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 toroidal 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 toroidal 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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