Analysis of Kerr comb generation in silicon microresonators under the influence of two-photon absorption and free-carrier absorption

Kerr frequency comb generation relies on dedicated waveguide platforms that are optimized towards ultralow loss while offering comparatively limited functionality restricted to passive building blocks. In contrast to that, the silicon-photonic platform offers a highly developed portfolio of high-performance devices, but is deemed to be inherently unsuited for Kerr comb generation at near-infrared (NIR) telecommunication wavelengths due to strong two-photon absorption (TPA) and subsequent free-carrier absorption (FCA). Here we present a theoretical investigation that quantifies the impact of TPA and FCA on Kerr comb formation and that is based on a modified version of the Lugiato-Lefever equation (LLE). We find that silicon microresonators may be used for Kerr comb generation in the NIR, provided that the dwell time of the TPA-generated free-carriers in the waveguide core is reduced by a reverse-biased p-i-njunction and that the pump parameters are chosen appropriately. We validate our analytical predictions with time integrations of the LLE, and we present a specific design of a silicon microresonator that may even support formation of dissipative Kerr soliton combs.


I. INTRODUCTION
Generation of optical frequency combs in high-Q Kerr-nonlinear microresonators [1,2] has the potential to unlock a wide range of applications such as timekeeping [3], frequency synthesis [4], optical communications [5], spectroscopy [6] and optical ranging [7,8]. Amongst different frequency comb states, dissipative Kerr solitons (DKS) [9] are particularly attractive, offering broadband optical spectra with hundreds of phase-locked optical tones spaced by free spectral ranges of tens of gigahertz. As a key advantage in comparison to conventional comb sources built from discrete components, Kerr comb generators offer small footprint and can be integrated into robust chip-scale photonic systems that lend themselves to cost-efficient mass production.
So far, integrated optical Kerr comb sources have mostly been realized using specifically optimized silica and silicon-nitridebased waveguides that offer ultra-low propagation loss down to 1 5.5 dB m − along with anomalous group-velocity dispersion [10,11], and that can bear high power levels. The functionality of these integration platforms, however, is still limited to merely passive devices. In contrast to that, silicon photonics offers a highly developed portfolio of active and passive devices that are specifically geared towards operation at near-infrared (NIR) telecommunication wavelengths between 1200 nm and 1700 nm.
These devices can be reliably fabricated at low cost on large-area silicon substrates [12][13][14][15] and lend themselves to cointegration with electronic devices [16]. Expanding the silicon-photonic integration platform by monolithically integrated Kerr comb sources could have transformative impact regarding functionality, performance, footprint and cost of comb-based optical systems. However, the current understanding is that silicon microresonators are inherently unsuited for Kerr comb generation at NIR telecommunication wavelengths due to strong two-photon absorption (TPA) and subsequent free-carrier absorption (FCA) [17,18]. While simulations for specific parameter sets confirm this notion [17], a broad theoretical investigation of Kerr comb formation under the influence of TPA and FCA is still lacking.
In this paper we present a theoretical analysis of the impact of TPA and FCA on Kerr comb formation. We build upon an analytical model that expands the Lugiato-Lefever equation (LLE [19,20]) to include TPA, relaxation dynamics of TPA-induced free carriers, as well as dispersion anomalies of the ring resonances that arise as a consequence of avoided mode crossings. In a first step, we use the TPA coefficient as well as the lifetime and the absorption cross-section of the associated free carriers as central parameters and formulate simple necessary conditions that must be fulfilled for achieving modulation instability (MI) and subsequent comb formation. We describe the dependence of the MI threshold pump power on TPA and FCA parameters, and we find an upper limit for the TPA coefficient, above which comb formation is impossible even in absence of FCA. The theoretical predictions are independently confirmed by numerical simulations that are based on time-integration of the Lugiato-Lefever equation. This model is general and can be broadly applied to different material platforms. In a second step, we use our model to investigate silicon-photonic microresonators, in which the free-carrier dwell time can be artificially reduced by a reverse-biased p-i-n-junction that is built around the respective waveguide core. We find that Kerr comb generation in silicon microresonators can be achieved within technically realistic parameter ranges for free-carrier lifetime and pump power. We further develop and numerically validate a design for a silicon-photonic Kerr comb source that has a free spectral range (FSR) of 100 GHz and a threshold pump power of 12 mW and that should even be suitable for DKS comb formation.

II. MODEL
In our model, we describe the time-and space-dependent electric field ± Ω = ± ± ′ … and thus given as The imaginary unit is i. The field and the FC density obey periodic boundary conditions, Disregarding temperature effects, self-steepening, higher-order dispersion and higher-order multiphoton absorption, the LLE and the FC equation read [17,18] Figure 1: Silicon-photonic microresonator as an example of a device suffering from both TPA and FCA. The bus waveguide as well as the resonator ring waveguide are surrounded by p + -doped (red) and n − -doped regions (blue) that form a p-i-n-junction. This junction allows reducing the dwell time of free carriers by applying a reverse bias voltage U through contact pads and vias. At the coupling section between the bus and the ring waveguide, the doping is locally inverted from an n − -doping to a p + -doping next to the bus waveguide to ensure maximum free-carrier removal in the microresonator [23]. For the geometrical dimensions indicated in the cross-section, the waveguide features anomalous group-velocity dispersion (GVD) at wavelengths near 1550 nm [24]. In this case a ring diameter of 230 µm leads to a free spectral range of FSR 100 GHz f = , corresponding to a round-trip time In these relations, the quantity in E denotes the electric field amplitude of the pump with power photon absorption coefficient, σ is the free carrier absorption cross-section, and µ the free-carrier dispersion parameter which describes the influence of FC on the real part of the refractive index. The reduced Planck constant is ℏ . The model additionally includes a field confinement factor c Γ which takes account of the fact that only a fraction of the optical mode field experiences the attenuation by FC generated in the resonator waveguide [25]. Since we consider only modes for which the field is strongly confined to the waveguide, we may assume c 1 Γ ≈ , whereas other waveguide designs, e. g., slot waveguides with a nonlinear organic cladding [26,27] may lead values of c Γ that are significantly smaller than 1. Finally, we include the possibility of local resonance shifts δω ′ Ω caused by avoided mode crossings (AMC) [28][29][30]. These resonance shifts lead to additional phase shifts RT t δω ′ Ω for the respective electric field envelopes ( ) E t ′ Ω . The impact of AMC can hence be described by an operator AMĈ Φ acting on the envelope field ( ) To investigate under which circumstances modulation instability can occur when pumping the resonator, we need to know whether any pair of resonator modes experiences a sufficiently high parametric gain to overcome the total resonator loss. This loss includes linear propagation loss and coupling loss, which are equal for critical coupling, i L and a pair of weak fields a ±Ω ("sidebands" for short) in resonator modes which are offset from the pump frequency P ω by ( )

FSR
π ω ±Ω ± × × = Ω ( + Ω ∈ ℕ ) [31]. The amplitude of these sidebands may change with time with a complex normalized gain rate i i λ λ λ = + , , i λ λ ∈ ℝ , leading to a three-wave ansatz of the form We assume that the sideband amplitudes â ±Ω are initially much smaller than the amplitude of the pumped mode 0 a , 0 a a ±Ω << , such that we can treat them as a weak perturbation by linearizing Eqs. (5) and (6)  To identify resonator and pump parameters for which modulation instability can occur, we first derive an expression for λ in terms of these parameters. To this end, we first solve Eq.
Both the real part and the imaginary part of the gain parameter show a dependence on the normalized power A of the pumped mode. For a given normalized pump power F , the normalized power A in the pumped mode can be determined by evaluating the expression A derivation of Eq. (10) can be found in Appendix A.

III. PARAMETER RANGES OF TPA, FCA, AND PUMP POWER LEADING TO MODULATION INSTABILITY
Using Equations (9) and (10)  and 100. Note that the following investigation aims at identifying the dominant terms in Eq. (9) and that the symbol " ≈ " used is to be understood as an order-of-magnitude quantification rather than as an approximate equality.
To identify the side bands at which MI will occur first, we need to find values of the side-band offset Ω that maximize the gain rate λ . To this end, we simplify Eq. (9) by reducing it to its dominant terms, assuming that the offset of the MI-generated side-bands from the pump is of the order of 10 Ω ≈ . With the above-mentioned parameters, this leads to see Appendix B for details. For simplicity, we treat Ω as a continuous non-negative real-valued variable even though it was originally defined as an integer parameter + Ω ∈ ℕ . The sidebands experiencing the highest gain rate are then obtained from 2 eff Note that Eq. (12) implies an appropriate choice of the detuning ζ such that ( Note also that a strict derivation of max Ω by computing d d 0 λ Ω = will yield 0 Ω = as an additional local extremum, specifically a local maximum for ( ) and a local minimum for ( ) This extremum is not considered further, since the associated gain parameter ( ) 0 λ is always smaller than ( ) max λ Ω , which is given by Note that in absence of free carriers, i.e., eff 0 τ σ Note that the upper limit for r of 1 3 is exact and is in agreement with results also obtained from a bifurcation study of the LLE including TPA [32]. For the values of r and eff τ σ ′ ′ specified by Eq. (14), we use Eq. (13) to compute the minimum threshold power th A of the pumped resonator mode that is required to achieve MI, i.e., max 0 ( ) λ Ω > . The forcing th F required to achieve th A is then determined from Eq. (10). For maximizing the power transfer from th F to th A , the detuning is chosen − − by appropriate adjustment of the pump frequency, thus eliminating the expression marked by a star (*) in Eq. (10). For this detuning, maximum gain is found for modes with offset ( ) can be found as long as , which includes also complete absence of AMC, 0 0 φ = . In contrast to that, normal dispersion, , i.e., a spectral shift of the resonance caused by sufficiently strong AMC, leading to a real-valued max Ω . For real max Ω , Figure 2 displays the color-coded threshold forcing th F that is required to achieve MI as a function of the normalized TPA coefficient r and the free-carrier influence eff ' τ σ ′ . The color-coded map is limited to the ranges within which MI can be achieved, see Eq. (14), while the remainder of the plot is kept in grey. We find that th F increases continuously with increasing r and eff ' τ σ r and eff τ ′ without the approximations involved in Eq. (13). In this investigation, we again assume that AMC is absent, i.e., 0 φ ′ Ω = ∀ ′ Ω . The relative deviation of the threshold forcing found by the numerical evaluation from its analytically approximated counterpart stays below 1 % as long as the threshold forcing is not significantly larger than 10. We hence conclude that the simplified procedure leading to Fig. 2 can be considered sufficiently accurate. ( 1 2dBcm − ) [21]. Reference [21] reports on silicon-photonic waveguides in which free carriers are actively removed by a reverse-biased p-i-n-junction, leading to a dwell time of eff 12 ps τ = . In all other cases, no free-carrier removal was used, leading to dwell times of the order of 1 3ns … according to [17,21,37]. Specifically, a value of 3ns was used for the data points related to the references [33][34][35][36], which do not specify values for eff τ . The value for the FCA cross-section is consistently found in various publications [17,21,34]. The operating wavelength is 1550 nm .  [17,21,[33][34][35][36] and stay below the limiting value of 1 3 as given in Eq. (14). For simple silicon-photonic waveguides without active free-carrier removal, dwell times eff τ are of the order of 1 … 3 ns [17,21,37] , thereby clearly inhibiting MI. However, active free-carrier removal by a reverse-biased p-i-n-junction can effectively reduce the dwell time to values of, e.g., 12 ps [21], such that modulation instability and frequency comb formation then become possible at telecommunication wavelengths. The lower limit for eff τ σ ′ ′ that is achievable by active free-carrier removal is dictated by the saturation drift velocity FC v of the free carriers, which is of the order of 5 1 10 m s − for electrons in silicon [38]. For a microresonator with a waveguide width of 480 nm w = as shown in Fig. 1

IV. SILICON MICRORESONATOR FOR COMB GENERATION AT TELECOM WAVELENGTHS
In Figure 1 we show the structure of a silicon microresonator along with geometrical parameters that lead to anomalous group velocity dispersion around 1550 nm [24] and to a free spectral range (FSR) of 100 GHz . The geometry of the cross-sectional design for fast carrier removal is similar to the one used in [21]. The waveguide is undoped and is part of the intrinsic zone of a p-i-n-junction [23,39]. A reverse voltage applied through vertical interconnect accesses (vias) to the p + -doped (red) and n − -doped (blue) regions of the p-i-n-junction leads to efficient removal of free carriers such that dwell times of the order of the round-trip time can be achieved. Finite-element simulations yield a second-order dispersion parameter of processes [40].

V. SIMULATING THE DYNAMICS OF COMB GENERATION
With a specific resonator design at hand, we next perform a time integration of the LLE to validate our theoretical predictions on comb formation. We start our consideration from the microresonator design described in Section IV, which features a ( Δ 0.9 ps t = ). In each simulation, we perform 100 000 time steps for the slow time. The initial field for each simulation is given by 1024 complex numbers with random phases between 0 and 2π and random amplitudes between 0 and 14 10 − . The maximum amplitude of the initial field is chosen such that the power of the initial field is negligible compared to the forcing but can still start the evolution of the differential equation system. More details on the integration of the coupled Eqs. (5) and (6) can be found in Appendix E. We analyze four different cases, see Columns (a)-(d) of Fig. 3 for the results.
In Column (a) (Inhibited modulation instability), the parameters are set as follows: corresponding to anomalous dispersion, i.e., 2 0 β < , see Table 1. The simulation parameters are listed in Row R1 of Fig. 3.
The dispersion profile of the 101 central frequency comb modes, represented in normalized terms by ( ) The above-mentioned choice of forcing and dwell time ensures that the normalized gain rate ( ) Ω λ (with Ω Ω′ = ), Eqs. (9) and (10), is always negative, see Row R3. As a consequence, modulation instability cannot occur and the only mode with nonzero power is the pumped mode at modal index Ω 0 ′ = . In Row R4, the color-coded power spectrum , which has previously been demonstrated in a comparable siliconphotonic waveguide [21]. In this case, the computed normalized gain rate is positive within a certain range of modal indices Ω′ , see Fig. 3 Note that there is a slow drift of this pulse train within the retarded time frame τ ′ , Row R6. This drift is attributed to the fact that the retarded time frame is defined via the group velocity g v of the "cold" resonator. In the presence of free carriers in the "hot" resonator, FC dispersion increases the actual group velocity and leads to a residual time shift, which accumulates over subsequent round-trips and therefore grows continuously with slow time t′ . Mathematically, this behavior can be also seen from the terms  Fig. 3(c). Such a procedure allows generating single-soliton states [42]. In Row R3, the gain rate is depicted for 0 t′ = , i.e., before the detuning sweep, showing a broad range of modes that experience parametric gain. In the simulated slow-time evolution of the spectrum, Row R4, the first sidebands become visible around 70 t′ ≈ , and the spectral position of these sidebands coincides with the maxima of computed gain rate in Row 3. The maximum of the gain parameter in Column (c) is slightly smaller than the one obtained for the scenario described in Column (b), and thus the sidebands become only visible at a later normalized time t'. The final power spectrum obtained at the end of the simulation, Row R5, is a very regular frequency comb with a smooth envelope, which is typical for a single dissipative Kerr soliton [9] circulating in the ring.
The evolution of the color-coded intracavity power, shown in Row R6, reveals the emergence of multiple pulses at modulation instability onset around 70 t′ ≈ . Due to the swept detuning, most of the pulses diminish over time, which is an experimentally well observed behavior [42]. Note that, while increasing the detuning, the pulses disappear consecutively in direction of decreasing τ ′ , starting to the left of the finally remaining pulse, Row R6. This is caused by the fact that, within a sequence of pulses, the last pulse is always subject to the highest accumulated FC concentration and thus experiences that highest FCA loss.
The final value of ζ was chosen such that a single pulse remains in the cavity, which can be seen from the plot of the IC power  Table 1 and Appendix D for details. In contrast to that, normal dispersion can be achieved for a rather large parameter range of waveguide widths and heights, including rather large cross-sections for which multi-mode propagation and avoided mode-crossings may occur [24]. Silicon-photonic microresonators corresponding to the scenario considered in Column (d) may therefore by realized for multiple different waveguide geometries.

VI. DISCUSSION AND CONCLUSIONS
The simulation results indicate that modulation instability in silicon-photonic microresonators at telecommunication wavelengths is most likely to be observed if the waveguide is designed for anomalous group-velocity dispersion, if FCA is mitigated by a reverse-biased p-i-n-junction which leads to a sufficiently small carrier dwell time, and if the pump power and detuning are chosen properly. If an avoided mode crossing induces local dispersion shifts, microresonators with otherwise normal dispersion can also exhibit modulation instability. In both cases, moderate on-chip pump powers in the range of 10 mW to 50 mW are sufficient to initiate modulation instability and to generate frequency combs. The required pump powers depend strongly on the actual values of the TPA coefficient TPA β , which is expressed by the normalized TPA parameter r , and on the actual FC dwell time eff τ and its normalized counterpart eff τ ′ . We illustrate this dependence in Fig. 4 (a), where the on-chip threshold pump power th P for modulation instability is displayed as a function of the TPA coefficient with the FC dwell time as a parameter. We assume critical coupling, a resonator design and waveguide properties as described in Section IV, a Kerr coefficient of  Fig. 4(a). For larger dwell times, FCA prevents comb formation, which is indicated by the fact that the green and magenta lines in Fig. 4(a) do not enter the range of reported values for TPA β . We assume a silicon microresonator with an at , see Section IV for details. The microresonator is assumed to have waveguide loss, anomalous dispersion and critical coupling to the bus waveguide. Reported values of for silicon range from to as indicated by the vertical dashed lines. (b) Soliton comb spectrum obtained in the bus waveguide after the microresonator. The resonator is identical to the one considered in Subfigure (a). We assume , , and as indicated by a red circle in Subfigure (a). The spectrum is derived from the normalized intra-cavity comb spectrum indicated in Fig. 3(c), Row R5. For illustration, we calculated the physical frequency comb spectrum that is obtained in the bus waveguide after the microresonator, see Fig. 4(b). We again assume a resonator design and waveguide properties as specified in Section IV, along with a Kerr coefficient of  Fig. 4(a). The physical spectrum shown in Fig. 4(b) is derived from the normalized intra-cavity comb spectrum indicated in Fig. 3(c), Row R5.
Our findings suggest that silicon microresonators may indeed be a viable option for comb generation also at telecommunication wavelengths. We expect that additional effects such as higher-order dispersion, Raman shift [44] and selfsteepening may lead to minor corrections of the quantitative predictions, but should not change the qualitative behaviour. More accurate data for TPA β and achievable ranges of eff τ will also help to obtain a more precise estimate of threshold powers.
Moreover, several approaches may allow to reduce the predicted pump powers. Further improvements in waveguide fabrication may allow to reduce the linear propagation losses to values of, e.g., 1 0.4 dBcm − [45], which is well below the 1 2 dBcm − assumed in this work, but still far above the intrinsic absorption of silicon of less than 1 0.01dBcm − at telecommunication wavelengths [46]. As an alternative or an addition to reverse-biased p-i-n junctions, silicon self-ion implantation may be used to reduce of the FC dwell time at the expense of slightly increased waveguide losses [47]. Alternative waveguide concepts, e. g., silicon-organic-hybrid (SOH) waveguides [26,27] with a high Kerr nonlinearity of order of may also be used in a microresonator.
In summary, we have presented a theoretical analysis of the impact of nonlinear loss mechanisms such as TPA and FCA on Kerr comb formation. We derive the maximum two-photon absorption and free-carrier lifetime that still permits to achieve modulation instability at sufficiently low pump powers and that can thus lead to frequency comb formation. We show that silicon microresonators are not necessarily unsuited for Kerr comb generation at NIR telecommunication wavelengths, provided that the dwell time of the free-carriers in the waveguide core is reduced by a reverse-biased p-i-n-junction and that the pump parameters are chosen appropriately, and we present a specific design of a silicon microresonator with anomalous group- The normalized difference between the pump frequency P ω and the closest resonance frequency R ω of un-pumped resonator is ζ . The second-order dispersion of the microresonator is considered through the normalized dispersion parameter d . The quantity r denotes the normalized TPA-coefficient. For the TPA-generated FC, the quantities eff , , σ µ τ ′ ′ define the normalized absorption cross-section, the normalized contribution to the refractive index, and the normalized dwell-time inside the microresonator waveguide, respectively. The relationship between normalized parameters and physical quantities are summarized in Table 1 of the main manuscript. We further include an operator AMĈ ′ Φ describing phase shifts φ ′ Ω experienced by individual components ( ) a t ′ Ω ′ of the optical field for frequencies ′ Ω . These phase shifts take into account local resonance frequency shifts caused by mode coupling of different transverse modes, an effect known as avoided mode crossing [43]. The operator can be written as In our simulations shown in the main manuscript, the strength of the mode coupling, which defines the exact values φ ′ Ω , was chosen to reflect typical experimental results.
To investigate the condition for the onset of MI in Eq. (A1) the periodicity of a and c N ′ is exploited to determine the solution for Eq. (A2), Next, we a assume that the optical field consists of three waves [18]. Specifically, we assume a constant pumped mode 0 a and two small sidebands a ±Ω ( 1, 2,3, Ω = …), the temporal evolution of which is described by complex gain parameter i i e e e e e e , .
This result is now substituted in Eq. (A1) along with the ansatz for ( ) , a t τ ′ ′ , Eq. (A5). Again, we keep only terms up to linear order in â ±Ω or * a ±Ω . We separate non-oscillating terms from terms oscillating with i e τ Ω ′ and i e τ − Ω ′ , and obtain ( ) ( ) We subtract iâ φ +Ω We back-substitute ,1 A a = to obtain expressions for the real part and the imaginary port of the gain parameter, see Eq. (9) in the main manuscript. From Eq. (A7) we derive the relation between the pump power F and the power of the pumped mode A , see Eq. (10) in the main manuscript.

B. APPROXIMATION OF THE GAIN PARAMETER FOR TECHNICALLY RELEVANT VALUES FOR TPA, FCA, AND PUMP PARAMETERS
In Section III of the main manuscript we simplify the expression for the real part of the gain parameter λ given in Eq. (9)  10 Ω ≈ . Here we specify the magnitude of specific terms occurring in Eq. (9) of the main manuscript for the given parameter ranges. We consider two different cases eff 2 τ π ′ ≈ (left value below the respective term) and eff 20 τ π ′ ≈ (right value below the respective term), and we neglect all terms that are either at least three orders of magnitude smaller than competing terms or that are at least two orders of magnitude smaller and that are approximately constant in eff τ ′ . The remaining terms are highlighted in blue. In the following relations, the symbol " ≈ " used is to be understood as an order-of-magnitude quantification rather than as an approximate equality.

C. THRESHOLD PUMP POWER FOR MODULATION INSTABILITY: APPROXIMATE ANALYTICAL VS. NUMERICAL EVALUATION
In Figure 2 of the main manuscript, we analytically evaluate an approximation of the threshold pump power th F need to achieve modulation instability, based on realistic values for eff , , , , , r A τ σ µ ′ ′ Ω , and for anomalous group-velocity dispersion 0 We further assume that mode coupling is absent or sufficiently weak such that avoided mode crossings do not need to be considered ( 0 φ ′ Ω = ∀ ′ Ω ). In the following, we validate this approximation by a numerical investigation. To this end, we find the threshold  Figure C1(a) shows the result of the analytic approximation and corresponds to Fig. 2 of the main manuscript, while Fig. C1 is smaller than 2 10 − in the whole region, and smaller than 3 10 − for 0.4 r ≤ , see Figure C1(c). the numerical method. The deviation is smaller than in the whole region, and smaller than for a TPA parameter .

D. MODE FIELD SIMULATIONS FOR NONLINEARITY PARAMETER, GROUP REFRACTIVE INDEX AND DISPERSION PARAMETER OF SILICON-PHOTONIC WAVEGUIDES
To determine the waveguide nonlinearity and the dispersion of the waveguide presented in Fig. 1  squared of the electric field of the quasi-TE mode field within the silicon waveguide is shown in Fig. D1. The undoped silicon portion of the waveguide is marked with a white outline, and the mode is essentially confined to this region such that the optical field will not be affected by the p-i-n-junction and by the vertical interconnect accesses (vias) shown in Fig. 1  Here, P ω is the angular frequency of the optical pump field, 0 Z is the free-space wave impedance, and z e the unit vector along a perimeter of the ring-shaped waveguide. We assume that the third-order nonlinear susceptibility of silicon is a scalar and can be expressed by corresponding Kerr coefficient 18

E. INTEGRATION OF THE LUGIATO-LEFEVER EQUATION AND THE FREE-CARRIER EQUATION
In Section V of the main manuscript, we numerically integrate the normalized Equations (5) and (6) of the main manuscript using the splitstep Fourier method [49]. The normalized temporal time step is t′ ∆ , the dispersion operator is denoted by D , and the nonlinear operator is N : In these relations, the discrete frequency index 0, 1, 2, 3, u ± ± ± = … accounts for the various comb tones, which superimpose to a waveform ( ) y τ ′ which is periodic in fast time . We substitute Eq. (E2) in Eq. (E1) and integrate stepwise.

F. DISPERSION PROFILE OF COUPLED MODE FAMILIES
In this section we describe the derivation of the dispersion profile used in Section V of the main manuscript to simulate comb formation in a normal-dispersive microresonator under the influence of an avoided mode crossing, see Column (d) of   The dispersion profile of M1 is a set of discrete points located on a parabola, which is defined by the second order dispersion coefficient of mode family M1. It allows to compute the phase deviation ( ) ϕ ′ Ω accumulated by comb modes ′ Ω , see Eq. (15) in the main text. In contrast to the parabolic dispersion profile of mode family M1, the points given by the dispersion profile of M2 are located on a strongly inclined line, which is indicated in red in Fig. 3(d), Row R2 of the main manuscript. Unavoidable deviations from the ideal resonator geometry lead to coupling of the mode families leads and hence to a hybridization, which is accompanied by a local shift of the resonance frequencies from ,M1/M2 ω ′ Ω to two hybrid mode resonances , ω ′ Ω ± for each mode index ′ Ω [43]. This shift depends on the coupling strength θ between the respective modes , such that the hybrid mode resonance frequencies are given by [43] ( )

G. LIST OF PHYSICAL QUANTITIES. PARAMETERS FOR THE SIMULATION OF FREQUENCY COMB DYNAMICS
Tables G1 summarizes both the physical and the normalized quantities used for describing the dynamics of the optical field in the Kerrnonlinear microresonators. In Table G2, we specify physical and normalized microresonator parameters along with their numerical values and the associated source references.