Fig. 3: Cancellation of the Raman self-frequency shift and Bloch oscillations of Kelly sidebands.

a Two-dimensional map of experimentally recorded spectra of a soliton trapped in a stationary intra-cavity phase modulation as a function of the detuning \(({P}_{{\rm{in}}}=100{\rm{mW}},{J}_{{\rm{RF}}}=1{\rm{rad}},{\omega }_{{\rm{RF}}}/2\pi =12\,{\rm{GHz}})\). The white dashed line shows the theoretical CS central wavelength and range of existence with Raman scattering (\(\bar{{\tau }_{{\rm{R}}}}=3{\rm{fs}}\)) but without modulation (\(V=0\)). b Two experimental spectra corresponding to different cavity detunings \({\delta }_{0}\) (solid lines), and predicted CS spectra by numerical integration of the mean-field model. c Intracavity peak power of stable CSs as a function of the detuning. The black curve represents the solutions of the mean-field model with SRS (solid), and the circles indicate the result of the motion equations, both for \({\Lambda }_{{\rm{e}}}=3\)%. The red and blue curves show the solutions for \(V=0\) with and without SRS, respectively. We note that the black curves are shifted by the modulation amplitude \({J}_{{\rm{RF}}}\) for small detunings, i.e., small soliton bandwidths, as the CS remains located close to the minimum local detuning \({\delta }_{0}-{J}_{{\rm{RF}}}\). The squares are the experimental values reported in (b). d Shortest cavity soliton duration as a function of the characteristic time \({\tau }_{c}\) with \(S=30\). Solid curves correspond to the results from numerical continuations of solutions of the mean-field model. The dots show the limit given by Eq. (10). The parameters for the potentials are \({J}_{{\rm{RF}}}/{\Lambda }_{{\rm{e}}}=105\), \({\omega }_{{\rm{RF}}}/2\pi =20{\rm{GHz}}\) and \(d=0\). We note that \({\tau }_{{\rm{c}}}=7\) ps in the experiments. e Spectrum of a CS displaying structured Kelly sidebands (50 pm resolution) due to periodic intracavity modulation. The shadowed regions highlight the extension of the Kelly sidebands. f Corresponding spectrogram of the intracavity field (gate function: 20 ps-wide super-Gaussian of order 6). The vertical band at the centre is the modulated background, while the soliton spectrum is seen as the horizontal band around the zero delay