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Nonlinear dynamics of Vertical-Cavity Surface-Emitting Lasers (VCSELs) induced by optical injection, optical feedback, current modulation and mutual coupling is reviewed. Due to the surface emission and cylindrical symmetry VCSELs lack strong polarization anisotropy and may undergo polarization switching. Furthermore, VCSELs may emit light in multiple transverse modes. These VCSEL properties provide new features to the rich nonlinear dynamics induced by an external perturbation. We demonstrate for the case of orthogonal optical injection that new Hopf bifurcation on a two-polarization-mode solution delimits the injection locking region and that polarization switching and injection locking of first-order transverse mode lead to a new resonance tongue for large positive detunings. Similarly, the underlying polarization mode competition leads to chaotic-like behavior in case of gain switching and the presence of two transverse modes additionally reduces the possibility of regular dynamics. The bistable property of VCSEL makes it possible to investigate very fundamental problems of bistable systems with time-delay, such as the coherence resonance phenomenon. We also demonstrate that the synchronization quality between unidirectionally coupled VCSELs can be significantly enhanced when the feedback-induced chaos in the master laser involves both orthogonal LP fundamental transverse modes.

The change of the semiconductor laser cavity geometry from facet to surface-emitting gave birth to the vertical-cavity surface-emitting lasers (VCSELs) [

Locking of the frequency of the injected (slave) semiconductor laser to the one of the injecting (master) laser has long been known [

Gain switching of semiconductor lasers has received a lot of attention, considering not only its potential to generate ultrafast sharp pulses but also because it can lead to complex dynamics such as period doubling and possibly chaotic pulsating [

Time-delayed optical feedback (OF) perturbs the steady state laser emission and leads to complex dynamics [

Synchronization of chaotic systems has been a subject of both theoretical and experimental studies [

In this paper we summarize recent results on nonlinear dynamics of VCSELs for the case of optical injection, gain switching, optical feedback, and unidirectional chaos synchronization. The paper is organized as follows: in Section

The experiments on orthogonal OI have been performed with oxide-confined GaAs quantum well VCSEL emitting at 845 nm as a slave laser (SL). The solitary VCSEL is polarization bistable in the injection current region 2.25 mA–4.60 mA, displaying type II PS (from the low to the high frequency LP mode [

In Figure

Polarization-resolved optical spectra of the VCSEL subject to optical injection at a frequency detuning of −4 GHz. (a)

Figure

Experimental mapping of bifurcations to qualitatively different polarization dynamics in a VCSEL subject to orthogonal optical injection. Labels and symbols are explained in the text. After [

In Figure

Experimental mapping of polarization switching and transverse modes competition extended towards large positive detunings. After [

It is possible to reproduce theoretically the experimental results on the base of a set of rate equations that accounts for the polarization properties of VCSELs, namely, the spin-flip model (SFM) [

In addition to the coupling of the polarizations states through the carrier density (

Figure ^{−1}, ^{−1}, ^{−1}, ^{−1}, and ^{−1}. Qualitative changes in the VCSEL dynamics are detected and followed using the continuation package AUTO 97. Different bifurcation curves are plotted: a saddle-node (SN), two Hopf (H_{1} and H_{2}), and a torus (TR). The supercritical and subcritical parts of each bifurcation curve are represented in black and gray, respectively. When increasing the injection strength, the VCSEL switches its polarization to that of the injected field; these “PS off” (_{1} are bifurcations on a stationary injection-locked state and have also been reported in the case of optically injected edge-emitting laser. In the conventional case of EEL, the locking region is then delimited by the codimension two-point G_{1} where SN and H_{1} intersect [_{1} but also by a new bifurcation H_{2} (see Figure _{1}. Apart from its effect on the locking, H_{2} also affects the PS mechanism. The smallest injection strength needed to achieve PS is located on H_{2} and corresponds to a dramatic change in the PS curve (_{1} and PD_{2}, also affect the switching mechanism leading to a second local minimum denoted by

Bifurcation diagram in the plane injection strength-frequency detuning calculated by the SFM model for ^{−1}, ^{−1}, ^{−1}, ^{−1}, ^{−1} and

Polarization dynamics of gain-switched VCSELs was first investigated in [

Bifurcation diagrams of the polarization-resolved output power ^{−1}, ^{−1}, ^{−1}, ^{−1}, ^{−1}. The modulation frequency

Bifurcation diagrams of the power of

The LP mode dynamics is detailed in Figure

Time traces of

These theoretical predictions have been experimentally confirmed in [

Experimental time traces of the intensities of the total (a),

The irregular behavior of individual polarizations, in contrast to the regular one of the total power, is also manifested in the corresponding RF spectra shown in Figure

Experimental RF spectra of the total,

Gain-switched multitransverse mode VCSEL has been considered in [_{11} mode is now excited with a significant power for all the modulation amplitudes. Time-periodic dynamics is now restricted to low values of

These theoretical predictions were experimentally confirmed by using a GaAs quantum well VCSEL with medium oxide aperture size, such that it can operate on two transverse-modes simultaneously but in one linear polarization only [

Time traces of the VCSEL output power for several values of the modulation frequency

Time-delayed optical feedback (OF) perturbs the steady state laser emission and leads to complex dynamics [

Numerically calculated bifurcation diagrams as a function of the injection current for external mirror reflectivity of

Experimental LI curve of a VCSEL subject to a weak optical feedback (thick lines), to be compared with the case without optical feedback (thin line). The

If we set the injection current close to a PS point light randomly alternates between emission of the

(a) Typical time-trace of mode hopping in the LP mode intensity of a VCSEL close to a polarization switching induced by weak optical feedback. (b) is an enlargement of (a) showing the two LP mode intensities when the light polarization hops between the two orthogonal directions. (c) shows a time-trace of the LP mode intensity (solid curve) together with the one of the total intensity (dotted curve). After [

The coexistence of stochastic (noise driven) process of polarization mode hoping with the deterministic oscillatory behavior at the delay time brings an interesting question about the possibility to observe a coherence resonance in such a system, that is, a resonance of the dynamical regularity of the system at a certain noise level. It is indeed commonly accepted now that noise can play constructive role in nonlinear dynamical systems. After the seminal paper of Benzi et al. [

A VCSEL subject to optical feedback is an example of a bistable time-delayed system and CR has been indeed observed in this system first numerically [

Numerical time traces of the

In order to observe CR experimentally, we have made use of the great possibility for tuning the PS current and hysteresis width by OF from an extremely short external cavity with a length of several 10

(a)–(c) Joint residence time distribution for three increasing noise levels, showing an optimal regularity at the delay time for an optimal noise level (case b). (d) is an enlargement of the JRTD around time values corresponding to the delay time, and for increasing values of noise added to the injection current. After [

The growing interest in implementing chaos communication using VCSELs motivates further investigations to better understand the influence of polarization dynamics on the synchronization properties. In this section, we present a theoretical investigation of the effect of polarization mode competition on the synchronization characteristics of two unidirectionally coupled VCSELs [

In isotropic optical feedback/injection configuration, the two VCSELs can exhibit either an anticipating or isochronous (injection-locking) type of synchronization [

Figure ^{−1} the lasers exhibit either weakly synchronized irregular dynamics or locked steady states (see the small range of

Evolution of the correlation coefficients related to isochronous (a1) and (b1) and anticipative (a2) and (b2) synchronization as a function of

In conclusions, we have shown that the nonlinear dynamics of vertical-cavity surface-emitting lasers (VCSELs) induced by optical injection, current modulation optical feedback and unidirectional coupling are drastically modified due to the underlying polarization and transverse mode competition. This provides new features to the rich nonlinear dynamics such as, for the case of optical injection, the new Hopf bifurcation on a two-polarization-mode solution that delimits the injection locking region and the new resonance tongue of polarization switching and injection locking of first-order transverse mode for large positive detunings. Similarly, the underlying polarization mode competition leads to chaotic-like behavior in case of gain switching and the presence of two transverse modes additionally reduces the possibility of regular dynamics.

Since the polarization selection is very weak, a small amount of optical feedback is enough to significantly modify the VCSEL polarization behavior and observe laser dynamics that significantly differ from those typically observed in edge-emitting lasers. Typical dynamical instabilities such as low-frequency fluctuations and self-pulsations induced by optical feedback still exist in VCSELs but with the additional complexity of the anticorrelated dynamics between polarization modes. The bistable property of VCSEL makes it possible to investigate very fundamental problems of bistable systems with time-delay, such as the coherence resonance phenomenon. We also demonstrated that the synchronization quality between unidirectionally coupled VCSELs can be significantly enhanced when the feedback-induced chaos in the master laser involves both orthogonal LP fundamental transverse modes.

Some of the works presented in this paper have been undertaken with other past and present colleagues and coworkers from other groups, as acknowledged via the relevant journal paper citations. The authors also acknowledge the support of COST MP0702 European Action, Conseil Régional de Lorraine, Fondation Supélec, IAP 6/10—Belgian Science Policy, FWO Flanders, GOA, and OZR of the VUB.