Directional reversals and multimode dynamics in semiconductor ring lasers

We investigate the dynamics of longitudinal modes in quantum-well semiconductor ring lasers by means of a spatiotemporal traveling-wave model. We report the existence of a multimode instability in such a system that provokes a periodic deterministic directional reversal involving jumps between consecutive longitudinal modes. The switching sequence follows the modal frequencies from blue to red, and every modal jump is accompanied by a reversal of the direction of emission. We characterize and analyze this instability via a bifurcation analysis of the full traveling-wave model as well as by performing a linear stability analysis of the monochromatic solutions.

Figure 1

Representation of the semiconductor ring laser. For the sake of simplicity in our model the input-output waveguide is supposed to be transparent and the coupling with the ring cavity described as one point (dashed black line) that imposes the boundary conditions described in Eqs. (6) where $t_{\pm }$ and $r_{\pm }$ are the transmission and reflection coefficients for the counterpropagating fields $E_{+}(z,t)$ and $E_{-}(z,t)$. $Y_{\pm }\left(t\right)$ are the injected fields for each direction.

Figure 2

Switching upon injection of a copropagating (left column) and a counterpropagating (right column) Gaussian pulse with $I_{\pm }=3$, ${\tau }_{\pm }=100$ ps, and $m=-1$ ($f=-78$ GHz). From top to bottom, the panels display the Gaussian pulse intensity, the total intensity, and the modal intensities for modes $m=0$ and $m=-1$, obtained by filtering the total intensity around each mode. The initial state was at $m=0$ ($\omega =0$). $J=2.5$.

Figure 3

Bifurcation diagrams of the dominant modes of a SRL with the parameters in Table 1. The solid (dashed) lines indicate stable (unstable) monochromatic solutions. Modes $m=0$ and $m=-1$ share a current region where they are stable in the UNI regime; thus there is longitudinal mode bistability. The other modes are unstable for the explored current density values.

Figure 4

Longitudinal mode switching instability at $J=5$. The period of the oscillation is 100 ns. Parameters as in Table 1 except $\mathcal{D}=2.32$ cm${}^2$ s${}^{-1}$.

Figure 5

Time trace for $J=6.5$ (upper panel) and the optical spectrum (lower panel). Parameters as in Table 1 except $\mathcal{D}=2.32$ cm${}^2$ s${}^{-1}$.

Figure 6

Time characterization of the longitudinal mode switching instability shown in Fig. 4 for different current density $J$ values. Parameters as in Table 1 except $\mathcal{D}=2.32$ cm${}^2$ s${}^{-1}$.

Figure 7

Bifurcation diagrams for four modes for a SRL with the parameters shown in Table 1 except $\mathcal{D}=2.32$ cm${}^2$ s${}^{-1}$. There is a short region of longitudinal mode bistability between modes $m=0$ and $m=-1$; however, both modes become unstable at different currents and finally there is a multimode instability.

Figure 8

Eigenvalue spectra corresponding to performing numerically the LSA of Eqs. (1, 2, 3, 4) for the monochromatic solutions for modes $m=0$ (a), $m=-1$ (b), and $m=-2$ (c) at $J=5$. We use the associated eigenvectors in order to find to what direction of emission the eigenvalues correspond.

Figure 9

Most unstable perturbation mode $m^{\prime }$ for UNI solutions in mode $m$ at $J=5$. Blue circles denote counterpropagating modes and red squares indicate copropagating modes. The red arrows indicate the cycle sequence arising from the instability.

Figure 10

Eigenvalue spectra obtained from Eqs. (14) and (15) for the monochromatic solutions for modes $m=0$ (a), $m=-1$ (b), and $m=-2$ (c) at $J=5$.

Figure 11

Effect of the absence of spontaneous-emission noise in the longitudinal mode switching instability shown in Fig. 4. $J=5$. Parameters as in Table 1 except $\mathcal{D}=2.32$ cm${}^2$ s${}^{-1}$ and $\beta =0$.

Figure 12

Bifurcation diagram of the mode $m=0$ for the parameter set from Table 1. The solid (dotted) black lines indicate stable (unstable) solutions for the branches Bi-CW and UNI. The unstable region around $J=1.5$ corresponds to the AO regime.

Figure 13

Eigenvalue spectra corresponding to Fig. 12 at $J=1.3$ (a),(b), and $J=1.4$ (c). The red line indicates the shape of the gain curve. (b),(c) Onset of the Hopf bifurcation by the crossing of the Im$\left\{\lambda \right\}$ axis of two complex conjugate eigenvalues. The other eigenvalue shown corresponds to phase invariance.