Phase Steps and Hot Resonator Detuning in Microresonator Frequency Combs

Experiments and theoretical modeling yielded significant progress towards understanding of Kerr-effect induced optical frequency comb generation in microresonators. However, the simultaneous interaction of hundreds or thousands of optical comb frequencies with the same number of resonator modes leads to complicated nonlinear dynamics that are far from fully understood. An important prerequisite for modeling the comb formation process is the knowledge of phase and amplitude of the comb modes as well as the detuning from their respective microresonator modes. Here, we present comprehensive measurements that fully characterize optical microcomb states. We introduce a way of measuring resonator dispersion and detuning of comb modes in a hot resonator while generating an optical frequency comb. The presented phase measurements show unpredicted comb states with discrete {\pi} and {\pi}/2 steps in the comb phases that are not observed in conventional optical frequency combs.

four-wave mixing obeying (with pump laser frequency and sideband frequencies ). At a later stage in the comb generation process, comb modes start to interact and generate new lines via non-degenerate four-wave mixing obeying energy conservation with . Each of these four-wave mixing processes is sensitive to the phases of the comb modes, which makes it difficult to predict the amplitudes and relative phases of the final comb state. Presently, it is not clear whether the comb phases can be aligned by a powerdependent mode-locking mechanism like in conventional mode-locked laser frequency combs.
Recent work shows the generation of solitons in microresonators [13,14,18], which would suggest a spontaneous phase-alignment mechanism.
Here, we show the measurement of phases of the modes in microresonator-based optical frequency combs. Most significantly, we find a variety of microcomb states in which the modes undergo discrete phase steps of and in different parts of the optical spectrum. The measured comb states are linked to pulses in time domain that are separated by rational fractions of the cavity round-trip time. As a step towards understanding these discrete phase steps, we introduce a new technique to measure the detuning of frequency comb modes from their respective resonator modes in a hot resonator (i.e. while generating optical frequency combs). This measurement utilizes a weak counter-propagating probe laser that does not disturb the comb generation process. We find comb states with all modes "red detuned" (comb mode at lower frequency than microresonator mode) from their resonances. This is a surprising result since the employed microresonators are thermally unstable in the presence of a slightly reddetuned laser, suggesting a nonlinear feedback mechanism on the resonator modes during comb generation.
In addition, the resonator mode detuning measurements provide important input for theoretical simulations of the frequency comb states. The last part of the paper shows these simulations, based on the Lugiato-Lefever equation (a nonlinear, damped Schrödinger equation) [19][20][21][22].
Many aspects of the measured comb spectra and phase profiles are in agreement with the simulations.

Results
The experimental setup for the measurement of phase and amplitudes of a microresonator-based optical frequency comb is shown in Fig. 1. The employed whispering gallery mode microresonator is a fused-silica microrod [23,24] with a mode spacing of ~25.6 GHz and a corresponding diameter of around 2.6 mm. The optical Q-factor is . Light is coupled into and out of the resonator using a tapered optical fiber. An optical frequency comb is generated by thermally locking [25] an amplified external cavity diode laser (~100 mW) to a microresonator mode. The high circulating power in the microresonator leads to the generation of an optical frequency comb via four-wave mixing [5,6], which is coupled out through the same tapered fiber that is used to couple light into the resonator. The generated optical frequency comb is sent through a liquid crystal array-based waveshaper that allows independent control of both the phase and amplitude of each comb mode. In a first step, a computer controlled feedback loop flattens the amplitude of the comb modes using simultaneously measured optical spectra as a reference. In the next step, a nonlinear optical intensity autocorrelator is used to measure the time domain response [26-28, 7, 29] and determine the peak optical power. Another feedback loop iterates the phases of the comb modes until the peak response of the autocorrelator is maximized (corresponding to all comb modes being in phase at a fixed time within each cavity round-trip time). We define the instantaneous field amplitude of the n-th comb mode at the frequency as The phase optimization works reliably by iterating the phase of one comb mode while maintaining all other phases constant. The iterated phase will be set to a value that increases the peak power of any existing amplitude modulation in the autocorrelation signal. This is done sequentially for all the comb modes. Running the optimization approximately 3 to 4 times for all the comb modes is sufficient to find the global maximum for the autocorrelation response, in which all modes are in phase [30,29]. In order to most efficiently use the waveshaper's dynamic amplitude range, it is advantageous to suppress excess pump laser light using a fiber coupled polarizing beam splitter. This pump suppression technique works by slightly misaligning the input polarization (using Pol1 in Fig.   1), such that only part of the pump light couples into the desired mode of the resonator. At the resonator output, the part of the light that did not couple into the resonator (because of mismatched polarization) interferes with the delayed pump light that couples back from the resonator, which in turn leads to an effective polarization change of the pump light. Thus, pump light and comb sidebands are in a different polarization state at the taper output. This polarization difference enables quenching only the pump light using another polarization controller and a fiber coupled polarizer (cf. Fig. 1).  In contrast, other comb states abruptly emerge from chaotic (not phase locked) comb spectra in a similar way as recently reported in soliton-like mode locking [13,14]. Although no complicated pump-laser frequency tuning algorithm is required to access them. One of these comb states is shown in     Fig. 4a. The optical frequency comb is generated in the same way as described in Fig. 1 by coupling an amplified external cavity diode laser (ECDL) into a microresonator. However, two optical circulators are added to the setup, which allows the light of an additional low power "probe ECDL" to propagate in backwards direction through the setup. The light of this backwards probe is frequency swept across a microresonator resonance (at a rate of ~20 Hz) and recorded with a fast photodiode.
Simultaneously, a small fraction of the comb light that is generated in the same resonance is backreflected (e.g. by small imperfections in the resonator and Rayleigh scattering). This backreflected comb light interferes with the probe laser, generating a beat note that changes in time depending on the current frequency offset between probe laser and comb light. The position of the microcomb mode follows from the position at which the beat note vanishes. An example for such a detuning measurement can be seen in Fig. 4b, showing a Lorentzian-shaped resonance dip combined with the beat note between probe laser and backreflected light from the resonator. This example shows backreflected pump laser light in the far blue detuned regime ("laser frequency" > "resonance frequency") at the beginning of thermal locking [25]. The time axis recorded on the oscilloscope has been replaced with a frequency axis that is directly calibrated by the beat note frequency.   However, we find that the detuning follows a parabola without obvious deviations, suggesting that the generation of phase-locked comb states is primarily governed by the cold cavity dispersion. More remarkable are the offsets of the detuning parabolas in Fig. 5b. While the pump laser detuning is close to zero in the case of the Munich Olympic Stadium comb (red squares), the offsets are getting smaller (red detuned) for the other two presented comb states with more random spectral envelopes and random phases (cf. Ref. [30]). The pump laser in the comb state in the lowest panel in Fig 5b is   The remaining part of the paper shows numerical simulations of comb states with phase relationships that are similar to those we experimentally measured. The comb simulations are based on the generalized spatiotemporal Lugiato-Lefever equation (LLE) that allows calculating the evolution of the optical field within a resonator [19][20][21][22]: where is the normalized field envelope, the cavity resonance linewidth, the normalized time, the normalized cold-cavity detuning between the pump laser frequency and the cold cavity resonance frequency , ̃ the dispersion parameter [20], the azimuthal angle along the resonator's circumference, and the pump laser amplitude normalized to the sideband generation threshold. Because of the Kerr effect, the resonance frequency is shifted, and the cold-cavity detuning cannot directly be compared to the experimental detuning measurements made at high intra-cavity power. Thus, we introduce the steady state "Kerr" detuning in units of cavity linewidths . The position of the Kerr-shifted resonance is given by the relation . A positive will therefore correspond to a blue detuning and a negative to a red detuning. In order to interpret the experimental results presented earlier in the paper, numerical simulations of the LLE were performed using a second order split-step Fourier method. In the anomalous regime of dispersion, the stationary solutions of the LLE are known to be either solitons or Turing patterns [36,37], the latter being referred to as "primary combs". In this framework, the experimental comb of Figure 3a-c can be interpreted as a pattern of 14 solitons on a 15-cell grid. Figure 6 shows such a comb simulated in the soliton regime (detuning and dispersion ̃ kHz/FSR) with an initial condition consisting of 14 Gaussian pulses positioned along the cavity circumference at angles , with . Experimentally, such a comb could be obtained from the initial primary comb corresponding to 15 sub-pulses in the time domain. When the pump laser is tuned into resonance, most of these oscillations turn into solitons, while one of them disappears.
While good qualitative agreement can be found between the measurement (Fig. 3a-c) and Fig.   6, it should be noted that this 14-soliton pattern is not stable when using the measured dispersion ( ̃ kHz/FSR). For such a dispersion value, the stable solitons are too long, and packing 14 of them in the cavity makes them interact and collide. This indicates the presence of an additional nonlinear mechanism that is not fully understood. ) which corresponds to a blue detuning in the hot cavity, while the solitons can be generated on both sides of the hot resonance. This latter remark is consistent with our experimental measurement of the detuning presented in Fig 5. To the best of our knowledge, such a linear combination of two solutions of the LLE has only been recently predicted in the case of a dual pumped cavity [38]. In our case, only one pump is used, and the specific mechanism leading to the generation of the observed combs is not fully understood. Further refinements have been proposed to the LLE, by including higher-order dispersion [39], or using the dispersion curve in the supplementary material of Ref. [30] to mimic the effects of modecrossings. None of these refinements, however, could explain the observed optical spectra.
Moreover, in the experiments we do not observe any mode crossings that would disturb the mode family used for comb generation (mode crossings would be also visible in the detuning measurement in Fig. 5b).

Discussion
In conclusion we have presented a scheme for measuring optical frequency comb phases, which reveals microresonator comb states with distinct phase steps across the optical spectrum.
Moreover, we introduced a method for in situ measurements of optical frequency comb mode detunings from the resonator modes in which they are generated. Surprisingly, our measurements show a red-detuning of all the comb modes in some of the phase locked states.
The combination of these measurements provides a comprehensive framework for the simulation of microcomb states using the nonlinear Schrödinger equation. We find that the nonlinear dynamics of the comb generation is mostly determined by the dispersion of the resonator, and we show that the LLE is able to qualitatively predict the main features of the complicated spectra and phase profiles of our measurements. The presented detuning measurement technique is promising for future studies in low dispersive resonators, in which it could be the key for experimental evidence on mode-pulling effects resulting from self-and cross-phase modulation between comb modes.