Bandwidth and conversion efficiency analysis of dissipative Kerr soliton frequency combs based on bifurcation theory

Dissipative Kerr soliton frequency combs generated in high-Q microresonators may unlock novel perspectives in a variety of applications and crucially rely on quantitative models for systematic device design. Here, we present a global bifurcation study of the Lugiato-Lefever equation which describes Kerr comb formation. Our study allows systematic investigation of stationary comb states over a wide range of technically relevant parameters. Quantifying key performance parameters of bright and dark-soliton combs, our findings may serve as a design guideline for Kerr comb generators.

Figure 1

Bifurcation maps and nontrivial comb states for bright solitons in anomalous-dispersion ($d>0$, top) and dark solitons in normal-dispersion microresonators ($d<0$, bottom). Quantities on axes are dimensionless. (a) Bifurcation map of the LLE for $f=2$ and $d=0.1$, indicating the normalized intracavity power ${\left|\left|a\right|\right|}_2^2$ vs. the normalized detuning $\zeta$. The constant solution is indicated in black, the single soliton state bifurcation branch ($m=1$) in red, while blue corresponds to other bifurcation branches of multisoliton states with $m=2,\dots ,8$ pulses circulating in the cavity. Circles indicate bifurcation points. (b) Spatial power distribution as a function of normalized intracavity position $x$ of single-soliton states corresponding to points A, B, and to the turning point C indicated in (a). (c) Spectral power distribution of single-soliton states corresponding to points A, B, and to the turning point C indicated in (a). Note that for illustrative purposes, a relatively low forcing $f=2$ was chosen, resulting in a quick drop of the power of spectral modes further away from the pump. Here the bifurcation-and-continuation method is sufficiently precise to correctly predict spectral components which are more than 150 dB below the pump, hence safely covering technical relevant power ranges. (d) Bifurcation map of the LLE for $f=2$ and $d=-0.1$, indicating the normalized intracavity power ${\left|\left|a\right|\right|}_2^2$ vs the normalized detuning $\zeta$. Black denotes again the constant solution, red the single-soliton state bifurcation branch, and blue corresponds to other bifurcation branches. Here, circles also mark bifurcation points. (e) Spatial power distribution as a function of normalized intracavity position $x$ of single-soliton states corresponding to points D, E, and to the turning point F indicated in (d). (f) Spectral power distribution of single-soliton states corresponding to points D, E, and to the turning point F indicated in (d).

Figure 2

Full width at half maximum (${\text{FWHM}}_a$) of bright solitons along the bifurcating branch for anomalous dispersion (red, $d=0.1)$ and full width at half minimum (${\text{FWHM}}_i$) of dark solitons for normal dispersion (blue, $d=-0.1$) for $f=2$. Horizontal axis shows normalized arc length along bifurcation branches. Quantities on both axes are dimensionless.

Figure 3

Bandwidths $2k^{★}$ and power conversion efficiencies (PCE) $\eta$ for bright-soliton combs (a), (c) and dark-soliton combs (b), (d) as a function of the forcing $f$ and dispersion $d=\pm 0.1,\pm 0.15,\pm 0.2,\pm 0.25$. Quantities on axes are dimensionless. (a) Bandwidth of bright-soliton combs obtained by numerical bifurcation and continuation (NBC, colored lines) along with an approximation according to Eq. (9a). The linear approximation is in good agreement with the numerical results and deviates only for a strong forcing. A stronger dispersion leads to a decreasing comb bandwidth. (b) Bandwidth of dark-soliton combs obtained by NBC. (c) PCE of bright-soliton states obtained by NBC (colored lines) along with an approximation according to Eq. (9b) (gray lines). The approximation is in good agreement with the numerical results, but deviates strongly for weak forcing. A weaker dispersion leads to a decreasing PCE. (d) PCE of dark-soliton states obtained by NBC. The PCE decreases with an increasing forcing, but is overall higher as for bright solitons. Here, weaker dispersion also leads to a decreasing PCE.

Figure 4

Connected components of bifurcation graphs (top) and selected bright solitons (bottom) in anomalous dispersion for $f=2$ and $d=0.1$. Quantities on axes are dimensionless. (a) Same graph as in Fig. 1 shows normalized intracavity power ${\parallel a\parallel }_2^2$ vs normalized detuning $\zeta$. Bifurcation points are marked as unfilled circles. Primary bifurcation points lie on the black curve of constant solutions, secondary bifurcation points lie off the black curve. Branches connected through secondary bifurcation points are plotted with the same color. (b) Detailed resolution of red connected component from (a) using the norm $\parallel a-a_{av}{\parallel }_2^2$ vs $\zeta$. The black curve of constant solutions is now on the $\zeta$ axis. Stable states lie on solid lines, unstable states on dashed lines. The curve of 1-solitons runs through the states A, C, B whose spatial power distribution was shown in Fig. 1. Following this branch further leads to states U, V, W marking the transition from 1-solitons to 2-solitons before meeting the branch of 2-solitons through A', C', B' at the secondary bifurcation point C'. Following the A', C', B' curve further up leads to another secondary bifurcation with the branch of 4-solitons that eventually meets the trivial curve again near ${\zeta }_0=0.3325$. (c) shows the spatial power distribution of selected 2-solitons and (d) shows how on the branch of 1-solitons the transition towards 2-solitons occurs before the two branches join at point C'.

Figure 5

Connected components of bifurcation graphs (top) and selected dark solitons (bottom) in normal dispersion for $f=2$ and $d=-0.1$. Quantities on axes are dimensionless. (a) Same graph as in Fig. 1 showing normalized intracavity power ${\parallel a\parallel }_2^2$ vs normalized detuning $\zeta$ restricted to the relevant range. Bifurcation points are marked as unfilled circles. There are six primary bifurcation points lying on the black curve of constant solutions, and three secondary bifurcation points lying off the black curve. All branches are connected to each other and are therefore shown in red. The zoom shows how the branch of 1-solitons starting on the trivial curve near $\zeta =4$ almost meets the curve of constant solutions near $\zeta =2.73$. Instead, it connects at a secondary bifurcation point near $\zeta =2.73$ with the curve of 3-solitons and immediately detaches from it again and then finally connects to the 2-solitons at the secondary bifurcation point near $\zeta =3.36$. (b) Detailed resolution of red connected component from (a) using the norm $\parallel a-a_{av}{\parallel }_2^2$ vs $\zeta$. The black curve of constant solutions is now on the $\zeta$ axis. Stable states lie on solid lines, unstable states on dashed lines. The curve of 1-solitons runs through the states D, F, E whose spatial power distribution was shown in Fig. 1. Following this branch further leads to states X, Y, Z marking the transition from 1-solitons to 2-solitons before meeting the branch of 2-solitons through D', F', E' at a secondary bifurcation point near $\zeta =3.36$. (c) shows the spatial power distribution of selected 2-solitons and (d) shows how on the branch of 1-solitons the transition toward 2-solitons occurs before the two branches meet near $\zeta =3.36$.