8 FIG. 6: (Color online) Central part of the dissipative soli- ton spectra perturbed by ˜ f 1 ( ω ) (open circles and crosses) or ˜ f 0 ( ω ) (solid gray curves) as well as the profiles of ℜ ( ˜ f 1 ) (solid black curve) and ℑ ( ˜ f 1 ) (dashed curve) from Eq. (12). Single absorption line with Ω = 1 GHz is centered at ω =0; ϵ = − 0 . 005 (black solid and dashed curves as well as open circles and solid gray curve 1) and ϵ = − 0 . 05 (crosses and solid gray curve 2). FIG. 7: (Color online) Central part of the dissipative soli- ton spectra perturbed by ˜ f 0 ( ω ) (black solid curve) and the profiles of ℜ ( ˜ f 0 ) (dashed curve) and ℑ ( ˜ f 0 ) (dotted curve) from Eq. (12). Triplet of absorption lines is centered at ω 1 = − 10, ω 2 =0, and ω 3 =10 GHz; ϵ 1 = ϵ 2 = ϵ 3 = − 0 . 005, Ω 1 = Ω 2 = Ω 3 = 2 GHz. 6. CONCLUSION 4088 cm − 1 (122.56 THz, marked with an asterisk), we obtain | ϵ/q | = 0 . 33 ± 0 . 02 for the maximum modulation, which is in perfect agreement with the observed value of 31.5% (Fig. 1b, black line). The agreement is remark- ably accurate given the less than optimal resolution of the spectrometer (0.25 cm − $^{1}$) and significant third-order dispersion of about +10 4 fs 3 that was not accounted for in the presented analysis. It is important to notice, that the expression (18) includes only the externally observable soliton band- width and relatively stable dispersion parameter. The alignment-sensitive values like saturated losses σ , nonlin- earity γ , nonlinearity saturation parameter κ , etc. which are in practice not known with sufficient accuracy, are all accounted for by the self-consistent soliton parameters. Another important point is the fact, that the signal amplitude 2 | ϵ/q | can be much bigger than that from con- ventional absorption spectroscopy χ$_{l}$L of the cell with the same length. The signal enhancement factor can be con- trolled by the pulse parameters and it exceeds an order of magnitude for the presented case ( χ$_{l}$L = 5% for selected line and a single-pass cell of a resonator size). For ad- ditional sensitivity improvement one can apply the well- developed intracavity multi-pass cell technique [26]. The expression (18) suggests, that ultimate sensitivity can be obtained at the expense of the reduced bandwidth cov- erage Δ ν . In this respect, the presented technique has the same quadratic dependence of sensitivity on spec- tral bandwidth as the conventional intracavity absorp- tion spectroscopy [5]. Further refinement of the presented theory should in- clude demonstration of its applicability to arbitrarily shaped absorption features. The superposition property provides a strong argument for such extension, but it has to be rigorously proven for Doppler- and more general Voigt-shaped lines, and also for the dense line groups in e.g. Q-branches. It would be interesting also to extend the theory to the absorber lines at the soliton wings (Fig. 1a). With the above issues resolved, the soliton-based spec- troscopy may become a powerful tool for high-resolution, high-sensitivity spectroscopy and sensing. Possible im- plementation include soliton propagation in a gas-filled holey fibers, as well as already presented intracavity spec- troscopy with femtosecond oscillators. The latter, being a natural frequency comb source, allows direct locking to optical frequency standards, providing for ultimate reso- lution and spectral accuracy. We have been able to derive an analytical solution to the problem of an one-dimensional optical dissipative soliton propagating in a medium with narrowband ab- sorption lines. We predict appearance of spectral mod- ulation that follows the associated index of refraction rather than absorption profile. The novel perturbation analysis technique is based on integral representation in the spectral domain and is insensitive to the diverging differential terms, inherent to the Taylor series represen- tation of the narrow spectral lines. The model is applicable to a conventional soliton prop- agation and to a passively modelocked laser with intra- cavity absorber, the only difference being the character- istic propagation distance (dispersion length and cavity GLYPH<s45>GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s53> GLYPH<s48> GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s45>GLYPH<s51> GLYPH<s45>GLYPH<s50> GLYPH<s45>GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s82> GLYPH<s101> GLYPH<s40> GLYPH<s102> GLYPH<s49> GLYPH<s41>GLYPH<s32>GLYPH<s97>GLYPH<s110>GLYPH<s100>GLYPH<s32>GLYPH<s73>GLYPH<s109> GLYPH<s40> GLYPH<s102> GLYPH<s49> GLYPH<s41>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<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<s49>GLYPH<s46>GLYPH<s57> GLYPH<s50>GLYPH<s46>GLYPH<s48> GLYPH<s50> 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<s49> GLYPH<s45>GLYPH<s52>GLYPH<s48> 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<s51>GLYPH<s48> GLYPH<s52>GLYPH<s48> GLYPH<s45>GLYPH<s51> GLYPH<s45>GLYPH<s50> GLYPH<s45>GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s82> GLYPH<s101>GLYPH<s40> GLYPH<s102> GLYPH<s49> GLYPH<s41>GLYPH<s32>GLYPH<s97>GLYPH<s110>GLYPH<s100>GLYPH<s32>GLYPH<s73>GLYPH<s109> GLYPH<s40> GLYPH<s102> GLYPH<s49> GLYPH<s41>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<s49>GLYPH<s46>GLYPH<s53>GLYPH<s52> GLYPH<s49>GLYPH<s46>GLYPH<s53>GLYPH<s54> GLYPH<s49>GLYPH<s46>GLYPH<s53>GLYPH<s56> GLYPH<s49>GLYPH<s46>GLYPH<s54>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s54>GLYPH<s50> 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>