Optical frequency combs (OFCs) consist of uniformly spaced discrete spectral lines. Frequency combs have seen a substantial development in the past decade, due to important applications in fundamental and applied physics, such as gas sensing [1], metrology [2], and attosecond physics [3]. Various schemes have been implemented for OFC generation, including Ti:sapphire mode-locked lasers [4], rare-earth-doped fiber lasers [5], quantum cascade lasers [6], electro-optic modulation comb generators [7], and externally driven passive resonators [8,9]. Among them, high resonance quality factor microresonators stand out, providing access to ultrahigh repetition rates in the gigahertz to terahertz range attributed to chip-scale integration. This approach, operating at the previously unattainable repetition rate, exhibits great potential in applications where access to individual comb lines is desirable, such as optical waveform synthesis [10], direct comb spectroscopy [11], high capacity telecommunications [12], and astrophysical spectrometer calibration [13,14]. Note that, however, there are still some technical challenges with the sophisticated control protocols to stabilize the comb repetition rate and carrier envelope phase. In addition, the repetition rate is difficult to tune with the fixed resonator optic-geometrical parameters. Such drawbacks limit the applications in ultrahigh-repetition-rate comb generation of the microresonator-based devices. Exploration of alternative concepts and methods of ultrahigh-repetition-rate comb production remains a fascinating topic in the field of ultrafast physics.

Interestingly, a novel class of alternative concepts has been proposed recently for ultrahigh-repetition-rate frequency comb generation in the sub-terahertz (THz) scale, i.e., the dissipative Faraday instability (DFI) in lasers [15–18], the self-induced Faraday instability in lasers [19], and gain-through-loss instability (GLI) in nonlinear systems such as parametric amplification and fiber resonators [20–22]. This provides supplements to the general family of modulation instabilities (MIs). MI is a ubiquitous mechanism in various nonlinear physical systems, where the symmetry breakup of homogeneous spatiotemporal background states leads to the formation of coherent patterns. There are several well-known classes of MI, including the Benjamin–Feir instability (BFI) in the framework of the nonlinear Schrödinger equation (NLSE) with anomalous group velocity dispersion (GVD) [23,24], the Faraday instability (FI) resulting from the periodic modulation of dispersive parameters of nonlinear systems [25–28], and other parametric instability in fiber amplified links and passive fiber loops related to periodic gain [29–32]. Note that the dissipative effects in resonators modify the features of parametric instabilities, which can also occur in conservative settings, leading to conservative parametric MI (non-BFI) in the normal dispersion regime [33].

In the mechanism of the DFI process, the periodic antiphase (zigzag) modulation of spectrally dependent losses leads to unequal instantaneous strength for the signal and idler waves, being symmetrically located with respect to the pump frequency [15]. This can result in counterintuitive energy transfer from the pump field to the symmetric signal and idler waves, where the phase-matching condition could be further modified via the filter phase associated to Kramers–Kronig relations [22]. Floquet stability analysis is usually utilized to reveal the MI spectrum evolution based on the well-known Ikeda map [22,25]. The concept of DFI has shown great potential in applications of various nonlinear optical systems [20], such as signal amplification processes, optical parametric oscillators, Mamyshev oscillators, and passive driven resonators. However, note that the research on the DFI mechanism is still insufficient, with only a few pioneering reports on theoretical or experimental investigations in recent years [15–22].

Very recently, the concept of GLI has been preliminarily realized in passive driven fiber resonators, adopting one intracavity asymmetric spectral filter to trigger the parametric amplification process and exhibiting a promising candidate in sub-THz frequency comb generation [21,22]. In Refs. [21,22], multiple sidebands are generated through the cascaded four-wave mixing process in the all-normal dispersion regime, where the BFI is inhibited. Dispersion management would significantly influence the nonlinear dynamics of resonators. For example, the scheme of the dispersion-management concept of fiber lasers has been currently introduced into passive driven fiber resonators, facilitating novel cavity soliton formation mechanisms, such as stretched-pulse solitons and chirped dissipative solitons [34,35]. However, in passive driven fiber resonators, the role of dispersion management in the GLI mechanism has not been investigated to date.

In the present paper, we investigate the influence of cavity dispersion on filter-induced instability (FII) formation mechanism, including DFI and GLI, in passive driven fiber resonators, with zigzag intracavity spectral filters. Sub-THz-repetition-rate coherent frequency combs are demonstrated by optimizing the cavity dispersion regimes. The Floquet spectra are calculated utilizing the Floquet stability analysis of the homogeneous solution. In the dominant FII mechanism, the maximally growing unstable mode frequency decreases with the field intensity, which contrasts with the typical BFI scaling. The comparison with parametric instability is also conducted in our passive driven resonators, beneficial to the further understanding of this novel modulation instability.

The scheme diagram of the passive driven resonator is depicted in Fig. 1. In the scheme, the 1550 nm continuous wave (cw) is divided into two parts to drive the passive resonator. As illustrated in Fig. 1, the driven frequency of the cw signal is piezoelectrically controlled by output of a proportional-integral-derivative (PID) control circuit, utilizing one of the cavity output signals as the error signal. In view of the experiment, the resonator is driven by a train of nanosecond square pulses, generated by the intensity modulator (IM), to trigger the parametric processes [34,35]. An erbium-doped fiber amplifier (EDFA) is utilized to amplify the square-pulse pump signal. In the resonator, zigzag intracavity spectral filters are introduced to facilitate the DFI mechanism formation. To investigate the role of dispersion on the DFI mechanism, the unidirectional resonator consists of different fibers for flexible dispersion management, with opposite signs of GVD (single-mode fiber with anomalous GVD and dispersion compensation fiber with normal GVD [36]).

The pulse evolution in the resonators can be modeled by the Ikeda map approach, with the following coupled equations [22,25]: $$\begin{gathered} \frac{\partial A_n(\tau ,z)}{\partial z}=-i\frac{{\beta }_2(z)}{2}\frac{{\partial }^2{A}_n(\tau ,z)}{\partial {\tau }^2}+\frac{{\beta }_3(z)}{6}\frac{{\partial }^3{A}_n(\tau ,z)}{\partial {\tau }^3} \\ +i\gamma (z){|A_n(\tau ,z)|}^2{A}_n(\tau ,z), \end{gathered}$$(1)$$A_{n+1}(\tau ,z=0)=\sqrt{1-T^2}\exp (-i{\delta }_0)A_n(\tau ,z=L)+{\mathit{TE}}_{\mathrm{IN}},$$(2)$${\tilde{A}}_{\mathrm{out}}(\omega )={\tilde{A}}_{\mathrm{in}}(\omega )\exp [-16\ln 2{\left(\frac{\omega -{\omega }_{1,2}}{\sigma }\right)}^4],$$(3)where $A_n$($\tau$, $z$) is the electric field slowly varying envelope of the $n$th round-trip propagation. The coordinate $\tau$ is the time in a reference frame traveling at the pulse group velocity, and the spatial coordinate $z$ measures the position inside the fiber ring cavity of length $L$. ${\beta }_2$($z$) and ${\beta }_3$($z$) are the GVD and third-order dispersion, respectively, and $\gamma$($z$) is the Kerr nonlinear coefficient of the fibers. $E_{\mathrm{IN}}$ is the amplitude of the cw pump, ${\delta }_0$ is the cavity phase detuning, and $T$ is the coupling coefficient of the couplers. ${\delta }_0=2\pi (f_0-f_P)/\mathrm{FSR}$, where $f_P$ represents the carrier frequency of the driving laser, $f_0$ represents the cavity resonance, and FSR represents the free spectral range.

The offset spectral filters, located before the couplers, without loss of generality, are set to have transmission function of super-Gaussian profiles. The central frequencies of the offset filters are ${\omega }_1$ and ${\omega }_2$. The frequency detuning between the two spectral filters and the spectral bandwidth are represented by $\mathrm{\Delta }\mathrm{\Omega }$ and $\sigma$, respectively.

First, we study the DFI formation in the all-normal dispersion regime in the passive driven fiber resonator. We use the following parameters for numerical simulations: $L_{\mathrm{DCF}1}=L_{\mathrm{DCF}2}=7.2\mathrm{m}$, ${\gamma }_{\mathrm{DCF}}=6.5{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, ${\beta }_{2,\mathrm{DCF}}=150.4{\mathrm{ps}}^2/\mathrm{km}$, ${\beta }_{3,\mathrm{DCF}}=-0.567{\mathrm{ps}}^3/\mathrm{km}$, $T=0.8$, and $\sigma =4\mathrm{nm}$. The total GVD is estimated to be $+2.17{\mathrm{ps}}^2$. In our simulations, the linear stability of the homogeneous state solution is investigated using the Floquet stability analysis method [15,16], where the stationary homogeneous state $A_{\mathrm{hs}}$ is described by [21] $$A_{\mathrm{hs}}(\tau ,z)=A_s{e}^{i\gamma P_{s}z},P_s={|A_s|}^2,$$(4)$$A_s=\frac{T}{1-\rho H{e}^{i\varphi }}{E}_{\mathrm{IN}},$$(5)$$P_s=\frac{T^2}{1+{\rho }^2{H}^2-2\rho H\cos (-{\delta }_0+\int \gamma P_s\mathrm{d}z)}{P}_{\mathrm{IN}},$$(6)where $H$ is the magnitude of the filters at the pump frequency, $\rho =\sqrt{1-T^2}$, and $E_{\mathrm{IN}}$ and $P_{\mathrm{IN}}$ represent the complex amplitude and power of the pump field, respectively.

The numerical analyses are performed by introducing two independent sufficiently small complex perturbation modes of $a_{+\mathrm{\Omega }}(z)\exp (i\mathrm{\Omega }\tau )$ and $a_{-\mathrm{\Omega }}(z)\exp (-i\mathrm{\Omega }\tau )$ to the homogeneous state $A_{\mathrm{hs}}$. The Ikeda map is calculated, and a 4 by 4 transfer matrix $M$ is obtained, relating each mode pair $+\mathrm{\Omega }$ and $-\mathrm{\Omega }$. The first and second rows entries are real and imaginary parts of mode pair amplitudes after round-trip evolution of real and imaginary perturbations to mode $+\mathrm{\Omega }$, respectively. Correspondingly, the third and fourth rows of the matrix contain the real and imaginary parts of the mode pair amplitudes, after the evolution of complex perturbations to mode $-\mathrm{\Omega }$. All resulting modes’ amplitudes are normalized to the initial perturbation’s absolute amplitude. The transfer matrix is numerically diagonalized to provide eigenvalues $F(\mathrm{\Omega })$ of the perturbation mode pairs. The modulus of the largest eigenvalue, namely, $\mathrm{Max}(|F(\mathrm{\Omega })|)$, indicates the stability of the perturbations.

The Floquet spectra are plotted in Fig. 2. Note that only the positive part of the spectrum is shown in our calculation results, considering the frequency symmetry of the instability spectrum. In Figs. 2(A), 2(B), and 2(D), the logarithm function of the largest eigenvalue, $G=10\times \log (\mathrm{Max}(|F(\mathrm{\Omega })|))$, is adopted to draw the Floquet spectrum. The mode $\mathrm{\Omega }$ is considered unstable, when the value of $G$ is larger than zero. The Floquet spectrum as a function of the pump is depicted in Fig. 2(A), with the remaining parameters of $\mathrm{\Delta }\mathrm{\Omega }=2\mathrm{nm}$ and ${\delta }_0=1.3\mathrm{rad}$. The maximum $G$ is described by the white dashed line, indicating the most unstable mode corresponding to each pump field level. The black solid line is also plotted in Fig. 2(A), corresponding to the critical value of $G=0$. As illustrated by the white dashed line, the frequency of the most unstable mode decreases with the pump field, which contrasts with the typical BF instability scaling, where the frequency of the maximally growing unstable mode usually increases with the field intensity [23,24].

The Floquet spectrum with $E_{\mathrm{IN}}$ value of 7.0 is depicted in Fig. 2(C), revealing the most unstable mode frequency of $\mathrm{\Omega }/2\pi =110\mathrm{GHz}$, equivalent to the $\sim 7.99\times {10}^3$ cavity harmonic, considering the fundamental frequency of 13.8 MHz. The Floquet spectrum is also shown in logarithmic coordinate, exhibiting several tongues of high-order extreme values. Figures 2(B) and 2(D) are Floquet spectra as functions of the frequency detuning of filters and the modulation period, respectively. The modulation period $T_f$ is the cavity length. In Fig. 2(B), the remaining parameters are ${\delta }_0=1.3\mathrm{rad}$ and $E_{\mathrm{IN}}=7.0$. Note that the frequency of the most unstable mode varies within a small range, different from the case in Ref. [21], where the frequency significantly varied with the detuning between the pump and filter frequency. This could be related to the induced phase part of the filter transfer function in Ref. [21], which is not the case in our investigations.

As shown in Fig. 2(D), we also find that the Floquet spectrum could be tailored to modify the number of extreme values by changing the dissipative function of modulation period $T_f$. The role of parameter ${\delta }_0$ on DFI generation is also checked in a certain range, indicating strict condition of cavity detuning, with inhibition of the DFI when ${\delta }_0$ is of the opposite sign. To give more insight into this DFI mechanism, the dynamics of complex amplitude of the most unstable modes $a(+\mathrm{\Omega })$ and $a(-\mathrm{\Omega })$ is further numerically investigated by direct integration of the Ikeda map. The results are plotted in Figs. 2(E) and 2(F). As shown in Fig. 2(E), the amplitudes of the most unstable modes increase exponentially during the circulation in the resonator described in the unit of $T_f$. Note that synchronous oscillations of the modes are observed, consistent with the external driving frequency. The absolute amplitudes of the mode pair are not equal at every instant point of the evolution. After interaction with the zigzag spectral filters, the attenuation of the absolute amplitudes depends on the function of each specific filter. The $a(+\mathrm{\Omega })$ mode experiences higher attenuation after interaction with the longer-wavelength filter, and vice versa. The exponential increase of the frequency symmetric unstable modes, with oscillations exhibiting alternate amplitude attenuation, indicates the nonlinear energy coupling between the unstable mode pair through this DFI mechanism. This could be explained by that, through the four-wave mixing process during pulse propagation after the filters, the higher attenuated unstable mode obtains energy transferred from the lower attenuated unstable mode, and the alternating attenuation degree of the unstable mode pair ultimately leads to the average growth of both modes.

Subsequently, the evolution dynamics of complex amplitude $a(+\mathrm{\Omega })$ is plotted in Fig. 2(F) in the phase space, with the direction of temporal evolution indicated by arrows. The complex amplitude evolution of the most unstable mode is demonstrated to be oscillatory and synchronized with the external modulation of dissipative parametric forcing, which is typical of the DFI mechanism [15].

To better understand the DFI mechanism, the spatiotemporal pattern formation is further investigated by solving the coupled Eqs. (1)–(3) of pulse evolution in the resonator. The results are illustrated in Figs. 3 and 4.

The spatiotemporal dynamics of pattern formation is depicted in Fig. 3. As shown in Figs. 3(A) and 3(C), ultrahigh harmonic mode locking is achieved, with temporal period of pulse train measured to be 9.12 ps. The pulse repetition rate is calculated to be 110 GHz, which matches well with the Floquet spectrum calculated with the same cavity parameters. The temporal and spectral evolutions during per cavity round trip are also displayed in Figs. 3(B) and 3(D). The envelope of the spectrum is observed to broaden during the pulse propagation in the normal dispersion fibers, which is attributed to the nonlinear self-phase modulation effect of each individual pulse in the fibers. The corresponding number of spectral comb lines increases during the evolution, revealing energy transfer to the spectral wings relating this DFI mechanism. Note that the temporal separations between adjacent pulses remain constant, indicating ultrahigh harmonic stability during the propagation. Figures 4(A) and 4(B) are individual pulse profiles before and after the spectral filter. The frequency chirping is typical of pulses after propagation in large positive GVD value fibers. The phase relationship of the pulse train is further tested, exhibiting an in-phase condition between the adjacent pulses, demonstrating the high coherence of the harmonic mode locking. The resonator will turn to chaos and collision operation if the wavelength distance between the two spectral filters is decreased under a certain degree, similar to Ref. [17].

Note that the role of dispersion management in DFI formation in passive driven fiber resonators has not been investigated to date. In this section, the Floquet linear stability analysis in the dispersion-managed regime is further conducted. To give systematic investigations of the DFI mechanism in different cavity dispersion regimes, i.e., weak and strong dispersion-managed and all-anomalous regimes, several results, under different sets of cavity parameters, are shown in Figs. 5–8. The adopted parameters for SMF are ${\gamma }_{\mathrm{SMF}}=1.43{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, ${\beta }_{2,\mathrm{SMF}}=-22.9{\mathrm{ps}}^2/\mathrm{km}$, and ${\beta }_{3,\mathrm{SMF}}=-0.187{\mathrm{ps}}^3/\mathrm{km}$ [36].

As in Fig. 5(A), the frequency of the maximally growing unstable mode is calculated to be 97.5 GHz, consistent with the 10.3 ps temporal period of the pulse train in Fig. 5(B). Results with near-zero GVD regimes are further calculated with the corresponding total GVD of $+0.026{\mathrm{ps}}^2$ and $-0.109{\mathrm{ps}}^2$. In the strong dispersion-managed regimes, the repetition rate of the stable frequency comb operations can be increased, due to the decreased average dispersion [17]. The pulse repetition rates are 275 GHz and 310 GHz for Figs. 5(D) and 5(F), respectively, which matches well with the Floquet stability analysis solved based on the Ikeda map.

To give more insight into this filter-induced instability, the dynamics of the complex amplitude of the most unstable mode must be revealed. Figure 6 provides an intuitive understanding of this nonlinear evolution dynamics. It is worth noting that, in the case of Fig. 6(A) with large net positive cavity GVD in the weak dispersion managed regime, the complex amplitude evolution of the perturbation remains similar profile to the all-normal dispersion regime, circulating anticlockwise in the phase space. The frequency scaling [see Fig. 6(B)] of the most unstable mode exhibits an inverse relationship with the pump field.

Note that there are some distinctive characteristics related to the near-zero GVD regimes. Figures 6(C) and 6(E) illustrate the corresponding evolution of the perturbation’s complex amplitude. The adjacent half-cavity periods, after the longer- or shorter-wavelength spectral filters, are represented by different colors of blue and red, respectively. The trajectories are rotating anticlockwise, except that the circulating centers are shifting along the horizontal coordinate of the phase space. After every cavity period, the trajectory returns to the origin of coordinates following interaction with the shorter-wavelength spectral filter. The Floquet spectra as a function of the pump are shown in Figs. 6(D) and 6(F). Similar to Ref. [27], the bending of the MI bands with power toward low frequency or high frequency depends on the sign of net dispersion. The relationship between the most unstable frequency and the pump deviates from Fig. 6(A), and the frequency barely changes over a wide intensity range. This could be explained by the combined influence with the BFI, a common phenomenon in the framework of NLSE with anomalous GVD [23,24]. In the strong dispersion managed regimes with near-zero GVD, the combination of the DFI and BFI leads to balanced solutions, in contrast with the case of the all-normal dispersion regime.

Moreover, the DFI formation is further investigated, extending to the limiting condition of the all-anomalous dispersion regime to give more insight into the role of dispersion. The results are illustrated in Fig. 7. Figure 7(A) shows the Floquet spectrum as a function of the pump. Note that the frequency of the most unstable mode gradually blueshifts with the increase of the pump, revealing influence of the BFI. The Floquet spectra as functions of the filter frequency detuning as well as the parametric modulation period are also depicted in Figs. 7(B) and 7(D), respectively, revealing slight variations. In the Floquet spectrum in Fig. 7(C) with the $E_{\mathrm{IN}}$ value of 6.6, the most unstable mode frequency is calculated to be $\mathrm{\Omega }/2\pi =385\mathrm{GHz}$, corresponding to the ultrahigh $\sim 7.75\times {10}^4$ cavity harmonic. The evolution of absolute values of amplitude of the most unstable mode pair is numerically calculated, as shown in Fig. 7(E), plotted by blue and red lines, correspondingly. The amplitudes of the mode pair significantly increase during the propagation described in the unit of $T_f$.

In the case of traditional BFI illustrated in Ref. [15], the growth of absolute amplitudes is continuous during the whole evolution process due to the absence of periodic external forcing. However, note that in contrast with the characteristics of BFI, in our simulations, oscillations of the absolute amplitude are still obtained, synchronized with the zigzag spectral filtering periods. Meanwhile, compared with the all-normal dispersion regime, after interaction with the spectral filters, the most unstable modes experience much higher attenuation of the absolute amplitude. Finally, the evolution of complex amplitude $a(+\mathrm{\Omega })$ is plotted in Fig. 7(F). After every cavity period evolution, the circulating trajectory returns to the origin of the coordinates. In the phase space, the trajectory of the amplitude evolution experiences oscillation in a zigzag way, revealing a much more complicated process than in other dispersion regimes.

The spatiotemporal pattern formation is also investigated as shown in Fig. 8. Figures 8(A) and 8(C) illustrate the temporal and spectral evolution of the ultrahigh harmonic mode locking operation, respectively. The pulse repetition rate coincides with the frequency value of the most unstable mode calculated in Fig. 7(C). The spectrum exhibits high-contrast modulation with period of 385 GHz between the adjacent spectral comb lines. Nonlinear self-phase modulation leads to the spectral broadening along propagation, with the pulse train energy transferred into frequency comb lines in the spectral wings. We also test the phase condition of the harmonic mode-locking pulse train, revealing a fixed phase relationship between the adjacent pulses, demonstrating high mode-locking coherence. Note that Kelly sidebands can be generated in a fiber laser operated in the anomalous dispersion regime [37], which is not our case where the dispersive wave components are restrained due to the action of detuned spectral filters in the resonator.

In our simulations, filters with different kinds of profiles are also tried, and we find that the filter profile does not qualitatively affect the results.

In addition, further simplification of the passive driven resonator structure has also been explored. As illustrated in Fig. 9, modeling without loss of generality, only one offset shorter-wavelength super-Gaussian spectral filter is adopted in the system for the excitation of the FII, also named GLI. In this modified structure, the frequency detuning, between the shorter-wavelength spectral filter and the cw pump, is represented by $\mathrm{\Delta }\omega$, while the spectral bandwidth is represented by $\sigma$.

The all-normal dispersion regime is first investigated. The system parameters are chosen as follows: $L_{\mathrm{DCF}}=7.2\mathrm{m}$, ${\gamma }_{\mathrm{DCF}}=6.5{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, ${\beta }_{2,\mathrm{DCF}}=150.4{\mathrm{ps}}^2/\mathrm{km}$, ${\beta }_{3,\mathrm{DCF}}=-0.567{\mathrm{ps}}^3/\mathrm{km}$, $T=0.8$, and $\sigma =4\mathrm{nm}$. GLI-induced spatiotemporal patterns are also observed in the numerical simulations. Results are shown in Fig. 10. As illustrated by the white dashed line in Fig. 10(A), the frequency of the most unstable mode decreases with the pump field, consistent with the character of the dominant GLI generation in the all-normal dispersion fiber resonator with dual-offset spectral filters.

The remaining parameters for Figs. 10(B), 10(C), and 10(D) are $\mathrm{\Delta }\omega =1\mathrm{nm}$, $E_{\mathrm{IN}}=7.0$, and ${\delta }_0=1.3\mathrm{rad}$. Temporal evolution over 200 round trips at the output of Coupler2 is shown in Fig. 10(B), achieving 115 GHz repetition rate ultrahigh harmonic mode locking. Evolution of the absolute amplitude of the most unstable modes is depicted in Fig. 10(C), exhibiting an exponential increase during circulation in the resonator, described in the unit of $T_f$. Oscillations are obtained for the absolute amplitude increase, synchronous with the external forcing frequency. Within every modulation period evolution, despite that the $a(-\mathrm{\Omega })$ mode experiences higher attenuation after interaction with the shorter-wavelength filter, the mode pair ultimately obtains nearly equal absolute amplitude in the end. This means that the $a(-\mathrm{\Omega })$ mode, after higher attenuation, experiences more rapid growth, attributed to the strong nonlinear energy coupling of the frequency comb lines through four-wave mixing. Moreover, the evolution of complex amplitude $a(+\mathrm{\Omega })$ is plotted in Fig. 10(D), with arrows indicating the direction of evolution. In the phase space, the evolution trajectory of $a(+\mathrm{\Omega })$ experiences anticlockwise rotation with origin of coordinates as the center.

These results confirm the feasibility of system simplification with one offset spectral filter for GLI formation. However, it is worth noting that, as shown in Fig. 10(A), the MI gain spectrum exhibits much broader bandwidth than the dual-offset-filter system. Further optimization of the spectral filter profile might benefit to sharpen the MI gain spectrum as well as to increase the growth speed of the most unstable modes.

In the one-offset spectral filter structure, the GLI formation in dispersion-managed regime is also investigated, exhibiting similar characteristics to the dual-offset-filter system. In case of the small perturbation evolution dynamics, anticlockwise rotating trajectory of $a(+\mathrm{\Omega })$ is also observed. The GLI pattern ultimately leads to sub-THz-repetition-rate frequency comb generation.

To further elucidate the GLI mechanism in the one-offset filter fiber resonator structure, we conduct numerical analysis in the all-anomalous dispersion regime. The results are illustrated in Fig. 11.

The Floquet spectrum as a function of pump field is shown in Fig. 11(A), where the white dashed line, describing the maximum $G$, indicates that the most unstable mode frequency increases with the amplitude of the homogeneous field. This demonstrates the role of BFI scaling in the all-anomalous dispersion regime. The temporal pattern formation in Fig. 11(B), with pulse train repetition rate of 392.5 GHz, coincides with the maximum $G$ frequency, corresponding to $\sim 1.59\times {10}^5$ ultrahigh cavity harmonic. The absolute amplitudes of $a(+\mathrm{\Omega })$ and $a(-\mathrm{\Omega })$ increase with oscillation, which is synchronous with the external driving. The growth speed of the most unstable modes is higher than in the all-normal dispersion regime. Figure 11(D) illustrates the evolution of $a(+\mathrm{\Omega })$. Interestingly, some distinctive features arise in the all-anomalous dispersion regime. It is remarkable that the $a(+\mathrm{\Omega })$ trajectory gradually deviates away from the coordinates’ origin, instead of circulating around, in contrast with the evolution in the dual-offset-filter system.

Furthermore, the parametric instability is investigated for comparison within the same cavity structure as in Fig. 10, except the absence of the offset spectral filter. Attributed to the all-normal cavity dispersion, the BFI is here completely inhibited [23,24]. Thus, the resonator operates at the BFI stable regime, and we obtain parametric instability. The Floquet stability analysis results are illustrated in Fig. 12.

Figure 12(A) shows the Floquet spectrum as a function of the pump. Multiple extreme values are noticeably observed in the spectrum. The extreme values evolve toward lower frequency with the increase of pump field, which contrasts with the characteristics of the BFI scaling. The Floquet spectrum with $E_{\mathrm{IN}}=6.4$ is depicted in Fig. 12(B), calculated by integration of the Ikeda map. Up to fifth-order extreme values can be observed in the frequency range of 800 GHz. The instability extreme values are located upon a wide pedestal that extends over the entire frequency range, and the pedestal value is measured to be 0.6. The first extreme value corresponds to the maximally growing unstable mode frequency of 180 GHz, and the growth is oscillatory and synchronized with the external parametric driving. Compared to the FII mechanism, the side-mode suppression ratio of the parametric instability, between the fundamental unstable mode and the second extreme value, is relatively lower. This indicates the superiority of the FII mechanism in ultrahigh-repetition-rate pulse train stability. As shown in Fig. 12(C), the absolute amplitude growth curves of the most unstable pair converge together after several evolution periods, different from the FII situations. Furthermore, the growth speed of the most unstable pair in parametric instability is much slower than that of the FII mechanism. Figure 12(D) shows the small perturbation evolution dynamics. In the phase space, the trajectory of the amplitude $a(+\mathrm{\Omega })$ rotates in the anticlockwise direction indicated by arrows, similar to the FII mechanism in the net-normal dispersion region.

Moreover, the Floquet stability analysis is also conducted in the strong dispersion-managed regime with absence of filters. The results are shown in Fig. 13. As shown in Fig. 13(B), under the cavity pump level of $E_{\mathrm{IN}}=8.4$, the most unstable mode frequency is calculated to be 945 GHz, much higher than that of FII in strong dispersion-managed regime in Fig. 6. This significant value difference of the most unstable mode frequency indicates that this type of instability induced by dispersion map is negligible when strong offset filtering effect participates in the resonators, even in the strong dispersion-managed regime.

This comparison facilitates understanding of the general family of modulation instabilities.

In conclusion, we demonstrate the sub-THz-repetition-rate frequency comb generation induced by FII in passive fiber resonators at different cavity dispersion regimes. Zigzag spectral filters are introduced in the resonators to trigger the DFI formation. Coherent temporal and spectral patterns are obtained, where the repetition rate is consistent with the most unstable mode frequency, solved by the Floquet stability analysis method.

For the all-normal dispersion resonator with dual offset spectral filters, dominant DFI formation is obtained, demonstrated by the typical DFI scaling, where the most unstable mode frequency decreases with pump field intensity. The maximally growing modes correspond to the maximum peaks in the Floquet spectrum, and the absolute amplitude of these modes increases exponentially along propagation in the unit of modulation period $T_f$. The increase is oscillatory and synchronized with the external dissipative parametric forcing frequency. The complex amplitude of the most unstable mode exhibits circulating trajectory in the phase space, with gradually growing radius. Furthermore, with the net cavity GVD approaching the large negative values, i.e., the anomalous dispersion regimes, the influence of traditional BFI arises, leading to BFI scaling of the most unstable mode frequency. The absolute amplitude of the modes experiences an oscillatory increase during evolution, and the oscillations are synchronized with the external driving. The complex amplitude evolution in the phase space is much more complicated due to the complex combined influence of DFI and BFI mechanisms. Simplification of the passive driven resonator system design is further proposed. GLI pattern formation is also achieved with one offset spectral filter adopted in the cavity. Finally, comparison between the filter-induced instability and parametric instability is performed, revealing the intrinsic distinction in the family of modulation instabilities. Our investigations might benefit the understanding of the novel nonlinear process of FII, and provide an alternative method of ultrahigh-repetition-rate comb generation by passive driven fiber resonators.

Acknowledgment. P. Wang acknowledges the inspiration in the Philip. St. J. Russell Division, Max Planck Institute for the Science of Light, Erlangen, Germany.