1. Introduction

Going from the standard edge-emitting heterostructure to the innovative vertical-cavity surface-emitting laser (VCSEL) was a major step forward in laser technology providing a circular beam, significant reduction of the threshold and possibilities for on-chip testing capability. However their circular geometry resulted in one of their main drawbacks: laser emission is accompanied by polarization instabilities inducing polarization switching [1–7], and polarization chaos [8]. For a recent review see Ref. [9]. These instabilities are attributed to the competition between the two preferred orthogonal and linearly polarized (LP) modes of the VCSEL. To explain the resulting dynamics, various mechanisms such as thermal lensing [10], gain dependency with the current [2], and spin-relaxation mechanisms (so-called spin-flip model or SFM) have been proposed [11–13].

Among these instabilities, the emergence of elliptically polarized (EP) states has been studied [5, 14], and they later appear to be the roots of chaotic mode hopping dynamics [6–8, 13]. Although the bifurcation scenarios observed experimentally are in complete agreement with the theoretical predictions, the mechanism selecting the EP states remains to be clarified. Indeed, we experimentally observe that EP states always appear with the same orientation [7], whereas in theory, e.g. in the SFM framework, symmetrical EP with respect to LP modes are strictly equivalent and the selection is made at random [12].

Here we report on a peculiar phenomenon calling for a closer look on the bifurcations from elliptically polarized states: we observe a bistability between two limit cycles (LCs) oscillating around two EPs with main axes symmetrical with respect to the LP at threshold. The two LCs exhibit slightly different frequencies and a clear hysteresis cycle is observed. Theoretically, we demonstrate that these features can, in fact, be accurately reproduced in the SFM framework when including a misalignment between phase and amplitude anisotropies [15,16]. In this contribution, we therefore bring new light about the importance of EP state asymmetry for VCSEL nonlinear dynamics and provide further evidence validating the SFM model.

The paper is organized as follows: we first present the setup and experimental observations in section 2. Then we describe the theoretical model we use including the anisotropy misalignment in section 3. In section 4, we give the simulation results and we conclude in section 5.

2. Experimental observation: bistability between two limit cycles

The VCSEL used in this experiment is a single longitudinal mode InGaAs submonolayer quantum dot laser emitting at 990 nm that has already been described in detail elsewhere [7]; the laser and its temperature are controlled by a Profile LDC 1000 controller. The laser beam is focused using an aspheric lens and coupled into a multimode fiber for the measurements. Reflections from the fiber-facet are avoided using an optical isolator and polarization resolved measurements are achieved by rotating a half-waveplate located just before the optical isolator. In this contribution, the axis of the linear polarization at threshold is considered as the reference for polarization orientation, and as such, will be identified as the polarization at 0°. To record time-series we use a fast-photodetector (NewFocus 1554-B, 10kHz – 12GHz bandwidth), an electronic amplifier (NewFocus 1422LF, 20 GHz) and an oscilloscope (Agilent DSOX92504A, 25 GHz, 80 GS/s). The frequency content of the measured time-series are obtained through a Fast Fourier Transform (FFT).

The device temperature is set to 22°C and, in this case, the laser turns into chaos at high injection current [8]. At threshold, around 0.3 mA, the laser emits LP light but, similarly to what is described in Ref. [17], it is quickly destabilized towards an EP state shown in Fig. 1(a). The first limit cycle (LC) then appears oscillating around this EP with a frequency of about 6.25GHz, see the time-series in Fig. 1(b) and the frequency evolution in Fig. 1(e). The amplitude of the LC increases along with the current but, around 1.95 mA, it is destabilized and a new limit cycle appears. The latter exhibits a slightly larger frequency around 6.55 GHz but, even more importantly, the cycle oscillates around the second, symmetrical EP with respect to the polarization of the laser at threshold, see the inversion of the +45° and −45° curves shown in Fig. 1(a). At this point, decreasing the current unveils the region of bistability between the two LCs - in Fig. 1(a), 1(d) and 1(f) - until the laser settles back on the first LC for currents below 1.77 mA. Moreover, before the switching, we observe that the cycle fades out to an EP steady-state as can be seen very clearly in Fig. 1(f) where the FFT peak vanishes before the switching. Due to a high sensitivity of the device to experimental conditions, e.g. stress from the probes, the range of current at which the bistability appear can change between different measurements. In particular the current value at which polarization switchings occur in Fig. 1(a) are slightly different from what can be concluded from panels 1(c)–1(d) or 1(e)–1(f).

 

Fig. 1 Experimental observations of the limit cycle bistability. (a) Polarization resolved LI curve at +45° and −45°, with respect to the LP at threshold, in red and black respectively, for increasing (solid) and decreasing (dashed) injection current. (b) Time-series recorded at a current of 1.85 mA when increasing (blue) and decreasing (red) current. (c)–(d) Plot of the time-series extrema versus increasing (top, c) and decreasing current (bottom, d). (e)–(f) FFT of the time-series versus increasing (top, e) and decreasing current (bottom, f). Time series considered for panels b to f are recorded at 0° with DC removed.

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3. Asymmetric SFM model: misaligned phase and amplitude anisotropies

To describe the behavior of the investigated VCSEL, we use the SFM model [11, 12]. In this framework we consider the left and circular polarizations as two competing emission processes with two separated carrier reservoirs coupled through the spin-flip relaxation processes. However, as will be shown in the following, to reproduce the reported behavior, we need to consider misaligned phase and amplitude anisotropies as described in Ref. [16]. As a result, the SFM model writes as follow:

With E_± the electrical field for the right (+) and left (−) circular polarization, N the normalized total carrier population and n the normalized carrier population difference between the two reservoirs. κ is the electric field decay rate in the cavity and γ_s the spin-flip relaxation rate that accounts for the spin homogenization of the spin up and spin down carrier populations. α is the linewidth enhancement factor, μ is the normalized injection current. Finally, the phase anisotropy or birefringence is γ_p whereas γ_a is the amplitude anisotropy. θ is defined as the angle between the axis of maximum frequency and the axis of maximum losses. All parameters and variables are dimensionless and the time is normalized by the carrier lifetime.

At this point it is convenient to define: $\overline{{\gamma }_p^{\pm }}={\gamma }_p\mp \mathit{sin}(2.\theta ).{\gamma }_a$ and $\overline{{\gamma }_a}=\mathit{cos}(2.\theta ).{\gamma }_a$. Thus, similarly to what is done in Refs. [13, 18], we can transform Eq. 1 into three real equations for the amplitude of the two circular polarizations R₊ and R₋ and their phase difference ϕ:

4. Simulation results: asymmetry impact on laser dynamics

With the given set of parameters and without any asymmetry, i.e. θ = 0, we obtain the dynamical evolution given in Fig. 2. At threshold the laser emits LP light - see Fig. 2(a) - but an increase of the injection leads to a pitchfork bifurcation at μ ∼ 3.4 which creates two symmetrical EP states - see Fig. 2(b). Because the system is completely symmetrical, the two EPs are strictly equivalent, the polarization selection is therefore only made at random depending on the noise and initial conditions. With a further increase of the current, these steady-states are both destabilized by two identical Hopf bifurcations at μ ∼ 3.5, hence creating two symmetrical LCs oscillating around the now unstable EP states - see Fig. 2(c). Finally, for μ > 5, the system experiences a cascade of period doubling bifurcation and enters a large region of chaotic dynamics [8,13]. In addition, we observe no difference for the described scenario when decreasing the injection current.

 

Fig. 2 Evolution of the emission without asymmetry, i.e. with aligned phase and amplitude anisotropies. (a)–(d) Polarization of the emitted light given by the trajectory of the system in the (Re(E_X), Re(E_Y)) phase plane in the 4 cases identified in (f). (e) Polarization resolved LI curve for −45°, 0°, +45° and 90° with respect to the polarization at threshold in black, green, red and blue respectively. (f) Extrema of the polarization resolved output power time-series at +45° and −45° in red and black respectively.

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Thus, without asymmetries in the SFM model, we observe an identical scenario for both symmetrical EP states. In other words, we already have a bistability between the two elliptical steady-states and the limit cycles born on those states, but no deterministic switching can occur between them: the jumps between the two symmetrical evolution can only occur through a noise-induced process. The reproducible hysteresis loop and frequency splitting therefore cannot be explained in this framework.

On the other hand, when considering a small misalignment between the phase and amplitude anisotropies, i.e. considering a small asymmetry in the model, we observe a very different behavior. In Fig. 3, we present the results obtained with θ = −0.023; this very specific angle between the two anisotropies has been chosen to provide a good match with the features observed in the experiments. For simplicity, we call EP₊ (EP₋) the EP state exhibiting a dominant emission at +45° (−45°). We also designate the two limit cycles oscillating around these states LC₊ and LC₋.

 

Fig. 3 Impact of the asymmetry on the VCSEL dynamics. (a)–(d) Polarization resolved LI curves for −45°, 0°, +45° and 90° with respect to the polarization at threshold in black, green, red and blue respectively. (b)–(e) Extrema of the polarization resolved output power time-series at +45° and −45° in red and black respectively. Inset in (e) shows a zoom of the transition between the two limit cycles. (c)–(f) Normalized radio-frequency spectrum of the polarization resolved time-series at +45°. Those results are given for increasing (top) and decreasing (bottom) normalized injection current. The blue dashed vertical lines show the limits of the bistability region.

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First, we observe that the pitchfork bifurcation destabilizing the LP at threshold has disappeared. As a result, for low injection current the polarization is not linear anymore but slightly EP₊ elliptical. For levels of current where the pitchfork bifurcation would appear when no asymmetry is considered, we observe a smooth transition towards an elliptical EP₊ emission with increasing ellipticity, see Fig. 3(a). This observation clearly confirms that the two EP states are no longer equivalent as the asymmetry obviously strengthens the EP₊ emission.

Secondly, we report an hysteresis cycle and therefore a bistability between LC₊ and LC₋, as can be seen from Fig. 3(a)–3(b) and 3(d)–3(e). Despite the misalignment induced asymmetry, a Hopf bifurcation still occurs around μ = 3.5 hence creating LC₊. The amplitude of the cycle largely increases along with the current until it is destabilized by a cascade of period doubling bifurcations at μ ∼ 4.9 (Fig. 3(b)). Without asymmetry the system would then enter in a region of polarization chaos [8], but here we find a polarization switching towards a second limit cycle which appears to be LC₋. The FFT of the output power time-series confirms a slight frequency shift, about 0.07 between the two cycles - i.e. about 70 MHz for a carrier lifetime of 1 ns. From this point, decreasing the injection current unveils a large bistability region between the two time-periodic solutions oscillating with different polarization and delimited by the two vertical dashed lines of Fig. 3(b)–3(e). The amplitude decreases along with the current until it reaches a steady EP₋ emission around μ = 3.5 (see inset of Fig. 3(e)). This transition is also visible on the FFT spectrum, Fig. 3(f), as the frequency peak vanishes close to the switching point. Indeed the EP₋ is stable only in a small region and the system quickly settles back on LC₊. The frequency shift we observe here is slightly larger around 0.1, i.e. about 100 MHz for a carrier lifetime of 1 ns, i.e. the same order of magnitude than in the experiment.

These results clearly demonstrate that misalignment as small as θ = −0.023 - which makes the $\overline{{\gamma }_p^{\pm }}$ differ by less than 3% - can induce completely different dynamical behavior than that of the aligned case. We described here only the bifurcation scenario that matches our experimental results but we also observed many other features that are left for discussions elsewhere.

5. Conclusion

To conclude, we experimentally unveil a peculiar behavior in a free-running VCSEL: bistability between two limit cycles. The two cycles exhibit different amplitudes, slightly different frequencies, and more importantly they oscillate around two symmetrical elliptically polarized states with respect to polarization at threshold. Theoretically we explain this behavior framework of the SFM model by taking into account a small misalignment between the phase and amplitude anisotropies. We clearly show that all the experimental features are reproduced, hence showing an excellent qualitative agreement between experiment and theory. This study therefore brings new light on the role and interplay of phase and amplitude anisotropies in the nonlinear dynamics of VCSELs.