1. Introduction

The concept of a dissipative soliton (DS) has an extremely wide horizon of application. It is useful in quantum optics and modeling of Bose-Einstein condensation, condensed matter physics and study of non-equilibrium phenomena, nonlinear dynamics and quantum mechanics of self-organizing dissipative systems [1]. Non-equilibrium character of a system, where a DS emerges, requires a well-organized energy exchange between the soliton and the environment. The energy flow forms a non-trivial internal structure of a DS, which provides the energy redistribution inside it [1]. As a result, the DS phase ϕ(t) becomes inhomogeneous with a nonzero chirp: Q ≡ ∂²ϕ/∂t² ≠ 0 (here t is the local time). A DS with the nontrivial phase distribution is called the “chirped dissipative soliton” (CDS) [2, 3]. The unique feature of a CDS is its capacity to accumulate energy E without stability loss: E ∝ Q [4, 5]. This results in the energy scalability of CDS [4–6]. Such phenomenon resembles a resonant enhancement of oscillations in environment-coupled systems and was named the “dissipative soliton resonance” (DSR) [7].

The capacity of a CDS to accumulate energy is of interest for a number of applications. For instance, it allows energy scaling of ultrashort laser pulses so that >10 MW peak powers and, respectively, >10¹⁴ W/cm² intensities become reachable directly from a laser operating at few MHz repetition rates [8–10]. Such pulse powers bring the high-field physics on tabletops of a mid-level university lab [11]. In particular, high-energy ultrashort pulse lasers allow nowadays such experiments as direct gas ionization and high-harmonic generation, pump-probe diffraction experiments with electrons and production of nm-scale structures at a surface of transparent materials, characterization and control of the electronic dynamics, a variety of biophotonic and biomedical applications, etc. [12–16]. Moreover, the MHz pulse repetition rates provide signal rate improvement factor of 10³ ÷ 10⁴ as compared to usual chirped-pulse amplifiers, with a corresponding signal-to-noise ratio improvement.

The mechanisms of the CDS formation and stabilization can be described qualitatively in the framework of the complex nonlinear Ginzburg-Landau equation [1,17,18]. The contribution of the group-delay dispersion (GDD) to a soliton phase can be expressed as (β/2)∂²a(t)/∂t² (here β is a GDD coefficient, a(t) is a complex slowly-varying envelope of DS). For a CDS, such a contribution can be compensated by the term −(β/2)a(t)(∂ϕ(t)/∂t)² if the dimensionless chirp parameter, also called the DS stretching parameter $\psi \simeq Q{T}^2=\sqrt{2+\gamma P_0{T}^2/\beta }$ (here γ is the coefficient of self-phase modulation (SPM), P₀ is the DS peak power, and T is the DS width). It is important, that such a phase compensation becomes possible in the normal GDD range (β > 0 in our notations), where a classical Schrödinger soliton does not exist.

Additional insight into the CDS stabilization mechanism can be obtained from the dispersion relations for DS and linear perturbation waves. In the normal dispersion range, a DS with the wave number q = γP₀ [2, 19] can interact resonantly with a linear perturbation wave if the wave number of the latter is k(ω) ≡ βω²/2 = q (ω is a frequency deviation from the DS carrier frequency). Hence, a stable DS must have the spectrum truncated at the frequencies ±Δ: k(±Δ) = q where Δ² = 2γP₀/β. On the other hand, a spectral loss ∼ αΔ² (α is an inverse squared bandwidth of a spectral filter) for the DS has to be compensated by a nonlinear gain ∼ κP₀ (κ is a nonlinear gain or, in another words, a self-amplitude modulation (SAM) coefficient). This condition in combination with the dispersion relation gives the DS stability criterion: 2αγ/βκ ≤ 1. More precise analysis [17] within the frameworks of complex nonlinear cubic-quintic Ginzburg-Landau model gives the conditions of DS asymptotical stability:

The DSR regime, i.e. a perfect energy-scalability of a DS requires extensive energy exchange with the environment followed by energy redistribution inside a soliton. As was mentioned above, such a redistribution results in the phase inhomogeneity, i.e. the strong chirp, which causes broadening of the DS in time. Since a DS peak power is fixed by the SAM saturation so that P₀ ∝ 1/ζ, the law lim_C→2/3 E^* = ∞ promises perfect energy scalability of a DS, for instance, by simply increasing the fibre laser length [20]. However, the recent studies have demonstrated that such scalability can be destroyed by gain saturation, stimulated Raman scattering (SRS), and noise amplification [21–23]. The two latter factors become especially important with the DS energy growth and are subjects of the study presented below.

2. Model

The main originality of our study is taking into account the effects of quantum noise and SRS on the DS stability and energy scalability. The presented analysis is based on the numerical solution of the generalized complex nonlinear Ginzburg-Landau equation, which is a testbed for modeling of nonlinear dissipative dynamics far from the thermodynamic equilibrium (for overview, see [1]). The master equation under consideration is the distributed complex nonlinear cubic-quintic Ginzburg-Landau one [2, 4]:

The unique property of Eq. (2) is that it has a CDS solution, which can be characterized by a two-dimensional parametric space, or “master diagram” [4]. The structure of such a diagram is defined by the set of the “isogain” curves with σ = const ≥ 0 on the (C, E^*)– plane. The zero-isogain σ = 0 (Fig. 1, black solid curve) defines both threshold of the CDS stability and its energy scalability law lim_C→2/3 E^* = ∞.

 

Fig. 1 Master diagram for a CDS. Black solid curve corresponds to the stability threshold in absence of the noise and SRS. The stability thresholds in the presence of the quantum noise only (Γ = 10⁻¹⁰/γ, f_R = 0) and noise with SRS (Γ = 10⁻¹⁰/γ, f_R =0.22) are shown by the red dashed and the blue dashed-dot curves, respectively. Spectral filter parameter α corresponds to ≈40 nm (α =366 fs²) at 1 μm central wavelength and to ≈20 nm (α =1460 fs²) for the magenta dotted curve. κ = 0.1γ, ζ = 0.05γ.

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However, the model based on Eq. (2) has two important disadvantages. The first one is that the quantum noises, which are inherent in any dissipative system [25], do not taken into account. The quantum noise can be introduced by addition of a complex white noise term s(z, t) with the correlation function

Another important generalization of Eq. (2) is taking into account the SRS. One may assume that a phonon basin acts as damped harmonic oscillator, which is nonlinearly coupled with a photon basin [24, 27]. The corresponding Raman response function can be written as

In fiber optics, the SRS contribution is often included as a first moment of the response function [24,28]: −iT_Rγa∂|a|²/∂t, where $T_R\approx f_R{\int }_0^{\infty }h(t)tdt$ and f_R is the fractional contribution of a Raman response to a nonlinear polarization. While this approximation is valid for the pulses with sufficiently narrow spectra (Δω ≪ 1/T₁), we use the more adequate and general approximation (4), thus including the Raman self-scattering shown to be important for femtosecond pulse dynamics [29]. The resulting generalized stochastic nonlinear Ginzburg-Landau equation has the following form:

This equation will be the subject of our analysis. The main difficulty here is the incommensurability of the temporal scales: from the femtosecond scale, corresponding to a Raman response, to over 100 ps duration of the strongly chirped CDS. We used a time-mesh with a step of 1 fs and up to 2²¹ points (2 ns time window) with a propagation distance up to z = 10⁴ round-trips. The numerical calculations have been realized by symmetrized split-step Fourier method with the convolution of the SRS term performed in the spectral domain.

3. Results

A master diagram structure is a network of “isogains” with σ = const > 0 [4]. The zero-level isogain σ = 0 defines the threshold of CDS stability against a vacuum excitation. Such excitation may cause, e.g., a collapsing DS instability or a multiple pulse generation [30]. The CDS stability border in the absence of noise and SRS is shown by a solid black curve in Fig. 1. A CDS is stable below this curve. A “plateau” in the dependence of the stability threshold on the energy for E^* ≫ 1 corresponds to the DSR. A typical Wigner function of a stable CDS in the vicinity of DSR are shown in Fig. 2(a). CDSs along the DSR curve shown in Fig. 1 have a characteristic finger-like spectrum (a truncated Lorenzian) and a flat-top temporal profile [2, 4, 31].

 

Fig. 2 Wigner function of CDSs in absence (a) and presence (b) of the quantum noise and SRS. The CDS temporal and spectral profiles are shown as the corresponding axis-projections. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.59, E^* =111.

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If the complete set of terms in Eq. (5) is taken into account, the stability of DS dramatically changes at high energies. If only the noise term in Eq. (5) contributes, then the DSR regime is suppressed. The corresponding stability border is shown by the red dashed curve in Fig. 1. The DSR disappearance means that higher energy requires smaller C–parameter (e.g., by increasing the dispersion β). Even more important, there might not exist a single-pulse CDS solution beyond some maximum E^* (see e.g. the break of the red dashed curve in Fig. 1).

The situation becomes more complicated in the presence of SRS. The corresponding stability border is shown by the blue dashed-dot curve in Fig. 1. Fig. 3 shows evolution of the CDS spectra with increasing energy, corresponding to the horizontal trajectory 1 in Fig. 1. With the E^* growth, the spectrum broadens and becomes Lorentz-profiled, truncated at the deviation frequencies ±Δ (see above). When the CDS spectral width reaches the resonant frequency ∼ 1/πT₁ = 0.026 fs⁻¹, the SRS causes self-scattering with the subsequent frequency self-Stokes-shift. This is a dissipative Raman soliton (DRS). It is chirped and possess a truncated Lorenz-profiled and Stokes-shifted spectrum. The spectrum also has an anti-Stokes component which corresponds to a perturbation of the DS trailing edge and causes the chaotic evolution of the DRS instantaneous power. Finally, the red spectrum shift due to the SRS in the normal dispersion regime results in the DRS acceleration (group velocity increase).

 

Fig. 3 Averaged CDS spectra along the trajectory 1 in Fig. 1. Spectral power profiles are averaged over z = 1000 round-trips.

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When spectral shift of a DRS becomes sufficiently strong, multiple pulse generation develops. The multipulsing is seen as modulated spectra in Fig. 3, which begin when the trajectory 1 in Fig. 1 intersects the stability border at E^* ≈ 30. We suggest, that the spectral shift induced by SRS acts an additional mechanism of nonlinear spectral filtering. These additional spectral dissipation initiate a multiple pulsing [23, 30]. The multipulsing instability has been experimentally identified as a typical regime of an all-fiber laser with the strong SRS [23, 32] and only stabilization of this regime has allowed generation of the DRS paired with the ordinary CDS [33]. Hence, the contribution of SRS and noise limits maximum DS soliton energies (blue dashed-dot curve in Fig. 1).

Let us consider a change of DS stability with the C–parameter variation (the vertical trajectory 2 in Fig. 1). This can be achieved e.g. by increasing the dispersion since C ∝ 1/β. At low energy values, the multiple pulse instability existing above the stability threshold shown by the solid black curve in Fig. 1 can always be suppressed by an appropriate decrease of the C–parameter. If the energy E^* is high, the noise contribution can prevent the DS stabilization (red dashed curve in Fig. 1).

The situation becomes more complicated in the presence of SRS, which causes pulse spectrum splitting with a C–parameter decrease (Fig. 2(b)). It is important, that an anti-Stokes component localized on the traveling edge of CDS is perturbed and it induces chaotization of CDS dynamics, in certain analogy to chaotic behavior of a DS interacting with a dispersive wave [19]. Such a chaotization increases with smaller C– values provoking earlier pulse splitting (intersection of trajectory 2 with the blue dashed-dot curve in Fig. 1). The Wigner function corresponding to the double-pulse CDS regime is shown in Fig. 4.

 

Fig. 4 Wigner function of double CDSs in the presence of the quantum noise and SRS. The CDS temporal and spectral profiles are shown as the corresponding axis-projections. The spectrum is modulated by the interferenge fringes with 100% visibility, which are not resolved. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.037, E^* =111.

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One should note “bugles” and “branches” on the Wigner function edges (Fig. 4). They can be considered as a manifestation of the CDS chaotization. Such a chaotization reveals itself in an irregular oscillation of the soliton instantaneous power, but may remain unnoticed in conventional experimental conditions when all measurements effectively average over many round-trips [19]. The double-pulse complex, despite the chaotisation at the trailing edges, remains very stable and mutually coherent, as confirmed by the interference fringes with 100% contrast in the spectrum (Fig. 4)

Further decrease of the C-parameter can suppress the multiple pulse instability (the trajectory 2 intersects again the blue dashed-dot curve in Fig. 1). The reason for this is that the inverse filter bandwidth parameter α enters the C-parameter in numerator, so smaller C equally represents more relaxed spectral filtering, allowing the soliton to shift away from the gain maximum. The resulting pulse is a chirped DRS with red-shifted finger-like regular spectrum (Fig. 5) that propagates faster than an ordinary DS. Simultaneously, the DRS width can be appreciably less than that of an ordinary CDS with the same C- and E^*-parameters. Note that the edges of the chirped DRS Wigner-function (Fig. 5) are bent in the inverse directions relatively to those of ordinary CDS (Fig. 2(a)), and the DRS fidelity (i.e., quality of its compressibility provided by a “flatness” of the spectral chirp frequency dependence [31]) is in fact higher than that of an ordinary CDS.

 

Fig. 5 Wigner function of a chirped DRS. The DRS temporal and spectral profiles are shown as the corresponding axis-projections. The top wavelength shift scale corresponds to an Yb:fiber laser centered around 1070 nm. The parameters along the trajectory 2 in Fig. 1 are C =0.021, E^* =111.

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The fingerprints of a DRS reported here are the pronounced perturbations of its envelope trailing edge (Fig. 5) and the chaotic evolution of its instantaneous power. With even smaller C parameter, the Stokes-shift decreases but the spectrum tends to the “triangular” shape, which correspond to a breezer-like dynamics of DRS [28].

Thus, the chaotization of high-energy DRS dynamics as well as the multiple pulse instability become the acute problem for a DS energy scalability. Since the main source of such instability is a perturbation of the anti-Stockes spectral component localized on the CDS trail, one may suggest a stabilization technique based on the manipulation with a spectral filter. The magenta dotted curve in Fig. 1 demonstrates the corresponding stability threshold for a twofold narrower filter bandwidth. One can see, that the while maximum dimensionless energy E^* becomes stringer confined, the stable range protracts towards smaller C values. Note, however, that confinement of E^* by increasing α parameter does not, in fact, decrease the physical energy E, since it scales as $E\propto E^{*}\sqrt{\alpha }$, while the extended range of the C parameter can be exploited e.g. by manipulation with dispersion only (β parameter). It thus seems that narrowing the spectral filter is one of the ways to counteract the Raman destabilization at high energies in the real-world systems.

4. Conclusion

We demonstrated the effects of quantum noise and stimulated Raman scattering (SRS) on the chirped dissipative soliton stability. These factors suppress the dissipative soliton resonance and reduces the soliton energy scalability pulse, by increasing the multiple pulse instability region. Simultaneously, the SRS induces the spectral Stokes-shift and perturbs the soliton trailing edge, where the anti-Stokes spectral components are localized. Such perturbation causes chaotic evolution of the instantaneous power and can initiate the pulse splitting with the dispersion growth. However, if the dispersion becomes sufficiently large, a stable dissipative Raman soliton develops. It has a finger-like and Stokes-shifted spectrum, higher group velocity, and shorter duration in comparison with an ordinary dissipative soliton. Narrowing of the spectral filter can further reduce the tendency to multipulsing caused by the stimulated Raman scattering.