1. Introduction

As first observed in 2007, microresonators can be used to generate optical frequency combs, i.e. regularly spaced optical laser lines, from a continous wave (c.w.) laser via parametric interactions [1–5]. Generation of such so-called Kerr frequency combs is the result of cascaded four-wave mixing processes [1,6]. The frequency spacing between comb lines is defined by the inversed round-trip time of light in the microresonator and typically may varies from tens of GHz up to THz. To date such frequency combs were demonstrated in many types of microstructures of different geometries made of different materials [7–17]. In contrast to the conventional mode-locked femtosecond laser based frequency combs [18], microresonator based Kerr combs are characterized, generally, by arbitrary phase relations between the comb lines that do not generally correspond to stable ultrashort pulses in time domain. It was demonstrated recently [19], that such parametric frequency combs can be operated in a regime where the formation of dissipative Kerr solitons (DKS) are possible. DKS correspond to a low-noise frequency comb having smooth envelope in the spectral domain [19–28]. This process was demonstrated experimentally in optical crystalline microresonators [19], integrated Si₃N₄ microrings [20] and silicon chips [21] with anomalous group velocity dispersion (GVD) by means of slow variation of pump frequency or by pump power pulsed modulation. However, it was found that the number of generated solitons is probabilistic and single-soliton generation is difficult to achieve directly (although techniques have recently been reported that enable to reduce deterministically the number of solitons, and prepare reliably the single soliton state). Single-soliton combs are of specific interest for many practical applications due to the highly coherent, spectrally smooth and low noise resulting frequency comb, and have already been succesfully applied to massively parallel terabit coherent communications, creating an RF to optical link using self referencing, dual comb spectroscopy [29] and dual comb coherent heterodyne receivers, as well as achieving self-referencing without external broadening.

Several approaches were proposed for deterministic single-soliton generation in optical microresonators. One method is based on the simultaneous tuning of pump frequency and power in order to avoid chaotic and unstable operating regimes [30]. However, the pump power variation may cause significant change of the thermal conditions that may complicate the process. Several methods of single-soliton generation based on thermorefractive effects [31–34] were demonstrated experimentally using a backward frequency scan [35] or an additional heater [36]. Single-soliton state was demonstrated also in a silicon integrated microresonator with electrical tuning of the free-carrier lifetime [37]. Harmonic pump modulation of the c.w. driving field was proposed as a tool for the manipulating of solitons [38]. It was also demonstrated that temporal cavity solitons in fiber ring cavities may be selectively excited or erased [39]. Since the round-trip time in a fiber loop is significantly larger than in a microresonator, the implementation of this approach in a microresonator meets significant difficulties. On the other hand, it was shown numerically that phase modulation of the pump at a frequency equal to the free spectral range of a microresonator provides a deterministic route toward soliton formation without undergoing a chaotic regime [40]. Our simulations reveal that this interesting method stands, unfortunately, only for a specific and very narrow frequency and power range. Generated pulses may also experience strong amplitude oscillations (breathers [24, 41, 42]). Amplitude modulation was shown to provide generation of platicons in microresonators with normal GVD [43].

Here we study the implementation of a harmonic phase and amplitude modulation for the reliable and efficient creation of single-soliton states but contrary to previous work we consider a simple modification of the frequency scanning approach that has already demonstrated its efficiency for soliton generation in optical microresonators [19]. The effect of phase modulation at frequencies much larger than the FSR on creation or annihilation of solitons was studied and experimentally demonstrated in fiber ring resonators [44], where it was used to suppress stimulated Brillouin scattering.

2. Modulated pump

We consider cases of harmonically phase f(t) = Fe^{iε sinΩt} and amplitude f (t) = F(1 + ε cos Ωt) modulated pump with modulation frequency Ω and modulation depth ε. Our numerical model is based on the system of dimensionless coupled nonlinear mode equations [45] modified to take into account the complexity of the pump field. Thermal effects are initially considered to be negligible. In simulations we neglect the frequency dependence of the nonlinearity, losses and mode-overlap, interactions with other mode families and any peculiarities of the resonator geometry. Assuming that pump modulation frequency is close to one free spectral range (FSR) we have:

Here a_μ is the slowly varying amplitude of the comb modes for the mode frequency ω_μ, $\tau =\frac{\kappa t}{2}$ denotes the normalized time, $\kappa =\frac{{\omega }_0}{Q}$ is the cavity decay rate, Q is the loaded quality factor, ω_p is the pump frequency; f_μ = FJ_μ (ε) for phase modulation and f_−1,0,1 = F[ε/2, 1, ε/2], f_{μ≠−1,0,1} = 0 for amplitude modulation with the dimensionless pump field amplitude F. J_μ (ε) is the Bessel function of the order μ, Δ = 2(D₁ − Ω)/κ is the normalized modulation frequency mismatch, D₁ is 2π×FSR. All mode numbers μ are defined relative to the pumped mode. We consider a Taylor expansion of the dispersion law ${\omega }_{\mu }={\omega }_0+D_1\mu +\frac{1}{2}{D}_2{\mu }^2+\dots$ and neglecting third-order dispersion we obtain the following expressions for the normalized detuning: ζ_μ = 2(ω₀ − ω_p)/κ + (D₂/κ)μ². Note, that D₂ > 0 corresponds to anomalous group velocity dispersion (GVD).

In our simulations we use initial weak noise-like inputs for seeding. Coupled mode equations (with optimally chosen number of simulated modes equal to 525 to balance simulation time and precision) are numerically propagated in time using the Runge-Kutta integrator. Nonlinear terms are calculated using the fast Fourier transform [46].

We’ve verified that results do not change with an increase of the number of modes taken into account (up to 1025) and smaller D₂ parameter, corresponding to wider frequency combs, although significant decrease of the considered modes noticeably changes statistics. For analysis we calculate average intracavity intensity $U={{\sum }_{\mu }\left|a_{\mu }\right|}^2$ for different values of normalized detuning ζ₀ and corresponding waveform $\psi (\phi )={\sum }_{\mu }{a}_{\mu }\text{exp}(i\mu \phi )$.

In order to observe soliton generation we scan adiabatically in time the pump frequency assuming that ζ₀(τ) = ζ₀(0) + ατ from the effective blue-detuned regime ζ₀(0) < 0, through the zero detuning frequency into the effectively red-detuned regime ζ₀ ≫ 0 to cover the whole range of soliton existence. As it was shown earlier soliton formation manifests itself in the appearance of characteristic step-like dependence. If frequency scan is halted within this step generated solitons continue to propagate in a stable manner. We take into account that maximal detuning value providing soliton existence may be estimated as [19] ζ_0max ∼ π²F²/8. In our simulations we consider F ≈ 4.11, D₂/κ ≈ 0.01 and ζ_0max ≈ 20.8. These numerical values correspond to real physical parameters of MgF₂ resonators with the effective mode volume V_eff = 5.6 × 10⁻¹² m³, dispersion D₂/2π = 0.398 × 10⁴ Hz, intrinsic quality factor Q₀ = 10⁹ at critical coupling for the wavelength λ = 1550 nm and pump power of 80 mW.

To collect statistics, we produce 100 realizations for each set of parameters and calculate the probability of the final generated number of solitons in each case.

It is known [19,35] that in the case of non-modulated pump (ε = 0) the number of generated solitons emerging from chaotic regime may vary from scan to scan. We confirm that single-soliton generation without special measures is quite a rare event [Figs. 1(a) and 1(b)] and the average number of generated solitons increases with increased pump power and decreased GVD value. The generated solitons propagate stably after frequency scan is halted. The most probable number of generated solitons may depend on scanning speed. Also using a single-soliton solution as an input we found that stationary solitons exist if ζ₀ > 8.46 and breather-like localized solutions appear for smaller values.

 

Fig. 1 Probability p of the final number of solitons distribution for different modulation amplitudes ε and scan speeds α. Final detuning is ζ₀ = 18 for the phase and ζ₀ = 15 for the amplitude modulation. In each case F ≈ 4.11 and 100 realizations were generated.

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Statistics changes radically if the weak resonant (Δ = 0) 1-FSR modulation is introduced. As it is shown at Figs. 1(a) and 1(b), if modulation depth is large enough (ε > 0.2), the probability distribution shifts radically to lower number of generated solitons. In the case of phase modulation only two nearly equiprobable results are possible for large depth: single-soliton generation and soliton absence. For amplitude modulation the alternative is mostly between one or two solitons (this may be attributed to the dominating influence on comb dynamics of neighboring sidebands).

This fact may be explained by soliton interactions under the impact of phase modulation. It may be shown using the LLE approach, that phase modulation results in soliton drift inside microresonator towards the equilibrium points [39]. Two solitons colliding at this point annihilate, so that the number of solitons decreases by 2. This behavior is clearly seen on Figs. 2(b)–2(d). Different scenarios of dissipative soliton interactions, including soliton merging and annihilation, were demonstrated earlier numerically and experimentally in nonlinear optical fiber resonator [47]. For our set of parameters practically in all cases we observed only the annihilation of colliding solitons in above-mentioned regime characterized by small modulation depths and moderate scanning speed. It means that even number of generated solitons results in the soliton absence, while odd number results in the single-soliton generation. In this way, to exploit this mechanism one may use phase modulation not during complete frequency scan, but only when the solitonic regime has been already reached, and, hence, obtains one or zero solitons depending on their initial number. Analogous annihilations of colliding solitons are observed in case of amplitude modulation Figs. 3(a)–3(c).

 

Fig. 2 Field distribution evolution inside microresonator for different values of phase modulation depth (α = 0.002, F ≈ 4.11): (a) ε = 0 (no modulation); (b,c) ε = 0.04; (d) ε = 0.08; (e) ε = 0.5; (f) fractional modulation frequency Ω = D₁/3, ε = 1.5, α = 0.0004, F = 5.

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Fig. 3 Field distribution evolution inside microresonator for different values of amplitude modulation depth (α = 0.002): (a,b) ε = 0.04; (c) ε = 0.08; (d,e) ε = 0.5; (f) ε = 1.0. Dashed line indicates the halt of frequency scan.

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Surprisingly, the situation changes if scanning speed α decreases. For phase modulated pump if α is small enough one may observe practically 100% probability of single-soliton generation if the modulation depth is large enough [Fig. 2(e)]. Moreover, critical value of the modulation depth diminishes with the decrease of scanning speed. This fact allows a solution for small diameter microresonator where FSR is too large for commercial phase modulator frequency range: one may use phase modulation with lower frequency but higher modulation depth so that odd multiple of modulation frequency coincides with the FSR (e.g. Ω = D₁/3). In that case resonant harmonics of modulation function (e.g. with numbers that are multiples of 3 for Ω = D₁/3) take the same form as in the case of resonant modulation. In that case one may estimate the effective modulation depth as ε_eff ≈ 2J₃(ε)/J₀(ε). However, in that case one should use higher pump power since only part of modulation harmonics are in resonance with the microresonator modes. One may estimate the effective pump value as $F_{\mathit{eff}}=F\sqrt{{\sum }_{\mu }{\left(J_{3\mu }(\epsilon )\right)}^2}$. The result of phase modulation with Ω = D₁/3 is shown in Fig. 2(f). The narrowing of soliton existence domain is the result of the decrease of the effective pump amplitude since in that case for ε = 1.5 F = 5 corresponds to F_eff ≈ 2.6.

It may be shown also using the LLE approach that the critical value of phase-modulation depth is inversely proportional to the square root of GVD value. Therefore, for smaller second-order dispersion stronger modulation for single-soliton generation is required.

In case of amplitude modulation the dynamics is more complicated. At weak modulation (for our set of parameters approximately up to ε ∼ 0.1 at α ∼ 0.001) it looks like the dynamics caused by the phase modulation with equilibrium point shifted from ϕ = π/2 to π [Figs. 3(a)–3(c)]. Number of generated solitons decreases with the growth of the modulation depth, but generated soliton decay at smaller values of ζ₀. In contrast to phase modulation case amplitude modulation changes the average value of pump power and may change the boundaries of soliton existence domain. One may notice in Fig. 3 that for different values of ε solitons decay at different values of ζ₀. For example, for the weak modulation subsequent growth of the modulation depth results in decrease of ζ_0max and moreover at some modulation depth value soliton generation is absent (e.g. ε = 0.3 at α = 0.001). If modulation depth increases more ζ_0max also increases and single-soliton generation may be observed in up to 80% of realizations. However, further growth of modulation depth results in appearance of two-soliton realizations and if ε > 0.6 the generation of two solitons is more probable than single-soliton generation [Figs. 3(d)–3(f)]. For high values of amplitude modulation depth one may observe soliton motion inside the microresonator [Figs. 3(d)–3(f)]. Surprisingly, this motion stops if frequency scan is halted [Fig. 3(f)].

Note, that for phase modulation increase of modulation depth results in more effective single-soliton generation, while for amplitude modulation one should use some optimal depth value (depending on scan speed) for preferable single-soliton output since large values may result in preferable two-soliton generation.

After stopping the frequency scan after stable soliton generation is reached, halting of the modulation preserves solitons if modulation depth value is small enough. If the modulation depth is significant, soliton profile is distorted and soliton extinction may be observed if detuning is close to the maximal value ${\zeta }_{0\mathit{max}}~\frac{{\pi }^2{F}^2}{8}$. To prevent such soliton decay at modulation halt adiabatic decrease of modulation depth was proposed [40].

One may notice that at low scan speed the number of transitions between states with different number of solitons decreases significantly [Figs. 2(d)–2(f)] and at large enough modulation depths such transitions are practically invisible. In this way for phase modulation the effect of single-soliton generation may be explained by the effect of soliton merging and not by soliton annihilation. Large modulation depths and small scanning speeds imply that all solitons reach the equilibrium point (and then interact) at comparatively small detuning values close to the lower boundary of the existence domain. As it was shown numerically and experimentally in fiber ring resonators [44] smaller detuning values provide greater probability of merging of colliding solitons than their annihilation. In this way, all generated solitons merge into one and single-soliton generation takes place. Note, that such merging is observed only in a very narrow frequency (detuning) range close to the transition from chaotic state to soliton/breather state. Therefore, to provide collisions inside this narrow range immediately after the formation of individual pulses, optimal combination of the scanning speed and modulation depth is necessary.

Importantly, such mechanism of single-soliton generation due to pump phase modulation may provide this effect for a wide frequency range (or wide range of detunings), which is impossible for the method without frequency scan proposed in [40]. Actually, for considered parameters scanning method with phase modulation provides single-soliton generation in a wide range of ζ₀ ∈ [8.5; 20.8] corresponding to nearly full soliton existence domain even for small modulation depths (e.g. for ε = 0.2 at α = 0.0005) while without frequency scan only localized breather-like states in a very narrow frequency range outside the soliton existence domain are possible. Namely, e.g. our simulations revealed ζ₀ ∈ [5.25; 5.28] for delayed phase modulation and ζ₀ ∈ [5.108; 5.122] for simultaneous phase modulation with ε = 1.0 (for larger modulation depths excitation domain becomes wider, e.g. ζ₀ ∈ [5.16; 5.75] for delayed phase modulation with ε = 2.0). To transform such breathers into solitons one should switch off the phase modulation adiabatically and then decrease the pump power gradually at fixed pump frequency [48].

Summing up, in order to obtain single-soliton regime by frequency scan method one should use scanning speed less than the critical value that depends on pump power and GVD value. One can easily calculate real scanning speed from normalized value α using simple formula: $\frac{d{\nu }_p}{dt}=\alpha \frac{\pi c^2}{2{\lambda }_p^2{Q}^2}$. For considered parameters normalized scanning speed α = 0.001 providing effective single-soliton generation at realistic values of phase-modulation depth corresponds to scanning speed of 236 MHz/s that is quite feasible. One should note, that since smaller values of GVD provide larger number of generated solitons, critical scanning speed decreases with the diminution of GVD (or larger modulation depths are necessary).

Results of numerical simulations show that for phase modulation further increase of modulation depth (above the value providing effective single-soliton generation) does not change the dynamics and does not decrease the efficiency of the process. However, for amplitude modulation in order to obtain more probable single-soliton generation one should use the modulation depth from the particular range.

3. Nonresonant modulation

We also studied nonresonant phase modulation introducing small mismatch Δ between modulation frequency and FSR (Δ is mismatch normalized to half-linewidth κ/2). It was found that effect of single-soliton generation is very sensitive to the mismatch value [Fig. 4]. Even for very small mismatch values (∼ one percent of linewidth κ or even less) one may observe not 1 soliton but 0, 1 or 2 solitons. The larger mismatch the more variants of possible soliton numbers exists. One may notice that large enough mismatch values (several percent of linewidth κ) prevent soliton collision and annihilation so that phase modulation becomes ineffective and multisoliton formation behaves in a similar way to the case without modulation [compare Figs. 4(a) and 4(c)]. In our simulations we found that critical mismatch value seems to be proportional to the modulation depth $\epsilon \left({\mathrm{\Delta }}_{cr}~\epsilon \left(\frac{D_2}{\kappa }\right)\right)$, that is quite similar with the formula obtained for the critical mismatch value for manipulation of temporal cavity solitons [38]. However, if detuning is present but significantly smaller than the critical value one may restore the single-soliton regime increasing the modulation depth [compare Figs. 4(b) and 4(c)] or decreasing the frequency scan velocity.

 

Fig. 4 Distribution of number of solitons generated by phase-modulated pump for different values of Δ and ε at α = 0.001, $\frac{D_2}{\kappa }\approx 0.01$. Final detuning is ζ₀ = 18: (a) ε = 0.3; (b) ε = 0.6; (c) ε = 1.0.

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The proposed method seems to be experimentally feasible mostly for microresonators with moderate quality factors, e.g. SiN microrings with κ ∼ 300 MHz than for high-quality crystalline microresonator with κ/2π ∼ 1 MHz. To control the modulation frequency value one may use the approach described in [49] or the sideband spectroscopy technique.

Numerical simulations also show that for amplitude modulation the effect is even more sensitive to the mismatch value. Two-soliton generation observed for large modulation depths [Figs. 3(e) and 3(f)] transforms in single-soliton generation even at mismatch values less than 0.002. For mismatch values above the critical value depending on modulation depth soliton generation was not observed.

4. Thermal effects

In real experiments Kerr nonlinearity is always overlaid by thermal nonlinearity caused by internal losses in combination with thermorefractivity and thermal expansion. Though this nonlinearity does not take part in soliton formation, it changes effective cavity detuning and in this way changes soliton existence range [35].

We study numerically the influence of the thermal effects on the efficiency of single-soliton generation by phase-modulated pump. To this purpose an additional equation for the normalized variation of temperature [33] $\mathrm{\Theta }=\left|\frac{1}{n}\frac{dn}{dT}+{\alpha }_T\right|\frac{2{\omega }_0}{\kappa }\delta T$, where α_T stands for thermal expansion coefficient [50], is solved simultaneously with conventional coupled mode equations. It was shown recently that this method provides adequate description of the soliton thermal dynamics [35]. Thus, the modified set of coupled mode equations reads:

In our simulations we consider positive sign of thermal nonlinearity coefficient n_2T > 0 providing thermo-optic locking [50]. At negative sign of this coefficient rapid growth of the average intracavity intensity leads to unstable regimes [32,33].

If thermal effects are strong enough, transition to solitonic regime may occur outside of single-soliton existence domain and in this case single-soliton regime may be obtained by backward scan only [35]. For example for considered parameters (F = 4.11) transition from chaos to breather/soliton regime occurs approximately at ζ₀ ∼ 6 and U_max ∼ 4.5, where U_max is intracavity intensity value before the transition to the soliton states [Fig. 2]. Thermally-induced shift may be estimated as $\theta \approx \frac{n_{2T}}{n_2}{U}_{\mathit{max}}\approx 4.5\frac{n_{2T}}{n_2}$ thus transition occurs at ${\zeta }_{0,T}\approx 6+4.5\frac{n_{2T}}{n_2}$. The boundary of single-soliton existence domain is ${\zeta }_{0,\mathit{max},T}=\frac{{\pi }^2{F}^2}{8}+\frac{n_{2T}}{n_2}{U}_{1\mathit{sol}}\approx 20.8+0.32\frac{n_{2T}}{n_2}$. Single-soliton generation is possible if ζ_0max,T > ζ_0,T thus thermal nonlinearity coefficient must be smaller than the critical value; for considered parameters the condition is the following: $\frac{n_{2T}}{n_2}<3.5$. We assume that this condition is satisfied.

However, even small values of the thermal nonlinearity results in decrease of single-soliton generation efficiency [Fig. 5(a)] and one may need larger modulation depths to overcome this obstacle [Fig. 5(b)]. We found that the efficiency of the process is very sensitive to the thermal relaxation time value [Figs. 5(c) and 5(d)]. Our simulations show that prevalent single-soliton generation may take place if this value is large enough and if thermal relaxation time is comparable with frequency scan time [Fig. 5(d)]. Fast thermal nonlinearity decreases the efficiency of single-soliton generation and results in two alternatives: one or zero solitons. Finite thermal relaxation time imposes limits on minimal scan speed rate and, consequently, on minimal possible modulation depth.

 

Fig. 5 Probabilities of realizations obtained via pump phase modulation resulting in generation of 1 or 0 solitons (a) vs. thermal nonlinearity coefficient, (b–c) vs. modulation depth and (d) vs thermal relaxation time. In all cases $\frac{D_2}{\kappa }\approx 0.01$, α = 0.002 and Δ = 0. Final detuning is ζ₀ = 18.

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If the relaxation time is large enough for effective single-soliton generation thermal effects may be compensated by higher modulation depths but smaller values of relaxation time makes this approach significantly less effective and may require significantly larger values of modulation depth (compare Fig. 5(b) for $\frac{n_{2T}}{n_2}=2$, 2/κτ_T = 0.001 and Fig. 5(c) for $\frac{n_{2T}}{n_2}=0.5$, 2/κτ_T = 0.1).

5. Experimental results

To verify our numerical simulations we fabricated MgF₂ microresonator using single point diamond turning. The resonator was then asymptotically polished with diamond slurries. Resonator major radius is 2.8 mm, minor radius – 35 microns. Light from a narrow linewidth fiber laser is coupled to microresonator with tapered fiber. The loaded cavity linewidth is 500 kHz. The experimental setup is presented on Fig. 6(a).

 

Fig. 6 Experimental measurement of soliton formation upon phase modulation and laser detuning. (a) experimental setup (AFG, arbitrary function generator; CW laser, continuous wave narrow linewidth tunable laser; FPC, fiber polarization controller, EDFA, Erbium-doped fiber amplifier; WGM, whispering gallery mode MgF₂ crystalline microresonator; FBG, Fiber Bragg grating filter; PD, photodiode; OSA, optical spectrum analyzer; OSC, oscilloscope); (b) statistics for 100 oscilloscope traces at scan repetition rate of 100 Hz for the output power versus laser detuning without modulation. (c) statistics of traces at 100 Hz with phase modulation, the zero soliton probability is 0.5, one soliton – 0.4, two solitons – 0.1; the inset shows one-soliton spectrum with sech²(x) envelope, spectrum width is 35 nm, line spacing is 12.1 GHz; (d) statistics of 100 traces with amplitude modulation, scan repetition rate is 5 Hz, zero soliton probability – 0.4, one soliton – 0.6.

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Pumping the resonator with 100 mW of power after EDFA at 1554 nm wavelength we observed characteristic steps in transmission corresponding to soliton formation. Corresponding single-soliton spectrum is shown as an inset on Fig. 6(c). We checked different laser sweep repetition rates (frequency scan speed) from 0.25 GHz/s to 25 GHz/s (α ≈ 0.0006...0.06). Simultaneously we applied either phase or amplitude modulation to the pump via EOM modulator. The depth of phase/amplitude modulation was approximately −26 dB (ε ≈ 0.1). For the analysis we collected 100 oscilloscope traces from photodetector for each measurement. Figures 6(b)–6(d) show overlaid traces demonstrating statistics of the final states as brighter curve colors correspond to larger number of overlaid traces. It was observed that the optimal value of modulation frequency providing the most effective single-soliton generation was set very close but not exactly to 12.1025 GHz – one “hot” cavity FSR which was measured by FSR beatnote measurements in soliton regime. The optimal detuning of modulation frequency from this FSR value was about 1 MHz. Deviation of the modulation frequency from the optimal value of 100 kHz leads to the disappearance of the effect. We explain this deviation by the change of the FSR due to thermal effects (thermorefractivity and thermal expansion): it means that the FSR of the “cold” microresonator considered in the model is different from the measured FSR of the “hot” microresonator. It can be seen from the obtained experimental data that both amplitude and phase modulation indeed radically change the probability distribution of final soliton states making single-soliton states accessible. In spite of good qualitative agreement with numerical simulations experimental results demonstrate some deviations from predictions due to not accounted for higher-order dispersion and thermal effects. Nevertheless, our experiments clearly showed that phase/amplitude modulation may provide more effective single-soliton generation that confirms the relevance of the proposed method.

6. Conclusion

In conclusion, we found using numerical simulations that effective single-soliton generation in optical microresonators with anomalous dispersion can be obtained by the frequency scanning method augmented with phase or amplitude pump modulation. We show that the efficiency of this approach depends on the frequency scan rate and the modulation depth. For effective single-soliton generation modulation frequency must be equal to one FSR, modulation depth must be large enough and frequency scan rate should be small enough. Thermal effects can significantly affect the efficiency of the process. The relevance of the proposed method was confirmed experimentally in MgF₂ microresonator.