2.3. Main obstacles to the DS energy harvesting

and, thereby, suppresses an instability caused by nonlinearity. This factor is especially important for all-fiber lasers, where the strong contribution of nonlinear effects is inevitable with

> <sup>τ</sup><sup>Γ</sup> <sup>p</sup> , E<sup>∝</sup> <sup>β</sup><sup>2</sup> � � � �<sup>=</sup> ffiffiffiffiffiffi

The chirped DS accumulates energy ∝ψ that allows using a so-called adiabatic theory for ψ ≫ 1 [24, 63, 64] which predicts a perfect energy scalability or a DS resonance [65] for the cubicquintic SAM (Eq. (14)). That means that energy can be scaled infinitely for C ¼ 1=3 due to pulse

<sup>Θ</sup> ! <sup>0</sup> }spectral condensation} � �

<sup>H</sup> <sup>Δ</sup><sup>2</sup> � <sup>ω</sup><sup>2</sup> � �

�=<sup>C</sup> ffiffiffiffiffiffi

:

,

<sup>ω</sup><sup>2</sup> <sup>þ</sup> <sup>Θ</sup><sup>2</sup> (20)

<sup>τ</sup><sup>Γ</sup> <sup>p</sup> (18)

(19)

Tres-growth. VA predicts the following energy-scaling laws4

for the SAM described by Eqs. (9) and (14), respectively [24, 25].

8 >>><

>>>:

Figure 9. A low-loss regime with δ ¼ �0:05=3, ζ ¼ 100γ, 1% output coupler, μ = 2%.

The negative sign of β<sup>2</sup> corresponds to a normal GDD in these notations.

stretching and simultaneous spectral condensation:

184 High Power Laser Systems

(here H is the Heaviside's function).

4

lim C!1=3 E∝ β<sup>2</sup> � � �

> E ! ∞ <sup>α</sup><sup>2</sup> ! <sup>1</sup>=<sup>ζ</sup> <sup>Δ</sup> ! ffiffiffiffiffiffiffiffiffiffiffiffiffi 2γ=βζ p

where the DS spectral profile is described by a truncated Lorentzian function:

pð Þ¼ ω

6πγ κζ

The DS energy harvesting in both chirped and chirp-free regimes has a common problem of ML self-starting. The DS stability is a necessary but not sufficient condition of its existence because it must develop from some stochastic process in a laser (eventually, from a quantum noise). Existing theories of the ML self-start [32–42] predict that a lot of effects are involved in a pulse formation. However, a spontaneous formation of the DS from noise (the DS self-start) as a general problem has not been studied in depth. In optics, this is often considered as a technical issue, because here one can use one of the proven ML techniques to guarantee self-starting. After the initial kick, however, the DS evolves by itself, and recent experiments have shown controversial results: in high-power solid-state lasers, the strong oscillations (Q-switching) during the DS buildup dynamics hinders the DS self-start [69], while in a fiber laser, such oscillations can accelerate the self-start [70]. That is obviously connected to co-existence of nonlinearities with different time scales: instantaneous non-dissipative SPM, and non-instantaneous dissipative nonlinearities like stimulated Raman scattering (SRS), saturable absorber losses, and gain saturation. This issue is especially intriguing, as the dynamic gain saturation can provide a supplementary mechanism of DS formation [71].

The growing nonlinearity results in quite nontrivial modification of dynamics [25] and causes whether DS stabilization or its chaotization [72–74]. For example, the practically relevant Yb-based thin-disk lasers possess reduced instantaneous nonlinearity and longer gain relaxation times as compared to a bulk Ti: sapphire laser. In the latter case, the enhanced dynamic gain saturation can destabilize a much-desired high-energy DS [75–77]. As another example, the experiments demonstrated, that DS energy scaling in all-fiber fiber lasers is limited by energy loss due to SRS [78]. Nevertheless, SRS could play a positive role providing the generation of dissipative Raman soliton and suppressing the optical turbulence [79–81]. The connection of this phenomenon to the general issues of the turbulence theory waits for its exploration [82].

<sup>5</sup> The adiabatic theory does not predict a spectral condensation near the carrier frequency ω ≈ 0 for this SAM law, but such a concentration is possible at spectrum edges. This phenomenon is clearly visible in the numerous experiments and can be explained by the DS perturbation theory [67].

<sup>6</sup> That is a measure of the DS fidelity.
