Bright-soliton frequency combs and dressed states in chi(2) microresonators

We present a theory of the frequency comb generation in the high-Q ring microresonators with quadratic nonlinearity and normal dispersion and demonstrate that the naturally large difference of the repetition rates at the fundamental and 2nd harmonic frequencies supports a family of the bright soliton frequency combs providing the parametric gain is moderated by tuning the index-matching parameter to exceed the repetition rate difference by a significant factor. This factor equals the sideband number associated with the high-order phase-matched sum-frequency process. The theoretical framework, i.e., the dressed-resonator method, to study the frequency conversion and comb generation is formulated by including the sum-frequency nonlinearity into the definition of the resonator spectrum. The Rabi splitting of the dressed frequencies leads to the four distinct parametric down-conversion conditions (signal-idler-pump photon energy conservation laws). The parametric instability tongues associated with the generation of the sparse, i.e., Turing-pattern-like, frequency combs with varying repetition rates are analysed in details. The sum-frequency matched sideband exhibits the optical Pockels nonlinearity and strongly modified dispersion, which limit the soliton bandwidth and also play a distinct role in the Turing comb generation. Our methodology and data highlight the analogy between the driven multimode resonators and the photon-atom interaction.

Ring microresonators break through the traditional barriers of frequency conversion in terms of the power efficiency, generated bandwidth, and compactness [1,2]. Together with the rise of the microresonator frequency conversion, the Kerr-soliton frequency combs are reaching unprecedented for optical solitons levels of practical relevance [1,2].
Second-order, χ (2) , nonlinearity is a viable alternative to the Kerr one. It provides a much stronger nonlinear response but comes with the caveats of the need to care about the phase and group velocity matching to take full advantage of it. Refs. [3][4][5][6] have been among the first arXiv:2107.09111v1 [physics.optics] 19 Jul 2021 ones to demonstrate frequency conversion in the highquality factor whispering gallery microresonators with quadratic nonlinearity. Since then, this area has made a significant progress, see Ref. [7] for the few-year old overview, and Refs. [8][9][10][11][12][13][14][15] for some of the recent experimental contributions. It is also important to mention the work on the χ (3) dominant frequency conversion in the significantly mismatched χ (2) resonators, see, e.g., [16][17][18][19], and Appendix B for the relative weighting of the χ (2) and χ (3) effects. Material wise, lithium-niobate (LN) remains the best-established platform choice for the smallfootprint χ (2) photonics [8], with the silicon [20] and aluminium [15] nitrides gaining grounds with accelerating pace. The whispering gallery system considered below achieves finesses F ∼ 10 4 . The high finesse, is an important prerequisite for achieving conditions when the rate of the sideband generating photon energy exchange driven by the sum-frequency nonlinear terms exceeds the loss rate [21,22] and the parametric gain rate [23], which corresponds to the strong coupling regime between the ω and 2ω photons. As we have reported recently [23] and investigate in depth below, this makes the frequency conversion and soliton generation mechanisms to depart significantly from what has been known about these effects in the relatively low finesse resonators, F ∼ 10 2 , which often have no resonator-feedback at one of the generated harmonics, see, e.g., Refs. [24,25] for an overview. A couple of most obvious and important features of the high-F resonators is the channelling of the parametric gain into the tongue-like instability domains [23] and the strong-coupling associated with the dressed states [21] and polaritons [23].
Experimental results on the χ (2) -driven microresonator solitons are limited for now by the outstanding recent report by Bruch and colleagues [15] on the solitons due to parametric down-conversion (PDC) in the aluminiumnitride microring with the finesse 10 3 . The numerical data reported in Ref. [15] show the exponentially localised pulse in the half-harmonic field and the delocalised wave-form in the pump. Ref. [15] poses several problems, in particular, what are the physical mechanisms facilitating the compensation of the large group velocity, i.e., the repetition rate, difference between the pump and half-harmonic fields allowing for the soliton to form, and if having a better shaped pulse in the pump field is possible. Identifying conditions for the multicolour multi-pulse modelocked solitons to compensate the large group velocity differences is one of the classic problems [26,27], that needs to be addressed in the context of the high-Q microresonators (see Ref. [28] for the no-resonator, i.e., bulk propagation, analysis).
Results on the bright PDC solitons in the resonators with the group velocity offset published two decades ago [27] provided a conceptual answer, that the compensation of the group velocity difference is achieved via the balancing interplay between the dissipative and nonlinear effects. However, Ref. [27] was published well before the frequency combs and solitons in the micro-resonators have become possible. Therefore, it could not anticipate a combination of the small sizes and high Q-factors of the modern day devices, which leads to the strong quantisation of the spectra of the operators underpinning the frequency conversion and soliton formation processes. The model that fully reflects on these aspects has been recently presented in Ref. [29], which introduction also includes sufficiently comprehensive coverage of the recent and historic work on the dissipative χ (2) solitons in resonators.
Following our recent work on the theory of χ (2) microresonators [12,23,[29][30][31], we present here the latest findings obtained in the second harmonic generation (SHG) setting. One of the prime results included below is the demonstration of the bright soliton pulses in a microresonator which has the large repetition rate difference between the fundamental, ω p , and 2nd harmonic frequencies, 2ω p , see Figs. 15-17. The resonator considered here has the 20GHz repetition rate and the 1GHz rate difference, which implies that the linear ω p and 2ω p pulses would be on the opposite sides of the resonator only after about seventy roundtrips. The dispersion of the resonator is normal and its deviation from the bulk dispersion is insignificant. The dispersion value is −100kHz so that the linear pulse becomes twice as broad after 10 5 round-trips, and hence the repetition-rate difference is by far the most dangerous for the bright soliton modelocking, which nevertheless will be shown to exist across the broad and practical range of the system parameters.
The soliton combs reported below represent a pair of the ω p and 2ω p modelocked, and hence the repetition rate locked, bright pulses which existence and properties are derived and interpreted by examining the details of how the comb teeth, i.e., the frequency sidebands, are generated and interact via the sum-frequency and PDC nonlinear mixing processes. The detailed understanding of this problem has become possible thanks to the dressed resonator theory [23]. The content of what follows is much wider than just reporting the soliton modelocking, and it is now useful to give it a brief overview.

II. CONTENT AND RESULTS OVERVIEW
The coupling between the sidebands, i.e., the resonator mode pairs or the comb teeth, underpins the formation of any frequency combs and modelocking in optical resonators. Below we consider the microresonator 2nd harmonic generation, where the sideband coupling mechanisms embrace the complex interplay of the parametric process that couples the µ and −µ sidebands with the two sum-frequency processes. One is responsible for the coupling between the µ sidebands of the fundamental and 2nd harmonic, while the other does the same but for the −µ sidebands. The sum-frequency processes can be phase-matched for the select sidebands with the positive and negative numbers, µ = µ * or µ = −µ * , where µ * is the ratio between the phase-mismatch parameter and the repetition rate difference. The key resonator parameters and characteristics, including µ * , are illustrated in Fig. 1, and defined in Tables I and II. If the microresonator is tuned to operate away from the µ = 0 phase-matching, then the sum-frequency sideband coupling rate exceeds the parametric gain. It creates an opportunity to redefine the resonator spectrum by including the sum-frequency associated nonlinear terms. The new spectrum, i.e., the dressed spectrum, can be calculated analytically, bringing a close analogy with the dressing of the atomic transitions and the Rabi theory [21,23,32].
The condition of the maximal parametric gain [33][34][35], should then be redefined using the dressed frequencies [23]. The Rabi splitting leads to the four distinct PDC conditions, see Eq. (43). Knowing them allows understanding the complex structures of the parametric instability tongues and the Turing-pattern like frequency combs across the parameter space spanned by the pump laser frequency, ω p , and the intra-resonator power. The dressed modes around either µ * or −µ * exhibit dispersion with the inverted signs and the values significantly exceeding the bare-resonator ones. Therefore, only by making µ * to be sufficiently large, i.e., by taking it away from the soliton spectral core, creates enough of the modal bandwidth around µ = 0 with the low dispersion allowing the formation of the two-color bright soliton pulses. The large values of µ * are achieved by making the frequency mismatch parameter between the fundamental and 2nd harmonic modes to exceed the repetition rate difference by a significant, 1, factor, which works out to be the µ * itself.

III. LINEARISED SIDEBAND EQUATIONS AND CW-STATE
We assume the intra-resonator electric fields of the fundamental and 2nd harmonic to be where |ψ f,s | 2 have units of power [29], see Appendix A. M and 2M are the absolute (physical) mode numbers with frequencies ω 0f and ω 0s . ϑ ∈ (−π, π] is the angular coordinate and t is time. ω p is the pump laser frequency tunable around ω 0f , so that, is the respective pump detuning. The envelopes of the fundamental, ψ f , and second harmonic, ψ s , are expressed via their mode expansions as FIG. 1. A schematic illustration of the spectrum of the linear, i.e., bare resonator, Ω = 0, around the pump laser frequency, ωp, and its 2nd harmonic, 2ωp. The frequency mismatch parameter ε0 is tunable and could be comparable with D 1ζ .
with ζ = f, s, and µ is an integer characterising the relative mode number. The resonator frequencies associated with ψ ±µζ are where D 1ζ /2π are the repetition rates (free spectral ranges) and D 2ζ are the dispersions, see Fig. 1 for a schematic illustration. The photon angular momenta corresponding to ω ±µf and ω ±µs are (M ± µ), and (2M ± µ).
Physical values of the parameters implemented throughout this work are shown in Table I, and are typical for a bulk-cut LiNbO 3 resonator. Detunings of the resonator frequencies ω ±µζ from the pump laser frequency, ω p , and its 2nd harmonic, 2ω p , are ω ±µf −ω p and ω ±µs − 2ω p . A Galilean transformation to the reference frame rotating with the rate D 1 /2π, converts these detunings to where ε 0 = 2ω 0f − ω 0s is the µ = 0 frequency mismatch parameter. Setting we get an immediate access to the difference of the repetition rates, D 1s − D 1f , inside ∆ ±µs . If c is the vacuum speed of light, n M +µ is the effective refractive index of the mode M +µ, and R is the resonator radius, then Hence, and requiring ε 0 = 0, yields the anticipated index matching condition, n M = n 2M . Refractive index, and hence ε 0 , can be fine tuned by, e.g., temperature or electro-optic controls. CW state of the resonator operation implies ψ ζ = ψ 0ζ = const ζ , i.e., ∂ t ψ 0ζ = 0, see Fig. 2 and Appendix C. During the initial stage of the comb development, the sidebands evolve and grow on top of the undepleted cwstate, so that the field envelopes can be sought as the cw plus small perturbations, with the latter taken as a superposition of the resonator modes, Here ψ ±µζ are the amplitudes of the growing sidebands. Complex conjugation of ψ −µζ was introduced to make the equations to follow less cluttered. If Eq. (4) could be referred to as the mode expansion in the bare resonator representation, then Eq. (11) is a step towards the dressed resonator theory. Substituting Eq. (11) to the envelope equations (A1), and linearising for small sidebands, we find that ψ µζ (t) are driven by the sideband combinations with the matched net momenta, see Appendix D. The resulting equations are Here, κ ζ are the loaded linewidth parameters, and γ ζ are the nonlinear coefficients measured in Hz/W 1/2 [29], see Table I. Eq. (12) are linear in the approximation of the undepleted cw-state, and they represent one of the fundamental models in nonlinear optics expressing the interplay of the PDC and sum-frequency processes. The sumfrequency terms are underlined once and the parametric ones, describing conversion to the ±µ sidebands (photonpair generation), are underlined twice. The momentum conservation laws corresponding to the parametric conversion and the two sum-frequency processes are and respectively. The eigenvalues and eigenvectors of the matrix acting on the vector ( ψ µf , ψ µs , ψ −µf , ψ −µs ) T in the righthand side of Eq. (12) are known in the explicit form in two cases: (i) for ∆ µf = ∆ −µf , ∆ µs = ∆ −µs and κ f = κ s , see, e.g., Refs. [36][37][38], and (ii) for κ f = κ s and ∆ ±µζ = 0, see Ref. [39]. While ∆ µf = ∆ −µf is satisfied exactly, and κ f = κ s could be assumed, the ∆ µs = ∆ −µs condition is typically far from being true. In most of the practical cases, the repetition rate difference, ∆ µs − ∆ −µs = 2µ(D 1s − D 1f ), creates one of the dominant frequency scales in Eqs. (7), which can be comparable only to ε 0 .

IV. RABI FREQUENCY, PARAMETRIC GAIN AND LINEWIDTH
To address the problem at hands, it is important to understand the balances between the characteristic frequency scales implicated in Eq. (12). ψ 0s , i.e., the cw 2nd harmonic amplitude, can be controlled by tuning the frequency mismatch, ε 0 , so that the parametric gain coefficient, γ f ψ 0s , can be made to be much less than the sum-frequency associated rate of the energy exchange, γ ζ ψ 0f , see Fig. 2 and compare the frequency scales along the vertical axes.
If the aim is to create the modal bandwidth sufficient for the soliton generation at both harmonics, then the moderation of the parametric gain is useful since it would then keep the energy of the 2nd harmonic pulse under control, and, hence, could be expected to ease, for the stronger fundamental pulse, the task of synchronising its repetition rate with the 2nd harmonic.
The sum-frequency driven energy exchange is illustrated in Fig. 3, where one can see the fast anti-phase oscillations of the ω µf and ω µs sidebands. The frequency of the oscillations is much larger than the decay rates, so that, in the leading order, the first two equations in Strong-coupling condition κ ζ |Ω| 8|ε0| Sum-frequency matching ε±µ * = 0, µ * ≈ |ε0|/|D 1f − D1s| TABLE II. Definitions of the key parameters, which complement the ones illustrated in Fig. 1 and listed in Table I. ωp is the laser photon frequency. ω µζ and ω (j) µζ are the bare and dressed resonator frequencies, respectively (j1,2 = 1, 2, 3, 4; ζ = f, s).
Eqs. (12) can be approximated with [23,40] The second pair of equations resemble the above but with µ → −µ. Parametric gain and losses should then come in the next to leading order, suggesting the development of a perturbation theory. The Rabi theory, well known in the semi-classical atom-photon interaction [32], is an obvious and best suited methodology to describe solutions of Eq. (15). The Rabi formalism was also previously applied for the resonator-free sum-frequency generation model [41]. The frequency of the oscillations in Fig. 3 would then be the effective Rabi frequency, It is controlled by the difference of the sideband detunings, i.e., the Rabi detuning, and by the coupling coefficient, i.e., by the off-diagonal terms in Eq. (15), characterised by the complex Rabi frequency, Ω, The complex 2nd harmonic amplitude can also be expressed via Ω and the auxiliary complex frequency Ω s , see Appendix C. |Ω|/2π ∼ 10 2 MHz gives |ψ 0f | 2 1W, which would be typical inside the resonator. Then, for κ ζ /2π ∼ 1MHz we have |Ω| κ ζ , i.e., the Rabi flops are indeed much faster than the decay rate. The frequency scale associated with the parametric gain is set by Arranging |ε 0 | to be close or larger than the repetition rate difference, Hence, the Rabi frequency is also much larger than the parametric gain rate. Thus, both the linewidth and the parametric terms are small relative to the right-hand side of Eq. (15), and the strong-coupling (SC) condition [23], is satisfied. Ref. [21] reported measurements of the χ (2) Rabi splitting (∼ 1GHz for the laser power W = 80mW) in the AlN resonators with |Ω|/κ ζ ∼ 1, which should be well out-performed by the bulk resonators considered here. The µ = 0 Rabi oscillations in the χ (2) resonators were looked at in Ref. [44], well before the multimode high-Q χ (2) microresonators have been demonstrated. The notations related to the Rabi theory and also the key quantities used below to characterise the matching conditions for the sum-frequency and parametric processes are summarised in Table II.

V. SUM-FREQUENCY MATCHING
While the cascade of the sum-and difference-frequency events engaging a sequence of µ's is critical for the generation of the fully blown combs, see Section X, the frequency matching for the one-step sum-frequency process in Eq. (13) is also very important and should be analysed in more details. As it is well known for the coupled oscillators, the full periodic power transfer between ψ µf and ψ µs is ensured by minimizing the Rabi detuning, i.e., ∆ µ → 0, Ω µ → |Ω|, see Eq. (16).
Here, we are dealing with a system possessing two different effective Rabi frequencies Ω µ and Ω −µ , and the respective Rabi detunings, ∆ ±µ , can be re-expressed as If one of ∆ ±µ = 0 is resolved by an integer, i.e., it implies the exact matching for one of the two coexisting sum-frequency processes. The examples of the exact matching between ω −µf and ω −µs for µ = 10 and of the mismatched case are shown in Fig. 4(a). In the mismatched case, the SC condition, Eq. (23), is perfectly satisfied, but the power transfer during the Rabi flops is significantly reduced, cf., Figs. 3(b) and (c). If the real positive µ * solves one of the ∆ ±µ = 0 equations, i.e., then an integer, or two, nearest to µ * provide the sidebands order best complying with the frequency matching.

VI. DRESSED STATES
Equation (15) has an obvious and important class of solutions with the time-independent sideband powersdressed (eigen) states. The Rabi oscillations stem from a superposition of the dressed states. Dressing the states, i.e., working with the superpositions between ψ µf and ψ µs , rather than with the modes of the linear resonator, allows to develop a theory embracing the cases with the arbitrary (small, large or near one) ratios of the | ψ µs | 2 and | ψ µf | 2 powers [23].
In the SC regime, see Eq. (23), V is a perturbation to H µ , and therefore, before incorporating V , we look into the details of the dressed states, The eigenfrequencies, β The branches (1) and (2) describe the Rabi induced coupling between the +µ sidebands in the fundamental and 2nd harmonic, and (3), (4) do the same for the −µ sidebands. The corresponding dressed frequencies are Taking the explicit expressions for ∆ µ and ( µζ are the reference frame, i.e., D 1 , independent. Power distribution between the fundamental and 2nd harmonic sidebands within a given branch and for a given µ, is determined by the dressing parameters, ∆ 2 ±µ /|Ω| 2 . If the Rabi detuning is relatively large, i.e., a particular mode is far from being the sum-frequency matched, see the points away from the zero line in Fig. 4(a), then the corresponding dressed state, |b (j) µ , tends towards an eigenstate of the bare, i.e., Ω = 0, resonator. The bare states are where, for the sake of clarity, we explicitly associated each of the state vectors to the corresponding resonator mode.
The maximal dressing condition, ∆ ±µ = 0, involves frequencies of the bare resonator, while the matching points are replaced by the avoided crossings in the dressed resonator, see Fig. 4(b). The avoided crossing between ω (1) ±µζ and ω (2) ±µζ exist for every µ, however, most of them do not come to the practical, tens of MHz, proximity of the δ = 0, apart from the ones nearest to ±µ * .
Noting the symmetries we conclude that there are two ways to proceed from this point. First, the problem could be formulated using the four dressed frequencies, e.g., ω µf [23]. Second, one could switch to using the two-branch formulation and deal with ω −µf . The latter approach is slightly more intuitive and we choose to follow it here. In either case, the four frequencies and eigenstates have to be traced. The ω ±µf frequencies in the dressed spectrum are illustrated in Fig. 6. The ω ±µs spectra characterize the same dressed states as ω (1) ±µf , ω (2) ±µf . If the former are plotted then they would make the same spectrum as in Fig. 6 apart from being centred at 2ω p , see Eq. (39).

VII. DRESSED SPECTRUM AND ENERGY CONSERVATION IN PARAMETRIC-DOWN CONVERSION (PDC)
In any parametric system, the parametric resonance is achieved for, usually, a sequence of the resonance values The blue lines (numbered 5 to 11) correspond to the cross-branch PDCs, which are ω of the drive frequency [42]. In optical resonators in general, and in the dressed χ (2) system, in particular, this is done by tuning ω p to the mid-point between the desired sidebands, e.g., ω (1) µf and ω (1) −µf . One peculiar feature of our case is that the dressed resonances depend on the pump power and frequency. Another is that for the two pairs of frequencies, ω ±µf , there could be four different mid-points for the same µ, and hence four conditions providing the maximum of the parametric gain [23,43], 2 ω p = ω The auxiliary frequency ω µ = ω p +µD 1f + 1 2 (∆ µf +∆ µs ) makes the role of the Rabi splitting in the above more transparent, The first pair of conditions, Eqs. (42a), (42b), corresponds to the intra-branch PDC, which is satisfied by tuning the pump frequency to the mid-point between the ω (j) µf and ω (j) −µf . The second pair, Eqs. (42c), (42d), are the cross-branch PDC conditions. They are satisfied by ω p being tuned to the mid-point between the µ and −µ sidebands from the two different branches of the dressed spectrum. Figure 6 illustrates achieving frequency matching for the intra-and cross-branch cases, and also shows how the effective Rabi frequencies, Ω ±µ , come into play. Ω µ and Ω −µ are generally very different, and coincide only for µ = 0, see Fig. 5.
To gain further important insights into the PDC conditions in Eq. (42), we rearrange them as and plot ε (j1,j2) µ vs µ, see Fig. 7. The sideband numbers grouped around the zero lines correspond to the MHz level mismatches that are more easily compensated by the nonlinear effects providing ε (j1,j2) µ = 0. The sidebands with ε (j1,j2) µ detuned away from the zero by the GHz offsets (see the black dots in Figs. 7(a), (b)) are cutting-off from the groups of the PDC capable mode numbers. The cut-off is happening for µ's around µ * , corresponding to the sum-frequency matching, see Eq. (30). Figure 7(a) shows that the intra-branch condition in Eq. (42a) can be satisfied for 0 ≤ µ µ * , while ε 0 is set to be positive. The second intra-branch condition, Eq. (42b), see Fig. 7(c), is shifted away from zero by ≈ 2ε 0 , and will be swapped with the first one for ε 0 → −ε 0 . Figures 7(b) and 7(d) show the cross-branch PDC conditions, with one of them being satisfied for µ µ * .
The challenge with resolving Eq. (42) analytically for either Ω or δ is in the occurrences of them under the square root sign in the equation for Ω µ , see Table II. However, the algebra is proceedable [23,43], and leads to finding that all four PDC conditions are resolved by Plots of |Ω pdc | vs δ for positive and negative ε 0 , and their associations with the PDC conditions are shown in Fig. 8. These plots in Fig. 8 could be compared with the temperature tuning diagrams of the parametric oscillators, see, e.g., [33][34][35], but in our case, the temperature FIG. 9. Parametric down-conversion (PDC) instability tongues mapped onto the parameter space span by the laser detuning, δ, and the Rabi frequency, |Ω| (see the left axis), or, and equivalently, by the intra-resonator cw power, |ψ 0f | 2 (see the right axis). (a) is for ε0/2π = 5GHz, and (b) is for ε0/2π = −5GHz. The grey shaded tongues correspond to the intra-branch PDC conditions marked by the red lines in Fig. 8, and the blue tongues -to the cross-branch PDCs. Some of the tongues are marked with the respective µ's, see Fig. 8 for the complete illustration of the µ ordering. The magenta lines show |Ω| = |ψ 0f | √ 8γ f γs vs δ achieved for the laser powers W = 66µW, 10mW, and 100mW.
is assumed fixed, while the tuning parameters are the pump power expressed via |Ω|, and the pump frequency. The range of |Ω|s considered by us provides the relatively small intraresonator powers, order of mW to < 1W, cf., the left and right axes in Fig. 9.
If |ε 0 | dominates over all δ's, and µ * falls between the two nearest integers, i.e., the exact sum-frequency matching point has been missed, then Eq. (45) simplifies to see Appendix E for details.
Eq. (46) reveals what has been described above based on the numerical plots. First, one can see that the resonances converge to points δ = δ µf for Ω → 0, which corresponds to the zero of the first bracket in the numerator of Eq. (45). Second, the direction of the nonlinearity induced tilts of the resonances depends on the sign of ε 0 and the value of µ, see Fig. 8. If ε 0 < 0, then the tilt is towards δ > δ µf for 0 ≤ µ < µ * , and towards δ < δ µf for µ > µ * . ε 0 > 0 changes the tilt direction for the two groups of modes. Figure 8 also shows a good agreement between the exact and approximate |Ω pdc | 2 vs δ dependencies. Analytical approximation for |Ω pdc | in the case when the sumfrequency process is either nearly or exactly matched, i.e., µ = µ ≈ µ * , is considered in Appendix E.
To summarise -the intra-branch PDC conditions are satisfied for a compact group of the sideband numbers, while the cross-branch ones are engaged for the unbound set of sidebands, Thus, the sum-frequency matched sideband, µ * or −µ * , defines the transition between the two different PDC scenarios.

VIII. PDC INSTABILITY TONGUES
While the PDC frequency matching provides conditions for the maximal parametric gain, the latter still needs to overcome the dissipation in order to trigger the exponential growth of sidebands, i.e., to induce the cw instabilities. Regions of the PDC instabilities for every ±µ pair of sidebands can be computed numerically by solving the eigenvalue problem in Eq. (36) and plotting the lines λ µ = 0 [23]. Every µ-specific instability area is represented by a tongue-like domain shaped around the respective |Ω pdc | vs δ line, cf., Figs. 8 and 9.
The intra-branch instabilities, Eqs. (42a) and (42b), are coloured in grey in Fig. 9. While, the cross-branch ones, Eqs. (42c), (42d), are shown in blue. The magenta lines show |Ω|'s vs δ corresponding to the cwstate achieved for three representative values of the laser power, W = 66µW, 10mW, and 100mW. The cw-state is expressed via Ω as per Eq. (18), while Ω itself is a solution of where W is the laser power in Watts, and W * is its scaling, see Eq. (C5). Taking modulus squared of Eq. (49) we find a real cubic equation for |Ω| 2 , that can have either one or three positive roots, with the latter case signalling the cwbistability, see Fig. 2, and further details in Appendix C.
Thus, a scan of the laser frequency, ω p , would go along an individual power-defined path crossing the different PDC domains, see the magenta lines in Fig. 9. The yellow shading marks the µ = 0 tongue embracing the middle branch of the bistability loop, see Fig. 2. For the Kerr resonators, the similar tongue diagrams were recently reported in Refs. [46,49].

IX. PARAMETRIC THRESHOLDS
PDC thresholds, i.e., the minimal intra-resonator powers triggering the exponential growth of the ±µ sideband pairs, happen at the tips of the instability tongues, see Fig. 9. To find the threshold when the system is confined to the |µ|-specific PDC lines, see Figs. 8, 9, we apply the degenerate state perturbation theory to Eq. (36) by treating V as a perturbation to H µ , which is valid in the SC regime. The generic condition for the parametric gain to overcome losses has been introduced in Ref. [23], where V Eqs. (51) express the balance between the PDC gain (right) and the net loss (left). The 2nd harmonic losses, κ s are weighted by the coefficients characterising the power distribution between the components in the state vectors, see Eq. (38). The explicit threshold condition presented in Ref. [23] transforms to Eq. (51c) after making use of the identity |Ω| Ωµ−∆µ = Ωµ+∆µ |Ω| . We now note that in the SC regime |Ω s | ≈ 8|ε 0 |, see Eq. (22), and hence the right-hand sides in Eq. (51) are approximated with |Ω| 4 /16|ε 0 | 2 . To simplify the threshold conditions we make use of the approximations in Eq. (31) related to the case of ε 0 > 0 and Ω − µ ≈ |Ω|, see Fig. 5(a). Then, for µ < µ both coefficients after κ s in Eq. (51a) are small and can be omitted in the leading order, so that the threshold is determined by κ f only and is well approximated by, see the grey tongs in Fig. 9(a). For µ > µ the left-hand side of Eq. (51a) becomes ≈ 4κ f κ s |∆ −µ | 2 /|Ω| 2 , which creates prohibitively large power thresholds, see the cutoff transitions in Fig. 7. For µ = µ, the threshold is approximated by κ s is now also impacting the threshold, but still in a way that is not equally important with κ f . This is because the powers of the fundamental and second harmonic are balanced only for ω − µf and ω − µs , but not for ω (1) µf and ω (1) µs sidebands. In the latter pair, the sum-frequency condition is mismatched, and hence the 2nd harmonic sideband is still very weak and can be disregarded, i.e., ∆ µ ≈ Ω µ in Eq. (51a). While, the approximations that work in the minus bracket are ∆ − µ ≈ 0, and Ω − µ ≈ |Ω|.
The analytical estimates for the detuning values where the µ-specific instabilities first happen, i.e., locations of the tips of the instability tongues and the respective laser powers are derived in Appendix F.

X. ENVELOPE AND COUPLED-MODE EQUATIONS FOR MODELOCKED COMBS
We anticipate that the frequency comb solutions bifurcate from the µ-specific boundaries of the instability tongues in Fig. 9, cf., Ref. [46]. Since we are going to continue to number the sidebands within the combs with µ, we use below the letter ν = 1, 2, 3, . . . to mark the comb states with the sideband spacing given by ν. We seek the modelocked combs as the solutions of the equation that couple all the modes through all the allowed nonlinear coupling terms, see Eq. (A1) in Appendix A.
The modelocked combs are assumed to have the period 2π/ν, i.e., ψ ζ (t, ϑ) = ψ ζ (t, ϑ + 2π/ν), and, therefore, we use the substitution Here θ ν is a new auxiliary coordinate, such that the period 2π in θ ν corresponds to the period 2π/ν in θ, Ψ νζ (θ ν ) = Ψ νζ (θ ν + 2π). D 1ν is an unknown comb repetition rate generally different from either D 1f or D 1s . If the reference frame is chosen to rotate with D 1f , then D 1ν = D 1f would imply the relative rotation with the D 1ν − D 1f rate, leading to the tilted spatio-temporal profiles like in the bottom row in Fig. 10 The selection mechanisms of the velocity of the dissipative χ (2) solitons, equivalent to the selection of D 1ν , have been discussed before the microresonator combs came into the existence [27]. This selection is a generic aspect also encountered in, e.g., the equations with the higher- order dispersion terms [45], and in the cases showing the spontaneous symmetry breaking effects [46]. Substituting Eq. (54) into Eq. (A1) we find As one can see, the use of θ ν as an argument has allowed to conveniently sort the modelocked combs by the spacing, ν, their sidebands make in the momentum space, since ν enters Eq. (55) explicitly. In fact, ν plays a role of the Bloch momentum which is now quantised, unlike the one that varies continuously in the theory of the unbound crystal lattices [47] and resonators [48].
For the frequency combs with the spatial period 2π/ν, the modes making non-zero contributions to the Ψ νζ have numbers µ = νm, where m = 0, ±1, ±2, ±3, . . . is another integer, Here Ψ mνζ are constants satisfying an algebraic system of equations, see Eq. (57). The repetition rate with which the 2π/ν state is reproducing itself while rotating in the resonator is νD 1ν /2π.
The coupling between the different mν sidebands is provided by the sequence, i.e., cascade, of the sumfrequency and difference-frequency events, which become  Fig. 11(a). The bottom row is the turbulent state at the point 'e' in Fig. 11(a). The 1st and 2nd columns compare the spatio-temporal dynamics (fundamental) and snapshots of the spatial profiles. The 3rd and 4th columns show the RF spectra of net powers (3rd), |F F T µ |ψ µf (t)| 2 |, and the per-mode RF spectra (4th), |F F T {ψ µf (t)}| 2 . One can see that the breather state consists from the five coherent sub-combs with the different offsets and the same repetition rates, while the turbulent state shows no modelocking signs. evident on substituting Eq. (56) into Eq. (55), The sideband detunings, ∆ mνζ , are defined in Eq.

XI. TURING-PATTERN COMBS
The comb equations, Eq. (55) or Eq. (57), have been solved by us with a Newton method allowing to selfconsistently find the sideband amplitudes, Ψ mνζ , and the comb repetition rate, D 1ν . Figure 10 shows how the comb branches with different periods, 2π/ν, emanate from the boundaries of the respective instability tongues.
A typical bifurcation scenario that we found is that the two Turing-comb branches split from the two edges of the tongues and extend well beyond the tongues. The branch crossing into the tongue area can be stable (full lines in the second row in Fig. 10), while the one deviating outside the tongue (dashed lines) is always unstable. These two branches coalesce for some detuning value, see Fig. 10(e). The Turing combs emerging from the intrabranch PDC conditions for ν = 4, 5 are shown in the first two columns in Fig. 10, and the ν = 16 cross-branch case is shown in the third column.
The ν = 5 tongue in Fig. 10(b) corresponds to the ω p + ω −5f − ω −5s = 0 sum-frequency matching, see Eq. (24), and therefore, the power of the −5th sideband in the 2nd harmonic is either comparable or even exceeds the one in the fundamental, see Figs. 10(e), (h), and compare the y-axis scales in (d),(e), and (f). The bottom line in Fig. 10 shows how the Turing patterns loop around the resonator in the reference frame rotating with D 1f , so that the most efficient generation of the 2nd harmonic leads to the largest differences between the linear and nonlinear repetition rates, D 1f − D 1ν , cf., the pattern angles in Figs. 10(j), (k), and (l). Figure 11 shows a more global outlook on the frequency conversion processes happening across the PDC tongues when the laser frequency is scanned and its power is fixed, and ε 0 > 0. The stable Turing-combs are typically generated for the relatively small intra-resonator powers achieved for large positive detunings. When detunings are reduced and the intra-resonator powers are increasing the instabilities bring the breather states producing the denser combs as in Figs. 11(f),(g),(i), and then more developed chaotic states as in Figs. 11(d), (e).
The difference between the breathers and chaos is further elucidated in Fig. 12. The first and 2nd columns in Fig. 12 compare the spatio-temporal and spatial profiles of the two. The coherence of the breather and the incoherence of the turbulent state are confirmed by comparing the RF spectra of the net powers (3rd column in Fig. 12), and the per-mode RF spectra (4th column). The latter shows |F F T {ψ µf (t)}| 2 plotted vs the mode number, µ, and the RF frequency. The top, i.e., the breather, panel has five visible tilted lines that are vertically separated by the breather period. These are the five sub-combs having different offset frequencies.
In the example shown in the top line of Fig. 12, the maximal parametric gain comes to the sidebands with µ = ±15 and to the associated sub-combs, see the top panel in the last column, so that the breather could be interpreted as a quasi-soliton created via the nondegenerate PDC and dominated by the three groups of modes centred around µ = −15, 0, 15. The power and per-mode RF spectra in the 2nd line of Fig. 12 are unambiguous about the absence of the inter-mode coherence in the turbulent state. For studies of the transitions between the breather and chaotic states associated with the rogue-wave turbulence in the Kerr resonators, see, e.g., Refs. [59][60][61].

XII. BRIGHT SOLITON PREREQUISITES
The strong-coupling regime and the dressed state that naturally emerge from it, imply that the parametric gain is relatively small, and therefore, to the leading order, the soliton existence problem could be approached through the prism of the nonlinearity vs dispersion balance. In this section, we work through the dispersive and nonlinear properties of the dressed states, and the stability of the cw state that underpins the existence of the bright soliton frequency combs. For normal dispersion, D 2ζ < 0, as we have here, these properties suggest the soliton existence for ε 0 > 0, which is the case we describe in details below. Solitons themselves are introduced in Section XIII.

A. Dispersion of dressed states
The microresonator dressed states are parametrised by momentum µ, and represent families of the quasiparticles, photon-photon polaritons, with the effective mass inversely proportional to the polariton dispersion [23]. Our focus here is on the frequency conversion, and therefore the dispersion terminology is more natural. Dispersions of the first and second branches in the   dressed spectrum are calculated as where, Ω µ = Ω µ+1 + Ω µ−1 − 2Ω µ . Ω µ depends on |Ω| (i.e., on the sum-frequency nonlinearity), D 2ζ , and on the repetition rate difference, D 1f − D 1s , which means that the dressed state dispersion is determined by the interplay of all three factors. This is unlike the barestate dispersion, trivially given by ∂ 2 µ ω ±µζ = D 2ζ . For ε 0 > 0, none of the ω (1) µζ frequencies is sumfrequency matched, while ω (1) −µζ are quasi-matched for µ around µ * , see Section V and Fig. 5. Therefore, the dressed states corresponding to ω (1) µζ are the quasi-bare states, |b (1) µ ≈ |1 , with the dispersion ≈ D 2f , see the red line in Fig. 13(a). The dressed states corresponding to ω  −µ ≈ |3 for µ < µ * , then, in the proximity of µ * , they hybridise to the maximally dressed state, |3 e iφ − |4 , and then transform to the quasi-bare |4 states.
It is instructive to evaluate the dressed dispersion for µ around µ * , where ∆ −µ ≈ 0, Ω −µ ≈ |Ω|, see Eqs. (16) ,  FIG. 14. (a) Parametric instability tongues in the (δ,|Ω|) plane, for ε0/2π = 25GHz, cf., ε0/2π = 5GHz in Fig. 9(a). The colorbar shows the number of the simultaneously unstable sidebands, N . The black doted line embraces the µ = 0 instability range, i.e., the middle branch of the bistability loop. N is clamped by µ * = |ε0|/|D 1f − D1s| = 25, N ≤ µ * . (b) N vs δ for three values of W (laser power) computed along the upper branch of the cw-state. The magenta line (W = 333µW) terminates at the tip of the cw resonance, i.e., at the end of the bistability range. The other two lines are for the higher powers and terminate before the tip, thereby marking stabilisation of the cw-state. (17), The above strongly dominates over D 2ζ , so that, the dispersion of ω −µζ becomes large anomalous. Thus, dressing the ω (1) −µζ branch creates two zero dispersion points around µ * and puts a group of modes into the range of anomalous dispersion, see Fig. 13(a).
The dressing induced dispersion changes are very large, see Eq. (60), so that for the nonlinearity to compensate the dispersive pulse spreading it would be preferential to engage the modes with the bare resonator dispersion around µ = 0, and away from µ * . Thus keeping µ * 1 and increasing it by tuning the index to make |ε 0 | larger, see Eq. (30), is expected to increase the bandwidth of modes suitable for the bright soliton modelocking regime. In other words, the repetition rate difference can be large, but the sum-frequency matching to the appropriately large sideband number puts the undesirably large dispersion this difference creates away from the spectral core of the soliton.

B. Optical Pockels and cascaded-Kerr nonlinearities
Since we are expecting to work with the normal dispersion, the bright solitons would require the negative, i.e., defocusing, nonlinearity to compensate for it. Derivatives of ω (j) ±µζ in |Ω| describe the rate of the nonlinear shifts of the frequencies in the dressed spectrum with the intra-resonator power, The signs of the above expressions determine the signs of the effective nonlinearities experienced by the sidebands.
Since the resonator frequencies are inversely proportional to the refractive index, see Eq. (9), we conclude that the frequencies ω (2) ±µζ in the dressed spectrum experience the net positive (self-focusing) nonlinearity, and ω (1) ±µζ -the net negative (defocusing) nonlinearity.
Though the branch nonlinearities do not change signs, their dependence on |Ω| varies profoundly. Indeed, Ω ±µ admits two qualitatively different Taylor series expansions, see Table II, Thus, if µ is near to µ * , then the nonlinear shift of ω (j) ±µζ frequency is proportional to the amplitude of the cwstate, |Ω| ∼ |ψ f |, which corresponds to the optical Pockels effect, see Eq. (62a). While, for µ away from µ * , the nonlinear shifts are proportional to the power, |Ω| 2 ∼ |ψ f | 2 , and hence are the Kerr-like, i.e., correspond to the cascaded-Kerr effect, see Eq. (62b).

C. CW-state stability and instability
From the above, one should conclude that, for ε 0 > 0, the first branch of the dressed states should be considered as the one able to form the bright solitons. Since it provides a combination of the defocusing nonlinearity and the relatively small normal dispersion, except several µ's around µ * , where dispersion is anomalous.
The same condition, as the one just stated, leads to the intra-branch instability of the cw-state, see and compare Figs. 9(a) for ε 0 /2π = 5GHz, and 14(a) for ε 0 /2π = 25GHz. Recalling the results of Section VII, see Eq. (47), the number, N , of the sidebands that can be simultaneously unstable under these conditions is clamped by µ * . Therefore, and also in line with the previous subsection, ε 0 /2π = 25GHz brings more of the unstable sidebands. Fig. 14 shows that there exists the optimal laser power, W, achieving the maximal N .
N going up and then down with δ tuned more negative and powers increasing, see Fig. 14(b), is due to the nonlinear shifts becoming saturated by the higher order terms in the expansions of the root function in Ω ±µ . The weakened nonlinear shifts gradually bring the parametric gain below the threshold first for some and finally for all the sidebands. The N vs δ dependencies in Fig. 14(b) would be very different in the Kerr resonators, where the number of the unstable sidebands tends to infinity with the simultaneous increases of δ and W [49].
It is now important to note that, for δ < 0, the above discussed instabilities happen to the upper state of the cw bistability loop, while the low branch is either exclusively or largely stable, see Fig. 14(a). The stable lowamplitude cw-state makes the background for the bright solitons reported in the next section.

XIII. BRIGHT SOLITON FREQUENCY COMBS
The results of the previous Section let us to conclude that, for the normal dispersion of the bare resonator modes, D 2ζ < 0, the bright solitons are expected providing one arranges the index/frequency matching parameter between the M and 2M resonator modes to be positive, and sufficiently large, so that the sum-frequency matching or near-matching, happens for The detuning should than be tuned to δ < 0, i.e., 'blue' detuning. µ * is well approximated by the ratio between |ε 0 |, and the repetition rate difference, |D 1f − D 1s |, see Eq. (30). We now do as prescribed and select the laser power, W, and detuning, δ, corresponding to a large number of the simultaneously unstable sidebands, see Fig. 14. Initialising Eq. (A1) with the cw-state we immediately observe its instability and the subsequent formation of the stable two-colour soliton, see Fig. 16.
To trace the soliton families in the parameter space we solve the comb equations in Section X. Since the teeth of the single soliton combs follow with step one, i.e., their spatial period is 2π, we set ν = 1 in Eqs. (55), (57). Families of the bright solitons traced in δ for the laser power W = 333µW and for a range of frequency mismatch parameters, ε 0 /2π ∈ [21,29]GHz are shown in Fig. 15. The unstable soliton branch (dashed red and green lines) split from the middle branch of cw state (dashed magenta lines) at the point of its µ = 1 instability. The stable soliton branches extend outside the cw-bistability towards more negative δ's.
The soliton profiles along the resonator circumference shown in Figs. 15, 17 are characterised by the tails oscillating with the period 2π/µ * . The corresponding spectra have pronounced peaks at µ = −µ * , where the powers of the fundamental and 2nd harmonic sidebands are balanced due to the sum-frequency matching. The background of the soliton in the fundamental field is set primarily by the µ = 0 sideband, see spectra in Fig. 15(d), where µ = 0 is stronger than µ = −µ * . Therefore, the fundamental background drops with δ becoming more negative, cf., the correlated changes of the full CW line in Fig. 15(a) and of the soliton background in Fig. 15(c). The background of the soliton in the 2nd harmonic is, however, set primarily by the µ = −µ * sideband, see spectra in Fig. 15(f), where µ = −µ * is the strongest and its power correlates with the soliton peak power. This explains why the 2nd harmonic soliton background goes up with δ becoming more negative, cf., the dropping full CW line in Fig. 15(b) and the increasing soliton background in Fig. 15(e).
The oscillatory soliton tail should be interpreted as due to the inability of the nonlinear effects to compensate for the sharp rise and the sign change of the dressed state dispersion around µ * , see Fig. 13. Therefore, the nature of the tails here is similar to the soliton Cherenkov radiation in the resonators [40,62,63], fibers [64], and bulk crystals [65]. We note, that, in the present case, the dispersion can be significantly altered by the pump power dependent state dressing, which provides a more flexible tool to control the radiation frequency. Figure 17 shows how the soliton families change with the tuning of ε 0 . Here the structure of the families appears to be more complex. The period of the oscillations of the soliton tail, 2π/µ * , can change only discretely while |ε 0 | is tuned continuously, so that the solution is forced to accommodate itself, as much as it can, to the rigid period of its tails, which leads to the discrete set of the soliton families. Each family is centred around the value of |ε 0 | = µ * |D 1f − D 1s |, we recall that |D 1f − D 1s |/2π = 1GHz. The tail oscillations of the Kerr solitons in microresonators with the large higher-order dispersions also lead to somewhat similar 'quantized' behaviour of the soliton parameters [45].

XIV. DISCUSSION
There are numerous open problems left for a researcher tempted to understand the multimode dynamics of the high-Q χ (2) microresonators by looking into properties of the individual modes, in line with the present-day exper- imental capabilities. The extension of our results to the case of the exact index matching ε 0 = 0 while keeping D 1f − D 1s large requires separate consideration. Extending the bandwidth of the soliton combs by taking the shorter resonators with the higher repetition rates and lower quality factors, and, perhaps, larger µ * needs to be investigated.
The power induced dispersion engineering of the dressed states offers a new method to control the comb widths and shapes. Observations of the solitons, the Rabi splitting and the sum-frequency matching associated spectral features predicted here are of course important and can be attempted with the available resonators and the index matching control tools. We note, the relatively high conversion efficiency from the pump to the soliton-comb spectra seen in Fig. 17, which could be an important practical aspect requiring further exploration.
We should recall here the prior theoretical work on the spatial [55,57] and temporal [30,38] resonator solitons achieved via the 2nd harmonic generation arrangements requiring the exactly or near matched phase, ε 0 = 0, and group, D 1f = D 1s , velocities. While a combination of these assumptions appears as the desirable idealisation in the contexts of the currently available ring microresonators, future studies along these lines are warranted. The relevance of the strong-coupling and dressed states for the microresonator half-harmonic generation arrangement remains to be analysed, including their links to the results on the half-harmonic bright-bright, dark and dark-bright soliton pulses [15,27,31,58,[66][67][68][69].

XV. SUMMARY
(1) The theoretical framework, i.e., dressed-resonator method, to study frequency conversion and solitons is formulated by including the sum-frequency nonlinearity into the definition of the resonator spectrum.
(2) The Rabi splitting of the dressed frequencies leads to the four distinct PDC conditions, see Eq. (43), which are used to explain the existence and generation of the sparse non-soliton, i.e., Turing-pattern-like, frequency combs.
(3) The effective nonlinearity and dispersion of the dressed states have been used to demonstrate that the microresonator with the normal dispersion and naturally large difference of the repetition rates at the fundamental and 2nd harmonic frequencies, D 1f − D 1s , supports a family of the bright soliton frequency combs, see Figs. 15, 17. Conditions for this are provided by tuning the index/frequency matching parameter, ε 0 = 2ω 0f −ω 0s , to be positive and large, so that it exceeds the repetition rate difference by a significant factor, µ * = |ε 0 |/|D 1f − D 1s | 1. µ * or −µ * approximate the mode number associated with the phase-matched sum-frequency process and set limits on the soliton bandwidth.

XVI. ACKNOWLEDGEMENT
This work was supported by the EU Horizon 2020 Framework Programme (812818, MICROCOMB).
Here, Ω f,s are defined after Eq. (49), and Taking the modulus squared of Eq. (C4) we find the real cubic equation for |Ω| 2 , A useful insight into the cw properties is provided by taking the limit when |ε 0 | is large relative to |δ|, and |δ| is large relative to κ ζ , so that Ω s ≈ −8ε 0 and Ω f ≈ δ. Then, bistability of the cw-state requires δε 0 < 0 (the square bracket in Eq. (C4) can be zero). In this regime, implying that the resonance shifts proportionally to the pump power, i.e., in the same way as it happens in Kerr effect. It means, that the cw-state behaves as it would in the Kerr resonator. For ε 0 = 0 or small, Ω f ≈ δ, Ω s ≈ 16δ. Then, the cw becomes simultaneously bistable for δ > 0 and δ < 0 [30,38], and max δ |Ω| 2 δ 2 , i.e., max δ |Ω| |δ|.
This is the Pockels regime of the cw-state, when the nonlinear change of the refractive index is proportional to the first power of the field amplitude, which we do not consider in this work. Conditions for the µ = 0 sidebands to be in either cascaded-Kerr or Pockels regimes are different and discussed in Section XII B.
Appendix D: Linearization around the cw-state In order to develop a theory of the growth of the multisideband signal, i.e., frequency comb, on top of the cw solution, ψ 0ζ , we extend Eqs. (A1) by a pair of the complexconjugated equations [70], and seek a solution in the form If ϑ is the angle measured along the resonator circumference, then θ = ϑ − D 1 t is the coordinate in the rotating frame, see Eq. (6). ψ ±µζ (t) are the sideband amplitudes. Summing up the first and second lines gives the net signals in the fundamental and 2nd harmonic, respectively, with the third and fourth lines being their conjugated, see Eq. (11). Substituting Eq. (D1) into the extended Eq. (A1), we assume smallness of the sideband amplitudes, linearise, and then derive equations for the individual sidebands using the angular momentum matching. For a given µ this leads to a coupled system of the four ordinary differential equations for ψ µf , ψ −µf , ψ µs , and ψ −µs , see Eq. (12).
Eqs. (F1a), (46), (52) For µ = µ, the procedure is the same. In order not to overcomplicate the answer, we impose a plausible condition (D 2f + D 2s ) = 3D 2f , leading to δ µs + δ µf = 3δ µf . Then, Eqs. (F1b), (E5), (53) yield Transparent analytic estimates for the laser powers W at the tips of the instability tongues can be worked out after observing that along the tails of the nonlinear resonances, see Fig. 2, the square bracket in Eq. (C4) is ≈ 1, see Eq. (C5) for W * . The balance of terms in Eq. (F3) is such that the root term dominates and δ µf can be neglected, which gives the following estimate for the power In Eq. (F2), the two terms are balanced for the moderate µ's leading to a longer equation not included here, but for µ 2 µ 2 * , the second term can be neglected, so that Thus, the powers to generate the sum-frequency matched sideband, µ = µ, scale with |ε 0 | 2 , and the ones for µ = µ with |ε 0 |, cf., Eq. (F5) and Eq. (F6).