Solitons and frequency combs in silica microring resonators: Interplay of the Raman and higher-order dispersion effects

The influence of Raman scattering and higher order dispersions on solitons and frequency comb generation in silica microring resonators is investigated. The Raman effect introduces a threshold value in the resonator quality factor above which the frequency-locked solitons cannot exist, and instead, a rich dynamics characterized by generation of self-frequency-shifting solitons and dispersive waves is observed. A mechanism for broadening the Cherenkov radiation through Hopf instability of the frequency-locked solitons is also reported.

Figure 1

(a) Soliton branches $\left(B_3=0\right)$, with and without the Raman effect, and bistability for $\delta =0.1,\gamma =0.02$ $\left(Q=8.5\times {10}^5\right)$, and $B_2=2.38\times {10}^{-7}$. Thick traces denote soliton stability. (b) Spatial and (c) spectral representation of solitons with $h=0.0045$ (gray) and $h=0.0089$ (black). Inset in (c): Several cavity modes around the pump wavelength. (d) Hopf instability growth rate for the solitons in (a). Inset: Raman soliton and Hopf mode spectra (linear scale) for $h=0.0089$. (e, f) Spatial soliton propagation for $h=0.0045$ (stable) and $h=0.0089$ (unstable), respectively. Dashed lines show soliton velocities calculated with the Newton method, demonstrating exact coincidence.

Figure 2

Soliton momentum branches vs loss with $\mu =0.18,B_3=B_4=\cdots =0$ for single-peak (black line) and double-peak [medium-gray (orange) line] solutions. Thick solid lines show stability regions and ”H” marks the onset of Hopf instability. The dashed vertical line at low $\gamma$ marks the threshold quality factor separating the unstable existence [see Fig. 3] and nonexistence [Figs. 3 and 3] regions. T's indicate turning points (local minima of $\gamma )$ that are not further explored in this work. Inset: Real (dashed lines) and imaginary (solid lines) spatial profiles of solitons around the double turning point: $\mathrm{A}\to B1,B2$.

Figure 3

Intracavity field after seeding with a pulse plus background: $\mu =0.18,h=0.003,\delta =0.1,B_2=2.38\times {10}^{-7}$, higher order dispersion = 0. (a–c) Space-time evolution for $\gamma =0.012,\gamma =0.005$, and $\gamma =0.0001$ (or $Q=1.4\times {10}^6,Q=3.3\times {10}^6$, and $Q=1.7\times {10}^8)$, corresponding to the excitation of stable FLR solitons, Hopf-unstable FLR solitons and SFSR solitons, respectively. (d) Spectral representation of (c). (i–iv) Spatial profiles corresponding to dashed lines in (c), illustrating the growth of the nonlinear-Schrödinger-like accelerating solitons. The field modulus, $\left|\psi \right|$ (black line), and its imaginary part, $\mathrm{Im}\left(\psi \right)$ [gray (red) line], are plotted. Positive [dark-gray (blue) area] and negative (light-gray) values of $\mathrm{Im}\left(\psi \right)$ illustrate where localized amplification and absorption are possible, respectively.

Figure 4

FLR soliton profiles in spatial (a–c) and (d–f) spectral domains for $\mu =0.18,\gamma =0.02,\delta =0.1,h=0.007$. (a,d) For the positive TOD, $B_3=2.48\times {10}^{-10}$, giving the quiescent soliton. (b,e) For the negative TOD $B_3=-2.48\times {10}^{-10}$. Insets in (a,b) show the Raman oscillator field, $w$. Solid lines in (d) and (e) correspond to spectra without the Raman effect. ${\mathrm{RR}}_{1\left(2\right)}$ label the resonant radiation (RR) roots. FWM peaks resulting from the interaction of pump and ${\mathrm{RR}}_1$ are also labeled. (c,f) $B_3=0.5\times {10}^{-10}$, corresponding to the oscillatory soliton in Fig. 6 at $T=920$. Solid vertical lines represent the predicted resonances; dashed vertical line, the zero GVD. Numbers in (f) correspond to $m$ in Eq. (9).

Figure 5

(a) Soliton velocity and (b) Hopf instability growth with $[\mu =0.18$; gray (blue) lines] and without $(\mu =0$; black lines) the Raman effect as a function of the third-order dispersion parameter. The curves in (a) and (b) emerge from those in Fig. 1 for $h=0.0055$ (dashed lines) and $h=0.007$ (solid lines): $\delta =0.1,\gamma =0.02$. (c) Phase-matching diagrams from Eq. (9) for $m\in [-5,5]$ (see Fig. 6 for parameters).

Figure 6

(a) Oscillations in time $T$ of the spectral maximum of a Hopf-unstable FLR soliton. (b) Fourier spectrum of the oscillations shown in (a), revealing multiple frequency components separated by the Hopf frequency ${\mathrm{\Omega }}_H$. (c) Periodic temporal dynamics of the spectrum of the Hopf-unstable FLR soliton core and of its radiation tails. The dashed vertical line shows the zero-GVD wavelength. (d) Spectrum of the radiation tail showing the formation of a frequency comb. Vertical lines show the prediction of new resonances from simple theoretical considerations [see. Eq. (9)]. $B_3=5\times {10}^{-11},\delta =0.1,\gamma =0.02,h=0.007,\mu =0.18$.

Figure 7

(a) Bistability vs detuning and soliton branches for $B_2=2.38\times {10}^{-7}$ and $\gamma =0.01$ $\left(Q=1.7\times {10}^6\right)$: $B_3=\mu =0$ (black line), $B_3=0,\mu =0.18$ [gray (red) line], and $B_3=\pm 2.48\times {10}^{-10},\mu =0.18$ (light-gray line). (b) GVD for two silica strands with cross-section diameters, $d=1.2\mu \mathrm{m}$ and $d=4\mu \mathrm{m}$ (see Figs. 8–10). Numerical simulations presented below use the full dispersion profiles in (b).

Figure 8

Comparison of chaotic (a, c, e, g) and solitonic (b, d, f, h) regimes for $\mu =0.18$ in the two waveguide profiles in Fig. 7: (a,b,e,f) $d=4$ $\mu \text{m},{\lambda }_p=1000$ nm, $\gamma =0.01$ $\left(Q=2.3\times {10}^6\right)$; (c,d,g,h) $d=1.2$ $\mu \text{m},{\lambda }_p=1470$ nm, $\gamma =0.01$ $\left(Q=1.7\times {10}^6\right)$. (a, c, e, g) ${\delta }_0=0.1$; (b, d, f, h) ${\delta }_0=0.16$.

Figure 9

Comb generation in the microresonator with a high quality factor $\left(Q=2.3\times {10}^8\right)$ and silica strand diameter $d=4\mu \mathrm{m}$: ${\lambda }_p=1000$ nm, $\delta =0.1,\mu =0.18,\gamma ={10}^{-4},h={10}^{-4}$. (a) Spatial intensity distribution and (b) spectrum evolution.

Figure 10

Comb generation in the microresonator with a high quality factor $\left(Q=1.7\times {10}^8\right)$ and silica strand diameter $d=1.2\mu \mathrm{m}$; ${\lambda }_p=1470$ nm, $h=\gamma ={10}^{-4},\mu =0.18,\delta =0.1$. (a) Spatial intensity distribution and (b) spectrum evolution.