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We show a new class of complex solitary wave that exists in a nonlinear optical cavity with appropriate dispersion characteristics. The cavity soliton consists of multiple soliton-like spectro-temporal components that exhibit distinctive colors but coincide in time and share a common phase, formed together via strong inter-soliton four-wave mixing and Cherenkov radiation. The multicolor cavity soliton shows intriguing spectral locking characteristics and remarkable capability of spectrum management to tailor soliton frequencies, which would be very useful for versatile generation and manipulation of multi-octave spanning phase-locked Kerr frequency combs, with great potential for applications in frequency metrology, optical frequency synthesis, and spectroscopy.
© 2016 Optical Society of America
1. Introduction
Optical solitons represent a fascinating manifestation of nonlinear optical phenomena in nature [1]. In an optical cavity, a balanced interaction among Kerr nonlinearity, group-velocity dispersion (GVD), optical gain, and loss results in temporal dissipative solitary waves that underlie a variety of passive mode locking [2]. With a particle-like nature, dissipative solitons exhibit profound nonlinear dynamics that has been extensively explored in the past decades [2–10]. One intriguing example is the formation of soliton molecules [2,5] in which multiple identical solitons are bound together with discrete values of phases and temporal separations [2,5,7–9].
Recently, temporal dissipative solitons become relevant in the context of optical Kerr frequency comb generation in high-Q microresonators [11], where the soliton formation was shown to be a major mechanism responsible for the phase locking of Kerr frequency comb [9,10,12–14]. The extremely highly nonlinear nature of the systems results in complicated interplay between the underlying four-wave mixing (FWM) process and the device dispersion, whose exact nature is currently under intensive investigation [15–35]. Recent studies show that high-order dispersions of the devices have significant impact on the comb generation, leading to Cherenkov radiation that considerably modifies the comb spectrum [22,23,26,29–31,33].
Here we show a new class of complex solitary wave that forms during Kerr comb generation. The cavity soliton consists of multiple soliton-like spectro-temporal components that exhibit distinctive colors, coincide in time, and share a common phase, formed together via strong inter-soliton FWM and inter-soliton Cherenkov radiation, in opposite to conventional solitons and soliton molecules that consist of a single color while separated in time [1,5]. As we will show below, the spectral locations of multicolor solitary waves can be tailored to far separate frequencies without sacrificing their amplitudes, in strong contrast to current Kerr comb generation where the soliton spectrum is primarily located around the pump frequency with a bandwidth dependent on the GVD and intracavity pump power [12,14,22]. The generation of such a multicolor cavity soliton offers an elegant solution to produce multi-octave ultra-broadband phase-locked Kerr frequency combs that are essential for broad applications such as frequency metrology, precision spectroscopy, and photonic signal processing [36,37].
2. Conceptual illustration
The dynamics of an optical wave inside a driven cavity is described by the generalized Lugiato-Lefever equation [3,4,12,38]
where E(t,τ) is the intracavity optical field, with t and τ representing the slow and fast times, respectively. A0 is the input continuous-wave (CW) pump field launched at frequency ω0, with a laser-cavity detuning of Δ0 ≈ M02π − ω0tR, where M0 is the order of cavity resonance closest to the pump field and tR is the round-trip time determined by the free spectral range (FSR). The cavity has a circumference of L, and a nonlinear parameter , where n2 and Aeff are the Kerr nonlinearity coefficient and effective mode area, respectively, and c is the velocity of light in vacuum [4,12]. To describe the optical wave with a broadband spectrum, Eq. (1) includes the self-steepening effect and high-order dispersion where βm denotes the m-th order dispersion coefficient at the pump mode. κt = κ0 + κe represents the total power loss per round trip, where κ0 and κe are the intrinsic loss and pump power coupling coefficients, respectively.We search for a steady-state solution of multicolor solitary wave in the following form
where E0 is the CW field background inside the cavity. En, Tn, and ϕn are the amplitude, temporal width, and phase, respectively, of the n-th soliton at frequency ωn. When N = 1, it reduces to the case of single-color solitons which are supported by a constant device GVD [12,14]. The fundamental reason underlying the single-color soliton formation lies in that a constant GVD leads to group index (and thus the mode spacing of the cavity resonances) dependent linearly on the resonance frequency. Only a certain amount of mode spacing mismatch can be compensated by the nonlinear phase modulations of the intracavity field, which leads to a limited soliton spectrum centered around the pump frequency [22].For a multicolor soliton state, the solitary waves with different colors should have group velocities close enough such that they can remain overlapping in time for strong inter-pulse interaction. As bright solitons generally form in the anomalous dispersion regime, this implies that the device should exhibit a dispersion that is anomalous in multiple spectral regimes where the soliton spectra are located while with group velocities matched with each other. We thus speculate that the optimal condition for a multicolor soliton state would be a cavity dispersion oscillatory around zero, as schematically shown in Fig. 1. A simple example of oscillatory GVD is a sinusoidal function of frequency given as with B < 0, which corresponds to a group index ng(ω) and a propagation constant β(ω) given as
For the degenerate FWM process 2ω0 → ωs + ωi responsible for the primary comb generation, such a device dispersion results in a linear phase mismatch of which is small for cavity modes around ω0 + M2πΩ (M is an integer) and can be compensated by the nonlinear phase shifts induced by the pump wave. Consequently, the pump would produce a primary frequency comb in these anomalous dispersion regions with uniform efficiency that would evolve into multiple solitary waves with different colors but similar amplitudes.
Fig. 1 Schematic of a multicolor cavity soliton, composed of soliton-like spectro-temporal components with distinctive colors. The figure also shows the the corresponding sinusoidally oscillatory GVD (red), group index (green), phase mismatch for FWM (orange), and phase mismatch for Cherenkov radiation (blue).
In particular, it is easy to show that the phase matching condition is satisfied for FWM among these solitary waves located around ωj ≈ ω0 + Mj2πΩ (j = 1–4), i.e., for the process ω1 + ω2 → ω3 + ω4: [39]
which would lead to strong inter-pulse FWM among the solitary waves. Moreover, Eqs. (3) and (4) show that these solitary waves also directly match their phases since [39] This would result in strong Cherenkov radiation between the solitary waves [40,41]. As these solitary waves match their group indices (Eq. (3)), they overlap with each other all the time, resulting in significant inter-soliton FWM and Cherenkov radiation which not only balance soliton energies, but also lead to strong phase locking among the multicolor solitary waves, eventually forming a state of a multicolor cavity soliton.3. Results
To show this concept, we present a numeric example with details shown in Figures 2 and 3. We construct an example of a device GVD as shown in the blue curve of Fig. 3(d), which oscillates sinusoidally between 130 and 260 THz. Such a dispersion curve leads to phase matching, i.e. ΔβFWM ≈ 0, in the spectral regions around 100, 190, and 280 THz for the FWM process (Fig. 2(b)). Consequently, pumping at a cavity mode around 194 THz produces a primary comb in these three regions, as shown in Stage I of Fig. 2(c). With a red detuning of the pump frequency, the primary comb introduces significant modulational instability that extends the spectrum into a broadband frequency comb (Fig. 2(c), Stage II). Of particular interest is that, with further red tuning of the pump frequency, the frequency comb evolves into three distinctive spectral components centered around 104, 194, and 284 THz, respectively, that are stable over time (Fig. 2(c), Stage III).
Fig. 2 Formation dynamics of a three-color cavity soliton. (a) Laser-cavity detuning Δ0 increases linearly from 0 to 0.4 within 9.29 ns and stays constant afterwards. (b) Phase mismatch ΔβFWM for FWM process. (c) Spectral growth of the multicolor cavity soliton. The scaling of the time axis changes at t = 5 ns (the vertical dashed line) to show the formation details of soliton and its stability, which is also responsible for the kink in (a). (d) Schematic of the mode locking mechanism. The device parameters are given in [42].
Fig. 3 Temporal and spectral structure of a three-color cavity soliton. (a) Pulse waveform. (b) Spectrogram. (c) Spectrum (blue) and phase (magenta). The arrow indicates the discrete phase of the pump mode. The dashed curves are hyperbolic-secant fittings of the three distinct spectral components. (d) GVD (blue) and corresponding ΔβCR(ω0,ω) (red), where ω0 is the pump frequency.
These three spectral components correspond to three soliton-like waves with different colors, as shown clearly in Fig. 3. All three solitary waves are free from chirp, coincide in time (Fig. 3(b)), and exhibit hyperbolic-sechant-like spectral shapes (Fig. 3(c)). As the solitary waves match their phases, i.e. ΔβFWM ≈ 0 and ΔβCR ≈ 0, as well as group velocities (Fig. 2(b) and Fig. 3(d)), strong inter-soliton FWM and Cherenkov radiation are expected to occur among them (Fig. 2(d)), which bond the solitary waves together to form a multicolor cavity soliton. Such a state is not simply a sum of individual solitary waves, but is supported by strong inter-soliton interactions. This is evident in the valley regions of the comb spectrum which connect the soliton spectra (Fig. 3(c)), where the spectral amplitudes are considerably higher than the sum of individual solitary waves. The multicolor soliton state is also directly reflected in the phase of the solitary waves. As shown in Fig. 3(c), a common phase is shared across the entire spectrum that is 0.758π different from that of the pump, corresponding to ϕn = 0.758π in Eq. (2). In the time domain, the multicolor cavity soliton manifests as an ultrashort pulse with a full-width at half maximum of 4 fs, as shown in Fig. 3(a). The sidelobes of the pulse indicate the phase coherence and temporal beating among solitary waves. It is important to note that, over each round-trip time, there is only one single pulse, the multicolor cavity soliton, that cycles inside the cavity.
Figure 3(c) shows that individual soliton-like components inside the multicolor soliton exhibit similar spectral amplitudes while their frequencies are separated far apart. Consequently, the whole comb spectrum extends over a broad bandwidth of two octaves, significantly beyond what can be produced by conventional Kerr comb generation mechanisms [11–35]. A slight difference among the spectral amplitudes of solitary waves arises from the self-steepening effect: a stronger optical Kerr effect at higher frequency leads to a higher spectral amplitude and a broader spectral width of a solitonic component. The uniformity of the energies of the solitonic components results from the regenerative nature of high-Q cavity which allows multicolor solitary waves to interact over enough time to balance their energies. This feature is distinctive from conventional resonant wave interaction introduced by simultaneously phase and group velocity matching [31,40,41,43], where the strength of nonlinear interaction decreases quickly with increased frequency separation. It is also in strong contrast to the Cherenkov radiation in current Kerr frequency combs which only produces narrow band dispersive waves [22,23,28–33].
The multicolor cavity soliton exhibits intriguing spectral locking characteristics. Fig. 4 shows this feature. Nearly identical comb spectra are produced even by tuning the pump frequency considerably from 189.96 to 198.97 THz, across 40 cavity modes. Of particular interest is that the frequencies of the three-color solitonic components remain intact, as indicated by the dashed lines in Fig. 4. The resilience of the multicolor soliton to the pump frequency variation is another consequence of the strong inter-soliton nonlinear interaction, which locks the frequencies of soliton-like components to where the phase matching of inter-soliton FWM, that of inter-soliton Cherenkov radiation, and group velocity matching are the best satisfied. In this sense, the device GVD functions as a potential array to spectrally trap the spectrum of the multicolor cavity soliton to the most stable region where the inter-soliton locking is maximized.
Fig. 4 Spectra of three-color cavity solitons when the pump frequency varies from 189.93 to 198.97 THz, with a step of 0.904 THz (corresponding to 4 FSRs). Each spectrum is relatively shifted by 20 dB for better comparison. The stars denote the spectral peaks of the solitons. Right panel: zoom-in spectra in the range of 187 – 203 THz.
The idea of multicolor cavity soliton can be applied to produce cavity solitons that constitute a larger number of solitonic components with different colors. Figs. 5 and 6 show an example of a four-color cavity soliton. The situation is slightly different from the previous case since the phase matching can only be satisfied among Region 1-3 for the initial FWM process driven by the pump mode (Fig. 6(b)), resulting in primary comb generated in Region 1-3, but not in Region 4 (Fig. 6(c), Stage I). However, since ΔβCR(ω0,ω) ≈ 0 among all these four regions (Fig. 5(b), red), inter-soliton Cherenkov radiation would transfer energy from other three regions to Region 4 (Fig. 6(c), Stage II), eventually forming a four-color soliton with a common phase of 0.785π but tilted amplitudes of solitonic components (Fig. 5(a)). Interestingly, Solitonic Component 1 exhibits a spectral amplitude even higher than Solitonic Component 2 located around the pump mode, which results from the spectral recoil effect [41]. The imbalanced energy transfer towards Soliton Component 4 produces a spectral recoil towards lower frequency regions. However, as the frequencies of the solitonic components are locked due to the spectral trapping induced by the device dispersion, the spectral recoil effect manifests here as the amplitude increase of Solitonic Component 1, rather than soliton spectral shifting appearing in conventional systems [40,41].
Fig. 5 (a) Spectrum (blue) and phase (magenta) of a four-color cavity soliton. The arrow indicates the pump phase, and dashed hyperbolic-secant curves fit the four spectral components. (b) GVD (blue) and ΔβCR(ω0,ω) (red). Device parameters are given in [42].
Fig. 6 Formation dynamics of a four-color cavity soliton. The figure structure is the same as Fig. 2. In (a), Δ0 increases linearly from 0 to 0.4 within 13.27 ns and stays constant afterwards. Note again the change of time scaling at t = 5 ns.
We have shown that multicolor cavity solitons can be generated in a device with an oscillatory GVD curve. The question now is what type of real device structures is able to exhibit such GVD. Recent advance on dispersion engineering has shown promise towards this direction [44, 45]. Here we show an example with two anomalous dispersion regions, which produces a two-color soliton through Cherenkov radiation. The device structure is a suspended microring whose waveguide cross section is shown in Fig. 7(b). The waveguide structure consists of an aluminum oxide (Al2O3) layer sandwiched by a partially etched silicon carbide (SiC) bottom layer and a SiC top layer. The waveguide is suspended in air. Such type of waveguide structure is able to produce an oscillatory GVD curve shown in Fig. 7(c) for the fundamental quasi-transverse magnetic (quasi-TM) mode, with two anomalous dispersion valleys centered at 1100 nm and 1650 nm, separated by a normal dispersion region in between and surrounded by two normal dispersion regions outside. As a result, for a pump wave at λ0 = 1600 nm, Cherenkov radiation is expected to transfer energy to three different spectral regions around λ1 = 750 nm, λ2 = 1150 nm, and λ3 = 2000 nm where βCR(λi,λ0) = 0 (i = 1,2,3) (see Fig. 7(c), red curve). However, only the spectral region around λ2 = 1150 nm shares the same group velocity with the pump wave and exhibits anomalous dispersion at the same time. Consequently, the pump wave would produce a second solitary wave with a broad bandwidth around λ2 and two narrow-band dispersive waves around λ1 and λ3. Fig. 7(d) shows the spectrum simulated by Eq. (1) that agrees excellently with the discussion above, featuring a two-color cavity soliton along with two dispersive waves located around the predicted wavelengths. The spectrum covers a wavelength range from 700 nm to 2100 nm, which is well over one octave. This example shows that the proposed ultra-broadband multicolor soliton can be achieved by real waveguide structures. We expect that the spectra of such solitons can be tailored in a very large spectral space by appropriate dispersion engineering.
Fig. 7 (a) Schematic of a suspended microring coupled with a phase-matched bus waveguide. (b) Cross section of the suspended microring made of SiC and Al2O3, and the field profile of its fundamental quasi-TM mode (at pump wavelength). (c) GVD (blue) and ΔβCR (red). The GVD curve is simulated by the finite element method. The dimensions of the waveguide structure are: h1 = 200 nm, h2 = 100 nm, h3 = 254 nm, h4 = 200 nm, and w = 500 nm. The material dispersions are from Refs. [46] and [47]. (d) Spectrum of the two-color cavity soliton produced inside the microring, simulated via Eq. (1). The dashed curves are hyperbolic-secant fittings of two solitons. The pump is launched at 1600 nm with Pin = |A0|2 = 1.1 W. Other parameters are Δ0 = 0.086, κ0 = κe = 0.0029, γ = 2.5 W−1m−1, FSR = 200 GHz, and L = 522.2 μm. (e) Schematic of the mode locking mechanism. The figure is shown as a function of wavelength instead of frequency, simply in a direct correspondence to (d).
Current waveguide design [44,45] can only produce four zero-dispersion wavelengths with two anomalous dispersion regimes. In practice, there also remains a certain challenge to fabricate such waveguide structures due to potential difficulty in etching heterogenous material structures with high quality. On the other hand, at this moment, there exists no waveguide structure that is able to produce more complex oscillatory dispersion profiles, which will be left for future exploration. Very recent development on dispersion engineering [48] shows promise towards this direction, whose waveguide structure is not only capable of producing complex dispersion characteristics, but is also potentially easy to fabricate in practice.
4. Conclusion
To date, it remains an open question what is the optimal device dispersion characteristics for generating broadband phase-locked Kerr frequency combs. We hope that our study here provides an answer to this question. Note that the formation of multicolor cavity soliton is universal as long as the conditions for inter-soliton FWM and Cherenkov radiation are satisfied. The exact values of the amplitude and period of the oscillatory GVD (B and Ω in Eq. (4)) are not critical. |B| affects primarily the spectral widths of individual solitonic components, which broaden with decreased |B| and eventually smear together to form a flat comb spectrum. By changing Ω, the frequencies of solitonic components in a multicolor soliton can be tailored to desired spectral regions without degrading their amplitudes.
In conclusion, we have demonstrated by numerical modeling a new class of complex solitary wave generated in a driven Kerr nonlinear cavity. The formation of multicolor cavity soliton would have profound impact on Kerr frequency combs. On one hand, it enables producing solitary waves with frequencies separated by multiple octaves, ideal for f -to-n f interference that is critical for frequency metrology and optical frequency synthesis [36,37]. On the other hand, it might function as a universal scheme to produce Kerr comb in the spectral regions that are challenging to access for other approaches. Although we focus here in the context of temporal dissipative solitons, given the space-time duality, the concept of multicolor soliton can be applied to spatial solitons as well. In this case, a certain periodic modulation of spatial frequencies (say, with a gradient-index coupled waveguide array) together with the optical Kerr effect might result in a soliton that consists of solitonic components with intriguing spatial and directional characteristics.
Acknowledgments
The authors thank Govind P. Agrawal for helpful discussion. This work was supported by the DARPA SCOUT program through grant number W31P4Q-15-1-0007 from AMRDEC.
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