Phase dynamics in vertical-cavity surface-emitting lasers with delayed optical feedback and cross-polarized reinjection

We study theoretically the nonlinear polarization dynamics of vertical-cavity surface-emitting lasers in the presence of an external cavity providing delayed optical feedback and cross-polarized reinjection. We show that, far from the laser threshold, the dynamics remains confined close to the equatorial plane of a Poincaré sphere with a fixed radius. It entails that the evolution of the system is described by two phase variables: the orientation phase of the quasilinear polarization and the optical phase of the field. We explore the complex modal structure given by the double reinjection configuration and how it evolves between the cases of single cross-polarized reinjection and single optical feedback, hence disclosing the relationship with the Lang-Kobayashi model. We also reinterpret the square-wave switching observed by J. Mulet et al. [Phys. Rev. A 76, 043801 (2007)] in terms of phase kinks.

Figure 1

Angular representation of the VCSEL dynamics on the Poincaré sphere.

Figure 2

Stability diagram of the solitary VCSEL far from threshold. The condition ${\gamma }_a+\alpha {\gamma }_p=0$ separates the two parameter regions where the stable emission is along the LP-$y$ or the LP-$x$ axis. Stable (unstable) emission is depicted by a minus sign (plus sign).

Figure 3

Monochromatic solutions of Eqs. (35) and (36) for $\eta =0$, $\beta =0.05$, and ${\tau }_r=1000$. The solutions are arranged around a tube defined by the function $\Phi \left(\omega \right)$ and $\theta \left(\omega \right)$ assuming $\omega$ is a continuous variable. Stable and unstable solutions are represented in green (light gray) and red (dark gray), respectively. The number of stable and unstable solutions $\left(S,U\right)$ is $\left(18,19\right)$.

Figure 4

Monochromatic solutions of Eqs. (35) and (36) for $\eta =0$, $\beta ={\beta }^{☆}+0.01$, and ${\tau }_r=1000$. There are 127 unstable solutions. Stable and unstable solutions are represented in green and red, respectively.

Figure 5

Square-wave switching as phase kinks. (a1) and (a2) The intensity of the $X$ and $Y$ polarization over different time scales reconstructed from the polarization orientation $\Phi .$ The traces are shifted for clarity and are actually in perfect antiphase. (b1) and (b2) The polarization orientation $\Phi$ and (c1) and (c2) the optical phase $\Sigma$ from which we subtracted a drift ${\omega }_{y}t$. The parameters are $\eta =0$, $\beta ={\beta }^{☆}+4\times {10}^{-3}$, and ${\tau }_r=1000$.

Figure 6

Monochromatic solutions of Eqs. (35) and (36) for $\eta =0.025$, $\beta =0.05$, and ${\tau }_r=1000$. (a) $\Delta \tau ={\tau }_f-{\tau }_r=0$, (b) $\Delta \tau =20$, (c) $\Delta \tau =40$, and (d) $\Delta \tau =100$; the numbers of stable and unstable solutions $\left(S,U\right)$ are $\left(24,31\right)$, $\left(14,55\right)$, $\left(21,36\right)$, and $\left(15,36\right)$, respectively.

Figure 7

Monochromatic solutions of Eqs. (35) and (36) for $\eta =0.025$, $\beta =0.05$, and ${\tau }_r=1000$. (a) ${\tau }_f=2{\tau }_r$, (b) ${\tau }_f=3\tau {}_r$, (c) ${\tau }_f=4{\tau }_r$, and (d) ${\tau }_f={\tau }_r/2$; the numbers of stable and unstable solutions $\left(S,U\right)$ are $\left(17,52\right)$, $\left(55,36\right)$, $\left(82,47\right)$, and $\left(15,30\right)$, respectively.

Figure 8

Square-wave switching dynamics in the presence of XPR. Time series (inset) and power spectrum for $\eta =0$, ${\gamma }_a=-0.09$, $\beta =0.25$, and ${\tau }_r=500$. The signal is very irregular, and the period is noticeably slower than $2{\tau }_r$. The power spectrum does not show the signature of a square-wave signal with only odd harmonics.

Figure 9

Square-wave switching dynamics in the presence of both XPR and PSF. Time series (inset) and power spectrum for $\eta =0.05$, ${\gamma }_a=-0.09$, $\beta =0.25$, ${\tau }_r=500$, and ${\tau }_f=1020$. The signal is much more regular, and the period is much closer than $2{\tau }_r$. The power spectrum shows the signature of a square-wave signal with only odd harmonics.