Giovanna Tissoni, Franco Prati, Luigi A. Lugiato, Reza Kheradmand INFM, Dipartimento di Fisica e Matematica, Università dell’ Insubria, Como, Italy Massimo Brambilla, INFM, Dip. Di Fisica Interateneo, Università e Politecnico di Bari, Italy Igor Protsenko Lebedev Physics Institute, Moscow, Russia Xavier Hachair, Massimo Giudici, Emilie Caboche, Francesco Pedaci, Stephane Barland, Jorge Tredicce Institut Non-Linéaire de Nice, France Cavity solitons in driven VCSELs above threshold: theory and experiment Nonlinear Guided Waves and Their Applications, NLGW 2005 Dresden, 6-9 September 2005

CAVITY SOLITONS Cavity solitons persist after the passage of the pulse, and their position can be controlled by appropriate phase and amplitude gradients in the holding field Phase profile Intensity xy Intensity profile In a semiconductor microcavity: Brambilla, Lugiato, Prati, Spinelli, Firth, Phys. Rev. Lett. 79, 2042 (1997). Nonlinear medium  nl Holding beamOutput field Writing pulses Possible applications: realisation of reconfigurable soliton matrices, serial/parallel converters, etc

The experiment at INLN First solid experimental demonstration of CS in semiconductor microresonators. S. Barland et al, Nature 419, 699 (2002)

Outline CS in driven VCSELs above threshold: existence, stability and control (theory and numerical simulations) Model for a VCSEL above threshold, beyond the rate-equation approximation CS in driven VCSELs above threshold: existence, switching on/off (experiment)

In presence of diffraction, for a free running laser (FRL), the simple adiabatic elimination (AE) of material polarization is not a good approximation see Jakobsen, Moloney, Newell and Indik, Phys. Rev. A 45, 8129 (1992) (case of two-level laser, no injected signal) Complete model, able to describe the material polarization dynamics.

The Model Equation for P: same structure as in Agrawal’s model: a complex parameter multiplies all the r.h.s. BUT the nonlinear term has the form (1+ia)ED, where D=N/N 0 -1, while in Agrawal’s model it was kEN Starting point: Yao, Agrawal et al., Opt. Comm. 119, 246 (1995) The rate equations (PRA model) are readily recovered with a standard adiabatic elimination of P:

Presence of “effective” damping  and detuning  in the macroscopic polarization equation. They depend on N and then on D.  (D) and  (D) can be derived: - from direct microscopic calculations (Agrawal’s model) - from fitting the calculated gain curves

In the parameter  a constant term   is added, to account for the injection frequency. The Hopf instability has different characteristics: For      i.e.  input frequencies on the left (right) of the gain maximum, a non-homogeneous (K  0) (homogeneous (K = 0)) emission is favoured. Linear stability analysis of the homogeneous steady-state: the lower intensity branch is Hopf unstable for K = 0 up to (stationary state of the free-running laser above threshold)

Depending on current injection level two different scenarios are possible: 1. Only a portion of the lower intensity branch of the homogeneous steady-state curve is unstable. Numerical simulations demonstrate that usual stable CS can be found, exactly as below the lasing threshold. The background is stable, and they can be written and erased in the usual way 2. The whole lower intensity branch of the homogeneous steady-state curve is unstable. Numerical simulations demonstrate that stable CS can be found, but they are sitting on unstable oscillating background. Nevertheless, they can be written and erased in the usual way

Depending on current injection level two different scenarios are possible: 2. The whole lower intensity branch of the homogeneous steady-state curve is unstable. Numerical simulations demonstrate that stable CS can be found, but they are sitting on unstable oscillating background. Nevertheless, they can be written and erased in the usual way 5% above threshold

Depending on current injection level two different scenarios are possible: 20% above threshold 5% above threshold

Despite the background oscillations, it is perfectly possible to create and erase solitons by means of the usual techniques of WB injection

Experimental setup S: Broad-area (150  m VCSEL); M: High power tunable laser ( nm); I: Power supply stabilized better than 1 0 / 00 ; T: Thermal stabilization better than 0,01°C; CCD: Ccd camera for near-field detection of the output intensity profile; PD: Photodiode for fast detection of local output intensity; PZT: piezo electric translator; FP: Fabry Perot interferometer (FSR 2.5 THz, Finesse 140); BC: Beam expander; AOM : Acousto Optical Modulator; OSA: optical Spectrum Analyzer; SA: power spectrum analyzer.

The VCSEL  Realised by ULM University (PIANOS project) (R. Jaeger, T. Knoedl, M. Miller) Grabherr Jaeger, Miller Thalmaier, Heerlein, Michalzik, Ebeling, Phot. Tech.Lett. 10, 1061 (98)  Emission wavelength: around 970 nm  Diameter 150  m  Configuration: “Bottom emitting” Bragg Mirror GaAs Substrate Definition of threshold Bragg Mirror Important feature: cavity length gradient along one axis, due to fabrication process => lasing frequency gradient along the cavity of 60 GHz over a diameter of 150  m

Solitary laser emission profile as a function of injected current 200 mA 750 mA 300 mA 350 mA 400 mA 550 mA

Emission profile with HB injection, as a function of injected frequency FRL, 350 mA nm + 34 GHz + 71 GHz+ 111 GHz+ 148 GHz GHz+ 225 GHz+ 279 GHz

injected frequency ab f c de Emission profile with HB injection, as a function of injected frequency, numerical simulation the cavity gradient has been taken into account the finite pump profile has been introduced

Control of two CS Region of Existence of CS CS no HB (FRL) CS with HB Patterns Hom. Experiment Spontaneous CS formation (decreasing the pump current)

Experiment, J = 450 mA Numerical Simulation Bistable behavior of CS

Intensity spectra Experiment Theory, J = 1.2 J = 316 mA J = 352 mA J = 362 mA J = 420 mA J = 498 mA

Numerical simulations predict that CSs persist above the laser threshold, possibly sitting on unstable background, and can be switched on and off in the usual way Effective two-level model describing the material polarization dynamics Conclusions Experimental results show the spontaneous formation of CS, their bistable character, and the possibility of switching them on and off, exactly as below threshold This Research is performed in the framework of the european project FunFAcs This work has been submitted to the Jour. of Sel. Top. in Quant. Elect. Special Issue on Nonlinear Optic (September 2005)

Profil d’émission Sans Injection I=300 mAInjection I=300 mA  La répartition du courant fait que les bords du VCSEL « lasent » avant la partie centrale  Sous injection la partie centrale peut s’accrocher à la fréquence du laser maître. Profil transverse d’émission, résolu spectralement Étude expérimentale Introduction théorie expérience conclusion

Gradient de fréquence de résonance de la cavité En-dessous du seuil nominal Puissance d’injection constante Pas de 50 GHz in Étude expérimentale Introduction théorie expérience conclusion Le VCSEL présente un gradient de fréquence de résonance le long de sa section transverse (40 GHz/150 µm). Le long de cet axe le désaccord en fréquence entre la résonance de la cavité et le FM change. Formation de structures de fréquences spatiales différentes le long de cet axe. La ligne verticale séparant la région homogène de celle possédant des structures spatiales correspondant à la frontière d’instabilité modulationnelle

Adaptation du modèle aux conditions expérimentales Prise en compte de différents éléments suivants Le désaccord en fréquence entre le champ injecté et la résonance de la cavité est fonction de l’espace Les inhomogénéités introduites lors la fabrication du VCSEL (générées de manière aléatoire) Profil spatial du courant injecté : I I(x,y) Expérience  (x,y)  (x,y) = (  C -  in ) (x,y)/  +  (x,y) Distribution stochastique gaussienne de moyenne nulle Simulation numérique Étude expérimentale Introduction théorie expérience conclusion

Interpretation théorique  x (  m) structures spatiales Solitons de cavité En fixant la fréquence d’injection, on peut associer une valeur de désaccord en fréquence pour chaque point de l’axe horizontal. Domaine d’existence des SC La ligne verticale: séparation entre la région homogène et celle présentant des structures spatiales ensemble des points où  remplit les conditions de l’instabilité modulationnelle (IM). Les SC se développent au voisinage de la frontière de IM et de plus leurs existence est limitée à une bande verticale prés de cette ligne. Étude expérimentale Introduction théorie expérience conclusion  structures spatiales solitons ce cavité

Observation expérimentale Gradient de fréquence de résonance : 40 GHz/ 150 mm Faisceau de maintien (FM) : ~ 3 mW Faisceau d’écriture (FE) : ~ 1 µW Points d’application du FE Champ proche avant l’application du FE Champ proche après l’application du FE Résultats expérimentaux Introduction théorie expérience conclusion