1. Introduction

The microcavity optical frequency comb (microcomb) is a kind of coherent spectrum of equally spaced spectral lines in the frequency domain generated in a nonlinear high-Q optical microcavity through the third-order nonlinear effect. Due to its compactness, solid stability and good coherence property, it can be used in many fields such as coherent communication [1], low-noise microwave generation [2], optical atomic clocks [3], etc.

In recent years, the method of generating soliton microcombs using laser diode (LD)-nonlinear Kerr microcavity butt-coupled system has received extensive attentions [4–10]. Compared with the traditional pump laser frequency sweeping method, this method has the advantages of system simplicity and compactness and does not require additional amplifiers and complex feedback control systems [11–16].

Experimentally, the heterogeneous integration of a pump LD with a high-Q nonlinear optical microcavity has been successfully achieved [10]. And it has been demonstrated that the system can generate soliton microcombs by simply changing the LD driven current either in forward or backward directions for LD output laser frequency sweeping, or even simpler, by directly setting the current in a certain interval to generate soliton microcombs in a so-called “self-starting” manner [9]. Theoretically, the researchers established some theoretical models to describe the dynamical process of the system by studying the relationships among the microcavity forward propagation mode, the microcavity backward propagation mode, the LD pump mode, and the LD carrier concentration [9,10,17]. These theoretical models and numerical simulation results are mainly used for analyzing the evolution of the system when the LD pump power level is close to the parametric oscillation threshold of the microcavity four-wave mixing (${\textrm{E}_{th}}$). Under such condition, the evolution of the system fits well with the soliton self-injection locking model [4,17]. However, the LD pump power (typical level of ∼100 mW) used in the actual experiment is often much higher than ${\textrm{E}_{th}}$. Under the changed condition, there would be distinct differences between the dynamics of the system with LD pump power much higher than ${\textrm{E}_{th}}$ and close to ${\textrm{E}_{th}}$.

In this paper, by combining the laser diode rate equation [18] and the microcavity coupled mode equation [19] with the introduction of resonant Rayleigh backscattering (hereinafter referred to as backscattering) [20], we obtain a system of rate equations satisfied by the four parameters describing the microcomb generation system. The numerical simulation of the model shows that when the LD pump power is much higher than ${\textrm{E}_{th}}$, the mechanism of the system generating the breathing soliton comb is not the self-injection locking of the LD main pump mode lasing frequency as predicted by the traditional theoretical models [12,21,22], but a newly discovered microcavity resonant excitation mechanism by the pump mode modulation sideband which is closely related to the breathing characteristics of the generated soliton microcomb.

The structure of this article is arranged as follows: In section 1 we introduce the research background and motivation of our work with a brief summary of our findings. Section 2 exhibits the theoretical model and numerical analysis, which can be divided into four sub-sections. Sub-section 2.1 builds the nonlinear optical Kerr microcavity-LD butt-coupled system model and formulates a set of rate equations describing it. Sub-section 2.2 deals with the numerical simulations and gives three running results with different conditions (i.e., ignoring the nonlinear effect, ignoring the backscattering effect, and the full model with LD pump power close to ${E_{th}}$). We compared them with the published works [4,10,22] to validate our numerical model. Sub-section 2.3 shows the detailed numerical results under high pump power injection condition with the analyses of those results given in sub-section 2.4. In section 3 we conclude our work.

2. Theoretical model and numerical analysis

2.1 Model overview

Figure 1 shows the butt-coupled system between the laser diode and the nonlinear optical microcavity. The parameters $\textrm{A}$, ${R_e}$ and ${R_0}$ are the mode amplitude of the LD resonator, the amplitude reflectivity of the LD rear facet and the output facet, respectively. ${F_{in}}$ and ${b_{out}}$ are the pump mode coupled into the microcavity and the backscattering mode coupled out of the microcavity, respectively. a and b represent the time-domain slowly varying amplitude envelopes of the forward (i.e., count clockwise direction) and backward (i.e., clockwise direction) propagation modes in the microcavity, respectively. The frequency domain distribution characteristics of the forward and backward propagation modes (i.e., the amplitudes of each order comb-tooth modes of the microcomb) can be readily obtained by performing the inverse Fourier transform on them respectively.

 

Fig. 1. Model diagram of butt-coupled system between laser diode and nonlinear optical microcavity with resonant Rayleigh backscattering.

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We introduce the microcavity backscattering effect into the Coupled Mode Theory (CMT) and combine with the rate equation of the LD to obtain the normalized system of rate equations [18–20,23]. It should be noted that, for the convenience of simulation, we make a frequency-shift to the amplitude of the LD pump mode: $A = \textrm{A}{e^{ - i({\omega - {\omega_0}} )t}}$, so that the change rate of its argument with time $\frac{d}{{dt}}\arg (A )={-} ({\omega - {\omega_0}} )$ is the frequency detuning between the LD pump lasing frequency $\omega$ and the cold cavity resonant frequency ${\omega _0}$ of the microcomb central mode (i.e., the pump mode) of the microcavity with no pump light injection [4]). At the same time, all of the parameters a, b, and $A$ are normalized by the photon number corresponding to ${E_{th}}$, that is, ${|A |^2} = 1$ represents the number of photons in the LD pump mode reaching ${k_m}/2{g_0}$.

All simulation parameters (see Table 1) are derived or adopted from data presented in the published literatures and related books [8,10,17,18,24]. Note that according to these parameters, the microcavity pump power ${|{{F_{in}}} |^2} \approx 600$, which is determined by the injected current of the LD, is much larger than ${\textrm{E}_{th}} = 1$, that is, the regime we deal with here is the so-called large-signal injection state which is significantly different from the small-signal injection regime where the LD pump power is just approaching or slightly higher than ${E_{th}}$ as analyzed in previous literatures [9,10,17].

In addition, it should be noted that the nonlinear thermal effect of the microcavity is not considered in our analysis model. On the one hand, it is for the sake of simplifying the analysis model. On the other hand, in some microcavities made of neglected thermo-optic coefficient materials (such as thin film lithium niobate) [25,26] or microcavities composed of athermalized optical waveguides which can effectively minimize the thermal effect [27], the influence of thermal effect during the generation of microcombs can be safely ignored.

Table 1. Normalized simulation parameters and adopted values

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The equation system (1) is integrated using the split-step Fourier method and the fourth-order Runge-Kutta method. It should be noticed that in the experimental system, the primary optical combs are generated by the quantum noise parametric amplification and the mode selection of the microcavity. However, our model does not introduce any white noise for seeding the initial parametric process. We define $a({\theta ,t} )$ as the time-domain slowly varying amplitude envelopes of the forward propagation modes in the microcavity. If only the microcomb central mode exists in the microcavity, then the curve ${|a |^2} - \theta$, $\theta \in ({ - \pi ,\pi } )$, is a horizontal line. According to coupled mode theory [19], the curve ${|a |^2} - \theta$ will always be in the form of a horizontal line (i.e., the state in which higher-order sideband microcomb modes have not yet initiated). But due to the truncation errors (introduced by the finite sideband comb modes considered in our model and the non-infinitely small step in discretization) in the numerical calculation, the curve ${|a |^2} - \theta$ would appear extremely small jitter. And in effect this jitter can take the role of noise seeding each simulation to evolve from the initial zero state (equivalent noise state) to the microcomb state.

In order to conveniently represent the simulation results, we introduce a set of normalized detuning quantities

2.2 Model validation and program verification

Before presenting the results of the full model (i.e., the model including both the backscattering feedback and the Kerr nonlinearity of the microcavity), we begin with the validation of the proposed model and our program’s credibility. First, we verify the LD frequency self-injection locking characteristics of our model under linear backscattering feedback condition. Let $\textrm{g} = 0$ (i.e., ignoring the nonlinearity of the microcavity) and adopt the simulation parameters provided by the literature [4]. Then we make a comparison of the simulation result with the following equation.

Equation (3) is the steady-state relationship formulated in the literature [4], where K, $\psi$ represent the combined coupling coefficient and the delay phase, respectively. We compare the simulation results of our model with the curves obtained by Eq. (3) in Fig. 2, indicating that our model is completely reliable when ignoring Kerr nonlinearity. Note that the simulation curves of our model would not fully follow the unstable regions (i.e., the bistability regions) of the theoretical steady-state curve, which is consistent with the results presented in the literature [17], where it is found that the locked state region for continuous one-way scanning may not fully coincide with the locking range (see Fig. 1(c) of the literature [17]) and even be different for different scanning directions.

 

Fig. 2. LD frequency self-injection locking characteristics of our model under linear backscattering feedback condition. The red line represents the steady-state curve, and the blue line represents the simulation results of our model. The delay phases of (a), (b), (c), (d) are Ψ = 0, π/2, π, 3π/2, respectively.

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Next, we ignore the backscattering feedback by setting ${\beta ^{\prime}} = 0$ but maintain the Kerr nonlinearity and adopt the parameters of the soliton evolution example in the literature [22] to perform the nonlinear characteristics validation. The simulation results shown in Fig. 3 are basically the same as the soliton evolution example reported in the literature [22], demonstrating that our program is also completely credible when ignoring backscattering feedback in our model.

 

Fig. 3. Nonlinear characteristics verification of our model and program. The parameters adopted from the example of soliton evolution in the literature [22].(a) Transient dynamics. (b) Final pattern in the azimuthal direction. (c) Corresponding Kerr comb in frequency domain.

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Finally, we validate the case of the full model by giving out the simulation results under the condition of the LD pump power ($J = 0.7785$) just nearing the threshold ${E_{th}}$. The results presented in Fig. 4 are highly alike with those reported in the literature [10]. The simulation results mainly include the following features, 1.The detuning curves $\zeta - \xi$, ${\zeta _a} - \xi$ and ${\zeta _b} - \xi$ basically coincide with each other, that is, $\omega$ and ${\omega _{{a_0}}}$ have the same values in most of the frequency sweeping regions (see Fig. 4(c)). 2. A frequency-locking region indeed appears in the detuning curve $\zeta - \xi$, and in the frequency-locking band, the energy in the microcavity is maintained at a constant state with a relatively high pump-comb conversion efficiency (see Fig. 4(a)). 3. During the frequency sweeping the microcomb does not change their dynamical states (see Fig. 4(b) and Fig. 4(d)), such as the evolution from bi-soliton to single soliton under high pump condition.

 

Fig. 4. When the laser diode pump power is close to the threshold E_th, the forward frequency sweeping results of our model. (a) The normalized intracavity power evolution of microcavity forward propagation modes. (b) The spatiotemporal intracavity power evolution in the Kerr microresonator. (c) Detuning relationship curves, ζ - ξ, ζ_a - ξ, and ζ_b - ξ. (d) The spatiotemporal intracavity power evolution in the Kerr microresonator reported by literature [10].

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2.3 Nonlinear dynamics under LD high pump power condition

Now we consider the case of the full model under LD high pump power condition. First, we show the simulation results of the microcomb excitation by pump frequency sweeping method. We set the frequency sweeping step size to $\Delta \xi = 0.5$. And in order to obtain stable $\zeta - \xi$, ${\zeta _a} - \xi$, ${\zeta _b} - \xi$ detuning relationship curves, for each detuning, before sampling, the system evolves sufficiently long time (50 normalized time units, or 100 microcavity photon lifetimes) to allow the system to reach steady states.

The results in Fig. 5(a) show that all the detuning relationship curves $\zeta - \xi$, ${\zeta _a} - \xi$, and ${\zeta _b} - \xi$ indeed exist frequency-locking regions. Curves ${\zeta _a} - \xi$ and ${\zeta _b} - \xi$ coincide with each other in most regions and their frequency locking region is much wider than that of curve $\zeta - \xi$. This behavior shows that the exact meanings of ${\omega _{{a_0}}}$ and $\omega$ are different in the frequency sweeping process. Therefore, the method which treats ${\omega _{{a_0}}}$ as $\omega$ in the theoretical analysis of the literature [10] may not applicable anymore in the case of LD high pump power condition. In Fig. 5(b), we define all the Turing state, multi-soliton state and single soliton state with breathing characteristics as the stable microcomb states. It can be clearly seen that the generation regions of the stable microcomb states are within the frequency-locked regions of curves ${\zeta _a} - \xi$ and ${\zeta _b} - \xi$, however, they are outside the frequency-locked region of curve $\zeta - \xi$, indicating that the reason for the system can produce stable microcomb states under the condition of LD high pump power is not due to the self-injection locking of the LD main pump lasing frequency, however, which is the conclusion of several literatures [9,10,17]. Those researches deal with the same nonlinear system under the LD pump power just above parametric threshold.

 

Fig. 5. Microcomb forward frequency sweeping excitation simulation results. (a) Detuning relationship curves ζ - ξ, ζ_a - ξ, and ζ_b - ξ, (b) The red and blue line represent the total number of photons in the microcavity and the average amplitude of LD pump mode varying with the detuning ξ, respectively, the color-coded regions represent the pump frequency band that can produce stable microcomb states with breathing characteristics, with orange-red, orange-yellow, purple, green, and sky blue ones representing the regions where Turing state, quadruple-soliton state, tri-soliton state, bi-soliton state and single soliton state microcombs are generated, respectively.

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The red curve in Fig. 5(b) shows the change of the energy in the microcavity during the frequency sweeping. The energy in the microcavity reaches the maximum when the detuning $\xi$ is near 0 and the corresponding dynamical state is the chaotic state shown in Fig. 6(b), 6(f) and Fig. 7(b), then the energy in the microcavity decreases rapidly with the increase of detuning $\xi$. As shown in Fig. 5(b), in the different color-coded regions where different stable microcomb states with breathing characteristics can be produced, the average total photons in the microcavity are maintained at a relatively low and constant level, which further indicates that the LD main lasing frequency in these regions is not locked by the microcavity backscattering feedback, otherwise, the total number of photons in the microcavity should be maintained at a relatively high level. The specific reason why the system can produce a stable microcomb state under the condition of LD high pump power will be explained below.

 

Fig. 6. Nonlinear dynamical evolution characteristics of microcomb at different frequency detuning points during forward frequency sweeping process, (a)-(d) respectively represent the evolution of the intracavity power |a|² at four specific frequency detuning points, (a) Turing state, (b) chaos state, (c) breathing bisoliton state, (d) breathing single soliton state, (e)-(h) respectively represent the amplitude variation of LD pump mode corresponding to (a)-(d). The insets in panel (e)-(h) are local zoom-in views of the corresponding images.

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Fig. 7. Frequency domain spectrum diagrams of the microcombs generated by forward frequency sweeping process, (a), (b), (c), (d) are the corresponding frequency domain spectrum diagrams of Fig. 6(a), (b), (c), (d) at t’=50, respectively.

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Figure 6 shows the evolution of the time-domain slowly varying amplitude envelopes a at four discrete frequency detuning points during the forward frequency sweeping process, and the corresponding periodic amplitude oscillation characteristics of the LD pump mode. More or less, all the stable microcomb states have the breathing characteristics, and the corresponding LD pump mode amplitudes are also in high-frequency periodic oscillation states.

Under the selective settings of our system model parameters, if the detuning parameter $\xi$ is adjusted to a certain frequency range, no matter what the initial state the system begins with, a specific stable microcomb state with breathing characteristic can be obtained directly and deterministically after the system evolves for a long enough time, such as the results illustrated in Fig. 8.

 

Fig. 8. Self-starting of single soliton with different initial states. (a), (b), (c), (d), (e) and (f) represent the self-starting of solitons with different initial states, whose initial states are 0, Turing state, quadruple-soliton state, tri-soliton state and bi-soliton state respectively.

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It should be noted that this excitation process does not require any frequency sweeping, so we call this special evolution process as the self-starting of the stable microcomb states.

As shown in Fig. 9, from the self-starting process of the system to generate different stable microcomb states, we can clearly see that when the system generates the stable microcomb states with breathing characteristics, the LD pump mode amplitude also oscillates periodically and synchronously with the corresponding breathing Turing or soliton microcomb. This phenomenon suggests that the two states (i.e., the breathing stable microcomb state and the LD pump mode amplitude oscillating state) may be mutually correlated and matched in phase. Based on this phenomenon, we speculate that the breathing period of the stable microcomb state should have some certain matching relationship with the oscillation frequency of the LD pump mode amplitude, which will be demonstrated in next sub-section.

 

Fig. 9. The self-starting of the stable microcomb states, (a)-(c) The self-starting of the microcomb stable stats at three specific frequencies, the final dynamical evolution states of (a), (b), (c) are Turing state, bisoliton state and single soliton state with breathing characteristics, respectively, (d), (e), (f) depict the LD pump mode amplitude oscillating states corresponding to (a), (b), (c) during the microcomb self-starting process, respectively. The inset in each panel is a local zoom-in view of the corresponding image.

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2.4 LD pump modulation sideband resonant excitation mechanism

Here we introduce the maximum value $\max ({|a |} )$ of the time-domain slowly varying amplitude envelope $|a |$, and use the oscillation period of this parameter to characterize the breathing period of the stable microcomb state. For simplicity we use a breathing soliton state for illustration.

As shown in Fig. 10(c) and (d), the relative positions of the intersections of the two curves always keep unchanged during the long-term evolution, indicating that the oscillation frequencies of the two parameters are the same and their relative phase relationship is fixed. This is consistent with the experimental results reported in the literature [28], where the experimental results show that when the pump amplitude injected into the microcavity is periodically modulated, and when the modulation amplitude is large enough, the breathing period of the breathing soliton generated in the microcavity would completely match the oscillation period of the modulated pump signal.

 

Fig. 10. When ξ =167, the system generates breathing soliton, the relationship between the soliton breathing and the LD pump mode amplitude oscillation periods, (a), the oscillation characteristics of the LD pump mode amplitude, (b), the evolution of the parameter max(|a|), (c), (d) are the local zoom-in views of the superimposed graphs of panels (a), (b) in different time intervals.

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We now try to explain the physics meaning of the numerically derived detuning value ${\zeta _a}$ when the system produces a stable microcomb state. For simplicity, we assume that when the system generates a breathing stable microcomb state, the LD pump mode amplitude is in a sinusoidal function oscillation state as shown in Fig. 10, and the normalized oscillation frequency is defined as ${\zeta _{|A |}}$, then the LD pump mode field A can be written as

Equation (4) indicates that the sinusoidally oscillating LD pump mode field will generate two modulation sidebands, and the normalized frequency detuning of the sidebands with respect to ${\omega _0}$ are $\zeta + {\zeta _{|A |}}$, $\zeta - {\zeta _{|A |}}$, respectively. We hereafter call these as “red-shifted” and “blue-shifted” modulation sidebands, respectively. In addition, the term ${\textrm{A}_1}{e^{i({\zeta {t^{\prime}}} )}}$ is defined as LD main pump mode.

Table 2 shows the simulation fitting results of the normalized oscillation frequency ${\zeta _{|A |}}$ and each detuning ${\zeta _a}$, ${\zeta _b}$ and $\zeta$ when the system generates a stable microcomb state with breathing characteristics and the LD pump mode amplitude is in the sinusoidal oscillation state. The fitting results show that in the case of ignoring the small errors caused by fitting and numerical calculation we can get ${\zeta _a} = \zeta - {\zeta _{|A |}}$, that is, when the system generates a microcomb stable soliton state, the average frequency ${\omega _{{a_0}}}$ derived by numerical calculation coincides with the “blue-shifted” modulation sideband frequency of the LD pump mode.

Table 2. Fitting results of normalized oscillation frequencies ζ _|A| and detuning ζ, ζ_a and ζ _b

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We define the argument of ${a_0}$ as ${\theta _a}$. In order to better explain the above phenomenon, we need to study, in the case of numerical simulation, the relationship between the mean of the argument change rate $d{\theta _a}/d{t^{\prime}}$ (in quantitative terms this value is exactly the mean of the normalized frequency detuning between the microcomb central mode ${a_0}$ and the cold microcavity resonant frequency ${\omega _0}$, i.e., ${\zeta _a}$) and the normalized microcavity injection pump light frequency detuning with respect to ${\omega _0}$, i.e., $\zeta$.

We begin with the frequency selection law of the microcavity central mode excitation under multi-frequency pump condition. For the sake of simplicity, we ignore the nonlinear effects (because the nonlinear effects such as Kerr mode pulling effect do not change the relative frequency composition of the microcavity central mode) and the weak backscattering effects. Under these assumptions, the variation of ${a_0}$ (called the microcavity central mode) with normalized time t′ can be expressed as follows

We first consider the case of single-frequency pump injection, let $F = B{e^{i\zeta {t^{\prime}}}}$, where B is the injected single-frequency pump amplitude, then the analytical solution of Eq. (5) is

From Eq. (6) we can see that, in the case of single-frequency pump injection, the frequency of the excited microcavity central mode is consistent with the pump light frequency. When the pump amplitude is fixed, the light energy which can be coupled into the microcavity decreases with the frequency detuning ζ. This conclusion of course complies with our common sense.

Next, we study the case of simultaneous injection of multi-frequency pump lights. Here It should be noted that all the excited frequencies under consideration are around the same cold microcavity resonant frequency i.e., ${\omega _0}$, and all their detuning with respect to ${\omega _0}$ are much smaller than the free spectral range (FSR) of the microcavity.

Equation (5) obviously satisfies the linear superposition principle, that is, when $F = \sum\limits_i {{B_i}\exp ({i{\zeta_i}{t^{\prime}} + {\psi_i}} )}$, where ${B_i}$, ${\psi _i}$ and ${\zeta _i}$ are the amplitude, initial phase and normalized frequency detuning of a pump light component of the multi-frequency pump lights, respectively, then the analytical solution can be written as

Taking the simplest case of dual-frequency pump lights injection as an example, that is, ${a_0} = {C_1}\exp ({i{\zeta_1}{t^{\prime}} + \psi_1^{\prime}} )+ {C_2}\exp ({i{\zeta_2}{t^{\prime}} + \psi_2^{\prime}} )$, and letting ${\zeta _1} = 166.5072$, ${\zeta _2} = 2.2908$, ${C_2} = 1$ to observe the change of argument ${\theta _a}$ with normalized time under different ${C_1}$ conditions.

Figures 11(a), 11(b), and 11(c) show the microcavity central mode frequency selective excitation characteristics under dual-frequency pump lights injection, the curve in each figure shows that based on linear change, it periodically oscillates with a certain amplitude, and the oscillation amplitude increases when ${C_1}$ approaches ${C_2}$. The slope of the linear fit of the curve represents the average normalized frequency detuning $\bar{\zeta }$ of the excited microcavity central mode ${\omega _{{a_0}}}$ from ${\omega _0}$. When ${C_1} \le {C_2}$, $\bar{\zeta } = {\zeta _2}$, but as long as ${C_1} > {C_2}$, then, $\bar{\zeta } = {\zeta _1}$, that is, $\bar{\zeta }$ is always consistent with the detuning of the relative stronger pump light component of the microcavity. Figure 11(d) shows the simulated result (note that it is not obtained from the calculation of Eqs. (4) and (6)) of the change of ${\theta _a}$ with normalized time in the case shown in Fig. 10, and we find it is quite alike with Fig. 11(a), that indicates, the system state described in Fig. 10 has the characteristics of dual-frequency pump light injection with the stronger pump frequency component (as ${C_2}$ in Fig. 11(a)) having smaller detuning (as ${\zeta _2}$ in Fig. 11(a)).

 

Fig. 11. (a), (b), (c) respectively show the change of θ_a with normalized time under dual-frequency pump excitation with C₁= 0.2, 1, 1.02, and (d) shows the simulated result of the change of θ_a with normalized time in the case shown in Fig. 10. The inset in each panel depicts the local zoom-in view of the corresponding graph.

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The cases for triple-frequency and multi-frequency pump injections are more complicated, but there is still a similar frequency selective excitation law for the microcavity central mode as the dual-frequency injection case even if they are not entirely the same. However, it should be noted that for the multi-frequency (three frequencies and above) pump injection condition, the actual value of $\bar{\zeta }$ depends not only on the relative intensity of each pump frequency component, but also on the initial phase of each pump frequency component (for brevity here we omit the detailed demonstration process and only give out the conclusion).

Through above simulations and comparison with the theoretical analysis, we have concluded the microcavity central mode frequency selective excitation characteristics under the multi-frequency pump lights injection condition. As mentioned above, by comparing Fig. 11(a) and 11(d), we can clearly see that the two curves have a very high similarity to each other, so we can make a judgement with confidence that the microcomb central mode in the system described in Fig. 10 should take the form of ${a_0} = {C_1}{e^{i{\zeta _1}{t^{\prime}} + \psi _1^{\prime}}} + {C_2}{e^{i{\zeta _2}{t^{\prime}} + \psi _2^{\prime}}}$, where ${\zeta _1} = \zeta$, ${\zeta _2} = \zeta - {\zeta _{|A |}}$ and ${C_2} \gg {C_1}$. This suggests that the microcomb central mode contains only two frequency components, namely the LD main pump mode with main lasing frequency and smaller amplitude, and other one is the frequency blue-shifted modulation sideband light with larger amplitude. Therefore, it can be judged that the blue-shifted modulation sideband frequency is well matched with the hot microcavity resonant frequency of the microcomb central mode, because the optical energy of the frequency blue-shifted modulation sideband with much smaller frequency detuning than the LD main pump mode can be effectively coupled into the hot microcavity.

From the above analysis we can see that the average value of the LD pump mode amplitude $|A |$ in Fig. 10 is 0.21, the equivalent pump intensity to be injected into the microcavity is ${|{{F_{in}}} |^2} = 600$, and the LD main pump frequency detuning is $\zeta = 166.5072$. Under such conditions, and if we insert an optical isolator between the LD and the microcavity to cancel the backscattering feedback effect, then the normalized total number of photons in the microcavity obtained by Eq. (6) is ${|{{a_0}} |^2} = 0.0216$, which is much smaller than the result shown in Fig. 12 (average value is 0.6). This further indicates that, by transferring the optical energy from the LD main pump to the blue-shifted modulation sideband first, the LD main pump light will find an effective route to couple into the microcavity in a self-organize manner through the microcavity backscattering feedback effect, so that the optical energy in the microcavity can reach a relative high level which can generate and maintain the breathing soliton stable state even if the frequency of the LD main pump mode has a much larger detuning with respect to the resonant frequency of the cold microcavity.

 

Fig. 12. The evolution of microcavity number of intracavity photons when the system under the condition corresponding to Fig. 10 generates a breathing soliton comb.

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Figure 13 is a schematic diagram of the pump modulation sideband resonant excitation mechanism of the butt-coupled system, which can explain the reason why the system can still generate and maintain the breathing stable microcomb states under the condition of high-power pump injection and large detuning between LD main pump frequency and cold microcavity resonant frequency. Specifically, the microcavity generates the counter-propagating light which carries the information of the hot cavity resonant frequency through the Rayleigh backscattering effect. And the counter-propagating light reversely injects into the LD causing its lasing mode amplitude to oscillate at an appropriate frequency. Then the oscillating LD pump light produces a blue-shifted sideband, whose frequency is well matched with the hot cavity resonant frequency of the microcavity central mode shifted by the nonlinear Kerr mode-pulling effect (the thermal effect is ignored in our model), so that more pump energy can be resonantly coupled into the hot microcavity to such a level to sufficiently generate and maintain the breathing stable microcomb states. In other words, it is exactly the LD pump modulation sideband, not the main pump itself, which is frequency mutually locked to the hot cavity resonant frequency of the microcomb central mode through the resonant Rayleigh backscattering feedback injection locking effect.

 

Fig. 13. Schematic diagram of the LD pump modulation sideband resonant excitation process of the LD and nonlinear optical microcavity butt-coupled system under the condition of high-power pump injection.

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3. Discussion and conclusion

Breather solitons are an important branch of Kerr soliton microcomb which can exhibit much complicated and rich nonlinear dynamics compared with stationary Kerr soliton microcomb, and attract considerable research attentions to explore their dynamics both in theoretical investigation and experimental characterization. However, up to now, most of the works regarding breather soliton dynamics are based on a microcavity pumped by an isolated narrow linewidth laser, nevertheless, in the platform of the LD-microcavity butt-coupled system with soliton self-injection locking mechanisms, the breather solitons dynamics remains largely unexplored. For the first time our work deals with this unexplored region and constitutes a meaningful contribution to understanding the breather soliton dynamics in the LD-microcavity butt-coupled system. Furthermore, some behavior of the breather solitons is still not fully understood, for example, in the works reported by Yu et al [29]., they find that the breathing frequency of the breather soliton increases for reduced detuning, however, E. Lucas et al. demonstrate an entirely opposite trend, that is, the breathing frequency increases with the cavity-pump detuning [30]. Although in Yu et al.’s works, their numerical simulations reveal that in the higher detuning range, a linear increase in the breathing frequency is expected with increasing detuning, however, this trend is not observed in their experiments. On the other hand, as reported by E. Lucas et al. in their article, numerical simulations predict an inversion of the trend over a narrow range of detuning, for small pump amplitudes and detuning. So, the dynamics of breather solitons in nonlinear Kerr microcavity is indeed complicated and deserves further investigations. The LD-microcavity butt-coupled system can provide a convenient testbed for investigating the dynamics of breather solitons, where the breathing period and the number of the generated breather soliton may be simply controlled by LD driven current or even by the feedback phase, and our work, although preliminary and insufficient, takes the first step for these purposes.

Through numerical simulations, we have analyzed the injection locking mechanism and the breathing microcomb generation dynamics of the Kerr nonlinear optical microcavity-LD butt-coupled system with resonant Rayleigh backscattering feedback, finding that when the high-power LD pump light is injected into the microcavity, the numerically derived microcomb central mode frequency is not equal to the LD main pump lasing frequency under the feedback injection effect from the microcavity. According to the simulation results, the numerically derived frequency of the microcomb central modes (both forward and backward propagation modes) indeed have a quite wide frequency locking region, but only a small part of it coincides with the frequency locking region of the LD main pump frequency. The range of the normalized LD main pump to microresonator frequency detuning ξ, where the stable microcomb state with breathing characteristics can be generated, is within the frequency locking region of the microcomb central mode, yet contrary to our intuition, it is outside the frequency locking region of the LD main pump mode. By comparing the theoretical frequency selective excitation characteristics of the microcavity resonant mode under multi-frequency pump injection with the numerical simulation results, we conclude that most of the frequency bands in the frequency-locking region of the numerically derived frequency (${\zeta _a}$ and ${\zeta _b}$) of the microcomb central mode represent the frequency-locking of the LD pump blue-shifted modulation sideband frequency ($\zeta - {\zeta _{|\textrm{A} |}}$) for breathing soliton microcomb generation or the red-shifted sideband frequency ($\zeta + {\zeta _{|\textrm{A} |}}$) for the breathing Turing microcomb generation, but definitely not the frequency locking region of the LD main pump mode itself. In fact, these frequency-locking regions of ${\zeta _a}$ and ${\zeta _b}$ include the frequency bands where the stable microcomb states with breathing characteristics can be generated and maintained. It is also verified that the microcomb central mode contains two frequency components of pump light, namely the LD main pump frequency light with much weaker amplitude and larger detuning, and the blue-shifted modulation sideband light with much higher amplitude and smaller detuning, therefore, the frequency of the LD blue-shifted modulation sideband light can be judged to be well matched with the hot microcomb central mode resonant frequency, and the intracavity intensity of the LD blue-shifted modulation sideband light is much higher than that of the LD main pump mode. Therefore, for the LD-nonlinear Kerr optical microcavity butt-coupled system under the condition of LD high-power pump injection, the generation of stable microcomb states with breathing characteristics is not caused by the self-injection locking of the LD main pump lasing frequency itself, but a newly discovered microcavity resonant excitation mechanism through pump mode modulation sideband, which is closely related to the breathing characteristics of the generated microcomb. Also, the discovered mechanism can illustrate the coupling process of the pump light with large pump-cavity frequency detuning to the breathing soliton microcomb in the microcavity. In fact, we find that it is exactly the LD pump modulation sideband, not the main pump, which is frequency mutually locked to the hot cavity resonant frequency of the microcomb central mode enabled by the resonant Rayleigh backscattering feedback injection locking effect.

Nowadays, the fabrication techniques of ultrahigh Q microcavities have made great progress such as the photonic Damascene reflow process which can assure Q-factors of 12 × 10⁶, and due to the fact that for microcomb generation in nonlinear Kerr microcavity, the parametric oscillation threshold ${E_{th}}$ is inversely proportional to the square of the microcavity quality factor Q, the threshold pump power can be lowered to the level of several milliwatts [10]. Meanwhile the output power of the DFB or FP LDs can be routinely as high as above one hundred milliwatts without amplification. So the situation where pump power is much higher than the parametric oscillation threshold for the microcomb generation in nonlinear microcavity would become quite normal. Therefore, our work can contribute to guiding experiment of the butt-coupled system where high-power pump LD and ultrahigh Q nonlinear microcavity are employed. At the same time, we provide a way for efficiently generating breathing soliton with adjustable breathing period simply by tuning the LD driven current.