Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers

In this contribution we provide an in depth theoretical analysis of the bifurcations leading to nonlinear polarization dynamics in a free-running vertical-cavity surface-emitting laser (VCSEL). We detail the sequence of bifurcations that occurs when increasing the injection current, and which brings the laser from linear to elliptical polarization emission and then self-pulsating or even more complex chaotic dynamics of the light intensity. Continuation techniques allow us to follow the stable and unstable limit cycle solutions emerging from Hopf bifurcations, and therefore to interpret the frequency of the self-pulsating polarization dynamics. The fundamental frequency of the pulsating dynamics is either close to the laser relaxation oscillation frequency or close to the linear-birefringence-induced polarization mode frequency splitting, depending on the laser parameters. A systematic analysis of the parameter space allows us to identify two scenarios that are in excellent qualitative agreement with those reported in recent experiments. Our results provide, moreover, evidence for an interesting polarization mode hopping mechanism, i.e., a so-called deterministic mode hopping where the laser exhibits a chaotic and therefore random-like hopping between two states that are elliptically polarized and nonorthogonal.

Figure 1

(a) Stability diagram in the (phase anisotropy, injection current) plane of the linearly and elliptically polarized steady states for a positive amplitude anisotropy ${\gamma }_a=0.7$. The (red) dashed-dotted curve is the Y-LP Hopf bifurcation, the black dashed curve is the X-LP pitchfork bifurcation, and the (blue) solid curve is the EP Hopf bifurcation. (b) Frequencies associated with Hopf bifurcation, the legend is identical to (a) except for the dashed black curve, which gives the birefringence-induced frequency splitting. (c) The same as (b) but normalized by the relaxation oscillation frequency given by the (green) horizontal solid line.

Figure 2

Same as Fig. 1 but for a negative amplitude anisotropy of ${\gamma }_a=-0.7$.

Figure 3

Direct numerical simulation for a type-II switching corresponding to case 6 described in Sec. 3. (a) Fractional polarization, (b) averaged instantaneous ellipticity, (c) average output power on the $X$ axis (black) and $Y$ axis (red) of polarization, (d), (e) bifurcation diagrams of the output power on the $X$ axis (black) and the $Y$ axis (red) polarization, i.e., extrema of the corresponding time series, for increasing and decreasing injection currents, respectively. Data are averaged over hundreds of nanoseconds. The two vertical dashed lines indicate the current range where the laser shows nonlinear polarization dynamics. The labeled arrows point to the position of the time series detailed in Fig. 4.

Figure 4

Detailed characteristic of dynamical states for (a) $\mu =1.223$, (b) $\mu =1.323$, (c) $\mu =1.381$, (d) $\mu =1.416$, (e) $\mu =1.549$, and (f) $\mu =1.650$. The first column gives a logarithmic plot of the complex field FFT versus frequency, the second one is a representation of the output polarization given by the system trajectory in the $[\text{Re}\left(E_X\right),\text{Re}\left(E_Y\right)]$ plane, and the third gives the time traces observed for the dynamical states. Both the FFT and the time traces are polarization resolved, hence the gray (red) and black plots are for the $X$ and $Y$ axes, respectively.

Figure 5

Continuation results for a type-II switching corresponding to case 6 described in Sec. 3: the stable (unstable) part of the branches are the thick (thin) lines. (a) Maximum and minimum of $A=R_{+}^2/(\mu -1)$ versus normalized injection current. (b) Frequency of the periodic solutions versus normalized injection current. The X and Y-LPs are in black and red, respectively, and the EP states are in blue. The periodic solution created on the EP states is in green (a similar branch is created at the lower Hopf bifurcation but not represented) while the one created on the Y-LP state is in orange. Diamonds are Hopf, squares are saddle-node, stars are torus, and crosses are period doubling bifurcations. On diagram (b) the black dashed line gives $F_{\text{split}}$ and the red one $F_{\text{RO}}$ which is not displayed for $\mu >1.4$.

Figure 6

Direct numerical simulation for a transition to a complex dynamical state without switching corresponding to case 7 described in Sec. 3: (a) fractional polarization, (b) averaged instantaneous ellipticity, (c) average output power on the $X$ axis (black) and $Y$ axis (red) of polarization, (d) bifurcation diagram of the output power on the $X$-axis (black) and the $Y$-axis (red) polarization, i.e., extrema of the corresponding time series. Data are averaged over hundreds of nanoseconds. The dashed lines indicate the current range where the laser shows nonlinear dynamics.

Figure 7

Detailed characteristic of dynamical states for (a) $\mu =2.910$, (b) $\mu =3.552$, (c) $\mu =5.960$, (d) $\mu =7.298$, (e) $\mu =8.475$, and (f) $\mu =9.786$. See Fig. 4 caption for more information.

Figure 8

Continuation results for the destabilization of a linearly polarized state without polarization switching corresponding to case 7 described in Sec. 3: the stable (unstable) part of the branches are in thick (thin) lines. (a) Maximum and minimum of $A=R_{+}^2/(\mu -1)$ versus normalized injection current. (b) Frequency of the periodic solutions versus normalized injection current. The X and Y-LPs are in black and red, respectively, and the EP states are in blue. The periodic solution created on the EP states is in green (a similar branch is created at the low Hopf bifurcation but not represented) while the one created on the Y-LP state is in orange. Diamonds are Hopf, squares are saddle-node, crosses are period doubling, and stars are torus bifurcations. In panel (b) the black dashed line gives $F_{\text{split}}$ and the red one $F_{\text{RO}}$.

Figure 9

Polarization-resolved output power versus time for projections at ${45}^{\circ }$ (red) and $-{45}^{\circ }$ (black) of the $X$ polarization axis. These time series have been obtained by direct numerical integration using the same parameters as in Sec. 4 and at an normalized injection current $\mu \sim 1.41$.

Figure 10

Representation of the chaotic attractor observed in Fig. 9: projection of the Poincare sphere on the $(s_2/s_0,s_3/s_0)$ plane, $s_x$ being Stokes parameters as defined in Ref. [16]. The inset gives the two elliptically polarized states around which the system is chaotically oscillating; they have been obtained by simulation at $\mu \sim 1.34$ with the same parameters. In the Poincare sphere the EP steady states are the two unstable fixed points around which the “butterfly” attractor is turning.