Stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers

A detailed stability and bifurcation analysis of spin-polarized vertical-cavity surface-emitting lasers (VCSELs) is presented. We consider both steady-state and dynamical regimes. In the case of steady-state operation, we carry out a small-signal (asymptotic) stability analysis of the steady-state solutions for a representative set of spin-VCSEL parameters. Compared with full numerical simulation, we show this produces surprisingly accurate results over the whole range of pump ellipticity, and spin-VCSEL bias up to 1.5 times the threshold. We then combine direct numerical integration of the extended spin-flip model and standard continuation technique to examine the underlying dynamics. We find that the spin VCSEL undergoes a period-doubling or quasiperiodic route to chaos as either the pump magnitude or polarization ellipticity is varied. Moreover, we find that different dynamical states can coexist in a finite interval of pump intensity, and observe a hysteresis loop whose width is tunable via the pump polarization. Finally we report a comparison of stability maps in the plane of the pump polarization against pump magnitude produced by categorizing the dynamic output of a spin VCSEL from time-domain simulations, against supercritical bifurcation curves obtained by the standard continuation package auto. This helps us better understand the underlying dynamics of the spin VCSELs.

Figure 1

Eigenvalues of the spin-polarized VCSEL equilibrium in the complex plane for ${\gamma }_s=80{\mathrm{ns}}^{-1}, \eta =2$, and $P=-1$. Red dots stand for numerical results, while black squares represent approximate results of the critical eigenvalues.

Figure 2

The real part (thick) and the imaginary part (thin) of the critical eigenvalues as a function of ${\gamma }_s$ for the (a) in-phase and (b) out-of-phase equilibrium. The dashed lines are the asymptotic approximations (16) and (23). Parameters are the same as those in Fig. 1.

Figure 3

The real part of the critical eigenvalues of the in-phase (red) and out-of-phase (blue) equilibrium as a function of ${\gamma }_s$, for different values of $\eta$ and $P$. (a1)–(a3) $P=-1$, (b1)–(b3) $P=-0.8$, (c1)–(c3) $P=-0.6$, and (d1)–(d3) $P=-0.4$. (a1)–(d1) $\eta =1.5$, (a2)–(d2) $\eta =1.2$, and (a3)–(d3) $\eta =1.1$. The dashed lines are the asymptotic approximations (16) and (23).

Figure 4

An example of comparison between the approximated and simulated oscillation frequency as a function of ${\gamma }_s$, with (inset) the rf spectrum giving the evidence of period one oscillation. Parameters are the same as those in Fig. 2.

Figure 5

Bifurcation diagrams of the RCP intensity as a function of $P$ for (a) $\eta =2$, (b) 4, and (c) 6. ${\mathrm{H}}_1$ and ${\mathrm{H}}_2$ denote two different Hopf bifurcations. The spin-flip relaxation rate ${\gamma }_s=30{\mathrm{ns}}^{-1}$.

Figure 6

The bifurcation diagram in the ($\eta , P$) plane. The period-one oscillation region is highlighted in yellow. The spin-flip relaxation rate ${\gamma }_s=30{\mathrm{ns}}^{-1}$ and all other parameters are the same as those in the preceding section.

Figure 7

Bifurcation diagram of the RCP intensity as a function of $\eta$ for (a) $P=-0.8$, (b) $-0.6$, (c) $-0.4$, and (d) $-0.1$. The spin-flip relaxation rate ${\gamma }_s=30{\mathrm{ns}}^{-1}$. Other parameters are specified in the text. The inset in (d) is a magnification of the period-doubling bifurcation region.

Figure 8

Period-doubling route to chaos; ${\gamma }_s=30{\mathrm{ns}}^{-1}, P=-0.1$ and, from (a) to (d), $\eta =3$, 3.15, 3.25, and 4, respectively.

Figure 9

Bifurcation diagram of the RCP intensity as a function of $P$ for $\eta =8.5$. Other parameters are the same as those in Fig. 6.

Figure 10

Quasiperiodic route to chaos; $\eta =8.5$ and, from (a) to (c), $P=-0.8, -0.587$, and $-0.3$, respectively.

Figure 11

(a),(c) The bifurcation diagrams and (b),(d) calculated color maps in the ($\eta , P$) plane. (a),(b) ${\gamma }_s=30{\mathrm{ns}}^{-1}$ and (c),(d) $60{\mathrm{ns}}^{-1}$. In (b) and (d), the cw, P1, P2, and complicated dynamics including chaos are marked in white, dark blue, light blue, and other colors, respectively.

Figure 12

Bifurcation diagrams of the RCP intensity as a function of $\eta$ for (a) $P=-0.04$, (b) $-0.1$, and (c) $-0.4$. The spin-flip relaxation rate ${\gamma }_s=30{\mathrm{ns}}^{-1}$. Other parameters are the same as those in Fig. 7. The red part is obtained for increasing $\eta$ and the blue part is obtained for decreasing $\eta$.

Figure 13

Bifurcation diagram of the RCP intensity as a function of $P$ for $\eta =4$. Other parameters are the same as those in Fig. 9. The red part is obtained for increasing $P$ and the blue part is obtained for decreasing $P$.

Figure 14

Bifurcation diagrams of the RCP intensity as a function of $P$ for (a) $\eta =4.12$, (b) 6, and (c) 10. The spin-flip relaxation rate ${\gamma }_s=60{\mathrm{ns}}^{-1}$. Other parameters are the same as those in Fig. 9. The red part is obtained for increasing $P$ and the blue part is obtained for decreasing $P$.