Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators

Using the Lugiato-Lefever model, we analyze the effects of third-order chromatic dispersion on the existence and stability of dark- and bright-soliton Kerr frequency combs in the normal dispersion regime. While in the absence of third-order dispersion only dark solitons exist over an extended parameter range, we find that third-order dispersion allows for stable dark and bright solitons to coexist. Reversibility is broken and the shape of the switching waves connecting the top and bottom homogeneous solutions is modified. Bright solitons come into existence thanks to the generation of oscillations in the switching-wave profiles. Temporal oscillatory instabilities of dark solitons are suppressed in the presence of sufficiently strong third-order dispersion, while bright solitons are never found to oscillate in time. As a result of third-order dispersion both bright and dark solitons are found to move with a velocity that depends on their width.

Figure 1

Solution profile at $t=150$ (black) after time evolution in the LLE (1) of an initial dip in the top HSS $A_t$ or a bump on the bottom HSS $A_b$ (red) in the (a) absence or (b) presence of TOD. Left panels show the time profile, while the right panels show its associated comb spectrum. Parameter set: (a) $(\theta ,\rho ,d_3)=(4,2.175,0)$; (b) $(\theta ,\rho ,d_3)=(4,2.3,0.7)$.

Figure 2

Switching waves (a) $S_u$ and (b) $S_d$, in each case for $d_3=0$ (top panel, red) and $d_3=0.7$ (bottom panel, blue). The spatial eigenvalues corresponding to $A_t$ and $A_b$ are shown in (c) and (d), respectively, both for $d_3=0$ (red crosses) and $d_3=0.7$ (blue diamonds). $(\theta ,\rho )=(4,2.325)$.

Figure 3

Bifurcation diagrams of HSSs (blue) and dark and bright solitons for $\theta =4$ and increasing values of $d_3$= 0 (black), 0.3 (yellow), 0.5 (green), 0.7 (red). Solid (dashed) lines correspond to stable (unstable) states.

Figure 4

Temporal profiles [left; $\mathrm{Re}(A$) in blue, $\left|A\right|$ in red] and spectral intensities (right, in dB) of asymmetric dark and bright solitons corresponding to the locations (i)–(iii) and (iv)–(vi) in Fig. 3. $(\theta ,\rho ,d_3)=(4,2.232,0.7)$.

Figure 5

Bifurcation diagrams for $\theta =5$ at different values of $d_3=0$ (a), 0.2 (b), 0.3 (c), 0.7 (d). ${\mathrm{H}}_{1,2}$ correspond to Hopf bifurcations (black dots). Solid (dashed) lines stand for stable (unstable) states. In solid purple (green) lines we refer to states with positive (negative) velocity. Solid red lines (for $d_3=0$) and red dots indicate zero velocity.

Figure 6

(a) Velocity of dark solitons as a function of the width at half minimum $\mathrm{\Delta }$ for $\theta =4$ and $d_3=0.3,0,5,0.7$. The velocity changes in a damped oscillatory way as a function of $\mathrm{\Delta }$. The stable (unstable) dark solitons correspond to the solid (dashed) lines. (b) Velocity of the bright solitons for the same parameter values.

Figure 7

Time evolution of two dark solitons traveling at different speeds for (a) $(\theta =4,d_3=0.3,\rho =2.55)$ and (b) $(\theta =5,d_3=0.2,\rho =2.55)$.