Bright-soliton frequency combs and dressed states in ${\chi }^{\left(2\right)}$ microresonators

We present a theory of frequency comb generation in 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 second-harmonic frequencies supports a family of bright soliton frequency combs provided 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 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 analyzed in detail. The sum-frequency matched sideband exhibits optical Pockels nonlinearity and strongly modified dispersion, which limit the soliton bandwidth and also play a distinct role in Turing comb generation. Our methodology and data highlight the analogy between the driven multimode resonators and the photon-atom interaction.

Figure 1

A schematic illustration of the spectrum of the linear, i.e., bare, resonator, $\mathrm{\Omega }=0$, around the pump laser frequency, ${\omega }_p$, and its second harmonic, $2{\omega }_p$. The frequency mismatch parameter ${\varepsilon }_0$ is tunable and could be comparable with $D_{1\zeta }$.

Figure 2

Cw state of the (a) fundamental and (b) second harmonic (see Appendix pp3). Pump laser power is $\mathcal{W}=0.1\mathrm{mW}$ [see Eq. (49)]. Mismatch parameter ${\varepsilon }_0/2\pi =-5\mathrm{GHz}$.

Figure 3

(a) The sum-frequency matched Rabi oscillations near to the $\mu =5$ avoided crossing (red, fundamental, left axis; green, second harmonic, right axis), ${\mathrm{\Omega }}_5\approx \left|\mathrm{\Omega }\right|\approx 2\pi \times 68\mathrm{MHz}$. (b) As in (a) but plotted over the time interval as is used in (c). (c) The mismatched Rabi oscillations for the $\mu =4$ sidebands, ${\mathrm{\Omega }}_4\approx \left|{\mathrm{\Delta }}_4\right|\approx 2\pi \times 1\mathrm{GHz}$. Parameters: (a, b) $\delta =15.15{\kappa }_{\text{f}}, {\varepsilon }_0/2\pi =-5\mathrm{GHz}, \mathcal{W}=1.7\mathrm{mW}$ (pump power), $|{\psi }_{5\text{s}}{|}^2=0.36\mathrm{mW}$ (sideband power) at $t=0$; (c) $\delta =6.45{\kappa }_{\text{f}}, {\varepsilon }_0/2\pi =-5\mathrm{GHz}, \mathcal{W}=2.11\mathrm{mW}, |{\psi }_{4\text{s}}{|}^2=0.33\mathrm{mW}$ at $t=0$.

Figure 4

(a) Frequency matching for the sum-frequency generation: The Rabi detuning ${\mathrm{\Delta }}_{-\mu }={\omega }_p+{\omega }_{-\mu \text{f}}-{\omega }_{-\mu s}$ vs the sideband order number $\mu , \delta =5{\kappa }_{\text{f}}$. ${\varepsilon }_0/2\pi =10.5$ and $10\mathrm{GHz}$ correspond to the mismatched and near-matched ($\mu =\widehat{\mu }=10$) cases, respectively. (b) Avoided-crossing diagram: The dressed frequencies vs $\delta$ for $\left|\mathrm{\Omega }\right|=20{\kappa }_{\text{f}}$. The straight lines correspond to $\mathrm{\Omega }=0$.

Figure 5

${\mathrm{\Omega }}_{\mu }$ and ${\mathrm{\Omega }}_{-\mu }$ Rabi frequencies vs $\mu$. (a) ${\varepsilon }_0/2\pi =10.5\mathrm{GHz}$, and (b) ${\varepsilon }_0/2\pi =-10.5\mathrm{GHz}, \delta =14{\kappa }_{\text{f}}, \left|\mathrm{\Omega }\right|=300{\kappa }_{\text{f}}$. ${\mathrm{\Delta }}_{-\mu }={\omega }_p+{\omega }_{-\mu \text{f}}-{\omega }_{-\mu \text{s}}$ is nearly matched for $\mu =10$ in (a), and ${\mathrm{\Delta }}_{\mu }={\omega }_p+{\omega }_{\mu \text{f}}-{\omega }_{\mu \text{s}}$ is in (b).

Figure 6

Diagram illustrating the dressed spectra. (a) Pump frequency tuned to satisfy the intrabranch PDC condition, $\hslash 2{\omega }_p=\hslash {\tilde{\omega }}_{\mu \text{f}}^{\left(1\right)}+\hslash {\tilde{\omega }}_{-\mu \text{f}}^{\left(1\right)}$. (b) How the cross-branch PDC condition is satisfied, $\hslash 2{\omega }_p=\hslash {\tilde{\omega }}_{\mu \text{f}}^{\left(2\right)}+\hslash {\tilde{\omega }}_{-\mu \text{f}}^{\left(1\right)}$. The red and blue arrows show how the PDC gain and Rabi flops redistribute power within the spectrum. The black dotted lines show the Rabi frequencies, ${\mathrm{\Omega }}_{\mu }$ and ${\mathrm{\Omega }}_{-\mu }$, and highlight their inequality.

Figure 7

Frequency matching parameters ${\varepsilon }_{\mu }^{\left(j_1{j}_2\right)}={\tilde{\omega }}_{{\mu }_1\text{f}}^{\left(j_1\right)}+{\tilde{\omega }}_{{\mu }_2\text{f}}^{\left(j_2\right)}-2{\omega }_p$ for the four types of the parametric down conversion (PDC) conditions. The sideband numbers grouped around the dashed horizontal lines, ${\varepsilon }_{\mu }^{\left(j_1{j}_2\right)}=0$, correspond to the MHz mismatches that can be compensated by the nonlinear effects and lead to the exact PDC frequency matching (see insets). (a, b) $\delta =2.55{\kappa }_{\text{f}}, \left|\mathrm{\Omega }\right|/2\pi =76\mathrm{MHz}$. (c, d) $\delta =8.1{\kappa }_{\text{f}}, \left|\mathrm{\Omega }\right|/2\pi =105\mathrm{MHz}$, and ${\varepsilon }_0/2\pi =10\mathrm{GHz}$. $\mu \approx 10$ (see the corner points) corresponds to the sum-frequency matching, ${\mathrm{\Delta }}_{-\mu }\approx 0$.

Figure 8

Lines corresponding to the exact PDC frequency matching conditions in the parameter space of the pump detuning, $\delta$, and of the Rabi frequency, $\left|\mathrm{\Omega }\right|$, characterizing the intraresonator cw power (see Fig. 2). The red lines (numbered 0 to 5) correspond to the intrabranch PDCs, which are $\hslash {\tilde{\omega }}_{\mu \text{f}}^{\left(1\right)}+\hslash {\tilde{\omega }}_{-\mu \text{f}}^{\left(1\right)}=\hslash 2{\omega }_p$ in (a) where ${\varepsilon }_0/2\pi =5\mathrm{GHz}$, and $\hslash {\tilde{\omega }}_{\mu \text{f}}^{\left(2\right)}+\hslash {\tilde{\omega }}_{-\mu \text{f}}^{\left(2\right)}=\hslash 2{\omega }_p$ in (b) where ${\varepsilon }_0/2\pi =-5\mathrm{GHz}, 0\le \mu \le 5$. The blue lines (numbered 5 to 11) correspond to the cross-branch PDCs, which are $\hslash {\tilde{\omega }}_{\mu \text{f}}^{\left(1\right)}+\hslash {\tilde{\omega }}_{-\mu \text{f}}^{\left(2\right)}=\hslash 2{\omega }_p$ in (a), and $\hslash {\tilde{\omega }}_{\mu \text{f}}^{\left(2\right)}+\hslash {\tilde{\omega }}_{-\mu \text{f}}^{\left(1\right)}=\hslash 2{\omega }_p$ in (b), $\mu \ge 5$. The lines in (a) and (b) use Eq. (45), while the dots in (a) are derived from the approximate Eq. (46) for $\mu \ne 5$ and Eq. (E5) for $\mu =5$.

Figure 9

Parametric down conversion (PDC) instability tongues mapped onto the parameter space spanned by the laser detuning, $\delta$, and the Rabi frequency, $\left|\mathrm{\Omega }\right|$ (see the left axis), or, and equivalently, by the intraresonator cw power, $|{\psi }_{0\text{f}}{|}^2$ (see the right axis). (a) ${\varepsilon }_0/2\pi =5\mathrm{GHz}$. (b) ${\varepsilon }_0/2\pi =-5\mathrm{GHz}$. The gray shaded tongues correspond to the intrabranch PDC conditions marked by the red lines in Fig. 8, and the blue tongues correspond to the cross-branch PDCs. Some of the tongues are marked with the respective $\mu$'s (see Fig. 8 for the complete illustration of the $\mu$ ordering). The magenta lines show $\left|\mathrm{\Omega }\right|=|{\psi }_{0\text{f}}|\sqrt{8{\gamma }_{\text{f}}{\gamma }_{\text{s}}}$ vs $\delta$ achieved for the laser powers $\mathcal{W}=66\mu \mathrm{W}, 10\mathrm{mW}$, and $100\mathrm{mW}$.

Figure 10

Examples of the Turing-pattern frequency combs for ${\varepsilon }_0/2\pi =5\mathrm{GHz}$. First and second columns show the combs emerging from the intrabranch PDC instability tongues [gray areas in (a) and (b)], and the third column shows the cross-branch PDC tongue [blue area in (c)] and the associated comb. Magenta lines in (a)–(c) show the cw states, $\left|\mathrm{\Omega }\right|=|{\psi }_{0\text{f}}|\sqrt{8{\gamma }_{\text{f}}{\gamma }_{\text{s}}}$ vs $\delta$. The respective laser powers (a) $\mathcal{W}=0.664\mathrm{mW}$, (b) $103\mathrm{mW}$, and (c) $10\mathrm{mW}$. $\left|\mathrm{\Omega }\right|/2\pi =0.2\mathrm{GHz}$ corresponds to the intraresonator power $|{\psi }_{0\text{f}}{|}^2\simeq 55\mathrm{mW}$. (d–f) Ratio of the second harmonic and fundamental powers in the indicated sidebands. The $\mu =-5$ case corresponds to the sum-frequency matching leading to the best conversion efficiency. The thick full lines are stable solutions, and the thin lines (full and dashed) are unstable. (g–i) Self-explanatory spectra of the Turing patterns (red, fundamental; green, second harmonic). (j–l) Space-time profiles of the corresponding Turing patterns. The tilt relative to the vertical axis characterizes deviation of the pattern repetition rate from $D_{1\text{f}}$.

Figure 11

(a) PDC instability tongues for $\mu =12–18, {\varepsilon }_0/2\pi =5\mathrm{GHz}$ as in Fig. 9. The magenta line shows the cw state for the laser power $\mathcal{W}=10\mathrm{mW}$. The left axis is the Rabi frequency, $\left|\mathrm{\Omega }\right|$, and the right axis is the intraresonator cw power, $|{\psi }_{0\text{f}}{|}^2$. (b) The spectrum computed from the scan of the cw frequency along the magenta line in (a). (c) The integrated intraresonator powers corresponding to (b). The red line (left axis) is the fundamental, and the green line (right axis) is the second harmonic. Panels in the two bottom lines show the comb spectra computed at the detunings marked with the respective letters in (a), and with the dashed white lines in (b). In the two bottom rows, the shorter bars are plotted in green and the longer ones are plotted in red.

Figure 12

The top row shows the breather state corresponding to the point “f” in Fig. 11. The bottom row is the turbulent state at the point “e” in Fig. 11. The first and second columns compare the spatiotemporal dynamics (fundamental) and snapshots of the spatial profiles. The third and fourth columns show the rf spectra of net powers (third), $|\text{FFT}{\sum }_{\mu }|{\psi }_{\mu \text{f}}\left(t\right){|}^2|$, and the per-mode rf spectra (fourth), $|\text{FFT}\left\{{\psi }_{\mu \text{f}}\left(t\right)\right\}{|}^2$. One can see that the breather state consists from the five coherent subcombs with the different offsets and the same repetition rates, while the turbulent state shows no mode-locking signs.

Figure 13

Dispersion of the dressed states [see Eq. (59)], normalized to $2\pi$: (a) ${\tilde{\omega }}_{\pm \mu \zeta }^{\left(1\right)}$ and (b) ${\tilde{\omega }}_{\pm \mu \zeta }^{\left(2\right)}$. The red lines correspond to the positive sidebands, ${\tilde{\omega }}_{\mu \zeta }^{\left(1\right),\left(2\right)}$, and the blue ones correspond to the negative sidebands, ${\tilde{\omega }}_{-\mu \zeta }^{\left(1\right),\left(2\right)}$. The values of $-100$ and $-200\mathrm{kHz}$ are the values of $D_{2\text{f}}$ and $D_{2\text{s}}$ in the bare resonator. The gray shading and stars show the dressed states with the anomalous dispersion induced by the dressing. In (a), the maximal anomalous dispersion of $\approx 1\mathrm{GHz}$ (not shown) is achieved at $\mu =25$ for the $|b_{25}^{\left(3\right)}\rangle \approx |3\rangle e^{i\varphi }-|4\rangle$ state. Parameters are $\left|\mathrm{\Omega }\right|/2\pi =116\mathrm{MHz}, {\varepsilon }_0/2\pi =25\mathrm{GHz}, \delta =-3.8{\kappa }_{\text{f}}$.

Figure 14

(a) Parametric instability tongues in the ($\delta ,\left|\mathrm{\Omega }\right|$) plane, for ${\varepsilon }_0/2\pi =25\mathrm{GHz}$ [see ${\varepsilon }_0/2\pi =5\mathrm{GHz}$ in Fig. 9]. The colorbar shows the number of the simultaneously unstable sidebands, $N$. The black dotted line embraces the $\mu =0$ instability range, i.e., the middle branch of the bistability loop. $N$ is clamped by ${\mu }_{*}=|{\varepsilon }_0|/|D_{1\text{f}}-D_{1\text{s}}|=25, N\le {\mu }_{*}$. (b) $N$ vs $\delta$ for three values of $\mathcal{W}$ (laser power) computed along the upper branch of the cw state. The magenta line ($\mathcal{W}=333\mu \mathrm{W}$) 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 stabilization of the cw state.

Figure 15

Families of the bright solitons computed for the laser power $\mathcal{W}=333\mu \mathrm{W}$ and plotted vs $\delta$. The first line data show the fundamental field, and the second line is for the second harmonic. (a, b) Soliton branches (red and green lines) and the cw state (magenta lines) vs $\delta$. The solitons are shown for a range of frequency mismatch parameters, ${\varepsilon }_0/2\pi =$ 21, 23, 25, 27, and 29 GHz. The full lines correspond to the stable solutions, and the dashed lines correspond to the unstable ones. The ${\varepsilon }_0/2\pi =25\mathrm{GHz}$ case in (a) and (b) is the one that shows how the unstable soliton splits from the cw state and then connects to the stable soliton. (c, e) How the spatial profile of the soliton changes with $\delta$ for ${\varepsilon }_0/2\pi =25\mathrm{GHz}$. (d, f) Like (c) and (e) but showing the envelopes of the discrete soliton spectra (see Fig. 17).

Figure 16

Instability of the cw state and spontaneous birth of the bright soliton. Laser power $\mathcal{W}=333\mu \mathrm{W}, {\varepsilon }_0/2\pi =25\mathrm{GHz}$, and $\delta =-3.8{\kappa }_{\text{f}}$. (a) Fundamental field. (b) Second harmonic.

Figure 17

Families of the bright solitons computed for the laser power $\mathcal{W}=333\mu \mathrm{W}$ and plotted vs ${\varepsilon }_0$ for $\delta =-3.9{\kappa }_{\text{f}}$. The first and second columns show the fundamental and second-harmonic data, respectively. (a, b) Multiple soliton branches centered around ${\varepsilon }_0/2\pi \approx {\mu }_{*}$ [see Eq. (30)]. (c–f) Spectra at the indicated values of ${\varepsilon }_0$. (g, i) Respective spatial soliton profiles. The shaded areas in (c)–(f) highlight the interval between the two zero dispersion points (see Fig. 13).