ar X iv : p hy si cs / 0 50 52 13 v 1 3 1 M ay 2 00 5 Self - similarities in the frequency - amplitude space of a loss - modulated CO 2 laser

Lasers with modulated parameters are arguably among the simplest and most accessible laser systems of interest for applications in science and engineering and for theoretical investigations. The intrinsic interest in practical applications and in the nonlinear dynamics of modulated lasers has spurred a wide range of studies after the remarkably influential work of Arecchi et al. [1] reporting the first measurement of subharmonic bifurcations, multistability, and chaotic behavior in a Q-switched CO2 laser. Since then, CO2 lasers have been fruitfully exploited in many situations. Recent applications include studies of stochastic bifurcations in modulated CO2 laser [2], multistability induced by periodic modulations [3]. Rich nonlinear response of CO2 lasers with current modulation and cavity detuning [4], and self-focusing effects in nematic liquid crystals [5]. In the last 20 years the CO2 laser was extensively studied theoretically, numerically and experimentally [6, 7, 8, 9], but focusing mainly on the characterization of dynamical behaviors in phase-space for specific parameters. While a detailed description of phase-space dynamics is already available in the literature [6, 10, 11, 12], no equivalent description exists for the parameter space, except for works by Goswami [13] who investigated analytically the first few period-doubling bifurcations for the Toda model of the CO2 laser [14]. The present Letter reports an investigation of the parameter space of a paradigmatic model of class B lasers, the CO2 laser. More specifically, we study a popular two-level model of a CO2 laser with modulated losses, focusing on the global stability of the laser with respect to the modulation, not the intensity. The remarkable discovery reported here is that stability islands of the continuous-time laser model emerge organized in a very regular network of self-similar structures called shrimps [15], illustrated in Figs. 1 and 2, and previously known to exist only in the parameter space of discrete-time dynamical systems [15, 16, 17, 18]. Thus far, all attempts to uncover shrimps in flows, i.e. in continuous-time dynamical systems modeled with sets of differential equations, have failed to produce them [19]. The single-mode dynamics of the loss-modulated CO2 laser involves two coupled degrees of freedom and a timedependent parameter which we write, as usual [3, 6, 8],