Optical frequency combs, which consist of precisely controlled spectral lines covering a wide range, have played a crucial role in enabling numerous scientific advancements. Beyond the conventional approach that relies on mode-locked lasers, microcombs generated from microresonators pumped at a single frequency have arguably given rise to a new field within cavity nonlinear photonics, which has led to a robust exchange of ideas and research between theoretical, experimental, and technological aspects. Microcombs are extremely attractive in applications requiring a compact footprint, low cost, good energy efficiency, large comb spacing, and access to nonconventional spectral regions. The recently arising microcombs based on fiber Fabry–Pérot microresonators provide unique opportunities for ultralow noise and high-dimensional nonlinear optics. In this review, we comprehensively examine the recent progress of fiber Kerr microcombs and discuss how various phenomena in fibers can be utilized to enhance the microcomb performances that benefit a plethora of applications.

## I. INTRODUCTION

Optical frequency combs (OFCs) are sets of equidistant spectral lines that were invented in the form of mode-locked lasers in the late 1990s.^{1} As a ruler made of light, OFCs signify a revolution in precision metrology and were recognized by the Nobel Prize in Physics in 2005.^{2} As an alternative to mode-locked lasers, microcombs are generated by pumping a passive high-quality (Q) tiny optical cavity called a microresonator with a resonant continuous-wave single-frequency laser, which can emit ultrafast pulses at astonishing rates (typically between 10 and 1000 GHz). Therefore, microcombs are extremely attractive for real-world applications due to their compact size and low power consumption, which are enabled by compatible photonic integration technology. Microcombs have rapidly approached the performance of traditional laser frequency combs since their invention in 2007^{3} and have become a competitive light source architecture for comb enabled research fields such as optical atomic clockwork and optical frequency synthesizers.^{4–10} Additionally, microcombs have also shown promise in applications including precision spectroscopy,^{11–14} astro spectrograph calibration,^{15,16} biomedical imaging,^{17–19} optical communications,^{20–27} coherent ranging,^{28–30} and quantum information science.^{31–36}

Microcombs based on dissipative Kerr soliton (DKS) have been realized in various platforms since the first demonstration in 2014,^{37,38} including whispering-gallery microresonators^{37,39–41} and microresonators based on planar waveguides^{42–47} and photonic structures^{48–52} fabricated in different material platforms, such as doped silica,^{39} calcium fluoride (CaF_{2}),^{41} magnesium fluoride (MgF_{2}),^{37} lithium niobate (LiNbO_{3}),^{42} aluminum nitride (AlN),^{43} aluminum gallium arsenide (AlGaAs),^{44} silicon nitride (Si_{3}N_{4}),^{45} silicon carbide (SiC),^{46} chalcogenide,^{47} and so on. Recently, fiber microresonators^{53–55} have received a lot of attention in the microcomb community, mainly due to their ultralow propagation loss, easy commercial access, compatibility with existing optical systems, ultralow noise enabled by the large mode volume, and multiple degrees of freedom for high-dimensional nonlinear optics. Here, we review the recent progress of fiber Kerr microcombs and discuss how various phenomena in fibers can be utilized to enhance the microcomb performances that benefit a plethora of applications.

The paper is organized as follows: Sec. II introduces fiber Fabry–Pérot (FP) microcavity, including the theory, factors affecting the cavity loss, and measurement of the cavity Q factor and dispersion. In Sec. III, we give a brief introduction to the DKS microcombs, including the governed theory and models, soliton dynamics, soliton properties, the experimental methods for soliton excitation and stabilization, and the soliton uniqueness in FP cavities. In Secs. IV and V, we focus on the fiber Kerr microcomb generation with the Brillouin effect and gain effect, respectively, from the basic mechanisms to the state-of-the-art DKS results. Of note, the discussion is not limited to the fiber microcombs but can be extended to other microresonators. Finally, in Sec. VI, a summary and outlook of fiber microcombs are given for future studies, including new physical mechanisms and potential applications.

## II. FIBER FP MICROCAVITY

As an ultralow-loss waveguide, fiber is naturally a good candidate for passive ultrahigh-Q resonators. Different forms of fiber resonators (Fig. 1) have been proposed, such as FP cavities made from fiber Bragg gratings [Fig. 1(a)]^{56} or free-space mirrors^{57} with high reflectivity, fiber ring resonators made from directional couplers [Fig. 1(b)],^{58} microfiber knot resonators [Fig. 1(c)],^{59–61} and monolithic FP resonators with high-reflectivity coating at both end faces [Figs. 1(d) and 1(e)].^{53–55,62–64} Among them, the last form [Figs. 1(d) and 1(e)] has been proved to have both ultrahigh Q and large comb spacing. Hence, we will focus on the dielectrically facet-coated monolithic fiber FP microresonator in the later discussion. Of note, in many published papers, fiber FP cavities are referred to as cavities where two fiber ends are coated with reflective mirrors and light propagates between them within an air gap.^{65} Despite the wide applications of air-gaped microcavities in optical sensing, they cannot be used for microcomb generation due to the small nonlinearity of air. Additionally, fiber microspheres^{66} are not considered here due to the large number of whispering gallery modes instead of the fiber structures that determine spatial modes.

### A. Theory of fiber FP cavity

A typical FP cavity is created by trapping optical energy in an enclosed optical path 2*nL* (*n* is the optical refractive index of the intracavity media, *L* is the cavity geometric length) with two highly reflective mirrors. The light field circulating back and forth between the mirrors experiences constructive interference when the roundtrip path covers an integer number of wavelengths, *N* · *λ* = 2*nL*. The wavelengths that exactly satisfy the above-mentioned equation are the longitudinal modes of the FP cavity, corresponding to the optical frequency through *ν* = *c*/*nλ*, where *c* is the speed of light. The frequency difference, Δ*v*_{FSR}, between adjacent longitudinal modes is called the free spectral range (FSR), or later the comb spacing, and can be calculated as Δ*v*_{FSR} = *c*/2*nL*. Of note, the complete symbol definition in this tutorial can be found in Nomenclature.

*A*

_{0}as

*α*is the propagation loss,

*T*and

*R*are the transmission and reflectivity coefficients of the optical power subject to the relationship

*T*+

*R*= 1 −

*α*and are assumed to be equivalent on both end faces.

*ϕ*= 4

*πnL*/

*λ*is the total accumulated roundtrip phase. As shown in Fig. 2, increasing the mirror reflectivity results in a narrower linewidth of the transmitted Lorentz curve as well as a higher extinction ratio (ER).

### B. Cavity loss

The monolithic fiber FP microresonator did not come back into the view of the nonlinear optics community until 2009,^{62} which was 30 years after its first application for optical sensing in 1979.^{67–69} During this long period, the development of fabrication techniques makes possible a fiber FP microresonator with ultralow loss and an ultrahigh Q factor. Monolithic fiber FP microresonators are usually fabricated through three steps:^{54,55,70} (i) fiber is carefully cleaved and encapsulated in a ceramic fiber ferrule; (ii) both fiber ends are mechanically polished to subwavelength smoothness; and (iii) both fiber ends are coated with an optical dielectric Bragg mirror (DBR) with high reflectivity.

*Q*, finesse

*F*, cavity linewidth Δ

*ν*, and cavity photon lifetime

*τ*. The definition and relationships between them are

Therefore, lower loss leads to a longer photon lifetime, narrower cavity linewidth, higher finesse, and higher *Q*. The intracavity power enhancement factor with respect to the pump is *F*/*π*, which verifies ultralow loss can lead to high intracavity power.

The total loss is divided into the output coupling loss *α*_{o} and the intrinsic loss *α*_{i}. According to the fabrication process, the output coupling loss comes from the designed DBR transmission loss. There are many factors affecting the intrinsic loss, for example, the absorption and scattering loss induced by defects in the fiber [Fig. 3(a)] or at the fiber end. There are three other dominant origins for intrinsic loss, such as thick coating layers [Fig. 3(b)], tilted layers [Fig. 3(c)], and imperfect angle-cleaving at the fiber end faces [Fig. 3(d)]. These errors can prevent 100% of the light from reflecting back into the fiber due to either diffraction or an optical path mismatch. The bending loss is usually neglected. Of note, a fiber FP resonator made from graded-index fiber has a better tolerance to the above-mentioned three factors thanks to the large core size and thus can achieve a finesse of >10^{4} and a measured loaded Q factor of >10^{8} with an FSR of ∼10 GHz.^{54}

To achieve efficient loading from an external pump into the resonator, the output coupling loss should match the intrinsic loss. Depending on the relationship between *α*_{o} and *α*_{i}, three cases can be classified: critical coupling (*α*_{o} = *α*_{i}), undercoupling (*α*_{o} < *α*_{i}), and overcoupling (*α*_{o} > *α*_{i}). For the critically coupled case, the light can be efficiently stored in the microresonator, and no pump light transmits through the coupling port. For the undercoupled case, part of the light transmits through the coupling port with no phase change compared to the pump. For the overcoupled case, part of the light transmits through the coupling port with a *π* phase change compared to the pump. As for FP resonators, if coupled from one port, the other port then contributes to the intrinsic loss. Therefore, only under coupling is possible in FP cavities when both end-facets have the same DBR transmission and there is non-zero intrinsic loss from other contributions.

### C. Measurement of Q factor

According to Eq. (2), there are two methods to measure the loaded Q factor (Q_{L}, including both the coupling loss and the intrinsic loss): (i) measure the cavity linewidth in the frequency domain; (ii) measure the photon lifetime in the time domain. In order to measure the cavity linewidth, the frequency of a low-power single-frequency laser is scanned across the cavity resonance at the center wavelength under test. At the same time, the scanned frequency is calibrated by a frequency reference such as a Mach–Zehnder interferometer (MZI) with a known FSR, converting the temporal signal into the frequency information. As illustrated in Fig. 4(a), the cavity linewidth is obtained from the full width at half maximum (FWHM) of the fitted Lorentz transmission spectrum. The other method is the cavity ring-down method, which measures the photon lifetime [Fig. 4(b)]. This is simply performed by loading the cavity at resonance, then removing the pump and recording the decay time of the intracavity intensity. Of note, the cavity photon lifetime is defined as the decay time of the stored energy to 1/e (≈37%) of its initial value.

### D. Measurement of dispersion

The group velocity dispersion (GVD) is defined as the frequency dependence of the group velocity in a medium, or quantitatively, the derivative of the inverse group velocity with respect to angular frequency. The GVD is one of the most important parameters for broadband DKS comb generation since it causes FSR differences for different center wavelengths and the temporal walk-offs of different frequency components for an ultra-short pulse in a resonator. The GVD is determined by both the material and geometry, such as the waveguide curvature, size, thickness, and angle, which can be numerically designed by commercially available software (e.g., COMSOL Multiphysics).

*L*

_{FS}is the difference in free space between the sample and reference arms, and

*L*

_{w}is the waveguide length. The phase can be Taylor expanded around

*ω*

_{0}as

The dispersion of the waveguide can be retrieved by fitting the spectral interference pattern, as shown in Figs. 5(b) and 5(c). According to this equation, a longer waveguide and a broader white-light source are required to measure lower dispersion.

*D*

_{j}corresponds to the order of the cavity dispersion;

*D*

_{1}is FSR, and

*D*

_{2}is GVD. To obtain the resonant frequencies, the wavelength of a low-power single-frequency pump laser is scanned across the cavity resonances. Several calibrated frequency references can be utilized to reveal the sweeping nonlinearity and determine the actual wavelength of the sweeping laser, such as a laser wavelength meter, a mode-locked fiber frequency comb, an electro-optic modulated frequency comb, or a fiber MZI.

^{73}Among these references, the fiber MZI is the most convenient and cheapest reference. However, the dispersion of fiber MZI should be compensated

^{55}across a wide wavelength range, and the FSR of the fiber MZI should be calibrated, for example, using a wavelength reference before use. Besides, it is better to enclose the MZI and conduct the dispersion measurement at a relatively fast speed to avoid environmental perturbation.

## III. INTRODUCTION TO DISSIPATIVE KERR SOLITON

### A. Principle of dissipative Kerr soliton generation in passive ring microresonators

In a conservative system, such as a loss-less fiber, a soliton is formed due to the balance between the Kerr nonlinearity and anomalous dispersion. In most materials, the Kerr nonlinearity leads to self-phase modulation (SPM), which causes the leading edge of a pulse to shift to longer wavelengths and the trailing edge to shift to shorter wavelengths. In addition, in a material with anomalous quadratic dispersion, blue-shifted frequencies travel faster than red-shifted frequencies. As a consequence, the newly generated frequencies on the leading and trailing edges from Kerr nonlinearity tend toward the center of the pulse due to the dispersion effect. This leads to a self-organized stable pulse, a conservative soliton that balances dispersion and nonlinearity. According to the theoretical analysis, a soliton has a sech^{2}-shape pulse profile and sech^{2}-shape spectrum. Of note, a pure-quartic soliton^{74,75} is feasible due to the balance between Kerr nonlinearity and negative quartic dispersion when the low-order dispersion is neglected through critical dispersion engineering.

^{76,77}

*A*is the intracavity field,

*A*

_{p}is the amplitude of the incident pump field,

*t*

_{R}is the round-trip time,

*t*is the slow time variable,

*τ*is the fast time that corresponds to the local time within the cavity,

*α*is the total round-trip loss,

*δ*is the detuning of the cavity resonance from the pump frequency,

*L*is the cavity length,

*β*

_{2}is the group velocity dispersion (GVD) of the microresonator (anomalous dispersion:

*β*

_{2}< 0, normal dispersion:

*β*

_{2}> 0),

*γ*is the Kerr nonlinear parameter, and

*κ*

_{ex}is the pump coupling coefficient. The LLE was first introduced in 1987 by Luigi Lugiato and Ren’e Lefever to analyze spatiotemporal pattern formation in a continuous-wave (CW)-driven, dissipative, diffractive, and nonlinear optical cavity.

^{76}This mean-field equation combines the well-known nonlinear Schrödinger equation with the boundary conditions under the good cavity approximation.

^{77}The LLE can also be normalized from Eq. (8) by replacing

*αt*/

*t*

_{R}→

*t*,

*δ*/

*α*→

*δ*, $\tau \alpha /L\u2192\theta $, $A\alpha /\gamma L\u2192\psi $, and $Ap\kappa ex/\alpha \gamma L\u2192S$. The normalized form reads as

^{78}

*ψ*is the intracavity field,

*S*is the amplitude of the incident pump field,

*t*is the slow time variable, and

*θ*is the cavity phase coordinate. The second term after the bracketed term on the right-hand side of Eq. (9) represents the self-phase modulation induced phase shift. In Eqs. (8) and (9), the pump can be either a continuous-wave or a pulse with a repetition rate synchronized to the cavity FSR. Of note, Kerr nonlinearity can be negative from cascaded quadratic processes

^{79,80}and be balanced by normal dispersion to form a DKS. Of note, there are some differences in LLE for FP resonators due to the counterpropagating fields (see Sec. III D for details).

Unlike the LLE, which describes the comb dynamics in the time domain, the coupled mode equations (CME) describe the mode interactions through four-wave mixing in the frequency domain.^{37} In theory, the LLE and CME are equivalent theoretical models.^{81} Numerical solutions of both the LLE and CME can be numerically solved through the split-step Fourier method, which allows computationally efficient modeling of octave-spanning microcombs even on a consumer-grade computer. However, it is more straightforward to include some nonlinear effects such as the Raman effect^{82} in the LLE, while it is beneficial to include avoided mode crossings (AMXs) or the Brillouin effect^{83} in the CME.

When $\delta >3$, single input leads to three possible system outputs, which is also shown in Figs. 6(a) and 6(b).

The relationship between the output and the detuning at a fixed pump power is also plotted in Fig. 6(c). Due to the Kerr nonlinearity, the curve is tilted toward the red side. According to the linear stability analysis,^{84,85} the middle branch is not accessible, while the upper branch is perturbation sensitive and modulationally unstable due to the interplay between Kerr nonlinearity and anomalous dispersion. Moreover, in the modulationally unstable regime for the homogeneous solution [Fig. 6(c)], a soliton solution can be found, as shown in Fig. 6(d). Therefore, it can be deduced that the soliton evolves from the modulation instability of the homogeneous solution, which is verified by the dynamics described in Sec. III B. Moreover, solitons exist only in the red-detuned regime (*δ* > 0), so the phase shift coming from the nonlinear terms can be balanced by the phase rotation from the detuning term as well as the dispersion-related term.

### B. Soliton dynamics

According to the bistability map in Fig. 6(c), the dynamics when the pump is scanned across the resonance from the blue-side [Figs. 7(a) and 7(b)] can be described using a phase diagram comprising continuous-wave, Turing pattern, chaotic combs, breathing soliton, and stable soliton regimes [Figs. 7(c) and 7(d)].

Turing patterns arise when the intracavity pump power exceeds the threshold of the optical parametric oscillator. A pair of symmetrical comb lines with respect to the pump is beneficial for squeezed light generation^{86} due to the degenerate four-wave mixing process. In addition, the large frequency space of a comb pair is a good candidate as a low-noise terahertz wave source^{87,88} through a large-bandwidth photodetector.

With the pump frequency close to the resonance peak, more comb lines arise from both degenerate and non-degenerate four-wave mixing processes, leading to non-phase-locked chaotic combs in the frequency domain and noise-like pulses in the time domain. The chaotic combs are useful in parallel LiDAR^{89,90} to break the temporal and frequency congestion and achieve both millimeter-level ranging accuracy and millimeter-per-second-level velocity resolution. Besides, the chaotic combs have recently found application in high random number generation rates of 3.84 Tbps for a scalable decision-making accelerator.^{91}

When the pump detuning reaches the edges of the soliton existing range, the noise-like pulses will be regulated into a breathing form with a smaller pulse number due to the interference with the pump laser and the interaction between pulses. The transition from chaotic combs to breathing solitons leads to a sudden drop in the intracavity average power, which may cause problems for soliton stabilization (see more details in Sec. III C). Additionally, breathing solitons are an interesting physical phenomenon, exhibiting intriguing dynamics like the Fermi–Pasta–Ulam recurrence between coupled modes in the frequency domain,^{92} and might be a good optical platform for the study of time crystals.^{93}

Finally, stable soliton can be achieved in the deep red-detuned regime, which is depicted as an up-tilted soliton step (see more details in Sec. III C). Of note, the final soliton number and their mutual phase relationships are random due to the stochastic transition from chaotic combs to breathing solitons. In addition, perturbation from other effects, such as the thermal effect,^{37} avoided mode crossings,^{94} dispersive wave emission, and the Raman effect,^{95} can lead to a reduction of the soliton number, more soliton steps, and deterministic soliton number generation^{94} during the scanning process.

### C. Soliton properties

According to the coherent pumping scheme, the external CW pump is required for DKS sustainment, and the generated DKS always sits on a CW pedestal, which is verified by the pump comb line standing out above the soliton spectral envelope in Fig. 6(d). Therefore, the power conversion efficiency from the pump to DKS is usually as low as 1%.^{96} Several schemes have been proposed to increase the power efficiency, including pulsed pumps,^{96} dark soliton,^{97} impedance matching using another coupled microresonator,^{98} nonlinear coupling using quadratic nonlinearity,^{99,100} and laser cavity soliton in microresonator-filtered lasers.^{70,101}

*P*

_{sol}and pulse width Δ

*τ*are given by the following expressions:

^{39}

*η*=

*Q*

_{L}/

*Q*

_{ext}characterizes loading efficiency from external coupling,

*Q*

_{ext}is the external or coupling

*Q*factor, and

*Q*

_{L}is the loaded

*Q*factor including both intrinsic loss and coupling loss,

*A*

_{eff}is the effective mode area,

*n*

_{2}is the Kerr coefficient, and

*n*is the refractive index. Hence, the soliton power and soliton pulse width are strongly dependent on the pump detuning

*δ*, which is verified by the up-titled soliton step in Fig. 7(a).

^{102}Therefore, narrow comb linewidth requires a soliton pump laser with low frequency noise, which can be realized by an internally excited Brillouin laser (see Sec. IV C for details) or by self-injection locking the pump laser to the microresonator.

^{103}However, there exists a fundamental limit for the comb linewidth, which results from thermo-refractive noise due to the thermal fluctuation induced refractive index change. The thermodynamic fluctuations $S\delta Tf$ come from the general temperature variance given by

^{104}

^{,}

*T*

_{em}is the temperature of the heat bath,

*k*

_{B}is the Boltzmann constant,

*ρ*

_{0}is the density,

*C*is the specific heat, and

*V*is the mode volume. Of note, the temperature fluctuations $S\delta Tf$ can be numerically obtained by commercial software such as COMSOL Multiphysics for irregular geometries

^{105}or theoretically obtained for regular waveguides such as single-mode fiber.

^{106,107}

^{106,107}

*dn*/

*dT*is the thermo-optic coefficient and

*f*is the offset frequency. Therefore, a large mode volume is beneficial for lower thermo-refractive noise but might come at the cost of an increased soliton pump threshold.

When a coherent soliton forms, the GVD induced FSR difference is compensated by the Kerr nonlinearity induced phase shift, leading to a narrowband beat note of comb spacings in the radiofrequency (RF) domain. The phase noise of the comb repetition rate and, equivalently the soliton timing jitter are the key parameters to characterize how equal the comb spacings are. According to Eqs. (11) and (12), pump detuning is one of the most important parameters affecting the soliton timing jitter through the detuning-dependent soliton peak power and the resulted change in nonlinear-phase shift from the Kerr effect. To reduce the detuning noise, the pump laser frequency should be tightly locked to the cavity, which can be accomplished by active locking methods such as the Pound–Drever–Hall (PDH) locking technique^{108} or locking the soliton power to an ultralow-noise voltage reference.^{109} Passive methods also exist, such as self-injection locking and Brillouin–Kerr soliton generation (see Sec. IV C for details). Compared to active methods, which are usually limited by the locking bandwidth, passive methods can achieve better noise performance, especially at high offset frequencies.

^{110}

^{,}

^{111,112}Due to the soliton coupling to the dispersive wave, hysteretic behavior can be observed in both the dispersive-wave power and the soliton properties,

^{113}such as the soliton center frequency shift and the soliton repetition rate. In particular, the hysteresis behavior can reduce the soliton timing jitter at the quiet point,

^{113}wherein the coupling of pump frequency noise into the soliton repetition rate is greatly weakened. However, there is a limit to this noise suppression approach due to the frequency fluctuations of the dispersive wave, which are attributed to the intermodal thermal noise from imperfect mode overlap between the dispersive-wave mode and soliton-forming mode.

^{105}

^{114}

*ν*is the FWHM cavity linewidth,

*g*is the frequency shift of a resonant mode per photon, and

*η*is the quantum efficiency of the detector. According to Eq. (16), quantum noise can be reduced by a smaller GVD

*D*

_{2}and a nonlinear frequency shift

*g*($g=\u210f\omega 02cn2/Vn2$, where

*ℏ*is the Plank constant,

*ω*

_{0}is the DKS center frequency).

Besides, the thermo-refractive noise (TRN) from thermal fluctuations will impose a noise limit on the repetition rate through two channels owing to the fluctuation of cavity resonance. First, TRN directly changes the resonator FSR, and the repetition rate fluctuation can be minified from TRN by a ratio between the soliton center frequency and the FSR. Second, the cavity resonance change will lead to detuning noise. For the repetition rate noise, the detuning noise limit from TRN is dominant at low offset frequencies (or at long time scales) and can be overcome by a strong active detuning control method, while the quantum limit is dominant at high offset frequencies (or at short time scales). According to Eqs. (13) and (16), a large mode volume is beneficial to lower both the quantum noise limit and the limits imposed by TRN. Therefore, a multimode fiber FP microresonator is a good candidate for both ultra-narrow comb linewidth and ultralow phase noise of the soliton repetition rate.

### D. Solitons in FP resonators

^{115}

*P*

_{cav}is the average power accumulated in the cavity, given by

Compared with the standard LLE for ring resonators, there is an additional nonlinear integral term representing phase modulation by twice the average intracavity power. This energy dependent detuning causes a cavity resonance frequency shift analogous to the thermal response, except instantaneously, due to the nature of the Kerr-nonlinearity. The theoretical comparison between ring-LLE and FP-LLE is shown in Figs. 6(a) and 6(b).^{116} More analysis has shown that the coupling between detuning and the cavity’s energy will cause a change in the soliton's existence range with either a continuous-wave or pulsed pump configuration.^{116–118} Despite the differences, DKS dynamics and properties in FP resonators can be well understood from the results for ring resonators.

### E. Soliton excitation and stabilization

Different from mode-locked lasers, DKS microcombs, either in ring resonators or FP resonators, usually cannot self-start and require excitation and stabilization techniques to achieve a stable microcomb for real-world applications. According to the analysis in Sec. III B, DKS is usually excited by scanning the pump frequency from blue detuning to red detuning, during which the modulation instability induced chaotic combs evolve into DKS combs at red detuning. However, the intracavity average power suddenly drops when solitons form [Fig. 7(a)], causing a blue shift of the cavity detuning due to the absorption-induced thermal nonlinearity in most microresonators. Therefore, the triggered DKSs, especially single solitons, are difficult to thermally stabilize even with a slow pump scanning speed.^{119} Either complex tuning schemes and active feedback loops [Fig. 8(a)],^{109} an additional auxiliary laser^{120} [Fig. 8(b)] or avoided mode crossings [Fig. 8(c)] are required for eventually stabilized DKS, which prevents the minimization and integration of microcomb systems.

In order to remove the additional components, other methods utilizing passive effects in the microresonator are proposed for self-starting DKS microcombs. The first is the nonlinear self-injection locking (SIL) method [Fig. 8(d)].^{121} The soliton is self-triggered by thermal nonlinearity and Kerr nonlinearity when the pump frequency is located on the red-detuned side. Since the SIL speed is much faster than the thermal effect, the triggered soliton can always stabilize itself. Despite the enabled turnkey operation of fully chip-based soliton microcombs, it requires deliberate debugging due to the small locking range, which strongly depends on both the frequency matching of the DFB-laser-microresonator pair and the environment-sensitive optical feedback phase. Moreover, the large soliton timing achieved jitter prohibits critical applications such as microwave photonics and timing distribution. The other methods utilize novel configurations of microresonator-filtered lasers,^{70,101,122} where soliton self-excitation is achieved by the self-organized behavior in the laser and soliton self-stabilization is achieved by thermal nonlinearity compensation from either gain nonlinearity or a coexisting intracavity Brillouin laser. Details can be found in Sec. IV C.

## IV. BRILLOUIN EFFECT ON SOLITON GENERATION

Various nonlinear effects can be utilized to enhance the microcomb performances, such as the Brillouin effect, the gain effect, and the Raman effect. Compared to microresonators based on planar waveguides, fiber microresonators have the unique advantage of including both the Brillouin effect and the gain effect. Therefore, we will focus on the fiber Kerr microcombs based on the Brillouin effect and gain effect in the next two sections. Readers can find more information on the Raman effect in microresonators in other papers, including the principle of Raman scattering,^{125–127} Raman microlasers,^{128–130} nonlinear competition between the Raman and Kerr effects^{131–134} and Raman–Kerr soliton generation.^{66,135–137}

### A. Principle of backward and forward Brillouin scattering

Brillouin scattering is a third-order nonlinear effect where the polarization intensity depends cubically on the field. From the perspective of quantum mechanics, Brillouin scattering is recognized as an interaction between an input photon, a scattered photon, and an acoustic phonon inside the medium. The phonon comes from the lattice vibration, which can be caused by thermally induced Brownian motion or intense pump field induced electrostriction, corresponding to spontaneous Brillouin scattering or stimulated Brillouin scattering (SBS). When the phase matching condition (or the conservation of energy and momentum) is fulfilled, the photon-phonon interaction is maximized, and considerable conversion efficiency can be achieved. Brillouin scattering can also be explained from a macroscopic perspective. The accumulated displacement and vibration of the lattice result in an overall material strain (elongation) and change the refractive index through the photoelastic effect. Therefore, the periodically oscillating acoustic field causes a periodic refractive index change, creating an effective index grating that scatters the input light with a Brillouin frequency shift [Fig. 9(a)].

The energy or frequency of the scattered photon can be the difference or sum between the photon and the phonon, leading to Stokes wave or anti-Stokes wave generation. Both the Stokes and anti-Stokes shifted processes can be visualized as a “two-step” event (Fig. 10). The first step is the annihilation of an incident photon due to the simultaneous excitation of a lattice from the ground vibration state (or excited vibration state) to the intermediate state. In the second step, this lattice returns to the excited state (or the ground state) with the simultaneous creation of a scattered photon with a Stokes (or anti-Stokes) frequency shift. Following the Bose–Einstein distribution, the scattered strength of the anti-Stokes wave is usually weaker than the Stokes wave since the population in the ground vibration state is larger than that in the excited vibration state.

According to the relationship of propagation direction between the pump field and the scattered field, there are two kinds of Brillouin scattering: the forward and backward Brillouin scattering. For the backward SBS [Fig. 9(a)], the Brillouin frequency shift depends on the acoustic velocity, the refractive index, and the pump wavelength as Ω_{B} = 2*π* · 2*nυ*_{s}/*λ*, which is usually around tens of GHz. The wavenumber of the generated hyper sound wave is at the same level as the optical wave [see the phase matching diagram in Fig. 11(a)]. The Brillouin gain bandwidth is usually tens of MHz, corresponding to a phonon lifetime of a few nanoseconds.

As for the forward SBS process, the acoustic modes no longer propagate along the light propagation direction but are guided in the transverse cross section [Fig. 9(b)], leading to near-zero wavenumbers [Fig. 11(b)]. The Brillouin frequency shift is determined by the transverse acoustic mode, which strongly depends on the material and geometry. For example, in fibers, the acoustic wave is well confined in the cylindrical geometry [Fig. 9(b)] and, therefore, the phonon lifetime can reach the level of a few microseconds and lead to a Brillouin gain bandwidth of hundreds of kHz, which benefits optical information processing with high spectral resolution. In a multimode system, both backward and forward Brillouin scattering can occur between different spatial modes with considerable mode overlap, which is a promising method to avoid cascaded Brillouin processes (see details in Sec. IV B). The frequency shift of forward Brillouin scattering is usually around hundreds of MHz, which is smaller than tens of GHz for backward Brillouin scattering in fibers.

Both the backward and forward Brillouin scattering can be realized between orthogonally polarized modes or different spatial modes in fibers, which are termed cross-polarized SBS or intermodal SBS. According to the principle in Fig. 9(a), there should be no cross-polarized gain due to no interference pattern between orthogonally polarized modes. For example, when the pump and probe are aligned to the fast and slow axes of a polarization-maintaining fiber, respectively, the measured cross-polarized SBS gain is near zero.^{139} However, in optical fibers with weak and random birefringence, cross-polarized SBS gain is possible^{139,141} despite lower gain compared to the SBS gain for the same polarizations. As for intermodal SBS, the imperfect mode overlap and induced lower SBS gain should be considered.

### B. Stimulated Brillouin lasers in microresonators

In an oscillator, when the intracavity gain compensates for the loss, a coherent laser stands out from the noise. The gain can arise from the active gain such as pumped Erbium-doped fiber, or the nonlinear gain such as Brillouin gain, Raman gain, Kerr gain and quadratic nonlinearity induced parametric gain.

^{142}

*g*

_{B}is the Brillouin gain coefficient, which is defined as

*γ*

_{e}is the electrostrictive constant,

*υ*

_{s}is the speed of sound, and Γ

_{B}is the loss rate of the density wave. The frequency shift between the Stokes mode resonance peak and the Brillouin gain spectrum will lead to an increased SBL threshold, non-zero SBL detuning, and degraded SBL linewidth, which will be discussed later.

Different ways have been proposed to achieve better alignment between the Stokes mode resonance and the Brillouin gain spectrum, depending on the characteristics of both modes. If the Stokes mode shares the same mode family with the pump, which is the intramodal SBS case, the cavity length should be carefully adjusted within only a few millimeters^{143} to ensure the Brillouin frequency shift is an integer multiple of the free spectral range (FSR) since Brillouin gain bandwidth is usually tens of MHz. Therefore, fine cavity length fabrication is one of the key challenges preventing efficient SBL generation in microresonators. Another method is the so-called intermodal SBS, which utilizes different mode families, such as cross-polarized modes, different spatial modes, or a combination. In an FP microresonator made from highly multimode fiber by our group,^{54} it is quite easy to achieve efficient intermodal SBL generation regardless of the cavity length. Of note, the intramodal SBL generation usually leads to cascaded high-order SBL generation, which limits the power scaling of the first-order SBL. A large group velocity difference induced by a large Brillouin frequency shift and/or a large GVD, such as in diamond,^{144} is required to increase the threshold of the cascaded SBL generation. As for the on-chip microresonators, the engineering of the avoided mode crossing can increase the loss of the cascaded Brillouin mode resonance and thus increase the threshold of high-order SBL.^{145} Besides, the gratings on the waveguide or microresonator can inhibit the cascaded SBS.^{146,147} Compared to the intramodal method, intermodal SBL generation is more general and efficient for cascading inhibition and SBL power scaling. Of note, the aforementioned two methods can be applied to both backward and forward Brillouin laser generation.

^{148}

*A*

_{F},

*A*

_{B}, and

*ρ*are the forward (pump), backward (SBL), and acoustic wave, respectively;

*δ*

_{F},

*δ*

_{B}, and

*δ*

_{ρ}are the detunings of pump, SBL, and acoustic fields with respect to their corresponding resonances; Λ

_{F}, Λ

_{B}, and Λ

_{ρ}are the mode overlaps of the forward, backward, and density waves;

*τ*

_{F}and

*τ*

_{B}are the lifetimes for the forward and backward waves; 1/

*τ*

_{ext}is the external coupling rate of the pump; Γ

_{B}is the loss rate of the density wave (Brillouin gain bandwidth); Ω

_{B}is the Brillouin frequency shift; and Ω is the actual frequency shift between the pump frequency and the generated SBL frequency.

Therefore, the detuning of the backward SBL is modulated by $1/1+\Gamma B\tau B$ times the pump detuning. For example, in a high-Q microresonator, the product of the Brillouin gain bandwidth and cavity photon lifetime is larger than 10, which means SBL detuning noise can be suppressed by more than 10 dB. This advantage can be utilized for generating solitons with ultralow-jitter since the soliton’s features are strongly detuning-dependent, which is discussed in Sec. III C.

Apart from detuning noise suppression, SBL frequency noise can also be effectively reduced by a narrowing factor $1+\Gamma B\tau B2$^{62,149} at the perfect phase matching condition. This noise suppression is achieved by the quick dampening of the pump noise through the acoustic wave because the phonon decay rate (Γ_{B}) is much faster than the intracavity photon decay rate (1/*τ*_{B}). The latest SBL fundamental linewidth is 53 mHz,^{54} which was achieved in 2022 by our group in a multimode fiber FP microresonator with Q factors of 384 × 10^{6}.

^{150}due to the increased amplitude-to-phase noise coupling at the imperfect phase matching condition (deviating from the maximum SBS gain). In this case, the SBL linewidth is given by

*ν*

_{0}denotes the non-broadened SBL linewidth and

*ζ*= 2

*δ*/Δ

*ν*

_{cav}determines the linewidth broadening factor (see Fig. 12).

Phonon noise can be squeezed if the phonon decay rate is slower than the intracavity photon decay rate, which can be realized by utilizing a guided acoustic wave in the forward Brillouin scattering process, such as in microresonators made from fibers or suspended waveguides. For example, in an on-chip silicon microresonator,^{151} the guided acoustic wave results in a longer phonon lifetime than the intracavity photon lifetime. Therefore, pump noise is directly damped to the generated forward SBL. A noise-suppressed phonon can be deduced from a noise-suppressed electrical beating signal between the two correlated optical fields. Recently, the first 20-dB noise-squeezed forward SBL has been achieved by our group by utilizing a short-lifetime phonon mode in a multimode fiber FP microresonator,^{140} reaching the noise level of backward SBL. Of note, the dynamics of forward SBL generation in microresonators can be similarly modeled by the three-wave coupled equations as in Eq. (22).

### C. Kerr soliton generation with backward Brillouin effect

The SBS effect can cooperate with other intracavity nonlinear effects, which provides opportunities to improve noise performance and benefits more applications that require ultrahigh laser coherence. The cascading between the Brillouin and Kerr effects dates back to 2009^{62} when Brillouin-enhanced Kerr combs were demonstrated in a monolithic highly nonlinear fiber cavity. After that, it is found that the interaction between the Brillouin and Kerr effects in a chalcogenide fiber resonator^{152,153} and later in an on-chip FP resonator^{146,154} can lead to phase locked frequency combs. Recently, our group showed for the first time that the controlled interaction between SBS and Kerr nonlinearity can result in two advantages of the generated SBL DKS: self-stabilization without active feedback control and ultralow noise, including ultra-narrow comb linewidth and ultralow DKS timing jitter.^{63} These demonstrations were first performed in a monolithic highly nonlinear fiber Fabry–Perot (HNLF FP) resonator^{63} and later extended to graded index multimode fiber Fabry–Perot (GRIN-MMF FP) resonators^{54} and other DKS platforms such as silica microdisk resonators,^{150} silica wedge resonators,^{155} and silica microspheres^{40} by other groups. In our original experiment, we introduced a novel two-step pumping scheme where the primary pump at one mode family triggers SBS lasing that served as the secondary pump for DKS generation at another mode family. The blue-detuned primary pump and the red-detuned SBL can work together to compensate for the detrimental cavity thermal nonlinearity that limits the reliability and robustness of SBL DKS generation [Fig. 13(a)], with the same principle used in the auxiliary-assisted soliton generation method,^{120,156} which is discussed in Sec. III E [Fig. 8(b)]. In principle, by adjusting the relative magnitude between the pump-Brillouin resonance offset frequency and the Brillouin frequency shift, either an SBL comb or a pump comb can be generated. In the experiment, we conveniently tuned the fiber stress, thus the Brillouin frequency shift, and realized DKS generation in the pump mode family [Fig. 13(b)] with a red-detuned pump and a blue-detuned SBL. Moreover, the two-step pumping scheme provides a feasible route toward high-dimensional nonlinear cavity dynamics such as dual comb or multiple comb generation using only a single pump! For instance, our group recently demonstrated the generation of spatiotemporal mode-locked (STML) DKS [Figs. 13(c) and 13(d)] for the first time in the GRIN-MMF FP microresonator with low modal dispersion enhanced intermodal nonlinear interaction,^{54} while the Yao group has achieved repetition-rate locked cross-polarized solitons in a monolithic highly nonlinear fiber FP resonator.^{64} Our group and the Yao group have also shown the feasibility of multiple DKS microcomb generation by high-order intermodal SBLs.^{40,140,157}

The two-step pumping scheme also takes advantage of noise suppression in the SBS process, as mentioned earlier in Sec. IV B, including comb linewidth narrowing and detuning noise suppression. Due to the degenerate four-wave-mixing process, the DKS comb linewidth is the same as the pump linewidth^{102} if other nonlinear effects are not considered.^{158} Therefore, soliton microcombs pumped by the intracavity SBL act as multiple single-frequency lasers with ultra-narrow linewidth. Besides this, SBL detuning noise suppression leads to large SBL soliton jitter reductions. In the first demonstration in an HNLF FP resonator, a microcomb linewidth of 22 Hz and quantum-limited DKS timing jitter of 995 attoseconds for averaging times up to 10 *µ*s were achieved [Fig. 14(a)].^{63} Such ultralow timing jitter can be measured by an all-fiber reference-free Michelson interferometer (ARMI) timing jitter measurement apparatus as described in Refs. 159 and 160. By implementing the two-step pumping scheme between two spatial modes in a GRIN-MMF FP resonator, our group recently demonstrated unprecedented microcomb linewidth of 400 mHz and DKS timing jitter of 500 attoseconds for averaging times up to 25 *µ*s [Fig. 14(b)].^{54} The enhanced performances were attributed to the improved Q factor, lower thermo-refractive noise,^{104} and lower quantum limit^{114} of the GRIN-MMF FP resonator.

Of note, all the reported SBL DKS generation utilizing the two-step pumping scheme is realized by intermodal SBS and/or cross-polarized SBS. In theory, the intramodal SBL DKS is also feasible with avoided cascaded SBL and high-power first-order SBL generation. Moreover, our proposed two-step pumping scheme should also be applicable to Raman–Kerr soliton generation in crystalline microresonators with narrowband Raman gain such as AlN microresonators,^{161} as well as the forward-Brillouin–Kerr soliton generation. Besides, the two-step pumping scheme can also be applied for Kerr soliton generation from either the forward SBL or the second harmonic field.

### D. Forward Brillouin scattering effect on DKS

Forward Brillouin scattering, also referred to as guided acoustic wave Brillouin scattering, was first studied by Shelby *et al.* in single-mode fibers^{162,163} due to the good acoustic waveguide enabled by the cylinder geometry [Fig. 9(b)]. The forward Brillouin scattering has been discovered to impose long-range conservative soliton interaction since 1989.^{164} In 2013, Jang *et al.* observed pairs of DKSs interacting over a range as large as 8000 times their width in a long passive fiber cavity.^{165} The interactions are so weak that, at the speed of light, an effective propagation distance on the order of an astronomical unit can be required to reveal the full dynamical evolution. The interactions are mediated by transverse acoustic waves generated in the optical fiber by the propagating solitons through electrostriction. First, the leading DKS excites the transverse vibration and acoustic waves through forward Brillouin scattering. Second, the acoustic wave is guided and reflected back-and-forth from the fiber cladding–coating boundary back into the fiber core, generating echoes with a period determined by the cladding diameter and the acoustic speed. Third, the trailing cavity soliton sees the refractive index change caused by the acoustic wave through electrostriction, resulting in the interaction between two solitons.

By exciting a large number of DKSs with random temporal separations in a continuously driven passive fiber resonator, the same group observed the real-time dynamics of DKS bunching from random sequence into regularly spaced aggregates (soliton crystal)^{166} (Fig. 15), where the acoustic wave played the key role. Therefore, it is quite possible that the forward SBS can impose additional noise on the soliton jitter and comb linewidth for the fiber microcombs, although it has not been well studied.^{167}

Besides these interesting optoacoustic phenomena, solitons in fiber resonators with active gain can be flexibly manipulated by enhancing optoacoustic interaction in photonic crystal fibers, where both the optical and acoustic fields are tightly guided in a small area.^{168} Interesting phenomena such as the formation and dissociation of optical supramolecular structures^{169} and novel laser sources such as GHz-repetition-rate mode-locked fiber lasers^{170} have also been demonstrated.

## V. GAIN EFFECT ON SOLITON GENERATION

To apply optical frequency combs for extensive real-world applications, three trends have arisen to push the comb repetition rate into microwave rates. The first is to shrink the size of the mode-locked laser, which is mainly limited by the requirement of high gain and a reliable saturate absorber. The second is to increase the size of the DKS comb in a microresonator, which is mainly limited by propagation loss and an increased pump threshold. The last is to combine the mode-locked laser and DKS in a novel configuration termed a microresonator-filtered laser, which will be discussed in detail later.

### A. Passively mode-locked monolithic fiber lasers

Besides the Kerr-lens mode-locked solid-state laser,^{171} harmonic mode locking,^{172} active mode-locking,^{173} and optoacoustic mode-locking,^{170} monolithic fiber FP lasers mode-locked by material-based saturable absorbers are another alternative to generate robust pulses at >5 GHz repetition rates with low cost and compact size. The keys to low-threshold mode-locking are the low-energy saturable absorbers and high gain (or low loss), which require low-transmission of the FP cavity coating, highly doped gain fiber, and relatively large pump power in a centimeter-long fiber. The demonstrations of mode-locked fiber FP lasers were first performed to achieve a 5-GHz pulse train by carbon nanotubes^{174} and later extended to realize a pulse repetition rate of 17.2 GHz.^{175} Other saturable absorbers such as graphene [Fig. 16(a)]^{176,177} and semiconductor saturable-absorber mirror (SESAM)^{178–180} are introduced to achieve mode-locked pulses with electrical tunability [Fig. 16(b)],^{177} high stability, and wavelength extension.^{180} Despite the remarkable results, more studies are necessary to increase the comb bandwidth and lower the pulse timing jitter for practical applications.

### B. Externally driven DKS generation with active gain

Active gain material can be incorporated to increase the Q factor and lower the pump threshold for DKS generation. Of note, these microresonators without saturable absorbers are configured to have a gain close to the lasing threshold, which is fundamentally different from mode-locked lasers. Such configuration ensures that DKS generation is still in the conventional framework where DKS is coherently driven by the external CW pump.

The first demonstration is performed in a long class B Erbium-doped fiber cavity,^{181} showing that spontaneous soliton formation is forbidden with a CW external pump but becomes accessible with a pulsed external pump depending on the gain saturation. In the regime of strong gain saturation under continuous-wave driving, the linear stability analysis shows (i) modulation instability of the homogeneous states does not exist; (ii) soliton exists in a large range of detunings. The soliton branch is very different from that of conventional DKS emerging from a homogeneous saddle-node bifurcation, as shown in Fig. 6(a). Since the spontaneous formation of solitons is prohibited due to the lack of MI, solitons can be observed by injecting a seed pulse. In the regime of weak gain saturation, the linear stability is similar to the conventional DKS, as in Sec. III A. This can be qualitatively understood with the modified LLE, where the loss term is compensated by a linear small-signal gain. The most significant finding is that, to the limit of the equipment, the so-called active cavity soliton does not suffer from the Gordon–Haus jitter resulting from the amplified spontaneous emission.

Another example is the gain-assisted chiral soliton microcombs in Er-doped microspheres.^{182} Two counterpropagating solitons with different repetition rates, which are assisted by backscattering and active gain, are demonstrated with a unidirectional continuous pump.

Recently, it was shown that a generalized LLE can also be used to describe the dynamics of a class B quantum cascade laser (QCL) with an external pump under weak saturation.^{183} Therefore, localized structures such as the Turing pattern and DKS in externally pumped QCL were predicted and numerically analyzed.^{183,184} Surprisingly, the first DKS generation in QCL was demonstrated without an external pump.^{185} Although not fully understood, the following three conditions might be the key conditions for QCL soliton generation: (i) a ring structure that removes instantaneous gain grating; (ii) unidirectional operation that minimizes optical backscattering through spontaneous symmetry breaking; and (iii) anomalous dispersion. Additionally, gain saturation may also be a reason why the results are different from previous theoretical analyses.^{186} Excitingly, without an external pump, higher soliton efficiency can be achieved, and it is more straightforward to extend the DKS wavelength range to the mid-infrared and terahertz regimes. Despite the remarkable results, more studies are necessary to better understand its physical mechanism, how it differs from saturable absorber enabled mode-locking, and what the consequences are.

### C. DKS microcomb generation in a microresonator-filtered fiber laser

The configuration of a microresonator-filtered fiber laser, where a microresonator is nested into a fiber laser cavity [Fig. 17(a)], is a novel configuration that combines the advantages of fiber lasers enabled self-starting and microresonators induced ultrahigh pulse repetition rate. In this configuration, the microresonator plays two roles: (i) a periodic intracavity filter and (ii) the nonlinear Kerr medium.^{70} The final lasing modes are the common modes of both the microresonator and the laser cavity. Similar to the self-injection locking method,^{187–190} such a configuration provides another method for locking the lasing mode to the microresonator resonance as well as narrowing the optical linewidth determined by the microresonator Q factor^{189} since only a high-coherent laser can circulate in the cavity.

Usually, this configuration requires an intracavity filter inside the fiber laser cavity to select the desired wavelengths, as well as a delay line to fine tune the relative position of lasing modes with respect to the microresonator’s resonance. With a broadband intracavity filter, multiple common modes can lase and experience amplification in the active fiber. When the lasing modes are blue detuned with respect to the microresonator’s resonance,^{101} phase locking can be achieved through the dissipative four-wave-mixing process and form a coherent localized Turing pattern.^{191}

Additionally, the so-called laser cavity soliton (LCS)^{101} can be achieved by fine tuning the delay line. According to our recent study,^{70} for LCS, the red-detuned nested microresonator has a nonlinear transmission that provides an effective saturable absorber to mode lock the fiber laser. The insight leads to the prediction of a new class of bright chirped LCS and the demonstration of a mode efficiency as high as 90.7%, approaching the theoretical limit of 96%. In addition, intriguing LCS interaction phenomena, including optical Newton’s cradle, were experimentally observed.^{70} Besides the free background and high mode efficiency, LCS is also found to be self-starting, which is attributed to the balance between thermal nonlinearity in the microresonator and gain nonlinearity in the fiber laser.^{192}

To improve the noise performance and long-term stability of microcombs for practical applications, our group recently introduced a novel two-step pumping scheme (Sec. IV C) in the microresonator-filtered laser configuration [Fig. 17(b)] and realized decoupling between the pump generation and the comb generation.^{122} Only a single-frequency laser is circulating in the large fiber laser cavity resulting from the narrowband filter, while the microcomb generation is realized by the pump-generated cross-polarized and intermodal SBL within the microresonator. Our approach for self-starting solitons [Fig. 17(c)] does not suffer from the vulnerability to feedback phase fluctuation^{121} and enables the deterministic selection of DKS soliton numbers. In addition, we demonstrate the self-healing capability of returning to the original comb state from instantaneous perturbations, excellent long-term stability, and modulation capability [Fig. 17(d)]. More importantly, our approach allows access to ultralow noise comb states. The turnkey Brillouin-DKS frequency comb achieves a fundamental comb linewidth of 100 mHz and a DKS timing jitter of 1 fs for averaging times up to 56 *µ*s. We claim that our approach (patent pending) is a universal topology for turnkey DKS generation and has the potential for fully on-chip integration. Moreover, SBS is not the only intracavity effect that can be used to decouple the pump and comb generation. Avoided mode crossing^{123,193,194} can also be utilized, and it relaxes the need to match the microresonator FSR with the Brillouin frequency shift, rendering AMX more flexible and user-friendly for mass production of user-friendly and field-deployable comb devices.

## VI. SUMMARY AND OUTLOOK

The fiber FP microresonator is an incredibly promising platform with boundless potential, waiting to be harnessed. This platform can leverage well-established fiber optics, which has been a cornerstone of modern optics since its inception in 1970. Researchers have explored the advantages of ultralow loss (<0.3 dB/km), large mode areas, and spatiotemporal mode-locking. The achievement of ultrahigh-Q microresonators (>10^{9}) with a large mode volume has enabled the development of novel comb sources with significantly improved performance, including ultra-narrow comb linewidth and minimal soliton jitter for applications with ultralow noise requirements, such as optical clockworks and optical synthesizers. Furthermore, fiber FP microresonators hold the potential to address practical challenges in microcomb utilization, including self-starting functionality and high-power efficiency with a continuous-wave pump. For instance, our group has demonstrated self-starting microcombs with high efficiency in a fiber laser nested with a fiber FP microresonator (see Fig. 17).

Beyond the examples discussed, numerous other exciting and innovative directions are anticipated. First, there is the potential for extending the wavelength range of microcombs in fiber FP resonators. For instance, it is feasible to generate mid-infrared microcombs using chalcogenide fibers through sophisticated dispersion engineering, while visible microcombs can be produced in periodically poled fibers^{195} via cascaded second-order nonlinearities.^{79,80,196} Second, the development of all-fiber integrated microcomb systems is worth exploring. For example, further integration of the fiber FP microresonator filtered lasers (Fig. 17) can enhance the applicability of comb-based technologies in the field. Furthermore, all-fiber integrated systems compatible with other fiber-based technologies can minimize coupling losses, which is crucial for applications like squeezed light generation.^{86} The third avenue of investigation involves exploring additional degrees of freedom offered by fibers, such as spatial modes and chiral modes. This exploration is likely to lead to a plethora of high-dimensional phenomena and applications, for example, optical vortex Brillouin lasers,^{197} temporal solitons with optical vortex in fiber,^{198} mode division multiplexing for optical communications, multi-core fibers for multiple combs,^{199} and so on. Additionally, ultrahigh-Q fiber microresonators can serve as a test bed for exploring interdisciplinary effects, such as integrating 2D materials for lab-on-fiber technology^{200,201} and optical sensing applications.

Despite these fascinating prospects, numerous challenges remain. The first challenge involves increasing the Q factor of fiber FP resonators. As previously discussed in Sec. II B, cavity losses primarily stem from imperfections in the fiber end facets, which vary depending on the type of fiber used. For example, chalcogenide fibers are very soft, while photonic crystal fibers feature numerous holes.^{202} Consequently, it is crucial to employ new techniques for highly precise mechanical polishing and monitoring, as well as precise coatings consistent with the designed parameters. The second challenge relates to dispersion engineering in conventional fibers, where weakly confined large modes might limit the engineering capability. However, it is still feasible to engineer the cavity dispersion not only through material dispersion but also by different sizes and micro- and-nanostructures such as photonic crystal fiber,^{202} chirped fiber Bragg gratings,^{203} and chirped coatings. The third challenge pertains to the scaling and control of cavity length. While it is relatively straightforward to fabricate longer cavities to meet the requirements of various applications,^{204,205} such as dual-comb spectroscopy and photonic analog-to-digital converters (ADCs), it is challenging to push the FSR of a fiber FP resonator to exceed 50 GHz. This limitation may arise from the overall thickness of the cavity becoming too thin for fabrication. Additionally, compared to their on-chip counterparts, new techniques are required to precisely control the FP cavity length for applications such as stimulated Brillouin laser generation and combs with precise repetition rate differences. The fourth challenge concerns the absence of fast-modulated cavity properties, which is particularly important for sensing applications like LiDAR.^{30,206,207} One feasible approach is to introduce nested cavities [as shown in Fig. 17(d)], while another involves integrating electrically tunable materials inside the fiber FP cavity. However, the latter approach may lead to a reduction in the Q factor. With continued efforts to understand and control new phenomena in fiber FP microcavities, the community will continue to improve the performance of fiber microcombs and expand the already impressive range of microcomb applications.

## ACKNOWLEDGMENTS

National Science Foundation (Grant Nos. ECCS 2048202 and OMA 2016244) and Office of Naval Research (Grant No. N00014-22-1-2224).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jonathan Musgrave**: Writing – original draft (equal); Writing – review & editing (equal). **Shu-Wei Huang**: Funding acquisition (lead); Writing – original draft (supporting); Writing – review & editing (lead). **Mingming Nie**: Conceptualization (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

All data generated or analyzed during this study are available within the paper. Furthermore, source data will be made available on reasonable request.

## NOMENCLATURE

- 1/
*τ*_{ext} pump external coupling rate

*A*intracavity field

*A*_{B}backward wave

*A*_{eff}effective mode area

*A*_{F}forward wave

*A*_{p}amplitude of the incident pump field

*C*specific heat

*c*speed of light

*D*_{1}free spectral range

*D*_{2}group velocity dispersion

*dn*/*dT*thermo-optic coefficient

*F*finesse

*f*offset frequency

*f*_{rep}repetition rate frequency

*g*frequency shift of a resonant mode per photon

*g*_{B}Brillouin gain coefficient

*k*_{B}Boltzmann constant

*L*cavity length

*n*optical refractive index

*n*_{2}Kerr coefficient

*Q*total quality factor

*Q*_{ext}external or coupling

*Q*factor*R*reflectivity coefficient

*S*normalized pump amplitude

- $S\delta Tf$
thermodynamic fluctuations

- $S\nu f$
frequency noise

*t*slow time

*T*transmission coefficients

*T*_{em}temperature

*t*_{R}round-trip time

*V*mode volume

- Γ
spatial overlap factor

- Γ
_{B} loss rate of the density wave (Brillouin gain bandwidth)

- Δ
*v*_{FSR} free spectral range

- Δ
*ν* cavity linewidth

- Δ
*ν*_{0} non-broadened SBL linewidth

- Δ
*τ* soliton pulse width

- Λ
_{B} mode overlaps of the backward wave

- Λ
_{F} mode overlaps of the forward wave

- Λ
_{ρ} mode overlaps of the density wave

- Ω
actual frequency shift (between pump and SBL)

- Ω
_{B} Brillouin frequency shift

*α*propagation loss

*α*_{i}intrinsic loss

*α*_{o}output coupling loss

*β*propagation constant

*β*_{1}group velocity

*β*_{2}group velocity dispersion

*γ*Kerr nonlinear parameter

*γ*_{e}electrostrictive constant

*δ*detuning

*δ*_{B}SBL detuning

*δ*_{F}pump detuning

*δ*_{ρ}acoustic wave detuning

*ζ*linewidth broadening factor

*η*loading efficiency

*η*_{q}quantum efficiency of the detector

*κ*_{ex}pump coupling coefficient

*λ*wavelength in vacuum

*ν*optical frequency

*ρ*acoustic wave

*ρ*_{0}density

*τ*cavity photon lifetime

*τ*fast time

*τ*_{B}backward wave lifetime

*τ*_{F}forward wave lifetime

*υ*_{s}speed of sound

*ϕ*optical phase

*ψ*intracavity field

*ω*,*ω*_{0}angular optical frequency

*ℏ*Plank constant

## NOMENCLATURE

- 1/
*τ*_{ext} pump external coupling rate

*A*intracavity field

*A*_{B}backward wave

*A*_{eff}effective mode area

*A*_{F}forward wave

*A*_{p}amplitude of the incident pump field

*C*specific heat

*c*speed of light

*D*_{1}free spectral range

*D*_{2}group velocity dispersion

*dn*/*dT*thermo-optic coefficient

*F*finesse

*f*offset frequency

*f*_{rep}repetition rate frequency

*g*frequency shift of a resonant mode per photon

*g*_{B}Brillouin gain coefficient

*k*_{B}Boltzmann constant

*L*cavity length

*n*optical refractive index

*n*_{2}Kerr coefficient

*Q*total quality factor

*Q*_{ext}external or coupling

*Q*factor*R*reflectivity coefficient

*S*normalized pump amplitude

- $S\delta Tf$
thermodynamic fluctuations

- $S\nu f$
frequency noise

*t*slow time

*T*transmission coefficients

*T*_{em}temperature

*t*_{R}round-trip time

*V*mode volume

- Γ
spatial overlap factor

- Γ
_{B} loss rate of the density wave (Brillouin gain bandwidth)

- Δ
*v*_{FSR} free spectral range

- Δ
*ν* cavity linewidth

- Δ
*ν*_{0} non-broadened SBL linewidth

- Δ
*τ* soliton pulse width

- Λ
_{B} mode overlaps of the backward wave

- Λ
_{F} mode overlaps of the forward wave

- Λ
_{ρ} mode overlaps of the density wave

- Ω
actual frequency shift (between pump and SBL)

- Ω
_{B} Brillouin frequency shift

*α*propagation loss

*α*_{i}intrinsic loss

*α*_{o}output coupling loss

*β*propagation constant

*β*_{1}group velocity

*β*_{2}group velocity dispersion

*γ*Kerr nonlinear parameter

*γ*_{e}electrostrictive constant

*δ*detuning

*δ*_{B}SBL detuning

*δ*_{F}pump detuning

*δ*_{ρ}acoustic wave detuning

*ζ*linewidth broadening factor

*η*loading efficiency

*η*_{q}quantum efficiency of the detector

*κ*_{ex}pump coupling coefficient

*λ*wavelength in vacuum

*ν*optical frequency

*ρ*acoustic wave

*ρ*_{0}density

*τ*cavity photon lifetime

*τ*fast time

*τ*_{B}backward wave lifetime

*τ*_{F}forward wave lifetime

*υ*_{s}speed of sound

*ϕ*optical phase

*ψ*intracavity field

*ω*,*ω*_{0}angular optical frequency

*ℏ*Plank constant

## REFERENCES

^{−16}relative uncertainty

*Q*silica microresonator

_{2}crystalline whispering gallery mode resonators with negative thermal-optical effects

_{3}soliton microcomb