Observation of Breathing Dark Pulses in Normal Dispersion Optical Microresonators

Breathers are localized waves in nonlinear systems that undergo a periodic variation in time or space. The concept of breathers is useful for describing many nonlinear physical systems including granular lattices, Bose-Einstein condensates, hydrodynamics, plasmas, and optics. In optics, breathers can exist in either the anomalous or the normal dispersion regimes, but they have only been characterized in the former, to our knowledge. Here, externally pumped optical microresonators are used to characterize the breathing dynamics of localized waves in the normal dispersion regime. High-$Q$ optical microresonators featuring normal dispersion can yield mode-locked Kerr combs whose time-domain waveform corresponds to circulating dark pulses in the cavity. We show that with relatively high pump power these Kerr combs can enter a breathing regime, in which the time-domain waveform remains a dark pulse but experiences a periodic modulation on a time scale much slower than the microresonator round trip time. The breathing is observed in the optical frequency domain as a significant difference in the phase and amplitude of the modulation experienced by different spectral lines. In the highly pumped regime, a transition to a chaotic breathing state where the waveform remains dark-pulse-like is also observed, for the first time to our knowledge; such a transition is reversible by reducing the pump power.

Figure 1

(a) Simulated breathing dynamics of optical power spectrum, showing complex breathing behavior. (b) Simulated breathing dynamics in the time domain. The switching waves (SWs) are modulated during breathing and the waveform hole breathes strongly. Above each panel we show snapshots of the spectrum and time-domain waveform at moments indicated by the dashed lines.

Figure 2

(a) Experimental setup and summary of comb states. A pulse shaper is used to select individual comb lines; their fast breathing dynamics are recorded individually using a fast photodetector (PD) and digital oscilloscope. When recording the fast breathing dynamics, a portion of the breathing comb is used to trigger the oscilloscope to provide a timing reference. By controlling the pump power, both periodic breathing and chaotic breathing are experimentally studied. The chip is placed on a temperature controlled stage. (b), (c) Optical and rf spectrum of the breathing dark pulse state under an on-chip power of 1.1 W. The simulated optical spectrum is in close agreement with the measured spectrum. The breathing dark pulse state has narrow rf tones on the rf spectrum, whose frequencies are outside the half-width-half-maximum (HWHM) of the resonance. The inset in (c) is the measured comb power change of the breathing dark pulse (including pump line) at the through-port, showing a weak modulation depth of $\sim 3\%$. (d), (e) Optical and rf spectrum of the stable dark pulse state under an on-chip power of 1.0 W. There is no significant difference in the averaged optical spectra between the breathing and stable state.

Figure 3

(a),(b) By line-by-line pulse shaping, stable dark pulses can be compressed into transform-limited pulses (the shaped comb lines contain 90% of the comb power). Applying the same phase on the pulse shaper, the breathing dark pulses can also be compressed into transform-limited pulses. The intensity autocorrelations (ACs) of the compressed pulses are plotted in the figure. Using the phase retrieved in the line-by-line pulse shaping, the actual intracavity field is deduced to have a dark-localized waveform for the breathing Kerr comb (inset). The averaged simulated AC trace of the pulse-shaped breathing dark pulse comb [see the Supplemental Material [36] for numerical methods] is close to the AC of the pulse-shaped stable dark pulse comb. The inset is an example of a snapshot of the breathing dark pulse (SW: switching wave). The AC in simulations is shorter, because it calculates the full spectrum while only a portion of the spectrum is shaped in experiments. (c),(d) The breathing depth (blue) and phase (red) of individual comb lines. The breathing depths tend to be larger at the wings of the spectrum; different lines also breathe with different phases. Both the breathing depth and phase can change abruptly between adjacent lines.

Figure 4

(a) The averaged experimental and simulated optical spectrum in a chaotic breathing state. (b) Measured broadband rf spectrum in the chaotic breathing state (red line is the noise floor); the inset shows the simulated broadband rf spectrum. (c) The AC of the Kerr comb in the chaotic breathing state after pulse shaping [the same amplitude and phase adjustment used in Fig. 3] is close to the calculated autocorrelation trace of the transform limited pulse. (d) The simulated comb in the chaotic state can also be shaped into transform limited pulse; the inset shows an example of the dark-localized waveform. The compressed pulse in simulations is shorter because it calculates the full spectrum [similar to Figs. 3 and 3].