Dynamics of two laterally coupled semiconductor lasers: Strong- and weak-coupling theory

The stability and nonlinear dynamics of two semiconductor lasers coupled side to side via evanescent waves are investigated by using three different models. In the composite-cavity model, the coupling between the lasers is accurately taken into account by calculating electric field profiles (composite-cavity modes) of the whole coupled-laser system. A bifurcation analysis of the composite-cavity model uncovers how different types of dynamics, including stationary phase-locking, periodic, quasiperiodic, and chaotic intensity oscillations, are organized. In the individual-laser model, the coupling between individual lasers is introduced phenomenologically with ad hoc coupling terms. Comparison with the composite-cavity model reveals drastic differences in the dynamics. To identify the causes of these differences, we derive a coupled-laser model with coupling terms which are consistent with the solution of the wave equation and the relevant boundary conditions. This coupled-laser model reproduces the dynamics of the composite-cavity model under weak-coupling conditions.

Figure 1

Sketch of the twin stripe laser geometry with two stripes of thickness $h$, widths $w_A$ and $w_B$, and the distance $2d$. $n_A$, $n_B,n_1$, $n_2$, and $n_3$ denote different refractive indices.

Figure 2

$x$ dependence of the two composite-cavity modes considered; the stripe-width difference is $\Delta w=-0.05\mu \mathrm{m}$ and from (a) to (c) the half distance $d$ is 0, 1, and $3\mu \mathrm{m}$.

Figure 3

(Color online) Off-diagonal matrix element ${\Delta }_{1,2}$ of Eq. (7) for the two composite-cavity modes as a function of the half distance $d$ and the stripe-width difference $\Delta w$.

Figure 4

(Color online) Magnitude of the wave vectors $k_1$ and $k_2$ (a) and the frequency detuning $\Delta \Omega =c(k_2-k_1)$ (b) as a function of the half distance $d$ and the stripe-width difference $\Delta w$.

Figure 5

(Color online) Modal gain coefficient $K_{22}^A$ (a) and composite-mode coupling coefficient $K_{21}^A$ (b) as a function of the half-distance $d$ and the stripe-width difference $\Delta w$.

Figure 6

One-parameter bifurcation diagrams of the composite-cavity model, shown as extrema of the intensity in composite mode 1, mode 2, and laser stripe $A$; throughout $\Delta w=0.02$ and from rows (a) to (c) $\alpha$ has the values 0, 0.4, and 2.0.

Figure 7

(Color online) Two-parameter bifurcation diagram in the $(d,\Delta w)$ plane of the composite-cavity model for different values of $\alpha$ as indicated in the individual panels. Shown are curves of saddle-node (S) and Hopf (H) bifurcations; codimension-2 saddle-node Hopf (SH) and degenerate Hopf (DH) bifurcations occur at isolated points; hatched regions indicate stable cw solution.

Figure 8

(Color online) Two-parameter bifurcation diagram in the $(d,\Delta w)$ plane of the composite-cavity model, showing also bifurcations of periodic solutions for three different values of $\alpha$ as shown in the panels. Additional curves are those of period doubling (P), torus (T), saddle-node (SL) of limit cycles, and homoclinic (hom) bifurcations; additional codimension-2 bifurcations are points of 1:2 resonance bifurcation and degenerate homoclinic points (B) points. Hatched regions indicate stable cw solution.

Figure 9

(Color online) Two-parameter bifurcation diagram in the $(d,\Delta \Omega )$ plane of Eqs. (31) for $\alpha =2.0$ with saddle-node bifurcation (S), Hopf bifurcation (H), and the cw singularity curve. The hatched region indicates the stable cw solution and the gray shaded region divergence of the model.

Figure 10

$x$ dependence of the electric fields of each individual laser (a) and the total electric field inside the coupled-laser device (b).

Figure 11

Inverse coupling length $p_G$ (a) and coupling rate $C$ (b) as a function of the laser half distance $d$, as calculated from the transcendental equation (45) for $n_b=3.61$, $w=4.0\mu \mathrm{m}$, and $n_G=3.6$.

Figure 12

(Color online) Two-parameter bifurcation diagram in the $(d,\Delta \Omega )$ plane of Eqs. (49) for $\alpha =0$ (a), 0.4 (b), and 2 (b), showing curves of saddle-node (S) and Hopf (H) bifurcations and codimension-2 saddle-node Hopf (SH), and degenerate Hopf (DH) points; the hatched regions indicate the stable cw solution.

Figure 13

(Color online) Two-parameter bifurcation diagram in the $(d,\Delta \Omega )$ plane of Eqs. (49) for $\alpha =0$ (a), 0.4 (b), and 2 (b), also showing curves of period doubling (PD), torus (T), saddle-node (SL) of limit-cycle, and homoclinic (hom) bifurcations, and codimension-2 saddle-node Hopf (SH) 1:2-resonance (1:2) points; hatched regions indicate stable cw solution.

Figure 14

Locking boundary in weak-coupling conditions given by the saddle-node bifurcation for different values of $\alpha$ in the individual-laser model (31) (gray) and the coupled-laser model (49) (black). The dots denote saddle-node Hopf points where the saddle-node curves change from super-to subcritical.