Deterministic polarization chaos from a laser diode

Abstract

Fifty years after the invention of the laser diode and fourty years after the report of the butterfly effect -i.e. the unpredictability of deterministic chaos, it is said that a laser diode behaves like a damped nonlinear oscillator. Hence no chaos can be generated unless with additional forcing or parameter modulation. Here we report the first counter-example of a free-running laser diode generating chaos. The underlying physics is a nonlinear coupling between two elliptically polarized modes in a vertical-cavity surface-emitting laser. We identify chaos in experimental time-series and show theoretically the bifurcations leading to single-and double-scroll attractors with characteristics similar to Lorenz chaos. The reported polarization chaos resembles at first sight a noise-driven mode hopping but shows opposite statistical properties. Our findings open up new research areas that combine the high speed performances of microcavity lasers with controllable and integrated sources of optical chaos.The discovery of deterministic chaos -i.e. the aperiodic deterministic dynamics of a nonlinear system showing sensitivity to initial conditions -has been a major paradigm shift overthrowing two centuries of Laplacian viewpoint of dynamical systems [1][2][3][4][5] . Looking into a system behavior with the use of chaos theory has helped to interpret and control many of such ordered or disordered behaviors in our present day life, such as the bifurcations leading to epilepsy and cancer 6 , the stabilization of cardiac arrhythmias 7 , and the improvement of complex behavioral patterns in robotics 8 .Soon after the invention of the laser, the possibility to observe light chaos raised attention. In 1975, Haken discovered an analogy between the Maxwell-Bloch equations for lasers and the Lorenz equations showing chaos 9 . The Maxwell-Bloch equations are three equations for the field E, the polarization P and the carrier inversion N, each with its own relaxation time. However, while in Lorenz equations the relaxation times of the dynamical variables are of similar order of magnitude, they may take very different values in lasers. If one variable relaxes much faster than the others, this variable is adiabatically eliminated, hence resulting in a reduced number of dynamical equations. Therefore, so-called class A (ex: He-Ne, Ar and Dye), class B (ex: Nd:YAG, CO2 and semiconductor) or class C (ex: NH3) lasers have dynamics governed either by a single equation for the field, two equations for the field and population inversion or the full set of equations, respectively. In class A or class B laser systems chaos cannot be observed unless one adds one or several independent control parameters 10 . Chaos has then been reported in, for example, free-running NH3 lasers 11 , He-Ne lasers with modulation of the external field 12 , CO2

Spiral attractor created by vector solitons

“…Lorenz, Rössler and Ikeda strange attractors. [22][23][24][25][26]40,42 The descriptions of new structures are found to be beyond the standard models of modelocked lasers based on coupled Schrödinger or Ginzburg-Landau equations. Therefore, we have developed a new model accounting for the dipole mechanism of the light absorption and emission in erbium and CNT, slow relaxation dynamics of erbium ions and absorption of erbium at the lasing wavelength.…”

“…In addition, a reconstruction procedure has to be applied to obtain the attractor in the embedded three-dimensional space from the measured one-dimensional data. 22,26 High signal-tonoise ratio (.40 dB) measurement in the case of mode-locked lasers and application of a polarimeter, gives an opportunity of direct observation of attractors embedded in three-dimensional space in terms of the Stokes parameters S 1 , S 2 and S 3 . 12,13 Though simple forms of polarisation attractors generated by VSs, fixed points and limit cycles, have already been observed 3-13 more complex types of polarisation attractors such as chaotic ones have not yet been demonstrated.…”

“…In addition, a reconstruction procedure has to be applied to obtain the attractor in the embedded three-dimensional space from the measured one-dimensional data. 22,26 High signal-to- …”

“…Quantitative characterisation of strange attractors is usually based on the determination of geometric properties such as the fractal dimensionality, entropy and Lyapunov exponents. [22][23][24][25][26] For experimental data obtained in the form of a one-dimensional waveform (output power vs. time), such analysis requires the accumulation of a large amount of low noise data that is very difficult to achieve in systems where the output power is corrupted by noise. In addition, a reconstruction procedure has to be applied to obtain the attractor in the embedded three-dimensional space from the measured one-dimensional data.…”

Polarization time of unpolarized light

“…The polarization time is a new quantity specifying the rate of the polarization-state fluctuations, and it can provide useful information on natural light sources and propagation media. Polarization dynamics may benefit research dealing with polarization fluctuations, e.g., polarization-mode dispersion in optical fibers [6], particle shape determination [7], polarimetric radar imaging [8], supercontinuum light [9], polarization beating for higher-order harmonic generation [10], vertical-cavity surface-emitting lasers [11], polarization chaos in laser diodes [12], and optical telecommunication [13], as well as polarimetry of stellar sources and thermal background radiation from the Big Bang [14][15][16]. Mapping the time-averaged polarization state of cosmic microwave background radiation has already provided us with new, groundbreaking information on the early universe, and the possibility of measuring dynamical characteristics of the polarization state of this radiation can open one more channel of information for further cosmological investigations.…”