Controlling intracavity dual-comb soliton motion in a single-fiber laser

Ultrafast science builds on dynamic compositions of precisely timed light pulses, and evolving groups of pulses are observed in almost every mode-locked laser. However, the underlying physics has rarely been controlled or used until now. Here, we demonstrate a general approach to control soliton motion inside a dual-comb laser and the programmable synthesis of ultrashort pulse patterns. Introducing single-pulse modulation inside an Er:fiber laser, we rapidly shift the timing between two temporally separated soliton combs. Their superposition outside the cavity yields ultrashort soliton sequences. On the basis of real-time spectral interferometry, we observe the deterministic switching of intersoliton separation arising from the interplay of attracting and repulsing forces via ultrafast nonlinearity and laser gain dynamics. Harnessing these insights, we demonstrate the high-speed all-optical synthesis of nano- to picosecond pump-probe delays and programmable free-form soliton trajectories. This concept may pave the way to a new class of all-optical delay generators for ultrafast measurements at unprecedented high tuning, cycling, and acquisition speeds.


Modelling the effects of laser gain dynamics onto relative soliton motion
In order to develop a simplified model for the dual-comb soliton motion in the presence of dynamically evolving laser gain, we evaluate the shaping of the pulse envelope via the SESAM and the time-dependent gain.In particular, the laser gain governs the relative dual-comb motion due to the coupling of intensity to soliton timing via the intensity-dependent shifts by the SESAM.The gain is simulated using the normalized coupled rate equations (Eq. 1) for an ideal three-level laser, assuming that the population of the third level is negligible.Eq. 1a describes the time evolution of the photon number ( ) and Eq.1b the population difference ( ). corresponds to a pump term and to the losses [42,43].
The SESAM is described as a two-level system, and the time-dependent amplitude absorption coefficient q( ) (saturable losses only) is calculated with the differential equation (2).q 0 is the unsaturated amplitude absorption coefficient corresponding to the maximum loss of the SESAM.
is the recovery time, denotes the time-dependent power of the incoming pulses, and E sat is the saturation fluence [44]. (2) Pulse shifts and amplitude changes due to the gain medium and SESAM are computed for each roundtrip.In this model, we do not include further pulse shaping mechanism, such as nonlinearity and dispersion, thus, we force stable pulse shapes upon temporal shifts by relauching Gaussian pulses at the current pulse energy and with shifted centre of gravity at each roundtrip.The modulation is implemented using a transmission function = 1 − ⋅ !" ( , $%& ) for one comb with different modulation strengths and a rectangular function of widths $%& .We compare an all-fiber laser and a fiber laser with free-space section.The timing jitter is significantly reduced for the all-fiber design.

Figure S1 :
Figure S1: Simulation results display the effect of the relaxation time of the absorber onto pulse propagation.a) Instantaneous (symmetric) relaxation results in an absence of temporal shifts.b) A relaxation time of 2 ps (similar to experiment) leads to accumulated temporal shifts.Insets: Sketched temporal responses of the absorption.c) An intensity change (3%) introduces variations in the timing, as evident after 200 roundtrips (parameters of b)).

Figure S2 :
Figure S2: Raw data and timing extraction from spectral interferograms obtained via timestretch dispersive Fourier transform.The data are underlying results in Fig. 4d.Interference fringes at the turning points are indicated, yielding highest and lowest modulation fringe periods.The absolute values of the Fourier transformation (lower panel) yield the soliton delays.

Figure S3 :
Figure S3: Long-term characterization of timing-jitter for two laser realizations.Data recorded without modulation and over a measurement interval of 2.5 seconds (grey area omitted due to low TS-DFT-sensitivity at zero overlap).We compare an all-fiber laser and a fiber laser with free-space section.The timing jitter is significantly reduced for the all-fiber design.

Figure S4 :
Figure S4: Characterization of short-term timing jitter and the influence of the AOM: Measured soliton trajectories (black) via TS-DFT for a scanning frequency 1 2& = 100 34.Deviations from straight lines (blue, orange) are displayed in the insets.Equivalent derivations are found for both scanning directions, indicating negligible influence of the AOM-action which is active only during the up-scan.

Figure S5 :
Figure S5: Illustration of feedback stabilization to regulate the scanning motion: a) Applying a modulation pattern with fixed scanning frequency 1 25 leads to the accumulation of temporal drift between modulated (orange) and unmodulated (purple) soliton combs.The relative timing of both combs can be inferred from cross-correlations (green).b) The envelope of the cross-correlations provides a temporal trigger reference as the active feedback to start the modulation at fixed soliton delay.Thus, the scanning motion is re-initiated to the same start value for every up-scan.