1. Experimental study of stochastic phenomena in vertical cavity lasers Istituto Nazionale Ottica Applicata Largo Enrico Fermi 6 50125 Firenze (Italy) www.inoa.it Giovanni Giacomelli (INOA), giacomelli@inoa.it COWORKERS: ● Sylvain Barbay, LPN-CNRS, Marcoussis (France) ● Vyacheslav Chizhevsky, B.I. Stepanov Institute of Physics, Minsk (Belarus) ● Stefano Lepri, INFM, UdR Firenze (Italy) ● Francesco Marin, Physics Dept., Univ. of Firenze (Italy) ● Ivan Rabbiosi, ENS, Lyon (France) ● Alessandro Zavatta, Systems and Informatics Dept., Univ. of Firenze (Italy) Key references: http://www.inoa.it/~gianni ● Polarization bistability: Quantum Semiclass. Opt. 10, 469 (1998) ● Stochastic resonance: - PRL 82, 675 (1999) - PRE 61, 157 (2000) ● Aperiodic SR: - PRL 85, 4652 (2000) - PRE 63, 051110-1 (2001) ● Phase synchronization: PRE 68, 020101(R) (2003) ● Vibrational resonance: PRL 91, 220602 (2003) The vertical cavity laser (VCSEL) ● Wavelength: near IR (~800 nm),..... ● Output power: ~mW.... ● Single longitudinal mode, multiple transverse modes (short and wide cavity – High Fresnel number) ● TWO LINEAR polarizations emission (symmetrical cavity) ● GOOD beam quality ● High frequency modulation Polarization bistability in VCSELs 2 polarization directions selected by the crystal axis AND symmetrical cavity (circular, rectangular) ===> laser action on both polarizations BUT impurities, inhomogeneities: ONE polarization lasing at threshold.....for particular values of the pump current, the symmetry can be restored: POLARIZATION BISTABILITY! Experimental setup Remark: very stable system (~hour), fast timescales (~ms) ==> GOOD STATISTICS Polarization bistability and noise driven polarization dynamics Scanning the pump current across a bistable point... Reduction to two level: residence time distributions (Kramers statistics) Kramers rates, changing the added noise power Periodic modulation I: stochastic resonance Time series (noise increases from bottom to top) Response at the modulation frequency. Curve: linear response theory with experimentally measured parameters Periodic modulation II: phase synchronization ● Definition of phase  of a (real) signal x(t) with the Hilbert transform H: X(t) = x(t) + i (Hx)(t) = A(t) exp( i  (t) ) (in the case of sinusoidal signal,  = Wt) ● Study of the phase difference: f =  OUT -  IN ● Feasible with experimental (or numerical...) time series Time series: SIN input Noise increases from bottom to top. Inset: at the resonant noise value, for a small modulation amplitude. Average output frequency and effective diffusion coefficient SIN: filled symbols; SQR: empty symbols. A SQR = A SIN /sqrt(2) Random modulation: aperiodic stochastic resonance A different input signal? Bit-stream Random sequence of ¨low¨ (0) and ¨high¨ (1) level amplitudes, with a fixed rate (clock) Advantages: ● Typical digital communication signal (applications...) ● Analytical evaluation of statistical indicators ASR: time series and in-out correlation Dots: experimental correlations Boxes: analytic theory with experimentally measured rates Vibrational resonance IDEA: (partially....) replacing the white, gaussian noise with high frequency modulation Left: laser response for increasing HF modulation amplitude; right: increasing the added noise power (SR)