4 FIG. 3: (Color online) Dissipative soliton in an oscillator: ( a ) central part of the spectrum and ( b ) power P ( t ). An oscillator is filled with an absorber described by Eq. (2) with the triplet of lines: ϵ 1 = ϵ 2 = ϵ 3 = − 0 . 005, Ω 1 = Ω 2 = Ω 3 = 2 GHz, and ω 1 = − 10, ω 2 = 0, ω 3 = 10 GHz. FIG. 4: (Color online) Central parts of the dissipative soliton spectra in an oscillator with the single absorption line centered at ω =0. ( a ) ϵ = − 0 . 05; Ω = 4 GHz (solid curve, open circles and crosses) and 1 GHz (dashed curve, open squares and triangles). ( b ) ϵ = − 0 . 005; Ω = 4 GHz. The solid and dashed curves correspond to the contribution of ˆ Γ described by Eq. (2). Open circles and squares describe the separate contribution of ℑ ( ˆ Γ ) . Crosses and triangles describe the separate contribution of ℜ ( ˆ Γ ) . tive chirp. In this study, we consider only the negative dispersion regime, the positive dispersion regime will be a subject of following studies. The results of the simulation are shown in Figs. 3 and 4, and they demonstrate the same dispersion-like modu- lation of the pulse spectrum. Fig. 3 demonstrates action by three narrow (Ω =2 GHz) absorption lines centered at -10, 0 and 10 GHz in the neighborhood of ω = 0. One can see (Fig. 3a), that the absorption lines do not cause spectral dips at ω$_{l}$ , but produce sharp bends, very much like the case of the true perturbed Schr¨ odinger soli- ton considered before. One can also clearly see the col- lective redistribution of spectral power from higher- to lower-frequencies, which enhances local spectral asym- metry (Fig. 3a). Such an asymmetry suggests that the dominating contribution to a soliton perturbation results from the real part of an absorber permittivity, which, in particular, causes the time-asymmetry of perturbation in the time domain. This asymmetry is seen in time domain as a ns-long modulated exponential precursor in Fig. 3b. The simulated effect of a single narrow absorption line centered at ω = 0 is shown in Fig. 4 for different values of peak absorption ϵ and width Ω. In Fig. 4a, ϵ = − 0 . 05 and Ω = 4 GHz (solid curve, open circles and crosses) and 1 GHz (dashed curve, open squares and triangles). The solid and dashed curves demonstrate the action of complex profile (2), whereas circles (squares) and crosses (triangles) demonstrate the separate action of ℑ ( ˆ Γ) and ℜ ( ˆ Γ), respectively. One can see, that the profile of per- turbed spectrum traces that formed by only ℑ ( ˆ Γ) (i.e. it traces the real part of an absorber permittivity). One can say, that the pure phase effect ( ℑ ( ˆ Γ), circles and squares in Figs. 4a,b) strongly dominates over the pure absorption ( ℜ ( ˆ Γ), crosses in Figs. 4a,b), like that for the Schr¨ odinger soliton. Such a domination enhances with a lowered ϵ (Fig. 4b, crosses), however the | ϵ | growth in- creases the relative contribution of ℜ ( ˆ Γ) and causes the frequency downshift of the bend (Fig. 4a). Amplitude of the bend traces the ϵ value, while its width is defined by Ω. The SAM considered above is modeled by the cubic- quintic nonlinear term ˆ Σ in Eq.(1). Such a SAM is typ- ically realized by using the self-focusing inside an ac- tive medium (Kerr-lens modelocking). For reliable self- starting operation of mode-locked oscillator it is often de- sirable to use a suitable saturable absorber (SA), e.g. a semiconductor-based SESAM [23, 24]. Such an absorber can be described in the simplest case by a single-lifetime two-level model, giving the time-dependent loss coeffi- cient Λ ( t ) as where Λ$_{0}$ is the loss coefficient for a small signal, A$_{eff}$ is the effective beam area on the SA, T$_{s}$ and J$_{s}$ are the SA relaxation time and the saturation energy fluency, respectively. Eq. (4) supplements Eq. (1) and the SAM term ˆ Σ in the latter has to be replaced by (Λ$_{0}$ − Λ ( t )). When the pulse width is longer than the SA relaxation time, one can replace Eq. (4) by its adiabatic solution so that ∂ Λ ( t ) ∂t = Λ$_{0}$ − Λ ( t ) T$_{s}$ − Λ ( t ) P ( t ) J$_{s}$A$_{eff}$ , (4) ˆ Σ = η$_{0}$ξP ( t ) 1 + ξP ( t ) , (5) GLYPH<s45>GLYPH<s51>GLYPH<s48> GLYPH<s45>GLYPH<s50>GLYPH<s48> GLYPH<s45>GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s50>GLYPH<s48> GLYPH<s49>GLYPH<s46> GLYPH<s52> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s50> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s52> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s54> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s56> GLYPH<s115>GLYPH<s112> GLYPH<s101> GLYPH<s99> GLYPH<s116>GLYPH<s114>GLYPH<s97> GLYPH<s108>GLYPH<s32>GLYPH<s112> GLYPH<s111> GLYPH<s119> GLYPH<s101> GLYPH<s114>GLYPH<s32>GLYPH<s40>GLYPH<s97> GLYPH<s114>GLYPH<s98> GLYPH<s46>GLYPH<s32>GLYPH<s117> GLYPH<s110> GLYPH<s105>GLYPH<s116>GLYPH<s115>GLYPH<s41> GLYPH<s32>GLYPH<s40> GLYPH<s71>GLYPH<s72>GLYPH<s122>GLYPH<s41> GLYPH<s45>GLYPH<s50> GLYPH<s45>GLYPH<s49> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s49> GLYPH<s55> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s49> GLYPH<s50> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s55> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s50> GLYPH<s40> GLYPH<s98> GLYPH<s41> GLYPH<s40> GLYPH<s97> GLYPH<s41> GLYPH<s32>GLYPH<s112> GLYPH<s111> GLYPH<s119> GLYPH<s101> GLYPH<s114>GLYPH<s32>GLYPH<s40>GLYPH<s97> GLYPH<s114>GLYPH<s98> GLYPH<s46>GLYPH<s32>GLYPH<s117> GLYPH<s110> GLYPH<s105>GLYPH<s116>GLYPH<s115>GLYPH<s41> GLYPH<s116>GLYPH<s32> GLYPH<s40>GLYPH<s110>GLYPH<s115>GLYPH<s41> GLYPH<s45>GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s46> GLYPH<s50> GLYPH<s49>GLYPH<s46> GLYPH<s51> GLYPH<s49>GLYPH<s46> GLYPH<s52> GLYPH<s49>GLYPH<s46> GLYPH<s53> GLYPH<s49>GLYPH<s46> GLYPH<s54> GLYPH<s49>GLYPH<s46> GLYPH<s55> GLYPH<s49>GLYPH<s46> GLYPH<s56> GLYPH<s40> GLYPH<s98> GLYPH<s41> GLYPH<s32> GLYPH<s32> GLYPH<s115>GLYPH<s112> GLYPH<s101> GLYPH<s99> GLYPH<s116>GLYPH<s114>GLYPH<s97> GLYPH<s108>GLYPH<s32>GLYPH<s112> GLYPH<s111> GLYPH<s119> GLYPH<s101> GLYPH<s114>GLYPH<s32>GLYPH<s40>GLYPH<s97> GLYPH<s114>GLYPH<s98> GLYPH<s46>GLYPH<s32>GLYPH<s117> GLYPH<s110> GLYPH<s105>GLYPH<s116>GLYPH<s115>GLYPH<s41> GLYPH<s32>GLYPH<s40> GLYPH<s71>GLYPH<s72>GLYPH<s122>GLYPH<s41> GLYPH<s45>GLYPH<s49>GLYPH<s48> GLYPH<s48> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s49> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s50> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s51> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s52> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s53> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s54> GLYPH<s49>GLYPH<s46> GLYPH<s52>GLYPH<s55> GLYPH<s32> GLYPH<s32> GLYPH<s32>GLYPH<s40> GLYPH<s71>GLYPH<s72>GLYPH<s122>GLYPH<s41> GLYPH<s40> GLYPH<s97> GLYPH<s41>