1. Introduction

Microresonator frequency combs [1], commonly known as microcombs, are emerging as critical technologies with the potential to transform fields such as spectroscopy [2], LiDARs [3–5], and THz wave generation [6]. Among these, microcombs generated by silicon nitride (SiN) microring resonators (hereafter microresonator) are gaining significant traction in optical communications [7–11]. Their potential is emphasized by their capacity for efficient integration and compatibility with CMOS fabrication techniques. The intrinsic properties of microcombs enable high-speed, high-capacity data transmission, manifested by their ability to handle concurrently multiple data streams across different frequency channels. This proficiency is pivotal in meeting the escalating bandwidth requirements of contemporary communication infrastructures.

Although soliton microcombs [12–15] are predominantly utilized for these applications, under certain conditions where the continuous wave (CW) pump is detuned to a frequency above the microresonator resonance (i.e., blue-detuned), microresonators can generate modulation instability (MI) combs [16]. MI combs are known to exhibit greater output power and chaotic temporal waveforms [11,16]. Such a chaotic behavior denotes a system’s output sensitivity to minor fluctuations in its initial parameters [17–19]. Their untapped potential can be utilized especially in the context of secure communications.

Laser chaos generation and its applications have been studied separately, emerging as a promising platform for information and communication technologies [20,21]. Through optical feedback mechanisms, these lasers can be induced to oscillate chaotically, presenting utility in ultrafast random number generation [22,23] and secure chaos communications [24–26]. The chaotic outputs, which are intrinsically unpredictable, can function as encryption keys [27–29], bolstering defense against potential eavesdropping [30]. In the paradigm of chaos-secure communication, the synchronization of chaos signals between the sender and receiver is paramount, and lays the foundation for operationalizing such a communication mechanism.

Since Huygens’ groundbreaking experiments on the synchronization of two pendulum clocks in 1665 [31], the concept of chaos synchronization has been extensively explored, and a multitude of contributions have emerged over the centuries. These contributions have applied the principle of chaos synchronization to various domains, such as laser technologies [20], secure communication [32], and spectroscopy [33].

Our objective is to broaden the transmission bandwidth of chaos-secure communication by incorporating MI combs. While previous research employing semiconductor and microchip lasers has demonstrated chaos communication capabilities, these endeavors have predominantly been limited to a single wavelength channel. Although there is a comprehensive body of work addressing the synchronization of various microcomb types, such as soliton combs [34] and dark-pulse combs [35], the realm of chaotic MI comb synchronization remains largely uncharted. Besides, chaotic microcombs have been explored for ranging applications [36] such as LiDAR [37]. Furthermore, recent studies have exploited chaotic microcombs for random number generation and optical decision making [38] in the context of chaos-based systems [39]. Venturing into this unexplored territory, our research constitutes a comprehensive numerical study dedicated to the synchronization of filtered chaotic MI combs.

2. Model

Figures 1(a) and 1(b) show our proposed configuration where SiN microresonators are integrated with waveguides. Within this architecture, one microresonator assumes the role of the leader, while the other is designated as the follower. Each microresonator is pumped by a CW laser light via a waveguide, with the same power and phase but different random initial noises. By meticulously scanning the wavelength of the CW pump from a shorter to a longer wavelength, we modulate the detunings between the frequencies of the pump laser and the microresonator resonance. Crucially, the MI comb is generated when the pump wavelength is blue-detuned as this is notably shorter relative to the microresonator resonance.

 

Fig. 1. (a) Two independent microcombs operating in a chaotic regime. (b) Concept of synchronization connecting the output of the leader microresonator to the follower microresonator. This injection has been delayed and attenuated before the input of the follower microresonator. A stopband filter is used to filter part of the injection MI comb spectrum.

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Figure 1(a) depicts a scenario where the leader and follower cavities operate autonomously, devoid of any coupling. By scanning both pump lights from a shorter wavelength to a longer wavelength, we can strategically capture the wavelength of the pump lasers within the blue-detuned wavelength region. This action readily leads to the emergence of MI combs in both cavities. Given the unique vacuum noise attributes that are inherent to each system independently, the generated MI combs remain entirely uncorrelated.

In contrast, Fig. 1(b) outlines our leader-follower arrangement. Here, a fraction of the signal emanating from the leader microresonator is relayed to the follower microresonator. An attenuator with a transmittance factor $\beta _{\mathrm {inj}}$ is implemented to regulate the power injection. Furthermore, an auxiliary attenuator with a transmittance $\alpha _{\mathrm {pump}}$ controls the pump power. This additional component is designed to modulate the state of the follower microresonator, affording an enhanced degree of parameterization flexibility. It is noteworthy that the signal injection ensues after propagation through a designated length of optical fiber. Beyond simulating real fiber characteristics, this delay proves instrumental in delineating synchronization nuances and waveform timings.

We model the synchronization in the leader-follower arrangement using the Lugiato-Lefever equations. This approach is conventionally employed to simulate the response of a nonlinear microresonator subject to a pump. In our model, the output from the leader microresonator is injected into the follower microresonator after exhibiting some delay $D$. The master equations are given as follows [40],

We assume that two SiN microresonators operate under identical parameters. Both resonators have the same free-spectral range of $200\,\mathrm {GHz}$, which corresponds to the cavity length of $757.1$ µm. Their intrinsic quality factors ($Q_i$) are $Q_\mathrm {i} = 2\times 10^6$, while the coupling $\theta$ between the rings and the waveguides is $\theta = 0.003$ (i.e., critical coupling condition). The nonlinear coefficient $\gamma$ is $0.645\,\mathrm {W^{-1}m^{-1}}$, and the second-order dispersion $\beta _2$ is set at $-100\,\mathrm {ps^2 km^{-1}}$ for both rings. The pump power $|E_\mathrm {in,1}|^2 =|E_\mathrm {in,2}|^2$ is maintained at $85\,\mathrm {mW}$. The pump wavelength and the phase delay $D$ are set at 1558 nm and $\pi$/3 rad, respectively.

3. Results

3.1 Chaos synchronization

The synchronization of the MI comb evolves through two primary stages. In the inaugural stage, each microresonator generates the MI comb independently by employing the blue-to-red wavelength scanning. At this juncture, we stop the wavelength scanning when the detuning reaches $\delta _0 = 1.9$. This equates to a normalized effective detuning of $\delta _{\mathrm {eff}} \sim -0.6$, wherein $\delta _{\mathrm {eff}}$ represents the normalized effective pump detuning. It should be noted that a negative $\delta _{\mathrm {eff}}$ signifies that the CW pump wavelength is positioned at a shorter wavelength compared with the microresonator resonance (see Appendix A). Crucially, until this point, the microcavities remain unconnected. As a result, each resonator generates independent the chaotic waveforms, seen in Figs. 2(a) and 2(b).

 

Fig. 2. Temporal waveforms in the leader and follower cavities under various conditions. The insets depict the corresponding spectra. (a) Snapshot of the waveform and spectrum for the leader microresonator at detuning $\delta _0=1.9$ with no injection locking. (b) Follower microresonator waveform and spectrum at the same instant as (a) without injection locking. (c) Leader microresonator waveform snapshot taken 5000 round trips after start of injection locking. (d) Follower microresonator waveform at the same moment as (c), when the waveform from (c) is injected.

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In the second stage, the two resonators are interconnected as depicted in Fig. 1(b), with the conditions from the first stage serving as the initial conditions. The parameters $\alpha _{\mathrm {pump}}$ and $\beta _{\mathrm {inj}}$ are set at 0.03 and 0.55, respectively. These values have been identified as optimal for injection locking, a detail we look into later. Here, the full spectrum from the leader microresonator is injected. Once the injection process from the leader to the follower microresonator is underway, the simulation is allowed to progress. With time, the follower microresonator waveform becomes increasingly similar to that of the leader microresonator. This convergence is further corroborated by monitoring the detunings and waveforms of the cavities as a function of round trip time. By approximately the $\sim$1000-th round trip, the waveforms of the two cavities are closely aligned. Figures 2(c) and 2(d) are snapshots of these waveforms at the 5000 round trip mark. A closer examination of the temporal waveforms reveals that they now exhibit relative peaks occurring at the same time instance, in contrast to the previous frame.

To offer a more lucid visualization of the injection locking, we juxtapose both the field amplitude and phase of the two waveforms in Fig. 3. Although Fig. 2 shows signals with power units to ease the peak visualization, Fig. 3(a) presents an instantaneous field amplitude comparison of the waveforms referenced in Figs. 2(a) and 2(b) at corresponding time points. The utilization of field units in Fig. 3(a) is chosen because closer maximum and minimum values makes it more convenient for observing the synchronization progress. The vertical axis represents the field amplitudes of the leader microresonator and the horizontal axis represents those of the follower microresonator. This highlights the correlation between the waveforms of the leader and follower cavities. As expected, we discern no evident correlation, which is consistent with their chaotic behavior.

 

Fig. 3. Comparison of microresonator signals before and after synchronization. (a) Field amplitude comparison of waveforms shown in Figs. 2(a) and 2(b). It represents the field values of the leader microresonator (horizontal axis) compared with those of the follower microresonator (vertical axis) at the same timings. (b) A continuation of (a), but after the follower microresonator has synchronized with the leader microresonator. The inset is the output spectrum from the leader microresonator. This comparison pertains to the waveforms shown in Figs. 2(c) and 2(d). (c) Depicts the phases of the leader (blue dots) and follower (red dots) waveforms relating to Figs. 2(a) and 2(b). (d) is analogous to (c) but for the waveforms for Figs. 2(c) and 2(d). It should be noted that the phase mentioned here is the phase delay of the slowly-varying envelope of the output pulses, not the optical phase of the fast-carrier electric field.

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In contrast, Fig. 3(b) shows an instantaneous field amplitude comparison of the waveforms in Figs. 2(c) and 2(d). Here, a pronounced correlation between the two waveforms emerges. Notably, we identify a full linear relationship, whereby the field amplitude in the follower microresonator aligns almost perfectly with its counterpart in the leader microresonator. The correlation coefficient $C$ (see Appendix B) between the leader and follower output is 0.97 (Fig. 3(b)), of which value is close to zero before synchronization ($C$ is $-0.22$ for the Fig. 3(a)).

Given that the phase usually serves as an even more sensitive quantitative measure, contrasting the phases between the leader and follower waveforms can furnish a superior visualization. In Fig. 3(c), we display the phases associated with the fast time shown in Fig. 2(a) and 2(b). As anticipated, a discernible phase difference emerges between the leader and follower waveforms, attributable to their independent operations. Thus, once the output of the leader ring is channeled into the follower ring, phase locking is significantly observed between the two rings as illustrated in Fig. 3(d). For a more comprehensive understanding of the origin of the phase calculation, additional explanations can be found in Appendix C. This conspicuous alignment serves as evidence of injection locking, indicating its effectiveness even in chaotic regimes.

These findings unequivocally demonstrate the synchronization of two MI combs, specifically chaotic combs. This represents a pioneering step towards chaos synchronization and secure communication.

3.2 Synchronization progression

To synchronize the system, we feed the output from the leader into the follower. Given this influence of the leader on the follower, we expect $\Delta \delta _{\mathrm {eff}}$ to converge towards a reduced value. This convergence is depicted in the temporal evolution of $\Delta \delta _{\mathrm {eff}}$, which signifies the differential $\delta _{\mathrm {eff}}$ between the two microresonators ($\Delta \delta _{\mathrm {eff}}=|\delta _{\mathrm {eff, leader}}-\delta _{\mathrm {eff, follower}}|$), as shown in Fig. 4(a). After several oscillations, this value settles around 0.5. Although it does not converge to zero, likely due to power imbalances between the resonators, the evident stability is a clear marker of the frequency attraction intrinsic to the injection locking process.

 

Fig. 4. Synchronization progression after injecting the leader output into the follower microresonator. (a) Evolution of normalized effective detuning difference $\Delta \delta _{\mathrm {eff}}$. (b) Evolution of correlation, $C$. (c) Evolution of the temporal waveform of the leader microresonator. (d) Evolution of the temporal waveform of the follower microresonator.

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To gain deeper insights into the synchronization progression, we further investigate the correlation between the temporal waveforms across simulated round trips. Fig. 4(b) highlights the convergence of correlation, moving towards a near-zero value, which suggests increasing similarity between the temporal waveforms of the two microresonators. The 2D heatmaps in Fig. 4(c) and 4(d) show the evolution of time waveforms with the fast time axis shown horizontally and the round trip number vertically. Over time, these waveforms begin to exhibit analogous patterns, reminiscent of the effects seen with injection pulling.

A meticulous assessment of the power dynamics indicates that the two resonators operate at similar power levels. This observation suggests that the behavior of the follower microresonator is not merely a byproduct of its coupling with the leader. Instead, its behavior is largely governed by its own pumping mechanism, enabling it to mirror the operations of the leader microresonator.

3.3 Parameter range for chaos synchronization

To determine the optimal parameter range for chaos synchronization in MI combs, we analyzed the synchronization range using the cross correlation $C$ and power differences $\Delta P$. These metrics are defined based on the values of $\alpha _{\mathrm {pump}}$ and $\beta _{\mathrm {inj}}$. As we detailed in Eqs. (4) and (9) (in Appendix B), both $C$ and $\Delta P$ provide insight into the degree of similarity between the signals and validate their amplitudes for effective synchronization.

Figure 5(a) shows the cross correlation $C$ between the output waveforms of the leader and follower microresonators. Notably, two specific parameter regions exhibit $C$ close to 1, suggesting a similarity between the leader and follower microresonator outputs. However, this observation does not conclusively confirm synchronization. This is attributed to a potential scenario wherein the follower microresonator might merely passively and linearly filter the injected waveform, subsequently emitting a waveform analogous to the input. To circumvent this ambiguity and verify that the follower microresonator is operating in the nonlinear regime, we evaluate the mutual power correlation by using $\Delta P$. A $\Delta P$ value close to unity signifies comparable average intracavity powers across both microcavities, implying similar nonlinear Kerr effects. Consequently, this confirms that the follower microresonator is not merely in a linear filtering mode. Figure 5(b) presents $\Delta P$ values across various $\alpha _{\mathrm {pump}}$ and $\beta _{\mathrm {inj}}$ parameters. From the analysis, the black-circled region in Fig. 5(a) can be disregarded, narrowing our focus to the blue-circled region in Fig. 5(a) where the synchronization is likely, as evidenced by $\Delta P$ approximating unity. The optimal point has been found in $\alpha _{\mathrm {pump}}=0.03$ and $\beta _{\mathrm {inj}}=0.55$. We extended our investigation in Appendix D to explore the correlation progression for various injection delays with these optimal values to study analogous trends in detail.

 

Fig. 5. (a) Cross correlation $C$ of the output waveforms between the leader and follower microresonators across different values of $\alpha _{\mathrm {pump}}$ and $\beta _{\mathrm {inj}}$. The correlation close to the unity indicates similar waveforms for the leader and follower microresonators. Each data point is derived after 5000 round trips after the commencement of injection. (b) The average internal power discrepancy between the leader and the follower microresonators. The value close to unity signifies comparable power levels in the two microresonators.

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3.4 Chaos synchronization with injection of partial spectrum

In applications such as secure chaos communication, demonstrating chaos synchronization using only a subset of the output spectrum from the leader microresonator will introduce a novel avenue of possibilities. We can use a specific spectrum segment for chaos synchronization and allocate the remainder for data transmission. This strategy emphasizes our system’s flexibility and resilience in practical settings. Importantly, our technique guarantees effective synchronization even in less-than-ideal optical fiber networks where some spectral components might be diminished or lost.

We applied a square band-stop filter to the output of the leader microresonator (the optical spectrum is depicted in the inset of Fig. 6(a)). This modified spectrum was then used as the injection input for the follower microresonator. Snapshots of amplitude and phase comparisons, taken after 5000 round trips, are shown in Figs. 6(a) and 6(b). For a direct comparison, results without the band-stop filter are provided in Figs. 3(c) and 3(d). Notably, despite filtering, the leader and follower microresonators remain strongly synchronized, emphasizing that the follower can maintain chaos synchronization even when using just a portion of the leader’s output.

 

Fig. 6. (a) Field amplitude comparison of the output waveforms from both leader and follower microresonators under the same conditions, including parameters $\alpha _{\mathrm {pump}}$ and $\beta _{\mathrm {inj}}$, as specified in Fig. 3. In the inset figure, we see the filtered spectrum from the leader MI comb used for injection into the follower microresonator. (b) Phases of the leader (blue dots) and follower (red dots) for the case with filtered injection.

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At the final synchronization step, the correlation coefficients $C$ for cases with (Figs. 6(a) and 6(b)) and without (Figs. 3(c) and 3(d)) the inclusion of a stop-band filter were found to be 0.95 and 0.97, respectively. This observation suggests that the partial suppression of the injected spectrum results in a slight reduction in the similarity between the synchronized signals. Nonetheless, it is evident that both cases exhibit a notably high degree of synchronization.

4. Conclusion

In conclusion, through our numerical simulations, we have definitively demonstrated the synchronization of MI combs. This achievement was facilitated by carefully selecting the $\alpha _{\mathrm {pump}}$ and $\beta _{\mathrm {inj}}$ parameters, which were pivotal in terms of modulating the power of the injection locking light and the pump power to the second microresonator. Our findings are robustly supported by a thorough analysis spanning power levels, correlation, effective detuning, and waveform similarities between the resonators. A particularly notable outcome is the realization that even when we inject only a segment of the leader’s output spectrum, chaos synchronization can prevail, with the follower cavity reinvigorating the synchronization process. This research represents a seminal advance that elucidates the profound potential of merging chaos synchronization with microresonator optical frequency combs, thus paving the way for the next generation of high-powered, rapid, and efficient encrypted communication systems.

Appendix A: Detunings of cold and hot resonators

For microcomb generation, we tune the CW pump wavelength from shorter to longer wavelengths. The detuning between the laser wavelength and an initial resonance wavelength (referred to as a cold wavelength) is represented by $\delta _0=t_R(\omega _0-\omega _p)$, where $\omega _0$ is the resonance frequency and $\omega _p$ the pump frequency. As we see in the LLE equation of Eq. (2), we compute the normalized pump detuning over a round trip, $\delta _0$, which also represents the detuning with the closest cavity mode.

Because of the Kerr effect in the microresonator, the resonance of the hot (or pumped) microresonator also shifts towards a longer wavelength as we scan the pump laser across the resonance. As an outcome, the effective detuning defined as $\delta _{\mathrm {eff}}$, between the CW pump and the resonance of the hot microresonator differs from $\delta _0$. We plotted the relationship by computing $\delta _{\mathrm {eff}}$ for microresonators based on $\delta _0$ [41], as shown in Fig. 7(a) and 7(b). Notably, after reaching a $\delta _0$ of 1.7, random oscillations are observed, marking the MI comb’s inception. We stop the scan at $\delta _0=1.9$, corresponding to a $\delta _{\mathrm {eff}}$ of approximately $-0.6$, where the detuning corresponds with the MI comb profiles in Figs. 2(a) and 2(b). A negative $\delta _{\mathrm {eff}}$ indicates a blue-detuned state, confirming that the CW pump position is on the shorter-wavelength side of the hot microresonator resonance, which is ideal for MI comb generation.

 

Fig. 7. (a) Relationship between the detuning $\delta _0$ and the effective detuning $\delta _{\mathrm {eff}}$ for the leader cavity. (b) Corresponds to (a) but for the follower cavity, which, at this stage, remains separate from the leader cavity.

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Appendix B: Analysis method

To gauge the degree of synchronization between the leader and follower microresonators, it is crucial to define parameters that capture the congruence of their output signals [21]. We define the cross correlation, $C$, between the waveforms of the two resonators at each round trip, $j$, as:

(4)$$C_j = \frac{\langle(a_{i}-\bar{a})(b_{i}-\bar{b})\rangle}{\sigma_a\sigma_b}.$$

First, we refer to $a$ and $b$ as the module of the slowly varying electric field $A$ and $B$, $a=|A|$ and $b=|B|$. Considering $a$ and $b$ as the leader and follower signals at the calculated correlation round trip, $j$, $\bar {a}$ and $\bar {b}$ represent the mean values of the respective signals so $a_i$ and $b_i$ are the $i$-th sample points of the signal. Thus, $\bar {a}$ and $\bar {b}$ can be calculated as follows:

(5)$$\bar{a}=\frac{1}{N}\sum_{1}^{N}a_i,$$

(6)$$\bar{b}=\frac{1}{N}\sum_{1}^{N}a_i.$$

The standard deviation $\sigma$ of each signal can be calculated as follows:

(7)$$\sigma_a=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(a_i-\bar{a})^2}$$

(8)$$\sigma_b=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(b_i-\bar{b})^2}$$

Further, we compute $\sigma ^2$ across various values of $\alpha _{\mathrm {pump}}$ and $\beta _{\mathrm {inj}}$ to identify the conditions optimal for achieving chaos synchronization.

However, relying solely on Eq. (4) may not conclusively prove synchronization. There is a possible scenario where the follower microresonator simply filters the injected waveform in a passive, linear manner, producing an output waveform similar to its input. To confirm that the follower operates within a nonlinear regime and mirrors the internal state of the leader, and to validate the occurrence of injection locking, it is pertinent to compare the power levels of the two resonators. This relation is given by,

(9)$$\Delta P = \frac{\sum_{j=0}^{N} a_j^2}{\sum_{j=0}^{N} b_j^2},$$

Such an analysis ensures that both microresonators are governed by comparable optical nonlinearities.

Appendix C: Phase analysis

As elucidated in Ref. [20], it is essential to note that the term "phase" in this context refers to the phase of chaotic oscillations of the output intensity, corresponding to the slow-envelope component, not the optical phase of the fast-carrier electric field. The phase is determined from the time series of the output signal by linearly interpolating two adjacent crossing points of a predetermined threshold value, so the phase gains $2\pi$ with each crossing of the threshold value [42].

(10)$$\phi (t) = 2\pi\frac{t-t_\mathrm{n}}{t_{\mathrm{n+1}}-t_\mathrm{n}}+2\pi \mathrm{n}, t_\mathrm{n} \leq t \leq t_{\mathrm{n+1}}.$$

Here, Fig. 8 illustrates the calculation of the interpolated phase. Due to real-time calculation, zeros are observed at the start and end of the phase. We initialize with zeros as the signal has not crossed the threshold yet. Towards the end, all values are set to zero since the next crossing in that roundtrip is still unknown.

 

Fig. 8. (a) Leader ring signal. The black dots indicate the crossing with the threshold of 10 in the up-direction. (b) Follower ring signal. The black dots indicate the crossing with the threshold of 5 in the up-direction. (c) The phase of both signals. The vertical axis shows the phase in rad multiples of $2\pi$.

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Consequently, it is evident from the analysis that following five successive crossings of the threshold in Figs. 8(a) and 8(b), the phase attains a value of 5, signifying the completion of five $2\pi$ cycles. Additionally, it is notable that within each roundtrip, there exist as many as 10 oscillations, representing the maximum count of oscillations observed in the phase profile corresponding to the quantity of $2\pi$ radians. Furthermore, it is noteworthy that the black dots discernible in each signal align with identical instants during the roundtrip time, notwithstanding the subtle variations in amplitude and oscillation shape between the two signals.

Appendix D: Iterating over different injection the delay

In this section, our objective is to illustrate various iterations of synchronization with distinct delay times. The incorporation of delay serves as a formalism to mitigate the conceptualization of simultaneous occurrences. Nevertheless, our findings indicate that the influence of this parameter is not as pronounced as we could anticipate. This can be attributed to the inherent chaotic nature of both signals, characterized by unpredictable oscillations during the roundtrip time, which are contingent upon the initial values. Consequently, in each iteration, we observe a consistent tendency toward synchronization, but with slight variations in the correlation value in the last roundtrip sample. This is likely to persist upon repeating the measurements, potentially yielding a consistent trend yet with distinct slopes and values in both phase and correlation.

As we can see in Fig. 9, the correlation curves across samples within each iteration exhibit discernible variations. Notably, these deviations do not exhibit a straightforward correlation with the introduced delay.

 

Fig. 9. Synchronization monitoring of correlation and phase over different injection delays. (a) 0 rad. (b) ${\pi }/{3}$ rad. (c) ${\pi }/{2}$ rad. (d) $\pi$ rad. (e) $1.6\pi$ rad. f) $2\pi$ rad. (g) $15\pi$ rad. (h) $123.4\pi$ rad.

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Additionally, it is noteworthy to discern that the phase in the instances depicted in Figs. 9(d) and 9(h) are not exactly identical. This disparity can be attributed to the varying timing of the crossing threshold for one of the ten oscillations during the roundtrip, given the distinct slopes of these oscillations. Despite this mismatch in crossing points, the phase profile consistently remains parallel, indicating a correlation in peaks timing. This observation underscores the challenge in establishing a threshold that accurately reflects the phase relationship. Furthermore, it is essential to highlight that this mismatch cannot be ascribed to any phase delay effect, as the slope of the chaotic oscillations can change in each iteration, allowing for the possibility of achieving an exact phase match.

Funding

Japan Society for the Promotion of Science (JP19H00868, JP19H00873, JP22H05195, JP22K14625).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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