Phase synchronization and polarization ordering IMEDEA Palma de Mallorca, Spain Phase synchronization and polarization ordering in VCSELs arrays Alessandro Scirè, Pere Colet and Maxi San Miguel http://www.imedea.uib.es/PhysDept

Motivation.I Synchronization, phase transitions and cooperative effects IMEDEA http://www.imedea.uib.es/PhysDept

Motivation.II IMEDEA K Kuramoto model: analogy between synchronization and phase transition IMEDEA Order parameter is a measure of the degree of synchronization Critical behavior 2nd order phase transition K http://www.imedea.uib.es/PhysDept

Motivation.III IMEDEA Phase transitions in optics 1. Laser oscillation “A complete analogy of the laser light distribution function to that of the Ginzburg-Landau theory of superconductivity is found mathematically which allows us to interpret the laser threshold as a quasi-second-order phase transition”. [Cooperative phenomena in systems far from thermal equilibrium and in non-physical systems H. Haken Rev. Mod. Phys. 47 67–121 (1975)] 2. Many-modes laser dynamics Introduction of a theory for the ordering of many interacting modes in lasers is presented. By exactly solving a Fokker-Planck equation for the distribution of waveforms in the laser in steady state, equivalence of the system to a canonical ensemble is established, where the role of temperature is taken by amplifier noise. Passive mode locking is obtained as a phase transition of the first kind. [Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers Ariel Gordon and Baruch Fischer, Physical Review Letters 89 103901 (2002)] http://www.imedea.uib.es/PhysDept

Lasers with polarization degree of freedom IMEDEA Amplitude equation for lasers with polarization degree of freedom m net gain w detuning b nonlinear frequency shift. g cross saturation term. g<1 => Linearly Polarized solutions are stable. ga dichroism gp birefringence d preferential polarization direction M. San Miguel, Phys. Rev. Lett. 75, 425 (1995). A. Amengual, D. Walgraef, M. San Miguel & E. Hernández-García, PRL, 76, 1956 (1996). Array of lasers with global coupling: http://www.imedea.uib.es/PhysDept

Phase model.I IMEDEA Disregarding polarization (2yj(t)=dj=d0, gp=0): Kuramoto model for limit cycle oscillators. Y. Kuramoto, Chemical Oscillations, Waves & Turbulence, Springer (1984). G. Kozyreff, A.G. Vladimirov & P. Mandel, PRL 85, 38095 (2000), PRE 64, 016613 (2001), Europhys. Lett. (2003) http://www.imedea.uib.es/PhysDept

Phase model.II Mean field vs local coupling IMEDEA Kuramoto model in a square lattice: with local coupling. The synchronization process is shown to be the same, but for a rescaling of the coupling K [arXiv:Cond-mat.0201508] MF 3rd 2nd 1st http://www.imedea.uib.es/PhysDept

Order parameters.I IMEDEA Pw distribution of frequencies definitions IMEDEA Pw distribution of frequencies Pd distribution of polariz. angles Two order parameters to characterize the degree of collective synchronization: phase synchronization polarization ordering http://www.imedea.uib.es/PhysDept

Uncoupled case. IMEDEA C=0 Stationary solutions: Then Continuous limit: P(d): distribution of natural polarization angles http://www.imedea.uib.es/PhysDept

Global phase synchronization transition IMEDEA sw<< sd Take yj=dj/2 then: Small disorder: define an effective coupling as: Finally: Effective Kuramoto model => Apply standard techniques. Pw distribution of natural frequencies http://www.imedea.uib.es/PhysDept

Polarization ordering transition IMEDEA sw<< sd Assume phase synchronization Not a Kuramoto-like model! However, we can apply similar techniques. Using order parameter definition: Implicit eq. for stationary values Introducing stationary values in order parameter definition provides a self-consistent eq. P(d): distribution of natural polarization angles http://www.imedea.uib.es/PhysDept

Order parameters.II IMEDEA sw=0.01 sd=0.91 numerical results Coupling term in polarization eq.

Synchronization transitions Coherence lowering due to polarization ordering IMEDEA sw=0.01 sd=0.91 http://www.imedea.uib.es/PhysDept

Synchronization transitions IMEDEA sw~ sd sw=0.12 sd=0.91 http://www.imedea.uib.es/PhysDept

Phase synchronization and polarization ordering in VCSELs arrays IMEDEA Palma de Mallorca, Spain Phase synchronization and polarization ordering in VCSELs arrays Conclusions We introduced a prototype model to study phase synchronization and polarization ordering in oscillators with polarization degree of freedom, with a possible application to VCSELs arrays. We derived a self-consistent theory to provide the order parameters describing the transitions to phase synchrony and polarization ordering. http://www.imedea.uib.es/PhysDept