## Abstract

Similar to edge-emitting lasers, vertical cavity surface emitting lasers (VCSELs) subjected to optical feedback are known for exhibiting erratic fluctuations of their optical power at slow and fast time scales; these are called low-frequency fluctuations (LFF). Here, we demonstrate both experimentally and numerically that the chaotic itinerancy in phase space associated with LFF has a deep connection with the creation of non-local correlations at multiple time scales between the two linear polarization modes. Our result provides a novel framework to interpret the unknown origin of spectral signatures in the optical power of chaotic lasers with optical feedback, which were observed in the past two decades.

© 2017 Optical Society of America

## 1. Introduction

Thanks to their numerous advantages, vertical cavity surface emitting lasers (VCSELs) overcome edge-emitting lasers (EEL) in several applications such as optical communications and sensing [1]. In addition, VCSELs are known to exhibit interesting properties related to polarization competition: although the light emitted at threshold is linearly polarized (LP), increasing the current may lead to polarization switching [2–4] and instabilities including deterministic polarization chaos [5].

When VCSELs are subjected to optical feedback, they undergo similar regimes of instabilities to those encountered in EELs [6], however with an additional interplay between the two competing linearized polarization modes (X- and Y-LP) and the optical feedback. This can unlock new dynamical behaviors that cannot be observed in conventional EELs, such as bistable polarization mode hopping [7], polarization coherence resonance induced by time-delay [8], polarization self-modulation leading to square waveforms [9,10], and vectorial dissipative solitons [11].

A chaotic dynamical regime of interest is the so-called low-frequency fluctuations (LFF). This consists of irregular power dropouts at a slow time scale [12]. The peculiarity of LFF in VCSELs is that two types of LFFs (Type I and II) can be observed depending on the competition between the X and Y-LP modes [13, 14]. In Type-I LFF, both modes shares approximately the same mean power and are correlated at the average LFF modulation time scale [14, 15]. In the case of Type-II LFF, however, the two modes are anti-correlated at that time scale, implying that one mode is dominant and shows sudden drops, while the other one is depressed and bursts from its quiescent state [13,16].

These correlation properties occurring on a slow time scale manifest themselves by a strong peak in the cross-spectral density (CSD) between the two LP modes. In addition, a double-peak structure appears in the vicinity of the external cavity frequency and seems to be also a generic feature of LFF regimes [17, 18], since it has been also documented in the RF spectrum of EEL with optical feedback [19–22, 27]. These time (resp. spectral) correlations are referred to as *non-local in time* as they results from the nonlinear interaction of optical fields and their delayed versions [23]. Multiple assumptions have been made as for the origin of the double-peak non-local correlation [18, 19, 27]. In Ref. [19], the double-peak power spectrum is explained by the influence of multiple round-trips in a slightly misaligned external cavity. In Ref [27], the same feature is explained by a competition between longitudinal modes. However, to the best of our knowledge, a detailed physical explanation of this deterministic phenomenon has remained elusive for more than a decade, thus motivating further theoretical and experimental exploration. In particular, the link between the double peak power spectrum and the properties of the deterministic chaotic dynamic remains to be elucidated.

We demonstrate here that the origin of the double-peak non-local correlation between the two LP modes in the LFF regimes results from chaotic itinerancy in phase space and depends on the type of LFF and hence the type of competition between the two LP modes. Finally, we provide a physical interpretation of these particular features and connect them with the trajectory of the system in the vicinity of ruins of stable and unstable external cavity modes. We perform a detailed study of the parametric influence (pumping current and time delay) and show that our interpretation for the double-peak holds also for EELs.

## 2. Experimental results

We design an experiment to measure the polarization dynamics within a large frequency range that encompasses the bandwidths of the chaotic dynamics and the relaxation oscillation frequency. The experimental setup, shown in Fig. 1(a), comprises an AlGaAs/GaAs VCSEL emitting at 850 nm with a solitary threshold current of ${J}_{\mathit{th}}^{s}=0.45\hspace{0.17em}\text{mA}$ at 15°C, which was measured under controlled thermal fluctuations within ±0.01°C. The output beam shows a single transverse mode with Gaussian profile in the range of currents where we operate the VCSEL i.e. up to 0.9 mA. The laser beam is collimated by a microscope objective before reaching a first 50/50 beam splitter. The first arm corresponds to the external cavity and uses a 99.9% reflectivity mirror and a variable attenuator to control the feedback strength. The measurement arm comprises a 50/50 beam-splitter, two polarizers and optical isolators to separate the two LP modes before coupling into two single-mode optical fibers. The measurement arm is designed so that we can guarantee a balanced coupling efficiency and identical traveling distances from the laser source. Finally, the two LP modes are measured by two identical photodiodes LeCroy OE425 with 4.5 GHz bandwidth and two Newport 1554-B with 12 GHz bandwidth which are connected to either a 4 GHz Lecroy Waverunner 640Zi oscilloscope with 40 GSamples/s or a 26.5 GHz Rohde & Schwarz FSW26 spectrum analyzer. The VCSEL emits in its Y-LP mode at threshold and switches to X-LP at pumping current 0.75 mA. X-(Y-) LP mode is the low (high) frequency mode hence the switching is of type I [2]. In the following, we shall operate in the range 0.48–0.56 mA, well below the switching point.

We select an external cavity length of *L _{ext}*≈75 cm corresponding to a time-delay of

*τ*= 2

*L*/

_{ext}*c*≈ 5 ns. This ensures the so-called long cavity regime, where the external cavity frequency

*f*= 1/

_{EC}*τ*is much smaller than the relaxation oscillation

*f*of the free-running VCSEL, which has been estimated to be about 1.37 GHz at 0.48 mA by relative intensity noise (RIN) measurements. In these experimental conditions with a pumping current range [0.43 mA, 0.6 mA], the VCSEL exhibits LFF dynamics, as shown in Fig. 1(b) and 1(c). We observe a typical instance of Type-II LFF which we can most likely attribute to the large dichroism of our device [13]. Anti-correlation between the LP mode dynamics manifests itself on a slow time scale as confirmed by CSD calculation in Fig. 2(d) at ∼20 MHz, which is mathematically defined by

_{RO}*Ĩ*(

_{x}*f*) (resp.

*Ĩ*(

_{y}*f*)) the Fourier Transform of the X-LP (resp. Y-LP) intensity and * denotes the complex conjugate. At a frequency

*f*≈

*f*, the CSD also shows an anti-correlated behavior. Most importantly, the CSD shows a double-peak structure with two peaks labeled

_{EC}*f*

_{−}and

*f*

_{+}in the spectral vicinity of

*f*. The frequency separation of Δ

_{EC}*f*=

*f*

_{+}−

*f*

_{−}is about 20 MHz. The CSD confirms that the dynamics at these two frequencies are anti-correlated between the two LP modes.

This result differs from previous studies [18] on Type-I LFF, where it was found that the X-LP and Y-LP modes are correlated at the second frequency *f*_{+}. This positive correlation was interpreted as the result of an unspecified mixing between *f _{EC}* and

*f*for VCSEL [18].

_{RO}The double peak structure observed in the CSD can also be retrieved from the respective RF spectra of each LP mode. In Fig. 2(a–c), we analyze the dependence of the structure at (*f*_{−}, *f*_{+}) by measuring the experimental power spectrum of the Y-LP mode (dominant LP mode) as a function of the external cavity delay [Fig. 2(a)] and the pumping current [Fig. 2(b) and 2(c)]. The spectrogram in Fig. 2(b) and 2(c) shows that the lowest frequency peak *f*_{−} is not affected by an increase of pumping current, while the highest one at *f*_{+} shifts towards higher frequencies when increasing the current. Interestingly, the frequency deviation of *f*_{+} evolves similarly to that of the average modulation LFF frequency (close to ∼20 MHz) with varying pumping current. The two *f*_{±} evolve similarly when varying the time-delay [Fig. 2(a)].

## 3. Numerical results

To gain additional insight, we have first verified that our experimental findings are reproducible by numerical simulations. Our model is based on the spin-flip model (SFM) [24], which is modified to account for optical feedback:

*E*are the complex slow-varying amplitudes of the (X,Y)-LP modes,

_{x,y}*N*is the carrier difference between the conduction and valence bands, and

*n*is the difference between the population inversions of the spin up and down radiation channels.

*κ*is the field decay rate,

*γ*is the decay rate of

_{n}*N*,

*γ*is the spin-flip relaxation rate,

_{s}*α*is the linewidth enhancement factor,

*μ*is the normalized injection current,

*γ*is the linear dichroism and

_{a}*γ*is the linear birefringence.

_{p}*τ*is the external cavity delay,

*η*is the feedback rate, and

*ϕ*

_{0}=

*ω*

_{0}

*τ*is the feedback phase, where

*ω*

_{0}is the optical frequency of the polarization mode in the absence of birefringence and feedback. Spontaneous emission noise is taken into account through the terms

*F*= [

_{x}*β*(

_{sp}*N*+

*n*)/2]

^{1/2}

*ξ*

_{1}+ [

*β*(

_{sp}*N*−

*n*)/2]

^{1/2}

*ξ*

_{2}and

*F*=

_{y}*iF*.

_{x}*β*is the spontaneous emission rate and

_{sp}*ξ*

_{1,2}are two complex uncorrelated white Gaussian noises with zero mean and unitary variance.

It is known that the VCSEL dynamics and its the polarization properties with optical feedback depend both on the laser and feedback parameters. Here, we choose *τ* = 5 ns to match the experimental condition. *γ _{p}* = 5 rad.ns

^{−1}corresponds to the experimentally measured frequency splitting between the X- and Y-LP modes,

*i.e.*2

*γ*/2

_{p}*π*= 1.59 GHz. We consider

*γ*= 0.5 rad.ns

_{a}^{−1}and

*γ*= 5 ns

_{s}^{−1}so that the model reproduces LFF Type-II which is the dynamic observed experimentally. Finally,

*η*and

*μ*are our bifurcation parameters. The other parameters have numerical values that can be found in the literature [13–15,28] :

*γ*= 1 ns

_{n}^{−1},

*β*= 10

_{sp}^{−4}ns

^{−1},

*α*= 3. LFF Type-II polarization dynamics is shown in Fig. 3(a)and 3(b). The corresponding power spectra are plotted in Fig. 3(c) and 3(d). The power spectra of the X- and Y-LP modes show also a double peak signature close to

*f*= 200 MHz. In Fig. 3(e), our model predicts that both peaks of the double peak structure leads to an anti-correlated polarization mode dynamics. Qualitatively similar observations can be made by adjusting the

_{EC}*γ*(resp.

_{a}*γ*) value between 0.01 rad.ns

_{s}^{−1}and 0.7 rad.ns

^{−1}(resp. 2 ns

^{−1}and 11 ns

^{−1}). For larger values of

*γ*and

_{a}*γ*, the system still operates in LFF Type-II regime as defined in [13]. However, is that case, the X-LP mode is too depressed and therefore the computed CSD doesn’t show a discernible double-peak. We are also able to induce LFF Type-I (not shown here) by adjusting the VCSEL parameters (for example with

_{s}*γ*= 0.001 rad.ns

_{a}^{−1}and

*γ*= 0.01 rad.ns

_{p}^{−1}and the other parameters identical to those of Fig. 3). In LFF Type-I, we find a strong correlation at low frequency and also at

*f*

_{+}, in agreement with Ref. [18].

Additional insight into the physical origin of the double peak can be provided by analyzing the VCSEL dynamics in the projection of the infinite dimension phase space. Figure 4(b) shows the projection of the trajectory in the plane (|*E _{y}*(

*t*)|,

*ϕ*(

_{y}*t*) −

*ϕ*(

_{y}*t*−

*τ*) +

*ϕ*

_{0}) where

*ϕ*(

_{y}*t*) is the phase of the dominent Y-LP mode and

*ϕ*(

_{y}*t*) −

*ϕ*(

_{y}*t*−

*τ*) +

*ϕ*

_{0}is the phase difference between the Y-LP mode and its time-delayed version. As demonstrated theoretically for EELs [12,25,26], the LFF trajectory in phase space corresponds to a chaotic itinerancy among destabilized external-cavity modes (ECMs). Numerically, the clearest insight is obtained when the system shows a limited set of ECMs and when the trajectory stays long enough around each ECM. These conditions can be achieved for a value of

*η*(

*γ*) slightly smaller (larger) than the one used in Fig. 3. As mentioned in the previous paragraph, for the value of

_{s}*γ*= 50 ns

_{s}^{−1}, the dynamic is still an LFF Type-II but the X-LP mode is highly depressed. Therefore, we focus the following study on the dominant mode only, i.e. the Y-LP mode. Its power spectrum associated with this complex trajectory in phase space shows clearly a double-peak structure at frequency close to

*f*[Fig. 4(a)]. We observe that the chaotic trajectory is attracted towards the ECM with maximum gain corresponding to the largest phase difference but visits a set of other ECMs. While pulsating around the position of the maximum gain mode (represented by a diamond in Fig. 4(c)), the trajectory is repelled by the close proximity of a saddle-type ECM called anti-mode [represented by a square in Fig. 4(c)].

_{EC}To give a dynamical interpretation to the observed double-peak structure, we analyze the frequency content of phase space trajectories that remain in the vicinity of either the maximum gain mode or the antimode. Figure 5(a) shows a typical drop-out event of LFF dynamics associated with the trajectory in the projected phase space of Fig. 4(b). Similar time-traces can be found at different time-locations in the numerical simulation as the system encounters many times similar trajectories in phase space after a drop-out. All of these typical time traces have two distinct dynamical behaviors marked by the orange and green time-windows : the orange window displays a dynamics corresponding to the trajectory around the maximum gain mode [orange trajectories in 4(c)] whereas the green window shows the typical time evolution around the corresponding antimode. In Fig. 5(b), we show the RF spectrum obtained by computing the concatenation of a large number of such orange time-windows. We notice the complete suppression of the peak at *f*_{−} in the double peak structure. In contrast by considering the additional green time-windows, the *f*_{−} component of the double-peak structure arises in Fig. 5(c). In Fig. 5(d) and 5(e), we further increase the window size in the phase-space and, thus, consider more points in the time-domain. Doing so, we recover the original double-peak structure shown in Fig. 4(a).

Finally, we show in Fig. 6 that the double-peak structure studied numerically behaves similarly as in the experiment when varying the parameters *τ* and *μ*. Indeed, in Fig. 6(a), the double peak structure follows the external cavity frequency when varying the delay. Furthermore, Fig. 6(b) show that when increasing the pumping current, the first frequency component *f*_{−} of the double peak structure remains unchanged while the second frequency component *f*_{+} shifts to higher values. This is also the case for the low frequency modulation. For higher values of current *μ* > 1.025, the dynamic of the system moves from LFF regime towards the so called coherence-collapse regime with no clear low frequency signature which is translated in the spectrogram by a slow fading of the low frequency component. Numerical simulations and experiments confirm that there is a strong link between the low frequency component corresponding to power dropouts and the the highest frequency *f*_{+} of the double peak structure. Indeed, (anti-)correlation at the low frequency leads to (anti-)correlation at *f*_{+} and when the low frequency shifts with varying the parameters, *f*_{+} also shifts in frequency.

## 4. Discussion

In this work, we have analyzed in detail how polarization mode competition in VCSEL interplays with optical feedback induced instabilities. The polarization mode competition manifests itself as a correlation or anticorrelation at the low frequency of LFF dynamics, as already evidenced in separated experiments [16, 18], but we unveil a new feature, which is the dependence of the correlation (or anticorrelation) between the slow-time scale of the LFF and a faster time scale close to the external cavity frequency. This complex interplay at multiple time scales yields a double-peak structure in both the CSD and the power spectrum with correlation properties depending on the type of polarization competition.

Appart from the correlation analysis, similar double-peak spectral signatures have been reported in previous studies on external-cavity laser diode in different dynamical regimes. This naturally raises the question of the generality of such spectral properties and their physical interpretation. For example, in Ref. [19], the explanation of the double-peak signature relied on multiple round-trips in the external cavity of a chaotic VCSEL. In Ref. [27], the authors based their explanation on the competition between longitudinal modes of an EEL subjected to a weak feedback. Finally, in Ref. [18], it was proposed that a still unexplained nonlinear mixing between the external-cavity frequency and the relaxation-oscillation frequency of a VCSEL was responsible for the appearance of the double-peak structure.

Here, we provide an alternative explanation relying on the chaotic itinerancy in the phase space due to the interplay between stable and unstable ECMs in LFF regime. Interestingly, and by contrast to these earlier explanations, this explanation does not require the inclusion of longitudinal mode competition or multiple round trips in cavity. This is evidenced in Fig. 7(b) where we simulate the single-mode LK equations with a single round-trip feedback [28,29] :

*E*(

*t*) =|E(t)|

*e*

^{iϕ(t)}is the slowly varying envelop of the complex electric field,

*N*is the carrier density in the active region,

*G*

_{N,|E|2}is the optical gain,

*α*is the linewidth-enhancement factor,

*∊*is the saturation coefficient,

*N*

_{0}is the carrier density at transparency,

*G*is the differential gain,

_{N}*ω*

_{0}is the angular frequency of the solitary laser,

*γ*is the feedback rate,

*τ*is the photon lifetime,

_{p}*τ*is the carrier lifetime,

_{s}*J*is the threshold current,

_{th}*p*is the pumping factor and

*τ*the external cavity delay. Used parameters are listed in the caption of Fig. 7 and lead to a chaotic LFF dynamics. Except for the specific correlation properties driven by the VCSEL polarization competition in Fig. 3, the double peak structure is well reproduced numerically.

Our work therefore confirms that the dynamical scenario leading to the double-peak structure in both the CSD and power spectrum is the manifestation of the combined attraction and repelling motions of ruins of modes and antimodes, respectively. In VCSELs the polarization competition however leads to an interesting unique feature: the correlation (anticorrelation) at the low-modulation frequency yields the correlation property at the shifted frequency *f*_{+}.

## 5. Conclusion

In summary, we have analyzed both numerically and experimentally the spectral properties of the linearly polarized modes of a chaotic VCSEL with optical feedback. The polarization mode competition, which interplays with the feedback instability, is known to impact the intermodal correlation properties at a slow time scale in the LFF regimes: (i) correlation with Type-I LFF or (ii) anti-correlation with type-II LFF. However, what we discover is that these correlation properties at the slow time-scale are transported to a faster time-scale located close to the vicinity of the external cavity frequency. This multiple time scales transport yields a so-called double-peak correlation signature in both the cross-spectral density (intermodal spectral information) and the individual power spectrum of each polarization modes. Furthermore, we show that the origin of this double peak could be explained by the chaotic itinerancy in LFF regimes occurring in the vicinity of the ruins of modes and anti-modes. Our study also shows that a simple model of an EEL with optical feedback is sufficient to observe the appearance of the double peak structure.

## Acknowledgments

C.H.U., D.R., S.B. and M.S. acknowledge the support of the Conseil Regional de Lorraine, the Préfecture de Lorraine, the Fondation Supelec, the Fond Europeen pour le Developpement Regional (FEDER) through the projects PHOTON (FEDER, 2014–2015) and APOLLO (FEDER and FNADT, 2015–2016), the Inter-University Attraction Pole (IAP) program of BEL-SPO through the project Photonics@be grant number IAP P7/35, and the Agence Nationale de la Recherche (ANR) through project TINO (ANR-12-JS03-005).

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