Two-color flat-top solitonic pulses in ${\chi }^{\left(2\right)}$ optical microresonators via second-harmonic generation

We studied numerically the generation of the coherent frequency combs at second-harmonic generation in ${\chi }^{\left(2\right)}$ microresonators via the conventional frequency scan method. It was demonstrated that under particular conditions it is possible to generate two-color flat-top solitonic pulses, platicons, using a pump amplitude modulation or a controllable mode interaction approach, if the signs of the group velocity dispersion coefficients at pump frequency and its second harmonic are opposite but absolute values of these coefficients are rather close. It was revealed that platicons may be observed on both sides of the linear microresonator resonance (at positive, as well as negative pump frequency detunings). For the efficient platicon excitation, one needs simultaneous accurate matching of both the microresonator free spectral ranges and resonant eigenfrequencies. Platicon generation processes were simulated numerically, excitation conditions and platicon generation domains were found for different generation methods, and the properties of generated platicons were studied for the different combinations of the medium parameters.

Figure 1

(Left column) The dependence $U_1\left({\alpha }_1\right)$ upon frequency scan at anomalous and normal GVD at both frequencies; (right column) soliton components profiles. All quantities are plotted in dimensionless units.

Figure 2

The dependence $U_1\left({\alpha }_1\right)$ upon forward (left column) and backward (right column) frequency scan at $b_{21}=-0.05$, $b_{22}=0.05$ (upper panels) and $b_{21}=0.05$, $b_{22}=-0.05$ (bottom panels). All quantities are plotted in dimensionless units.

Figure 3

The dependence $U_1\left({\alpha }_1\right)$ upon forward and backward frequency scan for different values of the modulation depth at $b_{21}=0.05$, $b_{22}=-0.05$ (top right, top left, bottom left panels) and $b_{21}=-0.05$, $b_{22}=0.05$ (bottom right panel). All quantities are plotted in dimensionless units.

Figure 4

(Left column) FW and (right column) SH field distribution (upper line) and spectrum (bottom line) evolution upon forward frequency scan at $b_{21}=0.05$, $b_{22}=-0.05$, $\varepsilon =0.3$. All quantities are plotted in dimensionless units.

Figure 5

Platicon generation domains for different values of the modulation depth and fixed pump amplitude (left panel) and for different pump amplitudes and fixed modulation depth (right panel) at $b_{21}=0.05$, $b_{22}=-0.05$ and $b_{21}=-0.05$, $b_{22}=0.05$. All quantities are plotted in dimensionless units.

Figure 6

Profiles of the platicon components for different detuning values at $\varepsilon =0.4$ for $b_{21}=0.05$, $b_{22}=-0.05$ (top panels) and $b_{21}=-0.05$, $b_{22}=0.05$ (bottom panels). All quantities are plotted in dimensionless units.

Figure 7

(Top left panel) The dependences $U_1\left({\alpha }_1\right)$ upon frequency scan at $b_{21}=0.05$, $b_{21}=-0.05$, $\varepsilon =0.4$ for different values of ${\kappa }_2/{\kappa }_1$; platicon components profiles at ${\alpha }_1=4.5$ for ${\kappa }_2/{\kappa }_1=1/3$(top right panel), ${\kappa }_2/{\kappa }_1=1$ (left bottom panel), and ${\kappa }_2/{\kappa }_1=3$ (right bottom panel). All quantities are plotted in dimensionless units.

Figure 8

Profiles of the platicon components for different FSRs mismatch values at $\varepsilon =0.5$ for $b_{21}=0.05$, $b_{22}=-0.05$. All quantities are plotted in dimensionless units.

Figure 9

(Top) The dependences $U_1\left({\alpha }_1\right)$ upon frequency scan at $b_{21}=-0.05$, $b_{21}=0.05$, $\varepsilon =0.4$ for different values of $\delta$; (bottom) platicon components profiles at $b_{21}=-0.05$, $b_{21}=0.05$, $\varepsilon =0.4$, $\delta =-6$ for different values of ${\alpha }_1$. All quantities are plotted in dimensionless units.

Figure 10

The dependence $U_1\left({\alpha }_{10}\right)$ upon forward and backward frequency scan for different values of pumped mode shift at $b_{21}=0.05$, $b_{22}=-0.05$. All quantities are plotted in dimensionless units.

Figure 11

(Left column) FW and (right column) SH field distribution (upper line) and spectrum (bottom line) evolution upon forward frequency scan at $b_{21}=0.05$, $b_{22}=-0.05$, $\mathrm{\Delta }={\kappa }_1$. All quantities are plotted in dimensionless units.

Figure 12

Platicon generation domains for different values of the pumped mode shift and fixed pump amplitude (left panel) and different pump amplitudes and fixed pumped mode shifts (right panel) at $b_{21}=0.05$, $b_{22}=-0.05$. Regions 1, 3, and 5 in left panel and 1 and 3 in right panel corresponds to the domains of the predominant single platicon generation. All quantities are plotted in dimensionless units.

Figure 13

Profiles of the platicon components for the different pumped mode shift values at $b_{21}=0.05$, $b_{22}=-0.05$. All quantities are plotted in dimensionless units.

Figure 14

(Left panel) The dependence $U_1\left({\alpha }_{10}\right)$ upon forward scan and (right panel) profiles of the platicon components for ${\alpha }_{10}=3.5$ at $b_{21}=-0.05$, $b_{22}=0.05$, $\mathrm{\Delta }=\kappa$. All quantities are plotted in dimensionless units.

Figure 15

Platicon generation domains for different values of the pumped mode shift and fixed pump amplitude (left panel) and for different pump amplitudes and fixed pumped mode shifts (right panel) at $b_{21}=-0.05$, $b_{22}=0.05$. Regions 1, 3, and 4 in the left panel and 1 and 3 in the right panel corresponds to the domains of the predominant single platicon generation. All quantities are plotted in dimensionless units.

Figure 16

Profiles of the platicon components at $f=10$, $d=0$, $\mathrm{\Delta }={\kappa }_1$, $b_{21}=0.05$, $b_{22}=-0.05$ for $\delta =-10$, ${\alpha }_{10}=5$ (left panel) and $\delta =2$, ${\alpha }_{10}=3$ (right panel). All quantities are plotted in dimensionless units.