1. Introduction

Optical frequency combs based on monolithic microresonators with Kerr nonlinearity have become the subject of intensive research. They represent a convenient and practical solution to applications where space and weight limitations are prohibitive for using conventional femtosecond laser based frequency combs. There are several phenomena in resonator based Kerr combs that have been observed experimentally (see [1, 2] for review), but have not as yet found a satisfactory theoretical explanation. In particular, the majority of current theoretical models [3–7] only predict combs in resonators with anomalous group velocity dispersion (GVD) [14–17], while multiple experimental observations with normal or nearly-zero GVD resonators [18, 19] contradict this prediction.

Several theoretical models have been proposed to describe generation of fundamental (repetition rate of the comb corresponds to free spectral range (FSR) of the resonator) frequency combs in resonators with small anomalous or normal GVD. If GVD is small and anomalous, the nonlinear process can start from generation of a higher order comb, where the repetition rate of the comb corresponds to several FSRs, with subsequent filling of frequency gaps [6, 7]. Several experimental observations of the frequency comb formation [7, 15, 17, 20] correspond well to the theoretical prediction.

It was also shown that hard [21] excitation of the frequency comb is possible independent of the sign and value of GVD [22]. However, in that study the approach for a practical realization of the hard excitation regime and the stability of the solution was not considered. Hard excitation occurs if a finite photon number is initially present in the resonator modes supporting frequency harmonics of the comb. If a soft excitation regime of the frequency comb takes place along with hard excitation, a cross over between the solutions may be in order. For example, an abrupt transition between a frequency comb having a narrow spectrum, to a frequency comb with a wide spectrum can be attributed to such a cross over [23]. However, this is impossible if the soft excitation regime is forbidden. The hard excitation regime can also be initiated by multimode Stimulated Raman Scattering (SRS) [24] and subsequent four wave mixing of the optical pump and SRS frequency harmonics, if such a scattering is present. An experimental study of the hard excitation regime is needed for a deeper understanding of the process that starts the comb in realistic experimental conditions.

Another explanation of the Kerr frequency comb generation in resonators with normal or nearly-zero GVD relies on the presence of several nearly equidistant mode families in a spheroidal microresonator [25]. The mode families can have significantly different morphology dependent FSR and GVD values, so that the frequency comb can only be observed in one mode family. Such a process can be easily distinguished because of the distinct spatial profiles of resonator modes belonging to a family.

Still, there is a whole class of Kerr frequency combs observations that cannot be attributed to the above specified models. These optical combs have comparably small repetition rates (below 100 GHz), have a distinct soft excitation regime, and are excited in well recognizable mode families that are not characterized with anomalous GVD. In this work we show that these observations originate from interaction of a set of resonator modes, belonging to different mode families, that overlap the same volume. Mode interaction modifies the GVD of the resonator so much that its local value changes significantly. Modal interaction is also capable of changing the sign of GVD. Thence, we argue below that mode crossing is the main process leading to observed frequency combs in larger monolithic resonators.

Mode interaction has been previously encountered in microresonators. It is directly observed as a disruption of the continuity of dispersion in smaller resonators [26], and is indirectly revealed by the asymmetry of comb spectra [18, 19] (spectrally narrow combs generated in resonators with no mode interaction have to be symmetric to fulfill the conservation of energy). The spectrum of a microresonator taken in a broad frequency range [27] clearly shows the presence of spurious modes changing their position with wavelength, with respect to the fundamental mode family. Since spurious and fundamental modes can interact due to the unavoidable imperfections in the resonator shape, and the coupling mechanisms that couple light into multiple modes, they change the dispersive properties of the resonator. Even if spurious modes are not seen in an experiment with a particular selection of the coupling technique, they still can exist in the resonator [28,29]. Only a resonator supporting a single mode family (see, e.g., [30]) can be considered free from mode interactions. Therefore, it is natural to expect that mode interaction plays a key role in a majority of observations of Kerr frequency combs in optical resonators, regardless of the size or intrinsic material dispersion.

2. Theoretical model

To reveal the importance of mode crossing we numerically simulate comb generation in 21 identical optical modes coupled through cubic nonlinearity, and in the case of normal or nearly-zero GVD find no example of excitation of the frequency comb from zero field fluctuations (soft excitation) [31]. We use the physical model presented in [6, 32] to derive a set of equations by solving:

The effective GVD value is negligible in a relatively large resonator that has no mode interaction, γ₀ ≫ |D|. Here $D={\omega }_{12}+{\omega }_{10}-2{\omega }_{11}=-{\beta }_{2}c{\omega }_{FSR}^2/n_0$, β₂ is the effective GVD of modes with no interaction taken into account, c is the speed of light in vacuum, and ω_FSR is free spectral range (FSR) of the resonator, e.g. 2ω_FSR ≃ ω₁₂ − ω₁₀. For example, the non-equidistance of modes can be estimated as D ≃ −2π × 39 Hz (D ≃ 2π × 1 kHz) in a 10 GHz FSR CaF₂ (MgF₂) whispering gallery mode (WGM) resonator pumped at 1550 nm. This value changes rather insignificantly compared with typical bandwidth of modes used in reported experiments (2γ₀ > 2π × 200 kHz, or Q < 10⁹) in smaller resonators (35 GHz FSR): D ≃ −2π × 5.2 kHz (D ≃ 2π × 8 kHz). On the other hand, D ≈ γ₀ is required for generation of a fundamental frequency comb [32] with soft excitation [31]. This condition is achievable in a small resonator [23]. For instance, for a fundamental TE mode of a 100 GHz FSR MgF₂ WGM resonator pumped with 1721 nm light, the GVD parameter is D ≈ 2π × 200 kHz. Any observation of soft excitation of the comb in a larger resonator is the evidence that mode spacing is somehow altered compared with its value in an ideal resonator. Such a modification can be caused by a different mode family, the GVD of which can be tuned with the morphology of the resonator [25]. Another way to modify the mode spacing is related to mode interaction, which will be described below.

Let us assume that the mode supporting the first higher frequency harmonic (a₁₂) interacts with another mode of the resonator (c) having the same loaded Q-factor, but belonging to a different mode family. While it is possible for many modes (c_j) to interact linearly with a₁₂, only one mode usually dominates. The interaction, described by Hamiltonian $\overline{h}\kappa \left(a_{12}^{†}c+c^{†}{a}_{12}\right)$, results in the well known splitting of the resonance for a₁₂. The splitting may be considered as a pure frequency shift of the mode a₁₂ expressed as ω̃₁₂ = ω₁₂ − κ²/Δ in the asymptotic case of a large difference between the eigenfrequencies of the interacting modes, Δ = ω_c − ω₁₂, compared with the interaction constant κ, |Δ| ≫ κ.

The shift of mode a₁₂ resulting from interaction with mode c leads to modification of the effective GVD value for the pumped mode and the first two sideband modes,

To validate the analytical calculations we performed a simulation for 21 modes under the conditions given above, but also took into account their interaction with a mode c. In our experiments we observe frequency shifts of modes exceeding κ²/Δγ₀ = 10³ occurring due to mode interaction. This means that the value of dimensionless parameter κ/γ₀ can exceed 30. By selecting κ/γ₀ = 20 we found that soft excitation of the fundamental Kerr comb is possible for a wide range of frequency detunings: −50γ₀ > Δ > −140γ₀. The Kerr comb for Δ = −70γ₀ and ω₁₁ − ω = −2.4γ₀ is shown in Fig. 1(a). It has a slightly asymmetric spectrum and a fast roll-off for higher order harmonics. The comb starts from zero fluctuations of the field with essentially zero initial conditions. Since such excitation regime of the comb is absent in the case where no mode interaction is available, we conclude that mode interaction is the actual cause, as predicted by the reasoning discussed above. It is worth noting that selection of κ/γ₀ = 20 is not critical; the comb is generated for a broad range of parameters. A detailed study of the dependence of the comb properties on parameters κ/γ₀ and Δ/γ₀ will be presented elsewhere.

 

Fig. 1 The spectra of frequency combs generated in a resonator with nearly zero normal GVD (D/γ₀ = −0.02) where one of the resonator modes is shifted due to interaction with another mode. The amplitude of the external cw pump of the central mode (a₁₁) is (|F₀|/γ₀)(g/γ₀)1/2 = 2. Three cases are shown: (a) mode a₁₂ is shifted due to the interaction; (b) mode a₁₃ is shifted; and (c) mode a₁₄ is shifted. Because the relative GVD of the resonator is small it does not influence the nonlinear process, so shapes of the comb envelopes simply scale from picture to picture.

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The interaction of modes also explains the experimentally observed generation of combs with different repetition rates in resonators with normal GVD [18]. The comb step changes depending on which comb-generating mode (in the sequence) is shifted by the interaction. The intrinsic GVD of the resonator by itself is too small to influence this process. We have simulated generation of higher repetition rate combs and validated the concept (Fig. 1). The simulation shows that detuning of the pump light from the corresponding mode of the resonator, as discussed in [18], is of less importance than mode interaction for generating high repetition rate frequency combs in soft excitation regime. The change of the comb step in normal GVD resonators is different from the mechanism for step change in resonators with anomalous GVD, where repetition rate strongly depends on intracavity power of the pump light [1, 15].

3. Experiment

Using the setup shown in Fig. 2, we performed an extensive experimental investigation of the GVD value in overmoded resonators to illustrate variation of the resonator FSR. Several experiments were performed using this setup. In the first experiment the laser was modulated at an RF frequency ω_RF nearly equal to the FSR of the resonator. The modulated light was fed into the resonator and then sent to a photodiode. The photocurrent was analyzed using an oscilloscope (channel 2 in Fig. 2).

 

Fig. 2 Schematic diagram of the experimental setup for observation of position of optical modes of an arbitrary resonator. A CaF₂ resonator (1) is evanescently coupled via a prism (2) to the 1549.7 nm emission of a Koheras Adjustik fiber laser (3). The laser light is modulated by a modulator (EO-space) (4) driven by an Agilent synthesizer (5) at a frequency close to the FSR of the resonator (10 GHz). The RF output of the synthesizer (5) is modulated by its internal modulator at 25 kHz (6). The light exiting the resonator is collected at the photodiode (PD) (7). The spectral bandwidth of the PD is selected to be larger than the frequency of oscillator (6). the PD (7) output is split with one part directly connected to channel 2 of a Tektronix oscilloscope (10). The other output of the PD (7) and the oscillator (6) are mixed and filtered with a Stanford Research lock-in amplifier (8). The output of the lock-in amplifier (8) is connected to the input channel 1 of the oscilloscope (10). The frequency of the Koheras laser (3) is swept with a saw-tooth voltage at 2–25 Hz rate (9) which also triggers the sweep of the oscilloscope (10).

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The power of light exiting the resonator dropped when either the carrier or the modulation sideband coincided in frequency with the corresponding resonator mode. As a result, we observed five absorption dips at the oscilloscope screen: three distinct and two shallow Fig. 3. The shallow dips correspond to higher order modulation sidebands, and are not considered because of their low signal to noise ratio. The resonances are separated by frequencies ω_l − ω_l−1 − ω_RF and ω_l+1 − ω_l − ω_RF. The difference of those frequencies corresponds to the dispersion coefficient D. The accuracy of measurement is approximately one tenth of the full width at half maximum (FWHM) of the resonator mode. According to Fig. 3, this accuracy is good enough to observe that some mode families have a nearly equidistant spectrum, while other mode families have significantly non-equidistant spectra.

 

Fig. 3 Absorption of the modulated light by modes of the resonator, versus carrier frequency of the tunable laser. The spectrum is measured using the setup shown in Fig. 2 (channel 2 of the oscilloscope). (a) Modes are nearly equidistant. (b) Modes are not equidistant. Data were taken using the same WGM resonator selecting different mode families with different quality factors.

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We manufactured a resonator with unloaded FWHM on the order of 4 kHz, so the accuracy of the measurement was about 0.4 kHz. To improve the accuracy we used a lock-in amplifier in our second experiment, as shown in Fig. 2. Laser light was modulated at a frequency close to the resonator FSR. The modulator was driven with a voltage to generate multiple sidebands with similar amplitudes. The RF output of the synthesizer was modulated by its internal source at 25 kHz so each optical modulation harmonic had three frequency components comprising of two high power components separated by 50 kHz and a low power component with unchanged frequency. The carrier frequency of the light was swept at 2 Hz rate and 200 kHz span centered at the mode of the resonator. The optical output of the resonator was demodulated at a photodiode with 40 kHz bandwidth. The output of the photodiode and the 25 kHz synthesizer tone were mixed at the lock-in amplifier, then coupled to channel 1 of the oscilloscope, which was triggered with the laser sweeping source. An example of the oscilloscope signal is shown in Fig. 4.

 

Fig. 4 Phase shift of the modulated light resulting from the interaction with modes of the resonator versus carrier frequency of the tunable laser. The resonant features are observed when the carrier or a modulation sideband interacts with a mode of the resonator. Experimental data are taken with a lock-in amplifier (see Fig. 2).

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The figure contains six sets of dispersive features, each of which is the response of the measurement system to phase rotation of the optical frequency component by the corresponding resonator mode. The frequency splitting between centers of the sets corresponds to the difference between particular frequency separation of two adjacent optical modes and the modulation frequency produced by the synthesizer. This is similar to the observation made in the previous experiment, with the difference that here we measure the phase shift and not the amplitude change of the frequency component. Each set of dispersive features contains three peaks separated by 25 kHz: the central feature appears when frequency of a low-power modulation sideband coincides with the frequency of the resonator mode. At this point the optical power circulating in the resonator is very low and the position of the mode is not distorted by nonlinear processes. The left and right peaks result from interrogation of the resonator with positive and negative 25 kHz high power sidebands. Positions of these features are distorted by nonlinearity and are not used in these measurements.

When the mode bandwidth is loaded up to 100 kHz the point of zero phase shift can be detected in our setup with an accuracy exceeding 10 Hz. Such an accuracy is sufficient to see if the GVD induced by mode interaction is large enough to generate a Kerr comb. Precision of this measurement is limited by i) nonlinearity of the laser carrier sweep, which can be compensated using the frequency spacing between identical 25 kHz sidebands; and ii) the lock-in gain, which is defined by the ratio between 25 kHz modulation frequency and lock-in bandwidth 25 Hz (30dB). We measured the GVD parameter in several calcium fluoride resonators with which frequency combs were generated. The result of measurements are summarized in Fig. 5. The soft excitation of the frequency comb was observed only when D/γ₀ exceeded unity. This observation supports our theoretical prediction.

 

Fig. 5 Group velocity dispersion measured using the setup shown in Fig. 4. The measurement accuracy is on the order of 0.01 kHz. (a) GVD parameter for a CaF₂ WGM resonator with 30 GHz free spectral range and 25 kHz loaded bandwidth. Two mode families are studied (data are shown by circles and squares). The comb generation is observed only in modes corresponding to filled squares. (b) GVD parameter for a CaF₂ WGM resonator with 10 GHz free spectral range and 78 kHz loaded bandwidth. No comb generation was observed. Red diamonds stand for theoretical value of the GVD parameter.

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Finally, we performed an experiment with a larger WGM resonator with nearly zero (calculated value) relative GVD (D/γ₀), and observed generation of frequency combs with envelopes similar to those predicted by the theory. In this experiment we used a CaF₂ resonator with a diameter of about 6.7 mm. The resonator had approximately 9.9 GHz FSR with loaded quality factor exceeding 10⁹. We pumped the resonator with 1545 nm light from a semiconductor laser. The optical power emitted by the laser was 15 mW, and 3.2-1 mW of the light entered the selected modes of the resonator (the value depends on the selected mode). The output light was collected using a fiber and introduced to an optical spectrum analyzer. The resultant optical spectra are shown in Fig. 6.

 

Fig. 6 Observed spectra of three optical frequency combs generated in the same over-moded WGM resonator when light is coupled to three strongly interacting arbitrary modes. The wavelength of the pumping light was changed to select the modes, and depending on mode selection we observed (a) single-FSR, (b) dual-FSR comb, and (c) triple-FSR combs. Generation of higher order frequency combs, not shown here, was also observed.

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As can be seen, the experimentally observed frequency combs have spectral shapes similar to those simulated numerically (compare Figs. 1 and 6). The comb properties change significantly when we pump different modes of the resonator. Such a modification cannot be explained by a change in the geometrical part of the GVD of the modes. The interaction with degenerate modes, however, explains the observation.

4. Conclusion

We have shown theoretically that generation of optical frequency combs in resonators with Kerr nonlinearity and small group velocity dispersion, as previously observed in several experiments, is the result of linear interaction of resonator modes. The interaction changes the frequency of resonator modes resulting in soft excitation of the comb. We also measured GVD of several WGM resonators and showed that mode crossing is significant enough to explain the observed comb generation. We emphasize that the theoretical framework and conclusions of this study are valid for any nonlinear microresonator.