Synchronization Between Kerr Cavity Solitons and Broad Laser Pulse Injection

Abstract

The synchronization of a soliton frequency comb in a Kerr cavity with pulsed laser injection is studied numerically. The neutral delay differential equation is used to model the light dynamics in the cavity. This model allows for the investigation of both cases where the pulse repetition period is close to the cavity round-trip time and where the repetition period of the injection pulses is close to a rational fraction $M/N$ of the round-trip time. It is demonstrated that solitons can exist in this latter case, provided that the injection pulses are of a higher amplitude, which is directly proportional to the number M. Furthermore, it is shown that the synchronization range of the solitons is also proportional to the number M. The solitons excited by pulses with a period slightly different from the M:N-resonance can be destabilized by the Andronov–Hopf bifurcation, which occurs when the injection level at the soliton position decreases to M times the injection amplitude corresponding to the saddle-node bifurcation in a model equation with uniform injection.

1. Introduction

The dynamics of solitons in Kerr microcavities has emerged as one of the central areas of research in nonlinear optics, driven by its profound implications for photonics and frequency comb technologies [1,2]. Kerr microcavities, which exploit Kerr nonlinearity to compensate the chromatic dispersion, enable the formation of so-called temporal cavity solitons (TCSs)—stable, self-reinforcing wave packets that propagate without changing shape [3,4]. These solitons are critical to many applications, ranging from high-precision metrology [5,6] to telecommunications [7] and beyond [8,9].

A novel approach to controlling soliton dynamics in optical microcavities is pulsed (synchronous) pumping [10]. Unlike continuous wave (CW) pumping, which maintains a constant energy input, pulsed injection is synchronized with the cavity round-trip time, introducing periodic energy perturbations and requiring much less energy to excite the TCS. This technique significantly affects soliton formation and their temporal position, stability, and bifurcations, resulting in diverse and complex dynamic behavior [11,12,13]. Pulsed injection has been shown to facilitate the generation of single solitons, soliton crystals, and chaotic states, expanding the toolkit for manipulating light in microcavities.

A standard method for modeling an optical microcavity with synchronous pumping is based on the application of the Lugiato–Lefever equation (LLE) [14]. The aforementioned approach was recently employed to demonstrate that in the absence of a frequency mismatch between the repetition rate of the broad injection pulses and the inverse cavity round-trip time, the stationary TCS is located at the top of the pulse when the pulse peak intensity is less than a specific pitchfork bifurcation threshold and is shifted to a position on the pulse periphery above this threshold [11,12,13,15]. The introduction of a frequency mismatch results in the stationary soliton position being determined by balancing the drift introduced by the mismatch with that due to the injection gradient. In order to describe the TCS motion in the presence of frequency mismatch and injection gradient, the soliton drift equation was proposed [16,17], which is valid when the limit of small frequency mismatch and gradient is applicable. Nevertheless, the drift equation is insufficient for describing the bifurcation mechanism of TCS desynchronization as a function of increasing mismatch. As demonstrated in Ref. [18], the desynchronization of the system occurs via an Andronov–Hopf bifurcation when the repetition rate of the broad injection pulse is close to the cavity round-trip time. This phenomenon typically precedes the saddle-node bifurcation predicted by the drift equation and occurs when the injection level at the TCS position is reduced to the value corresponding to the saddle-node bifurcation in the LLE with CW injection.

Although the LLE approach is widely used for the analysis of soliton dynamics in optical microcavities, this approach is not free from certain limitations. In particular, it is not applicable to the study of TCS synchronization when the injection pulse repetition period is close to a multiple of the cavity round-trip time. Therefore, here we use an alternative model of an optical microcavity based on the neutral delay differential equation (NDDE) [19]. In contrast to the LLE, this model incorporates a sole independent time variable and is applicable to any ratio M:N of cavity round-trip frequency and pulse repetition rate. In this study, we employ the NDDE model to investigate the impact of injection frequency detuning on TCS dynamics, with a particular focus on scenarios where the ratio M:N approaches a rational number, with the integers M and N. Our findings indicate that, akin to a 1:1 case, the TCS solutions can be exited with arbitrary M and N, although this process may necessitate much higher injection pulse peak powers. Starting from the NDDE model, we derive an analog of the drift equation describing the slow time evolution of the soliton position and investigate how the synchronization range depends on the M:N ratio.

2. Model Equation

The NDDE model of a Kerr cavity can be represented in the following form [19]:

$$\begin{array}{c}\left(A+ϵa{\partial }_{t}A+{ϵ}^2{\displaystyle \frac{a^2-ib}{2}}{\partial }_{tt}A\right)e^{-i{ϵ}^2\alpha {\left|A\right|}^2/2-i{ϵ}^2\theta /2}=\\ e^{-{ϵ}^2}\left(A_{\tau }-ϵa{\partial }_t{A}_{\tau }+{ϵ}^2{\displaystyle \frac{a^2+ib}{2}}{\partial }_{tt}{A}_{\tau }\right)e^{i{ϵ}^2\alpha {\left|A_{\tau }\right|}^2/2+i{ϵ}^2\theta /2}+{ϵ}^2\eta \left(t\right),\end{array}$$

(1)

where $A\left(t\right)$ is the complex field envelope and t is the time variable, $A_{\tau }=A(t-\tau )$, where $\tau$ is the cold cavity round-trip time. The coefficients a and b are the first- and second-order dispersion coefficients, respectively, $\alpha$ is the Kerr coefficient, $\theta$ is the detuning between the injection frequency and the frequency of a cavity mode, $\eta \left(t\right)$ is the laser injection term, and $ϵ$ describes small round-trip cavity losses. With the limit $ϵ\to 0$, the NDDE model Equation (1) can be reduced to the LLE [19]. Furthermore, it can be easily generalized to account for higher-order dispersion terms by including higher-order derivatives. We consider the injection in the form of pulses as follows:

$$\eta \left(t\right)=p{sin}^{2k(1+ic)}\left({\displaystyle \frac{\pi }{T_{inj}}}t\right).$$

(2)

where p is the amplitude of the pulses, $T_{inj}$ is the repetition period of the injection pulses, the integer k defines the width of the pulses, and the term c defines the chirp of the pulses. In this study, the width of the injection pulse was selected in a manner that ensured it was smaller than the cavity round-trip time and significantly larger than the TCS width.

As shown in [19], Equation (1) can exhibit stable TCS solutions under uniform pumping $\eta \left(t\right)=p_u$. In the mean-field limit, $ϵ\to 0$, these TCSs coincide with those of the LLE. Unlike the LLE, where the TCS period is fixed by the periodic boundary condition, in the NDDE model the TCS repetition period depends on the parameters of Equation (1). This period slightly exceeds the delay time $\tau$ and can be estimated as $T_s\approx \tau +2ϵa+\mathcal{O}\left({ϵ}^2\right)$. Synchronization of the TCS with the external pulsed injection can take place when the repetition rate of injection pulses becomes close to one of the resonances, in other words when ${\displaystyle \frac{1}{T_s}}$:${\displaystyle \frac{1}{T_{inj}}}\equiv T_{inj}$:$T_s\approx M:N$ with the integers M and N. In this context, the M:N notation is employed to denote the corresponding resonances under consideration. Figure 1 shows the TCS synchronization when M:N = 8:1 as an example. One can see that the solution train is $T_{inj}$-periodic and the TCSs repeat themselves for $M=8$ round trips in the cavity. It can be seen from Figure 1c that the TCS power decreases due to the cavity losses during the cavity round trips between the injection pulses, where the intensity of these pulses is very small. The calculation of Figure 1a involved a greater loss parameter than that used for Figure 1b. Consequently, the reduction in TCS power between the injection pulses is more pronounced in the former figure. The panel (c) demonstrates the spectra of the TCS from the panel (a) and the injection pulses.

3. Numerical Results

Let us start our analysis with a pulse period close to the resonance 1:1. The synchronization of the TCS with the pulsed pumping means that the soliton period becomes equal to the injection pulse period $T_{inj}$, so we looked for the periodic solution of (1) with the period $T_{inj}$. To find such solutions, we solved the boundary value problem with the periodic condition $A\left(0\right)=A\left(T_{inj}\right)$ using the package DDE-BIFTOOL [20], which contains MATLAB R2022b routines for numerical bifurcation analysis of the systems of delay differential equations. As an initial guess, the TCS solution under uniform pumping $A_u$ with the period $T_s$ is taken, which converges to the final solution by Newton’s iterations. The continuation of the TSC solution along the period $T_{inj}$ is then performed with the DDE-BIFTOOL, which uses the pseudo-arclength method for continuation along the chosen parameter.

The branch of TCSs synchronized with the injection pulses of the period close to the resonance 1:1 and amplitude $p=p_u$ is shown by the yellow line in Figure 2a. The horizontal axis shows the period mismatch $\Delta =T_{inj}-T_s$ in dimensionless units of time. To recalculate them in dimensional time, it is necessary to use the relation ${\Delta }^{\prime }={\displaystyle \frac{nL}{c\tau }}\Delta$, where n is the refractive index, L is the length of the resonator, c is the speed of light, and $\tau$ is the dimensionless cold cavity round-trip time. It can be observed that the TCS peak intensity is maximal when the two periods coincide, $\Delta \approx 0$. The reason for this is the injection gradient which pushes the soliton towards the center of the pulse, where the amplitude of the injection is at its maximum [11,12,13]. Strictly speaking, the maximal peak intensity is slightly shifted from $\Delta =0$. This discrepancy is due to the slight discrepancy between the TCS period and the calculated value of $T_s$. The latter was determined using a constant injection rate of $\eta =p$, which did not take into account the non-zero curvature of the injection at the top of the injection pulse. Given that the injection pulse is significantly broader than the TCS, the resulting change in period is negligible. A non-zero value of $\Delta$ results in the TCS drifting away from the top of the injection pulse. This leads to a decrease in the TCS peak power. Depending on the sign of $\Delta$, the soliton either lags or leads the pulse with each cavity round trip. The drift can be compensated by means of the injection gradient locking the TCS position. However, if the drift caused by $\Delta$ is too large, there is no periodic TCS solution. Therefore, the branch of solutions is limited by the minimum and maximum values of $\Delta$ corresponding to the saddle-node bifurcations. One can see that the branch of periodic TCS solutions is slightly asymmetric with respect to $\Delta \to -\Delta$ ($|min(\Delta \left)\right|>|max(\Delta \left)\right|$). This asymmetry is a consequence of the slight asymmetry inherent to the NDDE solitons. As the mean-field limit is approached, where the parameter $ϵ$ tends to zero, the asymmetry diminishes. Conversely, as $ϵ$ increases, the asymmetry becomes more pronounced.

In order to examine the stability of the synchronized $T_{inj}$-periodic solutions and to conduct an asymptotic analysis of the TCS motion, it is helpful to reformulate Equation (1) as a system of delay algebraic–differential equations:

$$\begin{array}{c}{\partial }_{t}A=B\\ {\partial }_{t}B=C\\ 0=\left(A+ϵaB+{ϵ}^2{\displaystyle \frac{a^2-ib}{2}}C\right)e^{-i{ϵ}^2\alpha {\left|A\right|}^2/2-i{ϵ}^2\theta /2}-\\ e^{-{ϵ}^2\kappa }\left(B_{\tau }-ϵa{B}_{\tau }+{ϵ}^2{\displaystyle \frac{a^2+ib}{2}}{C}_{\tau }\right)e^{i{ϵ}^2\alpha {\left|A_{\tau }\right|}^2/2+i{ϵ}^2\theta /2}-{ϵ}^2\eta \left(t\right).\end{array}$$

(3)

The system can be reformulated in a more general form, as follows:

$$\begin{array}{c}\mathcal{W}{\partial }_t\overrightarrow{X}=\overrightarrow{F}(\overrightarrow{X},{\overrightarrow{X}}_{\tau },\eta \left(t\right),ϵ),\end{array}$$

(4)

where $\mathcal{W}$ is a $3\times 3$ diagonal matrix with the diagonal elements $(1,1,0)$, $\overrightarrow{X}={(A,B,C)}^T$, and ${\overrightarrow{X}}_{\tau }={(A_{\tau },B_{\tau },C_{\tau })}^T$.

To determine the stability of the synchronized TCS solution ${\overrightarrow{X}}_0\left(t\right)={\overrightarrow{X}}_0(t+T_{inj})$, one must substitute a perturbed solution $\overrightarrow{X}\left(t\right)={\overrightarrow{X}}_0\left(t\right)+\overrightarrow{\varphi }\left(t\right)$ into Equation (3) and linearize this equation around ${\overrightarrow{X}}_0\left(t\right)$. Then, the resulting linear equation for the small perturbation $\overrightarrow{\varphi }$ is:

$$\begin{array}{c}\mathcal{W}{\partial }_t\overrightarrow{\varphi }\left(t\right)=\mathcal{L}\left(t\right)\overrightarrow{\varphi }\left(t\right)+\mathcal{M}\left(t\right)\overrightarrow{\varphi }(t-\tau ),\end{array}$$

(5)

where $\mathcal{L}\left(t\right)={\partial }_{\overrightarrow{X}}\overrightarrow{F}$ and $\mathcal{M}\left(t\right)={\partial }_{{\overrightarrow{X}}_{\tau }}\overrightarrow{F}$ are $T_{inj}$-periodic Jacobian matrices with respect to $\overrightarrow{X}$ and ${\overrightarrow{X}}_{\tau }$. In accordance with Floquet theory, the solutions $\overrightarrow{\varphi }\left(t\right)$ of the $T_{inj}$-periodic system (5), which satisfy the relation $\overrightarrow{\varphi }(t+T_{inj})=\mu \overrightarrow{\varphi }\left(t\right)$, are the eigenfunctions and $\mu$ are the corresponding complex multipliers. Floquet multipliers are essential for characterizing the stability of the solution ${\overrightarrow{X}}_0$. If the absolute value of the Floquet multiplier, $\left|\mu \right|$, is not greater than one, the perturbation, $\overrightarrow{\varphi }\left(t\right)$, will not grow with time. In the case of the uniform injection $\eta \left(t\right)=p_u=const$, Equation (1) exhibits the time shift symmetry and the linear stability of the TCS necessitates that, with the exception of the neutral multiplier $\mu =1$, which corresponds to this symmetry, all the absolute values of the Floquet multipliers remain below one. The nonuniform injection breaks the time-shift symmetry and eliminates the neutral multiplier from the TCS spectrum.

In Figure 2a, the stable (unstable) TCS solutions are represented by solid (dashed) curves. As can be observed, an increase (decrease) in $\Delta$ leads to a destabilization of the TCS before reaching the maximum (minimum) value of $\Delta$ corresponding to a saddle-node bifurcation. An examination of the Floquet multipliers reveals that the destabilization is due to an Andronov–Hopf bifurcation when two complex conjugate multipliers cross the unit circle $\left|\mu \right|=1$, as shown in Figure 2b. This result is similar to that obtained using the LLE model in Ref. [18].

The main advantage of the NDDE model (1) over the LLE is that it is applicable in the case where the soliton is not pumped on every round trip. In Figure 2a, different branches of synchronized TCS solutions correspond to the situation when the injection pulse arrives every Mth round trip with $M=1,2,3,4,8$ and is close to M:1 resonances. To maintain an approximately constant pulse width for varying integer values of M and N, the power k in Equation (2) was chosen so that the coefficient ${\pi }^{2}k/T_{inj}^2$ in the second term of the Taylor expansion ${sin}^{2k}\left(\pi t/T_{inj}\right)\approx 1-{\pi }^{2}k{\zeta }^2/T_{inj}^2$ at $\zeta =t-T_{inj}/2=0$ was independent of M and N. Consequently, for $T_{inj}\approx M/N{T}_s$, we chose $k=2M^2/N^2$ so that the injection stayed at a small $\zeta$ of the form $\left|\eta \left(\zeta \right)\right|\approx p\left(1-2{\pi }^2{\zeta }^2/T_s^2\right)$ with $T_s\approx 50.02$. As one can see in Figure 2a, the TCS can be synchronized to the injection pulse even with $M>1$. It follows from this figure that to excite TCSs with approximately the same peak power as that obtained with $p=p_u$ and $M=1$ using injection pulses with larger repetition periods $T_{inj}\approx M{T}_s$ ($M>1$), the amplitude of the injection pulses should be M times larger, $p=M{p}_u$. It is seen that all the curves for different M and $p=M{p}_u$ are similar and look like a scaling of the curve for $M=1$. Note that we considered only the periodic solutions in the vicinity of the M:N resonances with moderate M, which are the most interesting from a practical point of view. The complex dynamical regimes beyond the synchronization region are left for future investigation.

To gain insight into the rationale behind proportionally increasing the injection amplitude relative to the period $T_{inj}$ to achieve a TCS with equivalent peak power, we examined the case of $M=2$. The asymptotic approach was applied in a manner analogous to that utilized in deriving the LLE from the NDDE model, as detailed in reference [19] (see also Ref. [21]). Let us assume that the TCSs are synchronized to the injection pulses with a period $T_{inj}\approx 2T_s$. Furthermore, let the detuning from the resonance 2:1 be zero, which means that every second TCS sits on top of an injection pulse. In accordance with the aforementioned approach, we may express a stationary TCS solution to Equation (1) in the form $A\left(t\right)=A_0\left(t_0\right)+ϵA_1\left(t_0\right)+{ϵ}^2{A}_2\left(t_0\right)+\mathcal{O}\left({ϵ}^3\right)$, where $t_0=\omega t$ with $\omega =1-2ϵa/\tau +{(2ϵa/\tau )}^2+\mathcal{O}\left({ϵ}^3\right)$. These expansions are then substituted into Equation (1) and the resulting terms are collected according to their respective powers of the variable $ϵ$. In the zero order of the parameter $ϵ$, the solution is found to be $\tau$-periodic $A_0\left(t_0\right)=A_0(t_0-\tau )$. The first-order terms are collected, yielding the result that $A_1\left(t_0\right)=A_1(t_0-\tau )$. In the second order of the parameter $ϵ$, the result is

$$\begin{array}{c}{A}_2\left(t_0\right)-A_2(t_0-\tau )=\tilde{\eta }\left(t_0\right)-(1-i\theta )A_0\left(t_0\right)+i\alpha {\left|A_0\left(t_0\right)\right|}^2{A}_0\left(t_0\right)+ib{\partial }_{t_0{t}_0}{A}_0\left(t_0\right)\end{array}$$

(6)

with $\tilde{\eta }\left(t_0\right)=\eta \left(t\right)$. In the case of uniform injection $\tilde{\eta }\left(t_0\right)=const$, all TCSs are identical and have the period $T_s$ in the variable t corresponding to the period $\tau$ in the variable $t_0$. Hence, $A_2\left(t_0\right)$ is also $\tau$-periodic, the left-hand side of Equation (6) is zero, and the amplitude $A_0$ satisfies the stationary LLE obtained by equating the right-hand side of Equation (6) to zero. However, when the injection period becomes twice as large as $T_s$, the repetition period of the TCS also increases twice. Similarly, the second-order correction period $A_2$ also becomes twice as large, $A_2\left(t_0\right)=A_2(t_0-2\tau )$. Let the time moment $t_0=t_s$ correspond to the peak of the amplitude $A_0$ sitting on the top of the injection pulse. If we set $t_0=t_s$ and $t_0=t_s-\tau$ in Equation (6), we obtain:

$$\begin{array}{c}{A}_2\left(t_s\right)-A_2(t_s-\tau )={\eta }_1-(1-i\theta )A_0\left(t_s\right)+i\alpha {\left|A_0\left(t_s\right)\right|}^2{A}_0\left(t_s\right)+ib{\partial }_{t_0{t}_0}{A}_0\left(t_s\right),\end{array}$$

(7)

$$\begin{array}{c}{A}_2(t_s-\tau )-A_2(t_s-2\tau )={\eta }_2-(1-i\theta )A_0\left(t_s\right)+i\alpha {\left|A_0\left(t_s\right)\right|}^2{A}_0\left(t_s\right)+ib{\partial }_{t_0{t}_0}{A}_0\left(t_s\right),\end{array}$$

(8)

where ${\eta }_1=\tilde{\eta }\left(t_s\right)=p$ is the amplitude of the injection pulse, ${\eta }_2=\tilde{\eta }\left(t_s-\tau \right)$, and $A_0\left(t_s\right)=A_0(t_s-\tau )$ due to the $\tau$-periodicity of $A_0$. Summing up Equations (7) and (8), dividing the result by 2, using the $2\tau$ periodicity of $A_2$, and taking into account that the injection is present only on one cavity round-trip from two, ${\eta }_2=0$, we obtain

$${\displaystyle \frac{p}{2}}-(1-i\theta )A_0\left(t_s\right)+i\alpha {\left|A_0\left(t_s\right)\right|}^2{A}_0\left(t_s\right)+ib{\partial }_{t_0{t}_0}{A}_0\left(t_s\right)=0.$$

It follows from this equation that in order to obtain the same TCS peak amplitude $A_0\left(t_s\right)$ using one pulse per two cavity round trips, one needs twice the injection pulse amplitude p as in the case of one pulse per one cavity round trip. Note that this property holds in the mean field limit, $ϵ\to 0$, and assumes that the injection pulse is much wider than the TCS so that the injection is approximately constant across the soliton. With the deviation from the $ϵ\to 0$ limit, the TCS intensity decays on every round trip between the injection pulses. Figure 1 demonstrates how TCSs change between the injection pulses for $ϵ=0.1$ in panel (a) and for $ϵ=0.01$ in panel (b) when the TCS is pumped every eighth round trip. The figure shows that the TCS peak intensity stays almost the same for $ϵ=0.01$ and changes more noticeably at higher $ϵ=0.1$.

4. Asymptotic Drift Equation

As was shown in previous works [11,12], for zero periods of mismatch, the stationary position of the synchronized TCS is at the centre of the injection pulse (if $p<p_c$, where $p_c$ is the value of the injection amplitude at which pitchfork bifurcation occurs). A small period of mismatch shifts the stationary TCS position away from the center of the pulse and such a shift results in a decrease in the soliton intensity. The drift equation for determining the TCS position relative to the injection pulse, based on the given period mismatch, was derived for the LLE model in [12]. An analogous equation can be derived for the NDDE model. To obtain such an equation, we used a perturbative approach and looked for the solution of the Equation (3) in the form $\overrightarrow{X}\left(t\right)={\overrightarrow{X}}_0\left(\xi \right)$, where $\xi =t+t_s\left(t\right)-Vt$. Here, $t_s$ is a slowly varying TCS coordinate, ${\partial }_t{t}_s=\mathcal{O}\left(\delta \right)$, $V=\mathcal{O}\left(\delta \right)$ is the small drift velocity due to the frequency mismatch, and ${\overrightarrow{X}}_0\left(t\right)$ is the TCS solution of Equation (3) under uniform pumping $\eta =p_u$, corresponding to the injection value at the soliton position $p_u=\eta \left(t_s\right)/M$. The drift velocity can be expressed as $V=N-{\displaystyle \frac{M{T}_s}{T_{inj}}}$. Then, similarly to [12], we expanded the injection in the soliton position $t_s$ and assumed that the injection gradient ${\eta }_{t_s}^{\prime }={\partial }_t{\eta |}_{t=t_s}$ is of the order of $\delta$, while the constant injection term equals $p_u$. As a result, in the first order of $\delta$, we obtained a TCS drift equation similar to that reported in [11,12]:

$${\displaystyle \frac{d{t}_s}{dt}}=V+{ϵ}^2{\eta }_{t_s}^{\prime }{\displaystyle \frac{\langle {\overrightarrow{\psi }}^{†}|\overrightarrow{w}\xi \rangle }{\langle {\overrightarrow{\psi }}^{†}|\mathcal{W}\overrightarrow{\psi }\rangle }},$$

(9)

where $\overrightarrow{\psi }={\partial }_t\overrightarrow{X}$ is the column vector neutral mode associated with the time translation symmetry of Equations (1) and (3) with uniform injection $p_u$. This mode is the $T_s$-periodic solution of the linear Equation (5) and corresponds to the Floquet multiplier $\mu =1$. The adjoint neutral mode ${\overrightarrow{\psi }}^{†}$ is the $T_s$-periodic row vector solution of the equation ${\partial }_t{\overrightarrow{s}}^{†}\left(t\right)\mathcal{W}={\overrightarrow{s}}^{†}\left(t\right)\mathcal{L}\left(t\right)+{\overrightarrow{s}}^{†}(t+\tau )\mathcal{M}(t+\tau )$ adjoint to Equation (5) [22]. Since only the third equation in the system (3) contains the perturbed injection term, the first two components of the vector $\overrightarrow{w}$ are equal to zero, $\overrightarrow{w}={(0,0,1)}^T$. The stationary TCS solution exists when the right-hand side of Equation (9) equals zero. Therefore, the maximum and minimum values of the frequency mismatch parameter V, which can be compensated by the last term in Equation (9), define the range of stationary TCS existence.

As mentioned before, in our calculations the shape of the injection pulse remains approximately the same for different M, and only the injection amplitude is changed proportionally to its period, $\left|\eta \left(\zeta \right)\right|\approx M{p}_u\left(1-{\displaystyle \frac{2{\pi }^2{\zeta }^2}{T_s^2}}\right)$. Hence, the gradient term in Equation (9) is also directly proportional to the injection pulse amplitude $M{p}_u$. From this we can conclude that the width of the interval in V (and correspondingly in $\Delta$) where the TCS solution exists is also proportional to M. This can be clearly seen in Figure 2a, where the curves computed with $M>1$ look like the scaled in M versions of the curve for $M=1$. Furthermore, the points of TCS destabilization through an Andronov–Hopf bifurcation are also scaled by M, although the local injection value $\eta \left(t_s\right)/M$ remains the same; see Figure 3a,b. This value corresponds to the saddle-node bifurcation injection value in the uniform case $\eta \left(t_s\right)/M\approx {\eta }_0$, shown in Figure 3c. Such a result is in good agreement with Ref. [18], where this phenomenon was demonstrated and discussed for the LLE model.

Let us evaluate the quantitative accuracy of the predictions provided by Equation (9). For this, we need to compare the value ${\Delta }_n$ calculated numerically with the analytically predicted value ${\Delta }_a=V{T}_{inj}=-{ϵ}^2{\eta }_{t_s}^{\prime }{\displaystyle \frac{\langle {\overrightarrow{\psi }}^{†}|\overrightarrow{w}\xi \rangle }{\langle {\overrightarrow{\psi }}^{†}|\mathcal{W}\overrightarrow{\psi }\rangle }}{T}_{inj}$. As an example, we calculate ${\Delta }_n$ and ${\Delta }_a$ for the injection value $\eta \left(t_s\right)=1.7$ corresponding to the red dot in Figure 3b. First, we need to compute the neutral $\overrightarrow{\psi }$ and adjoint neutral ${\overrightarrow{\psi }}^{†}$ modes for the uniform injection case with $\eta \left(t\right)=p_u=1.7$. To unambiguously define the adjoint mode ${\psi }^{†}$, the DAE system (3) was reformulated as a delay differential equation (DDE) system by introducing a Lorentzian spectral filter of the width $d^{-1}$ into the cavity, as described in [19]. The resulting system of DDEs (Equation (32) in Ref. [19]) converges to the original system (3) in the limit $d\to 0$. Thus, the adjoint neutral mode ${\overrightarrow{\psi }}^{†}$ is calculated from the DDE system in this limit. The values of the position $t_s$ and the injection gradient ${\eta }_{t_s}^{\prime }$ are obtained from the shape of the injection pulse (2) for a given $p_u$. By calculating all modes and values for the parameters of Figure 3b with $p_u=1.7$, we obtain ${\Delta }_a\approx 1.143\times {10}^{-4}$, while the result of the numerical calculation is ${\Delta }_n\approx 1.239\times {10}^{-4}$. So, the results differ by less than $10\%$.

Let us now consider the case $N>1$, corresponding to the situation where there are N injection pulses in M cavity round trips. In the case of 1:2 resonance, the injection period is $T_{inj}\approx 1/2T_s$, so there are two injection pulses per period $T_s$. Two injection pulses can either excite two solitons or alternatively generate a single soliton, as shown in Figure 4b. These two scenarios correspond to the TCS solution branches that coincide with the graphical precision when plotted in $\Delta$ peak intensity coordinates. These branches are represented by the yellow line in Figure 4a. This figure was calculated with the same shape and amplitude of the injection pulse as Figure 2a. A comparison of the yellow branches for the resonances 1:2 in Figure 4 and 1:1 in Figure 2a reveals a striking similarity, with both branches closely coinciding. This observation can be easily confirmed using the drift Equation (9), which shows that the maximum value of V corresponding to a saddle-node bifurcation remains unchanged over 1:N resonances for any integer N. Moreover, the same result holds for any fixed value of M. In particular, the branch for the 3:2 resonance shown by the purple line in Figure 4 is similar to the one corresponding to the 3:1 resonance in Figure 2a.

In the results discussed above, only injection pulses with amplitudes below the pitchfork bifurcation threshold and without chirp are considered. However, it is important to emphasise that the synchronization range is proportional to M, regardless of the value of the pump amplitude and regardless of whether the pulses exhibit chirp or not. Figure 5a shows the solution branches for the case where the pulse amplitude p exceeds the pitchfork threshold $p_c$. Here, in contrast to the case $p<p_c$ where only a single stable TCS exists at $\Delta =0$, there are three TCSs at zero periods of mismatch above the pitchfork bifurcation threshold, one unstable and two stable. The TCS branch calculated for the resonances 1:1 and 2:1 are proportional to each other. The same proportionality is preserved even when the injection pulse is chirped; see Figure 5b.

5. Discussion

The synchronization of TCSs with pulsed laser injection in a Kerr cavity is considered using the neutral delay differential equation model. Although the numerical integration of this model is more time-consuming compared to a standard modeling approach based on the LLE, it allows us to study the more complex synchronization scenarios, in particular those where the soliton is not pumped on every round trip. We show that if the TCS is pumped by injection every Mth round trip, the amplitude of the injection pulses should be increased proportionally to the number M in order to excite a TCS with approximately the same peak power. This means that the intensity of each pulse is increased by a factor of $M^2$, while the repetition frequency is reduced by a factor of M compared to the case where a TCS is pumped on each round trip. On the other hand, the synchronization range is also increased by a factor of M. In other words, one would have to use M times more energy to ensure the TCS is pumped every Mth round trip, but the synchronization region is also increased by a factor of M. Furthermore, our results indicate that the synchronization region exhibits a proportional increase with respect to M, independent of the corresponding injection pulse amplitude at $M=1$ and the presence of the chirp.

We have derived an asymptotic drift equation similar to that previously reported for the LLE [11,12,17]. This equation predicts the stationary soliton saddle-node bifurcation when the mismatch between the injection period and the TCS period is sufficiently large, but is not sufficient to estimate the TCS synchronization range, which is limited by Andronov–Hopf bifurcations [18]. We have shown that when the injection pulse period is close to M cavity round-trip times, this bifurcation occurs when the injection level at the TCS position drops to the value $M{\eta }_0$, where ${\eta }_0$ is the injection rate corresponding to the saddle-node bifurcation of the NDDE with homogeneous injection. In addition, we considered a scenario with multiple injection pulses during the cavity traversal time and showed that in this case, the synchronization range remains unchanged regardless of whether each pulse generates a soliton or not.