Nonlinear-Fourier-Transform-Based Dynamic Analyses on the Multiple Cavity Solitons in Kerr Microresonators

The Kerr microresonators have aroused widespread interests for their ultrahigh integration, compatible fabrication and ultralow energy consumption. Similar to the traditional mode-locked fiber lasers, the microresontors can sustain the generation of dissipative solitons called cavity solitons relying on the double balance of gain and loss as well as nonlinearity and dispersion. The state of multiple solitons is common in the microresonator with anomalous dispersion. In particular, an ideal state named as soliton crystals is obtained when the multiple solitons have a equidistant arrangement. And recent experiments have successfullly observed soliton crystals in the Kerr microresonators. To intuitively reveal the formation and evolution dynamics of the multiple solitons and get insight into the underlying physical mechanisms of soliton crystals, we applied the nonlinear Fourier transform to project these states into nonlinear discrete spectrum. The discrete spectra of single soliton with different modulated background waves have been displayed, which can help distinguish different optical components. Then, the nonlinear spectrum evolution of normal multiple solitons state shows the typical nonlinear states of the intracavity field. And the equidistant solitons have the effects of power enhancement, whose spectral intensity is several times that of a single soliton state. Comparing with the discrete spectrum of the equidistant solitons, the solitons inside the soliton crystals interact with each other by the modulated background. Our results suggest that the nonlinear Fourier transform is a powerful technique to characterize soliton dynamics in the Kerr microresonator, which provides a new perspective to understand the interactions of cavity solitons.


I. INTRODUCTION
N ONLINEAR Fourier transform (NFT), as a powerful mathematical method for signal processing [1] and ultralarge capacity optical communication [2], [3], has recently aroused widespread attention. Originally proposed for solving integrable nonlinear dispersive partial differential equations [4], NFT has been applied to investigate the localized structures governed by the nonlinear Schrödinger (NLS) equation, such as the soliton propagating in the optical fiber [5]. Different from the traditional signal analysis based on Fourier transform (FT), NFT projects the temporal signal into the nonlinear spectrum containing both continuous and discrete spectra [6], [7] corresponding to the soliton and radiation components, respectively. Critically, NFT requires an conservative Hamiltonian system having infinite conserved quantities. However, external pumping system sustaining more complex nonlinear dynamics especially soliton generation could still be expected to be characterized and analyzed by NFT.
The mode-locked fiber laser, a typically dissipative resonant system with a periodic boundary condition, can sustain the dissipative soliton relying on the dynamical balance between nonlinearity and dispersion, as well as gain and loss. Recent researches have succeeded observing the transient soliton evolution of the break-up and collisions in the fiber laser through NFT in real time [8]. The subsequent investigation suggests that the NFT can be utilized to characterize the nonintegrable system, whose dynamics rely on a perturbed NLS equation [9], [10]. Based on the corresponding discrete spectrum, the number of soliton, relatively temporal drifting velocity, and the sideband component can be clearly distinguished according to the distinct distribution of their eigenvalues. The pure soliton is experimentally reconstructed by picking out the solitary eigenvalues with large imaginary part [11]. Apart from the incoherent sideband component, the symmetric Kelly sidebands resulted from the interference between the soliton and the dispersive waves are a usual phenomenon in the fiber laser [12]. Recent study has theoretically demonstrated that NFT can be used to separate the symmetric sidebands and solitons [11]. Besides, as an elegant characterization method, NFT has also been applied to analyze the temporal evolutions of the stable and unstable dissipative solitons in a fiber laser, whose soliton properties are susceptible to the cavity parameters and the pump power [13].
In addition to the above active fiber lasers, recent researches on the passive Kerr resonators have become a hot topic for its chip-scale integration [14], [15], [16] and complementary metal-oxide-semiconductor (CMOS) compatibility [17], [18]. The cavity soliton (CS), where a sech-shaped pulse sits atop an infinitely continuous background wave, is a stationary stable state sustained by the passive resonator. Distinct to the fiber laser, the spectrum of the CS usually contains a high spectral line at the pump frequency due to the residual pump. It is important to suppress the spectral line of pump to obtain a perfectly smooth comb for potential applications. Although the combination of the photonic crystal structure [19] and the mutually coupled microcavties [20] can be a potential method, the fabrication accuracy and the thermal control would be extremelly challenging. Recent theoretical derivation has been reported to show that NFT is applicable to the passive resonator, whose basic model can also be regarded as perturbed NLS equation [21], [22]. S. K. Turitsyn et al. have demonstrated the feasibility of the nonlinear Fourier transform (NFT) in applying it to the cavity soliton. Their rigorous mathematical derivations and comprehensive numerical simulations published in [21], [22] have helped extend NFT to dissipative nonlinear systems, particularly the Kerr nonlinearity resonator with an external pump. This original work has no doubt in paving the way for investigating the characteristics of coherent structures especially the cavity soliton in Kerr resonator. This provides an potential method to filter out the residual pump, thus reconstructing a pure soliton according to its eigenvalue. The above investigations mainly focus on the single soliton state in the passive resonator. It is necessary to consider another common state of multiple solitons [23], [24], [25]. To get deeper insights into the dynamics and interactions of multiple solitons in the passive dissipative system, NFT method is applied again to intuitively demonstrate these multiple solitons states in the discrete nonlinear spectrum.
In this article, with the inspiration and guidance of the analyses in [21], we report on the characterization of multiple-soliton states with non-equal and equal temporal intervals based on their eigenvalue distributions. In particular, when CSs equidistantly fill the entire temporal domain, the state of soliton crystals (SCs) in the passive Kerr resonator is discussed in detail. We compare the NFT-based nonlinear spectra of the avoided-mode-crossing (AMX) and the nonlinear-mode-coupling induced SCs, which can help understand the physical mechanism of the nonlinear mode coupling process. Our investigations provide an efficient characterization tool to investigate the dynamics and interactions of multiple solitons in the passive dissipative system. Referring to the published work in [21], [22] which indicates the NFTbased approach can reduce the number of effective degrees of freedom in describing cavity soliton, our work could be thought of as complementary research on multiple solitons generated in Kerr resonator.

II. THEORETICAL MODEL
The evolution of the slowly varying envelope of the optical field inside a microresonator is modeled by the mean-field Lugiato-Lefever equation (LLE) described as [25]: Where E(z, τ ) is the envelope of the electric field. z is the longitudinal coordinate along the resonator, τ is the fast time in a reference frame moving with the intracavity field. α is the total losses per unit length including the intrinsic and the coupling losses. δ is the phase detuning per unit length defined as δ = δ 0 /L, δ 0 = (ω 0 − ω p )τ R is the roundtrip phase between the pump frequency ω p and resonance frequency ω 0 . L is the length of the cavity, τ R is the round-trip time whose inverse is the free spectral range (FSR). γ and k are Kerr nonlinear coefficient and group velocity dispersion (GVD), respectively. η = √ θ/L is the coupling efficiency, θ is the power coupling coefficient between the bus waveguide and the cavity. E in is the pump amplitude, P in = |E in | 2 is the incident power.
With the split-step Fourier method, we then obtain the stationary CS state by linerly scanning the phase detuning δ. To decompose the CSs into the nonlinear spectral data, the solution of a linear scattering problem known as the Zakharov-Shabat problem (ZSP) is used to perform NFT [26]: where E(τ ) is the temporal waveform of CSs calculated from (1), which is normalized to a time scale approximately containing the total energy of the signal. λ is a spectral parameter, which plays a role of a nonlinear analog of frequency. ν 1,2 are the auxiliary functions and the scattering data a(λ) and b(λ) is connected with them: Then the nonlinear spectrum of signal E(τ ) is defined as whereẼ c (λ) is the continuous part of the nonlinear spectrum, which refers to the radiation component and converges to ordinary Fourier spectrum at the low-power limit, andẼ d (λ n ) is the discrete part corresponding to the soliton component of the signal. λ n is the eigenvalue in the upper half complex plane which is the root of a(λ), and a (λ n ) is defined as: By picking out the soliton eigenvalues, the true temporal waveform of pure solitons (PSs) can be obtained by the inverse NFT : where m corresponds to the number of solitons. λ m,I and λ m,R are the imaginary and real parts of the soliton eigenvalues, respectively. φ(z) is the spectrum phase, and τ 0 is the time center associated with λ m,I and spectrum amplitude. With the guidance of the normalized process reported in [9], we normalize the amplitude with a scale Q s , which is the squared root of the power of a hyperbolic secant signal with a time width equal to a time window containing 99% of the signal energy.

III. RESULTS AND DISCUSSIONS
For a better comprehension of multiple solitons, we first investigate the discrete nonlinear spectra of three states of single soliton state. As is shown with the blue curves in Fig. 1(a)-(c), the temporal waveforms correspond to the normal CS, CS with Kelly sidebands, and CS with the Cherenkov radiation (CR) sideband (the dispersive wave). It can be observed that their main difference is the background wave, which is continuous for the normal CS and quasi-continuous for the latter two CSs. For their corresponding spectra as shown with the blue curves in Fig. 1(d)-(f), the latter two CSs respectively have multi-order symmetric sidebands and single asymmetric sideband which can correspond to the temporal features of their background waves. The components of the PS, background wave, symmetric Kelly sidebands and asymmetric dispersive wave can be easily distinguished from the respective eigenvalue distributions as shown in Fig. 1(g)-(i). By selecting the CS eigenvalue only, we can reconstruct the corresponding PS shown as the orange curves in Fig. 1.
Based on the analyses of the single soliton state, we further investigate the dynamics of multiple solitons. The physical parameters of the microresonator are listed as follows: α = 3.038 m −1 , k = −100 ps 2 km −1 , γ=0.8 W −1 m −1 , η=75.51 m −1 , FSR=200 GHz. The incident power P in is set to be 22 mW to avoid the complicated evolution of the chaotic states. To deterministically obtain two or three solitons state, we first perform the numerical simulations with the initial condition described as [27]: where n is an integer representing the number of solitons, τ n is temporal position of soliton. The roundtrip phase δ 0 is fixed to be 0.8 × 10 −2 rad within the existence range of CS state. As shown in Fig. 2(a), it can be observed that two stable CSs co-exist after the oscillation. The corresponding eigenvalues shown in Fig. 2(b) display a similar damped oscillation. The curves of CSs and PSs shown in Fig. 2(c)-(d) deviate from each other due to the background wave. For the three solitons states shown in Fig. 2(e)-(h), besides above phenomena, we find that the coincidence of the soliton eigenvalues is not as high as that of two solitons, which could be explained by the closer interval leading to a stronger interaction of CSs. This could be further demonstrated in the Fig. 2(i)-(l). The close CSs attract each other, collide and finally annihilate, which corresponds to the increasing separation of the soliton eigenvalues. Then we investigate the dynamic evolution of multiple solitons with a scanning speed of 10 −7 rad roundtrip phase Δδ 0 (initial δ 0 = 0). The intracavity field experiences three typical states corresponding to the modulation instability (MI), Turing pattern and CSs, as shown in Fig. 2(m). The power of the intracavity field gradually increases before the appearance of Turing pattern. When further increasing the roundtrip phase, the Turing pattern transitions into CSs and then some CSs collide and merge into the same one at around 7 × 10 4 th roundtrip while the annihilations happen at the roundtrip larger than 8 × 10 4 . After that, three CSs stay at a fixed location and display the same temporal characteristics including the amplitude and pulse width. When projecting these states into the discrete spectrum shown in Fig. 2(n), we can find that the imaginary part of the eigenvalues of the MI state continuously increase with the roundtrip. At around 1.6 × 10 4 th roundtrip, the eigenvalues show the bifurcation paths to the higher and lower imaginary parts corresponding to the pulses and the background wave, respectively. The annihilating and merging after the collision display as the disappearance and transient oscillation of the eigenvalue. Three close eigenvalues with higher imaginary part survive till reaching the maximum phase detuning (δ 0 = 0.98 × 10 −2 rad). Besides, we also utilize the NFT method to characterize the temporal evolution of cavity solitons with a detuning δ 0 of more than 2π different from the previous work (δ 0 << 2π), as shown in Fig. 3. The related physical parameters except for the stimulated Raman scattering (ignored here) can refer to those listed in Fig. 4 of [28]. detuning δ 0 linearly increases to 2π within 500 round trips, then to 2.8π within 40 round trips, and finally remains constant at 2.8π in the following round trips, as is shown in the red curve in the left panel of Fig. 3(a). This detuning variation aims at ensuring the coexistence of different nonlinear states, particularly the nonidentical solitons. The eigenvalue distributions and temporal waveforms at specific roundtrips denoted as the white dashed lines in Fig. 3(a) are depicted in Fig. 3(b)-(k). The results shown in Fig. 3(j)-(k) are consistent with those in Fig. 2(b)-(c). With increasing δ 0 to be 2.4π at the 520th round-trip, the continuous background wave transforms to composite fields containing both continuous wave and Turing pattern as shown in Fig. 3(i). This feature can be obviously observed by the eigenvalues of continuous wave (blue hollow pentagrams) and Turing pattern (green circles and purple triangles) shown in Fig. 3(h). With further increasing δ 0 , the composite background becomes a combination of continuous and chaotic waves. The corresponding eigenvalue distribution depicted in Fig. 3(f) shows a significant distinction from those in Fig. 3(h) and (j). The pure solitons shown as the orange curves in Fig. 3(g) and (i) can still be reconstructed even if the solitons  are embedded in the background wave with high amplitude. The coexisted nonidentical solitons with different amplitudes, pulse widths, and drifting velocities emerge from the second chaotic transition, as shown in Fig. 3(d), and (e). And these two solitons approach each other due to their different drifting velocities, which correspond to the positive real parts of the soliton eigenvalues shown in the inset of Fig. 3(d). While getting close enough, these two solitons merge into a whole with two peaks drifting at the same velocity. The eigenvalues of this bound state shown in the inset of Fig. 3(b) have real parts with opposite signs, which can help distinguish this bound state from that in Fig. 3(d).
Comparing to the multiple CSs with stochastic interval, the equidistant CSs is more attractive for the power enhancement. To show the temporal waveforms and spectra of the single CS and five equidistant CSs, the initial conditions are determined according to the (7). The roundtrip phase δ 0 is fixed to be 0.8 × 10 −2 rad within the existence range of CS state. As shown in Fig. 4(a), the single CS overlaps with one pulse of the equidistant CSs indicating the same temporal profile. The spectral intensity of the equidistant CSs is 13.98 dB higher than that of the single CS at specific frequencies spaced by 5 × FSR, as shown in Fig. 4(b). This intensity difference exactly equals the square of the number of CSs denoted as 13.98 dB = −10log 10 5 2 . This connection is determined by the equation written as S CSs = 2 l=−2 S CS exp(−iωlτ R /5) → |S CSs | 2 = 5|S CS | 2 , where S CS and S CSs respectively represent the Fourier transforms of the single soliton and equidistant CSs, and lτ R /5 is the temporal position of the equidistant CSs. When concentrating on the discrete spectrum shown in Fig. 4(c), we can find the eigenvalues of the soliton components almost coincide with each other. In addition, we can also observe the eigenvalues at around R(λ) = ±1, which is similar to the condition of two symmetric sidebands shown in Fig. 1(h). After inverse NFT-based reconstructing shown in Fig. 4(d), the PSs deviate a lot from the equidistant CSs indicating significant interactions induced by the background wave.
The above equidistant CSs are obtained with an ideally initial condition, which is hard to generate such a high-repetition-rate initial pulse train. To achieve the SCs state in the microresonator, the method by introducing a proper AMX to alter the local dispersion has been numerically and experimentally demonstrated [29]. To investigate the evolution of SCs, AMX is included by introducing an additional detuning variation per round trip (Δ) for a certain mode with mode number m defined as Our previous work has studied the SCs containing 5 pulses [30], whose generation process and dynamics are representative to some extent. We still take the SCs with 5 pulses as an example which requires the mode number m = 5, Δ/2π = −12.7 MHz, and note that the SCs with more pulses has the similar phenomena. Before the complex dynamic evolution of SCs, we first observe the arrangement process of CSs induced by the modulated background wave. We fix δ 0 = 0.8 × 10 −2 rad and introduce the AMX at 8 × 10 4 th roundtrip to the five CSs with random intervals. As is shown in Fig. 5(a)-(b), the drifting velocity of each CS differs a lot when the AMX is just introduced. With the increasing of the roundtrips, they share the same left-drifting velocity and interval. The overlap of the soliton eigenvalues becomes higher after the AMX is introduced, which indicates the modulation of the background wave makes the interaction strength between CSs tend to be consistent. The temporal waveforms and the spectra shown in Fig. 5(c), and (d) are similar to that shown in Fig. 4(b) and (d).
For the SCs dynamics during scanning the detuning, typical states mentioned in the multiple CSs [cf. Fig. 2(m), and (n)] can also be observed in Fig. 5(e). The collisions almost all happening at around 7 × 10 4 th roundtrip only lead to the merging without annihilating, which is important to ensure 5 CSs left. After the frequent collisions, a perfect SCs state is formed with a left-drifting velocity. In the perspective of the discrete spectrum shown in Fig. 5(f), the chaotic distributions of the eigenvalues appear at around 7 × 10 4 th roundtrip corresponding to the frequent collisions. The eigenvalues corresponding to the soliton components survive after the collisions. The increasing of the soliton eigenvalues and the decreasing of the CW eigenvalues satisfy the physical properties of CS varying with the detuning δ [15]. In addition to the SCs generation induced by the AMX, our previous work based on the nonlinear mode coupling provides an alternative possible method to generate a perfect SCs state [30]. Because the coupling strength between the fundamental and the second harmonic waves is relatively weak, the second harmonic wave can be regarded as a perturbation and the fundamental wave still satisfies the NFT. The physical parameters are chosen according to the [30]. As is shown in Fig. 6(a), the colliding and merging of the adjacent pulses lead to the generation of SCs similar to that based on the AMX [cf. Fig. 5(e)]. Distinct to the situation shown in Fig. 5(f), The soliton and CW eigenvalues shown in Fig. 6(b) almost keep constant after the generation of SCs. In the single discrete spectrum at around 8.5 × 10 4 th roundtrip, an eigenvalue with a large real part appears indicating the generation of the asymmetric sideband. we could explain the above phenomena as that the energy exchange between the fundamental mode and second harmonic mode not only alters the local dispersion but also balance the power of the fundamental mode. The sideband generation induced by the local dispersion modulates the background wave and ensures the temporal equidistance. The reconstructed PSs shown in Fig. 6(c) and (d) interact with each other by the oscillated background wave.

IV. CONCLUSION
In summary, we first apply the NFT to characterize the dynamics and interactions of multiple CSs in the Kerr microresonator. The CS without any sidebands and CS with symmetric Kelly sidebands or asymmetric CR sideband display distinct eigenvalue distribution, which can act as a guidance to confirm the similar components. For the more usual multiple CSs state in the anomalous dispersive condition, the corresponding dynamics of the discrete spectrum can clearly show the typical nonlinear states in the Kerr microresonator. In particular, the equidistant CSs have the effects of power enhancement, whose spectral intensity is several times that of a single soliton state. Comparing to the ideally equidistant CSs, the discrete spectrum of the SCs based on AMX shows a sideband eigenvalue which will introduce a modulation on the background wave. The overlap between the reconstructed PSs and SCs is not as high as that of the ideally equidistant CSs, which indicates that the pulses inside the SCs interact with each other by the modulated background wave. Besides, the discrete spectrum of SCs based on the nonlinear mode coupling is similar to that of the AMX-based SCs, which can help understand the coupling process in a linear way. Our investigations broaden the NFT-based analyses to characterize the dynamics of multiple CSs, leading to the insight reinforcement of soliton dynamics in the Kerr microresonator.