LIS: Nonlinear Dynamics of Laser with Injected Signal

Keywords: chaos, Shilnikov phenomenon, degenerate global bifurcations, homoclinic and heteroclinic bifurcations, Hopf-saddle-node bifurcation.

Lasers with injected signals (LIS) have been studied for their wide range of application of nonlinear dynamics theory. On the one hand they may provide an indispensable method to extract information of all-optical computational devices. The model of laser with injected signal we have used is a 3-dimensional ODE, which accounts for the complex slowly-varying electric field, and the scalar population inversion (class B lasers). Previous studies did numerical work on attractors occurring in this system as parameters are tuned (amplitude of injection and two detunings). We have focused instead on global bifurcations and degenerate singularities to understand a larger portion of the parameter space of this problem.

The laser equations we have analysed in detail arise from an adiabatic elimination of the polarization from the 5-d Maxwell-Bloch equations [sola94]:

$\displaystyle E '$ $\displaystyle =$ $\displaystyle (1+i \theta) E W+i \eta E + \beta$ (1)
$\displaystyle W '$ $\displaystyle =$ $\displaystyle A^2-\chi W (1+g \vert E\vert^2) - \vert E\vert ^2$ (2)

where $ E$ is the complex electric field, and $ W$ is proportional to the inversion population. $ \theta$ is proportional to the cavity detuning with respect to the atomic eigenfrequency, while $ \eta$ is proportional to the detuning of the injected frequency with the unperturbed laser frequency. The parameter $ \beta$ corresponds to the rescaled amplitude of the injection, $ \chi$ corresponds to a dissipation constant, $ g\varpropto \gamma_\bot/(\gamma_\bot+k)$ comes from the adiabatic procedure to eliminate the material polarization (usually very small) and $ A$ is the pump with respect to the laser threshold.

Note that analogous equations may be deduced from models of semiconductors lasers [vant95], where the definitions of the above constants come in terms of the parameters relevant to these lasers, in particular $ \theta \rightarrow \alpha$, the linewidth enhancement factor. Thus, results obtained in our program are equally valid for those models under suitable rescalings.

Our scientific program is based on an analysis performed by Solari and Oppo [sola94], which analysed an averaged version of Eq. 2. They proposed a clasification in terms of the qualitative changes of a singularity which is found to organize the main features of LIS. The singularity arises when the locking regime (saddle-node bifurcation) collides in parameter space with the creation of the relaxation oscillations (Hopf bifurcation). In nonlinear dynamics this is called a Hopf-saddle-node singularity and needs two parameters to unfold it. The laser parameters used where $ \eta$ and $ \beta$, while it is well known that this sigularity has different cases (changes the stability of its invariant sets, and modifies the occurrence of some secondary bifurcations) based on the signs of nonlinear terms. Solari and Oppo found with their analysis that the cavity detuning parameter $ \theta$ controlled the Type of the singularity, and in particular the transitions occurs at:

\begin{equation*}\begin{aligned}[2]&\text{Type II} &:& \quad 0< \theta<1 &\te... ...ext{Type III}\; &:& \quad \sqrt{3}<\theta \nonumber \end{aligned}\end{equation*}

Apart from confirming the local phase portraits known for this singularity, we analysed the global bifurcations which arises. LIS has a global reinjection mechanism which provides new possible global bifurcations which cannot be found in local Hopf-saddle-node singularities. This is part of the exciting part of working with this system, in that new mechanism to generate chaos may arise. Thus our program analysed all regimes in detail and now we have a clear understanding of the organization of global bifurcations as the cavity detuning $ \theta$ changes.

Our initial results concentrated in studying the extremely large detuning case $ \theta>\sqrt{3}$. In this parameter regime a homoclinic orbit to a saddle-focus fixed point was identified. This is a known global bifurcation under the name of Shilnikov phenomena. The special thing we detected in the laser model, was that this homoclinic orbit interacted with a saddle-node bifurcation, destroying eventually the saddle-focus fixed point. This saddle-node bifurcation corresponds to the well known locking phenomenon, usually encountered in LIS. At this point the homoclinic orbit has become a degenerate global bifurcation, and we studied the periodic orbit structure around such a point in parameter space. From a pragmatic point of view one could argue that chaos is associated to an infinite number of periodic orbits, and under certain conditions, this is the case for the Shilnikov phenomenon. However we show that whenever one approaches this degenerate bifurcation point, chaos disappears [zimm97].

Next we analysed in detail the Type I regime, or $ 1<\theta<\sqrt{3}$. In this regime the main object is not a fixed point, as in the previous regime, but a small periodic orbit present in LIS. This periodic orbit is associated to what the laser physicist call relaxation oscillations. The important fact is that this orbit exists in general before locking occurs, thus the global bifurcations which could arise, could be distinguished from the previous regime where chaos appears after locking. We have confirmed by numerical simulations that the periodic orbit is involved in a homoclinic tangency bifurcation, before the locking regime, whose manifolds intersect transversally producing a chaotic set [guck83]. Furthermore we find that this tangency interacts with the Hopf-saddle-node local bifurcation in an unknown and novel higher codimension bifurcation.

Also in this paper we performed the rigorous normal form analysis to confirm Solari and Oppos clasification scheme. Our results show that the critical detuning are up to order $ O(\chi^2)$ correct.

Finaly, we analised the small detuning $ 0<\theta<1$ regime, where no global bifurcation involving the locking or the relaxation oscillation solution, was found. Instead, secondary Hopf bifurcations where found. A notorious sequence of resonance tongues between two torus bifurcations is found, which accumulate towards the Hopf-saddle-node bifurcation. Also, a homoclinic orbit to the laser-off state is found inside each resonance tongue and also accumulates towards the resonance tongue. This points to a higher codimension bifurcation between the Hopf-saddle-node invariant sets and the laser-off fixed point. Examples of this heteroclinic cycle have been found, and all of these features are belived to arise from, a yet unknown, codimension-3 bifurcation.

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