Stochastic bifurcation in noise-driven lasers and Hopf oscillators

This paper considers nonlinear dynamics in an ensemble of uncoupled lasers, each being a limit-cycle oscillator, which are driven by the same external white Gaussian noise. As the external-noise strength increases, there is an onset of synchronization and then subsequent loss of synchrony. Local analysis of the laser equations shows that synchronization becomes unstable via stochastic bifurcation to chaos, defined as a passing of the largest Lyapunov exponent through zero. The locus of this bifurcation is calculated in the three-dimensional parameter space defined by the Hopf parameter, amount of amplitude-phase coupling, and external-noise strength. Numerical comparison between the laser system and the normal form of Hopf bifurcation uncovers a square-root law for this stochastic bifurcation as well as strong enhancement in noise-induced chaos due to the laser’s relaxation oscillation.

Figure 1

(Color online) Projection onto the complex $E$ plane of (black) the limit cycle representing the “on” state of the laser and (blue) isochrones of (black dots) eight different points on the limit cycle as defined by Eq. (4). Each two-dimensional isochron is independent of $N$ and, hence, appears as a one-dimensional curve in the projection. $\Lambda =1$ and (a) $\alpha =0$ and (b) $\alpha =3$.

Figure 2

(Color online) Total average intensity as defined by Eq. (5) of $M=50$ independent lasers with common coherent external signal vs the external-signal amplitude $K$ for (blue) $\alpha =0$ and (red) $\alpha =3$; $\Lambda =5$ and ${\nu }_{\mathit{ext}}=\Delta$.

Figure 3

(Color online) Total average intensity as defined by Eq. (5) of $M=50$ independent lasers with common external white noise vs the external-noise variance $D_{\mathit{ext}}$ for (blue) $\alpha =0$ and (red) $\alpha =3$; $\Lambda =5$.

Figure 4

(Color online) The shape of the probability distribution of $I_M\left(t\right)$ from Fig. 3 for (blue) $\alpha =0$ and (red) $\alpha =3$. From (a) to (c) $D_{\mathit{ext}}={10}^0$, ${10}^1$, and ${10}^3$.

Figure 5

(Color online) Bifurcation diagram of a single laser with just external noise in the $(D_{\mathit{ext}},\Lambda )$ plane for (a) $\alpha =0$ and (b) $\alpha =3$. RS stands for random sink, RSA stands for random strange attractor, the two black curves denote stochastic $d$ bifurcation, and color coding is for the largest Lyapunov exponent. A noise-free laser has Hopf bifurcation at ($D_{\mathit{ext}}=0$, $\Lambda =0$).

Figure 6

The two different types of dynamics found in a single laser with external noise are shown as snapshots of 50 000 trajectories at different times $t$ in the projection onto the complex $E$ plane. The left column $(D_{\mathit{ext}}=0.1)$ shows convergence to a random sink and the right column $(D_{\mathit{ext}}=0.5)$ shows convergence to a random strange attractor; $\Lambda =1$ and $\alpha =3$. At $t=0$, the initial points are evenly distributed on the ellipse ${(E^R∕2)}^2+{\left(2E^I\right)}^2=1$, $N=0$.

Figure 7

(Color online) The largest Lyapunov exponent vs $D_{\mathit{ext}}$ in a single laser with $\Lambda =1$. In panel (a) $\beta =2.77$, $\gamma =500$ and from top to bottom $\alpha =$ (blue) 5.0, (green) 3.0, (red) 1.5, (black) 0.0. In panel (b) $\alpha =3$ and from top to bottom the cavity decay $\text{rate}=\left(\text{blue}\right)$ $2\times {10}^{12}{\mathrm{s}}^{-1}$ ($\beta =1.88$, $\gamma =1000$), (green) ${10}^{12}{\mathrm{s}}^{-1}$ ($\beta =2.77$, $\gamma =500$), (red) $5\times {10}^{11}{\mathrm{s}}^{-1}$ ($\beta =4.53$, $\gamma =250$), (black) $2\times {10}^{11}{\mathrm{s}}^{-1}$ ($\beta =9.82$, $\gamma =100$). In panel (c) $\beta =2.77$, $\alpha =3$ and from top to bottom the carrier decay $\text{rate}=\left(\text{black}\right)$ ${10}^9{\mathrm{s}}^{-1}$ $(\gamma =500)$, (red) $2\times {10}^9{\mathrm{s}}^{-1}$ $(\gamma =250)$, (green) $5\times {10}^9{\mathrm{s}}^{-1}$ $(\gamma =100)$, (blue) $8\times {10}^9{\mathrm{s}}^{-1}$ $(\gamma =62.5)$. The insets are expanded plots showing the first minimum.

Figure 8

(Color online) The two branches of the stochastic $d$ bifurcation in the Hopf normal form with external noise, Eq. (8), plotted in the $(\sqrt{2D_{\mathit{ext}}},\Lambda )$ plane for (two inner lines, red) $\alpha =6$ and (two outer lines, black) $\alpha =8$. See Fig. 10 for comparison with the laser model. Each dot was calculated numerically by detecting a change in the sign of the largest Lyapunow exponent from negative (positive) to positive (negative) for the left (right) branch.

Figure 9

The stochastic $d$ bifurcation in the Hopf normal form with external noise, Eq. (8), plotted in the $(\sqrt{2D_{\mathit{ext}}},\alpha )$ plane for $\Lambda =0.002$, $\Lambda =0.2$, and $\Lambda =20$. See Fig. 11 for comparison with the laser model. Each dot was calculated as in Fig. 8.

Figure 10

(Color online) The stochastic $d$ bifurcation in laser with no spontaneous emission noise and with external noise [equations (1, 2)] plotted in the $[\sqrt{2D_{\mathit{ext}}∕\left(\beta \gamma \right)},\Lambda ]$ plane. From inner to outer contour $\alpha =\left(\text{blue}\right)$ 1.5, (green) 3, (red) 6, and (black) 8. See Fig. 8 for comparison with the Hopf normal form. Each dot was calculated as in Fig. 8.

Figure 11

The stochastic $d$ bifurcation in laser with no spontaneous emission noise and with external noise [equations (1, 2)] plotted in the $[\sqrt{2D_{\mathit{ext}}∕\left(\beta \gamma \right)},\alpha ]$ plane for $\Lambda =0.002$, $\Lambda =0.2$, and $\Lambda =20$. Each dot was calculated as in Fig. 8

Figure 12

$\Lambda$ versus (solid) the two nonzero LEs of the deterministic laser system (1, 2) and (dashed) the nonzero LE of the deterministic Hopf normal form (8).

Figure 13

Projection onto the complex $E$ plane with snapshots of 5000 trajectories at times [panels (a), (c), (e)] $t=\tilde{t}=0.06$ and [panels (b), (d), (f)] $t=\tilde{t}=1.28$ for [panels (a), (b)] Eqs. (1, 2) with $\alpha =0$, [panels (c), (d)] Eqs. (1, 2) with $\alpha =3$, and [panels (e), (f)] Eq. (11) with $\alpha =3$. $\Lambda =1$ and at $t=0$ the initial points are evenly distributed on the ellipse ${(E^R∕2)}^2+{\left(2E^I\right)}^2=1$, $N=0$.

Figure 14

Phase-space stretching along the limit cycle shown as a change in the phase difference between two trajectories starting at $(E_1^R,E_1^I,N_1)=(2,0,0)$ and $(E_2^R,E_2^I,N_2)=(0,0.5,0)$, respectively. The three curves are obtained using (dashed) Eqs. (1, 2) with $\alpha =0$, (solid) Eqs. (1, 2) with $\alpha =3$, and (dotted) Eq. (11) with $\alpha =3$.