} \right],\) where A_{s} =\(\sqrt P\) exp(iγPz) and P is the power of the pump field, assuming for simplicity that χ_{2LS}(Ω_{p}) ≈ 0, e.g., that the pump field at frequency Ω = Ω_{p} is not affected by the presence of the 2LS, corresponding to the limit Ω_{a} » γ_{⊥}. A standard procedure leads to linearised equations for the complex amplitudes of perturbations a_{±}(z):
From Eqs. (3), in particular, from those for a_{+} and \(a_  ^ \ast ,\) it is straightforward to derive the 2by2 stability matrix from the coefficients which multiply the amplitudes of modes a_{+} and \(a_  ^ \ast .\) Diagonalization of the stability matrix provides two eigenvalues λ_{1,2}(Ω), from which the instability increment can be calculated as λ_{m}(Ω) = max{Re[λ_{1,2}(Ω)]}. If λ_{m}(Ω)>0, then the CW solution is unstable and modulation modes with frequency Ω experience exponential growth with a characteristic exponent given by λ_{m}(Ω).
GTLbased fibre amplifier
We discuss in detail the relevant implications of the stability analysis of Eq. (1).
In Fig. 2, the instability increment λ_{m}(Ω) and the results of full numerical simulations of Eq. (1) are depicted, showing the dependence of the MI gain on the position of the 2LS resonance. The main trends can be summarised as follows: the process efficiency increases when the filter frequency is close to the pump frequency. The process efficiency is also directly proportional to the nonlinearity (and consequently to the pump power) and to the loss strength and is inversely proportional to the GVD.
We have seen that the unbalanced losses for signal and idler waves lead to an MI of the homogeneous solution in normal dispersion without the standard requirement of anomalous dispersion in order to satisfy phasematching conditions.
In the temporal domain, the field exhibits periodic oscillations with frequency given by the position of the filter, but when the instability is in the linear stage, these oscillations have a very small amplitude and are indistinguishable from the homogeneous background. Examples of interesting temporal dynamics will be presented in more detail below, where we will describe how such dissipationinduced instability leads to the generation of pulses in a fibre ring resonator.
Note that in contrast to the other known MIs^{8,36,37,38,39,40,41,42,43}, where the growing modes are locked to the pump due to the balance between the nonlinearity and dispersion or due to the presence of a parametric periodic forcing, in our case, the unbalanced losses for signal and idler waves are responsible for the locking.
Considering two different 2LSs, with resonance frequencies positioned symmetrically with respect to the pump frequency, the wellknown stabilising effect of balanced symmetric dissipation is recovered, and modes with frequencies Ω ≈ Ω_{s} and Ω ≈ Ω_{i} are damped (see Supplementary Information). It is also worth stressing that the particular implementation of the GTL process discussed here potentially allows us to overcome, in terms of tunability, the limitation of the setup proposed in ref.^{84}, where the gain occurs for spectral modes separated from the pump frequency approximately by the Brillouin shift. It is also important to comment on the fact that the instability is not caused by the dispersive part of the absorber susceptibility; indeed, if the latter is removed, the analytical and numerical results are not significantly affected.
Imaging of losses into gain
The GTL process can also occur for field propagation in a standard passive fibre with lumped filters. This can be realised by considering a chain of fibre segments of length L_{s} with a filter located at the end of each segment and having a loss profile that peaks around the angular frequency ω_{f}. One farreaching consequence of this particular implementation of the GTL process is the possibility of designing arbitrary gain spectra: MI on demand. Indeed, the spectral shape of the lumped filter absorption/reflection profile, after the monochromatic electric field propagates through the chain, is imaged onto the evolved field spectrum in the form of a gain profile having the same original spectral shape. As anticipated, we call this powerful nonlinear photonics design tool the imaging of losses into gain (ILG). In Fig. 3, we show that the ILG allows successful tailoring of the MI gain spectrum in terms of the geometrical shape, width and frequency offset from the pump (more details about the filters can be found in the Supplementary Information). We underline that the cause of the sideband growth is not a parametric resonance due to the periodic action of the filters. Indeed, a hypothetical parametric resonance would produce gain in identical spectral regions for all three different filters considered in Fig. 3, as the spatial periodicity is the same in all three cases. Energy transfer occurs from the pump field exactly towards the spectral regions where the filters induce losses. Remarkably, losses can also be applied flexibly on the idler wave to achieve signal amplification, which is more convenient in a realworld setup design.
Pulses and frequency combs in a ring resonator
Another scenario where the GTL process can occur is the externally driven unidirectional ring fibre resonator with a lumped spectral filter, where the field’s dynamics is described by a generalised version of the wellknown Lugiato–Lefever equation^{45,48,89}. The generalised Lugiato–Lefever equation (GLLE) reads:
where T is the slow time, Δ the detuning, S the injection and f(τ) the inverse Fourier transform of the spectral filter reflectivity profile \(\tilde f\left( \omega \right) = \mu e^{  \left[ {\left( {\omega  \omega _{\rm f}} \right)^2/\sigma _{\rm f}^2} \right]}\), and μ, ω_{f} and σ_{f} are the filter strength, central frequency and width, respectively. A detailed derivation of Eq. (4) is provided in the Supplementary Information.
The GTL process leads to MI in the GLLE with associated pattern formation in the nonlinear stage. Generated patterns consist of a regular pulse train on a finite nonzero field background, with a repetition rate corresponding to the inverse of the frequency detuning between the pump and the filter, while the corresponding spectrum is a frequency comb (see Fig. 4).
GTL in OPOs
To stress the universality of the GTL process, we further provide an example of an optical system where this process can occur. OPOs are versatile coherent light sources with a variety of applications^{61}. Their operational principle relies on the frequency conversion process allowed by the nonlinear susceptibility of the medium through which the electromagnetic waves propagate.
It is considered a wellknown fact that in such systems, energy transfer from the pump wave to the signal can be achieved, provided that phasematching conditions corresponding to energy and momentum conservations are satisfied. When phasematching is difficult to achieve, the socalled quasiphase matching technique based on periodic poling can be successfully employed^{90}. Up to now, unbalanced dissipation for signal and idler waves has been considered in the literature as a mechanism to improve the performances of OPOs already operating under the phasematching condition, in particular for achieving higher beam quality and improved conversion efficiency by reducing back conversion to the pump^{91,92,93}. More recently, results have been reported in which, owing to a strong spectrally symmetric dissipation acting on the idler, a substantial signal amplification has been observed even for signal waves not satisfying the standard phasematching conditions^{94}. The results presented in ref. ^{94}, although amplification is observed on the signal while losses are applied to the idler, are a clear signature of the GTL process; indeed, this was also the case in the Tanemura and coauthors experiment^{84}. In the transient dynamics of the GTL process, the modes symmetric to the lossy ones with respect to the pump are amplified first, and only later do the amplitudes of the damped modes begin to grow (see also Fig. 2).
To better demonstrate the GTL dynamics in OPOs, we analytically show that dissipation can induce amplification in situations where standard parametric amplification should in principle be absent. Let us consider an OPO in the presence of unbalanced spectral losses for signal and idler waves.
The classical OPO evolution equations for media with χ_{2} nonlinearity are given in ref. ^{61} and, after a simple change of variables, in the limit of a powerful pump field and neglecting pump depletion, can be rewritten in the form:
where A_{s} is the signal and A_{i}^{*} the conjugate of the idler field amplitude. Δk_{s,i} are the detunings, which are considered equal and satisfy Δk_{s} = Δk_{i} = Δk/2, where Δk = k_{p}−k_{s}−k_{i} is the mismatch parameter (k_{p,s,i} being the wavenumbers of the fields involved in the process). α_{s,i} are the signal and idler losses, while the coupling constants M_{s} and M_{i} depend on the pump intensity such that their product can be written as M_{s}·M_{i} = ρA_{p}^{2}, with A_{p} being the pump field amplitude and ρ a proportionality factor. The linear stability analysis of the zero solution of Eqs. 5a and 5b gives the following eigenvalues:
If losses are neglected, then there is energy transfer from the pump to the signal and idler waves only if Δk^{2} ≤ 4ρA_{p}^{2}, while the presence of symmetric losses α_{s} = α_{i} just reduces the amplification, as one can naturally expect. Surprisingly, if losses are different for the signal and idler (α_{s} ≠ α_{i}), then both wave amplitudes can grow even if the standard amplification conditions are not satisfied. Let us consider for the sake of simplicity the case where α_{i} = 0 and α_{s} ≠ 0. The dependence of the growth exponent λ = max[Re(λ_{±})] as a function of α_{s} and phase mismatch Δk is given in Fig. 5. When λ>0, signal and idler amplification occurs. We have thus shown analytically, in agreement with previous findings present in the literature^{91,92,93}, that the GTL process can also occur in OPOs with secondorder nonlinearity, which can be useful in cases when standard phasematching conditions are difficult to achieve or to obtain a smoother gain profile.
Connection with dissipative Faraday instability
Recent results appeared in the literature concerning a novel MI induced by a modulation periodic in time, zig–zag in the wavenumber domain, of lumped losses, for spatially extended systems described by the CGLE^{95}. The peculiar periodic dissipative forcing leads to an MI of the homogeneous state of the system and to an associated pattern formation in space. Losses are applied using filters, which alternatively damp modes around signal (k_{s}) and idler (k_{i}) wavenumbers, with the filters transmission profiles centred around (k_{f+,−} = k_{s,i}).
In fibre optics, the dissipative Faraday instability is induced by a forcing that is periodic in space and alternating (zig–zag) in the frequency domain. A natural setting where the instability can be observed is a linear cavity fibre laser with spectral filters having blue and red frequencydetuned reflectivity profiles as cavity mirrors. In practice, such instability is excited using two frequencydetuned spectral filters, so damped modes in a certain frequency band experience gain; selection of the maximally unstable mode occurs through a parametric resonance condition, such as that in more classical configurations corresponding to Faraday instability, where the GVD or nonlinearity are periodically modulated along the propagation direction of the light in an optical fibre^{36,37,38,39,40,41,42,43}.
For the latter reason, such MI has been denoted as dissipative Faraday (parametric) instability. In this case, the mechanism that triggers the instability is a GTL process, so the damped modes themselves experience amplification due to losses. Most importantly, experimental observations using a Raman laser with detuned spectral filters and numerical studies have shown that the dissipative Faraday instability is a promising tool for achieving high repetition rate harmonic modelocking in fibre lasers^{96,97}.
Conclusions and future perspectives
In conclusion, we have reviewed the stateoftheart about experimental and theoretical studies of amplification and MI processes in nonlinear optics, with particular emphasis on those induced by a spectrally asymmetric dissipation function. The dissipation can be applied in a stationary or periodic fashion; in both cases, an average net growth of damped modes may occur.
We have furthermore included some novel results and extended the concept of MI induced by losses to various important systems and models: the NLSE coupled to a twolevel distributed system; the NLSE with periodic lumped filtering, leading to ILG (mapping of losses spectral profile into gain), which opens the possibility for a new class of fibreopticbased tuneable amplifiers; the coupled wave equations describing OPOs; and the GLLE that models an externally driven ring fibre resonator with an intracavity spectral filter. In the latter case, we have predicted for the first time the pattern formation associated with the MI, together with the possibility of frequency comb generation. We have finally mentioned the existing link between dissipationinduced MIs and the dissipative Faraday instability induced by periodic zig–zag modulation of spectral losses. All of the phenomena described in detail here share the following feature: gain for certain frequencies is available if losses are applied to the same frequencies.
Open challenges in this research direction include the development of a fundamental quantum theory of the GTL process, as well as a series of detailed studies aiming at understanding the concrete potentialities of the results summarised here for future photonics technologies.
Due to the universality of the NLSE, we expect that the GTL process could also be studied outside of nonlinear optics, for instance, in Bose–Einstein condensates, where the unbalanced spectral losses for different spectral components can be realised by means of dynamic optical lattices^{98}, and in other spatially extended nonlinear systems, such as fluids, plasmas, chemical reactions and biological systems.
References
 1.
Zakharov, V. E., L’vov, V. S. & Falkovich, G. Kolmogorov Spectra of Turbulence I: Wave Turbulence. (SpringerVerlag, Berlin, 1992).
 2.
Staliunas, K. & SánchezMorcillo, V. J. Transverse Patterns in Nonlinear Optical Resonators. (Springer: Berlin, 2003).
 3.
Lugiato, L., Prati, F. & Brambilla, M. Nonlinear Optical Systems. (Cambridge University Press, Cambridge, 2015).
 4.
Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).
 5.
Aranson, I. S. & Kramer, L. The world of the complex GinzburgLandau equation. Rev. Mod. Phys. 74, 99–143 (2002).
 6.
Agrawal, G. P. Nonlinear Fiber Optics. 4th edn, (Academic Press, San Diego, 2006).
 7.
Ablowitz, M. J. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons.. (Cambridge University Press, Cambridge, 2011).
 8.
Zakharov, V. E. & Ostrovsky, L. A. Modulation instability: the beginning. Physica D 238, 540–548 (2009).
 9.
Benjamin, T. B. & Feir, J. E. The disintegration of wave trains on deep water part 1. Theory. J. Fluid. Mech. 27, 417–430 (1967).
 10.
Bespalov, V. I. & Talanov, V. I. Filamentary structure of light beams in nonlinear liquids. JETP Lett. 3, 307–311 (1966).
 11.
Tai, K., Hasegawa, A. & Tomita, A. Observation of modulational instability in optical fibers. Phys. Rev. Lett. 56, 135–138 (1986).
 12.
Agrawal, G. P. Modulation instability induced by crossphase modulation. Phys. Rev. Lett. 59, 880–883 (1987).
 13.
Berkhoer, A. L. & Zakharov, V. E. Selfexcitation of waves with different polarizations in nonlinear media. Sov. Phys. JETP 31, 486–490 (1970).
 14.
Wabnitz, S. Modulational polarization instability of light in a nonlinear birefringent dispersive medium. Phys. Rev. A 38, 2018–2021 (1988).
 15.
Trillo, S., Wabnitz, S., Stolen, R. H., Assanto, G. & Seaton, C. T. et al. Experimental observation of polarization instability in a birefringent optical fiber. Appl. Phys. Lett. 49, 1224–1226 (1986).
 16.
Agrawal, G. P. Modulation instability in Erbiumdoped fiber amplifiers. IEEE Photon. Technol. Lett. 4, 562–564 (1992).
 17.
Van Tartwijk, G. H. M. & Agrawal, G. P. MaxwellBloch dynamics and modulation instabilities in fiber lasers and amplifiers. J. Opt. Soc. Am. B 14, 2618–2627 (1997).
 18.
Turitsyn, S., Rubenchik, A. M. & Fedoruk, M. P. On the theory of the modulation instability in optical fiber amplifiers. Opt. Lett. 35, 2864–2866 (2010).
 19.
Dudley, J. M., Goënty, G. & Coen, S. Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78, 1135–1184 (2006).
 20.
Kip, D., Soljacic, M., Segev, M., Eugenieva, E. & Christodoulides, D. N. Modulation instability and pattern formation in spatially incoherent light beams. Science 290, 495–498 (2000).
 21.
Zakharov, V. E. & Shabat, A. B. Exact theory of twodimensional selffocussing and onedimensional selfmodulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972).
 22.
Kuznetsov, E. Solitons in a parametrically unstable plasma. Sov. Phys. Dokl. 22, 507–508 (1977).
 23.
Akhmediev, N. & Korneev, V. I. Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69, 1089–1093 (1986).
 24.
Peregrine, D. H. Water waves, nonlinear Schrödinger equations and their solutions. ANZIAM J. 25, 16–43 (1983).
 25.
Dudley, J. M., Dias, F., Erkintalo, M. & Genty, G. Instabilities, breathers and rogue waves in optics. Nat. Photon. 8, 755–764 (2014).
 26.
Faraday, M. On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Philos. Trans. R. Soc. Lond. 121, 299–340 (1831).
 27.
Benjamin, T. B. & Ursell, F. The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. A 225, 505–515 (1954).
 28.
Coullet, P., Frisch, T. & Sonnino, G. Dispersioninduced patterns. Phys. Rev. E 49, 2087–2090 (1994).
 29.
Petrov, V., Ouyang, Q. & Swinney, H. L. Resonant pattern formation in a chemical system. Nature 388, 655–657 (1997).
 30.
Melo, F., Umbanhowar, P. & Swinney, H. L. Transition to parametric wave patterns in a vertically oscillated granular layer. Phys. Rev. Lett. 72, 172–175 (1994).
 31.
Drake, J. F., Kaw, P. K., Lee, Y. C., Schmid, G. & Liu, C. S. et al. Parametric instabilities of electromagnetic waves in plasmas. Phys. Fluids 17, 778–785 (1974).
 32.
Staliunas, K., Longhi, S. & De Valcárcel, G. J. Faraday patterns in BoseEinstein condensates. Phys. Rev. Lett. 89, 210406 (2002).
 33.
Engels, P., Atherton, C. & Hoefer, M. A. Observation of Faraday waves in a BoseEinstein condensate. Phys. Rev. Lett. 98, 095301 (2007).
 34.
Centurion, M., Porter, M. A., Pu, Y., Kevrekidis, P. G. & Frantzeskakis, D. J. et al. Modulational instability in a layered Kerr medium: theory and experiment. Phys. Rev. Lett. 97, 234101 (2006).
 35.
Szwaj, C., Bielawski, S., Derozier, D. & Erneux, T. Faraday instability in a multimode laser. Phys. Rev. Lett. 80, 3968 (1998).
 36.
Matera, F., Mecozzi, A., Romagnoli, M. & Settembre, M. Sideband instability induced by periodic power variation in longdistance fiber links. Opt. Lett. 18, 1499–1501 (1993).
 37.
Abdullaev, F. K., Darmanyan, S. A., Bischoff, S. & Sørensen, M. P. Modulational instability of electromagnetic waves in media with varying nonlinearity. J. Opt. Soc. Am. B 14, 27–33 (1997).
 38.
Smith, N. J. & Doran, N. J. Modulational instabilities in fibers with periodic dispersion management. Opt. Lett. 21, 570–572 (1996).
 39.
Abdullaev, F. K., Darmanyan, S. A., Kobyakov, A. & Lederer, F. Modulational instability in optical fibers with variable dispersion. Phys. Lett. A 220, 213–218 (1996).
 40.
Bronski, J. C. & Kutz, J. N. Modulational stability of plane waves in nonreturntozero communications systems with dispersion management. Opt. Lett. 21, 937–939 (1996).
 41.
Mussot, A., Conforti, M., Trillo, S., Copie, F. & Kudlinski, A. Modulation instability in dispersion oscillating fibers. Adv. Opt. Photon. 10, 1–42 (2018).
 42.
Staliunas, K., Hang, C. & Konotop, V. V. Parametric patterns in optical fiber ring nonlinear resonators. Phys. Rev. A 88, 023846 (2013).
 43.
Conforti, M., Mussot, A., Kudlinski, A. & Trillo, S. Modulational instability in dispersion oscillating fiber ring cavities. Opt. Lett. 39, 4200–4203 (2014).
 44.
Turing, A. M. The chemical basis of morphogenesis. Philos. Trans. R. Soc. B 237, 37–72 (1952).
 45.
Lugiato, L. A. & Lefever, R. Spatial dissipative structures in passive optical systems. Phys. Rev. Lett. 58, 2209–2211 (1987).
 46.
Copie, F., Conforti, M., Kudlinski, A., Mussot, A., & Trillo, S. Competing Turing and Faraday instabilities in longitudinally modulated passive resonators. Phys. Rev. Lett. 116, 143901 (2016).
 47.
Staliunas, K. & SánchezMorcillo, V. J. Turing patterns in nonlinear optics. Opt. Commun. 177, 389–395 (2000).
 48.
Haelterman, M., Trillo, S. & Wabnitz, S. Additivemodulationinstability ring laser in the normal dispersion regime of a fiber. Opt. Lett. 17, 745–747 (1992).
 49.
Coen, S. & Haelterman, M. Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber. Phys. Rev. Lett. 79, 4139–4142 (1997).
 50.
Perego, A. M., Smirnov, S. V., Staliunas, K., Churkin, D. V. & Wabnitz, S. Selfinduced Faraday instability laser. Phys. Rev. Lett. 120, 213902 (2018).
 51.
Guasoni, M. Generalized modulational instability in multimode fibers: wideband multimode parametric amplification. Phys. Rev. A 92, 033849 (2015).
 52.
Longhi, S. Modulational instability and space–time dynamics in nonlinear parabolicindex optical fibers. Opt. Lett. 28, 2363–2365 (2003).
 53.
Krupa, K., Tonello, A., Barthélémy, A., Couderc, V. & Shalaby, B. M. et al. Observation of geometric parametric instability induced by the periodic spatial selfimaging of multimode waves. Phys. Rev. Lett. 116, 183901 (2016).
 54.
Wright, L. G., Liu, Z. W., Nolan, D. A., Li, M. J. & Christodoulides, D. N. et al. Selforganized instability in gradedindex multimode fibres. Nat. Photon. 10, 771–776 (2016).
 55.
Haken, H. Analogy between higher instabilities in fluids and lasers. Phys. Lett. A 53, 77–78 (1975).
 56.
Arecchi, F. T., Giacomelli, G., Ramazza, P. L. & Residori, S. Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics. Phys. Rev. Lett. 65, 2531–2534 (1990).
 57.
Risken, H. & Nummedal, K. Instability of off resonance modes in lasers. Phys. Lett. A 26, 275–276 (1968).
 58.
Graham, R. & Haken, H. Quantum theory of light propagation in a fluctuating laseractive medium. Z. Phys. A 213, 420–450 (1968).
 59.
Pessina, E. M., Bonfrate, G., Fontana, F. & Lugiato, L. A. Experimental observation of the RiskenNummedalGrahamHaken multimode laser instability. Phys. Rev. A 56, 4086–4093 (1997).
 60.
Nakazawa, M., Suzuki, K. & Haus, H. A. The modulational instability laser. I. Experiment. IEEE J. Quant. Electron. 25, 2036–2044 (1989).
 61.
Boyd, R. W. Nonlinear Optics. (Academic Press, San Diego, 1991).
 62.
Armstrong, J. A., Bloembergen, N., Ducuing, J. & Pershan, P. S. Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939 (1962).
 63.
Cappellini, G. & Trillo, S. Thirdorder threewave mixing in singlemode fibers: exact solutions and spatial instability effects. J. Opt. Soc. Am. B 8, 824–838 (1991).
 64.
Ott, J. R., Steffensen, H., Rottwitt, K. & McKinstrie, C. J. Geometric interpretation of fourwave mixing. Phys. Rev. A 88, 043805 (2013).
 65.
Stolen, R. H. Phasematchedstimulated fourphoton mixing in silicafiber waveguides. IEEE J. Quant. Electron. 11, 100–103 (1975).
 66.
Marhic, M. Fiber Optical Parametric Amplifiers, Oscillators and Related Devices. (Cambridge University Press, Cambridge, 2007).
 67.
Torounidis, T., Andrekson, P. A. & Olsson, B. E. Fiberoptical parametric amplifier with 70dB gain. IEEE Photon. Technol. Lett. 18, 1194–1196 (2006).
 68.
Stephens, M. F. C., Phillips, I. D., Rosa, P., Harper, P. & Doran, N. J. Improved WDM performance of a fibre optical parametric amplifier using Ramanassisted pumping. Opt. Exp. 23, 902–911 (2015).
 69.
Hansryd, J., Andrekson, P. A., Westlund, M., Li, J. & Hedekvist, P. O. Fiberbased optical parametric amplifiers and their applications. IEEE J. Sel. Top. Quant. Electron. 8, 506–520 (2002).
 70.
Radic, S. & McKinstrie, C. J. Twopump fiber parametric amplifiers. Opt. Fiber Technol. 9, 7–23 (2003).
 71.
McKinstrie, C. J. & Radic, S. Phasesensitive amplification in a fiber. Opt. Exp. 12, 4973–4979 (2004).
 72.
Marhic, M. E., Andrekson, P. A., Petropoulos, P., Radic, S. & Peucheret, C. et al. Fiber optical parametric amplifiers in optical communication systems. Laser Photon. Rev. 9, 50–74 (2015).
 73.
Barthelemy, A. & De La Fuente, R. Unusual modulation instability in fibers with normal and anomalous dispersions. Opt. Commun. 73, 409–412 (1989).
 74.
Turitsyn, S. K., Bednyakova, A. E., Fedoruk, M. P., Papernyi, S. B. & Clements, W. R. L. Inverse fourwave mixing and selfparametric amplification in optical fibre. Nat. Photon. 9, 608–614 (2015).
 75.
Picozzi, A., Garnier, J., Hansson, T., Suret, P. & Randoux, S. et al. Optical wave turbulence: towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics. Phys. Rep. 542, 1–132 (2014).
 76.
Hall, B., Lisak, M., Anderson, D., Fedele, R. & Semenov, V. E. Statistical theory for incoherent light propagation in nonlinear media. Phys. Rev. E 65, 035602 (2002).
 77.
Churkin, D. V., Kolokolov, I. V., Podivilov, E. V., Vatnik, I. D., & Nikulin, M. A. et al. Wave kinetics of random fibre lasers. Nat. Commun. 2, 6214 (2015).
 78.
Babin, S. A., Churkin, D. V., Ismagulov, A. E., Kablukov, S. I. & Podivilov, E. V. Fourwavemixinginduced turbulent spectral broadening in a long Raman fiber laser. J. Opt. Soc. Am. B 24, 1729–1738 (2007).
 79.
Deng, L., Hagley, E. W., Wen, J., Trippenbach, M. & Band, Y. et al. Fourwave mixing with matter waves. Nature 398, 218–220 (1999).
 80.
Trippenbach, M., Band, Y. B. & Julienne, P. S. Theory of fourwave mixing of matter waves from a BoseEinstein condensate. Phys. Rev. A 62, 023608 (2000).
 81.
Vogels, J. M., Xu, K. & Ketterle, W. Generation of macroscopic paircorrelated atomic beams by fourwave mixing in BoseEinstein condensates. Phys. Rev. Lett. 89, 020401 (2002).
 82.
Karlsson, M. Modulational instability in lossy optical fibers. J. Opt. Soc. Am. B 12, 2071–2077 (1995).
 83.
Krechetnikov, R. & Marsden, J. E. Dissipationinduced instabilities in finite dimensions. Rev. Mod. Phys. 79, 519–553 (2007).
 84.
Tanemura, T., Ozeki, Y. & Kikuchi, K. Modulational instability and parametric amplification induced by loss dispersion in optical fibers. Phys. Rev. Lett. 93, 163902 (2004).
 85.
Trillo, S. & Wabnitz, S. Dynamics of the nonlinear modulational instability in optical fibers. Opt. Lett. 16, 986–988 (1991).
 86.
Xu, K., Liu, H. Y., Dai, Y. T., Wu, J. & Lin, J. T. Synthesis of broadband and flat parametric gain by idler loss in optical fiber. Opt. Commun. 285, 790–794 (2012).
 87.
Coen, S., Wardle, D. A. & Harvey, J. D. Observation of nonphasematched parametric amplification in resonant nonlinear optics. Phys. Rev. Lett. 89, 273901 (2002).
 88.
Sylvestre, T., Maillotte, H., Lantz, E. & Tchofo Dinda, P. Ramanassisted parametric frequency conversion in a normally dispersive singlemode fiber. Opt. Lett. 15, 1561–1563 (1999).
 89.
Castelli, F., Brambilla, M., Gatti, A., Prati, F. & Lugiato, L. A. The LLE, pattern formation and a novel coherent source. Eur. Phys. J. D. 71, 84 (2017).
 90.
Myers, L. E., Eckardt, R. C., Fejer, M. M., Byer, R. L. & Bosenberg, W. R. et al. Quasiphasematched optical parametric oscillators in bulk periodically poled LiNbO_{3}. J. Opt. Soc. Am. B 12, 2102–2116 (1995).
 91.
Lowenthal, D. D. CW periodically poled LiNbO_{3} optical parametric oscillator model with strong idler absorption. IEEE J. Quant. Electron. 34, 1356–1366 (1998).
 92.
Rustad, G., Arisholm, G. & Farsund, Ø. Effect of idler absorption in pulsed optical parametric oscillators. Opt. Exp. 19, 2815–2830 (2011).
 93.
Jauregui, C., Steinmetz, A., Limpert, J. & Tünnermann, A. Highpower efficient generation of visible and midinfrared radiation exploiting fourwavemixing in optical fibers. Opt. Exp. 20, 2495724965 (2012).
 94.
Ma, J. G., et al. Quasiparametric amplification of chirped pulses based on a Sm^{3+}doped yttrium calcium oxyborate crystal. Optica 20, 1006–1009 (2015).
 95.
Perego, A. M., Tarasov, N., Churkin, D. V., Turitsyn, S. K. & Staliunas, K. Pattern generation by dissipative parametric instability. Phys. Rev. Lett. 116, 028701 (2016).
 96.
Tarasov, N., Perego, A. M., Churkin, D. V., Staliunas, K. & Turitsyn, S. K. Modelocking via dissipative Faraday instability. Nat. Commun. 7, 12441 (2016).
 97.
Perego, A. M. Highrepetitionrate, multipulse allnormaldispersion fiber laser. Opt. Lett. 42, 3574–3577 (2017).
 98.
Staliunas, K. Removal of excitations of BoseEinstein condensates by space and timemodulated potentials. Phys. Rev. A. 84, 013626 (2011).
Acknowledgements
The authors acknowledge inspiring and stimulating discussions with Prof. E. A. Kuznetsov and Prof. A. Rubenchick. K. S. acknowledges support from a NATO SPS Research grant (SPS 985048), and from Spanish Ministerio de Ciencia e Innovación and European Union FEDER through project FIS201565998C21P. A. M. P. acknowledges support from the ICONE project through Marie Curie Grant No. 608099. The research of S. K. T. was supported by the Russian Science Foundation (Grant No. 177230006).
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Perego, A.M., Turitsyn, S.K. & Staliunas, K. Gain through losses in nonlinear optics. Light Sci Appl 7, 43 (2018). https://doi.org/10.1038/s4137701800429
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