Optical injection in semiconductor ring lasers

We theoretically investigate optical injection in semiconductor ring lasers and disclose several dynamical regimes. Through numerical simulations and bifurcation continuation, two separate parameter regions in which two different injection-locked solutions coexist are revealed, in addition to a region in which a frequency-locked limit cycle coexists with an injection-locked solution. Finally, an antiphase chaotic regime without the involvement of any carrier dynamics is revealed. Parallels are drawn with the onset of chaos in the periodically forced Duffing oscillator.

Figure 1

Schematic setup of the optically injected SRL ($m$ is assumed to be the amplitude coupling coefficient of the straight output waveguide into the ring cavity).

Figure 2

Orbit diagram depicting the local extrema of the intensity $\left|E_1\right|{}^2$ of the CW mode versus the injection amplitude $E_i$. Black (gray) dots indicate local maxima (minima). $\mu =1.704$, ${\varphi }_k=1.4966$, $k=0.4412$ ns${}^{-1}$, $\alpha =3.5$, $c=0.01$, $s=0.005$, $\kappa =100$ ns${}^{-1}$, $\gamma =0.2$ ns${}^{-1}$, $\Delta =-1$ ns${}^{-1}$.

Figure 3

Representative time trace of the respective intensities of the CW (dark gray, red online) and the CCW mode (light gray, green online), and the total intensity (black) inside the cavity of the SRL when operating in the chaotic regime. Transient behavior has been eliminated. $\mu =1.704$, ${\varphi }_k=1.4966$, $k=0.4412$ ns${}^{-1}$, $\alpha =3.5$, $c=0.01$, $s=0.005$, $\kappa =100$ ns${}^{-1}$, $\gamma =0.2$ ns${}^{-1}$, $E_i=6\times {10}^{-5}$, $\Delta =-1$ ns${}^{-1}$.

Figure 4

Bifurcation diagram in the $(\Delta ,E_i)$ plane, generated using the rate equations (1). Saddle-node (Hopf) bifurcations are depicted in black (light gray, red online). Supercritical (subcritical) bifurcations are depicted in full (dashed) lines.

Figure 5

Bifurcation diagram in the $(\Delta ,E_i)$ plane. This is a blow-up of Fig. 4 in order to show the three new saddle-node bifurcations (SN${}_2$, SN${}_3$, and SN${}_4$), which correspond to three additional resonance tongues. The curves SN${}_2$ and SN${}_4$ have their origin symmetrically around $\Delta =0$ (originating from the bidirectional solutions), while the SN${}_3$ curve has the same origin as the SN${}_1$ curve of Fig. 4 (originating from the unidirectional solutions). The notations $s_{\mathrm{cw}}$, $s_{\mathrm{ccw}}$, $s_{\mathrm{bi}}$, and ${\Gamma }_H$ indicate where these respective solutions (co-)exist. The shaded area indicates regions of chaotic behavior. Conventions as in Fig. 4.

Figure 6

Poincaré section corresponding to an orbit of approximately 100 $\mu$s ($\approx 16\hspace{0.5em}000$ points) in the reduced two-dimensional $(\theta ,\psi )$ phase space of the solitary SRL, when operating in the regime depicted in Fig. 3.