Oscillatory motion of a counterpropagating Kerr soliton dimer

Counterpropagating (CP) solitons generated in high-Q microcavities not only offer useful dual-comb sources, but also provide a new platform to study soliton interactions. Here, we demonstrate and theoretically explain a manifestation of soliton trapping that occurs between CP solitons in a silica microcavity introducing a Kerr soliton dimer. In conventional soliton trapping, the group velocities of two solitons can be synchronized by a Kerr-effect-mediated interaction. The solitons can then copropagate with a fixed temporal delay. However, as shown here, when counterpumping a microcavity using slightly detuned pump frequencies and in the presence of backscattering, the group velocities of clockwise and counterclockwise solitons undergo periodic modulation instead of being locked to a constant velocity. Upon emission from the microcavity, the solitons feature a relative oscillatory motion around a locked average relative displacement with an amplitude that can be larger than the soliton pulse width. This relative motion introduces a sideband fine structure into the optical spectrum of the CP solitons. Our observation provides insights on coherently pumped soliton dimers in microcavities.

Figure 1

Forced motion in a trapped CP soliton dimer. (a) In conventional soliton trapping without coherent pumps, both the phase velocity and the group velocity will be locked. In the frequency domain, both the repetition rate $f_r$ and the carrier envelope offset frequency $f_0$ of the corresponding combs will be equal. Note that in the illustration the pulse envelopes are plotted to be exactly overlapped and the phase is identical, which are not necessary in experiments. (b) For CP solitons in coherently pumped systems, backscattering enables a different trapping outcome. Due to the coherent pumping, the phase velocity and $f_0$ cannot be locked when using nondegenerate pumps. In this case, solitons will be locked in an averaged way, with the spectral center frequency (${\nu }_s$) and group velocity experiencing periodic modulation (upper shaded box). As a result, there is relative motion between the CP solitons after being emitted from the microcavity (lower shaded box). This relative motion repeats with a period of 1/$\mathrm{\Delta }{\nu }_P$. In the frequency domain, the motion will induce sidebands around the main comb lines. Note that the power ratio between the solitons and the backscattered counterpart is for illustration and does not reflect the actual ratio.

Figure 2

Observation of oscillatory motion in CP soliton trapping. (a) A balanced optical cross correlator (BOC) is used to experimentally measure relative CP soliton motion (left). The BOC output signal is shown when the delay is scanned (right). AOM, acousto-optical modulator; EDFA, erbium-doped fiber amplifier; PS, pulse shaper. (b) Optical spectrum of the CP solitons with a 3-dB bandwidth of 1.4 THz. The red line (spectral envelope) is the simulated spectrum. (c), (d) BOC-measured relative soliton motion when the pump frequency detuning is 10 and 100 kHz, respectively. The motion frequency is measured to be equal to $\mathrm{\Delta }{\nu }_P$. The dashed red (upper) and blue (lower) lines indicate the center of motion for 10- and 100-kHz detuning, respectively; and they are shifted by about 200 fs as shown in (c).

Figure 3

Simulation of forced motion in CP soliton trapping. (a), (b) Simulated relative soliton temporal motion when the two pumps are detuned by 10 and 100 kHz, respectively. Dashed red lines indicate the zero delay. (c) Numerically calculated relative spectral center frequency between two CP solitons for $\mathrm{\Delta }{\nu }_P=100$ kHz, showing periodic variation. Depending upon the sign of the relative frequency, the motion can be separated into two regions as indicated by the dashed green lines.

Figure 4

Measured electrical spectra of the beat between CP solitons. (a) $\mathrm{\Delta }{\nu }_P=10$ kHz. (b) $\mathrm{\Delta }{\nu }_P=100$ kHz.