Subharmonic locking and frequency combs in frequency-swept semiconductor lasers

We analyze a delay differential equation model for a swept semiconductor laser and demonstrate existence of various periodic solutions that are subharmonically locked to the sweep rate. These solutions provide optical frequency combs in spectral domain. We investigate the problem numerically and show that, due to the translational symmetry of the model, there exists a hysteresis loop formed by branches of steady states solutions, bridges of periodic solutions connecting stable and unstable steady state branches, and isolated branches of limit cycles. We discuss the role of bifurcation points and limit cycles embedded into the loop in the formation of the subharmonic dynamics.

Figure 1

The intensity (a) and logarithm of intensity (b) of two CW branches and periodic solutions for static $\varphi ={\varphi }_0$. Black (red) lines correspond to stable (unstable) steady state solutions in absence of any sweep ($\epsilon =0$). Green and purple circles indicate fold and Hopf bifurcation points, respectively. The parameters are $\gamma =0.25,\eta =1.0,\kappa =0.35,\alpha =5,J=10$. Intensity maxima of stable and unstable branches of the periodic solutions are represented by blue and orange lines, respectively. The solutions and all corresponding bifurcation points are $2\pi$-periodic with respect to $\varphi$; for simplicity, only two CW branches together with one of each type of periodic branches are shown.

Figure 2

Time traces and spectra for different $\epsilon$ values: $\epsilon =0.08$ [fundamental regime, (a)/(e)], $\epsilon =0.2$ [fundamental regime, (b)/(f)], $\epsilon =0.28$ [subharmonic 1/2, (c)/(g)], and $\epsilon =0.32$ [subharmonic 1/3, (d)/(h)].

Figure 3

Frequency comb line spacing vs sweep rate (0.005 step on $\epsilon$ with lines above $-10\mathrm{dB}$ level taken into account). Dark grey points correspond to increasing epsilon, red points to decreasing epsilon where it deviates. Dashed lines indicate several regions of linear dependency corresponding to distinct subharmonic regimes, where the slope of the lines is $1/\left(2\pi n\right)$ and $n=1$,2..5 defines the subharmonic number.

Figure 4

Overlay of the fundamental regime trajectories (blue lines) with the system's steady state solutions, bifurcation points, and max intensities of periodic solutions for $\epsilon =0.01$ (a) and $\epsilon =0.08$ (b). The bottom panel (c) shows both trajectories in logarithmic scale to emphasize the touches to a saddle at unstable low intensity branch.

Figure 5

Overlay of the 1/2 subharmonic regime trajectory (maroon line) with the system's steady state solutions, bifurcation points, and max intensities of periodic solutions for $\epsilon =0.28$ in linear (a) and logarithmic (b) scales.

Figure 6

Overlay of the 1/3 subharmonic regime trajectories (green lines) with the system's steady state solutions, bifurcation points, and max intensities of periodic solutions in linear (a) and logarithmic (b-c) scales for different $\epsilon$: (from highest to lowest intensity) $\epsilon =0.30,\epsilon =1.0$, and $\epsilon =1.5$.

Figure 7

Extended bifurcation diagram of CW branches and periodic solutions indicating other bifurcation points and both minimum and maximum intensities of periodic solutions given in Fig. 1. The colors and notations correspond to Fig. 1.

Figure 8

Extended bifurcation diagram of a bridge of periodic solutions ($H_1,H_0^{\prime }$) showing the evolution of its maximum and minimum intensity (a) and its logarithm (b) when $\varphi$ is varied. Cyan colored region marks a stable period doubling limit cycle arising from the bridge and then collapsing. The points of the stable region bifurcations are denoted by triangles (fold), squares (period doubling), and diamonds (torus). Other colors and notations correspond to Fig. 1. Color arrows indicate the locations of limit cycles used to give profile examples in Fig. 10.

Figure 9

Extended bifurcation diagram of an isola limit cycle $L{C}_0$ showing the evolution of its maximum and minimum intensity (a) and its logarithm (b) when $\varphi$ is varied. The points of the stable region bifurcations are denoted by triangles (fold), squares (period doubling), and diamonds (torus). The colors and notations correspond to Fig. 1. Color arrows indicate the locations of limit cycles used in Fig. 11.

Figure 10

Intensity profiles of the stable limit cycles at the bridge of periodic solutions ($H_1,H_0^{\prime }$) given for one period of oscillation at points (a) $\varphi =-1.6\pi$; (b) $\varphi =-2.12\pi$; (c) $\varphi =-3.13\pi$; (d) $\varphi =-3.16\pi$. The locations of the limit cycles are shown by correspondingly colored arrows in Fig. 8.

Figure 11

Intensity profiles of the stable limit cycles at the limit cycle isola $L{C}_0$ given for one period of oscillation at points (a) $\varphi =-2.65\pi$; (b) $\varphi =-2.29\pi$; (c) $\varphi =-2.25\pi$; (d) $\varphi =-1.33\pi$. The locations of the limit cycles are shown by correspondingly colored arrows in Fig. 9.