Nonlinear solutions for ${\chi }^{\left(2\right)}$ frequency combs in optical microresonators

Experimental and theoretical studies of nonlinear frequency combs in ${\chi }^{\left(3\right)}$ optical microresonators attracted tremendous research interest during the last decade and resulted in prototypes of soliton-based steadily working devices. Realization of similar combs owing to ${\chi }^{\left(2\right)}$ optical nonlinearity promises new breakthroughs and is a big scientific challenge. We analyze the main obstacles for realization of the ${\chi }^{\left(2\right)}$ frequency combs in high-$Q$ microresonators and propose two families of steady-state nonlinear solutions, including soliton and periodic solutions, for such combs. Despite periodicity of light fields inside microresonators, the nonlinear solutions can be topologically different and relevant to periodic and antiperiodic boundary conditions. The antiperiodic states are expected to be the most favorable for the comb generation. The found particular solutions exist owing to a large difference in the group velocities between the first and second harmonics, typical of ${\chi }^{\left(2\right)}$ microresonators, and to the presence of the pump. They have no zero-pump counterparts relevant to conservative solitons. The stability issue for the found comb solutions remains open and requires further numerical analysis.

Figure 1

Resonator geometry and radial poling. (a) Resonator shape and cross section; $\phi$ is the azimuth angle; the major radius $R$ is different from the rim curvature radius. The red spot shows localization of light, and different colors indicate the radial poling. (b) Micrograph of radially poled lithium-niobate-based resonator with the number of domain periods $N=72$. (c) Schematic of the azimuth dependence of the nonlinear coefficient $d\left(\phi \right)$ for perfectly periodic poling with $N=4$. (d) First harmonics of the Fourier spectrum $|d_j|$ for (c).

Figure 2

Wavelength dependencies of $v_{1,2}$ (a) and $v_{1,2}^{\prime }$ (b) for ${\mathrm{LiNbO}}_3$ crystals with the geometric dispersion neglected. The vertical line corresponds to the case $v_1=v_2$, where $v_1^{\prime }{v}_2^{\prime }<0$.

Figure 3

Dependencies $|{\overline{F}}_{\pm }{|}^2\left(\eta \right)$ for different values of $\mathrm{\Phi }=\arg \left({\mathrm{\Omega }}_1{\mathrm{\Omega }}_2\right)$. The $\pm$ branches correspond to positive and negative slopes. The two-branch black curve with $\mathrm{\Phi }=0$ corresponds to the limiting case ${\gamma }_{1,2}\to 0$ at ${\mathrm{\Delta }}_1{\mathrm{\Delta }}_2>0$, while the black line $\mathrm{\Phi }=\pi$ is relevant to the liming case ${\gamma }_{1,2}\to 0$ at ${\mathrm{\Delta }}_1{\mathrm{\Delta }}_2<0$ and to the case ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0$. The red segment shows the branch $\overline{F}=0$ for $\overline{S}\ne 0$.

Figure 4

Potentials $U(s,s_0)$ for different stopping points $s_0$ (circles) and values of $\eta$. (a)–(c) The cases $s_0=1, -1, \eta$ for $q=1$. (c) Shown also is the effect of small shifts of $s_0$ against $\eta$. (d) The case $s_0=\eta >1$ for $q=-1$. The horizontal dotted lines correspond to the soliton states.

Figure 5

Antiperiodic solitonlike solution for $s$ and $f$ at $q=1, \eta =0.3$, and $1-s_0\simeq 1.27\times {10}^{-2}$. The quantities $s$ and $\left|f\right|$ are periodic with the period ${\zeta }_0=16$ (a), while the phase $\arg \left(f\right)$ changes by $\pi$ within this period (b).

Figure 6

Periodic solitonlike solutions for $s, \left|f\right|$, and $\arg \left(f\right)$ for $q=-1$ within one period; (a) and b) correspond to $\eta =4, 1-s_0\simeq 0.0035$, and to the period ${\zeta }_0=6$, while (c) and d) refer to $\eta =16, 1-s_0\simeq 0.005$, and ${\zeta }_0=2$. Note the difference in the horizontal and vertical scales between (a) and (b), and (c) and (d).

Figure 7

Numerical results on stability of the antiperiodic solitonlike solution with $q=1$ and $\eta =0.22$ ($2\pi R=1$ cm, ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=3\times {10}^9{\mathrm{s}}^{-1}, v_1-v_2={10}^{-2}c, {\gamma }_{1,2}=0$, and $v_{1,2}^{\prime }=0$). The time $t$ is normalized to $t_0=R/c\simeq 5\times {10}^{-12}$ s. The left column shows the normalized intensities ${\left|f\right|}^2$ and ${\left|s\right|}^2$ versus $\phi$ and the right column shows the normalized modal powers $|f_{\delta j}{|}^2$ and $|s_{\delta j}{|}^2$ versus the deviations of the modal numbers $\delta j=j-j_0$ and $\delta l=l-l_0$.