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Abstract
We theoretically study the stability of mode-coupling-assisted frequency comb generation in normal-dispersion microresonators. With the aid of mode coupling, quantitative analysis of the modulational instability is explored in the parameter space of pump power and detuning. By exploring the coupled mode number, dispersion, and coupling strength in the normalized Lugiato-Lefever model, the modulational stability gain exists and yields extended spatial structures within the regime of eigenvalue bifurcations. Moreover, the dynamics and efficiency of microcombs are discussed, providing the accessibility of high-efficient, stable, and controllable combs. This work offers universal guidelines for operating mode-coupling-assisted combs in a normal-dispersion system.
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1. Introduction
Kerr frequency comb generation consisting of a serial of equidistant frequency components in high quality factor (Q) microresonators has been widely studied within the past decades [1–6]. This phenomenon is initiated by pumping the cavity mode with a continuous-wave (CW) laser and transferring the photon energy to different cavity modes through the cascaded four-wave-mixing (FWM) process. Models for Kerr comb generation have been developed from coupled mode equations with limited mode numbers [7,8] to the spatiotemporal formulation known as Lugiato-Lefever equation (LLE), deriving from a driven, detuned and damped nonlinear Schrödinger equation [9–11]. Early theoretical models predict microcombs in the anomalous-dispersion regime through the modulation instability (MI) in microresonators [8,12] while hard excitation offers the accessibility of microcomb generation in both normal- and anomalous- dispersion regimes [13–15]. Although Turing patterns and dark cavity solitons are possible nontrivial solutions in the normal-dispersion regime, it requires either large frequency detunings (from cavity resonance) or a suitable initial condition of cavity fields [16]. Lately, comb generation is also identified in normal dispersion with soft excitation [17–20]. This can be theoretically explained by the local dispersion change modified by backward-scattering, multiple spheroidal modes, or waveguide modes [21,22]. Since different mode families have morphologically dependent group index, the free-spectral range (FSR) and waveguide dispersion are significantly different, limiting comb excitation only from one mode family [21]. As two mode families approach each other around a specific frequency, mutual coupling induces significant frequency shift in the resonant frequencies, explaining by the mode-coupling model [23], and then the shift results in the avoided mode crossings (AMX) [5,19,21]. This effect not only changes the comb spectra locally [24,25] but also paves the way for microcomb initiation [5,25] in normal-dispersion microresonators. In addition, by controlling the position of AMX, this mechanism dominates the microcomb generation in the normal-dispersion regime, in which the initial comb lines (or primary combs) can be selectively generated in the form of Turing rolls or dark solitons [26]. Even with weak mode coupling instead of observing significant AMX, microcomb can be still generated in the large normal-dispersion regime [27]. Theoretically, by utilizing the coupled mode equations and LLE, model shows good agreement with the experimental results [5,28]. However, although these reports based on the LLE provide significant insight into the generation dynamics for Kerr microcombs in normal dispersion, the discussion on stability regime of threshold power and comb dynamics is still limited. Meanwhile, as for the conversion efficiency, previous literatures have analytically shown that the comb efficiency in anomalous dispersion can be > 40% as the pattern of Turing rolls [29]; while in normal dispersion, it was shown that a high-efficient, dark pulse is generated [30]. However, the conversion efficiency is mainly studied in the dark-soliton regime requiring high-power operation or a nontrivial initial condition of intracavity fields. There are limited works on comb efficiency with mode coupling, especially in the low-power, controllable Turing-roll regimes [26,28]. It is now essential to address these issues for practical operation in normal dispersion.
In this letter, we theoretically investigate the relation between mode coupling, MI gain, and conversion efficiency of combs by solving the normalized mean-field LLE. Similar to the previous identification in anomalous dispersion [31], the stability regime of coupling-mode-assisted microcomb generation in normal dispersion is explored in the space of detunings and pump powers. This work delivers several new findings. First, the stability regime of microcomb generation in normal dispersion is shown by introducing the mode coupling to the normalized LLE. We observe that the MI gain and stability regime are strongly correlated to the coupling strength between different modes. In addition, unlike that in anomalous dispersion, an upper bound on power for MI gain is identified. Second, we study the accessibility of microcomb. Through the normalized LLE, the dynamics of the mode-coupling-assisted combs is explored in the similar parameter space of detuning and pump power [16,29,31,32]. Due to the cubic dependency of CW steady-state solutions in a nonlinear cavity, hysteresis behavior is prominently identified by shifting the detuning in opposite directions. In addition, the threshold boundary is clarified with mode coupling. Last, the conversion efficiency of mode-coupling-assisted combs is discussed in the normal-dispersion regime. Stable, high efficient (≈ 50%) Turing rolls are observed with low operating power (down to a few tens milliwatt). This analytical work allows us to have more insights into practical experiments of low-power, stable, and controllable microcomb generation.
2. Theory
The generalized mean-field LLE can be written as [29,31,32]:
The normalized field envelope of a single mode family can be then expressed as
We should note this shift will be only applied to the field envelope (Al) of coupled mode l while other modes are remained unchanged as in Eq. (4). By introducing small perturbation $\delta {A_l}({t^{\prime}} )$ in ${A_l}({t^{\prime}} )$, the matrix of the linearized system can be then expressed as [16,31]:
3. Gain spectrum and stability analysis of mode-coupling-assisted Kerr microcombs
3.1 Mapping of MI gain with mode coupling
By solving the eigenvalues from Eq. (7), we show the maps of Re(λ) and the corresponding power spectra for different coupling strength in Fig. 1. The parameter space is in the range of mode number 0 < l < 15 and pump power 0 < S2 < 50 (0 < |Ein|2 < 210 mW). The pump detuning is set at $\Delta $ = 0, assuming pumping at the resonance of the cold cavity. First, for normal dispersion without mode coupling (${\Delta _{shift}} = 0$) as shown in Fig. 1(a), Re(λ) is always negative. This means that small perturbation $\delta {A_{ {\pm} l}}({t^{\prime}} )$ does not grow and the intracavity power is stable as a CW. When the mode coupling (${\Delta _{shift}}$) is applied, positive Re(λ) exists as shown in Fig. 1(b)–(c). The MI gain results in stable Turing patterns and the corresponding spectra in the cavity are shown in Fig. 1(e). This identification is in consistent with the previous experimental findings [5,26,28]. The central idea of the periodic Turing rolls is that they emerge from noise while a flat solution becomes unstable. One interesting behavior is that, although the mode coupling (${\Delta _{shift}}$) is applied to a specific mode l and MI gain is possible, the gain spectrum has a gain cutoff lmax. Both the available gain and gain bandwidth grow with the strength of mode coupling in analogy with the case of anomalous dispersion [33]. For instance, lmax increases from 7 to 10 by raising the strength ${\Delta _{shift}}$ from 6 to 8. This effect can be also identified from the relative microcomb power of the generated spectra in Fig. 1(e) and Fig. 1(f). It suggests stronger coupling strength is needed for the mode number farer away from the pump line.
Fig. 1. Mapping of MI gain versus the mode number and pump power for (a) ${\Delta _{shift}}$ = 0, (b) ${\Delta _{shift}}$ = 6, and (c) ${\Delta _{shift}}$ = 8. The corresponding spectra at S2 = 10 and l = 5 are shown in (d)-(f), respectively.
Next, for mode coupling at a particular mode, we see that the gain spectra only exist within a certain range of pump power. Unlike that a stronger pump power is favored in anomalous dispersion, providing more gain and broader bandwidth for microcomb generation, an upper boundary of pump power is now identified in normal dispersion. It implies that the pump power needs not only to satisfy the threshold condition of cavity loss [29] but also to work within the boundary of MI gain. For instance, the exemplary spectra are shown in Fig. 2 at S2 = 2, 10, and 50 by setting Δ= 0, Δshift = 8, and l = 5. Microcomb in the form of Turing-roll patterns can be initiated from the random noise fluctuation for S2 = 10, but no oscillation can be observed for the pump below threshold (e.g. S2 = 2) or above the upper boundary (e.g. S2 = 50). This poses a limitation on the required pump for microcomb formation in the normal-dispersion regime. Also, as stated above, the mode spacing is 5 FSRs, determining by the position of mode coupling.
Fig. 2. Spectra for different pump powers S2 = 2, 10, and 50. The microcomb is initiated from the random noise at Δ = 0, ${\Delta _{shift}}$ = 8, l = 5, and S2 = 10 but no oscillation can be found at S2 = 2 and 50.
In Fig. 3, we compare the MI gain without and with mode coupling in the parameter space of pump power and detuning. As shown in Fig. 3(a)-(c), for the gain spectrum without mode coupling (${\Delta _{shift}}$ = 0), microcombs can only be initiated with a large detuning and pump power (e.g. Δ= 3 and S2 = 6) [13–16], suggesting a much narrower gain regime in related to that in anomalous cases. An example of the microcomb spectrum is shown in Fig. 3(d), evidencing a periodic stable solution (Turing rolls). As for the one with mode coupling (${\Delta _{shift}}$ = 8, Fig. 3(c)), we can observe the positive MI gain is now accessible with a small detuning. Again, similar Turing-roll patterns can be identified. The corresponding spectrum is shown in Fig. 3(e). One significant difference is that the periodicity of microcombs (or the position of primary combs) is determined by the coupled mode number. This is in consistent with the previous findings [26,28] due to the fact that positive gain is only available from the mode coupling. It should be noted that, for microcomb generation without mode coupling, the oscillation induced by MI gain may come close to the unstable branch of the intracavity power (in-between the two stable equilibria, black-dashed lines). Therefore, unless well-split stable branches of pump with a relative large detuning are fulfilled, the oscillation may exhibit unstable Turing patterns and could be repelled to other stable equilibrium (flat solutions) [16]. As for the cases with mode coupling, MI gain can be generated very far away from the unstable point, providing a stable Turing-roll patterns.
Fig. 3. MI gain spectra without (${\Delta _{shift}}$ = 0) for (a) l = 5 and (b) l = 8 and (c) with (${\Delta _{shift}}$ = 8) mode coupling for l = 5. The corresponding spectra (d) without and (e) with mode coupling.
3.2 Stability regime analysis
To further study the effect on mode coupling, the maps of eigenvalue bifurcations are shown in Fig. 4, again in $\Delta $-S2 space. By introducing mode coupling and solving Eq. (7), we show the maps at different effective resonance shift $y^{\prime} = \; \frac{{{\Delta _{shift}}}}{2} - \frac{{\beta {l^2}}}{2}$, normalized dispersion parameters $\beta $, mode numbers l, and coupling strength ${\Delta _{shift}}$. We aim to find the MI gain regime in the presence of mode coupling. The curves in Eq. (7) suggest Re(λ) = 0 solutions in $\Delta $-S2 space. Since both mode coupling and dispersion shift the resonant frequencies, we first analyze the effect on effective resonance shift ($y^{\prime}$) in Fig. 4(a). Similar to that observed in Ref. [16], the normalized LLE yields a cubic equation in which multiple states emerge as the detuning is increasing. For instance, if we consider the effective resonance shift $y^{\prime} = 0$, only single solution of the eigenvalue exist and there is no net gain for $\Delta < \sqrt 3 $. For $\Delta > \sqrt 3 $, two boundary lines, corresponding to two critical pump powers, can be identified. However, still no gain is found and the cavity field remains as a stable CW. As increasing the effective resonance shift (e.g. $y^{\prime} = 4$), bifurcation now emerges at a smaller detuning. Meanwhile, for the eigenvalue solutions in-between the bifurcation, the intermediate Re(λ) is positive and therefore results in MI gain for microcomb generation (unstable perturbation $\delta {A_l}$).
Fig. 4. Maps of eigenvalue bifurcations in normal dispersion in related to different (a) effective coupling strength y’, (b) normalized dispersion $\beta $, (c) mode number l, and (d) coupling strength ${\Delta _{shift}}$.
Next, we study the effect on the magnitude of dispersion parameter β with fixed coupling strength (${\Delta _{shift}} = 6$) at l = 5 in Fig. 4(b). Significantly, as normal dispersion gets stronger, this results in the increase of critical detuning (${\Delta _{crit}}$) to achieve multiple solutions in the bifurcation maps, narrowing down the MI gain regime. For example, ${\Delta _{crit}}$ moves from −0.82 to 0.24 for β increasing from 0.042 to 0.126, suggesting that stronger detuning is needed to compensate the deep normal dispersion. In contrast to Fig. 4(a), the right boundaries are almost identical to different dispersion values. This can be explained by that the solution on the right boundary is dependent on $\Delta + \frac{{{\Delta _{shift}}}}{2} - \frac{\beta }{2}{l^2}$. Since $\Delta + \frac{{{\Delta _{shift}}}}{2}$ is order of magnitude stronger than $\frac{\beta }{2}{l^2}$ within the operation regime, the difference in the right boundary is negligible. As for the effect on the mode number l, similar trends can be identified as those in Fig. 4(c). Increasing the mode number effectively results in stronger normal dispersion. The quadratic dependency of mode number exhibits more significance on the power of the right boundary.
Lastly, we evaluate the effect on coupling strength. For weak mode coupling (e.g. ${\Delta _{shift}} = 3$), only a small intermediate regime ($\Delta $ from 0.73 to 3.57) of Re(λ) is positive. The area of MI gain is small; this regime expands as increasing the coupling strength. At the same time, stronger coupling contributes to lower ${\Delta _{crit}}$. For instance, ${\Delta _{crit}}$ shifts from 0.73 to −0.77 as doubling the strength ${\Delta _{shift}}$ from 3 to 6.
4. Accessibility, threshold power, generation dynamics, and efficiency of microcombs in normal dispersion
4.1 Comb accessibility and generation dynamics
The analysis of MI gain in Section 3 allows us to obtain the possible regimes for mode-coupling-assisted microcomb generation. However, beside the corresponding detuning and pump power in $\Delta $-S2 space, the normalized LLE in Eq. (2) with cubic nonlinearity exhibits the well-known hysteresis dynamics as adjusting either the detuning [16] or pump power [29]. The nontrivial equilibria are critical to the existence of microcombs in the bifurcation map [16]. We now study the accessibility of microcombs with mode coupling in the normal-dispersion regime. Figure 5 shows maps of comb efficiency in $\Delta $-S2 space for different modes l and coupling strength ${\Delta _{shift}}$, following the paths of red- (Fig. 5(a)–(d)) and blue-detuning (Fig. 5(e)–(h)). The dashed line shows the eigenvalue bifurcations as that in Figs. 4. Clearly, no matter which path we choose, microcomb is generally initiated as passing the MI gain regime, in the form of periodic stable solution (Turing rolls). This periodic wave remains stable into the regime where the CW is stable (no MI gain). This identification is in analogy to the soliton case in anomalous dispersion [31]. One significant difference is that, since the MI gain will only occur within the regime where mode coupling is introduced, the periodicity of Turing rolls is determined by the coupled mode number l for the entire path. With the determined periodicity by mode-coupling, the Turing patterns may later evolve into the soliton crystals (SCs) with a series of equidistant dissipative Kerr solitons (DKSs) while a stronger pump power and large detunings are studied [34,35]. We can also notice that, due to the higher available comb power on the right boundary of bifurcation curves, the hysteresis is more significant for red-detuning than blue-detuning. In addition, as stated in the previous section, strong mode coupling with mode number close to pump line (e.g. l = 1) leads to larger gain regime for comb generation, evidencing the trend in Fig. 5; as for weak mode coupling with large mode number (e.g. l = 5), the gain regime becomes relatively narrow.
Fig. 5. Maps of comb efficiency in $\Delta $-S2 space under red-detuning for (a) l = 1 and ${\Delta _{shift}}$ = 3, (b) l = 1 and ${\Delta _{shift}}$ = 6, (c) l = 5 and ${\Delta _{shift}}$ = 3, and (d) l = 5 and ${\Delta _{shift}}$ = 6, and (e)–(h) the corresponding maps under blue-detuning.
To further address the hysteresis dynamics, we discuss two examples in more detail for the blue-detuned cases. Although we have explicitly explored the MI gain regime with mode coupling, this scheme does not give the full explanation as operating the system, especially with large detunings. Figure 6(a) shows the blue-detuned case for l = 1 and ${\Delta _{shift}}$ = 6 while the white-dashed line indicates the eigenvalue bifurcation as that in Fig. 4, the red-dashed line shows the normalized intracavity pump ${|{{A_0}} |^2} = 1$, and the black-dashed line shows the two equilibria of the intracavity pump. The significant difference is that, when the pump is blue-detuned crossing the gain regime, the oscillation induced by MI is determined not by the right boundary of the eigenvalue bifurcations but close to the line of ${|{{A_0}} |^2} = 1$. This can be explained by that, although we show the gain regime with mode coupling, the eigenvalue solutions of pump (white-dashed line) in-between the two branches of pump equilibria (black-dashed lines) correspond to the unstable branch of the three equilibria. The negative slope of intracavity response is unstable with respect to CW perturbation [5]. Therefore, even with MI gain in-between the pump equilibria as shown in the zoom-in spectrum of Fig. 6(a), the intracavity pump may move to the stable equilibria and no stable microcomb can be identified. The oscillation regime is then determined by increasing the detuning and intracavity power as that in traditional normal-dispersion cavities [5]. Figure 6(b) shows another blue-detuned example for l = 5 and ${\Delta _{shift}}$ = 6. Since the mode number increases, it results in a smaller gain regime as explained in the previous section. We can see that again oscillation regime matches with the right side of the line of ${|{{A_0}} |^2} = 1$ when eigenvalue solution is in the unstable regime. We should note here that the tuning direction (blue-detuning) is opposite to the resonance red-shifting induced by the thermal effect. It generally results in unstable operation within this MI regime.
Fig. 6. Maps of comb efficiency in $\Delta $-S2 space under blue-detuning for (a) l = 1 and ${\Delta _{shift}}$ = 6 and (b) l = 5 and ${\Delta _{shift}}$ = 6. The corresponding zoom-in spectra are also shown on the right.
4.2 Comb efficiency in mode-coupling-assisted Kerr microcombs
Last, we discuss the efficiency maps in normal dispersion with mode coupling. First, unlike the frequency-sensitive MI gain in anomalous dispersion [33], the periodicity of Turing rolls is determined by the mode coupling. Therefore, as shown in Fig. 5 and Fig. 6, this provides a smooth map instead of showing abrupt boundaries in efficiency which is induced by the evolution of mode number in the anomalous-dispersion case [29,31]. Second, the microcomb efficiency can be up to ≈ 25% for the regime of interests. By red-detuning the pump, higher efficiency is obtained. This can be explained by that, while the detuning is increasing and sweeping across the cavity resonance, higher cavity power and enhancement factor are available, as shown in Figs. 7. Since the phase is red-detuned, the enhancement factor is determined by the upper branch of pump. When the comb threshold is reached, pump starts to saturate and transfer additional power above threshold to the microcombs [29,36]. Therefore, it gives better efficiency for microcomb generation. Third, the efficiency is also strongly dependent to the coupling condition 529]. Although a critical coupling scheme is applied to the above discussion, higher comb efficiency is available with an over-coupled device [29]. As shown in Fig. 8, by tripling the coupling coefficient ($\kappa $ = 0.0051) for over-coupling, ≈ 50% efficiency is achieved with the input power down to a few tens milliwatt (e.g. S2 = 3).
Fig. 7. (a) Normalized intracavity power versus phase detuning at S2 = 2, 5, and 8. (b) The enhancement factor of the intracavity power when the phase detuning is continuously increasing (under red-detuning). The red-dashed and black-dashed lines represent the normalized intracavity pump ${|{{A_0}} |^2} = 1$ and the two equilibria of the intracavity pump, respectively.
Fig. 8. Maps of comb efficiency in $\Delta $-S2 space under red-detuning for (a) l = 1 and ${\Delta _{shift}}$ = 1.5.
5. Conclusions
To conclude, by solving the normalized LLE, we theoretically study the MI gain, stability, accessibility, and dynamics for comb generation in normal-dispersion resonators with mode coupling. Considering the phase shift of mode coupling at a specific mode, parametric gain allows MI in the normal-dispersion resonators and shows strong correlation between the coupling strength, mode number, pump power, and detuning. We solve the bifurcation maps of MI gain to model the accessibility of microcomb generation. In comparison with the previous methods, our analytic work provides universal guidelines for operating microcomb in the normal-dispersion regime, especially with small detuning and low pump power. In addition, we investigate the comb dynamics under different detuning directions, showing good agreement with the bifurcation maps. With the aid of mode coupling, high efficient, stable microcombs can be generated in the form of Turing rolls with a controllable repetition rate.
Funding
National Science and Technology Council (NSTC), Taiwan (109-2221-E-008-091-MY2, 111-2221-E-008 -026).
Acknowledgment
We thank research financial support from the National Science and Technology Council, Taiwan.
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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