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# 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

$\left\{&space;\begin{array}{l}&space;\displaystyle&space;\dot{x}&space;=&space;\frac{xz}{2}&space;+&space;\left(&space;\displaystyle&space;\omega&space;-&space;\frac{\alpha&space;z}{2}&space;\right)&space;y&space;+&space;K&space;\\[0.3cm]&space;\displaystyle&space;\dot{y}&space;=&space;-&space;\left(&space;\displaystyle&space;\omega&space;-&space;\frac{\alpha&space;z}{2}&space;\right)&space;x&space;+&space;\frac{yz}{2}&space;\\[0.3cm]&space;\dot{z}&space;=&space;-2&space;\Gamma&space;z&space;-&space;(1+2Bz)&space;(x^2&space;+&space;y^2&space;-&space;1)&space;\,&space;.&space;\end{array}&space;\right.$

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, , , B=0.015 and , 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]

$\left\{&space;\begin{array}{l}&space;\dot{x}&space;=&space;z&space;\,&space;x&space;-&space;\alpha&space;z&space;\,&space;y&space;+&space;K&space;\\[0.1cm]&space;\dot{y}&space;=&space;\alpha&space;z&space;\,&space;x&space;-&space;\epsilon&space;z&space;\,&space;y&space;\\[0.1cm]&space;\dot{z}&space;=&space;1&space;-&space;(x^2&space;+&space;y^2)&space;\,&space;.&space;\end{array}&space;\right.$

With parameter values K=0.4, and , 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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