1999 A semi-conductor laser driven by a light beam
Christophe LETELLIER
31/07/2013
A single-mode class B laser with monochromatic external optical injection is described by Sebastian Wieczorek, Bernd Kraupskopf & Daan Lenstra [1].
The governing equations are
\[\left\{ \begin{array}{l} \displaystyle \dot{x} = \frac{xz}{2} + \left( \displaystyle \omega - \frac{\alpha z}{2} \right) y + K \\[0.3cm] \displaystyle \dot{y} = - \left( \displaystyle \omega - \frac{\alpha z}{2} \right) x + \frac{yz}{2} \\[0.3cm] \dot{z} = -2 \Gamma z - (1+2Bz) (x^2 + y^2 - 1) \, . \end{array} \right.\]
where \(B\) is the rescaled cavity lifetime of photons, \(F\) is the rescaled damping rate of the relaxation oscillations and \(K\) is the dimensionless injected field strength. Variables \(x\) and \(y\) are the components of the electric field \(E\) inside the laser and \(z\) is the normalized population inversion. For parameter values With parameter values \(K=0.3\), \(\omega=0\), \(\alpha=8\), \(B=0.015\) and \(\Gamma = 0.035\), it produces a toroïdal structure (Fig. 1) whose Kaplan-Yorke dimension is 2.76.
Fig. 1. Toroidal chaos solution to a semi-conductor laser driven by a light beam.
A simpler set of equations capturing most of the dynamical characteristics of the previous system was proposed by Konstantinos Chlouverakis & Julian Sprott [2]
\[\left\{ \begin{array}{l} \dot{x} = z \, x - \alpha z \, y + K \\[0.1cm] \dot{y} = \alpha z \, x - \epsilon z \, y \\[0.1cm] \dot{z} = 1 - (x^2 + y^2) \, . \end{array} \right.\]
With parameter values \(K=0.4\), \(\alpha = 3\), and \(\epsilon = -0.1\), it produces a toroïdal chaotic attractor (Figs. 2) whose Kaplan-Yorke dimension is 2.54.
Fig. 2. Toroidal chaos solution to the simple system with most of the dynamical characteristics observed on the laser system.
[1] S.Wieczorek, B. Krauskopf & D. Lenstra, A unifying view of bifurcations in a semiconductor laser subject to optical injection, Optics Communications, 172,
279295, 1999.
[2] K. E. Chlouverakis & J. C. Sprott, A comparison of correlation and Lyapunov dimensions, Physica D, 200, 156-164, 2004.
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